Capacity-Approaching Modulation Formats for Optical

Capacity-Approaching Modulation Formats for Optical
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Capacity-Approaching Modulation Formats for Optical Transmission Systems: Signal
shaping and advanced de/muxing for efficient resource exploitation.
Estaran Tolosa, Jose Manuel; Zibar, Darko; Tafur Monroy, Idelfonso ; Peucheret, Christophe
Publication date:
2015
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Citation (APA):
Estaran Tolosa, J. M., Zibar, D., Tafur Monroy, I., & Peucheret, C. (2015). Capacity-Approaching Modulation
Formats for Optical Transmission Systems: Signal shaping and advanced de/muxing for efficient resource
exploitation. Technical University of Denmark.
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Capacity-Approaching Modulation Formats for
Optical Transmission Systems:
Signal shaping and advanced de/muxing for
efficient resource exploitation
Ph.D. Thesis
José Manuel Estarán Tolosa
November 1st , 2015
DTU Fotonik
Department of Photonics Engineering
Technical University of Denmark
Ørsteds Plads 343
DK-2800 Kgs. Lyngby
Denmark
Preface
The work presented in this Thesis was carried out as a part of my Ph.D.
project in the period September 30th , 2012, to November 1st , 2015. The
work was conducted at various institutions including DTU Fotonik (Technical University of Denmark, Department of Photonics Engineering), Huawei
Technologies Co. Ltd., Shenzhen, China, and Alcatel-Lucent Bell Labs,
Villarceaux, France.
The Ph.D. project was partly financed by the VILLUM FOUNDATION,
Søborg, Denmark, within the Young Investigator Programme and supervised by
• Darko Zibar (main supervisor), Associate Professor, DTU Fotonik,
Technical University of Denmark, Kgs. Lyngby, Denmark
• Idelfonso Tafur Monroy (co-supervisor), Professor, DTU Fotonik,
Technical University of Denmark, Kgs. Lyngby, Denmark
• Christophe Peucheret (co-supervisor), Professor, FOTON Laboratory
(CNRS UMR 6082) - ENSSAT - University of Rennes 1, Lannion,
France
iii
Abstract
Aiming for efficient fiber-optic data transport, this thesis addresses three
scenario-specific modulation and/or multiplexing techniques which, leveraging digital signal processing, can further exploit the available resources.
The considered environments are: (i) (ultra) long-haul networks, where
we focus on improving the receiver sensitivity; (ii) metropolitan area networks, where the target is providing spectral and rate adaptability with
fine granularity and easy reconfigurability; and (iii) short-haul networks,
where facilitating more affordable throughput scaling is pursued.
Functioning essentials, signal attributes, and digital processing structures are discussed in detail; whereby we give grounds for their adoption
into the respective contexts of the three proposed systems. Eventually, we
conduct the corresponding experimental validations; overall proving their
potential. The reported results encourage the continuation of the research,
mainly towards algorithm refinement.
v
Resumé
Denne afhandlings mål er at præsentere effektive fiber-optiske data transport og tre scenerier adresseres – mere præcist modulation og/eller data
sammenfletnings teknikker som, ved hjælp af digital signal processer, kan
udnytte de givne ressourcer bedre. Følgende scenarier bliver betragtet. (i)
(ultra) lang distance netværk, hvor fokus er en forbedring af modtagerfølsomheden; (ii) metropol netværk, hvor målet er at forberede tilpasningen
af både spektrum og hastighed med en bedre opløsning kombineret med
nemme konfigurations muligheder ; (iii) kort distance netværk, med ønsket
om at understøtte en mere økonomisk skalering af den samlede trafik.
De essentielle netværks funktioner, signal egenskaber samt digital signal
proces strukturer bliver behandlet i detaljer, hvorefter der er basis for at
tilpasse dem til den respektive kontekst i form af de tre foreslåede scenarier.
Afslutningsvis bliver de tilsvarende eksperimentelle evalueringer udført, der
samlet set understreger deres potentiale. De præsenterede resultater opfordrer til en forsættelse af forskningen indenfor dette emne med primær fokus
på algoritme forbedringer.
vii
Ph.D. Publications
The following publications have resulted from this Ph.D. project.
Articles in international peer-reviewed journals: (8)
J1
D. Zibar, L. H. H. Carvalho, J.-M. Estarán Tolosa, E. Porto Da
Silva, C. Franciscangelis, V. Ribeiro, R. Borkowski, O. Winther, M.
N. Schmidt, J. F. R. de Oliveira and I. Tafur Monroy, “Joint Iterative
Carrier Synchronization and Signal Detection Employing Expectation Maximization,” in Journal of Lightwave Technology, vol. 32, no.
8, pp. 1608-1615, April 2014.
J2
J.-M. Estarán Tolosa, D. Zibar, I. Tafur Monroy, “CapacityApproaching Superposition Coding for Optical Fiber Links,” in Journal of Lightwave Technology, vol. 32, no. 17, pp. 2960-2972, Sept
2014.
J3
M. Piels, E. Porto Da Silva, J.-M. Estarán Tolosa, R. Borkowski,
I. Tafur Monroy, and D. Zibar, “Focusing over Optical Fiber Using
Time Reversal,” in IEEE Photonics Technology Letters, vol. 27, no.
6, pp. 631-634, March 2015.
J4
J.-M. Estarán Tolosa, M. A. Usuga Castaneda, E. Porto Da
Silva, M. Piels, M. Iglesias Olmedo, D. Zibar, and I. Tafur Monroy,
“Quaternary Polarization-Multiplexed Subsystem for High-Capacity
IM/DD Optical Data Links,” in Journal of Lightwave Technology,
vol. 33, no. 7, pp. 1408-1416, April 2015 (invited).
J5
D. Zibar, L. H. H. Carvalho, M. Piels, A. Doberstein, J. Diniz, B.
Nebendahl, C. Franciscangelis, J.-M. Estarán Tolosa, H. Haisch,
N. G. Gonzalez, J. F. R. de Oliveira, and I. Tafur Monroy, “Application of Machine Learning Techniques for Amplitude and Phase Noise
ix
Ph.D. Publications
Characterization,” in Journal of Lightwave Technology, vol. 33, no.
7, pp. 1333-1343, April 2015.
J6
S. Saldaña Cercos, M. Piels, J.-M. Estarán Tolosa, M. A. Usuga
Castaneda, E. Porto Da Silva, A. Manolova Fagertun, and I. Tafur
Monroy, “100 Gbps IM/DD links using quad-polarization: Performance, complexity, and power dissipation,” in Optics Express, vol.
23, no. 15, pp. 19954-19968, July 2015.
J7
A. Tatarczak, M. Iglesias Olmedo, T. Zuo, J.-M. Estarán Tolosa,
J. B. Jensen, X. Xu, and I. Tafur Monroy, “Enabling 4-Lane Based
400 G Client-Side Transmission Links with MultiCAP Modulation Advances in Optical Technologies,” in Advances in Optical
Technologies, vol. 2015, ID:935309, July 2015.
J8
M. A. Mestre, J.-M. Estarán Tolosa, P. Jennevé, H. Mardoyan, I.
Tafur Monroy, D. Zibar and S. Bigo, “Novel coherent optical OFDMbased transponder for optical slot switched networks,” in Journal of
Lightwave Technology (invited).
Contributions to international peer-reviewed conferences: (14)
C1
J.-M. Estarán Tolosa, D. Zibar, A. Caballero, C. Peucheret, and I.
Tafur Monroy, “Experimental Demonstration of Capacity-Achieving
Phase-Shifted Superposition Modulation,” in European Conference
on Optical Communication and Exhibition 2013, ECOC 2013, paper
We.4.D.5.
C2
D. Zibar, L. H. H. Carvalho, J.-M. Estarán Tolosa, E. Porto
Da Silva, C. Franciscangelis, V. Ribeiro, R. Borkowski, J. F. R. de
Oliveira, and I. Tafur Monroy, “Joint Iterative Carrier Synchronization and Signal Detection for Dual Carrier 448 Gb/s PDM 16-QAM,”
in European Conference on Optical Communication and Exhibition
2013, ECOC 2013, poster P.3.18.
C3
Y. An, M. Muller, J.-M. Estarán Tolosa, S. Spiga, F. Da Ros,
C. Peucheret, and M.-C. Amann, “Signal Quality Enhancement of
Directly- Modulated VCSELs Using a Micro-Ring Resonator Transfer Function,” in OptoElectronics and Communications Conference
2013 held jointly with International Conference on Photonics in
Switching 2013, OECC/PS 2013, paper ThK3-3.
x
C4
J.-M. Estarán Tolosa, M. Iglesias Olmedo, D. Zibar, X. Xu, and I.
Tafur Monroy, “First Experimental Demonstration of Coherent CAP
for 300-Gb/s Metropolitan Optical Networks,” in Optical Fiber Communication Conference and Exhibition 2014, OFC/NFOEC 2014, paper Th3K.3.
C5
T. Zuo, A. Tatarczak, M. Iglesias Olmedo, J.-M. Estarán Tolosa,
J. B. Jensen, Q. Zhong, X. Xu, I. Tafur Monroy, “O-band 400 Gbit/s
client side optical transmission link,” in Optical Fiber Communication Conference and Exhibition 2014, OFC/NFOEC 2014, paper
M2E.4.
C6
M. Iglesias Olmedo, A. Tatarczak, T. Zuo, J.-M. Estarán Tolosa,
X. Xu, and I. Tafur Monroy, “Towards 100 Gbps over 100 m MMF
using a 850 nm VCSEL,” in Optical Fiber Communication Conference and Exhibition 2014, OFC/NFOEC 2014, paper M2E.5.
C7
J.-M. Estarán Tolosa, M. A. Usuga Castaneda, E. Porto Da Silva,
M. Piels, M. Iglesias Olmedo, I. Tafur Monroy, “Quad-Polarization
Transmission for High-Capacity IM/DD Links,” in European Conference on Optical Communication and Exhibition 2014, ECOC 2014,
paper PD.4.3 (postdeadline paper).
C8
D. Zibar, L. H. H. Carvalho, M. Piels, A. Doberstein, J. Diniz, B.
Nebendahl, C. Franciscangelis, J.-M. Estarán Tolosa, H. Haisch,
N. G. Gonzalez, J. F. R. de Oliveira, and I. Tafur Monroy, “Bayesian
Filtering for Phase Noise Characterization and Carrier Synchronization of up to 192 Gb/s PDM 64-QAM,” in European Conference
on Optical Communication and Exhibition 2014, ECOC 2014, paper
Tu.1.3.1.
C9
M. Piels, E. Porto Da Silva, J.-M. Estarán Tolosa, R. Borkowski,
D. Zibar, I. Tafur Monroy, “DSP-Based Focusing over Optical Fiber
Using Time Reversal,” in European Conference on Optical Communication and Exhibition 2014, ECOC 2014, poster P.7.8.
C10
A. Tatarczak, Y. Zheng, G. A. Rodes, J.-M. Estarán Tolosa, C.-H.
Lin, A. V. Barve, R. Honore, N. Larsen, L. A. Coldren, and I. Tafur
Monroy, “30 Gbps bottom-emitting 1060 nm VCSEL,” in European
Conference on Optical Communication and Exhibition 2014, ECOC
2014, poster P.2.3.
xi
Ph.D. Publications
C11
J.-M. Estarán Tolosa, D. Zibar, I. Tafur Monroy, “Capacity and
Shaping in Coherent Fiber-Optic Links,” in IEEE Photonics Conference 2014, IPC 2014, paper TuG2.4 (invited).
C12
A. Tatarczak, J.-M. Estarán Tolosa, M. Iglesias Olmedo, J. B.
Jensen, J. J. Vegas Olmos, and I. Tafur Monroy, “Advanced digital
signal processing for high-speed access networks,” in SPIE Photonics
West 2015, SPIE PW 2015, (invited).
C13
J.-M. Estarán Tolosa, X. Lu, D. Zibar, and Tafur Monroy, “Stokes
Space in Direct-Detection Data Transmission Systems,” in Asia
Communications and Photonics Conference 2015, ACP 2015, paper
AS4D.1 (invited).
C14
J.-M. Estarán Tolosa, M. A. Mestre, P. Jennevé, H. Mardoyan,
I. Tafur Monroy, D. Zibar and S. Bigo, “Coherent Optical Orthogonal Frequency-Division Multiplexing for Optical Slot Switched IntraDatacenters Networks,” in European Conference on Optical Communication and Exhibition 2015, ECOC 2015, paper Tu.1.2.3.
xii
Contents
Preface
iii
Abstract
v
Resumé
vii
Ph.D. Publications
ix
1 Introduction
1.1 General motivation . . . . . . . . . . . . . . . . . . . . . . .
1.2 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2 (Ultra) long-haul networks
2.1 Work positioning . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Superposition Coded Modulation with Phase Shifted Mapping (SCM-PSM) . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Generation and signal properties . . . . . . . . . . .
2.2.2 Detection and de/coding . . . . . . . . . . . . . . . .
2.3 Experimental demonstration . . . . . . . . . . . . . . . . . .
2.3.1 Testbed . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
8
3 Metropolitan networks
3.1 Work positioning . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Multi-band carrierless amplitude/phase modulation for coherent communications (C-MultiCAP) . . . . . . . . . . . .
3.2.1 Generation and signal properties . . . . . . . . . . .
3.2.2 Detection . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Experimental demonstration . . . . . . . . . . . . . . . . . .
xiii
11
12
26
33
34
36
42
47
52
61
61
86
98
CONTENTS
3.4
3.3.1 Testbed . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 101
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 Short-haul networks
4.1 Work positioning . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Quaternary polarization-multiplexed direct-detection system
4.2.1 Conceptual description . . . . . . . . . . . . . . . . .
4.2.2 Mathematical description . . . . . . . . . . . . . . .
4.2.3 Transmitter . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Receiver . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Digital receiver - stages and practical considerations
4.3 Experimental demonstration . . . . . . . . . . . . . . . . . .
4.3.1 Testbed . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
112
117
117
118
121
124
127
135
136
137
144
5 Conclusion
149
5.1 (Ultra) long-haul networks . . . . . . . . . . . . . . . . . . . 149
5.2 Metropolitan networks . . . . . . . . . . . . . . . . . . . . . 151
5.3 Short-haul networks . . . . . . . . . . . . . . . . . . . . . . 152
A SCM-PSM 1D gaussianity
155
B Memoryless channel-capacity
161
C Frequency pilot tone
165
D DSP schema for 4-SOP IM/DD transmission system
169
Acronyms
171
Bibliography
177
xiv
Chapter 1
Introduction
Ever since the first steps back in 60s [1], fiber-optic data communications
have experienced thriving evolution. Along these years, the emerging capabilities have encouraged incessant development of new applications and
services with increasingly exigent technical requirements, in turn feeding
back the need for progress at all levels; that is, network management and
topology, degree of coverage and connectivity, and of course, achievable
throughput. Notwithstanding, as this technology gets relentlessly pushed
to the limits of performance, and the available resources to the edge of exhaustion, the evolution rate inescapably saturates. Broad-sense efficiency
has then become a figure of radical importance for the current research
force; such that, in the last years, it could be argued that a work paradigm
has been widely adopted whereby not just more is encouraged but also
better used. This is the big picture this investigation fits into.
1.1
General motivation
In this quest towards more efficient communications, the introduction of
cost-effective high-speed digital signal processing (DSP) fostered a remarkable changeover around 00s. Advanced modulation formats and coding plus
electronic compensation of linear and non-linear impairments became available, allowing for the simultaneous improvement of bandwidth use (spectral efficiency) and receiver sensitivity (power efficiency) with unmatched
flexibility. This features naturally led to countless scenario-specific opportunities to the extent that, from dual-polarization long-haul transmissions
to short-haul uncoded on-off-keying, transceivers implementing DSP are
presently found everywhere.
1
Introduction
The technical requirements on these transceivers, however, keep evolving with society’s needs; and with them, the disquieting hurry for adaptation. In this direction, the global purpose of this Ph.D. investigation is:
by leveraging the ubiquity and potential of DSP, elaborating solutions going
beyond existing ones in better exploiting the capacity of the medium and the
available resources in a scenario-specific manner.
1.2
Scope and outline
This thesis deals with the experimental validation of three concrete modulation and/or multiplexing techniques plus their associated transmitter/receiver algorithm structures. Aiming for more efficient communication,
these schemes are each devised or adopted based on the specific technoeconomical constraints and governing physics of their respective network
segments:
• (Ultra) long-haul (Chapter 2) : superposition coded modulation
with phase-shifted mapping [2–9] is proposed and experimentally
demonstrated for the first time in coherent optical communications.
This modulation scheme generates signals exhibiting two-dimensional
Gaussian-like power distribution with lower processing complexity in
both transmitter and receiver sides than similar alternatives. This
motivates its consideration and for improving the receiver sensitivity
of (ultra) long-haul lumped-amplified uncompensated links, where
the non-linear interference is accurately modeled as additive Gaussian [10, 11].
• Metropolitan area (Chapter 3) : coherent multiple-band carrierless amplitude/phase modulation is introduced and experimentally
demonstrated for the first time in coherent optical communications.
In this modulation and multiplexing scheme the spectrum is digitally sub-divided into a user-defined number of multi-dimensional
pass-band signals. High control of the spectral shape and line rate
is accomplished through the variation of the center frequency, bandwidth, modulation format and dimensionality of each individual band,
providing robustness against filtering effects without compromising
spectral efficiency. These factors become particularly beneficial in
metropolitan area networks, where the fast, diverse, and high-volume
traffic increase [12] is progressively favoring the migration to highspeed all-optical reconfigurable mesh architectures [13–16].
2
1.2 Scope and outline
• Short-haul (Chapter 4) : a technique enabling up to fourth-order
polarization-multiplexing is conceived and experimentally demonstrated for the first time in an optical transmission system employing intensity modulation and direct detection. The proposed digital
receiver operating on Stokes parameters performs blind and quasicontinuous tracking of characteristic Mueller channel matrix; hence
supporting the simultaneous transmission of ≤4 states of polarization
carrying independent data each. The use of complementary dimensions to wavelength and space in order to enhance short-haul networks
is steadily more justified as affordable scalability gets more and more
compromised with the increasing throughput demand [17].
These three self-contained chapters are all structured according to, first,
the introduction of the scenario-specific motivation and problem statement;
second, the contextualization of the work with respect to most-recent research activities plus explicit statement of our contribution; third, elaboration on the fundamentals that sustain the employed technique, as well as
signal properties and major implementation concerns from both transmitter and receiver perspectives; fourth, explanation of the experimental goals,
testbed, and discussion on the results; and fifth, synopsis, where the main
points and findings are highlighted and possible future work lines proposed.
In the last block, Chapter 5, the whole investigation is summarized.
3
Introduction
4
Chapter 2
(Ultra) long-haul networks
Carrying multi-terabit data traffic over reaches in the order of thousands
of kilometers, these big "pipes" represent the spine of the global data communication network. Technology-wise, long-haul (LH) and ultra long-haul
(ULH) optical data links employ wavelength-division multiplexing (WDM)
in various configurations. The use of lower-rate channels reduces the cost
per transmitted bit, which comes in addition to the functional throughput
scaling capabilities that WDM inherently features. For instance, assuming 50-GHz inter-channel spacing1 , up to 160 data-independent channels
may be multiplexed across the entire C and L bands. As far as transceivers
are concerned, most-recent implementation agreements define 100-Gbps coherent technology based on polarization-multiplexed quadrature phase-shift
keying (PM-QPSK) with 2 bit/s/Hz of spectral efficiency (SE) [19]. Several
technological alternatives for next-gen 400-Gbps coherent transceivers are
elaborated in detail in [13]. Channel coding blocks typically employ harddecision 7%-overhead Reed-Solomon forward error correction (FEC) [20].
More sophisticated recommendations with improved performance are described by the ITU-T under super-FEC denomination, covering mixed alternatives with 7%-, 11%- and 25%-overhead as well as hard- and soft-decision
approaches [20]. Software-configurable FEC options are also supported in
some commercially available cards.
These networks describe centralized point-to-point or few-node distributed mesh topologies in which assuring light-path integrity2 and sig1
Nominal inter-channel spacing as recommended by telecommunication standardization sector of ITU (ITU-T) include 12.5, 25, 50 and 100 GHz. ITU-T also defines the
nominal central frequencies for each grid [18].
2
Virtue of the light-path to remain operative.
5
(Ultra) long-haul networks
nal integrity3 becomes critical. Whereas the former requires standalone
improvements in the reliability of the link’s components (e.g. fibers, cables, optical amplifiers or optical add-drop multiplexers), enhancing the
signal integrity demands measures essentially opposed to those for increasing throughput (e.g. increasing inter-channel spacing or reducing launch
power to diminish nonlinearities). The latter hints one conclusion of practical importance [21]: the existence of an optimum operation point for each
system configuration which establishes the throughput-times-distance product performance beyond which reliable transmission is not possible. Faced
with need for longer distances and/or higher throughput [12] as well as economical restrictions that impede riotous fiber deployment [22], the scrutiny
of the boundaries of achievable capacity has gained great attention [23–30].
The optimum operation point results from trying to overcome the optical bandwidth scarcity and/or insufficient values of optical signal-to-noise
ratio (OSNR). This premise evinces the appropriateness of maximizing
both the SE and the receiver sensitivity, which directly relates to the optimization of the inter-channel spacing, the per-channel bandwidth, and the
per-channel modulation format plus de/coding scheme. Migrating from
coarse frequency grids, highly spectral-efficient and flexible solutions4 are
attained thanks to tighter channel bandwidth assignation and fine guardband reduction [32, 33], ultimately limited by detrimental effects like spectral clipping and inter-carrier overlapping. It is precisely in the limit of
channel packing density that advanced modulation formats (AMF) become
major drivers of the SE and the receiver sensitivity for a given transmission reach. Richer constellations provide greater information rates per Hz
at the expense of signal power and noise tolerance reduction [34], which may
not be acceptable. Interestingly, inspired by the progress in the design of
application-specific integrated circuit (ASIC) and digital-analog interfaces,
DSP has allowed for the relaxation of such compromise, bringing forward
the processing complexity as additional resource to trade on [35].
With coherent detection enabling digital manipulation of the optical field, uncompensated polarization-multiplexed transmission over long
distances signals is a reality. Continuous algorithm refinement has per3
Virtue of the signal to meet the quality criteria that enables successful data transmission and recovery.
4
Slot granularities of 12.5 GHz are being studied. This fragmentation allows for
fitting 100 Gbps within 37.5 GHz instead of the standard 50 GHz window (33% spectral
savings) [31]. Anticipating future needs, other research activities contemplate 6.25-GHz
slot granularity (see footnote 1).
6
mitted close to total compensation of linear distortions (e.g. chromatic
dispersion (CD) and polarization-mode dispersion (PMD)) to an extent
that, at present, non-linear signal-noise interaction totally governs the engineering of the optimum throughput-times-distance product [36]. This
outlook has fostered research activities towards exploring the underlying physics for accurate channel modeling, paving the way for the development of manifold DSP methods aiming nonlinearity compensation.
Some of the most explored algorithms for optical communications include
back-propagation [37] and variants5 , inverse Volterra series [43], phaseconjugated twin-waves [44], mid-span optical phase conjugation [45], frequency pilot tone conjugation [46] or non-linear Fourier transformation [47].
Unfortunately, despite the efforts to improve the trade-off between implementation complexity and performance, achievable sensitivity gains have
proved insufficient6 within acceptable cost and power consumption margins. As a result, techniques for nonlinearity mitigation (i.e. nonlinearity minimization during propagation [49, 50] and/or nonlinearity tolerance
maximization [51, 52]) are being considered to further boost the receiver
sensitivity; and in this direction, coded modulation combined with constellation shaping [53, 54] has the potential to achieve constrained capacity by
itself [55, 56].
Even though shaping the power distribution of the transmitted signal
can extend the achievable throughput to its upperbound [55, 57], practical restrictions (ranging from technological limitations to fundamental
non-linear noise understanding) have forced much humbler aspirations as
of today [23, 49, 50]. Nevertheless, the undisputed potential of this technique has motivated the study and perfection of existing candidates towards
their adaptation to different scenarios in fiber-optics data communications
(see Section 2.1). Among those, certain variations of superposition coded
modulation (SCM) [7, 58–61] have been proved to approach continuous
memoryless additive white Gaussian noise (AWGN) channel capacity (see
Appendix B) with low processing complexity, while providing convenient
modulation order adaptability and simplified demapping. Resorting to the
Gaussian noise model [29], the latter approach arises as a promising candidate for application in the scenarios covered in this chapter. Originally suggested by H. Schoeneich and P. A. Hoeher for wireless communications [2]
5
Some examples include stochastic [38], folded [39], weighted [40], perturbation [41]
and filtered [42] back-propagation.
6
in the sub-decibel range in typical densely populated WDM LH/ULH scenarios,
employing uncompensated lumped-amplified standard single mode fiber (SSMF) transmission links and free-running lasers [48]
7
(Ultra) long-haul networks
and successively researched [3–9], we propose and experimentally demonstrate for the first time the use of SCM with phase-shifted mapping (PSM)
in optical communications.
This chapter is subdivided into four blocks. First, this investigation is
positioned with respect to the latest research, narrowing down the scope to
coded modulation schemes employing Gaussian shaping for application to
optical data links where the Gaussian noise model holds strictly. Second,
the properties of PSM signals are elaborated together with the essentials
concerning digital generation, detection and decoding. Third, the experimental setup and results are presented and discussed, focusing on the implementation feasibility when standard DSP algorithms are employed for
signal recovery. Finally, the work is summarized, comprehending a motivation recap, main results and conclusions, and plausible future work lines.
2.1
Work positioning
Two aspects shall be used to classify diverse constellation shaping techniques: (i) the optimization criterion, and (ii) the statistical properties
of the signal impairments, in turn subject to the particular link components and transmission technology. In this chapter, the focus is on mutual information (MI) maximization for coherent LH/ULH uncompensated
lumped-amplified SSMF transmission links employing WDM. Customary
treated as noise, non-linear interference in these scenarios is accurately
modeled as additive Gaussian [10, 11], for which the optimum source distribution in MI sense has been known with exactitude for more than half
a century [62]. This fact has motivated the application of mature constellation shaping techniques to current fiber-optic communications and the
immediate extrapolation of certain results for performance estimation and
capacity lower-bounding.
In the following, we present the most relevant investigations on MIwise constellation shaping for the above-mentioned scenarios, subdivided
into probabilistic and geometric methods. Our work is positioned with
respect to the literature and major contributions highlighted.
Probabilistic shaping
In this family of techniques, the power distribution across the constellation space is altered by varying the individual probability of occurrence of
8
2.1 Work positioning
the symbols through the inducement of controlled redundancy or bit-tosymbol mapping ambiguities. The constellation grid remains unmodified.
Simulation and/or theoretical results include:
• [63]: adaptation of Trellis shaping for attaining Gaussian distribution
and high-speed efficient decoding.
• [64]: four-dimensional rate-adaptive coded modulation with shaping capabilities based on low-density parity-check (LDPC) codes and
Shell mapping.
• [65]: arithmetic distribution matching in combination with systematic LDPC codes for enabling bit-level shaping and decoding [66]. No
iterative decoding is needed. Maximum reach extensions of 8% and
15% are shown for 16-ary and 64-ary shaped quadrature amplitude
modulation (QAM) respectively.
• [67]: modification of the pragmatic shaping algorithm proposed by
D. Raphaeli which outperforms the original for high signal-to-noise
ratio (SNR). Theoretical SE gains around 0.35 bit/s/Hz are reported
in the optimum operation point for shaped 1024-ary QAM.
To the best of our knowledge, only one experimental demonstration of
probabilistic shaping has been reported in coherent optical communications:
•
[68]: employing arithmetic distribution matching, the authors
demonstrate shaped and rate-adaptive transmission with fixed
channel-code overhead. This is possible owed to the bit-level redundancy that ADM process introduces, then diminishing the effective
transmission rate without having to modify the channel code. The
results show showing 15% increase in the achievable throughput and
43% increase in the achievable reach with respect to uniform 200Gbps 16-ary QAM.
Geometric shaping
The power distribution across the constellation space is modified via direct
reallocation of the conforming symbols while maintaining uniform probability of occurrence. Simulation and/or theoretical results include:
• [23]: constrained-capacity increase by optimizing the inter-ring spacing and the ring occupation density of uniform continuous ring constellations. Gain in SE in the order of 0.2 bit/s/Hz is obtained in the
9
(Ultra) long-haul networks
corresponding point of optimum operation for shaped 2-ring constellation.
• [69]: a systematic method to find MI-wise optimized constellations
in minimum mean square error (MMSE) sense is presented within the
context of high-speed optical links.
• [70]: capacity-approaching (in memoryless AWGN channel) iterative
polar modulation (IPM) is introduced, a modulation format obtained
through the iterative quantization of a bivariate Gaussian distribution
while minimizing the quantization mean-square error (QMSE).
The experimental demonstrations cited below need clarification. Although the employed link infrastructures do not fully correspond to the
one of interest, the shaping optimization criterion and the assumptions on
the transition probability density function (pdf) of the channel converge to
the ones discussed. Because these investigations partly deal with general
implementation concerns, their relevance in our context is justified:
•
[71]: 256-ary IPM on orthogonal frequency-division multiplexing
(OFDM) signal is transmitted over 800-km uncompensated ramanamplified ultra-large-area fiber (ULAF). Up to 6 dB of power budget
margin increase compared to the same order QAM is shown.
• [72]: following up on [71], terabit transmission of OFDM signal over
Korean deployed legacy WDM link employing 256-ary IPM is demonstrated. Enhancements in the soft-based decoding process are introduced.
Constellation shaping techniques incorporated within coded modulation
schemes are meant to further increase the power budget7 , hence potentially
coping with the performance penalty induced when extending the reach
and/or increasing the symbol rate. Nonetheless, both longer reach and
boosting the baudrate worsen the noise severity via linear and non-linear
mechanisms, imperiling the sensitivity improvement and thus questioning
the utility of the typically "very high complexity" [72] and hard-scalable
shaping procedures. Because such gains are already diminished by the
assumptions on the transition pdf of the channel, functional schemes worth
consideration for implementation must accomplish non-negligible shaping
7
Understood as the range of propagated optical power values for which, given predefined link characteristics, signal integrity is preserved (see footnote 3)
10
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
gains maintaining low-complexity generation and optimization. This is
exactly the strength of SCM-PSM over other solutions presented to date, a
transmission model that approaches memoryless AWGN channel capacity
asymptotically with the number of bits per symbol, while offering enhanced
simplicity and flexibility at the transmitter and receiver sides.
Being SCM-PSM the focus of this investigation, in the next sections we
elaborate on the essentials and main properties of this modulation format,
and present the results of the first experimental generation, transmission
and detection in optical communications. Theoretical shaping gains with
respect to uniform QAM are provided for memoryless AWGN channel, as
well as the information rates for up to 12 bit/sym configuration. The system performance is experimentally investigated for dual-polarization (DP)
single-carrier 16-ary, 32-ary and 64-ary PSM in optical back-to-back (B2B)
configuration, and after 240-km uncompensated lumped-amplified SSMF
link.
Contributions of this work
• Original idea: proposal of SCM-PSM for its use in uncompensated
lumped-amplified SSMF transmission links employing WDM with the
goal of increasing constrained capacity.
• Theoretical characterization: the exact values of information rate,
shaping gain and absolute gap to capacity are calculated. Likewise,
a detailed normality test including two-dimensional (2D) Gaussian
fitting is realized.
• Experimental results: experimental demonstration of SCM-PSM in
coherent optical communications.
• Related first-author publications: [73] (conference), [74] (journal) and
[75] (invited conference).
2.2
Superposition Coded Modulation with
Phase Shifted Mapping (SCM-PSM)
Constellation shaping in SCM-PSM is passive. The superposition of independent streams of weighted binary phase-shift keying (BPSK) symbols
provides a simplified, flexible and effective means of fitting the capacityachieving Gaussian distribution with monotonically increasing precision.
11
(Ultra) long-haul networks
Non-bijective8 constellations may occur for certain configurations, which
together with the sub-optimal symbol allocation in symbol error rate (SER)
sense, suggest the use of maximum a posteriori probability (MAP) iterative demodulation. Interestingly, the superposition-based generation allows for considerable simplification of the soft-input soft-output (SISO)
demapper, thus conveniently alleviating the DSP complexity ratio between
receiver (RX) and transmitter (TX). These aspects are detailed in the
following subsections.
2.2.1
Generation and signal properties
Devised by L. Duang et al. in 1997 [58], the exploitation of the central
limit theorem to create discrete constellations with arbitrarily accurate
Gaussian power distribution is a mature concept. The passive generation of symbol distributions that asymptotically close the gap to the Shannon limit and the possibility of conducting algorithmic de-/mapping, motivated the study of this technique as part of sophisticated coded modulation
schemes. In our case, we employ PSM-based shaping approach within the
SCM framework with single-layer encoding strategy, bit-wise interleaving
and iterative decoding-demapping (ID) (identical to bit-interleaved coded
modulation (BICM) with ID).
In this subsection we spotlight the shaping mapper, i.e. PSM; a functional configuration that yields 2D constellations with near 1 average information rate per bit, yet effectively approaching capacity.
2.2.1.1
Digital transmitter
Rather than performing non-linear bit-to-symbol mapping, the output symbols are designed by linearly superimposing certain number of weighted
parallel data streams. This readily scalable and optimizable structure, facilitates information rate adjustment and the arrangement of diverse power
distributions by modifying the number of overlapped sequences and the
multiplicative factors. Besides, the fact that each branch may contain independent data with different coding strategies suggests its application as
medium access control (MAC) technique in multi-user scenarios. Mathematically, SCM with generalized mapping can be expressed as:
8
Bijective modulation: the constellation cardinality equals the number of unique bit
combinations for a fixed bit-tuple length (number of superimposed branches in SCM).
Non-bijective modulation: the constellation cardinality is lower the number of unique bit
combinations for a fixed bit-tuple length.
12
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
y = yI + jyQ =
N
X
ejπbn hn with hn = αn ejθn
(2.1)
n=1
where yI and yQ are the in-phase and quadrature components, N establishes
the number of superimposed branches (bits per symbol), bn corresponds to
the encoded binary bits and hn represents the complex weights that determine the symbol mapping scheme. It is now apparent that αn and θn
constitute the mapping parameters, thus their concrete selection fully dictates the properties of the output constellation. For instance, it is possible
to select between probabilistic shaping or geometric shaping. An example
of the former is the type-I sigma mapper [59], which reaches theoretical
memoryless AWGN channel capacity when N → ∞, and whose parameters
are given by:
αn = 1 for 1 ≤ n ≤ N
(
θn =
0
π
2
for 1 ≤ n ≤ N2
for N2 ≤ n ≤ N
(2.2)
The output constellations with type-I sigma for N = 4, 6 and 8 along
with the corresponding 2D histogram are shown in Figure 2.1. Even though
it is not surprising that the goodness of fit is inversely proportional to the
number of ambiguities, the information rate increase is only logarithmic
with N with type-I sigma mapper [3]; obviously slower than the number
of bits per symbol, N . Resolving the numerous uncertainties for large N
requires channel coding to induce equivalent redundancy9 for reliable transmission (>50% for N > 8), thereof drastically decreasing the throughput
under realistic implementation constraints (i.e. finite resources including
affordable latency and implementation penalties). For example, one important design constrain in LH/ULH systems is per-carrier throughput maximization which, in case of severe information rate penalties, leads to technologically unsupportable constellation expansion ratio (CER). One intuitive
solution to increase the cardinality for a given N is to assign different power
coefficients (αn ) to all the branches so as to force bijectivity. This mapper
is called type-II sigma [59] and belongs to the geometric shaping group.
9
Shaping techniques based on the superposition of independent data streams are
redundancy-free, thus the equivalent information rate reduction can be interpreted as
the necessary channel coding-induced redundancy that enables reliable transmission.
13
(Ultra) long-haul networks
(b) N = 6
(a) N = 4
(c) N = 8
Figure 2.1: Constellation diagrams (bottom) and 2D histograms (top) of
SCM with type-I sigma mapper for N ∈ {4, 6, 8}.
Unfortunately, the goodness of fit is strongly diminished, and the variable
power scaling across the branches poses undesirable energy inefficiency.
In the case of PSM configuration, each branch is applied a different
constant-amplitude complex weight. This simple procedure provides further symbol allocation diversity on the constellation diagram, increasing
the cardinality and maintaining equal peak magnitude in all the parallel
data streams. The latter turns convenient for analogue (digital-to-analogue
converter (DAC)-free) implementation. The mapper parameters are given
by:
αn = 1 for 1 ≤ n ≤ N
(2.3)
θn =
π(n−1)
N
for 1 ≤ n ≤ N
The output constellations with PSM for N = 4, 6 and 8 along with
the corresponding 2D histograms are shown in Figure 2.2. As previously
pointed out and inferred from the existence of ambiguities in the middle
case, the inter-layer interference may give rise to non-bijective mapping for
certain values of N , yielding a predominantly (but not totally) geometric
14
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
(a) N = 4
(b) N = 6
(c) N = 8
Figure 2.2: Constellation diagrams (bottom) and 2D histograms (top) of
SCM with PSM for N ∈ {4, 6, 8}.
shaping with 90%<rate≤100% (see Subsection 2.2.1.2). Nevertheless, it is
noteworthy that ambiguous mapping may be beneficial for some modulation
orders, where the constellation’s inter-symbol minimum Euclidean distance
is increased with respect to the bijective equivalent for the same average
power [7]. Consequently, the receiver sensitivity is nominally increased in
addition to the shaping gain10 . Assuming standard memoryless demapper,
10
Because the contribution to the error rate of each constellation symbol is directly
related to its probability of occurrence, such probabilities shall be taken into consideration
for estimating the performance and for enabling fair comparison among different mapping
approaches. One possible figure to coarsely observe the influence of those probabilities
is the statistically averaged minimum Euclidean distance, in which a simple probabilitydependent weighting of the per-symbol minimum Euclidean distance is applied. The
expression is given by:
N
N
X
X
min(d(xi , xj ))p(xi ),
i=1 j=1
where xi and xj are complex values representing concrete symbol locations from the
reference constellation, N is the constellation cardinality, min(•) is the minimum opera-
15
(Ultra) long-haul networks
inter-symbol minimum Euclidean distance is consistently related to SER
in channels exhibiting additive circularly symmetric noise. Needless mentioning that symbol labeling optimization is crucial for assuring low-error
binary data recovery, notably challenging task given irregular symbol grids.
2.2.1.2
Information rate
It is now clear that determinate mapping parameters in SCM produce ambiguous bit-to-symbol mapping that reduces the average entropy per bit,
then directly influencing the information rate, the code design and the receiver structure. Understanding and quantifying the achievable rates turns
of importance for the subsequent code design and thus determining what
configurations suit the systems’ requirements. In this subsection, the exact
numbers for SCM-PSM are presented. Despite other authors have carried
out similar exercise [3], our results introduce corrections in the accuracy
and add extra cases of study.
Following general information theory one can develop SCM’s theoretical information rate for any given mapping. Because ideal generation is
noiseless, the modulation information rate results in:
I(X; Y ) , H(Y ) − H(Y |X) = H(Y )
(2.4)
where I(X; Y ) is the mutual information between the output random variable Y and input random variable X to the SCM structure, and H(•) is
the discrete entropy operator. Now, applying 2.4 to SCM case:
I(y; α1 ej(πb1 +θ1 ) ,...,αn ej(πbn +θn ) ) = H(y)
(2.5)
where y ∈ y1 , y2 , . . . , yK represents the set of possible output complex values (K is the cardinality and it accounts for ambiguities caused by nonbijective mapping). Since the discrete entropy of the linear addition of the
constituent antipodal binary symbols will always be equal to the number
of conforming layers N , we can write:
H(y) =
K
X
P (ym ) log2 P (ym ) ≤ −
m=1
N
X
H(αn ej(πbn +θn ) ) = N
(2.6)
n=1
from which follows that the ideal transmitter output information rate is
upper-bounded as:
tor, d(•) is the Euclidean distance operator and p(x) is the probability of occurrence of
symbol x.
16
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
H(y)
N
Rate ≤
(2.7)
Inequality 2.7 makes explicit a well-known result: the achievable information rate is maximized when the output distribution from the mapper is
statistically uniform given a fixed input bit-tuple length (N ). In this regard,
it should be understood that pure geometric shaping is a redundancy-free
process that does not condition the channel code in the absence of noise.
Although this case does not strictly apply to all PSM cases, the average
rate never goes below 91.6% for up to 12 bits per symbol. Table 2.1 summarizes these and other results in a roster of the most relevant parameters
of the output constellations, including bit-tuple length, cardinality, discrete
output entropy and resulting information rate per bit.
Table 2.1: Roster of information theory data of PSM signaling. The constellation cardinality, its entropy and the average information rate per bit
are listed for {N ∈ N; 2 ≤ N ≤ 12}.
N
Cardinality
Entropy [bits]
Inf. rate per bit
2
3
4
5
6
7
8
9
10
11
12
4
7
16
31
49
127
256
343
961
2047
2401
2.000
2.750
4.000
4.937
5.500
6.984
8.000
8.250
9.875
10.999
11.000
1.000
0.916
1.000
0.987
0.916
0.997
1.000
0.916
0.987
0.999
0.916
Variations in the information rate can also be qualitatively evaluated
from the output constellation histograms (see Figure 2.2). Unlike N =
4 and 8, for which the probability of occurrence of all the symbols equals N1 ,
64-ary PSM shows clear non-bijective behavior. The lowest power symbol
(0 power in this case) is attained with up to 4 different combinations of 6
bits, and the symbols in the lowest-power middle ring are obtained with
2 unequal bit-tuples each. On this account, the total number of unique
discrete power locations (cardinality) is reduced to 49, which yields one
17
(Ultra) long-haul networks
of the lowest information rates in PSM; anyway imposing as low as 8.4%
worst-case channel coding redundancy, well within the 25%-overhead FEC
considered for high-speed LH/ULH coherent systems [20].
2.2.1.3
Gaussianity and shaping gain
Owed to diverse technological limitations (e.g. processing power cap and
partial knowledge of the interfering spectrum), it is customary to resort
to the average transition pdf of the optical channel for the calculation of
pragmatic capacity lower-bounds and the associated constellation shapes.
The harshness of this approximation is often relaxed assuming nonlinearity
compensation on the channel of interest, removing thereby great part of the
CD-induced memory and other intra-carrier effects. Interestingly, regardless of the use of compensation techniques, the time-averaged overall noise
interference (linear and non-linear) in the scenarios of concern has been long
proved additive Gaussian; for which the capacity-achieving input power distribution is known with exactitude: continuous zero-mean circularly symmetric normal. This entails two important observations, (i) Gaussian-like
constellation shaping can improve the receiver sensitivity (shaping gain)
both in the linear and non-linear (or mixed) regimes; and (ii) conditioned
to their cautious interpretation, well-known capacity results for memoryless AWGN channel can be extrapolated to the concrete LH/ULH optical
links discussed here, thus facilitating analogue analyses. Relying on these
points to support the relevance of the forthcoming study, this subsection
elaborates on the Gaussian-fitting accuracy of SCM-PSM (Part I), and its
influence on the theoretical SE and shaping gain over QAM modulation
(Part II) in the presence of memoryless AWGN.
Gaussianity : Any joint multi-dimensional distribution can be generated, and therefore studied, from the constituent marginals and the mathematical description of their dependence (copula). In our case the consideration framework is narrowed down to 2D output distributions, hence
two random variables corresponded to orthogonal quadrature components.
When general SCM is contemplated, the marginal distributions are given
by the superposition of antipodal symbols, singly scaled according to predefined mapping rules that also determine the copula. It follows from the
previous observation that not all the weight configurations enable the strict
application of the central limit theorem (CLT) and/or guarantee statistical independence. For instance type-I sigma mapper (see 2.2) ensures the
variables to be marginally Gaussian (classical CLT formulation holds) and
18
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
(a) Stat. indep. quadratures
(b) Stat. dep. quadratures (PSM)
Figure 2.3: 2D pmf of the N = 8 output signal when the same marginal
is uniformly sampled for each quadrature independently, and when those
marginals are dependent according to PSM equations.
independent, hence trivially leading to circularly-symmetric11 normal distribution (see evolution from Figure 2.1a to Figure 2.1c). In contrast, the
PSM structure (see in 2.3) couples the quadrature components via nonorthogonal phase rotations across the branches (notice the uniform phase
distribution between 0 and π(NN−1) ), whose unequal Cartesian projections
prevent classical CLT from holding. Although the asymptotic Gaussianity
of the marginals can be easily demonstrated (see appendix A), the proof
of joint 2D-normal distribution is not so immediate, as the degree of dependence changes for different values of N . As illustrative evidence of the
dependence between the quadrature components in SCM-PSM, Figure 2.3
shows the 2D pmf of the output distribution for N = 8 when identical
marginals are independently sampled with a uniform distribution (2.3a),
and when they are dependent according to PSM rules (2.3b). The observed differences confirm the dependency of the quadratures components
in PSM.
In order to evaluate the point-wise convergence to bivariate Gaussian
distribution of SCM-PSM constellations, a normality test is performed by
comparing theoretical and empirical cdfs for increasing values of N . The
metrics are both qualitative, through visual correlation of the overlapped
11
The orthogonality of the quadrature vectors imply uncorrelation when their individual mean equals zero.
19
(Ultra) long-haul networks
Cummulative Distribution Functions
Normal Fitting Error
(a) N = 4
(b) N = 6
(c) N = 12
Figure 2.4: 2D cdf of PSM output distributions plus ideal unit-variance
Gaussian distribution (top), and the corresponding fitting error (bottom)
for N ∈ {4, 6, 12}.
cdfs and the areas of discrepancy, and quantitative, with the percentage
of average fitting accuracy. Figure 2.4 groups together the results for
the visual assessment: each column is associated with a different value
of N (N = 4, 6 and 12 from left to right), the top row shows the empirical unit-variance cdfs together with the mesh-styled theoretical cdf of
the zero-mean unit-variance joint bivariate Gaussian distribution, and the
bottom row presents the fitting deviation magnitude12 for each grid position on the plane. As expected, increasing N has a strong influence on
the resulting 2D cdfs, whose resemblance with the theoretical normal cdf
becomes remarkable for N > 6 (see N = 12 in 2.4c). This is confirmed
by the evolution of the error magnitude, which becomes lower and more
uniformly distributed with N . In numbers, the normal fitting accuracy is
ij
ij
12
We define as errorij = |cdftheory
− cdfempirical
|, where i, j ∈ N and correspond to
grid positions on the plane.
20
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
expressed as the percentage of error magnitude reduction with respect to
uniform distribution (i.e. 0% means perfect uniform distribution and 100%
means perfect circularly-symmetric Gaussian distribution). The results are
collected in Table 2.2, all in all showing that despite the dependencies between the quadrature components, the output distribution in SCM-PSM
approaches capacity-achieving Gaussian as N increases (monotonically for
2 ≤ N ≤ 12).
Table 2.2: Goodness of fit of PSM output distributions with respect to ideal
Gaussian for {N ∈ Neven ; 2 ≤ N ≤ 12}. The values referenced to the ideal
uniform distribution and presented in percentage fashion (0% means perfect
uniform distribution and 100% means perfect Gaussian distribution).
N
Accuracy [%]
(uniform ref.)
2
4
6
8
10
12
0
2.3
50.8
71.8
86.2
90.0
Shaping gain : Although the asymptotic convergence to normal-like distribution surely increases the achievable capacity under the considered assumptions, practical constraints prevent N from reaching arbitrarily large
orders. Consequently, the goodness of fit is likewise bounded, diminishing the achievable gains now substantially lower than the ultimate limit.
The study of the potential benefits of constellation shaping for technologically feasible modulation levels is thence an important step in system optimization and design. In the following, we compare the capacity of SCM-PSM against QAM in terms of receiver sensitivity and SE for
{N ∈ N; 4 ≤ N ≤ 9}, which encompasses the lowest modulation orders
with positive Gaussian fitting (see Table 2.2). It is important to clarify
that, in this study, the contribution to the channel noise comes from both
linear and non-linear interference effects, which are treated as a single additive circularly-symmetric Gaussian random source. This consideration
facilitates the inclusion of the power dependent distortions in the SNR
parameter through an overall equivalent noise variance; and foremost, it
21
(Ultra) long-haul networks
makes possible to extrapolate the mutual information results in the linear
memoryless AWGN channel to the discussed optical transmission links.
Figure 2.5: Spectral efficiency per polarization versus signal-to-noise ratio
per symbol of PSM constellations for {N ∈ N; 4 ≤ N ≤ 9}.
Figure 2.5 shows the constrained capacity of SCM-PSM ∀N against
SNR for memoryless AWGN channel, the maximum SE of QAM for various orders is included for reference purposes. As explained in Section 2.2.1.2
and clearly visible in the highest-SNR regions in Figure 2.5, non-bijective
configurations (e.g. N = 6, see Figure 2.2b) cannot deliver the same maximum mutual information than same-order QAMs (always bijective). Note
that these values of maximum absolute SE coincide with the theoretical information rates previously calculated and collected in Table 2.1. Based on
the information provided by the capacity curves, shaping gains can be readily extrapolated by comparing the results for diverse modulation schemes.
Accordingly, we first present the gain in SNR for different values of SE
along with the associated shaping inaccuracies, providing thereby useful
data for calculating the power budget increase given a target bitrate and
modulation bandwidth. Secondly, the shaping gain is expressed in terms of
SE for specific values of SNR, which could be employed for improving the
channel code through the indicated extra redundancy.
In Figure 2.6a, the amount of extra SNR degradation that SCM-PSM
can endure over QAM for a range of SE values is shown ∀N . The information on this graph may be employed for calculating the transmission
22
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
reach extension of shaped signals for a pre-defined effective throughput.
Because the shaped power distributions are designed for concrete channel responses, SCM-PSM consistently outperforms QAM in low-to-medium
SE regions (peaking at ≈60% of N ), where the performance is principally
dominated by the channel noise. Beyond this threshold the gain decreases
rapidly, noticeably faster than other geometric shaping techniques (see [71])
for certain N . This is partly caused by the ambiguities and the reduced
Euclidean distance between symbols (e.g. Figure 2.2c) resulting from the
irregularly spaced clusters in SCM-PSM constellations, which dictate the
performance at high SNR. Notice that owed to the cross-like distribution
of QAM for N ∈ {5, 7, 9} (see Appendix A), the energy efficiency is slightly
increased hence the comparable shaping gains between N ∈ {5, 7, 9} and
the immediate inferior even orders.
Complementing Figure 2.6a, the remaining SNR gap to channel capacity for SCM-PSM is depicted in Figure 2.6b. This graph quantifies the
inefficiency (in SNR) of the output constellations as a result of their discrete nature and their imprecise Gaussianity (see Figure 2.4). Inferred
from the non-crossing curves progressively approaching capacity with N ,
the first conclusion is that the gap to capacity closes ∀SE as N increases.
In addition, it is noteworthy that despite the shaping gain increases and
even peaks at specific SE values ∀N , the gap to capacity of SCM-PSM
increases monotonically with SE. It follows that increasing the shaping
gain does not mean closing the distance to channel capacity and, therefore,
running the system at SEs where maximum shaping gains are attained
may not be the most power efficient option. As an example, consider 64ary PSM encoded with an ideal FEC with ≈35% total overhead. In that
case, the SE (≈3.9 bit/s/Hz/pol) achieves maximum shaping (≈0.5 dB,
see Figure 2.6a) though still missing ≈0.4 dB to Shannon limit (see Figure
2.6b). Given this configuration, setting the redundancy overhead to 50% (3
bit/s/Hz/pol equivalent SE) would reduce the shaping gain by negligible
0.06 dB while decreasing the gap to capacity by 0.3 dB extra (less than
0.15 dB remaining). These SNR savings add to the ≈3 dB obtained by
accepting 1 bit of extra redundancy. Nevertheless, it is not often that we
are willing to sacrifice effective throughput, so the next observation shall
be regarded. At 4 bit/s/Hz/pol of SE, the shaping gains for 64-ary and
128-ary PSM converge to the same magnitude (≈0.49 dB), nonetheless, the
gap to capacity is ≈0.27 dB lower for the second one (see superimposed intervals in Figure 2.6b) at the expense of ≈10% extra overhead. The process
of combining higher-order constellation with stronger coding schemes, so as
23
(Ultra) long-haul networks
(a) Shaping gain
(b) Gap to Shannon limit
Figure 2.6: Receiver sensitivity gain and remaining gap to Shannon limit
of PSM constellations for {N ∈ N; 4 ≤ N ≤ 9}.
to gaining extra sensitivity and at the same time maintaining the effective
bitrate is called coded modulation.
An alternative interpretation of the benefits of constellation shaping is
that, for the same SNR, higher information rate is possible as compared
24
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
Figure 2.7: Shaping gain in SE for {N ∈ N; 4 ≤ N ≤ 9}.
to non-shaped signals. A case where this information may be of pragmatic interest is when the transmission reach and link properties are given
(approximately fixed SNR) and the error rate requisites to guarantee successful error correction are not met. The gain values of SCM-PSM versus
uniform QAM ∀N are presented in Figure 2.7 up to 35 dB of SNR. The
top white-background part correspond to the area where PSM outperforms
QAM, and vice versa for the bottom grey-background part. Previously
hinted in the explanation of Figure 2.6a, the achievable performance, and
then the shaping gain, is dictated by the ambiguities and the constellation
grid irregularities for high values of SNR. This is apparent in Figure 2.7,
where the falling-slope trajectories change with N exactly as the number of
ambiguities and the constellation diagrams do. Note that the SE for those
ambiguous modulation orders stabilizes below zero (e.g. N = 5 and 6),
and it converges to zero gain in bijective cases (e.g. N = 4). Further information can be extracted from the existence of valleys and the depth of
their minima after crossing the zero-gain point. If the gain curve describes
fall-and-rise profile, it indicates that once the inter-symbol Euclidean distance start dominating the information rate, the regular grid of QAM is
favorable in the corresponding SNR interval. In this line, the depth of
the minima relative to the convergence gain tells about the severity of the
grid-driven penalties. For example, both 16-ary and 32-ary PSM exhibit
clear valleys in their profile, thus their grids entail error rate degradation
25
(Ultra) long-haul networks
as compared to same-order QAM; however, the average Euclidean distance
is larger for 32-PSM, given that the minima is the same for both orders,
but not the convergence gain (0 and -0.06 bit/s/Hz/pol for 16-ary and 32ary respectively). Especially attractive is the 64-PSM configuration, whose
gain profile indicates that the shaping process does not penalize the Euclidean distance (as explained in Section 2.2.1.1) but only the information
rate (see Table 2.1).
All these scattered examples demonstrate that shaping gain and power
efficiency are dependent but not trivially related, requiring a meticulous
optimization of the modulation parameters in which the coding system (indeed never ideal) plays a fundamental role13 . Particularly in SCM-PSM,
the coding approach is not only fundamental for maximizing the power efficiency but also to resolve the ambiguities caused by the mapping/shaping
architecture that, in short, prevent the successful transmission of uncoded
data irrespective of the SNR.
2.2.2
Detection and de/coding
As previously discussed, non-bijective symbol arrangements may occur for
certain constellation orders during the mapping procedure in SCM-PSM.
Even though we have shown that the information rate does not decrease by
more than 9% (see Table 2.1), successful signal demodulation is subjected
to the complete mitigation of the ambiguous inter-layer interferences. This
task becomes further challenging owed to the minute Euclidean distances
between some non-overlapping symbols, aggravating the uncertainty under
the low-to-medium SNR conditions in which shaped constellations are intended to operate (this holds for any power-constrained non-equally spaced
constellation grid). Consequently, highly inaccurate demapping decisions
are initially made, strongly suggesting iterative belief propagation between
demapper and decoder. The fundamental idea is that both blocks repeatedly exchange Bayesian information to mutually improve their extrinsic
information, thus progressively refining symbol detection, ambiguity resolution and error correction. Adjusting the parameters of such iterative
SISO receiver to efficiently use up the available shaping gain, demands
meticulous study of the extrinsic information flow and its dependence with
SNR, inter-symbol distance, symbol labeling and encoding scheme. In spite
of its conceptual simplicity, the latter is a convoluted optimization process
13
Implementation penalties, processing latency and demodulation robustness are other
aspects that determine the actual system design.
26
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
with big impact on the processing complexity and the error correction capabilities.
In this concern, we next elaborate on SCM-PSM from the digital receiver perspective, subdividing the analysis into (i) coded modulation structure (ii) demapping and (iii) channel code. The essential properties of SCMPSM regarding the design of the iterative SISO receiver and the associated
channel code are discussed, as well as general considerations of practical
interest.
2.2.2.1
Coded modulation and iterative SISO receiver
Two main reasons motivate the use of bit-oriented coded modulation together with PSM. First, PSM works autonomously when the superimposed
layers are statistically independent and present uniform data distribution,
thus generating asymptotically accurate 2D Gaussian-like constellations independently of the FEC and encoding strategy. Second, when linear mapping14 is used, symbol labels are typically built by concatenating the binary
values of the N conforming branches (one bit per layer). Even if different
bit-significance conventions are employed, the mapping function remains
systematic according to PSM equations (see 2.3), compromising the labeling freedom and increasing the optimization complexity. That is, the
adoption of a bit-oriented coded modulation framework is encouraged for
facilitating seamless integration of the PSM shaping module, and also the
joint de/coding and de/mapping optimization in spite of the rigid labeling.
In this regard, bit-interleaved coded modulation BICM has been widely
proved as enabling bit-oriented approach whose flexibility allows for the
coherent application of powerful families of binary codes on virtually any
modulation format [76]. The respective extension of BICM to support
iterative decoding, BICM-ID15 [77], was employed in this investigation. It
14
We refer to linear mapping as the process of generating the symbols based on the
linear superposition of weighted antipodal values. In contrast, non-linear mapping is the
process of assigning bits to a pre-defined constellation (i.e. 16-ary QAM) according to
certain rules that do not hold a linear mathematical relation with the input bits.
15
A posteriori probabilities (APPs) in the form of log-likelihood ratios (LLRs) per
coded bit are repeatedly circulated among the demapper and decoder. These factors are
calculated as:
LLRP OST (cm ) = log
p(y|cm = 0)
p(y|cm = 1)
(2.8)
where cm is the mth coded bit in the block being processed, and p(y|cm ) is the conditional
27
(Ultra) long-haul networks
(a) Transmitter
(b) Receiver
Figure 2.8: Digital transmitter and SISO iterative receiver of BICM-PSM.
should be noted that SCM with single-layer encoding strategy and bit-wise
interleaving is identical to BICM with superposition mapping. For reading
ease, and without loss of generality, we refer to SCM with single-layer
encoding strategy, bit-wise interleaving and PSM as BICM-PSM. Figure
2.8 shows the the schematic of the digital transmitter and iterative receiver
in BICM-PSM.
probability of y given cm . The extrinsic (new) information is extracted from the generated
APPs by removing the fraction of old a priori content (LLRP RIOR ):
LLREXT RIN SIC = LLRP OST − LLRP RIOR
(2.9)
Then, the LLREXT RIN SIC are passed to the next block (demapper towards decoder
or vice versa) as new a priori information. This continuous dialog progressively increases
the reliability of the transferred APPs and, in turn, the decoding accuracy and bit error
rate (BER) performance. After a predefined number of iterations, the LLRs of the
information bits are fed into a slicer that outputs the estimated binary sequence.
28
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
Figure 2.9: Path evolution on the tree-diagram model of PSM. The constellation diagrams for every state > 0 are included.
2.2.2.2
Demapping
In relation to the algorithms for SISO demapping, MAP methods are extensively applied as they give optimal estimation of the states, or outputs, of a
Markov process observed in AWGN [78]. Bridging to our case of study, after signal propagation in the target scenarios, optimum symbol estimation
is achieved if the mapping procedure behaves as a Markov process. This is
the case of superposition modulation, whose operation can be modeled as
a tree diagram [9] in which two branches emerge from every state, both accounting for each of the possible outputs of the weighted BPSK symbol (see
Figure 2.9). Despite attaining optimal layer separation and demonstrated
intra-channel nonlinearity mitigation [79], general MAP symbol estimation
suffers from scalability limitations due to the quadratic dependence of the
computational complexity with N . For this reason, various approaches
have been presented in the literature so as to alleviate this circumstance.
For instance, seizing upon the fact that superposition modulation modeled
as a Markov process, the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [80]
can be used instead of the general MAP estimator to strongly reduce the
computational load and processing latency without performance degradation.
The estimation accuracy of the average transition probability of the
channel is another significant factor to muse over when employing SISO
demapping. As argued in previous sections, the combined imprint of both
linear and non-linear impairments after propagation in the scenarios re29
(Ultra) long-haul networks
Figure 2.10: Recovered 16-ary PSM constellation after 240-km uncompensated SSMF transmission at 6 dBm launched power. The per-cluster covariances estimated by the expectation maximization algorithm are shown
at 99% confidence.
garded in this chapter is typically simplified to additive and circularlysymmetric Gaussian noise. Whereas this approximation has been widely
adopted, its validity has been likewise proved positively gradual with the
transmission reach (lumped amplification and WDM technology are invariants), originating an apparent noise correlation for insufficient propagation
distances. Albeit this fact would have an impact in symbol detection for any
constellation diagram, the extreme proximity of the symbols in the lower
