Institutionen för systemteknik Department of Electrical Engineering Examensarbete

Institutionen för systemteknik Department of Electrical Engineering Examensarbete
Institutionen för systemteknik
Department of Electrical Engineering
Examensarbete
On Data Compression for TDOA Localization
Examensarbete utfört i Informationskodning
vid Tekniska högskolan i Linköping
av
Joel Arbring, Patrik Hedström
LiTH-ISY-EX--10/4352--SE
Linköping 2010
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
Linköpings tekniska högskola
Linköpings universitet
581 83 Linköping
On Data Compression for TDOA Localization
Examensarbete utfört i Informationskodning
vid Tekniska högskolan i Linköping
av
Joel Arbring, Patrik Hedström
LiTH-ISY-EX--10/4352--SE
Handledare:
Anders Johansson, Ph.D
Informationssystem, Totalförsvarets forskningsinstitut
Harald Nautsch
isy, Linköpings universitet
Examinator:
Robert Forchheimer, Ph.D
isy, Linköpings universitet
Linköping, 15 June, 2010
Avdelning, Institution
Division, Department
Datum
Date
Division of Information Coding
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
Övrig rapport
2010-06-15
—
LiTH-ISY-EX--10/4352--SE
Serietitel och serienummer ISSN
Title of series, numbering
—
URL för elektronisk version
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-57274
Titel
Title
Datakompression för TDOA-lokalisering
On Data Compression for TDOA Localization
Författare Joel Arbring, Patrik Hedström
Author
Sammanfattning
Abstract
This master thesis investigates different approaches to data compression on common types of signals in the context of localization by estimating time difference of
arrival (TDOA). The thesis includes evaluation of the compression schemes using
recorded data, collected as part of the thesis work. This evaluation shows that
compression is possible while preserving localization accuracy.
The recorded data is backed up with more extensive simulations using a free
space propagation model without attenuation. The signals investigated are flat
spectrum signals, signals using phase-shift keying and single side band speech
signals. Signals with low bandwidth are given precedence over high bandwidth
signals, since they require more data in order to get an accurate localization estimate.
The compression methods used are transform based schemes. The transforms
utilized are the Karhunen-Loéve transform and the discrete Fourier transform.
Different approaches for quantization of the transform components are examined,
one of them being zonal sampling.
Localization is performed in the Fourier domain by calculating the steered
response power from the cross-spectral density matrix. The simulations are performed in Matlab using three recording nodes in a symmetrical geometry.
The performance of localization accuracy is compared with the Cramér-Rao
bound for flat spectrum signals using the standard deviation of the localization
error from the compressed signals.
Nyckelord
Keywords
Time difference of arrival, Localization, Data compression, Cramér-Rao matrix
bound, Electronic warfare
Abstract
This master thesis investigates different approaches to data compression on common types of signals in the context of localization by estimating time difference of
arrival (TDOA). The thesis includes evaluation of the compression schemes using
recorded data, collected as part of the thesis work. This evaluation shows that
compression is possible while preserving localization accuracy.
The recorded data is backed up with more extensive simulations using a free
space propagation model without attenuation. The signals investigated are flat
spectrum signals, signals using phase-shift keying and single side band speech
signals. Signals with low bandwidth are given precedence over high bandwidth
signals, since they require more data in order to get an accurate localization estimate.
The compression methods used are transform based schemes. The transforms
utilized are the Karhunen-Loéve transform and the discrete Fourier transform.
Different approaches for quantization of the transform components are examined,
one of them being zonal sampling.
Localization is performed in the Fourier domain by calculating the steered
response power from the cross-spectral density matrix. The simulations are performed in Matlab using three recording nodes in a symmetrical geometry.
The performance of localization accuracy is compared with the Cramér-Rao
bound for flat spectrum signals using the standard deviation of the localization
error from the compressed signals.
v
Acknowledgments
We would like to thank Swedish Defence Research Agency (FOI) for the opportunity to do this master thesis project for them. Special gratitude goes to Anders
Johansson, who supplied us with the main idea and supervised our work and
provided invaluable input, time and knowledge.
During the field data recording, we received a lot of help from the other members of the Electronic Warfare division at FOI, notably Daniel Henriksson who
assisted us well outside normal business hours.
vii
Contents
1 Introduction
1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Thesis Disposition . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Overview of TDOA
2.1 Introduction . . . . . . . .
2.2 Localization Using TDOA
2.3 Experiment Model . . . .
2.4 The Cramér-Rao Bound .
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3 Signals
3.1 Signal to Noise Ratio . . . . . . . . . . . . . . .
3.2 Signal and Noise Spectrum . . . . . . . . . . .
3.3 Signals of Interest . . . . . . . . . . . . . . . .
3.3.1 Flat Spectrum Signals . . . . . . . . . .
3.3.2 Phase-Shift Keying . . . . . . . . . . . .
3.3.3 Amplitude Modulated Signal Side Band
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13
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4 Compression of signal data
4.1 Transform Coding . . . . . . . . . . . . . . . . . . .
4.1.1 Illustration of Decorrelation . . . . . . . . . .
4.1.2 Karhunen-Loéve Transform . . . . . . . . . .
4.1.3 Discrete Cosine Transform . . . . . . . . . . .
4.1.4 Discrete Fourier Transform . . . . . . . . . .
4.2 Quantization . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Integral components . . . . . . . . . . . . . .
4.2.2 Partial Components . . . . . . . . . . . . . .
4.2.3 Compression Using Time-Frequency Masking
4.3 Entropy Coding . . . . . . . . . . . . . . . . . . . . .
4.4 Distortion . . . . . . . . . . . . . . . . . . . . . . . .
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x
Contents
5 Evaulation by simulation
5.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . .
5.1.1 SRP Grid Granularity . . . . . . . . . . . . . . .
5.1.2 Data Rate Reference . . . . . . . . . . . . . . . .
5.1.3 Block Effects . . . . . . . . . . . . . . . . . . . .
5.1.4 Bandwidth and Signal Length . . . . . . . . . . .
5.1.5 Compression Ratio . . . . . . . . . . . . . . . . .
5.1.6 Localization Ability Adjusted Compression Ratio
5.2 Evaluation of Compression Impact . . . . . . . . . . . .
5.2.1 Compression Using Integral DFT Components .
5.2.2 Compression Using Integral KLT Components .
5.2.3 Compression Using Partial Components . . . . .
5.2.4 Compression Using Time-Frequency Masking . .
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37
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53
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6 Recorded field data for evaluation
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
6.2 Signals Used . . . . . . . . . . . . . . . . . . . . . .
6.3 Transmitting and Receiving . . . . . . . . . . . . . .
6.4 Post Processing . . . . . . . . . . . . . . . . . . . . .
6.5 Localization Using Recorded Data . . . . . . . . . .
6.6 Evaluation of Compression Impact . . . . . . . . . .
6.6.1 Compression Using DFT Components . . . .
6.6.2 Compression Using KLT Components . . . .
6.6.3 Compression Using Partial Components . . .
6.6.4 Compression Using Time-Frequency Masking
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65
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7 Conclusion and discussion
7.1 Comments on Compression and Noise . . . . . . . . . .
7.1.1 Separate Signal from Noise . . . . . . . . . . . .
7.1.2 Noise Reduction . . . . . . . . . . . . . . . . . .
7.2 Comments on the Field Recordings . . . . . . . . . . . .
7.3 Proposed Use . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Localization Ability Reduction . . . . . . . . . . . . . .
7.5 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 Amplitude Data . . . . . . . . . . . . . . . . . .
7.5.2 Phase-amplitude Data Optimization . . . . . . .
7.5.3 Block Length and Ratio . . . . . . . . . . . . . .
7.5.4 Impact on Node-Base Transmission Redundancy
7.5.5 Other Areas to Look at . . . . . . . . . . . . . .
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85
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Lists
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
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Bibliography
92
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Acronyms
Notation
AM
CRB
Description
Amplitude modulation
Cramér-Rao bound
CRMB
DCT
DFT
Cramér-Rao matrix bound
Discrete cosine transform
Discrete Fourier transform
FIM
FOI
FSS
Fisher information matrix
Swedish Defence Research Agency
Flat spectrum signal
GPS
KLT
Global Positioning System
Karhunen-Loéve transform
PCA
PSK
Principal component analysis
Phase-shift keying
SNR
Signal to noise ratio
SRP
Steered response power
SSB
Single side band
STFT
SVD
TDOA
Short-time Fourier transform
Singular Value Decomposition
Time difference of arrival
TFM
Time-frequency masking
xi
Page List
17
iii, v, 3, 8, 10, 11, 12, 14, 15,
33, 35, 37, 40, 43, 45, 47, 49,
50, 52, 55–57, 60, 61, 63, 71,
89
iii, 11, 11
21, 21, 41, 43
iii, v, 21, 21, 25, 27, 41, 43,
48, 58, 73, 87, 91
xiii, 11, 11, 34
vii, 68
iii, v, 14, 15, 15, 25, 27, 34,
35, 37, 42–44, 53, 58, 68, 70,
71, 73, 74, 85, 86, 89–91
65, 66, 70
iii, v, 20, 21, 21, 33, 40, 43,
48, 76
20, 21
iii, v, 14, 16, 16, 30, 31, 43,
44, 48, 53, 58, 65, 68, 71, 73,
88
3, 6, 9, 12, 13, 13, 14, 15, 25,
27, 33–35, 37, 40, 43–63, 66,
68, 70, 71, 74, 76, 78, 80, 83,
85, 88, 90
iii, v, 9, 9, 10, 38, 39, 70, 86,
89
iii, v, 14, 17, 17, 20, 27, 30,
43, 46, 48, 53, 62, 65, 68, 71,
73, 86, 89, 91
xiv, 9, 9, 21, 30, 31
85
iii, v, xiii, 1–3, 5, 7–9, 11, 14,
19, 25, 31, 70, 85–87
31, 58, 62, 80, 87
xii
Notation
WGN
Acronyms
Description
White Gaussian noise
Page List
13, 14, 15, 27, 43, 65, 70, 85
Symbols
Notation
E[◦]
F{◦}
◦∗
◦T
ˆ
◦
◦i
◦q
◦∗◦
Description
Expected value of ◦
Fourier transform of ◦
Conjugate transpose of ◦
Transpose of ◦
Estimate of ◦
In-phase component of ◦
Quadrature component of ◦
Convolution of ◦ and ◦
Page List
6
7
9
11
5
16
16
6
B
c
D
δ
∆
G
g
H
h
J
j
K
k
κ
L
l
LB
M
N
ν
ω
P
p
p
q
q
Bandwidth
Propagation speed (of light)
Distortion
Dirac’s delta function
Time difference of arrival
Sensor array matrix
Sensor position matrix
System function
Impulse response
The Fisher information matrix (FIM)
Imaginary unit
Number of components chosen
Frequency index
Noise coefficient
Number of blocks
Block index