power areas in PSM output constellations makes the accurate calculation
of per-cluster centroids and covariances especially useful. In this investigation, we make use of a two-step iterative procedure called expectation
maximization (EM) [81] for optimum estimation in a maximum likelihood
sense (see Figure 2.10). The calculations are simplified by assuming a mixture of Gaussian (MoG) statistical model for each cluster, and restricting
the MoG to a 2D space (2x2 covariance matrix). See [82] for details on EM
algorithm applied to this task.
2.2.2.3
Channel coding
Indispensable for providing the required coding gain (virtually increasing
the Euclidean distance among symbols) and diversity gain (for ambigu30
2.2 Superposition Coded Modulation with Phase Shifted Mapping
(SCM-PSM)
Figure 2.11: EXIT profiles of PSM soft demapper for {N ∈ N; 4 ≤ N ≤ 9}
at 8 dB SNRb, and BCJR decoder for 0.5 channel code rate convolutional
code.
ous multi-layer interference resolution), the election/design of the adequate
code for ID receivers is inseparably linked to the modulation properties,
the SNR and the requirements of the application (e.g. processing latency
or bit error tolerance). It follows that claiming the ubiquitous superiority of any code is inexact. Joint optimization of the transmitter and receiver parameters for pre-defined SNR values is required to minimize inefficiencies in the SISO demodulation and thus assess appropriateness of
the proposed coded modulation architecture. This rigorous process starts
observing the evolution of the extrinsic information transfer flow between
demapper and decoder, intuitively carried out with the help of extrinsic
information transfer (EXIT) charts (see [83] for detailed information).
In Figure 2.11, the EXIT profile of the PSM BCJR demapper for
{N ∈ N; 4 ≤ N ≤ 9} is depicted at 8 dB of SNRb together with the one
of BCJR decoder for 50% overhead (0.5 channel code rate) convolutional
code. Due to the high symbol density in the lowest power area of Gaussianlike geometrically shaped constellations, the MI from the demapper in the
absence of a priori information (M IP RIOR;DEM ) is considerably weak as
compared to uniform distributions . Particularly in PSM, the symbols can
be arbitrarily close or even overlap (ambiguous mapping) resulting in EXIT
functions that become convex with the modulation order (clearly visible
31
(Ultra) long-haul networks
for M IP RIOR;DEM < 0.4). Already pointed out previously, this situation
forces the decoder to generate strong extrinsic information based on the
hardly reliable initial a priori MI from the demapper in order to prevent
the blockage of MI improvement at an early stage (examples of blockage
are illustrated as red points for N = 9 and 8). This represents a major
restriction imposed, to larger or smaller extent, by all power-constrained
geometrically shaped constellations (see [71]) and makes of zero-iteration
receivers an inappropriate choice.
Given that PSM symbols are built from the successive accumulation of
phase-shifted antipodal values, the MI profile for all the modulation orders
converges to that of BPSK when decoding reliability tends to 100%. Notice
that an alternative interpretation of the superposition of branches is that
N − 1 addends act as deterministic interference channels for the remaining BPSK-modulated one. When this interference noise is compensated,
equivalent performance to single-branch transmission is accomplished independently of N and the level of SNR. This has an important implication: since the maximum MI out of the demapper will always end at
a high value, the profile of the decoder can similarly be designed to end
at a very high point. This means that, unlike in high-order QAM for instance, strong coding gains16 are not strictly necessary in order to achieve
complete error correction given sufficient number of iterations. In addition,
because maximum output mutual information from the demapper is steady
∀N and fixed SNR, the transfer function will progressively steepen as the
N -dependent Euclidean distances reduce. This emphasizes the need for
iterative receivers under the assumption of well-fitted demapper-decoder
EXIT functions, which is the idea behind efficient code design.
Words on selected code : Even though each modulation order requires
specific code design, for the proof of concept a 0.5 channel code rate convolutional encoder with (5, 7)8 polynomial generator was implemented for the
forthcoming experimental demonstration. The BCJR simplification of the
MAP decoder is implemented for optimum performance [78]. This scheme
enables complete layer separation within 10 iterations for the three experimentally studied cases, and sets the number of effective bit/symbol very
close to the maximum shaping gain regions (see Figure 2.6a).
16
The MI that the decoder is able to deliver to the demapper for strongly reliable prior
extrinsic information
32
2.3 Experimental demonstration
2.3
Experimental demonstration
Up to this section, BICM-PSM has been revisited from the point of view
of digital communications and information theory. It has been concluded
that, in channels where the noise statistics within the channel of interest
can be modeled as memoryless additive Gaussian, BICM-PSM can effectively approach capacity for sufficiently high modulation levels when the
coded modulation scheme is designed/adjusted accordingly. These promising gains, however, assume that signal generation and recovery are, not
just possible, but ideal. This is truly harsh conjecture in coherent optical communications, further emphasized by the potential difficulties that
irregular constellation grids pose on blind phase-noise compensation and
channel equalization. The ensuing implementation penalties jeopardize the
sensitivity gain provided by the shaping process in different manners. In
the transmitter, insufficient quantization resolution (when DACs are used)
or non-linear electro-optical conversion make the output power distribution
divert from the optimum one. During propagation, varying the conditions
that support circularly-symmetric additive Gaussian noise approximation
will reduce the shaping effectiveness. In the receiver, non-linear effects
during field recovery and low digitization resolution will likewise distort
the optimum shape and thus the associated gain. Finally, unsuitable DSP
algorithms may produce noise undercompensation or enhancement thus
diminishing the effective SNR coming into the SISO receiver, formerly conceived for optimum operation in a different working point. Because of
all the above-mentioned aspects, studying the experimental feasibility of
BICM-PSM is indispensable for correct end-to-end power budget calculation. This is the context of the following section, where we have focused on
the evaluation of the implementation penalties with respect to ideal digital
transmission when standard DSP algorithms are used for signal recovery
and impairment compensation. This study represents the just a first steps
towards complete characterization.
Dual-polarization single-carrier 16-ary, 32-ary and 64-ary PSM are experimentally generated, transmitted and recovered. Numerical simulations
of digital transmission over purely AWGN channel for coded and uncoded
configuration are included for benchmarking optical B2B measurements.
Theoretical, simulated and experimental effective SEs are compared and
discussed for all the cases of study. Regarding transmission, we benefit from
EM-based cluster parametrization to achieve a slight sensitivity improvement after non-linear propagation over 240-km uncompensated SSMF.
33
(Ultra) long-haul networks
Figure 2.12: Schematic of the experimental setup for dual-polarization
single-carrier BICM-PSM transmission. Insets: optical eye diagrams for
N = 4 (a), N = 5 (b) and N = 6 (c).
2.3.1
Testbed
The schematic of the experimental setup is shown in Figure 2.12 along with
root-raised-cosine (RRC) pulse-shaped DP eyediagrams for N ∈ {4, 5, 6},
which were captured after ≈10-s accumulation time. As a hint of the shaping effect, it is distinguishable that the average point density concentrates
around the lowest power range as N increases. In the transmitter side, a
1550-nm external-cavity laser (ECL) with ≈100 kHz linewidth is used as
the light source. The output of the laser is modulated by a dual-drive optical I/Q modulator, which is differentially driven (≈2 Vpp) directly from the
outputs of a 10-bit resolution arbitrary waveform generator (AWGen) employed for electrical signal generation. The sampling rate and the baudrate
are fixed over the entire experiment at 12 GS/s and 6 GBd (2 S/symbol)
respectively. A pseudo-random bit sequence (PRBS) of length 215 − 1 were
digitally processed to generate the waveforms according to Figure 2.8a. The
digital signal processing consists of: encoding, interleaving, mapping, upsampling and pulse shaping. First, 150000 bits are encoded block-wise in
a single-level strategy with a 0.5 rate convolutional code and (5, 7)8 polynomial generator; afterwards, half-random interleaving is realized (lengths
vary within 10000-15000 bits depending on the modulation level) to break
the sequential correlation and increase diversity order. The sequence is
rendered parallel, offering further scrambling and preparing the data for
PSM (see Subsection 2.2.1.1 for details about 1 S/symbol signal generation). In the last digital block, 50 quadrature phase-shift keying (QPSK)
pilot symbols are inserted every 1000 data symbols (5% overhead) for estimating the channel response and compensating the frequency offset from
coherent intradyning; finally, the aggregated waveform is 2x oversampled
and pulse-shaped with 0.8-roll-off factor (β) RRC.
34
2.3 Experimental demonstration
After the optical I/Q modulator, dual-polarization multiplexing is emulated; the transmitter output is amplified and then launched into an uncompensated fiber link for transmission. The link is made up of 3 spans of
80-km SSMF and Erbium-doped fiber amplifiers (EDFAs) for compensating
the attenuation after each span.
In the receiver, standard coherent detection is carried out. The incoming optical data signal is coherently mixed with the local oscillator (LO)
(ECL, ≈100 kHz linewidth) and, subsequently, the outputs from the four
balanced photodiodes are digitized at 40-GS/s and stored for offline processing in a digital storage oscilloscope (DSO) with 13-GHz analogue bandwidth. The receiver DSP comprises I/Q imbalance compensation, CD compensation, low-pass filtering, clock recovery, joint polarization demultiplexing and equalization, carrier frequency and phase recovery and SISO BICM
receiver with ID.
DSP workflow and punctualizations : After standard front-end correction and frame synchronization, the 10-tap filters of the adaptive equalizer are first estimated with constant-modulus algorithm (CMA) [84, 85]
operating on the QPSK pilot sequence. After pre-convergence, PSMoptimized radius-directed equalizer (RDE) [85,86] (sometimes referred to as
multi-modulus algorithm (MMA)) is used for fine tap adaptation17 . Note
that the circular symmetry that PSM constellations exhibit results in fewer
discrete constant-magnitude levels than in the analogous square-QAM ∀N ,
hence showing faster convergence rate for the same number of taps and
SNR conditions. Table 2.3 presents the number of discrete power levels for
both QAM and PSM up to N = 12, as well as the multiplicative scaling
ratio associated with PSM for each N that makes the minimum distance
between magnitude levels be the same as in QAM18 .
Next, carrier recovery is realized with second-order digital phase-locked
loop (DPLL) [87], where minimum Euclidean distance metric is used. The
proportional loop gain is minutely adjusted for each modulation order and
SNR level. The information bits are recovered after 10 SISO iterations of
error correction plus demapping optimization. For the iterative receiver to
17
Despite knowing the QPSK sequence beforehand enables 2-rank complex channel
matrix calculation and inversion, blind CMA was chosen instead for reasons related to
code integrability. After running RDE, the equalizer filters’ taps showed less than 5%
average discrepancy between both methods in MMSE sense. The implications on the
computational complexity are disregarded.
18
Larger than 1 means that the minimum distance between magnitude levels/rings in
QAM is smaller than in PSM and vice versa.
35
(Ultra) long-haul networks
Table 2.3: Number of constant magnitude levels in QAM and PSM constellations for {N ∈ N; 4 ≤ N ≤ 12}. The rightmost column shows the scaling
factor to be applied on QAM in order to equalize the obtain similar minimum distance between constant magnitude levels than PSM constellation
for a given N . A value larger than 1 means that the minimum distance
between constant magnitude levels is smaller in QAM by as much as the
respective value indicates.
# of Mag. levels
N
QAM
PSM
Min. Inter-ring
distance ratio ( QAM
PSM )
4
5
6
7
8
9
10
11
12
3
6
9
20
32
56
109
235
398
2
4
5
9
12
17
34
63
68
2.24
2.89
4.02
3.75
2.86
2.52
2.89
1.13
4.77
work optimally, the demapper is fed with maximum likelihood covariance
estimations calculated through the EM algorithm (see Subsection 2.2.2.2).
This exercise boosts the performance of the ID receiver in situations where
the average pdf varies between clusters like, for instance, when random
phase noise is not totally compensated by the DPLL. In our case, phase
noise is intentionally introduced via self-phase modulation (SPM) for analysis purposes. Whenever the average transition pdf at the input to the
SISO receiver is well approximated by circularly symmetric additive noise,
per-cluster pdf estimation is meaningless and shall be sidestepped.
2.3.2
Results
The discussion on the results is subdivided into (i) optical B2B, (ii) spectral
efficiency, and (iii) transmission.
36
2.3 Experimental demonstration
Figure 2.13: BER as a function of SNR per information bit for optical
back-to-back measurements and simulated digital transmission over AWGN
channel. Coded and uncoded configurations are considered.
2.3.2.1
Optical back-to-back
Figure 2.13 shows BER as a function of SNRb for measured optical B2B
after soft-decision decoding for 16-ary, 32-ary, and 64-ary PSM. Numerically calculated BER curves for coded and uncoded PSM after AWGN-only
channel transmission are included as reference. All the BER measurements
result from averaging three samples of 150000 bits each. Accounting for
the possibility of using an additional outer decoder in a concatenated coding/decoding structure [28], the pre-FEC BER threshold (1.1 · 10−3 ) of a
0.9375 code rate Reed-Solomon code [29] is traced for analysis purposes.
The grayed area represents the numerically calculated BER boundary for
BICM-PSM. This error bound corresponds with the performance of 2-ary
PSM (BPSK), and it is independent of the number of iterations and the
modulation order (see Subsection 2.2.2.3).
As previously explained, the occasional information rate reduction in
PSM’s constellations requires using a proportional percentage of redundancy that enables sequence corroboration; otherwise, data recovery is impossible independently of the level of SNR. This is apparent in the uncoded results for 32-ary and 64-ary PSM, whose ambiguities cause BER
floors around 3.2 · 10−2 and 1.3 · 10−1 respectively. Demodulation is pos37
(Ultra) long-haul networks
Figure 2.14: Average-power normalized experimental constellation diagram
of 64-ary PSM for ≈30 dB OSNR (gray). The ideal symbol distribution is
superimposed (red). The histograms for each quadrature of both data sets
are included at the corresponding sides of the measurement box.
sible with SISO iterative decoding , showing strong coding gains (≈10 dB
for 16-ary PSM at the outer decoder pre-FEC BER threshold) with turbo
cliffs starting at 2 dB for N = 4, 4 dB for N = 5, and 6 dB for N = 6 for
the numerically simulated data. Experimental data exhibits similar behavior, showing the first successful demodulation and decoding of PSM signals
in coherent optical communications. This results confirm that ambiguities
and transmission errors can be simultaneously resolved employing BICM
structure with ID, as long as the channel code rate is higher or equal than
the constellation’s information rate (see Table 2.1) and the EXIT profiles
of both demapper and decoder do not cross at the desired SNR level [21].
Even though coded modulation makes demodulation possible for increasing SNR, it is visible that simulated coded 16-ary and 32-ary PSM
experience error correction limitation starting at ≈3.5 dB and ≈5.5 dB
SNRb respectively. The reasoning is intuitive from the analysis of the
EXIT chart (see Figure 2.11). As SNRb decreases beyond certain threshold, PSM’s demapper EXIT profile starts moving downwards to cross with
the decoder’s (static with SNR) at a point where M IEXT RIN SICDEM < 1,
38
2.3 Experimental demonstration
hence preventing complete error correction irrespective of the number of
iterations (see the convergence towards the system’s error-rate boundary).
Note that, in contrast to the demapper EXIT functions of the three considered cases, the convolutional decoder evolves from concave to convex as
the input MI increases. This causes that the iterative exchange of extrinsic
information gets blocked at an early stage for very low SNRb, thus producing worse BER than the system’s minimum error rate (clearly visible in
16-ary PSM for 0 dB ≤ SNRb ≤ 3.5 dB). These observations illustrate the
importance of joint demapper-decoder optimization in coded modulation
schemes for maximum computational efficiency and error correction capabilities. Regarding experimental curves, owed to implementation penalties
(1.7 dB, 3.3 dB and 6 dB for 16-, 32- and 64-PSM at the outer decoder preFEC BER threshold) requiring higher SNRb to obtain similar BER, such
error floor is not visible anymore for 32-ary PSM, whose turbo cliff starts at
7.5 dB SNRb. However, the tendency of the boundary is strictly followed
in the 16-ary PSM case, where the implementation penalty is not so severe.
Realize that implementation penalties change the demapper EXIT profile
for every N . In all cases, the major contribution to implementation penalties comes from the DPLL not totally compensating for the lasers phase
noise. Notice in Figure 2.14 the clear separation of magnitude levels (see
Table 2.3 and comments in Subsection 2.3.1), and that hardly any distortion is appreciated in the symbol locations with the exception of a slight
compression in the quadrature component. Despite the DPLL loop parameters are optimally adjusted for every measurement point and the use of
EM-based per-cluster pdf estimation, the low baudrate together with the
stringent SNR conditions in the maximum shaping gain ranges pose clear
sensitivity degradation.
2.3.2.2
Spectral efficiency
Figure 2.15 shows the conversion to effective SE per polarization versus
SNRs for the BER results at the outer FEC BER threshold (1.1 · 10−3 ) in
Figure 2.13. The BER to SE calculations are conducted under the assumption of brickwall power spectral density (PSD) for all the cases (see [88]
for details on the conversion method). Maximum SE-wise shaping gain regions for N ∈ {4, 5, 6} are shaded within the corresponding SNRs intervals
(see Figure 2.7), and effective SE curves of the simulated BER results after
inner decoding for N ∈ {4, 5, 6} (discontinuous lines) serve as benchmark.
Full-filled symbols represent the effective SE of each of the simulated cases
under study at above-mentioned pre-FEC BER threshold. The SE values
39
(Ultra) long-haul networks
Figure 2.15: Effective SE as a function of SNRs for optical back-to-back
measurements and simulated digital transmission over AWGN channel.
Color-matched shades indicate maximum shaping gain SNRs intervals for
each case.
account for both the 0.5 and 0.9375 channel code rates of the inner and
outer decoder respectively (0.5 · 0.9375 = 0.46875 equivalent channel code
rate). As expected from the analysis in Subsection 2.2.1.3 (see Figure 2.6a),
the receiver sensitivity for all the modulation orders fall within the ascribed
areas of maximum shaping gain for ≈50% effective information rate. This
means that the coded modulation configuration elaborated in Subsection
2.2.2 plus the suggested outer FEC decoder can perform at < 10−15 BER
with a SNRs tolerance that falls within SE-wise maximum shaping gain
intervals ∀N . However, it should be noted by comparing the simulated
results against those in Figure 2.6b that the achieved SE is clearly apart
from the capacity limit for all the cases. Taking as example 64-ary PSM,
we see in Figure 2.15 that gap to capacity is ≈4 dB, whereas the theoretical results in Figure 2.6b indicate that the achievable capacity at the
aggregate information rate (≈2.81 bit/s/Hz/pol) is approximately 0.1 dB.
This means that, even though the the concatenated code structure makes
possible to tolerate the stringent SNR levels at which shaping gain is maximized, the efficiency of the 0.5 convolutional code at the target SNR from
the EXIT perspective (understood as how well it fits the demmapers’) is
very low. This observation reemphasizes the importance of the joint opti40
2.3 Experimental demonstration
mization of the coding/decoding strategy and the modulation format, and
totally discards the combination of 0.5 code-rate convolutional codes with
PSM.
For their part, half-filled symbols stand for the effective SE of the experimental results at the outer decoder pre-FEC BER threshold. Unlike
simulated curves where the PSD was assumed rectangular (β → 0), experimental values account for β = 0.8 pulse-shaping in addition to the 0.46875
overall channel code rate. Therefore, the effective SE per polarization is
6 [GBd]
given by N · 0.46875 · 6 [GHz]·(1+β)
. The results prove that implementation
penalties can increase the necessary receiver sensitivity well beyond the
areas of shaping gain. Considering that most penalties are caused by incomplete/incorrect phase noise compensation, we conclude that standard
second-order DPLL is not an appropriate algorithm for phase noise compensation in the scenarios of interest when shaping gain is pursued; suggesting
the use of more robust alternatives such as data-independent frequency
pilot tones [46].
2.3.2.3
Transmission
Figure 2.16 presents the measured BER as a function launch signal power
for 16-ary, 32-ary and 64-ary PSM after 240-km uncompensated SSMF
transmission. Recovered constellations with EM-based covariances for each
cluster are shown as insets. The analyzed input powers range between 0
dBm and 8 dBm in steps of 0.5 dB, and the BER values are averaged
over three traces of 150000 bits each. Zero counted BER transmission is
measured below 2.5 dBm, 6 dBm and 7.5 dBm for 64-ary, 32-ary and 16ary PSM respectively. Because of the insufficient accumulated dispersion,
SPM manifests as phase noise on the constellation diagrams. The resultant
ellipticity and the noisy clusters and their centroids is correctly tracked by
the EM algorithm hence improving the performance of the ID routine and
making demodulation possible at considerably high launched power levels
(up to 7.5 dBm for 16-ary PSM). Compared to the approach when the
SNR is assumed circularly symmetric Gaussian with variance estimated
from the worst-case cluster, the receiver sensitivity experiences an average
improvement of 0.6 dB at the expense of ≈ (2N − 1) times more complexity
in the pdf calculation stage. The previous observation denotes the local
imprecision of the assumption about the circular symmetry of the noise,
and simply suggests considering non-Gaussian constellation shapes.
41
(Ultra) long-haul networks
Figure 2.16: BER as a function of launched power after 240-km uncompensated SSMF transmission. Insets: Recovered constellations with the
estimated covariances per cluster for N = 4 (7 dBm, right), N = 5 (4
dBm, middle) and N = 6 (1 dBm, left).
2.4
Summary
Current long-haul networks need higher throughput-times-distance product. The underlying limitation results from non-linear effects (random and
deterministic unknown) opposing to the power-driven SE gains that technologies like AMF or grid-less WDM provide to these finite-bandwidth
links. Urged by the need for more throughput and supported by the
progress in digital processing capabilities, manifold DSP techniques aiming
the compensation and/or mitigation of such fundamental nonlinearities are
being intensively looked into. In this context, coded modulation featuring constellation shaping is deemed an attractive approach to complement
existing compensation techniques for further sensitivity improvement.
Given our band-limited power-constrained channel, molding the power
distribution of the transmitted signal can close the gap to capacity. Unfortunately, such optimistic goal is subjected to the perfect knowledge of
channel’s transition pdf, which is computationally intractable; and the use
of infinitely large constellation size (continuous input), which is impractical from the implementation point of view. Consequently, it is accustomed
to approximate the overall signal interference as additive power-dependent
42
2.4 Summary
2D Gaussian noise as well as the use of pragmatic modulation orders. All
in all, the achievable capacity is constrained. Even if the receiver sensitivity is still improved, the shaping gains are diminished to a point where
the complexity-to-gain ratio turns unfavorable; then bringing the interest towards constellation shaping methods that exploit the assumptions to
simplify the process.
In this regard, SCM had been proposed as scalable and flexible coded
modulation scheme that allows for considerable complexity reduction owed
to its inherent linear bit-to-symbol mapping. In combination with PSM,
SCM can (i) passively generate circularly-symmetric Gaussian-like constellations that asymptotically approach (with modulation level) memoryless
AWGN channel capacity while maintaining high information rate; and (ii)
simplify the general MAP demapper which, over and above, becomes optimum as symbol generation in SCM can be modeled by Markov process.
These reasons motivate the consideration of SCM-PSM in the context of
long-haul scenarios, as well as studying its information transfer capabilities, and the experimental assessment of its implementation feasibility in
coherent systems.
This investigation
In this chapter, SCM-PSM is revisited as a promising coded modulation
scheme with constellation shaping capabilities for increasing the achievable
capacity of LH and ULH optical transmission systems at lower complexity
as compared to equivalent techniques. In the first block, signal generation
and detection are thoroughly covered. The analysis shows that PSM passively generates asymptotically accurate 2D Gaussian constellations with
less than 8.4% information rate penalty (minimum 0%) for up to 12 bits
per symbol. Resorting to the Gaussian noise model, PSM is thus able to increase the achievable capacity of the considered scenarios. These mapping
ambiguities have been proved beneficial in PSM for certain modulation
orders (i.e. 6 bit/symbol), where the average Euclidean distance among
symbols is higher than in QAM when the average power is equalized. The
asymptotic convergence rate towards Gaussian distribution as well as the
fitting inaccuracies are theoretically characterized based on the calculated
achievable shaping gains (both in terms of SE and SNR) and the absolute gap to Shannon limit. The ranges where PSM outperforms QAM are
reckoned, in turn establishing the base for the initial decisions on the code
strategy and the shaping-wise optimum transmission reach per modulation
order.
43
(Ultra) long-haul networks
As far as the coded modulation scheme is concerned, the use of bitwise approaches employing MAP detection is well justified in combination
with PSM. Coded signal generation and detection plus decoding is hence
carried out with BICM (SCM with single-layer encoding) with SISO ID
receiver. Based on the calculated maximum shaping gain ranges, 0.5 code
rate convolutional code is employed for its simplicity and the optimal decoding performance when BCJR algorithm is employed. Likewise modeled
as a Markov process, PSM operation enables the use of BCJR demapper
instead of general MAP, on the whole simplifying the iterative receiver’s
processing load. With this framework, the criteria for efficient code design
are then discussed with the help of the resulting EXIT chart. Among the
most relevant conclusions we note that efficient decoding of PSM constellations demands iterative routines, and that PSM requires lower coding gains
than QAM to achieve successful demodulation irrespective of the labeling.
In the second and last block, the experimental validation of the previously analyzed BICM-PSM scheme with SISO ID receiver is presented.
Coherent detection is employed to recover single-carrier DP 6-GBd (grossrate) 16-ary, 32-ary and 64-ary PSM constellations in B2B configuration
and after uncompensated 240-km SSMF transmission with lumped amplification. The contemplated figures of merit include BER and effective SE.
The SE results show that even though complete error correction and ambiguity resolution after ≤10 iterations is obtained, 0.5 code-rate convolutional
codes are highly inefficient in combination with PSM despite the relatively
similar EXIT profiles; penalizing the gap to capacity by more than 3 dB.
Thorough matching of the EXIT functions of both decoder and demapper
at the target SNR is mandatory for designing dedicated codes. The experimental measurements in B2B configuration hint the need to improve the
robustness of the algorithm for phase noise compensation under low SNR
conditions, which represented the main contribution to the implementation penalties. In this line, per-cluster covariance an centroid estimation
using EM algorithm was proposed and experimentally demonstrated to reduce the impact of phase noise on the performance of systems employing
soft-based receivers. By feeding the demapper with optimally estimated
cluster parameters in maximum likelihood sense, the average receiver sensitivity was improved by ≈0.6 dB in highly non-linear transmission regime
(SPM-dominated phase noise imprint).
Future work
The present investigation conveys the following items:
44
2.4 Summary
• Dedicated code design for PSM constellations : optimal code elaboration is an necessary step to truly exploit shaping gains and close
the gap to constrained capacity down to the theoretical limits [5].
Comparisons with most popular modulation formats or alternative
shaping models would then be possible. This is the way to make
PSM worth consideration for practical implementation.
• Multi-layer coding strategy plus PSM: coding gains could be further
improved with a multi-layer strategies with successive interference
cancellation [61]. This configuration could be appropriate in multiuser scenarios.
• Frequency pilot tone/s : due to the proximity of certain groups
of symbols for some modulation orders and the low SNR levels for
which maximum shaping gains are obtained, phase noise compensation poses great difficulty. Frequency pilot tones permit blind recovery while preserving the generation structure and the receiver simplicity [46].
45
(Ultra) long-haul networks
46
Chapter 3
Metropolitan networks
Metropolitan-area networks (MANs) are the bridge between access and
long-haul systems. Comprehending transmission reaches from ≈100 km to
≈1000 km, this network segment gives service to large metropolitan areas,
providing direct interconnection among cities, regions and even neighboring countries. Mainly due to the increasingly significant role of contentdelivery networks1 , MANs’ traffic is growing twice as fast as LH’s and
currently accounting for more than 50% of the global IP transport volume [12]. This proliferation combines with an exceptional diversity of
flows to motivate the use of (i) high-speed flexible WDM transceivers,
then enabling on-demand spectral allocation allowing for bandwidth savings and dynamic service support; and (ii) the use of all-optical adddrop multiplexers (OADMs), facilitating periodic spectral defragmentation,
and providing transparent routing/switching capabilities that reduce the
number of required transponders. As the ever-evolving connectivity and
throughput requirements shape MANs infrastructure from 100-Gbps channels and point-to-point WDM rings towards ≥400-Gbps super-channels and
all-optical reconfigurable mesh architectures, these two technologies rise as
1
Content-delivery networks or CDNs are systems of distributed datacenters that deliver content to the end-user based on geographical proximity. Two distances are considered: (i) end-user to the CDN’s point-of-presence, and (ii) the CDN’s point-of-presence
to the original server.
CDNs’ datacenters dynamically cache/store large amounts of data form various origins
(e.g. web and downloadable objects, applications, video streaming, social networks and
any kind of cloud service) and physically bring it closer to the end-user thus improving the
availability, security, and retrieval latency while offloading traffic from core backbones.
Therefore, it is apparent that the larger the network spreads the better the performance
will be, leading to CDNs with up to several thousands of interconnected datacenters
whose traffic is inherently loaded onto metro reach.
47
Metropolitan networks
the pillars of future metro networking [13–16].
From the implementation perspective, high-speed adaptive and multidegree OADM-based MANs pose various important challenges. On one
hand, photonic-layer networking leads to longer regenerator-free transmission distances, then worsening the end-to-end performance due to (i) filtering effects2 accentuated by high-order OADM cascadability (up to ≈10
nodes), (ii) larger PMD3 , and (iii) OSNR degradation. On the other hand,
sustainable increase of the per-channel bitrates requires using AMF to boast
the SE as a means of sidestepping the economical and technological challenges that high-baudrate signaling entails. Unfortunately, AMF aggravate
the OSNR sensitivity and reduce the system’s tolerance to filtering effects
and phase noise, hence clearly harmed by long-reach all-optical routing.
All of these challenges strongly encourage the use of coherent technology
supported with advanced DSP, suggesting the direct application of LH’s
transceivers as first approach. Nevertheless, the high (and growing) number of end-nodes in MANs imposes severe requisites in cost, footprint, and
power consumption; making such straight integration prohibitive, and underscoring the relevance of low-noise optical amplification, low-attenuation
large-effective-area fibers, intelligent optical spectrum managing (resource
sharing), and an efficient AMF-FEC-DSP combo. These measures are common factor in high-speed MAN demonstrations (see Section 3.1), where, unlike in point-to-point (P2P) networks, the most-effective techniques for nonlinear impairment compensation are inapplicable from both the conceptual
and the complexity-cost standpoint4 . Whereas renovating the network’s
amplification scheme and/or fiber type boast the performance with total
certainty, modulation format selection/design and bandwidth use optimization are both non-trivial actions tightly linked to the techno-economical requirements of the network. In this regard, sought solutions for MAN must
2
Current wavelength-selective technology is unable to manage 12.5-GHz granularity
without inferring notable crosstalk and spectral distortion of the signal of interest, hence
requiring the use of frequency guard-bands (≈18 GHz [18]) that partly dispel the spectral
savings [13, 31].
3
The minimum tolerance is estimated around 25-30 ps for MANs [13]
4
Approaches for nonlinearity compensation like digital back-propagation [37] and variants, inverse Volterra series [43], phase-conjugated twin-waves [44], or mid-span phase
conjugation [45], rely on the exact knowledge of the physical properties of the propagation medium; which, furthermore, should remain constant from transmitter to receiver
end (ideally). This fact prevents the use of those techniques to improve the communication performance across present and specially future MANs, showing many-node mesh
topology and dynamic signal routing. Other methods such as co-propagating frequency
tone conjugation [46] or sophisticated FEC schemes [89] may still be used.
48
feature [13]:
• Compact spectral occupancy : given a target bitrate, increasing the
SE per carrier reduces the cost per bit, allows for higher degree of
spectrum sharing, and reduces the crosstalk penalty from/to neighboring WDM channels. Pulse-shaping and AMF are key techniques
within this group.
• Robustness to OADM (filtering effects) : the non-rectangular filter
shape that realistic OADMs exhibit causes frequency-selective distortions whose severity aggravates as the number of traversed nodes
increases. Signal pre-distortion together with spectrally efficient techniques can considerably increase the cascadability tolerance [90]. Of
course, concomitant improvements in the OADMs’ flatness and rolloff sharpness are most desirable.
• Spectral granularity : super-channels (WDM channels conformed by
>1 lower-baudrate optical sub-carrier) offer enhanced intra-channel
spectral flexibility at the expense higher number of low-speed electrooptical components. An intelligent exploitation of such flexibility
shall make possible the use of finer-granularity WDM grids (see [18])
as compared to the analogous single-carrier configuration, thereby
leading to higher SE through the reduction of idle optical spectrum. Nonetheless, it is important to realize that only WDM
channels/super-channels are liable to optical routing (not the constituent sub-carriers), meaning that very broad super-channels pose
difficulties in terms of network managing for spectrum sharing, signal
protection and optical re/routing [13]. For its part, the lowerbound
for the spectral occupancy is primarily established by the selectivity
of the optical filters5 , where 37.5-GHz and 75-GHz slot granularity
represent the next milestones in the managing capabilities of OADM
technology.
Further flexibility can be gained by digitally splitting the optical subcarriers themselves into multiple low-baudrate signals. The ensuing
high-selective spectral control enables accurate and dynamic signal
pre-distortion, thus notably improving the SE per route, and the
resiliency to filtering effects. These ultra-fine granularity spectral
components are often separated and processed (in parallel) in the
5
We define filter selectivity as the ratio between the passband width and the transitionband width.
49
Metropolitan networks
digital domain, hence requiring additional power and computational
capability.
• Adhere to standard frequency grids : complying with standardized
frequency grids makes possible seamless (transparent) connectivity
across manyfold networks (e.g. MANs and access networks). This
shall permit unifying the control plane for more efficient resource
allocation (including hardware management/sharing) [91].
Consequence of a balanced accommodation of these various requirements, the implementation complexity, and the throughput demand, equipment suppliers have converged to 2×sub-carrier 400-Gbps super-channels
as the next evolutive step in high-speed WDM systems for MANs. This
solution relies on quadrature multiplexing, polarization multiplexing (PolMux), and spectral shaping to deliver 5.33 bit/s/Hz of net SE (WDM
operation on the 75-GHz grid) with a maximum transmission distance of
≈500-km over uncompensated SSMF fiber link with EDFA-only amplification (longer transmission requires improvements in the noise figure of
the optical amplification stages and/or the utilization of optical fibers with
very low attenuation and large effective area to mitigate nonlinearities).
At present, this configuration is deemed the optimum trade-off for attaining MAN’s throughput-time-distance products without compromising the
OADMs’ selectivity, and keeping the electronics within acceptable margins
of cost and stress6 [92]. As far as the modulation format per sub-carrier is
concerned, Nyquist pulse-shaped QAM has been commonly adopted. With
typical roll-off factors (β)7 ranging from 0.2 to 0.01, this option achieves
notable inter-symbol interference (ISI)-free spectral compactness; favoring
the spectral sharing (more nodes/transceivers can be allocated within the
same network), and improving the routing/switching manageability (signals’ spectrum can be readily accommodated within different standardized
WDM grids). Although desirable, it is precisely the spectral stiffness that
Nyquist pulse-shaped signals exhibit what prevents effective adaptability to
the channel response, making this modulation approach particularly sensitive to filtering effects. Despite this issue has been so far circumvented with
6
Other alternatives include 1×400-Gbps net rate with ≈100-GHz optical bandwidth
occupancy, where small footprint and low cost are the main values; and 3×133-Gbps
net-rate with ≈50-GHz optical bandwidth occupancy, focusing SE maximization.
7
The roll-off factor is a measure of the excess bandwidth of the filter beyond the
Nyquist bandwidth (Symbol rate/2). Accordingly, the total signal bandwidth is expressed as Signal bandwidth = (Symbol rate/2)(1 + β)
50
pre-distortion techniques and advanced DSP/FEC, such solution shows insufficient as MANs grow in size and traffic volume; thus increasing the
average number of intermediate nodes per route, and aggravating the overfiltering per node due to the compression of the WDM spectrum together
with the deficient selectivity of current OADM technology for fine granularity (≤37.5 GHz). This situation has engaged the attention of the research
force, resulting in diverse proposals in which further super-channel partitioning (<32-GBd sub-carriers) is contemplated with the main purpose of
enabling more precise spectral shaping and lower implementation penalties [15, 93]; nevertheless, at the expense of abundant optical parallelism,
definitely contraposing the aspirations for inexpensive and small-footprint
transceivers.
In this regard, and supported by the improvements in the speed and
resolution of digital-analog interfaces, digital sub-carrier de/multiplexing
is increasingly looked into for alleviating the mentioned trade-off and providing flexible software-defined control of the sub-carriers’ properties [94].
This is the context of the next investigation, where we report on a novel
DSP-mediate method for signal multiplexing and modulation called coherent multi-band carrierless amplitude/phase modulation (C-MultiCAP).
Unlike in other techniques such as OFDM [95], the ensemble spectral shape
and the total bitrate in C-MultiCAP are tailored in a sub-band by sub-band
basis, enabling individual adjustment of the central frequency, baudrate,
SE, as well as number of dimensions per sub-band and per polarization. As
a consequence of these numerous degrees of freedom, unmatched flexibilityrobustness ratio is achieved without compromising the SE at the expense of
processing complexity. Resulting from an adaptation of the original work
by M. I. Olmedo et al. [96] for operation in coherent communications, we
propose and experimentally demonstrate for the first time C-MultiCAP
in fiber-optic communications as potential multiplexing plus modulation
technique for future MAN’s transceivers.
This chapter is subdivided into four blocks. First, the most relevant
research work in the context of high-speed WDM MANs is presented,
making explicit distinction between single-carrier and super-channel approaches, as well as among diverse link configurations. This information
bases the subsequent positioning of the present investigation. In the second
block, the essentials of single- and multi-band multi-dimensional carrierless
modulation are thoroughly discussed as the fundamental building blocks
in C-MultiCAP. Signal properties and practical implications of using this
51
Metropolitan networks
multiplexing technique are the main focus. In the last part of this second block, C-MultiCAP detection is elaborated from the DSP perspective.
The core processes for enabling successful equalization and demodulation
are explained in detail, together with the constraints and assumptions that
these imply. In the third block, the experimental proof-of-concept of a
C-MultiCAP signal successfully transmitted over a MAN-reach scenario is
introduced. Finally, the work is summarized, comprehending a motivation
recap, main results and conclusions, and plausible future work lines.
3.1
Work positioning
In this section, the most relevant ≥4-bit/s/Hz net SE transmission demonstrations are presented in the context of WDM MANs. For concision
and taking into consideration MANs’ constraints and operational requirements, we restrict the covered literature to >100-Gbps net rate WDM
channel/super-channels with ≤175 GHz of spectral occupancy, standardized WDM frequency grid/slots [18], and employing compatible techniques
for impairment compensation (see footnote 4). The results are organized
into two subsections according the number of sub-carriers constituting
the WDM channel, that is single-carrier and multiple sub-carriers (superchannels). In addition, further subdivision is realized within each category
based on the span configuration, where the usual combination of SSMF or
non-zero dispersion-shifted fiber (NZDSF) plus EDFA is distinguished from
any other alternative. In the end, this investigation is contextualized and
the main contributions highlighted.
Single-carrier
High-baudrate single-carrier approaches target the minimization of the
transceivers’ footprint by reducing the number of electro-optical components and thereby the associated cost. On the other hand, the ensuing
broad spectrum often undergoes acute over-filtering (even in the absence
of OADMs), then requiring the use of sophisticated DSP, powerful FEC
and/or pre-emphasis techniques that entail extra computational load and
power consumption. Recent investigations include:
SSMF/NZDSF plus EDFA :
• [97–99]: 1 WDM channel modulated with DP 56-GBd 16-ary QAM
is transmitted over 12×75-km and 16×75-km uncompensated SSMF
52
3.1 Work positioning
spans (900 km and 1200 km in total respectively) with EDFA-based
amplification. Optical pulse-shaping with β = 0.2 suffices for fitting
the signal within one 100-GHz grid slot, yielding 4.48 bit/s/Hz gross
SE per channel. In order to counteract the severe ISI, the DSP at the
receiver features 3-, 5-, or 7-symbol memory MAP demapping. The a
priori estimations are calculated during system’s initialization (using
training sequence) and subsequently stored in a look-up table to minimize the computational load. Standard hard-decision 7%-overhead
Reed-Solomon FEC is considered [20]. This work outperforms preceding investigations for single-carrier DP 56-GBd 16-ary QAM in
terms of throughput-times-distance product [100–102].
• [103]: 5 WDM channels implementing DP 42.66-GBd 64-ary QAM
each are transmitted over 2×80-km uncompensated SSMF spans (160
km in total) and 3×100-km uncompensated ULAF spans (300 km in
total), both cases employing EDFA-only amplification stages. Digital
pulse-shaping with β = 0.15 (1.5 S/symbol) and pre-emphasis are
employed, allowing for 50-GHz frequency-slot granularity (one slot
per channel with ≈10.24 bit/s/Hz gross SE) while keeping the linear cross-talk within negligible margins. Soft-decision 24%-overhead
spatially-coupled LDPC convolution codes are used for FEC (≈5·10−2
pre-FEC BER threshold).
Other span configurations :
•
[104] and the follow-up [105]: 1 WDM channel modulated with
DP 124-GBd 32-ary QAM is transmitted over 12×55-km uncompensated low-loss ULAF spans (660 km in total) using EDFA-only
amplification. Four-fold spectral slicing is employed at the transmitter to synthesize the high-baudrate signal, thus relaxing the electrooptical bandwidth requirements and enabling near-Nyquist digital
pulse-shaping to fit the signal in a single 125-GHz frequency slot
(9.92 bit/s/Hz gross SE per channel). After detection with a single
coherent receiver, the signal is post-processed offline. Powerful softdecision spatially-coupled LDPC convolutional FEC code with 24%
overhead is implemented within standard BICM setting (≈4.5·10−2
pre-FEC BER threshold).
• [106]: 5 WDM channel modulated with DP 72-GBd 64-ary QAM
is transmitted over 4×100-km uncompensated ULAF spans (400 km
53
Metropolitan networks
in total) using hybrid Raman/EDFA amplification stages. The signal generation is performed with two differential-output DACs at 1
S/symbol (72 GS/s) per channel, and one programmable optical filter is employed to fit each channel in one 100-GHz frequency slot
(8.64 bit/s/Hz gross SE per channel) and pre-emphasize the spectrum
to compensate for bandwidth limitations. An adaptive real-valued
8×4 multi-input multi-output (MIMO) (40-symbol memory) feedforward equalization accounting for differences in the linear response
between quadrature components is implemented. Hard-decision FEC
with ≈35% redundancy is considered.
• [107]: 20 WDM channels modulated with DP electrical time-division
multiplexing (ETDM) 43-GBd 64-ary QAM are transmitted over
12×50-km uncompensated low-loss ULAF spans (600 km in total)
using EDFA-only amplification. Optical pre-emphasis is carried out
to alleviate filtering impairments while fitting the channels in the
50-GHz grid (≈10.32 bit/s/Hz gross SE). In order to compensate
for the ISI, triple stage equalization and MAP detector are implemented. Soft-decision 20%-overhead quasi-cyclic LDPC is employed
as reference FEC code [108].
• [109]: 1 WDM channel modulated with DP Nyquist optical timedivision multiplexing (OTDM) 160-GBd 64-ary QAM is transmitted over 2 spans constituted by 50-km super-large-area fiber (SLAF)
plus 25-km inverse dispersion fiber (IDF) (150 km in total) using
EDFA-only amplification. The advantage of using Nyquist pulses as
compared to other shapes is two-fold: fit the signal within the 175GHz grid (≈10.97 bit/s/Hz gross SE) and exploit their time-domain
orthogonality to facilitate pulse demultiplexing. Hard-decision 7%
overhead FEC is considered.
Multiple sub-carriers
These approaches increase the number of electro-optical components in
favor of routing flexibility, enhanced signal protection/robustness against
filtering impairments, and lower stress on the electronics; hence potentially
enabling higher throughput-times-distance products owed to the minimization of the implementation penalty. In addition, the milder bandwidth requirements allow for using commonplace DACs for signal generation, making possible near-Nyquist spectral compactness via digital pulse-shaping
54
3.1 Work positioning
with arbitrary-low β > 0; and the use of sophisticated modulation plus
multiplexing schemes (e.g. OFDM). Investigations in controlled laboratory environment include:
SSMF/NZDSF plus EDFA :
• [110]: 3 WDM super-channels constituted by 5×28-GBd pre-filtered
return to zero (RZ) (50% duty-cycle) DP 16-ary QAM (1.12 Tbps
gross rate per super-channel) are transmitted over 400-km uncompensated SSMF link with a total of 2 OADM passes using EDFA-only
amplification. Each super-channel is fit in one 162.5-GHz frequencyslot, delivering ≈6.89 bit/s/Hz gross SE per super-channel. The DSP
in the receiver side comprises duobinary filtering plus maximumlikelihood sequence estimation (MLSE) for boasting the SNR. Harddecision 7%-overhead Reed Solomon code is assumed for FEC [20].
• [111]: 1 WDM super-channel constituted by 2×32-GBd DP 16-ary
QAM sub-carriers is transmitted in real time along with another
58×25-GBd 50-GHz spaced DP-QPSK neighboring WDM channels
over 10×100-km uncompensated SSMF spans (1000 km in total),
and 5×100-km uncompensated NZDSF spans (500 km in total) using only EDFAs in both scenarios. The 200-Gbps sub-carriers are
spaced by 37.5 GHz after digital Nyquist pulse-shaping, and both fit
in 75-GHz optical bandwidth (≈6.82 bit/s/Hz gross SE per superchannel). Soft-decision 20%-overhead quasi-cyclic LDPC is employed
as reference FEC code [108].
•
[112]: investigation of optimum sub-carrier power and frequency
guard-band between super-channels to maximize SE and reach in
optically-routed flexible networks. Numerical study is performed for
terabit-class super-channel comprising 5×32-GBd sub-carriers modulated with DP 16-ary QAM (1280 Gbps gross rate per super-channel)
transmitted over 15×60-km uncompensated SSMF spans (900 km in
total) and EDFA-based amplification. Digital Nyquist pulse-shaping
with β = 0.15 is employed, permitting 35-GHz spacing between
sub-carriers and 175-GHz optical bandwidth occupancy per superchannel. The results prove 21% reach improvement for as low as 12.5
GHz guard-bands, and up to 62% increase in SE-times-reach product
for full optimization.
55
Metropolitan networks
Other span configurations :
• [113]: 1 WDM super-channel constituted by 2×28-GBd sub-carriers
using pre-filtered RZ (50% duty-cycle) DP 16-ary QAM (448 Gbps
gross rate per super-channel) is recirculated 10 times through a fiberloop made up of 1×72-km pure silica core fiber (PSCF) (720-km total
propagation distance) and one OADM; Raman amplification with
counterpropagating pump plus EDFA are employed for attenuation
compensation. The use of pre-filtered RZ pulse-shaping enabled the
use of 75-GHz slot granularity (5.97 bit/s/Hz gross SE per superchannel) while achieving high resilience to filtering effects without
either digital pre-emphasis or many-tap finite impulse response (FIR)
Nyquist filters. Standard hard-decision 7%-overhead Reed-Solomon
FEC is considered [20].
With the focus on inter-channel interference analysis, the same group
extends [113] by demonstrating 5 WDM super-channel transmission
over 3 turns on a recirculating loop comprising 4×50-km plus 1×26km PSCF (678 km in total) and one OADM using EDFA-only amplification [114]. Each super-channel comprehends 2 sub-carriers modulated with pre-filtered RZ (50% duty-cycle) DP 16-ary QAM, allowing
for 75-GHz of optical bandwidth occupancy per super-channel (same
configuration as in [113]) and 35-GHz spacing between all neighboring
sub-carriers.
• [115]: 4 WDM super-channels constituted by 2×32-GBd DP 16ary QAM sub-carriers (512 Gbps gross rate per super-channel) are
transmitted over 3×210-km uncompensated SSMF spans (630 km in
total) and amplified with hybrid stages comprising bidirectional Raman pumping and one EDFA. Digital pulse-shaping enables the use
of 37.5-GHz frequency-slot granularity (one slot per sub-carrier), thus
leading to ≈6.82 bit/s/Hz gross SE per super-channel. Soft-decision
24.5%-overhead spatially-coupled irregular LDPC code is concatenated with an outer BCH de/coding block (25.5% of total redundancy) for FEC [116].
• [117]: 3 WDM super-channels constituted by 3×46-GBd DP 32-ary
QAM sub-carriers (1.38 Tbps gross rate per super-channel) are transmitted over uncompensated SSMF with hybrid Raman-EDFA opti56
3.1 Work positioning
cal repeaters for attenuation compensation. The three optical subcarriers are packed in 140-GHz (allowing for 150-GHz frequency grid
including 1-GHz guard-bands) through digital Nyquist pulse-shaping
with β = 0.1, yielding a gross SE per super-channel of ≈9.2 bit/s/Hz.
The maximum transmission reach is evaluated with and without intermediate OADMs, showing successful signal recovery after 600-km of
propagation with 3 OADM passes and up to 800 km without filters.
Soft-decision 32%-overhead convolutional LDPC code is considered
for FEC, enabling error-free demodulation at BER≤ 5.3 · 10−2 .
• [118]: 5 WDM super-channels constituted by 5×9-GBd DP 32-ary
QAM sub-carriers (450 Gbps gross rate per super-channel) are transmitted over 8×100-km ULAF (800 km in total) using Raman amplification plus EDFA. Near-Nyquist digital pulse-shaping (β = 0.01)
permits fitting the signal within just 45.8 GHz optical bandwidth
(50-GHz WDM grid), which gives ≈9 bit/s/Hz gross SE per superchannel. Optical pre-emphasis is used to counteract the filtering effect
of a single OADM in the transmitter, thereby achieving 100% longer
reach as compared to the case without pre-compensation. Harddecision continuously interleaved BCH FEC code with 7% redundancy is contemplated [119].
• [120]: 5 WDM super-channels carrying 2×9-GBd DP 64-ary QAM
sub-carriers interleaved with 3×9-GBd DP hybrid 64-32-ary QAM
sub-carriers (504 Gbps gross rate per super-channel) are transmitted over 12×100-km uncompensated ULAF spans (1200 km in total) using all-Raman amplification. Band-limiting effects are alleviated via frequency-domain digital pre-distortion and Nyquist pulseshaping with β = 0.01. Operation in the 50-GHz frequency grid is
demonstrated (≈45.8-GHz optical bandwidth occupancy), resulting
in ≈10.1 bit/s/Hz gross SE per super-channel. The use of hybrid
QAM is key to permit successful post-FEC demodulation in the considered scenario, where the soft-decision 20%-overhead quasi-cyclic
LDPC code is employed as reference [108].
• [121]: 90 WDM super-channels constituted by 8×5.6-GBd DP 64ary QAM sub-carriers (538 Gbps gross rate per super-channel) are
transmitted over 3×80-km uncompensated PSCF spans (240 km in
total) using Raman amplification. The super-channel is fit within one
50-GHz frequency slot via digital Nyquist pulse-shaping with β = 0.1,
57
Metropolitan networks
providing ≈10.76 bit/s/Hz gross SE per super-channel. A radiofrequency pilot tone is digitally inserted in the transmitter side for
enabling robust and data-independent phase noise compensation, allowing for stable performance for up to ≈15 dBm total launch power.
Hard-decision continuously interleaved BCH FEC code with 7% redundancy is contemplated [119].
• [122]: 370 WDM super-channels constituted by 4×6-GHz DP OFDM
sub-carriers modulated with 128-ary QAM (294 Gbps net rate per
super-channel) are transmitted over 3×55-km uncompensated SSMF
(165 km in total) using all-Raman amplification. Frequency locking
among the sub-carriers permits using down to 1-GHz frequency guard
band without performance degradation and thus fitting the superchannel in one 25-GHz slot (≈11.76 bit/s/Hz gross SE per superchannel). The DSP comprehends standard OFDM TX/RX blocks
with the exception of a dual-stage phase noise compensation algorithm in the receiver side, which exploits the residual direct current
(DC) component that appears after modulation to realize coarse estimation followed by fine tracking employing pilot sub-carriers. Harddecision 7%-overhead Reed-Solomon FEC is considered [20].
•
[123]: 1 WDM super-channel constituted by 12×3.94-GHz DP
OFDM sub-carriers using offset 64-ary QAM for the payload (516.17
Gbps gross rate per super-channel) is transmitted over 5×80-km uncompensated SSMF spans (400 km in total) using Raman-only amplification. The use of offset-QAM technique in combination with
OFDM achieves remarkable side-lobe suppression, thus providing a
quasi-rectangular spectral shape with high confinement (≈49.8-GHz
total optical bandwidth occupancy) that avoids the need for timing or
frequency alignment among sub-carriers (OFDM sub-bands) to form
the super-channel. Considering 50-GHz frequency-slot granularity,
the net and gross SE per super-channel equals ≈8.6 bit/s/Hz and
≈10.32 bit/s/Hz respectively.
As far as field-trial demonstrations are concerned:
• [124]: 2 WDM super-channels constituted by 2×32.5-GBd DP 16ary QAM sub-carriers (520 Gbps gross rate per super-channel) are
transmitted in real time over a legacy multi-vendor 400-km uncompensated SSMF link with EDFA-only amplification in Turin area,
58
3.1 Work positioning
Italy. Each sub-carrier is electronically Nyquist pulse-shaped to fit
in one 50-GHz frequency slot, yielding 100 GHz per 400-Gbps superchannel (4 bit/s/Hz net SE and 5.2 bit/s/Hz gross SE). Stable and
successful post-FEC demodulation is reported over >65 hours, where
the FEC code is composed of an inner soft-decision LDPC convolutional block and an outer hard-decision Reed-Solomon block for a
total 23% redundancy.
• [125]: 1 WDM super-channel constituted by 2×32.5-GBd DP 16ary QAM sub-carriers (520 Gbps gross rate per super-channel) is
transmitted in real time over a commercial 550-km uncompensated
SSMF link with EDFA-only amplification between Paris and Lyon,
France. Uninterrupted post-FEC error-free demodulation is reported
in both directions for 24 hours with more than 2 dB average margin
in Q2 -factor at 193.1 THz center frequency. Soft-decision FEC is
employed.
Deprived of intermediate electronic signal regeneration and end-to-end
nonlinearity compensation, research activities within MAN context consistently evince the difficulty of attaining ≥5.33 b/s/Hz net SE over >500 km
of SSMF/NZDSF and EDFA amplification while complying with standard
≈2-dB OSNR industrial margins. This fact holds even without mid-link
OADM passes8 , becoming further challenging in large optically-switched
networks carrying high-baudrate signals, where filtering distortions predictably dominate the performance [13, 90]. Aiming for boosting the receiver sensitivity irrespective of the node-cascading degree, numerous studies opt to improve the link components, that is, amplification stages with
lower noise figure and/or fibers with lower attenuation-per-kilometer and
larger effective area. Albeit net SE-times-reach products as high as 9600
bit/s/Hz-km have been proved in such scenarios [120], network operators
rather deem this potential upgrade as an undesirable concern in terms of
cost and deployment time, prioritizing the reuse of the existing equipment
to the largest possible extent [111]. All of the mentioned motivates the development of de/multiplexing and de/modulation approaches focusing on
maximizing the receiver sensitivity at the same time that providing high
spectral efficiency and robustness to filtering effects; which partly justifies
8
Note that all cited investigations with the exception of [113, 114] employ WDM
technology in P2P configuration with ≤2 OADMs per path.
59
Metropolitan networks
the abundant research on WDM super-channels and alternatives to raisedcosine pulse shapes (e.g. RZ [110,114] or doubinary [126]). With such target
and considering modulation schemes, OFDM represents the most resorted
alternative to the rigid QAM in order to equip the transceivers with accurate digital control on the sub-carriers’ signal properties; featuring down to
sub-gigahertz granularity spectral control and rate adaptability, while delivering near-Nyquist SE. Regrettably, because OFDM’s functioning relies
on the frequency-wise orthogonality among typically hundreds of superimposed low-baudrate modulated sinusoids, it presents high sensitivity to
effects causing inter-carrier interference (ICI) and phase noise, as well as
it exhibits severe peak-to-average power ratio (PAPR) that increases the
quantization noise and makes it susceptible to nonlinearities.
It is thus clear that the selection/design of a modulation technique that
meets the requirements of future MANs still eludes a satisfactory solution as
of today, where the usually considered alternatives (i.e. QAM and OFDM)
show a series of complementary pros and cons that suggest looking into
approaches offering an intermediate compromise. That is the purpose of CMultiCAP, a flexible multiplexing plus modulation technique conceived for
coherent communications, that leverages on DSP in the transmitter and receiver sides to sub-divide the electrical bandwidth into a user-defined number N-dimensional pass-band signals. Besides supporting multi-dimensional
modulation format per-band, their individual center frequency, baudrate
and excess bandwidth (e.g. determined by β) are also liable to adjustment,
further allowing for near-Nyquist spectral compactness and improved robustness and sensitivity through vast spectral control.
In the following, we develop in detail on the generation and detection
of C-MultiCAP prior to discuss the results of the first experimental MANreach transmission demonstration. These results show successful signal
recovery after >200-km and >400-km SSMF EDFA-only link for 300-Gbps
and 200-Gbps net rate respectively employing standard 25G-class electrooptical components.
Contributions of this work
• Original idea: development of C-MultiCAP for enabling the advantages of carrierless amplitude/phase modulation (CAP) and multiband carrierless amplitude/phase modulation (MultiCAP) in coherent communications. Development of the RX DSP structure and the
multi-dimensional parallel equalization approach that makes possible
impairment compensation and correct signal recovery. Proposal of
60
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
C-MultiCAP for its use in optically-routed MANs.
• Experimental results: experimental demonstration of C-MultiCAP in
metro-reach coherent optical communications.
• Related first-author publications: [127] (conference).
3.2
Multi-band carrierless amplitude/phase
modulation for coherent communications
(C-MultiCAP)
C-MultiCAP results from the evolutionary adaptation of CAP, a pass-band
2D modulation approach introduced by D.D. Falconer in 1975 [128] and
later generalized to multiple dimensions; firstly to overcome the original
susceptibility to filtering distortions and relax the stress on the electronics
when digitally generated (MultiCAP, [96]); and finally making possible its
use in coherent communications to boost throughput and reach [127].
In this section, C-MultiCAP generation and detection are discussed and
contrasted with the predecessors’, paying special attention to the influence
of the signal properties on the general performance, as well as the associated
challenges from the implementation standpoint.
3.2.1
Generation and signal properties
The exceptional property in all CAP flavors is the ability to de/multiplex ≥2 independent data streams within a given bandwidth while maintaining end-to-end system linearity, that is, FIR filters suffices for multidimensional signal generation and data recovery in the absence of distortions. With such filters fundamentally determining all the signal properties,
understanding the restrictions and variables governing their design is crucial to assure correct adaptation to the available resources and channel
properties for maximum sensitivity and SE. This is the issue covered in
the following subsection.
Starting from it most basic form, CAP, the mathematical foundations
in linear multi-dimensional carrierless de/multiplexing are presented, as
well as the practical value and potential applications of the resulting signal
properties. Following the initial observations, the generalization to multiple
bands is briefly discussed as a bridge towards C-MultiCAP, whose characteristics are detailed along with the technical requisites to permit successful
demodulation.
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Figure 3.1: General schematic diagram of CAP-based systems. Transmitter
on the left-most side and receiver on the right-most side. The mathematical
nomenclature of the signal at different stages is included on the top part.
3.2.1.1
CAP
Figure 3.1 shows the schematic diagram illustrating the general principle
for CAP-type signal generation and reception. In the transmitter side, N
parallel streams of binary data9 are modulated in amplitude prior to their
convolution with respective filters, gs , where {s ∈ N; s ≤ N }. The output
from these filters is linearly superimposed and propagated over the channel.
The received waveform is N -fold split/copied and processed individually
with another set of N filters, fr ; followed by regular amplitude-to-binary
demapping according to the same labeling strategy as the transmitters’.
Mathematically, the noise-free characteristic equation from the input of
transmitters’ filters, xs , to the output of receivers’, x̂r , can be expressed as
(see Figure 3.1 for full nomenclature):
x̂r [n] = (fr ∗ y)[n] = fr [n] ∗
N
X
!
us [n]
s=1
= fr [n] ∗
N
X
!
gs [n] ∗ xs [n]
s=1
(3.1)
with xs [n] defined as:
(
xs [n] =
As [k] for n = kT with k ∈ Z
0
otherwise
(3.2)
9
For concision, the transition from serial source-coded binary to parallel and channelcoded binary is assumed and thus not depicted.
62
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
where T is the symbol period in samples (i.e. over-sampling factor) and As
is a real value that stands for the amplitude at symbol instants. Notice that
different mapping alphabets can be used for each branch (e.g. A0 and A3
may be obtained from 4-ary and 8-ary pulse-amplitude modulation (PAM)
respectively).
Then expanding the convolutions in 3.1 and considering 3.2, we obtain
the general expression governing the system depicted in Figure 3.1:
x̂r [n] =
∞
X

fr [m] 
m=−∞
=
∞
X
∞
N X
X

gs [l]xs [n − m − l]
s=1 l=−∞
N
X
fr [m]gs [l]xs [n − m − l]
m,l=−∞ s=1
=
=
N
X
s=1
N
X
xs [kT ]
(3.3)
∞
X
fr [m]gs [n − kT − m]
m=−∞
As [k] (fr ∗ gs ) [n − kT ]
s=1
whose impulse response hrs [n] (see10 ), defined in a MIMO system as the collection of impulse responses for each input channel , and the corresponding
frequency response Hrs (f ) are given by:
hrs [n] = (fr ∗ gs ) [n],
(3.4)
Hrs (f ) = Fr (f )Gs (f )
(3.5)
and
Now, by constraining 3.4 and 3.5 based on the desired functional qualities of the system in time- and frequency-domain, the implementation
10
Because linear MIMO systems are prone to matrix formulation, the matrix-form
equation equivalent to 3.3 is provided below for the readers’ convenience:
x̂ = F ∗ G ∗ x = H ∗ x,
where H = F ∗ G = hrs [n] is the N ×N matrix describing the input-output
response, with N equal to the number of transmitted parallel data streams and
F = {f1 [n], f2 [n], ..., fN [n]} and G = {g1 [n], g2 [n], ..., gN [n]}> .
For their part,
x = {x1 [n], x2 [n], ..., xN [n]} and x̂ = {x̂1 [n], x̂2 [n], ..., x̂N [n]} represent the instantaneous
transmitted and received real-valued signals respectively.
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Metropolitan networks
conditions on the TX/RX filters (gs , fr ) can be extrapolated. Next we
particularize for CAP signals’ properties as originally described in [128]
and later thoroughly detailed in [129], that is, a linear ISI-free multidimensional modulation and multiplexing technique delivering optimum
end-to-end SNR sensitivity in the presence of AWGN.
Time-domain constraints
ISI-free communication : zero end-to-end ISI avoids the need for equalization or reduces its complexity. In the system described by 3.3, the ISIfree condition is given by:
hrs [n] =



1
0


undefined
for n = 0
for n = kT with {k ∈ Z; k 6= 0} ,
otherwise
(3.6)
Accordingly, 3.3 results in x̂r [kT ] = N
s=1 As [k] ∀r. Note how crosschannel/dimension interference impairs the recovered signal if no further
action is taken. In the frequency domain, the ISI-free condition is equivalent
to (Nyquist ISI criterion [130]):
P
∞
X
Hrs f −
j=−∞
j
T
= T ∀f
(3.7)
As an example, Figure 3.2 shows one general type of frequency response
that meets 3.7. Raised-cosine pulse shape belongs to this class.
Figure 3.2: Spectral illustration of a general system response showing zero
end-to-end ISI.
64
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Multi-dimensionality : this quality plus 3.6 require the impulse response describe an N ×N MIMO system providing transparent communication from each of the N inputs to exclusively one of the outputs (zero
cross-channel/dimension interference) at the symbol instants (n = kT ). For
instance, the most immediate configuration is x̂r = xs ∀r = s (i.e. x̂ = x
in footnote 10), which is conditional upon:
hrs [n] =


1




0

0





undefined
for r = s and n = 0
for r 6= s and n = 0
for n = kT with {k ∈ Z; k 6= 0}
otherwise
(3.8)
The top case in 3.8 imposes transparent input-output communication
at symbol instants, the second forces the necessary orthogonality between
any other TX-RX filter-pair combination, and the third assures zero ISI.
In particular, this solution restricts the system to a static configuration in
which x̂r = xs if and only if r = s (color-matched pairs in Figure 3.1);
however, it should be noted that dynamic filter adjustment or interchange
would allow for N ×N transparent switching in whatever domain (optical or electrical) the filters are implemented. Herein, and without loss of
generality, we adhere to the former for simplicity.
SNR maximization : finally, assuming AWGN channel, employing
matched filters optimally maximizes the SNR in the receiver. With the
general TX-RX matched-filter pair being defined as hT X [n] = h∗RX [−n], we
obtain the following constrain in our system:
fr [n] = gs [−n], or fr [−n] = gs [n] for r = s
(3.9)
Because gs and fr are real-valued filters ∀r, s, conjugation has no effect
and can be neglected. Under 3.9, the impulse response 3.4 is restated as:
hrs [n] = gr [−n] ∗ gs [n]
(3.10)
In summary, 3.8 and 3.9 constitute the essential time-domain restrictions to be forced on 3.4 so as to enable CAP-type communication.
Frequency-domain constraints
As far as spectral requirements are concerned, it is highly desirable to
minimize the bandwidth occupancy of our signal such that the spectral
65
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components beyond certain frequency can be forced to zero while assuring
compliance with 3.8. From the implementation perspective, this translates
into higher SE and thus denser WDM packing. Mathematically:
(
undefined
0
|f | < BWmin
,
otherwise
(3.11)
undefined
0
|f | < BWmin
,
otherwise
(3.12)
|Hrs (f )| =
which after 3.10:
(
|Gs (f )| =
where BWmin is given by the dimensionality theorem [130], which is a
generalization of the Nyquist–Shannon sampling condition for N > 1. It
states that the number of independent pieces of information (dimensions,
N ) that can be conveyed by a band-limited waveform or communication
system is proportional to the product of such bandwidth (BWmin ) and the
time allowed for transmission of the information (T ):
N
N FB
=
(3.13)
2T
2
where FB = T1 is the symbol rate. After 3.13, 3.12 accepts the graphical
representation in Figure 3.3, which depicts instances of the minimum optical bandwidth occupancy of 1-, 2-, and 3-dimensional CAP signals as a
function of the symbol rate.
BWmin =
Figure 3.3: Minimum spectral occupancy of multi-dimensional CAP signals
for N ∈ {1, 2, 3}.
One relevant observation becomes apparent from 3.13 and Figure 3.3, it
is not possible to increase the SE only by adding further dimensions. Given
66
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
a fixed symbol rate, increasing the number of orthogonal filters requires a
proportional increase in the signal bandwidth; or alternatively, given a fixed
signal bandwidth, increasing the number of orthogonal filters requires a proportional decrease of the symbol rate. This makes the maximum achievable
SE is independent of N , resulting in a SE upperbound for CAP equal to
the best-case SE across all constituent one-dimensional (1D) signals. On
the other hand, these independent 1D signals remain operational for multidimensional modulation format design, whereby the receiver sensitivity is
potentially increased by inducing a controlled degree of correlation among
the amplitude modulators of the employed dimensions. In this way, by distributing the symbols on the ≤N-dimensional constellation space to maximize their relative Euclidean distance11 , achievable SNR sensitivity gains
up to 1-dB with respect to DP-QPSK have been theoretically demonstrated
at BER=10−3 for N = 8 and 1 b/symbol/dimension [131, 132].
The dimensionality theorem admits a second interpretation of practical
relevance. In this case, we generalize the expression by considering excess
bandwidth:
2BWmin = N FB (1 + β) = FS;min
(3.14)
where FS;min is the required sampling rate. Then normalizing by FB , the
minimum number of samples per symbol (Ssym ) is obtained (Ssym ∈ N
assumed):
Ssym = dN (1 + β)e
(3.15)
It becomes clear one major implementation challenge in N D-CAP system: ≥N S/symbol are needed for perfect signal generation/reconstruction
(no information loss). This means that, considering a fixed β, N D-CAP
requires N times longer FIR filters than PAM for achieving the same SE.
Over and above, N D-CAP employs N times more filters; resulting in a total
of N 2 times more processing complexity than the 1D counterpart, and only
for signal generation/multiplexing and detection/demultiplexing. In absolute numbers, the total amount of operations results from a compromise
between the target β and the desired pulse-shaping fidelity (to avoid premature error floors). Leveraging the β-versus-tap characterization for RRC
11
The minimum squared Euclidean distance is the optimum optimization criterion in
the presence of isotropic noise (e.g. AWGN). Different transition probabilities, such
as phase noise’s, need consequent revision of the criteria in order to obtain optimum
performance.
67
Metropolitan networks
(a) Frequency response
(b) Impulse response
(c) System matrix H = hrs [n].
Figure 3.4: Characteristics graphs for one solution to 5D-CAP employing
minimax rule and constraints 3.8, 3.12, and 3.10.
filters in [96] and empirical observations after [127,133–135], ≥10N -sample
FIR is required for β≥0.1 at BER=10−9 .
Filter calculation
Once the desired signal properties are mathematically characterized,
the basis functions corresponding to the series of discrete CAP filters gs
meeting those requisites is calculated according to certain decision method.
In this regard, the minimax rule has been extensively employed in the literature for this concrete purpose [136–140], aiming for the minimization of
the largest value of a set of functions (e.g. the spectral components beyond
BWmin for every filter) under a set of inequality, equality or even non68
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
linear constraints. As an example, Figure 3.4 shows the squared frequency
response magnitude, impulse responses, and the characteristic system matrix H = hr s[n] of one possible 5D-CAP solution employing minimax. The
excess bandwidth is fixed at a relaxed 185% (β=0.85) of the minimum
bandwidth N 2FB .
Further conditions shall be imposed on the frequency response to suit
the signal to particular scenario-dependent requisites, for instance, no DC
component [139]:
(
|Gs (f )| =
undefined
0
0 < f < BWmin
,
otherwise
(3.16)
or forcing identical spectra among all N frequency responses, hence obtaining homogeneous and power equalized frequency responses [140]:
< (|Gr (f )| − |Gr0 (f )|)2 >= 0 for {r0 ∈ Z; r0 6= r and r0 ≤ N }
(3.17)
The reader is referred to the cited literature for graphical examples with
constraints 3.16 and 3.17.
Original 2D-CAP [128, 129]: Initially, CAP was conceived as a concrete ISI-free 2D pass-band line code with adaptive bandwidth occupancy.
Essentially, a variant implementation to standard QAM’s in which high
simplicity was comparatively accomplished by linearizing signal generation
and detection without trading SE or performance. Unlike in the general
approach previously discussed, in this CAP form the two orthogonal filters exhibit close-form expressions, resulting from the combination of two
powerful tools: (root) raised-cosine filter type (rrc[n]) and the sinusoidal
Hilbert-pair basis ({cos, sin}):
(
gs [n] = fr [−n] =
rrc[n] cos(2πfc n)
rrc[n] sin(2πfc n)
for s = r = 1
,
for s = r = 2
(3.18)
where f c is the pass-band’s center frequency, and rrc[n] is given by [130]
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Metropolitan networks
(a) g1 [n]
(b) g2 [n]
(c) |G1 (f )|2
Figure 3.5: On the top, orthogonal filter pair in conventional 2D-CAP for
fc = FB and β = 0.1. On the bottom, G1 (f ) magnitude squared. See 3.18.