Block length
Number of receivers
Number of samples
Noise
Angular frequency
Signal power
Receiver
Receiver position vector
Transmitter
Transmitter position vector
2
5
33
6
5
11
11
6
6
11
6
25
9
14
9
9
9, 40
2
11
5
6
13
6
5
6
5
xiii
xiv
Symbols
Notation
Rx
R
r(τ )
s
σ
σ2
S(ω)
T
t
θ
u
w
x
χ
X(l, ω)
Description
The auto-correlation matrix of x
Data rate, bits per sample
Cross correlation function
Transmitted signal
Standard deviation
Variance
Cross spectral density function
Measured time
Time
Phase offset
Position vector
Hamming Window
Received signal
x arranged in blocks
STFT of received signal
Page List
20
29, 33, 53
5
5
11, 29
10
6
11
5
16, 71
9
9
5
20
9
Chapter 1
Introduction
In electronic warfare, one objective is localization of transmitters. One way of doing this is by sampling the signal with three or more receivers at different positions
and estimating the time difference of arrival (TDOA)1 of the signal. By doing this
it is possible to triangulate the location of the transmitter. A common system
setup used for localization is illustrated in figure 1.1 with three receivers and one
transmitter.
Receiver
Transmitter
Receiver
Receiver
Base node
Figure 1.1: A common system setup used for localization in electronic warfare
scenarios. Each receiver retransmits the signal to the base node for correlation.
1A
list of abbreviations can be found on page xi.
1
2
Introduction
Media
Radio link
Passive laser
Active laser
Electrical cable
Optical cable
Transmission rate
∼ 1 kbit/s – ∼ 50 Mbit/s
∼ 50 Mbit/s
∼ 1 Gbit/s
∼ 1 Gbit/s
∼ 100 Gbit/s
Table 1.1: Transmission rates in different media used in localization systems [16,
chapter 4].
To estimate the TDOA, data must be gathered and correlated from all receivers.
One problem is that the channel capacity between the receivers is limited, therefore
one may wish to reduce the amount of data needed to be sent for correlation.
Given M receivers and a signal bandwidth of B Hz the bandwidth of the
channel needs to be M · B Hz. A simple example can be calculated for 10 MHz
instantaneous bandwidth, this gives a receiving data rate of 107 · 8 · 2 = 160 Mbit
per second given an 8 bit word length and sampling done at the Nyquist frequency.
Table 1.1 gives the transmission rates of different connections used in localization
systems.
This thesis investigates the possibility of data reduction in the application of
estimating TDOA. If data reduction is possible the time it takes to triangulate the
transmitter position can be reduced, thus more transmitters can be localized and
the efficiency is increased.
1.1
Scope
The nodes in figure 1.1 on the previous page can have different capability levels, everything from just modulating the signal to another frequency and re-transmitting
it to time stamping and processing the data. The latter case is assumed, hence the
time needed for compression and transmission to the base node will not introduce
a relative delay between the nodes.
This thesis is about compression and localization, not detection and/or demodulation. Therefore, signals will be treated as baseband signals. Implementation
and evaluation have been done in Matlab, even though some actual signal transmissions have been sampled, this thesis has an exploratory focus rather than an
implementation focus.
Furthermore, the types of signals investigated have been limited to a few common types, see section 3.3 on page 14 for discussion and signal descriptions. The
signals are assumed to be transmitted during the entire sampling time.
Earlier work on the subject includes the works of Mark Fowler among others,
but the papers found are either very general (e.g. [10], saying compression can
be applied to localization systems) or very signal specific (e.g. [12], which looks
at a certain type of radar signals). This thesis aim to investigate rather straightforward approaches to data compression on common types of signals, and see if
they are beneficial.
1.2 Method
1.2
3
Method
To evaluate the different compression schemes utilized in this thesis a simulation
model is used. The localization performance using compressed signals is compared
against the performance using uncompressed signals. Each simulation is performed
100 times to get a reliable measure of the standard deviation. The simulation test
bench is written in Matlab.
The simulation uses three different signal types, see section 3.3 on page 14.
The simulations are also compared against the theoretical limit of the localization
performance, see section 2.4 on page 10.