β

√1
1
−
β
+
4


π
T 

 √β
2
π
2
π
1
+
sin
+
1
−
cos
π
4β
π
4β
2T
rrc[n] =
n
n
n

π T (1−β))
 √1 sin(π T (1−β))+4β T cos(



2
T
n
n

πT
1−(4β T )
for n = 0
T
for n = ± 2β
,
otherwise
(3.19)
Figure 3.5 shows an example of orthogonal pulses, gs [n], as described
in 3.18. Now, expressing the transmitter output, y, according to 3.18:
70
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
(a) CAP
(b) Raised cosine
Figure 3.6: Recovered eye-diagrams over one symbol-period of 4-level conventional 2D-CAP for fc = FB (a), and for raised cosine pulse-shaping (b);
both with β = 0.1.
y[n] =
+
∞
X
k=−∞
∞
X
A1 [k]rrc[n − kT ] cos(2πfc (n − kT ))+
(3.20)
A2 [k]rrc[n − kT ] sin(2πfc (n − kT ))
k=−∞
The above expression proves that CAP behaves as a regular quadrature
modulator (with the exception of a 2πfc kT phase shift, which turns irrelevant from the rrc[n] function perspective, as well as for maintaining the
orthogonality condition) without requiring analog mixers and with accurately adjustable center frequency and SE through β. Notice that the use
of raised-cosine is not strictly necessary for attaining zero ISI as long as f c
is an integer multiple of FB = T1 .
Eye pattern : as a linear system that CAP is, there is not TX/RX mixing element realizing up-/down-conversion of the data signal; but rather
the pulse-shaping filter itself defines a pass-band impulse response. Consequence of this, the output signal after matched filtering at the receiver
remains in pass-band, where the demodulation takes place, and hence displaying eye diagrams with unusually narrow horizontal opening (see comparison in Figure 3.6).
Highly accurate timing recovery and sampling point estimation are required if conventional slicing is used for signal demodulation, motivating
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Metropolitan networks
the study of alternative methods for CAP filter design that manage to relax
such stringency [141]. Nevertheless, considerably more robust demodulation is attained if statistical tools are contemplated for level identification.
The reason for this performance improvement is subtly explained by 3.20
for conventional 2D-CAP, making visible that the horizontal eye closure is
dictated by the 2πfc kT phase shift resulting from the convolution. Deviating from the maximum variance sampling instants thus translates into
cross-dimensional coupling, which shows as a standard rotation of the complex 2D constellation. Simple clustering algorithms based on Euclidean
distance such as k-means are able to double the sampling-deviation tolerance without requiring special filter design [96].
Peak-to-mean envelope power ratio : similarly to other multiplexing
techniques, a linear superposition of independent waveforms takes place in
N D-CAP for N > 1, leading to local interference instants with abnormally
high magnitudes. In the electrical domain, these power peaks make the
signal susceptible to non-linear effects in the power amplifiers, and exacerbate the quantization noise when the full dynamic range of the signal is
accommodated within the DAC/analogue-to-digital converter (ADC) resolution. In addition, because fiber nonlinearities are also dependent on
the instantaneous field magnitude, large values of PAPR reduce the nonlinear average-power threshold, then forcing to operate the system at lower
OSNRs. Accordingly, we next elaborate on the PAPR performance of N DCAP signals as an essential step in the design of any multiplexed transmission system. Without loss of generality, we employ the peak-to-mean
envelop power ratio (PMEPR) parameter, which is the base-band equivalent of PAPR12 [142, 143]. In all CAP cases, the filters are 14-symbol long
and the statistical analysis is performed over 20000 samples of 300 symbols
each.
Figure 3.7 shows the complementary cumulative distribution function
(CCDF) of the PMEPR for (i) a set of realizations of 1-to-5D-CAP systems, all of them calculated through the previously mentioned minimax
algorithm constrained to 3.12 with β = 0.85; and for (ii) uniformly bitloaded 256-carrier discrete multitone (DMT)13 , well-known for its extraor12
Consequence of the additional sinusoidal term after up-conversion to the carrier frequency, the PAPR is twice as high as the PMEPR. A ±3-dB correction term can be applied for back-forth conversion according to PMEPR(x[n])|dB = PAPR(x[n])|dB − 3dB.
13
DMT technique is a modified version of OFDM in which Hermitian symmetry is
forced to the input of the ifft block typically employed as the filter-bank for signal generation. As a result, the transmitter output is purely real, hence allowing for exploiting
72
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.7: CCDF of PMEPR for concrete realization of 1-to-5D-CAP systems with 2- (orange), 4- (green), and 8-level (blue) amplitude-modulation
per dimension, and for uniformly bit-loaded DMT.
dinary PMEPR [143] and serving as worst-case reference. In all scenarios,
2-, 4-, and 8-level amplitude-modulation per dimension are simulated, although some curves are intentionally removed due to the little additional
information provided, and for the ease of visualization.
A quick comparison with the 2-level 1D reference makes clear the negative influence of N > 1 CAP at all CCDF levels. Owed to the high
sensitivity of the PMEPR to the modulation order in single/few-carrier
systems, this difference substantially increases for 4- and 8-level modulation, as it is visible in 2D-CAP with up to ≈3-dB additional PMEPR
in the latter case at CCDF=10−3 . Predictably, the relative and absolute
magnitude of this penalty experiences saturation due to two phenomena:
the stabilization of the average power of the peak-normalized signal for
incremental constellation size, which governs the differential penalty; and
the fact that the PMEPR is predominantly dominated by the incoherent
waveform superposition for high values of N , making the penalty increasingly insensitive to the modulation level (see how the performance for all
orders converge in 5D-CAP). An extreme case of the latter observation is
DMT, whose PMEPR performance is primarily determined by the amount
the potential of OFDM in scenarios employing intensity modulation.
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of superimposed subcarriers (typically in the order of hundreds), appearing totally invariant with the modulation per carrier. Nevertheless, the
PMEPR clearly raises with N despite an apparent drop in the incremental
step; and consequently, 4/8-level 4D-CAP (not visible) and the three cases
in 5D-CAP follow DMT’s curve with notable proximity; evincing the need
to pay serious attention to PMEPR-driven performance degradation even
in single-carrier configuration when high-order modulation and multiple
dimensions are to be employed.
Furthermore, the CCDF profiles of CAP and DMT are noticeably disparate, where the former decreases more steeply. The consequence of this
observation turns of importance in clipping, a commonly used technique
to alleviate PMEPR-driven penalties by which all values surpassing a predefined threshold are clipped out [144]; and hence a proportional amount
of information is lost. In this regard, flatter curves are more prone to clipping, as the dynamic range of the signal can be reduced with hardly any
information loss and vice versa; demonstrating that clipping in N D-CAP
signals is not as effective as in DMT/OFDM, albeit the trend slightly equalizes with increasing N . Nevertheless, CAP exhibits lower average PMEPR
for all studied dimensions and any modulation order, being N = 4 the
largest dimensionality experimentally demonstrated in optical communications as yet [145]; and making CAP’s (or PAM’s) receiver sensitivity
provenly superior in conditions where nonlinearities are fundamental limiting factors [146, 147]. However, the picture changes substantially when
band-limited or frequency-selective channels are taken into account, where
spectral flexibility is most-wanted property instead [90, 96].
Finally, we observed that the PMEPR is heavily dependent on the
band’s center frequency; potentially changing by more than 2 dB between
certain implementations. As illustrative example, Figure 3.8 shows the
PMEPR performance of 4-level original 2D-CAP for 200 center frequencies (fc in 3.18) ranging from 1 to 2 times the symbol rate FB (right);
and the CCDF of the respective average power (left), evincing that it is
exclusively the peak power what drives the extra degradation. The filters
are 25-symbol long and the statistical analysis is performed over 20000
samples of 500 symbols each per frequency offset case. We observed that
the offset-driven PMEPR penalty is periodic every FB with mirrored symmetric every 0.5×FB at exactly FB periods, thus evaluating 1-to-0.5×FB
would anyways cover the possible dynamic range entirely. For example,
the curve at 1.1×FB precisely coincide with the ones at 0.9, 1.9 or 2.1×FB .
Even though locking the band to the best-performing frequency range is
74
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
(a)
(b)
Figure 3.8: In b, CCDF of PMEPR of 4-level original 2D-CAP system for
200 different center frequencies ranging from 1 to 2 times the symbol rate
FB ; in a, the CCDF of the average power.
definitely doable, it should be noted that CAP bands are often applied a
β-dependent offset to place them near DC with the purpose of maximizing
the electrical bandwidth usage and avoid over-filtering effects to the largest
extent. In these cases, and those where bands may exhibit arbitrary offsets
like in next-described MultiCAP systems, this impairment factor may be
worth consideration.
3.2.1.2
MultiCAP
One relevant conclusion from the previous section is that CAP bands are
not restricted to a particular center frequency, but rather to a minimum
spectral occupancy determined by 3.13. That is, on condition that 3.14
is satisfied, several independent multi-dimensional bands can be arbitrarily
arranged across the available bandwidth. This is the core idea in MultiCAP,
where the electrical spectrum is subdivided into a user-defined number of
low-baudrate bands, enabling highly flexible spectral control and hence
proven robustness against frequency-selective distortions (e.g. CD [96]) or
in strong band-limited channels (e.g. visible light communications [148], or
cost-effective fiber-optic short-haul systems [133,135]) without jeopardizing
CAP’s fundamental signal properties. Figure 3.9 illustrates a basic example
of channel-dependent power-loading in a 3-band 2D-MultiCAP with same
bandwidth per band.
Mathematically, each of the bands in MultiCAP accepts identical end75
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Figure 3.9: Frequency response of one realization of 3-band 2D-MultiCAP.
The spectral occupancy for all the bands is 12.5% of the Nyquist rate, and
they are centered at 12.5%, 50% and 75% of the Nyquist rate. Their power
gain is adjusted to compensate for the loss induced by an emulated channel
response (red dashed line) at the respective center frequencies.
to-end formulation as standard CAP (see 3.3), and consequently, the full
system can be viewed as an M -fold concatenation of independent CAP instances. This drives the attention to the major implementation challenge
in MultiCAP system with large number of bands and dimensions, the processing complexity; directly related to the required number of filters and
their taps. To quantify the bare minimum number of coefficients per filter,
we start by adapting the sampling theorem in 3.14 to multiple bands:
FS;min = 2BWmin =2
=
M X
FB;b
b=1
M
X
2
Nb (1 + βb ) + γb
FB;b
2
=
(3.21)
FB;b (Nb (1 + βb ) + γb )
b=1
where M is the total number of bands, FB;b is the symbol rate of the band b
with b ∈ N, βb is the respective roll-off factor, Nb the number of dimensions,
F
and γb is the fraction of B;b
2 standing for the frequency guard-band on the
lower frequency side of the corresponding band (custom frequency offsets
shall be included in this parameter). Now dividing 3.21 by the symbol rate
76
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
per band FB;b and rounding to the immediate superior integer, we obtain
the associated number of samples per symbol:
PM
Ssym;b =
b0 =1 FB;b0
(Nb0 (1 + βb0 ) + γb0 )
FB;b
(3.22)
which in the case where all parameters are identical for all bands, 3.22
simplifies to:
Ssym;b = dM (N (1 + β) + γ)e
(3.23)
The influence of sub-band partitioning (M > 1) and possible guard intervals (γ > 0) on the number of samples per symbol is clearly revealed in
3.22, becoming particularly explicit for the scenario described by 3.23. Considering 3.23 for simplicity, we observe that even when maximum spectral
compactness is assumed for Ssym;b minimization (γ = β = 0), the number
of samples per symbol is increased M -fold per additional band. Therefore,
reworking the calculations in Section 3.2.1.1, M -band N D-CAP uses M N
times longer pulse-shaping filters than 1-band 1D-CAP for achieving the
same SE (M N S/symbol); resulting in a total of (M N )2 times more filter
coefficients to be processed for signal de/multiplexing. For example in a
6-band 2D-CAP configuration, assuming 15-symbol long filters so that the
BER error floor keeps ≤10−9 for β = 0.1 in all bands [96], the total number
of extra taps disregarding equalization equals (6·2)2 ·15 = 2160; approximately 7-to-8 times the amount required for bulk frequency-domain CD
compensation of 1000-km SSMF transmission [149].
Besides the additional complexity, two further pertinent conclusions
are drawn from the above equations concerning digital signal generation.
First, the over-sampling factor per band, M , is not strictly needed for
its perfect generation, but rather a consequence of the required Nyquist
sampling rate for the ensemble/multi-band spectrum. Thereby the number
of samples per symbol and hence the spectral occupancy is variable and
potentially different among bands. Assuming a band-limited channel for
which BWmin is wished invariant, the eligible range of symbol rates is
directly proportional to M ; providing the system with an additional tool
(not present in OFDM/DMT) for rate adaptation and spectral shaping with
M -driven granularity. The quantitative influence of M on the per-band
spectral occupancy is made explicit through the symbol rate by imposing
FB;b Ssym;b = 2BWmin to be constant and resolving for 3.22.
Second, the minimum sampling frequency satisfying 3.22 given a fixed
BWmin converges to the Nyquist rate (2BWmin ) as M N increases. This is
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Figure 3.10: Excess sampling rate versus M N product for the system configuration described by 3.23 with γ = 0. Hundred values of β ranging from
0.01 (highest values) to 1 (flat bottom line at 1 excess sampling rate) are
evaluated.
because the deviation from exact number of samples per symbol induced by
the ceiling function in 3.22 becomes progressively negligible as the operand
grows. The excess sampling rate is then given as the ratio between Ssym;b
with and without ceiling function; which is illustrated in Figure 3.10 for the
simplified configuration in 3.23 with γ = 0 (maximum SE). For example,
we observe that conventional 1-band 2D-CAP with β = 0.01 requires 1.5
times higher sampling rate than theoretically necessary; however, if the
number of bands is increased to 3 while maintaining the bi-dimensionality
(M N = 6), the excess rate goes down to 1.15. This means that MultiCAP
allows for signal generation across the full-bandwidth of the DAC with more
relaxed sampling clocks; or from a different perspective, more efficient use
of the available bandwidth is attainable.
Peak-to-mean envelope power ratio : exactly like multiplexing subcarriers or increasing the CAP’s dimensions do, assembling independent
sub-bands increases the PMEPR. Whereas this is a readily foreseeable
point, the complete characterization of all the possible combination of parameters (M , Nb , βb , γb , amplitude modulation order, and symbol rate)
is an intractable endeavor; hence some pragmatic selectivity is mandatory.
In this regard, we have demonstrated the quadratic growth of the processing complexity with M and Nb in the optimum-SE scenario (see 3.23 and
78
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.11: CCDF of PMEPR for 1-to-10-band (plus 15-band case) 4-level
original 2D-CAP system. Uniformly bit-loaded DMT is included for reference.
subsequent elaboration), from which Nb has been demonstrated to raise
the per-band PMEPR to equivalent ranges as DMT beyond three dimensions (see Figure 3.7). On the other hand, because the primary purpose of
MultiCAP is providing spectral flexibility, the number of bands is wished
M ≥ 2 where the upperbound is determined by the desired granularity and
the associated processing complexity. Therefore, because of these observations, we reduce the analysis to zero guard-band (γ = 0) for maximizing
the SE, a practical β = 0.1 for which 10 symbols per filter suffice for BER
error floor at 10−9 , N = 2 in order to have enough PMEPR budget to accommodate M > 2 incurring neither excessive degradation nor complexity,
and up to M = 10 for allowing independent spectral shaping in fractions
of 10% of the total bandwidth. For simplicity, FB;b is considered the same
in all bands (≈3% of the sampling rate), bit- and power-loading are uniform distributed across all bands, and conventional 2D-CAP is employed
for signal generation.
Referencing single-band CAP, ≈4.5 dB PMEPR penalty is observed at
CCDF=10−3 when the number of bands is increased to 10, with an absolute
maximum of ≈14 dB. Besides the expected behavior, three other observations are worth mentioning. First, the curves’ profile are independent of M
(except for the outlier at M =2) and remarkably steep (≈2 orders of mag79
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nitude in CCDF per dB in PMEPR); underscoring the little efficiency of
clipping for 2D configuration in both CAP (see Figure 3.7) and MultiCAP
(the tendency appears mainly dependent on N ). Second, the incremental
penalty per additional M is variant, proving the existence of an optimum
∆PMEPR-to-∆M ratio. For example 5, 6, and 7 bands perform in less
than 0.25 dB from each other at CCDF=10−3 and up to ≈1.5-dB better
than 256-carrier DMT; strongly encouraging M ∈{5, 6, 7} as long as the
corresponding spectral partitioning granularity (from 14.2% to 20% of the
total bandwidth) and the processing complexity meet the target requirements. Third, unlike in DMT/OFDM, the degree of spectral flexibility in
MultiCAP systems is traded on PMEPR and complexity without jeopardizing the SE. This enables efficient system designs better adjusted to the
needs in spectral control. For instance, if 25% partitioning granularity was
hypothetically enough in a given scenario, we could use the above-described
configuration with just M =4; achieving similar SE than 256-carrier DMT
with more than 2 dB better PMEPR performance at CCDF=10−3 . Further
performance gains could be achieved by optimally adapting the bandwidth
per band to the end-to-end response (e.g. finer granularity to combat higher
attenuation selectivity).
3.2.1.3
C-MultiCAP
Elementally based on MultiCAP system, C-MultiCAP enables its use in coherent communications, aiming for longer transmission distances and higher
throughput. Besides the LO-driven sensitivity improvement, quadrature and polarization components14 are now available to carry M -band
N D-CAP signals; providing 4-fold extra dimensions whereby the SE is multiplied, and multi-dimensional modulation formats of higher order are made
possible.
Irrespective of the application, MultiCAP signal represents the building block in C-MultiCAP generation, and so the same mathematical formulation, signal properties and implementation constraints per coherentdimension hold (see Section 3.2.1.2). Differently however, C-MultiCAP
transceivers are devised to compensate for a handful of essential distortions typically associated with polarization-multiplexed coherent communication: carrier frequency offset plus phase noise, and quadrature plus
polarization mixing. Whereas electronic mitigation of such impairments
14
To avoid confusion with CAP dimensions (N ), quadrature and polarization components (4 in total) are referred to as coherent-dimensions.
80
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
is already a mature subject in both single-carrier configurations [85, 150]
and OFDM systems [151], C-MultiCAP presents original characteristics
that impede the straight application of existing DSP structures. Adapting
those individual algorithms and structures according to the signal properties is thus obligatory, and as it is often the case, it implies taking specific
actions in the transmitter side. Next, we elaborate on the transmitter-side
modifications and generation constraints adopted in C-MultiCAP to enable
impairment compensation and successful demodulation.
Frequency offset and phase noise
Commonly operated in intradyne configuration, coherent detection results in a time-varying phase distortion that couples both quadrature components in a random and dynamic manner. Typically in single-carrier approaches employing AMF, this residual phase variation is compensated via
constellation-driven second-order DPLL [85]; whereas in OFDM, special
training symbols plus pilot carriers are transmitted for carrier frequency
and common phase noise estimation respectively [151]. Notwithstanding,
none of these approaches suit the functional requirements of MultiCAPbased signals for the following reasons:
• High-SE systems featuring spectral fragmentation rely upon exceptionally selective frequency demultiplexing.
Particularly in
C-MultiCAP, band separation is performed through a set of nonoverlapping matched filters per coherent-dimension, which are in turn
necessary for sustaining the Nb th order dimensionality and ISI-free
communication (see Section 3.2.1.1 and Section 3.2.1.2). That is,
RX CAP filtering is not just a digital means of isolating the subbands, but also indispensable front computation to enable any subsequent processing towards successful demodulation. Failing to maintain exact TX-RX filter matching in CAP-based systems along with
quadrature coupling, causes the combined effect of: (i) intra- and
cross-coherent-dimension inter-band interference (IBI), (ii) asymmetric band attenuation, and (iii) incorrect cancellation of the ISI inherent to CAP signals. As a result, in the absence of prior compensation
of carrier’s frequency offset and phase noise, the standard real-valued
CAP filters work as 4×2×Nb ×time-dimensional15 coupling functions;
15
The dimensionality is given by: 4 coherent-dimensions (polarization-mixing is assumed), 2-bands per coherent dimension (worst-case cross-dimensional IBI), Nb CAP
81
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compelling to process the entire signal spectrum in an intractable
blind adaptive equalizer. This represents the major difference with
single-carrier digital coherent receivers, where filtering and equalization is first step to boost the SNR and thereby improving the estimation accuracy of the residual phase.
• Owed to reasons relating to electro-optical generation stability and
digital processing ease plus performance, training symbols are designed so that the occupied bandwidth and the average power remain as constant as possible. In this regard, because OFDM/DMT
modulation is performed directly in the frequency domain, timesensitive training symbols with different properties for frequency recovery [152] can be readily inserted and, at the same time, jointly
satisfying method-specific and scenario-dependent spectral and power
constraints. For example, odd sub-carrier suppression plus power
loading to create even symmetry [153] while compensating for the
power loss associated to the nulling process and adapt to the channel
response. Moreover, the PMEPR in OFDM/DMT systems is insensitive to the modulation order as long as the number of carriers is
in the order of hundreds; in turn making feasible the use of very
low-bandwidth dummy sub-carriers spread across the spectrum for
tracking the common phase noise with affordable overhead [151].
Modulation in MultiCAP, on contrary, is carried out in timedomain on a band-by-band basis, whose center frequency is to remain adjustable, and whose estimated minimum spectral occupancy
is bounded by processing complexity and PMEPR between 14.2%
and 20% of the electrical modulation bandwidth (see Section 3.2.1.1).
Clearly on this account, the design of effective training symbols for
frequency offset compensation meeting such requisites in spectral flexibility is utmost challenging, and the use of dedicated sub-bands for
tracking the phase noise is prohibitive from the overhead standpoint.
In summary, coherent DP-MultiCAP transmissions need blind estimation of the residual frequency offset and phase noise at the beginning of
the RX DSP chain, thereby increasing the isolation ratio among intra-band
CAP dimensions; and minimizing the random/noisy contributions from
intra- and cross-dimensional IBI. As it will be reasoned in the next point,
these benefits facilitate per-band equalization with standard single-carrier
dimensions per band, and time (ISI).
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3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
tools when the characteristic parameters of the constituent MultiCAP signals are identical.
For this matter, we propose inserting one or multiple unmodulated frequency pilot tones (FPTs) as low-overhead gauge for unambiguous, continuous and blind phase tracking; a simple method whose effectiveness and
robustness as compared to other alternatives has been demonstrated in
several scenarios [46, 122, 154–158]. The reader is referred to Appendix C
for detailed description of the FPT technique and general implementation
concerns.
Quadrature and polarization mixing
Even after ideal frequency offset and phase noise compensation, it exists certain degree of cross-dimensional coupling originating from polarization mixing plus PMD, residual CD (prior bulk CD compensation in
understood), and possible filtering effects specially on the outer sub-bands.
Compensating for those effects demands the transmitted signal to present
certain deterministic relations among coupled dimensions, which can later
be exploited at the receiver as reference to recover all independent signals. Clear examples are the stationary characteristics of the constellation
diagrams employed in popular blind equalization techniques like CMA or
RDE [85], space-time codes for polarization demultiplexing [159], or the use
of 2D training sequences jointly carried by both quadrature components
(e.g. n-ary QAM or CAZAC) [160, 161]. For this reason, the parameters of
the constituent MultiCAP signals must be neither random nor mutually independent, but rather describe a signal structure which facilitates effective
equalization with as little complexity as possible.
Accordingly, we constrain MultiCAP’s parameters (M , Nb , βb , γb , amplitude modulation order, and symbol rate) to be variable but homogeneous
across all bands, a decision resulting from various other assumptions and
requirements: (i) end-to-end filtering impairments are similar in all coherent dimensions, so that the same spectral arrangement equally applies in
an optimum manner; (ii) all the operative frequency ranges (not disabled
by filtering restrictions) within the available electro-optical bandwidth are
to be used for SE maximization; and (iii) lessening the maximum absolute
number of overlapping bands is desirable to cut down the plurality of interfering dimensions, and thus to simplify equalization. In addition, we impose
CAP filters’ impulse responses to be identical in overlapping bands which,
leveraging the structural constraints, conveniently allows for building Nb
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Figure 3.12: Conceptual illustration of C-MultiCAP signal spectrum as
devised in this investigation. Both polarization and quadrature components
are represented.
equivalent DP complex base-band signals per band. The use of well-known
algorithms in standard single-carrier digital coherent receivers [85, 150] is
thereby enabled per CAP dimension (see Section 3.2.2), making possible
the scalability of Nb at a proportional increase in processing load.
Final C-MultiCAP transmission system : the spectrum of
C-MultiCAP signal is conceptually depicted in Figure 3.12, incorporating the above-mentioned structural requirements. All coherent-dimensions
carry identical MultiCAP structures, where M , Nb , βb , γb , amplitude modulation order, and symbol rate are allowed to vary within margins determined
by a compromise between spectral flexibility, rate adaptation granularity,
processing complexity and PMEPR performance. It is important to remark
that the structural restriction does not prevent transmitting independent
data per dimension.
One single-sideband radio-frequency FPT is digitally inserted (see Appendix C) between DC and the first spectral component of band #1 in
one polarization. A frequency guard-band of ≤500-MHz is allocated for
its insertion16 , representing the only efficiency loss per coherent-dimension
as compared to standard MultiCAP; less than 3% overhead considering
≥20-GHz modulation bandwidth. The reasons for this layout are:
• Near DC (carrier frequency after up-converion) the accumulated passband and low-pass filtering distortions are decreased, thus improving
16
The concrete value depends on the phase noise characteristics and the filter employed
for FPT isolation.
84
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.13: Simulated spectrum of 2D-CAP signal with ≈300-MHz guardband with respect to DC (grey); and one 100-MHz FPT (orange), right
between sidelobes for interference minimization.
the estimation performance over longer transmission distances. Additionally, because DC is the symmetry center of the signal spectrum,
the maximum distance to the highest modulated frequency is minimum, which reduces the gravity of the phase misestimation caused
by group-delay mechanisms such as CD.
• Strong phase noise and broad filters increase the sidelobes-FPT interference, hence it is convenient to allocate ample guard-band range
(up to 500 MHz in our case) to shift in the lower-magnitude sidelobes
of band #1 whenever necessary. However, because the sidelobes’
attributes vary with different MultiCAP parameters, the FPT may
undergo severe interference even for high offsets. To solve this, we digitally aligning the FPT center frequency to the spectral zero closest
to DC, thus minimizing the cross-talk, stabilizing the phase estimation reliability, and allowing for better pilot-to-signal ratios (PSRs).
Figure 3.13 shows a clarifying example.
• Because the FPT period is hundreds of times longer than the expected
differential group delay (DGD) in MAN networks (see footnote 3),
the equivalent polarization-coupling matrix acting on the FPT can
be approximated by a real-valued rotator. By transmitting only one
FPT in one polarization, coarse π-ambiguous estimation of the mixing
factor can be performed through blind search upon the criterion of
maximum constructive interference (see Section 3.2.2). In addition to
the prior compensation of the phase noise and the residual frequency
85
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offset, this operation potentially improves the performance and the
tap-convergence speed of the adaptive equalizer.
Considering the guard-band for FPT insertion, the maximum gross
throughput is thus ≈97% of 4×MultiCAP, exactly the same 4-fold factor
by which multiplexing complexity is multiplied. The extra computational
cost of FPT insertion should also be added, since it represents a core functional element in C-MultiCAP. In terms of PMEPR, and benchmarking
MultiCAP, overall reduction is experienced consequence of the addition
of the orthogonal coherent-dimensions. The relative performance against
DMT is thus expected to be maintained within the same range (see Figure
3.11); although their disparate PMEPR statistical properties, and the influence of the amplitude modulation level in C-MultiCAP suggest scenariospecific analysis. In this regard, although the influence of the FPT itself on
the PMEPR is negligible with typical PSR around -15 dB, the particular
frequency offset induced on the MultiCAP signal for allocating the FPT
guard-band may be determinant and shall be contemplated.
3.2.2
Detection
Enabled by FPT technique and structural restrictions, multi-dimensional
and spectrally flexible C-MultiCAP signals can solely leverage DSP for
their successful transmission in coherent communications. Whereas the
algorithms in C-MultiCAP and standard single-carrier systems are very
much alike, their succession is disposed similarly to OFDM’s in the former;
resulting in a mixed organization worth detailing. In the following subsection, the arrangement of RX DSP algorithms for C-MultiCAP detection
is discussed as employed in this investigation; from which the FPT-based
phase tracking method and the multi-dimensional adaptive equalizer are
further developed as the primary modifications with respect to standard
single-carrier’s DSP.
3.2.2.1
DSP structure
Figure 3.14 shows the schematic diagram of the RX coherent DSP for singlecarrier (a), C-MultiCAP (b), and OFDM (c) systems. It is visible that CMultiCAP processing represents an intermediate compromise with strong
resemblance with both other alternatives.
The first four blocks in C-MultiCAP are similar to single-carrier’s, that
is, after initial front-end corrections, bulk CD compensation is applied to re86
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
(a) Single-carrier
(b) C-MultiCAP
(c) OFDM
Figure 3.14: Digital coherent receiver structures for single-carrier,
C-MultiCAP and OFDM signals. Color-matched blocks across structures
perform operations of similar purpose.
lieve the complexity of the posterior equalizer, followed by broad-band lowpass filtering that removes out-of-band noise and hence improves the overall
SNR. At this stage, the residual frequency offset from intradyne detection
and the phase noise is estimated by means of FPT. Conditional upon
correct timing recovery17 , FPT processing mitigates the inter- and intracoherent-dimension IBI potentially occurring in the successive 4×M ×Nb
matched filtering processes, and permits the correct separation of the Nb
dimensions per band. In addition, when the FPT is inserted in one polarization exclusively, coarse estimation of the polarization mixing angle is
possible via blind search; with the cost function determined by the peak
magnitude of the field’s Fourier transform on the employed polarization,
17
Because clock recovery is needed before having access to the modulated waveforms,
frequency-domain approaches are mostly adequate. In this concern, techniques exploiting
FPTs have been long proved successful [162] also in similar multi-carrier scenarios [163],
although their utilization in C-MultiCAP systems may require adaptation given the variability of the bands’ allocation.
87
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and maximum absolute as break condition. See Subsection 3.2.2.2. Coarse
polarization de-rotation speeds-up the convergence time of the following
adaptive equalizer.
Afterwards, relying on the the fact that all 4 coherent-dimensions employ the same MultiCAP parameters and filters per band (see the explanation after Figure 3.12), every CAP dimension in every band is grouped
up with the corresponding one from the other coherent-dimensions. The
aggregate entity is treated as an equivalent DP complex base-band signal,
and then transferred to the equalization block (M ×Nb in total) for bandby-band linear impairment mitigation. Three alternative approaches may
be employed: (i) the algorithm is run on only one equivalent DP complex
base-band and then applied to the remaining dimensions, then reducing
the processing complexity; (ii) the channel is individually calculated for
each dimension, improving the performance and robustness; and (iii) the
estimated impulse responses of ≥2 CAP dimensions are averaged and the
outcome is applied to the entire band, which diminishes the influence of the
noise in low SNR conditions. Irrespective of the option, because the input
waveforms to the equalizer have been previously compensated for frequency
offset and phase noise, the use of decision-directed error functions is possible; which can further reduce the convergence time [164]. See Subsection
3.2.2.3.
3.2.2.2
FPT processing
Depending on the number of pilots and their distribution, the processing
steps for phase estimation and the associated performance will change, as
well as their availability for additional applications like clock recovery or
coarse polarization decoupling. For concision and generality, in this part we
elaborate on the DSP blocks and applications particularizing to one singlesideband radio-frequency FPT digitally inserted into one of the orthogonal
polarization states. This arrangement is employed in the experimental
demonstration discussed Section 3.3.
Coarse polarization decoupling : Along propagation over optical
fiber, the polarization state of the field changes. When this field impinges
upon the static front-end polarization splitter, inescapable signal mixing
occurs, which for the case of an FPT with periodDGD can be well modeled with a 2×2 real-valued rotation matrix. This being the case, if the
FPT is inserted only in one polarization, the mixing angle can be estimated by searching over a set of test rotators the one that yields maximum
88
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.15: Graphical representation of the iterative algorithm for coarse
estimation of the signal’s polarization mixing angle. Top-right inset figure conceptually illustrates the power-leakeage effect of an arbitrary lowfrequency FPT on the spectra of both orthogonal polarization fields when
the mixing is not corrected for. Likewise, the middle inset figure represents
the target power coupling.
peak magnitude on the target polarization. The inherent π ambiguity can
be compensated after or during equalization by means of a correlated DP
training symbol. Figure 3.15 shows a clarifying illustration of how the algorithm proceeds, the effect of mixing on the magnitude spectra of both
polarization, and the target coupling ratio. Note that for typical PSRs, the
tone’s peak prominently rises over the maximum signal level perfect polarization alignment (e.g. Figure 3.13), hence the maximum-peak criterion
holds without requiring specific FPT isolation.
Because this technique relies on keeping a very high ratio between the
FPT period and the DGD, it is important to assure compliance given the
expected scenario-specific PMD values (see footnote 3) and the concrete
FPT allocation restrictions (≤500 MHz). With that purpose, we studied
the evolution of the ratio between the peak magnitude of the Fourier trans89
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(a) 200-MHz FPT
(b) 10-GHz FPT
Figure 3.16: Ratio between the peak magnitude of the field’s Fourier transform in X and Y polarization versus the mixing angle. Two FPT center
frequencies are evaluated, 200 MHz and 10 GHz, for 31 DGD cases ranging
from 0 to 30 ps. The insets conceptually illustrate the alignment of the
FPT with respect to the reference axis which, in this case, is arbitrarily
chosen as the X polarization.
form in both polarizations (X and Y ) against the mixing angle. This is
performed for two different FPT center frequencies (200 MHz and 10 GHz)
in the absence of data signal, and for 31 DGD cases (from 0 to 30 ps).
The process is: initially, the FPT power is equally split between both polarization, which represents the worst-case mixing; then, certain degree of
DGD is induced; and finally, the polarization rotation angle is swept and
the ratio evaluated. Figure 3.16 presents the results. For 200 MHz and
center frequencies in the same order of magnitude, PMD has hardly any
effect in the ratio. This indicates that the angle at which the maximum
peak magnitude is found, unequivocally tells the polarization mixing angle,
thereby allowing for its pre-compensation before regular equalization. On
the contrary, the interference of GHz-range FPTs appears highly sensitive
to most of the considered DGD values; such that, from ≈15 ps on, the
ratio ratio profile flattens out, preventing peak detection and thus reliable
polarization de-rotation.
The use of one ≤500-MHz FPT if further justified as a means of realizing coarse estimation of the polarization mixing angle, accelerating the convergence speed in blind adaptive equalizers, and so reducing the required
training time that partly determines their computational complexity. In
addition, most of the FPT power will converge to one polarization state,
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3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.17: Conceptual schematic of 0-Hz FPT processing to compensate
for residual frequency offset and phase noise.
which leads to higher pilot-to-noise ratios and therefore more precision in
the next phase noise and frequency offset estimation process
Phase noise and frequency offset compensation : Figure 3.17 depicts the processing stages of the algorithm under discussion, where all
computations preceding the correction of the estimated phase (i.e. peak
detection, filtering, phase conjugation and tone reconstructions) are exclusively carried out on the polarization state containing the largest fraction
of the FPT power. For simplicity, the example in Figure 3.17 concerns an
FPT initially located at DC.
After Fourier transformation of the field, the maximum peak of the
magnitude is detected. Maintaining frequency-domain operation, filtering is then applied centered around this spectral component to separate
the FPT from the signal. The filter parameters and its type are selected
based on empirical performance evaluation. In this investigation, rectangular or Gaussian profile is used, whose typical bandwidth ranges from
15 to 30 MHz primarily depending on the SNR level. At this point, the
FPT is assumed perfectly isolated, and so the time-varying phase distortions are extracted by directly reading the signal argument. The measurement noise in the output phase vector may be attenuated through
various smoothing tools, from which statistical techniques such as particle filtering [165], or extended Kalman filtering incorporating the laser’s
dynamics [166] have demonstrated remarkable performance. Finally, the
equivalent phase-conjugated tone is reconstructed in the time domain, and
multiplied sample-by-sample with the full detected C-MultiCAP signal. If
the FPT is not initially inserted at DC, a simple correction factor corresponding to the default center frequency must be included in the estimated
phase vector.
Figure 3.18 shows a simulated example of the FPT tracking performance
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(a) 35-dB SNR
(b) 15-dB SNR
Figure 3.18: Results of an FPT-based phase-noise tracking simulation at
-15-dB PSR, and 35 and 15 dB of SNR. The reference vector is generated through a random walk process with Gaussian-distributed transitions
exhibiting zero mean and 200-kHz equivalent variance.
at -15-dB PSR, and 35 and 15 dB of SNR (adjusted through AWGN loading). The data signal is a 4-level 6-band 2D-C-MultiCAP with 400-MHz
guard-band for FPT allocation, βb =0.1 and γb =0 ∀b. The isolating filter is
Gaussian with 20-MHz bandwidth, and the smoothing technique is a simple moving average with ≈10-ns window. Frequency offset is not simulated
in order to facilitate the visualization of the phase fluctuations, which are
generated by means of a random walk process with Gaussian-distributed
steps presenting zero mean and 200-kHz equivalent variance. The effectiveness of the FPT for phase-noise tracking is observed, as well as the
apparent accuracy improvement that a simple moving-average smoothing
may provide.
3.2.2.3
Blind channel equalization
Succeeding FPT processing and timing recovery, the output sequences from
CAP matched filtering are decimated prior to blind channel equalization;
generally entrusted to reverse all remaining intra- and inter-dimensional
linear distortions. For instance in single-carrier transmissions, the joint
compensation of time, polarization, and quadrature coupling (e.g. ISI,
polarization mixing, and CD) is a well-known procedure, typically performed through a set of four complex-valued adaptive filters disposed in a
2×2×time MIMO butterfly structure [85], and possibly governed by different updating rules and error functions. In C-MultiCAP, however, the di92
3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
mensionality is variable and M Nb times higher than single-carrier’s, bringing the processing load well beyond pragmatic limits. Decreasing such
complexity first requires identifying the least relevant coupling mechanisms,
thus laying the ground for selectively grouping the independent sequences
into lower-dimensional structures which can be later equalized in a simpler
parallel fashion. In particular, we make the following observations:
• Because the residual frequency offset and phase noise are compensated beforehand, the IBI within each coherent-dimension will hardly
degrade the performance. Inaccuracies in the estimation of the phase
vector will lessen the validity of this consideration, nonetheless, we
note that the magnitude of the deviation does not exceed 1% in normalized mean square error (NMSE) sense for >10-dB SNR under optimum PSR configurations. For instance, the smoothed curve in Figure
3.18b presents ≈0.5% deviation. In the presence of polarization and
quadrature coupling, cross-dimensional IBI is conveniently reduced to
its minimum by making all coherent-dimensions carry parameter-wise
MultiCAP replicas. As advanced in Subsection 3.2.1.3, this arrangement assumes that the end-to-end filtering degradation is similar in
the four coherent-dimensions, so that the same structure optimally
applies; and the desire to exploit all operative spectral regions for SE
maximization.
• Finer spectral partitioning makes the residual CD (after bulk compensation) and other filtering effects asymptotically equivalent to
complex-valued scaling; such that, in the convergence limit, the ISIdriven coupling among CAP dimensions becomes suppressed. This results in CD and filtering impairments which only manifest as quadrature mixing from the bands’ perspective.
• Imperfect FPT-based estimation of the polarization mixing angle,
and the required PMD tolerance margins18 compel to ≥2×2×timedimensional MIMO equalization.
In short, CAP bands and dimensions are, to a large extent, exclusively
coupled through quadrature and polarization mixing. This statement holds
18
With average symbol rates per band as low as units of GHz, the mean √
DGD after
1000-km SSMF propagation with a characteristic PMD parameter of 0.1 ps/ km represents less than 2% of the symbol period. However, tolerance margins recommended for
high-speed MAN (see footnote 3) increase this fraction to approximately 20%-25% of the
symbol period.
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upon the assumption that the phase noise and frequency offset are neutralized, and that per-band filtering and CD degradation are well approximated
by a flat complex-valued scaling. In such case, and given the imposed CMultiCAP’s structural restrictions, the joint equalizer can be simplified
without risking performance by trimming off its dimensionality (i) M -fold,
enabled through parallel processing per band; and (ii) Nb -fold, enabled
through parallel processing per CAP dimension. Accordingly, the final
equalization approach comprises a maximum of M Nb independent complexvalued 2×2×time-MIMO adaptive filter banks, for what the efficient and
well-understood algorithms commonly employed in DP QAM single-carrier
transmissions can be utilized. For facilitating the understanding, the reader
is referred to Figure 3.19; a conceptual graphic showing the distribution of
all available dimensions in C-MultiCAP systems, as well as their interaction before and after equalization under the above-mentioned assumptions
and structural restrictions.
Comments on collective equalizer : besides decreasing the maximum
number of jointly processed dimensions, this parallel approach also provides modular flexibility and diversity, then opening for strategic resource
utilization through selective equalization; as well as the implementation of
cooperative modes of operation with different purposes. In this investigation, three modes are devised:
• Share : channel estimation is performed on a single CAP dimension
and the outcome is applied to itself and the remaining ones. Because the adaptive algorithm only needs to be run once per band, the
complexity is minimized.
• Average : the channel is individually estimated for ≥2 CAP dimensions, the impulse responses are averaged for partial suppression of
intra-band noise, and finally the resultant response is equally applied
to all CAP dimensions. Intermediate compromise between complexity
and performance.
• Independent : no cooperation between CAP dimensions. Linear impairments are individually corrected for each of them. Best performance and maximum complexity.
In addition, because the proposed method is not fundamentally limited by the number of bands or CAP dimensions, the system is potentially
scalable to any M and Nb conditional upon satisfying the two core assumptions.
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3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.19: Conceptual distribution of the available dimensions in
C-MultiCAP and their relation after equalization (Band #1 and Band
#2), with all blocks clearly separated indicating orthogonality; and before
equalization (Band #M), with the blocks vertically overlapped indicating
coupling among coherent-dimensions only. On the top right, equivalent
collapsed spectrum of the conceptual diagram; on the left-most side, the
legend for each CAP dimension per band; and on top left, reference constellations for 4-ary PAM CAP dimensions (color-matched with the legend).
The workflow of Share and Average equalization modes is also sketched,
where CAP dimensions actively employed in the estimation of the equivalent channel response are colored in green, whereas dummy dimensions are
colored in grey.
Comments on individual equalizers : blind compensation of the linear impairments per CAP dimension is therefore simplified to resolving
polarization-, quadrature-, and time-wise coupling. To carry that out, we
build two complex-valued sequences by assembling the two independent
PAM signals of each polarization component, and treat them as the two
complex-valued sequences of an equivalent DP QAM single-carrier transmission undergoing similar impairments. By doing so, we likewise make
available these aggregate structures for their processing with mature tech95
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niques such as CMA, RDE, or their decision-directed variants [84–86, 164];
whose proved simplicity and robustness when applied to blind equalization
of QAM-like signals has fostered their pervasive acceptance in coherent optical communications. The interpretation of each CAP dimension as an
individual DP QAM entity is grounded on the following observations:
• The amplitude modulation order is the same for all sequences within
each CAP dimension; resulting in ensemble constellations outlining
square-QAM-like distributions. Note that, however, the order is allowed to change among CAP dimensions (indicated by the different
subscripts in Figure 3.19).
• Identical matched-filter pairs are used per CAP dimension across
all coherent-dimensions. This configuration keeps 1:1 coupling proportion between polarization components and also between quadratures; and makes the above-mentioned square-QAM-like constellations remain valid as reference without requiring their modification
to consider ISI effects (inherently induced when CAP filters are not
matched).
In terms of implementation, certain simplification and performance increase can be obtained by contextualizing the above-mentioned algorithms
to C-MultiCAP’s DSP chain (see Figure 3.14b):
• After coarse compensation of the polarization mixing angle as well as
the phase noise and frequency offset, decision-directed error functions
can be directly applied on the equalizer’s input. A reduction of the
computational load and time is then possible by skipping the preconvergence period, while improving the channel estimation accuracy
by incorporating phase sensitivity.
• Due to precise matched filtering ensured by the prior FPT processing, symbol-spaced operation (in contrast to fractionally-spaced) is
possible with negligible performance difference; further reducing the
processing complexity if desired.
Simulation example : so as to validate the good functioning of the
proposed equalization approach, we simulate the coherent reception of a
6-band 2D-C-MultiCAP system, operating at 20 dB SNR per band. The
bands run at 4 GBd, and feature flat 4-ary bit-loading in all dimensions,
flat power-loading, and βb =0.1 plus γb =0 ∀b. A guard-band of 400 MHz is
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3.2 Multi-band carrierless amplitude/phase modulation for coherent
communications (C-MultiCAP)
Figure 3.20: Evolution of the constellation diagram along the DSP chain
for a simulation of 6-band 2D-C-MultiCAP detection. The constellations
are all extracted from the X polarization of a single CAP dimension of the
second band on the lowest-frequency side.
allocated for the FPT, whose PSR is adjusted for optimum performance.
The FPT is employed for coarse polarization de-rotation, as well as for
compensating 200-kHz variance phase noise (generated as indicated in Figure 3.18) and 200-MHz residual frequency offset. Concerning the individual
equalizer, 4 filters with 10 symbol-spaced taps each are disposed in butterfly
structure and iteratively updated by means of standard decision-directed
algorithm [85] without pre-convergence interval. The equalizer is meant
to compensate for 20-km residual CD (SSMF considered) and polarization
mixing. Low-pass filtering effects were not simulated. The collective equalizer is set at Independent operation.
Demodulation below BER=10−3 (zero counted) was observed for all
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bands over 30000 symbols per band. Figure 3.20 shows the constellations
of the X polarization of one CAP dimension of the second band from the
lowest-frequency side at different stages of the DSP chain. We selected
this allocation for simplicity, as all constellations in all bands were alike
in the absence of filtering effects. It is subtly appreciated how, after FPT
processing, the raw constellation tends to square; hinting the satisfactory
correction of the frequency offset. The constellations diagrams following
matched filtering evince the usefulness of coarse polarization mixing estimation for facilitating the tap-updating convergence towards the reference
16-ary QAM constellation in the subsequent equalization stage; whose output confirms the feasibility of the proposed RX DSP scheme for its application in this concrete scenario.
3.3
Experimental demonstration
We have seen that C-MultiCAP systems, as conceived in this investigation,
rely on a set of assumptions and generation restraints for enabling digital
impairment compensation. Here summarize the most relevant ones:
• If frequency walk-off effect is disregarded, a single FPT can be used to
compensate the whole signal for all time-varying phase fluctuations.
• If sub-bands are sufficiently narrow, the degradation caused by filtering effects and CD simplifies to complex-valued scaling, and thus fading out the coupling among CAP dimensions within the same band.
• If the post-detection residual frequency offset is ideally counteracted,
so is the IBI. This leads to the complete isolation of all bands and
CAP dimensions.
• If all coherent-dimensions carry parameter-wise MultiCAP replicas
and use the same matched filters, the separate equalization of all CAP
dimensions is possible by means of the standard algorithms employed
in DP QAM single-carrier systems. This is also conditional upon
satisfying the two previous points.
Imperfect compliance unequivocally ends in performance degradation
or even total inability to recover the data depending on the deviation degree. Therefore, verifying the plausibility of C-MultiCAP for actual use in
coherent communications first requires consolidating the soundness of these
points in the intended experimental conditions. In our case, MANs are the
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3.3 Experimental demonstration
Figure 3.21: Schematic of the experimental setup for C-MultiCAP transmission. Insets: digital spectrum of the in-phase X polarization component
of the 200-Gbps net rate case of study (a), its respective optical spectrum
(b), and the post-digitization end-to-end system’s frequency response from
one of the input ports to the DSO.
targeted scenario, a network segment whose requirements and constraints
naturally harmonize the potential features of C-MultiCAP. In this direction, we conducted the first proof-of-concept transmission of two high-rate
C-MultiCAP configurations over a straight MAN-reach EDFA-only SSMF
P2P link, employing just standard 25-GHz electro-optical components, and
without nonlinearity compensation. The performance is evaluated in terms
of BER for both B2B and transmission.
3.3.1
Testbed
Figure 3.21 shows the schematic of the experimental setup. The three insets
on the bottom part show, on the left, the digital spectrum of the in-phase
component on the X polarization for one C-MultiCAP case of study. Note
the effect of power-loading targeting BER equalization among bands. The
middle inset figure depicts the optical spectrum of the same case of study
before the transmission link. And the right-most inset figure, which shows
the end-to-end frequency response of the system at one of the RX ports
after digitization.
At the transmitter, a 1544.5-nm ECL with ≈100-kHz linewidth is used
as the light source. The output of the ECL is modulated by a DP I/Q
modulator with a 3-dB bandwidth of 23 GHz. The electrical inputs to the
modulator are pre-amplified with a set of four linear broadband amplifiers.
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For data signal generation, one 4-port 64-GS/s DAC with 8-bit nominal
resolution and 3-dB bandwidth of ≈12 GHz is used. The TX DSP consists of a C-MultiCAP signal generator as detailed in Subsection 3.2.1.3.
The detailed description of two cases of study is included at the end. Linear M -shape digital pre-distortion is implemented to partly counteract the
transmitter’s low-pass attenuation, and ≈95% clipping factor is used to
mitigate the quantization noise.
The output from the optical modulator is boosted with an EDFA and
then launched into the fiber link for transmission. This link consists of 3 or
6 spans of ≈75 km of SSMF resulting in total transmission distances of 225
km and 450 km respectively. The propagation attenuation is compensated
after every span employing EDFA-only amplification stages with ≈6 dB of
noise figure. In the receiver side the signal is pre-amplified, filtered with a
reconfigurable wavelength-selective switch, and subsequently detected in a
standard polarization-diversity coherent receiver.
The integrated coherent receiver presented a 3-dB opto-electrical bandwidth of ≈22 GHz. One 100-kHz ECL is used as LO. After coherent
mixing, the four outputs from the balanced photodiodes are digitized for
offline processing by an 80-GS/s, 32-GHz bandwidth DSO. The RX DSP
follows the steps described in Subsection 3.2.2. The demodulation is supported with k-means clustering as justified in Subsection 3.2.1, under the
discussion on CAP’s eye pattern.
C-MultiCAP generation : Two different sets of parameters are studied.
Both are based on 2D-MultiCAP, where the original 2D-CAP technique
(see 3.18) is employed for band generation and multiplexing19 :
• Targeting 200 Gbps net rate, 3-band system with 32-, 16- and 16ary QAM respectively20 at 4.27 GBd each (221.88 Gbps gross rate),
roll-off factor of 0.1 (15 taps per CAP filter) and 12% of the symbol
19
In conventional 2D-CAP, both dimensions are typically paired to conform QAM
constellation, which are not necessarily square (e.g. 32-ary QAM). This possible mutual
dependence slightly complicates symbol generation and demodulation; nevertheless, it
also serves for finer rate adaptation without opposing to the required assumptions or
structural constraints to enable C-MultiCAP detection. Herein, adhering to this convention, we refer to the joint constellation traced by the 2 CAP dimensions within each
band of each coherent-dimension as a K-ary QAM. This should not to be confused with
the square-QAM-like constellation described by the same CAP dimension across the
coherent-dimensions, which serves as reference for equalization (see Subsection 3.2.2.3)
20
Bit-loading is initially adjusted for optimum BER performance in B2B configuration
and left invariant throughout the experiment.
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3.3 Experimental demonstration
rate as inter-band spacing for all bands. This relaxed spectral distribution is chosen so as to minimize possible IBI while maintaining
the entire signal within only 5-dB low-pass attenuation (see Figure
3.21). The electrical modulation bandwidth is ≈14.7 GHz, resulting in a total spectral occupancy after optical up-conversion of 29.4
GHz. This numbers yield ≈7.5 b/s/Hz of gross SE considering the
actual occupied bandwidth, and ≈5.92 b/s/Hz gross SE resorting to
the standard 37.5-GHz frequency slot.
• Targeting maximum throughput, 336 Gbps gross rate is attained with
5 bands of 4 GBd each using 32-, 32-, 16-, 16- and 8-ary QAM respectively. The roll-off factor is maintained in 0.1 (15 taps per CAP filter),
however, in this case the inter-band spacing is necessarily reduced in
order to squeeze the signal within the end-to-end 20-dB bandwidth.
Accordingly, the resultant electrical modulation bandwidth is ≈21.5GHz (≈15-dB low-pass attenuation at this point); translating into
7.81 b/s/Hz gross SE within the actual occupied optical spectrum (43
GHz), and 6.72 b/s/Hz gross SE in the standard 50-GHz frequency
slot.
In both schemes, sub-band power-loading is implemented to minimize
uncoded BER performance according to the end-to-end channel response.
Concerning FEC, a hard-decision 7%-overhead product code with shortened
BCH components is considered [167], achieving post-FEC BER of 10−15
at a pre-FEC of ≤4.4·10−3 contemplating iterative decoding [168]. The
reader is referred to Subsection 3.2.1.3 for detailed description of the FPT
properties and insertion methodology.
3.3.2
Results
The discussion on the results is subdivided into (i) optical B2B, and (ii)
Transmission.
3.3.2.1
Optical back-to-back
Figure 3.22 shows the BER performance as a function of OSNR for both CMultiCAP generation cases in B2B configuration. Theoretical BER curves
of equal-rate QAM signals are included for reference, where gray mapping
and AWGN-only channel is considered for their calculation. Every mea101
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Figure 3.22: BER as a function of OSNR for optical back-to-back measurements. Inset: one instance of the recovered constellations in the in-phase
component of the X polarization at 25-dB OSNR for the 221.88-Gbps gross
rate case.
sured BER value is obtained from one sequence of ≈15000 symbols21 . Performance below the FEC threshold is attained for both schemes, proving
successful demodulation in the linear regime of operation.
The receiver sensitivity approximates 24-dB OSNR for 221-Gbps, and
around 29 dB for 336 Gbps; which stands for 5 dB and 8 dB of implementation penalty respectively at the FEC BER threshold. The relative
degradation difference of 3 dB is owed to two main factors. First, the bands
in the 336-Gbps case are packed denser together, leading to IBI increase.
And second, the outer part of the signal in the 336-Gbps case undergoes
strong non-flat filtering; resulting in ISI-driven coupling between CAP dimensions in the highest-frequency bands. Both impairments partially break
two of the fundamental assumptions that assure correct equalization, and
therefore, strong degradation is understandable. Likewise, we estimate that
other effects, such as PMEPR or partial noise correlation in the 336-Gbps
case, are the reason for the slight discrepancy observed between both slopes;
which is also believed to be the reason why both schemes exhibit disparate
21
This is equivalent to 15000·4· 5+4+4
= 259980 bits in the 221-Gbps generation scheme
3
and 15000·4· 5+5+4+4+3
= 252000 bits in the 336-Gbps one.
5
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3.3 Experimental demonstration
Figure 3.23: BER as a function of optical launched power for the two
considered C-MultiCAP signals after 225-km and 450-km EDFA-only uncompensated SSMF transmission. Inset: one instance of the recovered
constellations in the in-phase component of the X polarization at 2-dBm
launched power for the 336-Gbps gross rate case.
error-floors (≈6·104 for 336 Gbps, while not apparent error stabilization is
observed for 221 Gbps down to ≈1·104 ).
Regarding the 5-dB base penalty of the 221-Gbps configuration, we
simply attribute it to a general breach of compliance with the conditions
for faultless equalization in combination with quantization noise. A clear
violation example is visible in the inset of Figure 3.22, where even the
lowest-frequency bands are non-flatly attenuated. These results and observations evince the need to (i) reduce the spectral occupancy per band
below ≈4-GHz, to further mitigate non-flat filtering at the expense of
processing complexity (see 3.23); (ii) trade SE for broader inter-band frequency guard interval, thus reducing IBI; and/or (iii) perfect the equalization technique, perhaps multiple stage equalization with alternative crossdimensional grouping, which also requires extra processing power.
3.3.2.2
Transmission
Figure 3.23 presents the BER as a function of launched power. For
221 Gbps, the performance is evaluated after 225 km and 451 km of trans103
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mission. For 336 Gbps, the BER results are only addressed for the 225-km
transmission, since successful demodulation was not achieved in the longest
link with the considered FEC. One example of recovered constellations at
2 dBm for the 336-Gbps case is shown for reference. BER values are each
obtained from individual sequences of ≈15000 symbols (see footnote 21).
For 221 Gbps, transmission below the FEC BER threshold is observed
for the full range of probed launched powers after 225-km transmission;
and between -2 dBm and 5 dBm after 451 km. The non-linear threshold
(best performance) is measured at 2 dBm for both propagation distances, a
reasonable value for single-carrier systems that hints an acceptable level of
PMEPR partly controlled with clipping technique. Interestingly, the transmission penalty between 225 km and 451 km at BER=10−3 equals ≈5 dB
on the linear regime, approximately 2 dB higher than expected since we
are just doubling the propagation distance22 . This can be explained by the
greater group delay that CD induces for longer uncompensated links, which
increasingly worsens the accuracy of the FPT-based phase noise estimation
for the highest frequency components (note that the FPT is allocated near
DC). Imprecise phase noise estimation gives rise to an error accumulation
along the subsequent DSP blocks (see Figure 3.14), not only translating
in time-varying quadrature coupling but also imperfect equalization. This
penalty manifests the importance of performing as exact carrier recovery
as possible at the begining of the DSP in C-MultiCAP systems.
Concerning power margins, 4 dB and 3 dB are observed in the 451 km
case for the linear and nonlinear regimes respectively; where the steeper
non-linear slope as compared to 225-km transmission is simply owed to the
higher accumulation of nonlinearities. In this context, despite the ample
margins seen after 225 km of propagation plus the B2B results invite to
think that the 336-Gbps system will also widely fit below the FEC threshold, up to 5 dB of penalty is measured at BER≈10−3 . This is consequence
of the broader bandwidth aggravating the CD-induced walk-off, and again
suggesting the consideration of an improved method for carrier estimation.
One option consists of spreading 2 or more FPTs across the spectrum leveraging inter-band spacing (see Appendix C). The non-linear threshold for
336 Gbps is also found at 2 dBm input power, allowing for 4 dB power
margin allocation (-2 dBm sensitivity).
22
The OSNR over 12.5 GHz bandwidth around 1550 nm of wavelength is approximated
by: OSNR|dB = 58 + PIN |dBm − NF|dB − αL − 10 log N , where PIN is the launched
power, NF is the equivalent noise figure of the lumped amplifier (EDFA in our case), α is
the fiber’s attenuation coefficient in dB/km, L is the span length, and N is the number
of spans.
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3.4 Summary
Overall, transmission results confirm B2B’s in the need to perfect both
C-MultiCAP parameters and RX DSP to guarantee robust transmission in
coherent MAN scenarios. Particularly in this last analysis, the imprecision
of the FPT-based phase noise compensation has been revealed; consequence
of CD decorrelating the phase vectors among spectral components.
3.4
Summary
The burgeoning role of content-delivery networks combines with an unprecedented traffic volume and diversity to prompt the ongoing evolution
of MANs towards ≥400-Gbps all-optical reconfigurable mesh architectures.
These infrastructures enable high order of transparent and flexible connectivity which, nevertheless, come along with important technical challenges
including pronounced low-pass end-to-end frequency responses caused by
manyfold OADM cascadability plus electronics, larger PMD, and extra
OSNR degradation given the absence of intermediate regeneration. In this
context, high-baudrate single-carrier approaches have given way to superchannels and flexible-rate spectral partitioning techniques, proving extraordinarily convenient for counteracting these limitations [13]. Many-carrier
super-channels result in lower symbol rates, thus relaxing the stress on
the electronics and lowering the susceptibility to PMD and CD; as well as
they enable precise spectral shaping, which improves the signal robustness
and the per-channel SE. Nevertheless, the required degree of parallelism
raises proportionally with the technical benefits, which given the stringent
techno-economical requisites in MAN [91], poses a limit in the achievable
granularity for hardware-based implementations.
In order to alleviate this trade-off, digital de/multiplexing has been
contemplated as an powerful complement to parallel optics. Leveraging
TX/RX DSP, these techniques provide software-level control of the signal’s
properties per carrier, translating not only into much finer spectral granularity but also into easy-reconfigurable spectral shape and throughput; all of
which naturally suits the requirements of flexible transponders. This is the
scope this investigation belongs to, where a novel DSP-based multiplexing
and modulation technique is proposed and experimentally demonstrated.
This investigation
In this chapter, C-MultiCAP has been introduced; a P2P DSP-based
multiplexing and modulation scheme which brings together the multi105
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dimensionality of CAP and the spectral flexibility of MultiCAP for application in coherent communications. First, the essentials concerning signal generation and detection are explained in detail. Initially, the mathematical fundamentals and implementation concerns in multi-dimensional
carrierless multiplexing are covered. We have seen that raising the dimensionality causes a quadratic increase of the total number of taps required for
de/multiplexing such dimensions and, nevertheless, does not improve the
SE and worsens the PMEPR. This leaves aside this feature for, perhaps,
implementing multi-dimensional modulation formats which can potentially
improve the receiver sensitivity by ≤1 dB within pragmatic complexity. We
then extend the analysis to multiple bands (MultiCAP) where it is found
that the total number of taps increases quadratically with both the number of bands and the number of dimensions per band. Accordingly, a set of
practical generation parameters is proposed; whereby a spectral granularity down to ≈14.3% of the total bandwidth is possible with two dimensions
per band and ≈1.5 dB PMEPR margin with respect to 256-carrier DMT
without jeopardizing SE. These findings base the final elaboration on CMultiCAP’s generation and detection. It is argued why intradyne detection
of C-MultiCAP signals requires early carrier recovery in the RX DSP as
well as identical generation parameters in the four coherent dimensions to
enable blind equalization taking advantage of conventional algorithms (e.g.
CMA or RDE). Based on this, C-MultiCAP’s RX DSP is conceived and
explained along with its most distinctive blocks; that is, early FPT-based
coarse polarization mixing compensation plus carrier recovery, and channel
estimation per CAP dimension.
Following the characterization of C-MultiCAP generation and detection, we perform an experimental proof-of-concept with the purpose of verifying the validity of the assumptions supporting the proposed RX DSP in
the target scenarios; and given imperfect compliance, evaluate the deviation
tolerance. Two different 2D-C-MultiCAP configurations are implemented:
3-band case at 221.88 Gbps occupying 29.4-GHz optical bandwidth, and
5-band case at 336 Gbps occupying 43-GHz optical bandwidth. The BER
performance is measured for B2B and transmission, where 225-km and
450-km EDFA-only SSMF links are considered. The main conclusion is
that, under this combination of experimental conditions and generation
parameters, the partial fulfillment of the basal assumptions translates into
notorious implementation penalties (≈5 dB and ≈8 dB for 221 Gbps and
336 Gbps respectively) and transmission penalties (≈5 dB after distance
doubling for 221 Gbps).
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3.4 Summary
The results indicate the need to perfect both C-MultiCAP parameters and RX DSP to guarantee robust transmission. Narrower bands and
multiple stage equalization with complementary parallelization approach
stand for possible options in this direction, although the entailed increase
in processing complexity partially questions their feasibility. Alternatively,
certain increase in performance is achievable if looser inter-band spacing is
applied. Despite this measure automatically leads to broader total bandwidths, we have demonstrated ≈8 GHz and ≈7 GHz optical spectral occupancy margin with respect to the tightest-fit standard grid for the 3-band
and 5-band cases respectively; permitting this more relaxed allocation of
bands without sacrificing per-channel SE. FPT processing has proved satisfactory for both polarization mixing angle estimation and frequency offset
compensation. Nonetheless, with regard to phase noise estimation accuracy, strong penalty has been measured owed to CD-induced walk-off effect
for long distances and broad signal bandwidths; which definitely motivates
the use of multiple FPTs strategically inserted across the spectrum, and
perhaps exploiting the above-mentioned looser inter-band spacing. On the
other hand, the potential spectral and rate adaptability of C-MultiCAP
have been demonstrated with up to 336-Gbps line rate at 7.8 b/s/Hz gross
SE in a system exhibiting less than 6-GHz end-to-end 3-dB bandwidth.
In addition, the convenience of C-MultiCAP’s reconfigurability has also
been shown, allowing for >100-Gbps line rate increase in the same link
by simply adjusting the TX/RX DSP. Finally, recovery and equalization
of multiple correlated CAP dimensions (e.g. those bands featuring 32-ary
QAM) has been presented for the first time in coherent systems, proving
the plausibility of their joint exploitation for implementing power-efficient
multi-dimensional modulation formats.
Future work
The present investigation conveys the following items:
• Optimization of generation parameters and RX DSP : ensuring robust signal recovery is necessary first stage, and in this direction,
several aspects still remain uncovered. For instance, per-band bandwidth and inter-band spacing have shown determinant in the equalization performance, and thus realizing scenario-dependent optimization
can be particularly beneficial. Additionally, while the effectiveness
of FPT-based techniques has been demonstrated, great transmission
penalty reduction can be obtained by upgrading the proposed scheme
107
Metropolitan networks
to two or more spread tones. This counteracts the CD-driven decorrelation of the phase noise, resulting in accurate matched filtering
and thus error propagation avoidance. Finally, data-aided equalization should be contemplated. Because active polarization rotation
tracking is performed by the FPT (anyways requiring data-aided disambiguation), the extra overhead owed to the training sequence could
be substantially small.
• Beyond 2D-C-MultiCAP : after ensuring robust signal demodulation,
further features can be contemplated. Despite processing complexity and PMEPR are proven limiting factors in C-MultiCAP that call
for careful control, investigating the pragmatic benefits of those affordable >2D system configurations may be of interest (e.g. multidimensional modulation formats). Cross-band modulation/correlation also represents a plausible alternative.
• Non-linear tolerance : besides the well-known capability of FPT conjugation for partial nonlinearity compensation [46], recent studies suggest that sub-carrier/sub-band multiplexing employing very low symbol rates (2-10 GBd) partly reduces the non-linear noise interference
in uncompensated lumped-amplified transmission links [169, 170].
Both techniques are present in C-MultiCAP, encouraging an interesting demonstration and characterization.
108
Chapter 4
Short-haul networks
Client-side networks comprehend the most varied technological ecosystem
in the global communications hierarchy; showing reaches ranging from few
meters to tens of kilometers, bit-rates from Megas to hundreds of Gigas,
and propagation channels from wireless to SSMF. In this network segment, seamless communication is conditional on the punctual preparation
of market-driven standards enabling equipment interoperability. The concrete implementation agreements vary according to the technical requirements of the service to support (e.g. reach, throughput or latency) and
the environment of application (e.g. enterprise, residential, datacenters or
service-providers’ networks), all in all giving room to a broad spectrum
of functional alternatives. For example, with the vast majority of wired
local-area networks (LANs) operating on Ethernet1 , InfiniBand2 and Fiber
Channel3 standards, more than 40 unique transmission speeds are considered in the aggregate roadmap including currently deployable and prospective configurations [171–173].
Within wired solutions, fiber-optic systems are unquestioned in scenarios beyond 100-m transmission and 10-Gbps rates4 [171]; at the same
time that their penetration in the ≤100-m market5 is progressively favored by attributes like electromagnetic interference immunity and lightness [171, 174, 175]. Accentuated by the relentless increase of throughput
1
http://www.ethernetalliance.org/
http://www.infinibandta.org/
3
http://fibrechannel.org/
4
Router-to-transport and router-to-router applications where ≥10-Gbps 2-km, 10km and 40-km duplex SSMF are industry modes; or warehouse-scale computing with
thousands of 25-Gbps and 50-Gbps servers.
5
In-house connectivity, airborne/automotive datacom or supercomputing.
2
109
Short-haul networks
demand, the technological panorama appears predominantly fiber-based;
evincing the impact of optical LANs’ design on the characteristics of the
entire network. Fundamentally driven by cost and power consumption,
optical LANs are typically built from duplex transceiver units employing intensity-modulation (IM) and direct-detection (DD) and applicationspecific combinations of fiber types and frequency bands6,7 . In terms of
modulation and capacity scaling, serialization (as opposed to parallelization) is preferred for reasons relating to simplicity of implementation, cost
and power consumption [179, 180]. This premise governs the evolution of
data communication standards, where resource-to-performance ratio optimization (e.g. electrical bandwidth, power, physical space and cost) is
prioritized over the incorporation of new dimensions or its extension (e.g.
wavelength, space, phase or polarization). For instance, enabled by the
progress in ASIC design and DSP, digital equalization and FEC are now
possible within acceptable latency margins. This has spur the recent inclusion of AMF into the scope of ≥100-Gbps standardization processes in order
to boost the SE8 . Nevertheless, as line-rate demand exceeds the capabilities of the affordable electrical bandwidth9 , parallelization and multiplexing
are inescapably resorted to; on the whole yielding the hybrid technological
variety that current transceiver modules exhibit.
Particularly in DD systems, WDM and fiber-pair parallelization have
been favored as techniques that do not require retaining the field’s phase
information after photodetection, hence relieving the system from the extra complexity that it entails. Unfortunately, sustainable scalability is not
exempt from saturation either, as tolerable power consumption, cost and
per-blade transceiver density are surpassed for certain line rates and be6
850-nm single-λ/fiber or 850-to-940-nm WDM for ≤500-m transmission over laseroptimized multimode fiber and wideband multimode fiber respectively [176]. WDM Oband for ≥100-m SSMF transmission [177].
7
Technological synergy among different modules is pursued during standardization for
facilitating common testing and deployment, as well as the use of economies of scale for
additional reduction in costs and time to market [178].
8
In the second-to-third quarter of 2015, the IEEE 802.3bs 400 GbE task force moved
to baseline adoption for 500-m reach employing 50-GBd 4-ary PAM (100 Gbps serial)
and for 2-km and 10-km reach employing 25-GBd 4-ary PAM (50 Gbps serial) [177]. The
100GBASE-KP4 standard developed by the IEEE P802.3bj 100 Gbps task force will be
used on the backplane (4 × 25-GBd 4-ary PAM lanes) [181]. Concerning FEC, harddecision Reed Solomon (544,514,10) has been agreed on, with estimated total latency
ranging from 80 ns to 120 ns depending on the concrete chip architecture and predecoding BER threshold ≈10−6 [182]
9
This is especially pronounced in sectors like hyperscale datacenters and service
providers’ networks [171].
110
yond [17]. For instance, an excessive number of wavelengths requires strict
wavelength stability control and makes packaging challenging; while too
many fiber-pairs would likewise suffer from packaging and fiber-transceiver
interfacing issues. Over and above, C-band transmission is avoided in clientside telecom scenarios10 despite the ≈40% lower attenuation/kilometer with
respect to O-band due to CD, hence tightening the power budget margins.
All the latter naturally put the research spotlight on the previously precluded phase diversity, possibly allowing for quadrature multiplexing and
effective CD compensation in the digital domain. On this subject, selfcoherent approaches have been long studied as a means of enabling full-field
recovery in standard single-detector systems by exploiting the photodiodes’
quadratic response [184]. Despite proving successful for full field recovery,
signal-signal beating terms cause notorious electrical spectral inefficiency,
encouraging the development of solutions that trade, without exception,
computational load [185, 186] and/or structural complexity [187, 188]. Interestingly, polarization diversity has been intensively investigated to circumvent the limitations of self-coherent approaches, resulting in numerous
methods that partly counteract the mentioned inefficiencies [189–192], or
even match the SE of single-polarization coherent transmissions [193, 194].
Both as complementary or alternative dimension, polarization-diversity has
the potential to double the spectral efficiency, improve the power efficiency
or provide the transmission systems with extra functionality without requiring explicit phase recovery (see Section 4.1).
Being optical fiber a transmission medium where polarization mixing
changes over time, continuous or periodic tracking of the received field’s
state of polarization (SOP) is primordial in implementations of practical
value. Often leveraging DSP capabilities, recent proposals conduct SOP rotation compensation in the digital domain, where computational load and
latency become the bargaining chip. Emphasized by the stringent criteria
on those very same resources, clear cause is given for SOP manipulation
through simple real Mueller calculus on inherently phase-insensitive Stokes
vectors. Unlike Jones calculus, Stokes parameters allow for (i) unequivocal SOP characterization under any degree of polarization (ii) in absence
of phase information. Initially meant for polarization monitoring, Stokes
receivers and Mueller calculus have lately burst into the research activities
of DD data communications; showing especially suitable for DSP simpli10
Based upon China Unicon data, 60% of core router to core router applications are
covered with 10-km links, and 80%-90% with 40-km links. For its part, 95% of router to
transport applications are covered with 10-km links [183].
111
Short-haul networks
fication in comparison to analogous Jones-based approaches. Following
this work line, we present and experimentally demonstrate a DSP-mediate
technique for polarization-multiplexed incoherent IM/DD systems, allowing
blind tracking of the entire Stokes space. This enables multiplexing beyond
two SOPs, while maintaining the simplicity of Mueller calculus and making possible zero-overhead continuous channel estimation without requiring
matrix inversions.
This chapter is subdivided into four blocks. First, this investigation is
positioned with respect to published literature concerning PolMux in DD
optical transmission systems; distinguishing between phase-sensitive selfhomodyne (coherent), and full-incoherent approaches. Second, the proposed fourth-order PolMux system is thoroughly explained, including its
mathematical formulation, design requirements for the transmitter and receiver, and description of the employed Stokes-based DSP for channel estimation and SOP demultiplexing. Third, the experimental validation is
presented, paying special attention to the receiver sensitivity and the robustness of the polarization-tracking algorithm. Finally, the work is summarized, comprehending a motivation recap, main results and conclusions,
and plausible future work lines.
4.1
Work positioning
Stokes-based transmission systems admit as diverse classification as the
purpose of polarization-diversity does, encompassing from SE increase to
DD-channel linearization and receiver sensitivity improvement. However,
for simplicity and without loss of generality, we employ the field’s phase
preservation after photodetection as the organizational factor. Accordingly,
in this section we present the most recent investigations on DD transmission
systems operating on Stokes space for SOP tracking and rotation compensation categorized as incoherent or coherent. Our work is contrasted with
the literature and the major contributions highlighted.
Incoherent
The absence of phase information and the wish for low processing complexity motivates the use of incoherent detection in scenarios where CD
and PMD hardly influence the system’s performance. Within this family
of techniques, we subdivide the published work according to the domain/s112
4.1 Work positioning
pace where symbol design and detection is realized. An alternative perspective of this subdivision is that we differentiate between DD systems
that operate on the Stokes space strictly for SOP rotation compensation;
and those which, additionally, decide on the transmitted symbols directly
on Stokes space. These options are referred to as Jones-Stokes and Stokes
respectively.
Jones-Stokes
Simulation results:
• [195]: a blind DSP algorithm for SOP tracking and demultiplexing for on-off keying (OOK) dual-polarization signals is introduced
and characterized. A standard Stokes analyzer [196] is employed in
the receiver side. The simulations account for phase noise, amplified spontaneous emission (ASE) noise and random fiber birefringence
(polarization-dependent loss (PDL) is disregarded). The BER results
against different carrier-to-noise ratios per polarization show negligible sensitivity from digital demultiplexing even under very fast SOP
fluctuations.
• [197]: the same author adapts the algorithm in [195] to support dualpolarization PAM signals by adjusting the decision thresholds of the
SOP tracker. The system structure is preserved.
Experimental results:
• [198]: single-laser dual-polarization 4-ary PAM SSMF transmission
is demonstrated using 1310-nm silicon photonic intensity modulator.
Successful demodulation is reported at a maximum line-rate of 224
Gbps after 10-km link. The Stokes receiver is made up of a 2×4-90o
optical hybrid, a variable optical delay line and 6 photodiodes. The
DSP algorithm achieves polarization demultiplexing plus ISI cancellation through a set of 4 multi-tap adaptive filters applied on the
respective Stokes parameters. The updating rule is MMSE on 2 different error signals.
•
[199]: Extending the their work in [198], single-laser dualpolarization 4-ary PAM plus 4-ary inter-polarization phase modulation (6 bit/symbol) is demonstrated in B2B configuration at a maximum gross rate of 350 Gbps (soft-decision 25% overhead FEC is con113
Short-haul networks
sidered). DSP comprises two stages, first multi-tap 4×4 MIMO processing inverts the induced polarization rotation and partially compensate the end-to-end ISI. The taps are updated using MMSE on
training symbols prior to decision-directed mode. Second, 4 extra
multi-tap adaptive filters (interpreted as one per Stokes parameter)
remove the residual ISI.
Stokes
Symbol distribution design for optimum detection on the polarization space
increases the modulation flexibility and, potentially, the receiver sensitivity at the expense of transmitter complexity and cumbersome equalization.
This modulation scheme is known as polarization-shift keying (PolSK) or
polarization modulation, and the initial investigations involving Stokes receivers date back to late 80’s [200]. From that moment, a series of reference
investigations by S. Betti et al. [201,202], S. Benedetto et al. [203–207], and
M. Nazarathy et al. [208, 209] thoroughly explored this modulation format and its potential benefits from different perspectives; e.g. theoretical
fundamentals, multilevel and multipower PolSK performance for different
noise sources, optimum receiver structures, optimum constellations in SER
sense, or the development of optimization frameworks. These results pave
the way for today’s understanding11 which, seizing on the evolution of DSP,
has given rise to newly-published 100-Gbps multilevel PolSK proposals exhibiting tolerable complexity [210]. Continuing the work in [211], in [210]
the authors elaborate on the receiver structure and the blind DSP algorithm
for SOP tracking and demultiplexing. Via simulations, the BER of up to
32-ary PolSK targeting 100-Gbps line-rate is assessed accounting for phase
noise, ASE noise and random fiber birefringence (PDL is disregarded). Particularly for 16-ary PolSK, as low as ≈3-dB sensitivity penalty is attained
with respect to single-polarization IM binary scheme at BER=10−5 despite
the 400% greater SE.
Coherent
Stokes receivers have also been employed in conjunction with self-coherent
detection for simplifying the linearization of the DD channel, thence making
possible to retrieve the field’s phase information. In addition to quadrature
11
Note that the literature on PolSK extends well beyond the alluded work, however,
we only considered fundamental investigations and/or those related to Stokes-based detection.
114
4.1 Work positioning
multiplexing, phase preservation permits digital manipulation for effective
CD compensation, allowing for C-band transmission with ≈0.14 dB/km
power-budget increase with respect to O-band in SSMF. The main idea
is that, in the absence of polarization mixing or complete compensation,
a simple linear combination of the Stokes parameters S2 and S3 bears the
electrical field along one Jones vector’s elements (e.g. X or Y ) when an
unmodulated carrier copy is transmitted on the orthogonal complementary.
Due to the spectral superposition of data signal and optical carrier, the electrical SE is not diminished and the optical bandwidth is minimized (same
SE as single-polarization coherent transmission). These are the main advantages with respect to polarization-diversity self-homodyne alternatives
without Stokes detection [212]. Recent results include:
• Repeatered uncompensated 160-km SSMF transmission of OFDM
with flat 16-ary and 64-ary QAM bit-loading is reported at 160-Gbps
and 80-Gbps gross rate respectively [213]. Four months later, the
same group demonstrated single-carrier 32-ary QAM at 62.5 Gbps
transmitted over the same link configuration [214]. One month later
from the latter, the case of OFDM with flat 16-ary QAM bit-loading
in [213] was subsequently extended to WDM configuration for ≈1
Tbps gross-rate over repeatered uncompensated 480-km SSMF transmission [215]. All these experimental validations operated on the
C-band and employed Stokes receiver.
• [212, 216, 217]: based on the above-mentioned investigations, the
same group revisited the topic in a series of publications. These
contain details on the Stokes-based DD channel linearization concept,
analysis on PMD-induced performance degradation, DSP algorithm
elaboration, as well as discussions on experimental results.
• [218]: DSP algorithms for PMD tolerance improvement when OFDM
signal is employed are experimentally evaluated based on the simulation work in [219]. Under 1 dB Q-factor penalty is observed up to
10-ps DGD12 compared to the original DSP chain in [213, 215].
12
When the spectral occupancy of the polarization-multiplexed signal the central
frequency, the impact of PMD is customary estimated from
√ the average DGD; directly
related to the propagation distance as hDGDi ≈ 0.92Dp L, where L is the √
reach in
kilometers and Dp is a characteristic fiber parameter in the order of 0.1 ps/ km for
modern SSMF.
115
Short-haul networks
The technique presented in this chapter belongs to the incoherent JonesStokes group, and it goes beyond the state of the art by enabling up to
quaternary (four-fold) polarization-multiplexed fiber transmission with low
DSP complexity. The software does continuous and blind tracking of the
full channel’s Mueller matrix (in contrast to tracking only a particular vector projection onto one of the components of the Stokes-space basis [195]),
making possible the compensation of very fast polarization mixing effects
on any number of simultaneously transmitted SOPs constituting a nonsingular ensemble. We exploit this capability to simultaneously transmit
four independent data streams carried by four wavelengths with different
SOPs each, and employ polarization as the selective factor for demultiplexing the carriers without employing wavelength-selective devices or phase
information. Note that dual-polarization configuration for SE doubling
shall be understood as a lower-dimensional particularization of the system
under discussion, implicitly demonstrating its feasibility for such purpose
as well.
In the next sections we elaborate on the proposed four-fold polarizationmultiplexed subsystem including conceptual description, mathematical
model, detailed explanation of the software employed for SOP tracking
and demultiplexing, and experimental validation after 2-km SSMF at 100
Gbps (128-Gbps gross rate) and non-return-to-zero (NRZ) modulation.
Contributions of this work
• Original idea: developing a general PolMux technique for simultaneous transmission and demultiplexing of up to 4 disparate SOPs in DD
communication systems. Proposal of such system for its use alongside WDM technology to perform wavelength demultiplexing leveraging polarization diversity, thereby avoiding the need for wavelengthselective devices.
• Experimental results: experimental validation for short-reach (≤2
km) C-band DD transmission systems.
• Related first-author publications: [220] (post-deadline conference),
[221] (invited journal) and [222] (invited conference).
116
4.2 Quaternary polarization-multiplexed direct-detection system
Figure 4.1: Conceptual scheme of the proposed quaternarty polarizationmultiplexed data transmission DD system employing Stokes analyzer.
4.2
Quaternary polarization-multiplexed
direct-detection system
The traditional approach to PolMux is to encode independent data streams
into two linear and orthogonal polarization components (X and Y ). The
orthogonality avoids the interference between the polarized waves in the
absence of polarization mixing or after compensation, thus leading to SE
doubling. Nonetheless, it does not limit the multiplexing order to two.
Herard and Lacourt [223] were the first to take advantage of this fact in
an optical communication system, experimentally demonstrating a freespace IM/DD link with three linear SOPs and analog demultiplexing. This
was followed by [224], with an extended theoretical analysis accounting for
the impact of various factors on the system’s performance including PMD
and mode coupling. At that time, no more than three SOPs were simultaneously transmitted, and no algorithm for real-time channel estimation
was presented. In this work, we demonstrate for the first time a quaternary
polarization-multiplexed (4-SOP) transmission over a medium in which polarization mixing/rotation occurs, i.e. optical fiber. The consideration of
circular SOPs enables forth-order multiplexing, and Stokes-based DSP permits blind and simple polarization rotation compensation without requiring
the field’s phase information.
4.2.1
Conceptual description
The proposed 4-SOP system is illustrated conceptually in Figure 4.1. Four
independently IM optical signals modulate four different C-band carriers
(distinguished by color) in four different SOPs (arrows on the left/trans117
Short-haul networks
mitter side) for later simultaneous transmission over SSMF, where polarization rotates over time [225] (indicated by the vectors in the Poincaré
sphere). At the receiver, the field’s intensities along SOPs X, 45° and rightcircular (RC) (or orthogonal counterparts) are independently detected,
together with the instantaneous total intensity in a standard Stokes analyzer [196]. Subsequently, polarization rotation compensation, demultiplexing and demodulation (encompassed within 4-SOP MIMO) are carried
out in the digital domain.
4.2.2
Mathematical description
In matrix form, the noisy13 system up to photodetection (inclusive) can be
expressed as:
Out = Hsys In + Noise,
(4.1)
where Hsys is a N ×N matrix, with N equal to the number of transmitted SOPs, which includes both the channel linear distortions and the predefined input-output SOP configuration (e.g. standard dual polarization).
In and Out constitute the transmitted and received signals respectively. In
is constructed from the N independently modulated intensity (real) waveforms. Out represents the observed system outputs after photodetection
(real), and it results from the transformation of the transmitted intensity
signals (In) through Hsys . And Noise is the system’s noise in matrix form.
At the receiver the distortions are compensated and the polarizations
demultiplexed (using 4.1):
−1
Inrec = H−1
sys Out = In + Hsys Noise = In + W,
(4.2)
Because, in our case, the necessary polarization diversity for SOP demultiplexing is obtained by detecting the field’s intensity along various polarization orientations14 , the real transfer function of the proposed IM/DD
13
Given that optical amplification is not considered in this application, the dominating
noise is predominantly AWGN from the electrical domain (mostly from transimpedance
amplifiers). In cases were optical amplification is employed (e.g. in some self-homodyne
system proposals, see Section 4.1 under coherent group), the noise statistics becomes a
mixture of the electrical AWGN and non-negligible squared ASE. It is then foreseeable,
though not proved in this report, that the quadratic noise component will limit the maximum reach and system performance through accuracy degradation of the SOP tracking
process, which always operates on intensities (Stokes parameters). This suggests using
multilayer non-linear equalizers (MLP equalizer [226]) for prior SNR enhancement.
14
Alternatively, some receivers attain polarization diversity by optically manipulating just orthogonal polarization components of the field (e.g. X and Y ) prior to
118
4.2 Quaternary polarization-multiplexed direct-detection system
system (Hsys ) can be readily calculated in two steps. First, the field transfer function is constructed:
~vR,1
 .. 
>
>
· · · ~vT,N
,
=  .  Hdist ~vT,1
~vR,M