To evaluate how the compression schemes works on recorded data using a
real channel, a field experiment was conducted. Two vehicles were used, one
equipped with a transmission system and one equipped with a receiving system.
The recorded data was used in a modified version of the simulation test bench for
evaluation.
1.3
Thesis Disposition
Chapter 1 is this chapter, and aims to briefly explain what this thesis is about,
the problem it investigates and why the problem is of interest.
Chapter 2 describes the theory behind TDOA and how it can be used for localization. The localization system used in this thesis is presented and finally
a theoretical lower bound for the accuracy of the estimated transmitter position is discussed.
Chapter 3 characterizes the signal types investigated and defines how signal to
noise ratio is calculated.
Chapter 4 deals with compression techniques, the transforms and quantization
methods used in this thesis. It presents the transform coding model that the
thesis rely on. The chapter also depicts decorrelation and compaction. The
Cramér-Rao bound is also extended with the distortion from compression.
Chapter 5 shows simulation results and how they where obtained. It presents ratio calculations for the different compression schemes and localization ability
adjusted ratio.
Chapter 6 describes a field experiment performed where signals where transmitted and received. It presents how this experiment was conducted, the
problems that occurred, and the localization result using the collected data.
Chapter 7 draws conclusions and looks forward.
Chapter 2
Overview of TDOA
2.1
Introduction
Signal localization can be done in a wide variety of ways [2, 3], but this thesis
will focus exclusively on time difference of arrival (TDOA)1 . This method relies on
the finite propagation speed of the measured signals, resulting in different time of
arrival at differently located receivers.
Given two sensors and one transmitted signal, the received signals can be
modeled as2 [17]
x1 (t) = s(t) + ν1 (t) and x2 (t) = s(t + ∆) + ν2 (t),
(2.1)
with the noise ν1 (t) and ν2 (t), uncorrelated to s(t), and the time delay ∆.
From two real-valued signals, the cross correlation function can be calculated
as the expected value
r1 2 (τ ) = E[x1 (t)x2 (t + τ )]
(2.2)
ˆ as the τ that maximizes r1 2 (τ ). Given the
to find the time delay estimator ∆
ˆ the position of the
geometry for the sensor pair together with the estimated ∆,
transmitter is known to be somewhere along a hyperbola [3] with equation
ˆ
|p1 − q| − |p2 − q| = ∆c,
(2.3)
where p1 and p2 are the receiver positions, q is the sought transmitter position
and c is the propagation speed. This is illustrated in figure 2.1 on the next page,
ˆ
where each hyperbola corresponds to an estimated ∆.
2.2
Localization Using TDOA
In general, localization is a difficult problem. Therefore, some simpler models are
needed, which provide useful tools to investigate localization properties. In free
1 Sometimes
2 Without
referred to as time delay of arrival.
regard to attenuation.
5
6
Overview of TDOA
Figure 2.1: Two receivers, each hyperbola corresponds to an estimated time delay
ˆ
∆.
space propagation without attenuation, which is the only propagation model used
in this thesis, the channel can be modeled as a function of time delay. Attenuation
can be thought of as a lowered signal to noise ratio (SNR) level, and is therefore
not necessary for the setup described in section 2.3. Between a signal transmitter
q and a receiver p, the channel impulse response becomes [15, part 4]
kp − qk
hq p (t) = δ t −
= δ (t − ∆)
(2.4)
c
in the time domain and the channel system function
−jωkp − qk
Hq p (ω) = exp
= exp (−jω∆)
c
(2.5)
in the Fourier domain. Here, p and q are the position vectors of p and q, respectively, δ(t) is the Dirac’s delta function, and c is the propagation speed. In this
case, c is the speed of light. The measured signal at p can then be modeled as
x(t) = s(t) ∗ hq p (t) + νp (t),
(2.6)
where s(t) is the transmitted signal and νp (t) is the noise3 .
Given two received signals xm and xn from a sensor pair pm and pn , the cross
spectral density function between the signals can be formulated as
Sm n (ω) = F {rm n (τ )} = F {E[xm (t)xn (t + τ )]} ,
(2.7)
where rm n (τ ) is the cross-correlation function between xm and xn . The reasons
to take the Fourier transform are practical. One might wish to do Fourier domain
signal processing before sending the signal to the localization system, and the
3∗
denotes convolution, see symbol list on page xiii.
2.3 Experiment Model
7
computation complexity of a cross-spectral density is significantly less than that
of a convolution.
Using (2.6) and (2.7) on the facing page together with the cross spectral density
output of a linear system4 it can be rewritten as
Sm n (ω) = F {E[s(t)s(t + τ )] ∗ hm (t) ∗ hn (−t) + E[νm (t)νn (t + τ )]} .
(2.8)
The Fourier transform is then applied and the result is
Sm n (ω) = Ss (ω)Hm (ω)Hn∗ (ω) + Sνm νn (ω).