GN

(4.3)
where ~vT,j and ~vR,i are the 1×2 Jones vectors15 of the jth and ith transmitted and detected SOPs respectively, and Hdist is the 2×2 Jones matrix
that represents the polarization rotation undergone by the field along propagation over fiber. Note that number of individual receiver outputs (i.e.
number of individual output photocurrents), M , is not necessarily equal to
the number of multiplexed SOPs, N .
Second, the intensity (real) system matrix is obtained from GN applying
the Hadamard product operator:
Hsys = GN ◦ G∗N
(4.4)
In the absence of polarization mixing or after compensation (Hdist =
I2 ), Hsys strictly describes the transfer function of the ideal polarizationmultiplexed transmission system; herein referred to as mapping matrix. In
this case, if ~vT,j = ~vR,i ∀j = i and when linear SOPs are used, Hsys can be
conveniently constructed by direct evaluation of Malus’s law16 [223].
It is apparent from 4.2 that Hsys must be invertible in order to make
demultiplexing possible . Since the channel properties cannot be changed
as part of the design procedure, the careful choice of ~vT,j and ~vR,i is the
only method to assure det(Hsys ) 6= 0. Therefore, by inserting Equation 4.3
into Equation 4.4 for different SOP configurations, the existence of H−1
sys
shall be evaluated under the assumption of perfect compensation of the
channel distortions (Hdist = I2 ). This matrix (inverse of mapping matrix)
is herein referred to as demapping matrix.
photodetection. Examples recently suggested for data communications can be found
in [198, 214, 217, 227].
15
Its mathematical expression is given by ~v = E0x ·eiφx E0y ·eiφy , where E0 and
φ represent the complex amplitude and the absolute phase of the electromagnetic field
along the polarization components X and Y. Patently, any Jones vector is fully described
by no more than four independent variables (one complex number per dimension).
16
The total intensity of a light beam going through a perfect polarizer is dependent on
the incident angle θ as follows: Iout = Iin cos2 θ
119
Short-haul networks
Stokes space : The four real Stokes parameters (S0−3 ) are the elements
~ satisfying S 2 ≥ S 2 + S 2 + S 2 (see17 ) and unequivoof the Stokes vector (S),
0
1
2
3
cally describing the polarization characteristics of the electromagnetic field
at the desired time instant. Therefore, the explicit detection of any combination of field’s information variables enabling the reconstruction of the
Stokes parameters (e.g. coherent receiver, Stokes receiver or Stokes analyzer) suffices for assuring successful demultiplexing of any non-singular set
of SOPs. Particularly in the considered DD system (described by 4.3 and
4.4), this avoids the need to determine ~vR,i , now fixed according to the concrete receiver structure and constituting a full polarization-diversity basis
(see Section 4.2.4). By fixing ~vR,i to such a basis, corroborating the existence of demapping matrix can be simplified to find those ~vT,j that satisfy
det(Hsys ) 6= 0 for Hdist = I2 when ~vR,i in 4.3 are replaced by the ~vT,j
themselves. This reduces the number of open variables to N (instead of
N +M ).
Figure 4.2: Poincaré sphere with highlighted SOPs (in color). Their projection on the X ⊥Y plane is included for reference (color matched).
For its part, the Stokes space is the four-dimensional vector space
~ often represented in a S0 -normalized three-dimensional Euspanned by S,
clidean space with S1−3 as coordinate system for intuitive visualization
and analysis (Poincaré formalism). Figure 4.2 shows the Poincaré sphere
17
The inequality becomes equality when light is perfectly polarized, i.e. the degree
of polarization (℘) equals 1 [196]. Otherwise, we talk about partially polarized (0 ≤
℘ ≤ 1) or unpolarized (℘ = 0) light, which can be interpreted as unstable (noisy) and
completely random SOPs over time respectively. When light is fully polarized, both
Jones calculus (Jones vector matrix manipulation) and Mueller calculus (Stokes vector
matrix manipulation) can be employed, otherwise Mueller calculus is mandatory.
120
4.2 Quaternary polarization-multiplexed direct-detection system
with the coordinates of arbitrary SOPs, as well as their equivalent representation on the X ⊥Y plane (along the direction of propagation). It
becomes visible that any SOP distortion maintaining the field’s intensity
unmodified translates to a simple real three-dimensional (3D)-rotation of
the Stokes vector. This fact enables tracking of the SOP changes occurring
during propagation over optical fiber employing cost-effective DD systems
without compromising processing complexity.
4.2.3
Transmitter
Recalling the findings in [223], when linear SOPs are used as selective agents
for signal de/multiplexing, the maximum achievable diversity equals three.
This is intuitively explained from the reformulation of the Jones vector in
footnote 15 as:
h
~v = |E0x |· cos α |E0y |· sin α·eiδ
i
(4.5)
where |E0x | and |E0y | represent the magnitude18 of the field along SOPs
X and Y ; α is the azimuthal rotation of the field’s SOP (often referenced
to the SOP X ); and δ is the phase difference between SOPs X and Y (i.e.
δ = φy − φx ) which equals 0 for linear SOPs. Two decisive conclusions
are extracted from 4.5: (i) at each time instant, the superposition of any
number of linear SOPs traces an equivalent composite linear SOP:
Xh
i
|E0x |· cos α |E0y |· sin α =
hP
|E0x |· cos α
i
|E0y |· sin α
(4.6)
and (ii) any linear SOP is unequivocally described by no more than three
independent parameters (i.e. |E0x |, |E0y | and α). An illustrative example
is shown in Figure 4.3. Therefore, when more than three linear SOPs are
superimposed, their mutually-independent intensities (e.g. data) cannot
be recovered from the measured linear SOP (N independent inputs by 3
independent outputs). This leads to an impractical system with irreversible
generation process and then, to the misconception that no more than three
SOPs can be simultaneously transmitted and demultiplexed [223, 224].
In order to scale the reversible multiplexing order beyond three, the
number of total independent parameters required to describe the composite SOP at all time instants must be strictly greater than three. Mathe~vcomp =
P
18
It is important to realize that the signed amplitude of the field is not available in
incoherent DD systems, but rather the magnitude is detected.
121
Short-haul networks
Figure 4.3: Example of superposition of three linear SOPs (0°, 30° and
120°, in blue solid) and the ensuing aggregate SOP (45°, in green solid).
The direction of the electric field oscillations are displayed as double-ended
arrows scaled by the magnitude of the corresponding SOP along the traced
angle. The unit-magnitude circle (dashed black) is included for reference.
matically, this enables the design of input-output characteristic matrices
(whose coefficients are associated with physically meaningful independent
variables) defining consistent systems of more than three equations with
unique solution (introduced in Section 4.2.2 as mapping matrices with nonzero determinant). Based on 4.6 it is apparent that, besides |E0x |, |E0y | and
α, the relative phase (δ) constitutes a fourth independent variable to fully
describe the SOP of any polarized electromagnetic wave. In this work, we
exploit this extra degree of freedom by encoding part of the information on
a circular SOP, thus resulting in a composite SOP unequivocally described
by (and no more than) four independent parameters. Particularly, the four
transmitted Jones vectors (~vT,j ) employed in the experimental demonstration are X, Y, 45° and LC.
X
X
1
~vT,N
π
45o 
 .. 
 cos 4
Tx =  .  =

Y  0
~vT,N
LC cos π4



122
Y
0
sin π4
1
i· sin π4





(4.7)
4.2 Quaternary polarization-multiplexed direct-detection system
As indicated in Section 4.2.2, when the use of a receiver structure offering full polarization-diversity is assumed, the existence of demapping matrix
for a set of given ~vT,j vectors can be evaluated from det(Hsys ) 6= 0 by making ~vR,i equal to 4.7 in the calculations (the actual ~vR,i may be different,
and thus the mapping matrix too):

Hsys
1.0
0.5

=
0.0
0.5
0.5
1.0
0.5
0.5
0.0
0.5
1.0
0.5

0.5
0.5


0.5
0.0
(4.8)
with det(Hsys ) = −0.25 and inverse:

H−1
sys

1.0 −1.0 0.0
1.0
−1.0 2.0 −1.0 0.0 


=

 0.0 −1.0 1.0
1.0 
1.0
0.0
1.0 −2.0
(4.9)
The above matrices prove that quaternary PolMux is possible in IM/DD
systems by confirming the reversibility of one particular example of multiplexing scheme (4.7).
Practical considerations : It is conspicuous from 4.8 and 4.9 that the
fact that multiplexing is reversible does not imply that the transmitted
SOPs are uncoupled. Consequently, multilevel signals are formed at the
output of all the photodiodes owed to the incoherent superposition of the
~vT,j -weighted intensities even without polarization rotation. This contrasts
with orthogonal polarization multiplexing. Therefore, assuming that the
receiver’s electrical AWGN is the dominating noise source (see footnote 13)
and the perfect channel estimation, the receiver sensitivity with respect to
single-polarization 4-λ WDM with ideal optical filtering is mainly penalized by the SOP-demultiplexing process (W in 4.2 for Hdist = I2 ); which
unavoidably couples the diverse independent noise contributions of each
th
SOP according to H−1
sys . For instance, in N -order orthogonal multiplexing
−1
Hsys = IN after compensating channel distortions, evincing the mentioned
signal degradation when compared with 4.9. Likewise, it becomes immediate from 4.9 that not all the channels undergo the same penalty, and that
the fewer the number of coupled ~vT,j are transmitted (either by imposing
mutual orthogonality or by reducing the multiplexing order), the lower the
sensitivity degradation will be. This in turn justifies the use of X and Y
on the transmitted set of SOPs (see 4.7).
123
Short-haul networks
Despite this inherent degradation, det(Hsys ) 6= 0 indicates that the
set of SOPs defining Hsys can be successfully employed as demultiplexing selective agents of any other multiplexing technology, hence enabling
the interchange of the respective de/multiplexing apparatus. This idea inspired the application of PolMux along side WDM in order to replace the
wavelength-selective components, thus considerably relaxing the requirements on lasers’ frequency stability, and on the power consumption due to
the thermal control of the gratings.
4.2.4
Receiver
Random fluctuations of fiber’s birefringence along the link cause a timevarying oscillation of the aggregate SOP (aSOP)19 . In other words, fiber
propagation20 may transform (usually expressed as rotation) the transmitted SOP into any other state, which translates into polarization mixing after detection with a static polarization-diversity receiver21 . Consequently,
compensating such effect needs acting on all four independent parameters
of the received SOP, and provided that phase information is lost after
photodectection, a quaternary polarization-diversity receiver is necessary
irrespective of the multiplexing order. This represents an important rule
for the receiver design of polarization-multiplexed DD systems, being the
4-SOP setting the only 100% hardware efficient (4 independent inputs by
4 photodetectors22 ).
Hardware-wise, the simplest receiver structure comprises a one-to-four
power splitter prior to a fixed set of polarizers, which provide the necessary polarization diversity between the various signal copies. Each of the
branches ends with respective photodetectors for subsequent digitization
and processing in the digital domain (see Section 4.2.5). The polarization filtering stage constitutes the only optical processing in the receiver,
becoming the sole degree of freedom for field manipulation before acquisition. This makes the concrete ~vR,i configuration (in relation to ~vT,j ) have
19
Composite SOP conformed by the superposition of SOPs described by ~vT,j .
With the exception of polarization-maintaining fibers
21
PDL is assumed negligible and hDGDi ≪T, where T is the symbol
period.
p
22
∗ 2
Subjected to |Eox |2 |Eoy |2 = <((Eox Eoy
) ), the equality S0 = S12 + S22 + S32 holds.
This means that when the field projection along either X or Y is 0 or constant, or when
both polarization components are identical, it is possible to detect S1−3 exclusively and
thus reduce the number of of measured outputs to 3. Although in our case none of these
requirements are met (note that all of them imply no more than 1 independent input),
this receiver structure can be equally employed in all the coherent approaches introduced
in Section 4.1, e.g. [227].
20
124
4.2 Quaternary polarization-multiplexed direct-detection system
determinant influence on the robustness and simplicity of the DSP for polarization tracking and demultiplexing.
The ~vR,i vectors must be selected to assure that the restoration of the
Stokes parameters is possible; meaning that all potential received SOPs
have a unique imprint and can be discerned. In our case, a standard Stokes
analyzer was employed. In this structure, three of the polarizers are aligned
to X or Y, 45° or 135° and RC or left-circular (LC) respectively. The
remaining photodiode captures the instantaneous total field intensity (no
polarizer). Given this setting, the Stokes parameters are calculated from
the measured field intensities through the following linear transformation:


S0 = IT OT AL


(




1,


S1 = 2(−1)K1 IX/Y − S0 , with K1 =



0,


(
if Y
if X
1, if 135o
K2 I

S
=
2(−1)
−
S
,
with
K
=

2
0
2
45/135


0, if 45o


(



1, if LC


K3


S3 = 2(−1) IRC/LC − S0 , with K3 = 0, if RC
(4.10)
where for the experimental demonstration, the polarizers are fixed to X,
135° and RC for S1 , S2 and S3 respectively:
X

−
1
~vR,1

X  1


Rx =  ...  =

135o  cos 3π
4
~vR,M
RC
cos π4


Y
1
0
sin 3π
4
−i· sin π4





(4.11)
It is clear from 4.10 that operating in the Stokes space implies a change to a coordinate system in which orthogonal SOPs
(X /Y ⊥45° /135° ⊥RC /LC ) are not X ⊥Y ; or alternatively, multipolarization signal design for Euclidean-distance-wise optimum symbol decision employing Jones basis (e.g. uniform angular distribution of linear
SOPs [223, 224]) is different to that for optimum decision in the Stokes
space. Besides being the essence of PolSK modulation, the latter fact
evinces that orienting the transmitted SOPs along orthogonal Stokes co125
Short-haul networks
(b) ~vR,3 =45°
(a) ~vR,3 =135°
Figure 4.4: Simulated received eyediagrams in the absence of polarization
mixing when four independent IM NRZ signals are transmitted according
to 4.7. The eyediagrams are detected at the output of the photodetector
when the preceding polarizer is aligned to 135° (a) and 45° (b). The
inter-channel spacing is fixed at 100 GHz, the OSNR is 60 dB and the
receiver bandwidth is 90% of the baud-rate. Both cases are presented in
1:1 proportion (horizontal axis: time; vertical axis: signal amplitude). The
reduction in levels is apparent for the case where ~vT,j ⊥~vR,i (6 levels against
4).
ordinates23 avoids undesired intermediate vector transformations24 , as well
as it improves the robustness of blind SOP-tracking algorithms based on
Stokes-parameter manipulation. This motivates the particular set of transmitted Jones vectors ~vT,j in 4.7, placing all the SOPs along the Stokes coordinates and maximizing the number of transmitted SOPs being mutually
orthogonal in Stokes space. In addition, the number of orthogonal transmitted and detected Jones vectors (~vT,j ⊥~vR,i ) is intentionally maximized,
thereby minimizing the order of the resulting multilevel signals under perfect alignment conditions. For instance, by orienting one of the ~vR,i along
135°, the coupling from the transmitted component along 45° (90°degrees
apart) is null (see Figure 4.4). The latter turns very convenient in the
initial ~vT,j -~vR,i alignment and for system optimization.
23
We refer to Stokes coordinates as those unit vectors describing the linear space
spanned by the Stokes parameters. Being S0 the aggregate scaling magnitude
C(see
~1C = 1 0 0 , S
~ =
equation under footnote 17), Stokes coordinates are given by S
2 C
~
0 1 0 , and S3 = 0 0 1 ; whose respective Jones vectors are ~vS C = 1 0 ,
1
~vS C = cos π4 sin π4 and ~vS C = cos π4 −i· cos π4 .
2
3
24
Representation of whatever ~vT,j according to the orthogonal Stokes basis ~vR,i in
which polarization tracking and rotation compensation is done.
126
4.2 Quaternary polarization-multiplexed direct-detection system
4.2.5
Digital receiver - stages and practical considerations
The DSP chain described in this subsection corresponds to one of the possible solutions for satisfactory polarization tracking and data demodulation
(considering FEC codes) employing the proposed fourth-order polarizationmultiplexed IM/DD system. We subdivide the elaboration in three main
blocks: (i) general front-end correction, (ii) polarization tracking plus channel rotation compensation and (iii) demultiplexing. The full DSP schema
is included in Appendix D for the readers’ convenience.
4.2.5.1
Front-end correction and signal conditioning
The purpose of the first set of algorithms within this block is to prepare the
digitized photocurrents in order to maximize the accuracy of the subsequent
calculation of Stokes parameters (see 4.10) for best-possible SOP tracking.
The comprehended steps are:
Resampling and Filtering: The four sequences are resampled to an
integer number of samples per symbol (2 in our case). Out-of-band noise
and the remaining undesired spectral components caused by signal-signal
beating in the photodetector are attenuated for SNR maximization.
Time skew : Skews originating both at the transmitter and receiver sides
have different though critical impact on the system’s functioning. The
skew at the transmitter cannot be easily corrected by DSP at the receiver,
and it requires precise symbol synchronization between the independent
signal generators up to the fiber input. It determines the quality of the
received multilevel signals, affecting both the cleanliness of the levels and
the sampling-point deviation tolerance. Note that the relative delay between carriers caused by chromatic dispersion appears as transmitter skew,
evincing that every frequency grid requires specific compensation.
The skew at the receiver does not alter the multilevel signals’ quality
as described above, but their respective time alignment before digitization.
Because the calculation of all the Stokes parameters depends on the measured instantaneous total field intensity (S0 ), misalignments of sufficient
magnitude between symbol periods can severely affect the performance.
For instance, a relative skew of S0 with respect to S1−3 of >30% the symbol time prevents successful demodulation owed to the errors induced by
127
Short-haul networks
the SOP tracking algorithm25 . In addition, because the total propagated
power is not affected by channel-induced polarization mixing, S0 becomes a
robust reference for common optimum sampling selection irrespective of the
severity of the distortion. This underscores the need for accurate symbol
synchronization across all the four sequences, where preceding frame/coarse
synchronization is understood.
Scaling : The amplitude ratios between the measured photocurrents are
crucial for the precise determination the polarization-mixing coefficients.
Therefore, the individual normalization of the captured photocurrents is a
mistake that leads to distorted Stokes parameters when the intensity waveforms are combined (4.10). It follows that differences in the responsivities of
the four photodiodes must be equalized a priori, in order to avoid incorrect
impairment compensation and demultiplexing.
DC offset : Different photodiodes may also show disparate output DC
components for the same input optical power. Even with equalized responsivities, uncompensated DC values may accumulate when calculating the
Stokes parameters, then producing biased waveforms that modify the symmetry of the decision boundaries. Unless addressed, this effect compromises
the robustness of the algorithm for SOP tracking and demultiplexing. Note
that the response of the photodiodes may also vary with the input power
level. In such case, scaling factors and DC compensation values should be
adjusted according to the received optical power and/or power-dependent
adjustment of the slicers’ thresholds should be implemented for improved
decisions.
After the first set of corrections on the digitized photocurrents, the
transformation of the intensities to Stokes parameters is done as indicated
in 4.10. From this point on, the DSP operates on the Stokes parameters:
Timing recovery and sampling point : As previously mentioned, the
invulnerability of S0 to polarization mixing enables its use as robust reference for common clock recovery and optimum sampling point estimation. This is conditional upon precise skew compensation and negligible
deviation between the individual sampling clocks across the four branches.
In the experimental demonstration, standard Gardner error detector with
25
The number is estimated from simulated noiseless B2B configuration with 100-GHz
inter-channel spacing and ideal signal-signal beating removal.
128
4.2 Quaternary polarization-multiplexed direct-detection system
proportional-integrator loop is employed for clock recovery followed by
maximum-variance decimation [87].
4.2.5.2
Polarization mixing compensation
In this stage, the time-varying Mueller matrix of the channel Mdist is
blindly estimated (Hdist in Jones calculus26 ) for the subsequent compensation of possible polarization mixing, thus enabling effective SOP demultiplexing and demodulation. Because channel-induced polarization mixing
reflects as a 3D rotation (see footnote 21) of the sub-space defined by S1−3
(graphically represented by the Poincaré sphere), the channel’s Mueller
matrix reduces to the standard rotator form:


Mdist
1 0
0
0

0 S
S
S

1,1
1,2
1,3 
=
,
0 S2,1 S2,2 S2,3 
0 S3,1 S3,2 S3,3
(4.12)
where Si,j for i, j ∈ Z are real-valued coefficients which quantize the normalized coupling factors between the Stokes parameters i and j.
The next algorithm is inspired by the work in [195], where a blind
method for compensating polarization-mixing on a single Stokes coordinate was presented (2nd , 3rd or 4th row in 4.12, see footnote 23); thereby
enabling PolMux of up to two orthogonal Jones vectors undergoing very
fast polarization rotation. The main DSP-novelty introduced in this work
resides in the capability to measure the rotations on all three Stokes coordinates, that is, tracking Mdist entirely. This extension maintains the
simplicity and blindness of the original algorithm while making possible
polarization tracking and de/multiplexing beyond two SOPs which, additionally, are not restricted to be orthogonal.
In the following, the details of the tracking process are elaborated. This
is supported with Figure 4.5, where simulated (color) histograms of the
normalized Stokes parameters S1 (a) and S2 (b) are shown for perfect
For Jones-to-Mueller conversion: M = A(J ⊗ J∗ )A−1 , where M and J are the
Mueller and Jones matrices respectively and A is given by:
26

1
1
A=
0
0
0
0
1
i
129
0
0
1
−i

1
−1
0
0
Short-haul networks
(a) S1
(b) S2
Figure 4.5: Histogram of the Stokes parameters S1 (a) and S2 (b) normalized by S0 for perfect transmitter-receiver polarization alignment in
the polarization-multiplexed IM/DD data transmission system described
by 4.7 and 4.11 employing NRZ modulation. In color, simulated data for
40-dB SNR B2B configuration. In grey, experimental results for 4×27 GBd
at 6 dBm optical power into the receiver. Insets: Poincaré spheres. The
Stokes parameters of both (a) and (b) are highlighted.
polarization-mixing compensation. Experimental results (grey) are presented for comparison. Note that the asymmetry around zero of Figure 4.5b
~ C . This
is due to the fact that 45° is the only ~vT,j explicitly lying along S
2
~ C , since both X and Y are transmitted.
contrasts with the projection on S
1
In this regard, S2 and S3 show similar histograms.
SOP tracking : First, the trivial zero-power cases are detected and sliced
by thresholding on S0 ’s lowest level. The samples below threshold can already be demapped to the corresponding bit tuple (e.g. [0 0 0 0] with each
column associated with one of the four independent data streams), and
the respective instantaneous samples are removed from the four Stokesparameter sequences. This avoids mathematical indeterminations in the
following normalization by S0 , and slightly reduces the processing load
by discarding trivial cases to track on. For its part, normalizing by S0
compresses the Stokes space to the unit-radius sphere irrespective of the
modulation format, the transmitted power or the polarization-dependent
distortions. Since the characteristic matrix of the channel is assumed unitary, common S0 -normalization becomes convenient to simplify the channelestimation algorithm by removing the influence of the irrelevant magnitude
130
4.2 Quaternary polarization-multiplexed direct-detection system
variable.
Because the number of possible magnitudes and orientations (Euler angles) of the aSOP is finite and deterministic given the modulation format per SOP, the representation of the output combinations in the Stokes
space shows well-determined locations in the absence of distortion or after
polarization-rotation compensation. Therefore, the deviations with respect
to such pre-defined positions serve as quantifiable figures of the experienced
polarization rotation. This is the observation governing the algorithm for
estimating the channel’s characteristic matrix (Mdist ).
As first step, the projection of all the aSOP combinations upon the
Stokes coordinates in the absence of polarization mixing must be identified
and studied. In this regard, we subdivide the cases into two groups:
~ C and conformed by 1 non-zero
(A) Impinging aSOP aligned to one S
1−3
SOP : all the power is projected onto the respective Stokes coordinate (the magnitude of one Stokes parameter equals 1 and the others’ equals 0), thus resulting in the maximization of the Euclidean
distance between the magnitude levels. The latter is visible in the
histograms in Figure 4.5, where the discussed (signed) cases are displayed in green color. For example, the observations around 1 in the
S1 histogram correspond to those instants when only ~vT,1 was excited (e.g. [1 0 0 0] input symbol-tuple), in turn making the received
aSOP be oriented along X. Those very same observations contribute
to the point accumulation around zero in the S2 histogram. In these
instances aSOP=SOP.
(B) Impinging aSOP conformed by any number >1 of non-zero SOPs : at
these time instants the magnitude of the individual Stokes parameters
describing the received aSOP vector will be <1 (blue observations in
Figure 4.5). Unlike in the previous group, these magnitude levels vary
according to the relative power ratios of the individual SOPs, hence
requiring dedicated detection-thresholds for each modulation format.
Furthermore, the higher proximity between the projections as compared to the previous group, increases the overlapping region among
the noisy clusters (see Figure 4.5); hence hindering their detection
and thus the robustness of SOP tracking.
For improved robustness in blind channel estimation, and for enabling
the generalization of the SOP tracking algorithm to other modulation formats, the observations belonging to group B are disregarded in this work.
131
Short-haul networks
As a result, the channel matrix Mdist can only be estimated conditional
on ~vT,j defining a collection of SOPs with at least one oriented along each
Stokes coordinate. In the proposed 4-SOP system, ~vT,1 ⊥~vT,3 , ~vT,2 and ~vT,4
are aligned with each of the three Stokes coordinates (see 4.7 and footnote
23). It is important to emphasize that because only group-A observations
are used for error signal estimation, polarization tracking is not totally
continuous. The probability of occurrence of these specific aSOPs will determine the tracking accuracy around the respective coordinate for a given
observation time interval (more samples for filter adaptation); whose duration is dictated by a compromise between the baud-rate, the polarization
rotation speed, and the tolerable penalty caused by the polarization mixing. Nonetheless, given uniform probability distribution of the input data
streams, the speed at which the three necessary kind of group-A observations occur at the considered baud-rates is much faster than the speed of
polarization changes in regular fiber communications (below microsecond
scale [225]). Given the proposed 4-SOP system with ~vT,j defined by 4.7,
the probability of transmitting a group-A aSOP along each axis:

2

PS~ C = 16 (X and Y )

 1
P
=
1
(45° )
~
16
S

2


P ~ C = 1 (LC )
16
S3
C
where the probability of receiving all three is:
PT OT AL = PS~ C · PS~ C · PS~ C =
1
2
3
1
2048
(4.13)
The conclusion is now immediate: after locking, the entire coordinate
system can be updated every 2048 symbols. At a transmission speed of
25 GBd, the updating periodicity ≈0.082 µs. Accordingly, the updating
assiduity will be deemed quasi-continuous, as it is in the order of 100 times
faster than required.
As previously stated in Section 4.2.4, polarization rotation appears as
polarization mixing when the signal is detected with a static polarizationdiversity receiver. That being so, the total received power spreads among
the three Stokes coordinates even when the transmitted aSOP is perfectly
aligned with one of them. This effect reflects as a unitary 3D rotation of
~ C . Particularizing to observations
the real vector space determined by S
1−3
belonging to group A, the deviation from the unit magnitude along the
132
4.2 Quaternary polarization-multiplexed direct-detection system
~ C at every instant (together with the
corresponding Stokes coordinate S
1−3
leakage on the others and their relative ratios) represents a time-varying
quantification of the observed polarization rotation around that particular
axis. This information (error signal), shall be employed to update the
coordinate system accordingly so that, under these new reference axes, the
received Stokes vector appears uncoupled. Therefore, polarization mixing
compensation starts with the blind estimation of the signed error signal
for each reference vector, whose mathematical expression is given by (after
[195]):

~
sign(S
~ E [n]) S[n]
r1−3 [n],
1−3
S0 [n] − ~
~e1−3 [n] =
0,
~ E [n]| > K
if |S
1−3
(4.14)
otherwise
with
~
S[n]
~r1−3 [n]
S0 [n]
!
E
~1−3
S
[n]
=
(4.15)
~
where S[n]
is the received Stokes vector, S0 [n] is the total intensity, ~r1−3 [n]
are the unitary reference vector (progressively updated according to the es~ E [n] are the projections of the normalized received
timated rotation) and S
1−3
Stokes vector along ~r1−3 [n]. For its part, K provides static threshold for
selecting group-A observations exclusively which, employing Figure 4.5 as
example, |K| ≈ 0.8. It is likewise apparent from Figure 4.5 that, although
group-B observations are disregarded for updating the error signal, they
may hinder the detection of group-A instants under the influence of noise
(notice the overlap between the equivalent green and blue regions in the
experimental data). This emphasizes the importance of accurate threshold
calculation, as well as it evinces its dependence on ~vT,j and the modulation
format [197].
The three ensuing signed error signal are then included in respective
normalized least mean square (NLMS) adaptive filter calculations targeting
~
the adaptation of ~r1−3 to mimic the time-varying rotation undergone by S
C
~ :
with respect to the Stokes coordinates S
1−3
~r1−3 [n + 1] =
~r1−3 [n] + µ~e1−3 [n]
k~r1−3 [n] + µ~e1−3 [n]k
(4.16)
where µ stands for the convergence step size and k•k is the norm operator.
133
Short-haul networks
These calculations are executed for each of the three coordinate vectors in series, where the alternation is triggered by a zero-power exception
indicating that any other orthogonal Stokes coordinate has been transmitted instead (green-colored observations around 0 in Figure 4.5). These
serial operation is embedded in an iterative loop which breaks when the
error signal is converged (for which a user-defined threshold must be preestablished) or when the group-A observations within the captured window
are exhausted. It is worth remarking that the algorithm described here is
a tracking tool and, as such, assumes prior initialization (e.g. via training sequences [199, 212, 228]). Assuming initialization, the algorithm locks
after convergence, then enabling simple, quasi-continuous, and blind characterization of the polarization rotation for all the Stokes coordinates. The
process ends by building 4.12.
Channel distortion compensation : In the SOP-tracking process, the
three-dimensional rotation of the Stokes space is characterized and quantified. Equivalently, a polarization-mixing-free coordinate system is obtained.
Assuming the inability to change ~vT,j and/or ~vR,i dynamically, we realize
~
digital derotation of the received Stokes vector S[n].
Given that rotation
does not break the orthogonality among the coordinate vectors, the matrix
inverse is readily calculated through transposition:

>
M−1
dist = Mdist

1 0
0
0
0 S

S
S

11
21
31 
=

0 S12 S22 S32 
0 S13 S23 S33
(4.17)
Afterwards, the matrix is applied on the received Stokes sequences.
From this step on, transmitter and receiver are assumed perfectly aligned
(Hdist = I2 , Mdist = I4 ).
4.2.5.3
Demultiplexing
The third and last block deals with static polarization demultiplexing. This
stage comprises Stokes-to-intensity transformation and demapping matrix
application. Because both mentioned transformation matrices are known a
priori, no real-time inversion is involved. The two aforementioned processes
can be done separately, or they can be collapsed into a single pre-defined
and static matrix multiplication. In both cases, the four SOPs are decoupled and the estimated transmitted intensity waveforms are retrieved. The
134
4.3 Experimental demonstration
inputs are the Stokes parameters at the output of the channel compensation
~ E [n]) irrespective of the approach:
step (S
1−3
• Two steps
First, the input Stokes parameters are transformed into intensities
through 4.10. After that, the demapping matrix is applied.
• One step
The matrix ~vT,j is converted to Stokes parameters, and its inverse is
~ E [n] without intermediate conversion to intendirectly applied on S
1−3
sity waveforms. For the particular configuration 4.7, the demultiplexing matrix is given by:

Mdemux

0.5 0.5 −0.5 −0.5
 0
0
1
0 


=

0.5 −0.5 −0.5 −0.5
0
0
0
1
(4.18)
The DSP is ended with a slicer and error counting before FEC. Note
that other features/technologies can be added on top of the presented stages
(e.g. nonlinear equalization for quadratic noise compensation or support
for higher-order modulation formats) or in place (e.g. multi-dimensional
clustering algorithms for joint Stokes vectors tracking). However, the proofof-concept experimental demonstration was realized with the strictly necessary modules for simplicity and worst-case evaluation.
4.3
Experimental demonstration
Critically driven by cost and power consumption, client-side optical
transceiver modules are designed upon stringent requirements on the maximum average launch power. For instance, Ethernet task forces IEEE
P802.3ba (40 Gbps and 100 Gbps) and IEEE P802.3bs (400 Gbps) specify
quaternary WDM configurations with ≤4 dBm per lane [177,229]. Together
with the various power-budget margins accounting for manifold purposes,
those restrictions establish an upper bound for the receiver sensitivity beyond which communication is not sustainable. Therefore, characterizing
the receiver sensitivity is essential for assessing the practical interest of
data-communications systems intended for application in closed short-reach
135
Short-haul networks
networks. This is the purpose of the following section, where the experimental validation of the 4-SOP IM/DD data transmission system described
in Section 4.2 is elaborated.
Targeting 100 Gbps net bit-rate, 4×27 GBd and 4×32 GBd NRZ configurations are considered along with 7% and 20% overhead FEC codes
respectively. The performance of both B2B configuration and 2-km SSMF
transmission is evaluated in terms of BER versus received optical power,
whereas the robustness of the SOP-tracking algorithm is analyzed in terms
of time-averaged BER under time-varying polarization changes far more
demanding than expected in the considered scenarios [225, 230]. Throughout the demonstration, only the strictly necessary DSP blocks are employed
(see Section 4.2.5).
4.3.1
Testbed
Figure 4.6 shows the schematic of the experimental setup. At the transmitter side, four C-band distributed feedback lasers (DFBA-D ) with 100GHz spaced center frequencies (193.2-5 THz) are used as light sources.
Frequency spacing is used to ensure incoherent power addition after their
beating in the receiver’s photodiodes27 . The outputs of the distributed
feedbacks (DFBs) are externally modulated with four integrated MachZehnder modulators (MZMs) with ≈30-GHz eletro-optical bandwidth.
Note that the system employs IM, opening for the use of other externalmodulation devices or direct-modulation techniques. The MZMs are electrically driven by two pulse pattern generator (PPG) modules, delivering a total of four independent 27 GBd (108 Gbps gross rate) or 32 GBd
(128 Gbps gross rate) NRZ data streams with ≈3 Vpp. Standard PRBSs
of length 215 − 1 are used, and fine symbol synchronization is attained with
electrical delay lines. No DSP is employed at the transmitter. Lastly, the
SOP of each modulated carrier is aligned along the respective ~vT,j (i.e. X,
27
For a given bandwidth per transmitted carrier, the laser frequency spacing will determine the degree of interference from the higher-frequency beatings to the baseband
component. Insufficient separation will make these higher-frequency beatings fall within
the photodiodes’ electrical bandwidth, thus partially overlapping the baseband signal
with the consequent distortion (the modulation bandwidth is assumed greater or equal
than the photodiodes’ electrical bandwidth).
Interference between non-orthogonal Jones vectors occurs when the lasers’ frequency
spacing ranges from DC to twice the electrical bandwidth of the photodiode. In nonsevere cases, digital filtering can minimize the impact of the distortion. Severe cases may
prevent correct signal demodulation, hence requiring the redesign of the frequency grid
as first approach.
136
4.3 Experimental demonstration
Figure 4.6: Experimental setup of the considered 4-SOP data transmission
system.
Y, 45° and LC, see 4.7) with polarization controllers, and combined with
one 4×1 optical coupler.
The output from the transmitter is passed through a power equalization
stage before fiber transmission. This stage comprises an EDFA operated in
saturation regime (≈5 dB of noise figure) plus a variable optical attenuator
(VOA). This structure allows for compensation of the disparate insertion
losses among the MZMs in our transmitter, in no case providing inputoutput optical gain. After equalization, the signal is launched into 2-km
SSMF for transmission. One polarization scrambler is installed at the end
of the link for the evaluation of the polarization tracking algorithm.
At the receiver side, the signal is detected with a standard Stokes analyzer (see ~vR,i in 4.11) with ≈30-GHz electrical bandwidth. The top
branch in Figure 4.6 measures the instantaneous field intensity (S0 ), the
other three include polarization controllers (PolCs) and polarization beam
splitters (PBSs) for providing the required polarization diversity (Section
4.2.4). After photodetection, the four electrical signals are digitized and
stored for offline processing in a DSO with 25-GHz bandwidth input ports
and 80-GS/s sampling rate. The DSP at the receiver consists of front-end
correction, Stokes-based compensation of the polarization mixing plus polarization demultiplexing, and error counting (see Section 4.2.5 for detailed
explanation).
4.3.2
Results
The discussion on the results is subdivided into (i) receiver sensitivity analysis for both B2B and transmission configuration, (ii) polarization tracking
performance, and (iii) comments on power consumption and processing
complexity.
137
Short-haul networks
Figure 4.7: BER (≈2.9 million bits per point) versus total and per-channel
optical power at the input of the receiver for B2B and 2-km SSMF transmission.
4.3.2.1
Receiver sensitivity
Figure 4.7 shows the BER performance of the system versus optical power
at the input of the receiver for perfect polarization rotation compensation
(polarization tracking algorithm always operational). Measured curves for
both B2B and transmission after 2-km SSMF are presented for the two
cases of study, i.e. 4×27 GBd (108 Gbps) and 4×32 GBd (128 Gbps)
gross rates. The pre-FEC BER thresholds for post-FEC BER of 10−15 are
included for hard-decision 7%- and 20%-overhead BCH codes (4.4·10-3 [231]
and 1.1·10-2 [232] respectively). Aiming 100 Gbps net bit-rate, lower baudrate is associated with lower code-rate in this work; yielding net bit-rates
of ≈100.93 Gbps for the 4×27 GBd case, and ≈106.66 Gbps 4×32 GBd.
Besides overhead, processing complexity and latency, and de/coding power
consumption are other important FEC-dependent aspects that should be
considered for optimum transceiver design. These are, however, hardly
relevant from the receiver sensitivity perspective.
Below-FEC threshold BER is attained for both 27 GBd and 32 GBd
irrespective of the transmission configuration. For 27 GBd, insignificant
transmission penalty is observed, what makes the overall receiver sensitivity ≈4.4 dBm (≈-1.6 dBm per wavelength). For its part, 32 GBd B2B
138
4.3 Experimental demonstration
exhibits similar performance to 27 GBd below ≈5 dBm, showing a receiver sensitivity of ≈3.7 dBm (≈-2.3 dBm per wavelength). The reason
for the divergence beyond ≈5 dBm is that, as SNR increases with average
launch power (nonlinearities are negligible due to the short propagation),
the ISI originating from signal over-filtering starts dominating the error
rate; thus causing a slower BER decay rate in the high baud-rate signal.
Note that zero-tap filters are employed for channel equalization (see 4.17),
thus intra-channel ISI is not compensated in this experiment. The penalty
after transmission is ≈0.5 dB, what worsens the pre-FEC sensitivity to a
range comparable to that of 27 GBd at the 7%-overhead FEC threshold.
This leads to similar power budget margin between both baud-rate cases,
which could be argued to favor 27 GBd over 32 GBd in scenarios targeting
100 Gbps (despite the 2 Gbps lower transmission rate for 27 GBd) based
on the lower processing complexity of the higher code-rate, and the lower
requirements in electrical bandwidth.
Benchmarking : For putting into perspective the above-described sensitivity results, we resort to the investigation in [233]; where we compare the
4×32-GBd case against analogous transmission systems employing parallel
optics (one fiber per carrier) and WDM technology (filter-based wavelength
demultiplexing). The modulation format, the inter-channel spacing, and
the considered FEC are therefore kept in NRZ, 100 GHz and hard-decision
20%-overhead BCH respectively. As far as DSP is concerned, only static
front-end corrections and timing recovery are preserved for parallel optics
and WDM configurations as they do not feature polarization diversity (see
Section 4.2.5). Figure 4.8 presents the sensitivity results after [233] in BER
versus total optical power (four channels) at the input of the receiver28 .
For brevity and because the difference between the curves of WDM and
parallel optics reduces to a constant ≈1-dB sensitivity degradation owed
to optical filtering, we will only compare the results of the 4-SOP system
with those of parallel optics.
The experimental data for B2B configuration shows ≈14.9 dB sensitivity penalty at the pre-FEC BER threshold. This number results from (i)
the different power loss from the input of the receiver to the input of the
photodiodes between both cases29 and (ii) the demultiplexing process (see
28
For the case of parallel optics, the individual power of each carrier is individually
measured and added a posteriori due to the physical separation of the channels.
29
The insertion losses owed to the extra filter in the WDM case are already corrected
139
Short-haul networks
Figure 4.8: BER versus received optical power at the input of the photodiode for parallel optics and WDM technology considering both B2B and
2-km SSMF transmission [233].
Section 4.2.5.3), which enhances the noise per channel by aggregating different scaled combinations of the independent AWGN components in the
receiver branches:
• Insertion loss: referencing to parallel optics, the 4-SOP system
presents one extra 1×4 optical power splitter, plus PolCs and PBS
for the three bottom branches in Figure 4.6; all in all yielding: 7.1 dB
(S0 ), 8.8 dB (X ), 9.1 dB (135° ) and 8.9 dB (RC) extra losses. Because
the field intensity along Y is reconstructed from S0 (low loss) and S1
1
(high loss) through IY = S0 −S
2 , it exhibits slightly better average
performance than the explicitly detected SOPs (high loss). Nevertheless, for simplicity and taking into account that the BER curves
in Figure 4.7 result from the average of all four channels, the sensitivity degradation associated to this contribution is approximated to
9 dB for all branches.
• Demultiplexing: so as to quantify noise enhancement, we refer to
4.2 and 4.9, and assume independent zero-mean and same-variance
AWGN across the four branches Nj (0, σ 2 ). Note that the unitary
for in Figure 4.8
140
4.3 Experimental demonstration
rotator matrix employed for channel distortion compensation (M−1
dist )
does not have any influence concerning noise enhancement, hence it
is made equal to I4 for simplicity:
1.0 −1.0 0.0
1.0
NX (0, σ 2 )
NX (0, 3σ 2 )
−1.0 2.0 −1.0 0.0  N o (0, σ 2 ) N o (0, 6σ 2 )

  45
  45

W=

=

 0.0 −1.0 1.0
1.0   NY (0, σ 2 )   NY (0, 3σ 2 ) 
1.0
0.0
1.0 −2.0
NLC (0, σ 2 )
NLC (0, 6σ 2 )
(4.19)





Four new and independent noise processes with apparent increased
variance are defined for each intensity signal. The values correspond
to a sensitivity of ≈4.7 dB for X and Y and ≈7.8 dB for 45° and
LC. Again considering that the BER curves in Figure 4.7 result from
the average of all four channels, ≈6-dB sensitivity penalty shall be
expected.
Summing both average contributions, ≈15 dB sensitivity penalty is obtained, precisely agreeing with the outcome from the experimental results
and confirming that, in 4-SOP systems, the receiver sensitivity suffers notorious and unavoidable degradation even under ideal polarization rotation
compensation (or absence of channel distortions). This extra optical power
requirement greatly contributes to the overall power consumption through
lasers driving, in turn representing up to ≈40% of the total [233].
For its part, the difference between the transmission penalty for parallel
optics (negligible) and the 4-SOP system (0.5 dB) reports about the inaccuracies caused by CD on the channel estimation process. This gap to B2B
can be reduced via multi-tap signal equalization, which along with pulseshaping techniques, can effectively reduce the impact of CD and any other
form of linear ISI (e.g. signal over-filtering at the electro-optical and optoeletrical interfaces) at the expense of processing complexity. In addition,
improvements on the insertion loss of the optical components (e.g. power
splitter/combiner or modulators) will increase the power-budget margin,
thus enhancing the protection against aging, imperfect connectors and splicings, and fiber attenuation. In turn, this would motivate operating in the
O-band as a means of canceling out the influence of CD and then reducing
the DSP complexity.
141
Short-haul networks
Figure 4.9: Time-averaged BER versus time in B2B configuration for 4-SOP
32-GBd transmission with (blue) and without (red) polarization-rotation
compensation algorithm.
4.3.2.2
Polarization tracking
The experimental assessment of the polarization tracking algorithm is done
with the help of an inline polarization scrambler. The fastest rotation
rate the scrambler can achieve is on the order of radians per second (≈18
rad/s mode and ≈200 rad/s after [225]), which is much slower than what
can be detected within the maximum observation window of the employed
DSO. This means that one single frame could not capture such polarization
changes even using the maximum-available storage memory. To overcome
this practical impediment, short traces are periodically stored, where the
time-period determines the rotation magnitude to be tracked given fixed
angular speed and how often such estimation is done. After calculating the
rotation matrix, it remains constant for the entire period until re-update.
This routine allows for testing the software for diverse rotation magnitudes
(i.e. time-periods), with a limit determined by the minimum necessary
number of symbols for successful tracking (≤2048 symbols in our case, see
SOP tracking in Section 4.2.5.2) and the baudrate. Note that this approach
is equivalent to fixing the time-period and increasing the angular rotation
speed.
In our experimental characterization, the angular speed is fixed at max142
4.3 Experimental demonstration
imum, which is around one order of magnitude faster than expected in
buried-fiber scenarios and four times faster than aerial routing [230]. A
total of 120 frames of 16000 bits each were processed every 75 ms (total of
9 seconds). The measurement process starts with and initialization period
comprehending ≈1 second during which the polarization scrambler remains
idle and ~vT,j are kept perfectly aligned to ~vR,i . Figure 4.9 presents the timeaveraged BER over time with and without channel estimation (the same
data sequence is used) for the 4-SOP×32-GBd configuration at 25.5 dB
SNR subjected to the above-described polarization rotation. This case is
deemed representative of poor performance case.
After the initialization interval, the received SOP progressively deviates
from ideal alignment. In the case without polarization tracking, the instantaneous BER quickly converges to ≈0.5 (with the exception of those coincidental moments in which the received SOP is perfectly aligned with the
receiver). Consequently, the time-averaged BER shows an apparent monotonically increasing tendency. However, when the same data is processed
with polarization tracking the performance changes considerably. After
initialization, the instantaneous BER starts oscillating within a maximum
of ≈8·10−3 and a minimum of 0 (see30 ). This instability suggests using
shorter updating periods for the channel matrix (<16000 symbols) and dynamic (impairment dependent) threshold control (see K in SOP tracking
in Section 4.2.5). The resulting time-averaged BER shows stable around
≈10−3 , denoting successful demodulation after 20%-overhead FEC processing31 . The presented curves prove (i) the need for rotation compensation to
accomplish correct signal recovery, and (ii) successful post-FEC demodulation with the algorithm proposed in Section 4.2.5 under conditions much
more demanding than can be expected short-reach transmission links [230].
4.3.2.3
Power consumption and processing complexity
Despite ≈40% of the total power consumption comes from lasers driving
in the proposed 4-SOP system, there is a considerable remaining fraction
that cannot be neglected. Based on the results collected in [233], it is
precisely the contribution from this remaining part what makes the major difference, yielding around ≈37% higher estimated power consumption
30
Because 16000 symbols are employed for each measurement point in Figure 4.9, the
reliable BER is ≈ 6·10−4 . Zero counted BER must be interpreted cautiously.
31
This can be assured because the worst-case instantaneous BER does never exceed
the threshold for 20%-overhead FEC. Not being the case, successful demodulation is
conditional upon error spreading via bit-interleaving or similar techniques.
143
Short-haul networks
per channel when referenced to parallel optics. This is because in 4-SOP,
four-fold more sequences need to be digitized through respective ADCs for
recovering a single channel. Note that the number of ADCs is identical in
terms of hardware; that is, if the channels are employed to carry in parallel
mode a single higher-speed serial data stream belonging to the same user
(in contrast to one user per carrier), the additional power consumption
vanishes.
Interestingly, the polarization-dependent DSP only constitutes 2.47%
additional power consumption, confirming the good effectiveness-tosimplicity ratio of the proposed algorithm. Translating to complexity, both
the extra number of real multiplications and additions keep below 1%. Enhanced equalization, or the adaptation to AMF will surely increase the
consumption which, nevertheless, is not expected to exceed 5-7%.
In order for polarization-based wavelength demultiplexing to be worth
consideration, all additional contributions to the power consumption must
not exceed the one associated to standard WDM-demultiplexing apparatus.
Although this balance is not included in this report, the reader is referred to
[233] for a detailed breakdown of the power consumption and the processing
complexity.
4.4
Summary
Fundamentally driven by cost and power consumption, throughput scaling
in client-side optics is sustained by, first, optimizing the resources for serial
rate maximization; and subsequently, incorporating new dimensions when
possible and needed. This routine has based the upgrade and development of transmission standards across all currently considered reaches and
line-rates, giving rise to hybrid serial/parallel IM/DD transceiver modules
featuring both sustainable power consumption and very compact form factors.
Primarily owed to the simplicity of the demultiplexing process and the
cost-effectiveness of the associated equipment, WDM and parallel optics
have been the preferred approaches for throughput aggregation. Albeit
these technologies have sufficed to overcome electrical-bandwidth limitations as yet, the efficiency of their scalability saturates as the terabit traffic
range is approached; excessively increasing the number of parallel lanes
and leading to yield degradation, challenging packaging, and aggravating
the requirements on lasers’ frequency stability and thermal control. In
consequence, the incorporation of techniques that allow for increasing the
144
4.4 Summary
aggregate throughput without employing additional lasers or fiber pairs
becomes of great interest. This has recently put the research focus on
quadrature- and/or polarization-multiplexing, from which the latter constitutes the area covered in this investigation.
This investigation
In this chapter, we have described in detail a PolMux IM/DD technique enabling transmission and demultiplexing of up to four distinct SOPs in fiberoptic communication systems, and with the potential to be used alongside
or in place of existing technologies (e.g. WDM or AMF). In the first
block the essentials of this technique are discussed, covering from general mathematical description of N -fold PolMux, to requirements on the
transmitter plus receiver structures. The study intuitively proves that the
maximum number of simultaneously transmittable SOPs equals four, and
elaborates on their optimum orientation with respect to the polarizationdiversity receiver for (i) facilitating system optimization, (ii) increasing the
robustness and simplicity of the polarization tracking algorithm operating
on Stokes parameters and (iii) improving the overall error rate by minimizing noise enhancement. Additionally, one potential DSP chain for blind
polarization-rotation tracking and signal demultiplexing based on Stokes
parameter manipulation is thoroughly explained, allowing quasi-continuous
(microsecond-periodic) channel estimation at low computational complexity.
Besides the traditional X ⊥Y dual-polarization approach for SE doubling, non-orthogonal (≥3-SOP) PolMux configurations are suggested as
alternative demultiplexing method in other multiplexing technologies; then
permitting the interchange of the respective de/multiplexing apparatus. In
this regard, we propose and experimentally demonstrate the combination
of 4-SOP with 4-λ WDM (one SOP per λ) without employing wavelengthselective devices; thus considerably relaxing the requirements on the lasers’
frequency stability and removing the power consumption ascribed to thermal control of the passive optical filters. The second block covers such
experimental validation, where two distinct transmission schemes are considered aiming 100 Gbps net-rate, that is 4×27 GBd (108 Gbps) and 4×32
GBd (128 Gbps) with hard-decision 7%- and 20%-overhead BCH FEC respectively. Three different system figures are discussed: receiver sensitivity,
channel estimation robustness under fast polarization rotations, and power
consumption plus complexity benchmarking.
The receiver sensitivity performance is studied in terms of uncoded
145
Short-haul networks
BER versus optical power at the receiver input for B2B configuration and
C-band 2-km SSMF transmission. Below-FEC threshold demodulation is
observed in all cases, proving successful communication employing fourthorder PolMux in IM/DD fiber-optic systems. Particularly for the 4×32GBd case, ≈15-dB sensitivity penalty is measured with respect analogous
WDM approach with ideal optical filtering. This number is theoretically
argued, manifesting the strong base power penalty that that non-orthogonal
PolMux entails.
By inducing polarization changes around one order of magnitude faster
than expected in buried fiber and around four times faster than aerialrouted scenarios, ≈10−3 time-averaged BER is achieved with maximum
instantaneous BER ≈8·10−3 for the 4×32-GBd case at 25.5 dB of SNR.
Despite successful below-FEC threshold demodulation, the instabilities in
the instantaneous BER suggest using shorter period (<16000 symbols) between channel matrix updates.
Finally, a brief discussion on the power consumption and the DSP complexity is included. It is shown that in WDM scenarios where the carriers
transport a single higher-speed serial data stream, the additional power
consumption with respect to parallel optics is mainly caused by the extra
driving current to overcome the sensitivity penalty. In the alternative scenario where every carrier belongs to a different user, the three extra ADCs
(four in total) necessary to recover a single channel represent the highest
fraction of the total power consumption. Along with the lasers driving, the
power consumption in the latter case turns ≈37% higher than parallel optics. For its part, and due to its simplicity, the additional DSP as described
in Section 4.2.5 constitutes 2.47% of the total consumption. Equivalently,
up to fourth-order PolMux in IM/DD systems is possible by employing a
Stokes-based set of algorithms implying as few as <1% extra summations
and multiplications.
Future work
The present investigation conveys the following items:
• ISI cancellation : the accuracy of the polarization-tracking algorithm
is hindered by ISI, thus enhancing current channel estimation to support multi-tap filters adaptation represents an intuitive solution [198].
The processing complexity and its impact on the power consumption
must be considered.
146
4.4 Summary
• Polarization-tracking robustness and stability : the proposed algorithm requires reliable and consistent initialization when (i) the system operation is started and (ii) when the tracking is lost due to
extremely-fast polarization changes. Data-aided channel estimation
can be employed together with the presented algorithm for robustness increase [217]. Additionally, dynamic threshold adjustment in
4.14 (K) and shortening the time-period between channel matrix updates, can help with the precision of channel estimation and the BER
stability.
147
Short-haul networks
148
Chapter 5
Conclusion
As the ever-evolving requirements in data throughput and connectivity
keep pushing the existent electro-optical components plus transport networks to their functional limits, solutions facilitating efficient resource allocation gain proportional interest. In this line, DSP represents a powerful
cost-effective tool, capable of concurrently improve spectral and power utilization through advance modulation and multiplexing schemes, coding,
and electronic compensation of linear and non-linear impairments.
In this Thesis, we go beyond the state of the art by further exploiting
the potential of DSP. We propose, develop, and experimentally validate
for the first time in optical communications three advanced modulation
and/or multiplexing techniques, each formulated according to the technical
requirements of their specific scenarios. The target was to adapt the characteristics of the signal and the digital TX/RX features to better employ
the available capacity. These three studies are initial steps towards the
complete feasibility evaluation of the respective techniques for their use in
actual systems.
In the following, we summarize the major findings and conclusions of
each individual chapter.
5.1
(Ultra) long-haul networks
The need : currently, nonlineartities represent the major hindrance to
increase the throughput-times-distance product in these scenarios. Customary treated as noise, it has been demonstrated that nonlinearities in
>1000-km lumped-amplified uncompensated links show an additive Gaussian imprint on the constellation clusters; whereof we can leverage to apply
149
Conclusion
mature constellation-shaping techniques tracing capacity-achieving Gaussian distributions to boost the receiver sensitivity.
Proposal : SCM-PSM has been proposed as scalable and flexible coded
modulation scheme with shaping capabilities featuring lower complexity
in both transmitter and receiver sides than other alternatives. While this
scheme has been long explored for wireless systems, it had not been demonstrated for optical communications.
Less than 8.4% information rate penalty for up to 12 bits per symbol
is calculated while monotonically increasing the wellness of fit to normal
distribution. The occasional ambiguous mapping for certain modulation
orders (i.e. 6 bit/symbol) has been proved beneficial, resulting in larger
average Euclidean distance among symbols than same-order QAM when
the average power is equalized. This non-bijectivity in combination with
the circular-symmetry of PSM also results in better defined power levels,
whereby typical modulus-driven equalization techniques (e.g. RDE) base
their operation on. This potentially improves the accuracy of the channel
estimation. As far as decoding is concerned, we conclude that iterative
demapping/decoding routines are necessary in order to achieve successful demodulation without strongly penalizing the effective rate; for which,
nevertheless, lower coding gains than the analogous QAM are required.
Experimental proof of concept : successful decoding and demodulation is achieved after coherent detection of single-carrier DP 6-GBd 16ary, 32-ary and 64-ary PSM constellations in B2B configuration and after
EDFA-only uncompensated 240-km SSMF transmission. According to the
SE results, 0.5 code-rate convolutional codes are shown highly inefficient
despite the relatively similar EXIT profile with respect to PSM’s; with more
than 3 dB SNR gap to capacity. For their part, B2B measurements confirm
the unsuitability of DPLLs for conducting carrier recovery at the low-SNR
regions where shaping gain is achieved, as it caused the majority of the implementation penalties. Finally, the use of EM for cluster parametrization
in order to feed the soft-based receiver with more accurate statistics proved
≈0.6 dB sensitivity improvement in highly non-linear transmission regime.
The use of EM (or similar options) turns then interesting to allocate certain
performance margins in those cases where the compliance with circularly
symmetric Gaussian noise is not perfectly met. This is improvement is not
related to the shaping gain.
150
5.2 Metropolitan networks
Future : the results convey putting efforts in the code design and the
robustness of carrier recovery; where multi-layer coding strategies and the
use use of frequency pilot tones are concrete recommendations.
5.2
Metropolitan networks
The need : seeking for transparent data transport with high degree of
connectivity, MAN networks progressively converge to all-optical reconfigurable mesh architectures. Besides the benefits, this class of network
infrastructure poses challenges including pronounced over-filtering caused
by manyfold OADM cascadability plus electronics, larger PMD, and extra OSNR degradation owed to the absence of intermediate regeneration.
Consequently, reconfigurable software-driven sub-band/sub-carrier multiplexing is reasonably contemplated to increase the robustness against filtering effects and reduce the susceptibility to PMD without trading SE.
Proposal : C-MultiCAP has been conceived and proposed a scheme for
sub-band de/multiplexing and de/modulation which groups together the
multi-dimensionality of CAP and the spectral flexibility of MultiCAP for
application in coherent communications.
We have seen that the use of ≥2 CAP dimensions per band gets relegated to, perhaps, implementing power-efficient multi-dimensional modulation formats of ≤5 order. This is due to the quadratic growth of total
number of required filter taps in the corresponding de/multiplexing, and
the strong increase on the PMEPR. This observation directly maps to the
multi-band case, where the number of required filter taps is observed to
increase approximately with the square of the product between the number
of CAP dimensions and bands. In addition, the PMEPR may widely exceed
DMT’s for certain combinations of bands and CAP dimensions; all in all
setting a pragmatic limit to both spectral granularity and dimensionality
(≈15-20% of the total bandwidth for two dimensions per band). The above
observations base the decisions taken on the conception of the proposed
C-MultiCAP system. We note that, by realizing early carrier recovery in
the RX DSP (FPT technique is suggested) and forcing identical generation parameters in the four coherent dimensions, blind equalization taking
advantage of conventional algorithms (e.g. CMA or RDE) is possible.
Experimental proof of concept : aiming for verifying the soundness
of the assumptions supporting the proposed C-MultiCAP system, we im151
Conclusion
plement and transmit two different 2D-C-MultiCAP configurations over
225-km and 451-km EDFA-only SSMF links. The two generation schemes
are: 3-band partitioning at 221.88 Gbps occupying 29.4-GHz optical bandwidth (≈7.5 b/s/Hz gross SE), and 5-band partitioning at 336 Gbps occupying 43-GHz optical bandwidth (≈7.8 b/s/Hz gross SE). We conclude
that this combination of experimental conditions and generation parameters translates into notorious implementation penalties (≈5 dB and ≈8 dB
for 221 Gbps and 336 Gbps respectively) and transmission penalties (≈5
dB after distance doubling for 221 Gbps) owed to the imperfect fulfillment
of the basal assumptions. If the proposed equalization technique is to be
used, narrower bands and looser inter-band spacing are solutions to improve the performance at the cost of more complexity and bandwidth. For
its part, owed to CD-induced group-delay, FPT-based phase-noise estimation has been argued to be the main reason for transmission penalties and
it shall be improved (e.g. spreading multiple tones across the spectrum) or
alternative approaches.
On the other hand, the spectral and rate adaptability of C-MultiCAP
have been demonstrated by achieving >300 Gbps in a system with less
than 6-GHz end-to-end bandwidth; as well as the ease of reconfigurability, allowing for >100-Gbps line rate increase only adjusting the TX/RX
DSP. Finally, recovery and equalization of multiple correlated CAP dimensions has been shown, proving the plausibility of their joint exploitation for
implementing multi-dimensional modulation formats.
Future : generation parameters, phase noise estimation, and equalization
need to be perfected for ensuring robust demodulation and higher sensitivities (approximately from 3 to 5 dB better is possible). The reduction of
processing complexity should be pursued.
5.3
Short-haul networks
The need : with short-haul subsystems mainly driven by cost and power
consumption, WDM and parallel optics have been preferred methods to
keep pace with throughput growth due to the apparent de/multiplexing
simplicity. Nevertheless, as this throughput demand grows steadily, the
scaling efficiency gets jeopardized. For instance, an excessive number of
wavelengths requires strict wavelength stability control and makes packaging challenging; while too many fiber-pairs would likewise suffer from
packaging and fiber-transceiver interfacing issues. This outlook encourages
152
5.3 Short-haul networks
the incorporation of polarization-multiplexing to make better use of the
available bandwidth and/or space.
Proposal : we have introduced a polarization-multiplexed technique enabling simultaneous transmission of ≤4 distinct SOPs in IM/DD fiber-optic
communication systems, and usable alongside or in place of existing technologies (e.g. WDM or multilevel modulation).
During the initial explanation of the fundamentals, the limitation to 4
as the maximum number of demultiplexable SOPs is argued; as well as their
optimum orientation with respect to the receiver’s polarizers to improve the
robustness and simplicity of the channel estimation and minimizing noise
enhancement. This is followed by the elaboration on one possible RX DSP
chain for blind low-complexity quasi-continuous SOP tracking and signal
demultiplexing. Finally, exploiting the possibility to go beyond traditional
dual-polarization for SE doubling, we propose ≥3-SOP configurations as
alternative demultiplexing method in other multiplexing technologies (e.g.
WDM); thereby enabling the interchange of the respective apparatus.
Experimental proof of concept : based on the last-mentioned idea,
we conduct its experimental validation in a 4-λ WDM system, where wavelength demultiplexing is performed without the need for optical filters. This
potentially relaxes the requirements on the lasers’ stability and avoids the
power consumption associated to the thermal control of such filters. Targeting 100-Gbps net rate, two WDM systems are evaluated: 4×27 GBd
(108 Gbps gross) and 4×32 GBd (128 Gbps gross) considering 7%- and
20%-overhead FEC respectively.
Besides achieving successful wavelength demultiplexing and signal demodulation, the results in terms of receiver sensitivity evince the strong
base power penalty that that non-orthogonal polarization-multiplexing
(≥3-SOP) entails (≈15-dB sensitivity penalty with respect to ideally filtered WDM). However, below-FEC average BER performance is observed
even under abnormally fast polarization changes; still motivating the application of the proposed DSP for other configurations (e.g. standard dualpolarization OOK). In terms of power consumption, the proposed system
consumes ≈37% more than parallel optics, from which only 2.47% belongs
to the software; or equivalently, as few as ≈1% extra summations and multiplications.
Future : main recommendations relate to improving the robustness of the
153
Conclusion
polarization tracking algorithm; certainly susceptible to ISI and imprecise
initialization.
154
Appendix A
SCM-PSM 1D gaussianity
This appendix provides comprehensive and complementary study on the
convergence evolution of SCM-PSM output constellations’ distribution towards Gaussianity. The metrics used permit adaptation to any reference
distribution (not necessarily Gaussian), hence constituting a straight and
generalizable method for analyzing the shaping gain evolution for any given
memoryless channel and comparing the results among various modulation
alternatives.
Tools
The evaluation of PSM constellations’ distribution is performed through
numerical and visual analysis of the cdf of the individual quadrature components in the absence of noise. The visual test is performed with a statistical tool named quantile-quantile (Q-Q) plot, which is a graphical method
for the diagnosis of fitting deviations between two probability distributions
based on percentile comparison. These percentiles are measured or estimated from the cdf of the respective random variables and then plotted
against each other in an unequivocal scatter plot. Given that we want to
check upon the Gaussianity of certain data sample, one of the probability
distributions acts as the reference model with a theoretical normal profile N (µ, σ), while the other corresponds to the cdf of PSM constellations’
quadrature projections. This constitutes the most basic plotting format
and it is referred as one-sample Q-Q plot. It is noteworthy that, since the
reference distribution is continuous and known, all possible quantiles are
calculable through simple cdf inversion. This means that any value from
the discontinuous cdf can be directly represented thence avoiding inaccurate
155
SCM-PSM 1D gaussianity
data interpolation. The descriptive parametric equations of the one-sample
Q-Q plot are given by:
−1
Yi ≡ Data quantiles = FDAT
A (pi )
−1
Xi ≡ Reference quantiles = FREF
(pi )
(A.1)
−1
−1
where FDAT
A and FREF represent the quantile functions of the cdfs for the
data and the reference model respectively, and pi ∈ (0, 1) determines the
order of the ith quantile. In order to normalize the axes’ scales, the mean
(µ) and variance (σ 2 ) of the theoretical normal reference are equalized to
the estimated mean and variance of PSM quadrature components for every
N . In that case, the distributions being analyzed will be identical if ∀i:
−1
−1
FDAT
A (pi ) = FREF (pi )
(A.2)
The evaluation of the resulting curves will provide qualitative information on the distribution shapes, tail behavior, mean/variance deviations,
outliers and symmetry unbalances. However, in order to obtain numeric
evaluation of the goodness of fit between the empirical and theoretical cdf,
certain statistical criteria must be associated with each case of study. In this
research, the fitting variance (σF IT ) and the Kolmogorov-Smirnov statistic
factor (D) are used as quantitative indicators. The mathematical expressions:
h
−1
−1
2
σF2 IT = E |FDAT
A (pi ) − FREF (pi )|
h
i
−1
−1
D = sup |FDAT
A (pi ) − FREF (pi )|
i
(A.3)
where σF2 IT , D ∈ [0, ∞). Whereas σF2 IT gives information about the
average Gaussian purity of the empirical distribution, D shows possible
outliers and symmetry disturbances. Both figures of merit denote good
agreement for values approaching zero.
Results and discussion
Figure A.1 shows the visual test based on Q-Q plotting for N ∈ {4, 6, 9}.
The corresponding cross-sectional probability mass functions of the constellations are included for the reference. Figure A.2 displays the numerical
results of goodness of fit based on the normal fitting variance and the onesample Kolmogorov–Smirnov (KS) statistic for both QAM and PSM up to
156
Figure A.1: Q-Q plot Gaussianity test on the quadrature projections of
SCM-PSM constellations for N = 4 (a), N = 6 (b) and N = 9 (c).
Figure A.2: Gaussian-fitting variance and Kolmogorov-Smirnov factor of
QAM and PSM for {N ∈ N, 4 ≤ N ≤ 21}.
N = 21. The fitting variance and KS statistic are chosen because of their
complementarity, simplicity of calculation, and ease of understanding. Tendency lines are also presented for reference purposes.
It is observed in Figure A.1 that the dispersion of the Q-Q points around
the reference normal distribution decreases as the modulation order increases. As N increases good fitting is observed around quantile zero, where
the average deviation from Gaussianity is dominated by disagreements on
the tails (see Figure A.1c). Notice that only one cross-section of the powernormalized constellation is analyzed, hence the probability distribution of
the Gaussian reference that yields one-to-one quantile scaling is given by
µ = 0 and σ 2 = 0.5 (σ ≈ 0.7).
Figure A.2 shows that the average Gaussianity for every N and modulation format is determined by the KS statistic with some exceptions in
157
SCM-PSM 1D gaussianity
SCM-PSM comprising N6,8,14,15,18. The difference in the fitting variance
between PSM and QAM exceeds one order of magnitude for N = 9 and two
orders for N = 16. For low number of bits per symbol, the quantization
error governs the deviation with respect to Gaussianity. This fact explains
the convergence to normal distribution of QAM for N ∈ [0, 10], and the fast
convergence of PSM (two orders of magnitude within N = 4 and N = 11).
Once the quantization error stops dominating, QAM shows flat tendency
and PSM shows ≈0.65 dB/N convergence rate to unit normal distribution.
Note that owed to the square-distribution of QAM constellations’ for N
even, σF2 IT is 2 dB higher than cross-distributed orders, i.e. N odd. The
monotonic decrease of the fitting curve for PSM which partially proves the
asymptotic tendency to Gaussianity.
Central limit theorem
Because in SCM signal generation is carried out via the superposition of N
data-independent sequences, the convergence to Gaussianity can be readily tested through the evaluation of the CLT. Particularly in PSM, the
phase-shift induced by the complex weights before superposition makes the
random independent data streams not be identically distributed, hence preventing the strict application of the traditional CLT definition. In this case,
Lyapunov condition can be evaluated instead.
The Lyapunov condition states that, given a sequence of independent
but not necessarily identically distributed random variables Xi , with finite
mean µi and variance σi2 , and
rn2+ =
n
X
h
E |Xi − µi |2+
i
(A.4)
i=1
if for some > 0
rn2+
=0
n→∞ s2+
n
lim
(A.5)
where
s2n =
n
X
i=1
then the CLT holds.
158
σi2
(A.6)
Figure A.3: Conceptual illustration of the proposed quadruple polarization
system for IM/DD optical data links.
Figure A.3 shows the evaluation of A.5 versus the number of added
random variables n (i.e. number of bits/sym N ) for = 10 (arbitrary).
For the ease of visualization, the increment magnitude is show instead;
meaning that, if the curve stabilizes, the convergence to zero (Gaussianity)
is relentless and invariant. The results prove positive convergence up to
N = 12 with constant increment tendency, presumably signifying that CLT
applies and thus the asymptotic convergence to Gaussianity of the marginal
quadrature distributions.
159
SCM-PSM 1D gaussianity
160
Appendix B
Memoryless
channel-capacity
In the optical communications society, the word capacity is often employed
alluding to two related but distinct concepts. The first one is the traffic
volume or amount of information per unit time, which is referred to as
throughput across the entire document. And secondly, as a characteristic
parameter of any noisy transmission medium or channel, by which the
maximum amount of information that the observed output can ever give
about the input gets quantified. We use capacity for the latter meaning. In
the following, we provide the mathematical fundamentals behind capacity
calculation for memoryless channels.
In information-theory terms, the capacity of a given a noisy channel is
expressed as the supremum of the mutual information across all possible
input distributions:
C , sup I(X; Y )
(B.1)
pX
where pX is the input signal’s pdf and the mutual information is given by:
I(X; Y ) = I(Y ; X) , H(Y ) − H(Y |X) = H(X) − H(X|Y )
(B.2)
where H(•) and H(•|•) are the entropy and the conditional entropy operators respectively1 . Figure B.1 intuitively illustrates the relation among
1
For simplicity, no distinction is made between discrete and differential entropy concerning the operator’s symbol.
161
Memoryless channel-capacity
Figure B.1: Venn diagram relating various information measuremets.
important measurements of information, making I(X; Y )=I(X; Y ) apparent. Considering continuous complex input/output signals:
H(X) , −
Z
pX (x) log2 pX (x)dx
(B.3)
C
H(X|Y ) , −
ZZ
pX,Y (x, y) log2 pX|Y (x|y)dxdy
(B.4)
C
Substituting B.3 and B.4 into B.2 and leveraging Bayes’ theorem2 :
ZZ
I(X; Y ) ,
pX,Y (x, y) log2
C
pX,Y (x, y)
dxdy
pX (x)pY (y)
(B.5)
or alternatively, expressing B.5 as a function of the input distribution and
the channel’s transition probability:
pY |X (y|x)
dxdy
C pX (x)pY |X (y|x)dx
ZZ
I(X; Y ) ,
C
pX (x)pY |X (y|x) log2 R
(B.6)
Discrete-input continuous-output
If the input to the channel is discrete with finite alphabet Υ, B.6 becomes:
I(X; Y ) ,
X
x∈Υ
2
pY |X (y|x)
dy
x∈Υ pX (x)pY |X (y|x)
Z
pX (x)
C
pY |X (y, x) log2 P
pX,Y (x, y) = pX|Y (x|y)pY (y)
162
(B.7)
The resulting mutual information is called constrained capacity, making
direct allusion to the fact that the optimization is performed on a subset of
possible input distributions. Note that given a continuous-output channel,
conditioning the mutual information optimization to finite-alphabet input
unavoidably results in a capacity lowerbound.
The link between the purely statistical expression B.7 and physically
meaningful parameters turns apparent when the finite input alphabet is related to a particular M-ary constellation, e.g. the 16 points of 16-QAM or
the 4 points of QPSK. Upon such interpretation, the relation between constrained capacity and the symbols’ relative position (present in pY |X (y|x))
and/or their probability of occurrence (pX (x)) gets intuitively revealed,
as well as the basis of constellation shaping. Accordingly, the following
inequality holds in scenarios where the input power is limited:
X
|x|2 pX (x) ≤ P
(B.8)
x∈Υ
In the case where the channel’s noise is zero-mean complex AWG,
pY |X (y|x) is given by:
pY |X (y|x) = pZ (y − x)
(B.9)
where
pZ (z) =
1 |z|22
eσ
πσ 2
(B.10)
where σ 2 is the total complex-noise
power. Because under AWGN the inR ∞ −x2
tegral in B.7 is of the kind −∞ e f (x)dx, the Gauss-Hermite quadrature
numerical approximation shall be employed to accelerate the calculation:
Z ∞
2
e−x f (x)dx ≈
−∞
n
X
ωi f (xi )
(B.11)
i=1
where n is the number of sample points used and ωi is given by:
√
2n−1 n! π
ωi =
n2 [Qn−1 (xi )]2
(B.12)
where Qn (x) is the Hermite polynomial. In this investigation we employed
a 16-point (n) approximation as the optimum quadrature for polynomials of
degree 31 or less, widely sufficing for the accurate computation of capacity
curves.
163
Memoryless channel-capacity
AWGN channel capacity :
Under AWGN, B.2 reduces to:
I(X; Y ) = H(Y ) − H(X + Z|X)
= H(Y ) − H(Z)
(B.13)
inserting B.12 into B.13:
I(X; Y ) = H(Y ) − log2 πeσ 2
(B.14)
Because the differential entropy maximizes when the random variable
describes a Gaussian distribution, and knowing that the noise and the input
are statistically independent:
I(X; Y ) ≤ log2 πe(P + σ 2 ) − log2 πeσ 2
(B.15)
where P is the input signal power. Then resolving for the maximum, the
2D (complex) continuous-input continuous-output channel capacity is obtained:
C = log2
P
1+ 2
σ
164
(B.16)
Appendix C
Frequency pilot tone
The principle of FPT-based frequency offset and phase noise estimation
is very intuitive. Let the noiseless transmitted field of the FPT in one
polarization be expressed as:
Et (t) = Et;0 ei2π(fF P T +fC )t+φC (t)
(C.1)
where fC and φC (t) are the carrier frequency and phase respectively, fF P T
is a user-defined intermediate frequency (DC is also considered), and Et;0
is the constant-valued field’s amplitude function, in turn dependent on the
FPT insertion technique. After ideal transmission and coherent detection,
the field is given by:
Ed (t) =Ed;0 ei2π(fF P T +fC −fLO )t+φC (t)−φLO (t)
=Ed;0 ei2π(fF P T +∆f )t+∆φ(t)
(C.2)
where fLO and φLO (t) are the frequency and phase of the LO, and Ed;0
is the constant received field’s amplitude. Because fF P T is known beforehand, a simple evaluation of the argument of C.2 yields ∆f , the frequency
offset after coherent beating; and ∆φ(t), the combined phase-noise function
of both transmitter and receiver lasers. The frequency tone is thus able to
blindly and continuously track the evolution of the phase fluctuations experienced by the electric field at the exact frequency where it is inserted. In
the receiver, such phase distortions are uniformly corrected for across the
entire detected spectrum; which includes the co-propagating data signal.
However, the accuracy of this technique relies on two important assumptions made on the above mathematical elaboration and whose validity
weakens in real transmission scenarios:
165
Frequency pilot tone
• Absence of noise plus the perfect isolation of the FPT upon reception;
which not being the case, requires careful optimization of the PSR,
and allocating enough guard-band around the FPT. Both directly
related to the phase estimation fidelity.
By increasing the PSR, the influence of the noise in the phase estimation becomes weaker, while excessively high PSR results in OSNR
degradation for the information signal and quantization noise when
digital FPT generation is employed. The optimum ratio depends on
the system’s noise power, the signal bandwidth, the phase noise characteristics, and the properties of the filter employed for FPT isolation.
Experimental characterization and empirical observations in various
demonstrations [90, 157, 158] indicate optimum PSR of ≈-15 dB.
Then FPT guard-banding, dictating the trade-off between total effective rate and the maximum frequency excursions that can be tracked
within affordable performance degradation. Given a fixed observation time, time-varying phase fluctuations show as signal broadening
in the frequency domain; therefore, proportional margin between the
FPT and the closest frequency components of the information signal
must be allocated so as not to incur spectral overlapping, and improve
FPT-signal isolation ratio. The particular guard-band arrangement
depends on the characteristics of the spectral tails of the neighboring
signal, and on possible constraints on the filter design.
• The phase distortion that the FPT reads is common to all spectral
components; which is false in the presence of any mechanism inducing
certain degree of group delay, such as some analog filters or CD. For
instance, CD-induced walk-off effect will cause a frequency-dependent
decorrelation of the phase vectors by which neighboring components
to the FPT will exhibit better performance than those far apart [156].
Possible solutions to alleviate this imprecision include the use of multiple FPTs in different frequency regions [122], enabling more dedicated phase-noise compensation (for instance, one FPT per sub-band
in C-MultiCAP systems) and/or measurement noise averaging; and
placing the FPTs such that the maximum distance to all transmitted
spectral components and other FPTs is minimized.
With regard to implementation, FPT insertion can be realized in three
alternative ways. Figure C.1 shows a conceptual illustration of the corresponding output spectra:
166
Figure C.1: On the top, FPT-less information signal. On the bottom from
left to right, conceptual representation of the ouput spectra of the three
main techniques for FPT insertion: digital, electrical and optical.
• Digital (e.g. [46]) : the FPT is inserted at the end of the transmitter’s DSP. Depending on the FPT allocation, the insertion stage
may vary between prior or posterior to high-frequencies enhancement
pre-distortion to minimize the manipulation of the tone before digitalto-analog conversion. This approach permits highly accurate centerfrequency allocation, readily allowing for radio-frequency FPTs and
hence a desirable null DC component at the expense of quantization
noise. Care should be taken to ensure an integer number of period
within DAC’s repetition memory so as not to incur in windowing
effects that degrade the later phase estimation.
• Electrical analog (e.g. [90]): certain DC level is intentionally induced
during electro-optical conversion. Precise PSR control, distortion-less
FPT insertion, and hardware minimization is obtained at the expense
of higher average power plus consumption, and potential non-linear
signal distortions according to the modulator’s response.
• Optical analog (e.g. [155]) : in this method a copy of the carrier
is inserted without intermediate DSP or electro-optical interfaces.
Whereas the FPT phase is not distorted, additional pieces of optical hardware are necessary.
167
Frequency pilot tone
168
Appendix D
DSP schema for 4-SOP
IM/DD transmission system
% INTENSITY SIGNAL MANIPULATION
% Signal conditioning
Resampling % Outputs 2 samples per symbol
Filtering % Low−pass filtering for SNR improvement
% Front−end correction
Skew % time skew compensation
Responsivity % Photodioide's responsivity equalization
DC Compensation % DC offset correction
% STOKES PARAMETER MANIPULATION
Intensity to Stokes conversion % Stokes parameter generation
% based on 1.10
Timing Recovery % time−domain Gardner error estimation
% and decimation
% SOP tracking
Zero−power discriminator % Trivial zero−power case is detected,
169
DSP schema for 4-SOP IM/DD transmission system
% demodulated and removed.
Stokes parameters normalization % S1 to S3 are normalized by S0
Initialization of coordinate system and decision thresholds % In
% the absence of initialization algorithm (e.g. training sequence),
% the coordinate system may be initialized with the identity matrix.
while not(convergence)
Estimate Stokes parameter (4.15)
switch estimated Stokes parameter
case +/−1
Calculate the sign−dependent error signal (4.14)
Update the coordinate system (4.16).
Update decision thresholds % dependent on noise,
% modulation format, and transmitted plus detected SOPs
Stokes parameter index = Stokes parameter index;
symbol to analyze = symbol to analyze + 1;
case 0
Stokes parameter index = Stokes parameter index + 1;
symbol to analyze = symbol to analyze;
end
end
% Channel distortion compensation
Transpose estimated channel matrix (4.17)
Compensate distortion (4.2)
% Demultiplexing
SOP demultiplexing (e.g. 4.18)
% Demodulation
Demodulation and error counting per channel
170
List of Acronyms
1D one-dimensional
2D two-dimensional
3D three-dimensional
ADC analogue-to-digital converter
AMF advanced modulation formats
ASE amplified spontaneous emission
ASIC application-specific integrated circuit
aSOP aggregate SOP
AWGen arbitrary waveform generator
AWGN additive white Gaussian noise
B2B back-to-back
BCJR Bahl-Cocke-Jelinek-Raviv
BER bit error rate
BICM bit-interleaved coded modulation
BPSK binary phase-shift keying
CAP carrierless amplitude/phase modulation
CCDF complementary cumulative distribution function
CD chromatic dispersion
171
ACRONYMS
cdf cumulative distribution function
CER constellation expansion ratio
CLT central limit theorem
CMA constant-modulus algorithm
C-MultiCAP coherent multi-band carrierless amplitude/phase modulation
DAC digital-to-analogue converter
DC direct current
DD direct-detection
DFB distributed feedback
DGD differential group delay
DMT discrete multitone
DP dual-polarization
DPLL digital phase-locked loop
DSO digital storage oscilloscope
DSP digital signal processing
ECL external-cavity laser
EDFA Erbium-doped fiber amplifier
EM expectation maximization
ETDM electrical time-division multiplexing
EXIT extrinsic information transfer
FEC forward error correction
FIR finite impulse response
FPT frequency pilot tone
IBI inter-band interference
172
ICI inter-carrier interference
ID iterative decoding-demapping
IDF inverse dispersion fiber
IM intensity-modulation
IPM iterative polar modulation
ISI inter-symbol interference
LAN local-area network
LC left-circular
LDPC low-density parity-check
LH long-haul
LO local oscillator
MAC medium access control
MAN Metropolitan-area network
MAP maximum a posteriori probability
MI mutual information
MIMO multi-input multi-output
MLSE maximum-likelihood sequence estimation
MMA multi-modulus algorithm
MMSE minimum mean square error
MoG mixture of Gaussian
MultiCAP multi-band carrierless amplitude/phase modulation
MZM Mach-Zehnder modulator
NLMS normalized least mean square
NMSE normalized mean square error
173
ACRONYMS
NRZ non-return-to-zero
NZDSF non-zero dispersion-shifted fiber
OADM optical add-drop multiplexer
OFDM orthogonal frequency-division multiplexing
OOK on-off keying
OSNR optical signal-to-noise ratio
OTDM optical time-division multiplexing
P2P point-to-point
PAM pulse-amplitude modulation
PAPR peak-to-average power ratio
PBS polarization beam splitter
pdf probability density function
PDL polarization-dependent loss
PMD polarization-mode dispersion
PMEPR peak-to-mean envelop power ratio
pmf probability mass function
PM-QPSK polarization-multiplexed quadrature phase-shift keying
PolC polarization controller
PolMux polarization multiplexing
PolSK polarization-shift keying
PPG pulse pattern generator
PRBS pseudo-random bit sequence
PSCF pure silica core fiber
PSD power spectral density
174
PSM phase-shifted mapping
PSR pilot-to-signal ratio
QAM quadrature amplitude modulation
QMSE quantization mean-square error
QPSK quadrature phase-shift keying
Q-Q quantile-quantile
RC right-circular
RDE radius-directed equalizer
RRC root-raised-cosine
RX receiver
RZ return to zero
SCM superposition coded modulation
SE spectral efficiency
SER symbol error rate
SISO soft-input soft-output
SLAF super-large-area fiber
SNR signal-to-noise ratio
SNRb signal-to-noise ratio per bit
SNRs signal-to-noise ratio per symbol
SOP state of polarization
SPM self-phase modulation
SSMF standard single mode fiber
TX transmitter
ULAF ultra-large-area fiber
175
ACRONYMS
ULH ultra long-haul
VOA variable optical attenuator
WDM wavelength-division multiplexing
176
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