(2.9)
Then, using (2.5) on the preceding page,
Sm n (ω) = Ss (ω)e−jω∆m ejω∆n + Sνm νn (ω)
= Ss (ω)e−jω(∆m −∆n ) + Sνm νn (ω)
= Ss (ω)ejω∆m,n (q) + Sνm νn (ω)
(2.10)
where ∆m,n (q) is the TDOA, for the signal s(t) from transmitter q, between pm
and pn .
It is not sufficient using only two receivers to locate a signal source on a two
dimensional plane using TDOA. Figure 2.2 shows a linear array setup with three
receivers. Linear arrays setups are seldom used in reality, in part because there
is a false mirror interception point, and the accuracy along an imagined y-axis is
rather poor.
Figure 2.2: Three receivers locating one transmitter using TDOA.
2.3
Experiment Model
Matlab is used to establish a test bench for verification. The setup, a rather
simple symmetrical case, is shown in figure 2.4 on the next page, with receivers p1 ,
4 See
e.g. [18, page 68].
8
Overview of TDOA
h1
s
h2
h3
ν1
+
STFT
ν2
+
STFT
ν3
+
STFT
XC
Sx
q̂
SRP
Figure 2.3: Localization system using TDOA and cross correlation (XC).
p2 and p3 and the transmitter q. All receivers are equidistant from the transmitter and distributed evenly around the transmitter, forming a circle. The reason
for such a symmetric setup is to introduce symmetry of the error distribution,
simplifying the Cramér-Rao bound (CRB) (see section 2.4 on page 10). TDOA
localization works well in an asymmetrical setup as well, as long as the problem
illustrated in figure 2.2 on the previous page, that the localization ability becomes
one-dimensional, do not arise. This experimental model will not consider multipath propagation.
√
The distance from the transmitter to the receivers is 3 · 10 km. This distance is arbritrary, but a sensor array small enough to avoid multiple solutions is
desirable, a localization scheme looking at the phase data will have a hard time
differentiating between solutions separated by a whole wavelength. A signal bandwidth much smaller than the propagation speed c divided by the largest distance
in the receiver array, is desirable [6]. The minimum distance is limited by the time
synchronization and self localization ability of the system.
Figure 2.4: Experiment setup displaying the three receivers as circles and the
transmitter
as a triangle. The distance between the transmitter and receivers are
√
3 · 10 km and the receivers are evenly distributed around the transmitter forming
a circle.
The transmitted signal s is filtered with the channels impulse response described in (2.4) on page 6. When the signal has passed the channel, uncorrelated
2.3 Experiment Model
9
noise is added independently to each signal, amplitude scaled to get the desired
SNR (see section 3.1 on page 13).
To estimate the cross-spectral density the signal is first transformed using shorttime Fourier transform (STFT) described below
∞
X
STFT {x(n)} = X(l, ω) =
x(n − LB l)w(n)e−jωn .
(2.11)
n=−∞
The STFT divides the signal into L blocks of length LB with the Hamming window
w,
2πn
,
(2.12)
w(n) = 0.54 − 0.46 cos
LB − 1
and performs the discrete Fourier transform of every block. Each block overlaps
the preceding one with 50%, this is ideal when using the Hamming window since
the sum of all windowed blocks will produce the original signal.
The cross-spectral density between the signals is then estimated for every sensor
pair according to
L
1X
Xm (l, ωk )Xn∗ (l, ωk ).
(2.13)
Sm,n (ωk ) =
L
l=1
where the indexes m and n denotes the received signal at pn and pm , respectively.
X ∗ denotes the conjugate transpose5 of X. ωk is the angular frequency with
index k.
The last step is to locate the position of the transmitter from the estimated
cross-spectral density. This is done by using the steered response power (SRP).
Different approaches can be taken here, see e.g. [15, part 4] for alternate methods.
Steered response power (SRP) is a well used and robust method for localization
[8] and suitable to be used in this experiment. Given the estimated cross-spectral
density the SRP at the position u is in this case
P (ωk , u) =
M
M X
X
Sm,n (ωk )e−jωk ∆m,n (u) .
(2.14)
m=1 n=1
∆m,n (u) is the TDOA for transmitting a signal from position u to receivers pm and
pn . By using (2.10) on page 7 and an ideal noise free scenario the SRP is
P (ωk , u) =
3 X
3
X
Ss ejωk ∆m,n (q) e−jωk ∆m,n (u)
m=1 n=1
=
3 X
3
X
Ss ejωk (∆m,n (q)−∆m,n (u)) .
(2.15)
m=1 n=1
It can be shown that in an ideal noise free scenario max P (ωk , u) can be found at,
u
and only at, u = q [15]. Figure 2.5 on the next page shows the amplitude scaled
5 Sometimes
called the Hermitian transpose.
10
Overview of TDOA
magnitude of the frequency average SRP in the experiment scenario, i.e.
X
P (ωk , u) .
(2.16)
k
The peak is found at the position of the transmitter, where
u = q =⇒ ejωk (∆m,n (q)−∆m,n (u)) = 1.
(2.17)
0.1
0.3
0.5
0.7
0.9
0.5 0.7
0.1
0.3
0.3
0.1
Figure 2.5: The magnitude of a SRP scaled between 0.0 and 1.0 illustrated with
contour lines. The receivers are illustrated as circles and the transmitter as a
triangle.
In section 2.4, it will be shown that there is a fundamental limit to how good an
estimator can be, called the Cramér-Rao bound (CRB). This bound is used in this
thesis to determine the number of time samples needed for a reasonable precision
of flat spectrum signal localization (see section 3.3.1 on page 15).
2.4
The Cramér-Rao Bound
There exists a theoretical lower bound for the variance of the error in the estimated
location of a transmitter [17]. When estimating the position from a finite recorded
2.4 The Cramér-Rao Bound
11
data set the variance of the estimation error is dependent on a matrix J, which
is called the Fisher information matrix (FIM)6 [16]. This matrix can be used
to set a theoretical lower bound for the accuracy of a localization system. The
inverse of FIM is called the Cramér-Rao matrix bound (CRMB) and can be used to
geometrically describe the variance of the localization error in a system. The FIM,
with respect to TDOA, for a flat spectrum baseband signal is [16, chapter 11]
J=
(SNR)2
1
T B3
· GGT ,
·
·
·
2π 6 1 + M · (SNR) c2
(2.18)
where B is the bandwidth of the signal, SNR is the signal-to-noise gain ratio, and
GGT is a matrix dependent on the sensor array. If N is the number of signal
samples, the measured time becomes
T =
N 2π
B
so J can be also be expressed as
J=
N B2
(SNR)2
1
·
·
· GGT ,
6
1 + M · (SNR) c2
(2.19)
For the propagation model used in this thesis GGT is
GGT = g(M I − 1)gT ,
(2.20)
where I is the identity matrix, 1 is a matrix of ones and the matrix g is a function
of the positions of the transmitter q and receivers pm . In the simulation scenario
with 3 receivers pm and one transmitter q the matrix g, with the direction to each
transmitter along its rows, is
q − p1
q − p2
q − p3
g=
.
(2.21)
||q − p1 || ||q − p2 || ||q − p3 ||
Note that this matrix does not contain any information about the distances, only
the direction to receivers from the transmitter.
The Cramér-Rao matrix bound (CRMB) follows as
CRMB
= J−1 .
(2.22)
As mentioned, the CRMB geometrically describes the variance in the localization
error. In the symmetric simulation scenario described in section 5.1 on page 37,
this matrix will not only be diagonal, but have the same value in all positions
along the diagonal. Hence, it is useful to speak of the Cramér-Rao bound (CRB),
referring to one of these identical elements. The standard deviation, the square
root of the variance, can be expressed in meters, and is therefore the most useful
measurement to relate evaluation results to.
6 The Fisher information matrix (FIM) is a general measure of variance of the score of the
given estimator. For more details, see [7, chapter 11].
12
Overview of TDOA
The CRB is only a theoretical bound that rarely is reached in real life scenarios,
however it tells a lot about the trend of the localization error. One can note that
the error variance will be proportional to the reciprocal of T , B 3 , and have a
proportional trend for SNR (in gain). In this thesis the CRB is used together
with the calculated standard deviations from simulation to compare the trends.
Figure 2.6 shows the lower standard deviation bound for a 1 kHz signal using
different measured times, T .
1 kHz, 1s
105
1 kHz, 4s
1 kHz, 16s
σ [m]
104
1 kHz, 64s
103
102
101
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 2.6: CRB for FSS, 1 kHz signal bandwidth at different signal lengths using
SNR levels from −10 to 30 dB.
Chapter 3
Signals
The signals used for compression in this thesis are baseband signals, i.e. have been
demodulated before entering the system. This assumes that there is some kind
of intelligence prior to the localization, e.g. an operator or an automatic system
that can identify the presence of a signal. The aim of this thesis is localization,
not detection. The channel is assumed to be a Gaussian channel [7, chapter 9], as
depicted in figure 3.1
ν
s
+
x
Figure 3.1: Gaussian channel.
3.1
Signal to Noise Ratio
Signals with different signal to noise ratio (SNR) is generated by adding white
Gaussian noise (WGN) to the original signal. WGN is generated and the power is
estimated as
N
1 X
|x(n)|2
(3.1)
Px =
N n
for both the original signal s and and the noise ν. The noise is then multiplied
with an appropriate coefficient κ and added to the signal to get the desired SNR.
The SNR is defined as
Ps
(3.2)
SNR = 10 log10
Pν
13
14
Signals
and the coefficient, κ, is calculated using
r
Ps
κ=
· 10−SNR/10 .
Pν
(3.3)
The SNR range used is between −10 dB and 30 dB, this approximate range is
commonly used when illustrating the performance of TDOA1 .
3.2
Signal and Noise Spectrum
Throughout this thesis, the noise will have more or less the same energy distribution, WGN, but the signals will not always have a distribution that resembles
the noise. This is important to take into consideration when interpreting signal
bandwidth and SNR. When considering a single node receiving a single side band
(SSB) signal with 0 dB SNR, the signal might be easy to distinguish from the noise,
while a flat spectrum signal (FSS) will not.
3.3
Signals of Interest
There is a practically unlimited numbers of signals that can be sent, so there is
a need to limit the range of signals to study. First, there is the limit mentioned
above, that only baseband signals are considered. Secondly, the study is limited
to the following signal types, more on that in the appropriate sections.
• Flat spectrum signals (FSSs), see section 3.3.1 on the next page. This is a
very general signal model.
• Phase-shift keying (PSK) signals, see section 3.3.2 on page 16. Commonly
used by data modems.
• Single side band (SSB) signals, see section 3.3.3 on page 17. Used by some
walkie-talkie systems.
Furthermore, narrow bandwidth signals are more interesting because they require
more data to be used when estimating TDOA (this follows from the CRB, see
section 2.4 on page 10). Table 3.1 on the next page shows the number of samples
needed for signals at different bandwidth and different SNR. It can be seen that
for higher bandwidth signals the number of samples needed is small, therefore no
compression is needed.
1 See
e.g. [9], [11], or [15].
3.3 Signals of Interest
SNR
-10
0
10
20
30
dB
dB
dB
dB
dB
2 kHz
9.9 · 106
3.0 · 105
2.4 · 104
2.3 · 103
230
15
10 kHz
3.9 · 105
1.2 · 104
940
91
< 10
20 kHz
9.9 · 104
3.0 · 103
240
23
< 10
30 kHz
4.4 · 104
1.3 · 103
100
10
< 10
300 kHz
440
13
10
< 10
< 10
Table 3.1: The number of samples needed for flat spectrum signals at different
bandwidths and SNR levels for 100 meter standard deviation CRB. Note that data
compression on wide bandwidth signals with high SNR is not necessary due to the
small amount of data needed.
3.3.1
Flat Spectrum Signals
A signal resembling white Gaussian noise (WGN) is the most general signal model
there is. Many communication channels can be modeled as Gaussian channels,
and it can be shown that the most efficient use (achieving channel capacity) of
such a channel is to send signals with a Gaussian distribution [7, chapter 9].
f
(a) White Gaussian noise
f
(b) Bandwidth limited flat spectrum signal
Figure 3.2: Frequency energy distributions for WGN and bandwidth limited FSS.
The FSSs used in this thesis are complex-valued and bandwidth limited, generated by Matlab’s pseudo-random randn by generating the real valued and
complex valued parts separately.
16
3.3.2
Signals
Phase-Shift Keying
A modulated bandpass signal can be modeled as [5]
x(t) = Ac [xi (t) cos(ωc t + θ) − xq (t) sin(ωc t + θ)]
(3.4)
with the coded message contained in the in-phase component xi (t) and the quadrature component xq (t).
Phase-shift keying (PSK) is a way to modulate digital signals by placing a
number of code words around the unit circle in the in-phase/quadrature plane.
8-PSK and 4-PSK are common signal types used in simple data modems, where 4
and 8 is the number of code words used. Figure 3.3 shows the signal constellation
for such signals.
Q
Q
2
1
1
0
3
0
I
I
4
2
7
3
5
(a) 4-PSK (Q-PSK)
6
(b) 8-PSK
Figure 3.3: Examples of phase-shift keying signal constellations in the inphase/quadrature plane.
3.3 Signals of Interest
3.3.3
17
Amplitude Modulated Signal Side Band
An AM-SSB is constructed from an amplitude modulation (AM) signal, suppressing
its carrier wave and one of the sidebands. This is an energy and bandwidth efficient
use of AM, but is harder to properly synchronize on the receiver side [5, chapter
4.5]. This type of signal is commonly used by radio amateurs and for long distance
voice radio transmission in some systems.
A simple baseband representation of a single side band (SSB)-like signal with
bandwidth B can be constructed by “demodulating” an ordinary wave audio file by
B/2, run through a low-pass filter with ±B/2 cut-off frequencies and resample it.
This is illustrated in figure 3.4. Note that the resulting signal is complex valued.
The SSB signals used in this thesis are created in this manner from an audio book
source.
f
−B
−B/2
B/2
B
(a) Audio file (mixed sound channels)
f
−B
−B/2
B/2
B
(b) Demodulated signal
f
−B/2
B/2
(c) Low-pass filtered SSB ready for use
Figure 3.4: Schematic frequency distributions of stages in single side band signal
creation.
Chapter 4
Compression of signal data
To estimate the location of a transmitter each sampled signal needs to be transferred to a shared resource where the signals can be correlated. The capacity of
the channel between the receivers is limited [16], see table 1.1 on page 2.
A compression algorithm is sought that keep data relevant for TDOA and discard irrelevant data. If such an algorithm is found it is possible to reduce the
amount of data needed to be sent from the receiving nodes and this gives the possibility to estimate the position of more transmitters for the given communication
resources.
4.1
Transform Coding
Transform coding is a technique in which the data is transformed into components.
These components can then be sent over the given channel; the receiver can then
transform the components back to its original base and thus recreate the signal.
Since Hotelling’s paper in 1933 [13], the first to describe principal component
analysis, different transforms have been used to better capture the characteristics
of a given signal.
By choosing the used transform base carefully it is possible to achieve results
where most of the signal information is compacted into only a few components.
These components are then sent over the link, but the components with little or
no information are discarded. A good estimate of the signal can then be recreated
at the receiver. Transform coding can also achieve decorrelation, so that the
redundancy between data points is reduced. Reducing redundancy might be a
problem if the communication from the nodes to the base system suffers data loss.
This is probably better handled by the use of error correcting codes rather than
skipping compression, but is outside the scope of this thesis.
Figure 4.1 on the next page describes a system using transform coding for data
compression. The signal x(t) is transformed and then quantized in a way that
preserves the characteristics of the signal most interesting to the application, e.g.
signal variance, signal energy, or some other measure of perceived quality. The
quantized signal is then entropy coded, with e.g. Huffman coding, and sent over
19
20
Compression of signal data
x(t)
transform
X
quantization
X̃
entropy encoding
Y
channel
x̂(t)
inverse transform
dequantization
entropy decoding
Y
Figure 4.1: Transform coding compression.
the channel. The signal is then decoded, dequantized and re-transformed to its
original base to get the estimate x̂(t).
4.1.1
Illustration of Decorrelation
The auto-correlation matrix of a sampled signal x is defined as
Rx = E[xx∗ ].
(4.1)
A nice way to illustrate transform decorrelation efficiency is by looking at the
auto-correlation matrix Rx for the signal at hand.
Algorithm 4.1: Transformation of Rx
1. Generate signal x
2. Rearrange x into blocks of length LB , placing them as rows of χ
3. Calculate the auto-correlation matrix Rχ = E[χχ∗ ]
4. If desired, change basis using the transform T , Ry = T Rχ T −1
5. Normalize and plot Ry
Figures 4.2 to 4.5 on pages 23–24 serves as examples of such an illustration, using
a SSB speech signal. Decorrelation shows up as diagonally dominant matrices,
complete decorrelation is a diagonal Ry . The Karhunen-Loéve transform (KLT)
achieves this [1], see figure 4.3 on page 23. In this figure we also have high energy
compaction to few components, indicating a compression-friendly signal.
4.1.2
Karhunen-Loéve Transform
Principal component analysis (PCA) of a signal is performed by looking at the
eigenvalue decomposition of the auto-correlation matrix. This captures many characteristics of the given signal and effectively ranking them, the transform puts the
most energy in the fewest components [19, chapter 3]. Then, we can reduce the
amount of data by only transmitting the eigenvectors belonging to the largest
eigenvalues, and one symbol for each such component per signal block.
4.1 Transform Coding
21
The Karhunen-Loéve transform (KLT), with the eigenvectors of the auto-correlation matrix along its columns, is an example of a PCA transform.
This transform provides the largest transform coding gain of any transform
method. It minimizes the mean squared error for the given number of components,
but has a large overhead [20, chapter 13.4], since each used eigenvector has to be
transmitted. This overhead can remove the advantages of this optimal transform
and make it impractical to use for compression. It is still interesting from a
theoretical point of view but other transform methods may be needed that do not
depend on the data being sent.
4.1.3
Discrete Cosine Transform
The discrete cosine transform (DCT) type 21 transforms x(k) to X(m) as
X(m) =
N
−1
X
x(k) cos
k=0
1
π
m k+
N
2
(4.2)
where N is the block length2 , with an inverse transform
x(k) =
n−1
π
1
X(0) X
X(m) cos
+
m k+
.
2
N
2
m=1
(4.3)
The big advantage of the DCT compared to the KLT (see section 4.1.2) is that
the transform matrix is not signal dependent, and therefore does not need to be
transmitted. Asymptotically, if the signal is Markovian, the DCT is equivalent
to KLT as the correlation coefficient ρ → 1 (or ρ → 0, since the auto-correlation
matrix is then diagonal). Similarly, as the matrix size N → ∞ (and thus the block
length), the DCT is equivalent to the KLT. [19, chapter 3.3–4]
4.1.4
Discrete Fourier Transform
The DCT, described in 4.1.3, is a special case of the discrete Fourier transform
(DFT) for real-valued functions. The DFT is defined as
X(k) =
N
−1
X
x(n) exp
n=0
and its inverse as
x(n) =
−2πjnk
N
N −1
1 X
2πjnk
X(k) exp
.
N
N
(4.4)
(4.5)
k=0
The short-time Fourier transform (STFT) is a time limited DFT, essentially
performing the transform at the signal one block at a time, see section 2.3. This
1 Only
the DCT type 2 is considered. For different types of DCT, see [19].
that N indicates the number of samples transformed rather than the block length, but
since a transform is often applied to one block at a time, the number of samples N is the same
as the block length, from the transform’s point of view.
2 Note
22
Compression of signal data
scheme is used in the localization system so some synergy can be found here.
Instead of applying the inverse transform before localization, we can skip this
step. Although of little importance for localization properties in theory, it helps
to rid simulations of block effects (see section 5.1.3 on page 38) and will save
computation time.
4.1 Transform Coding
23
position within block
250
200
150
100
50
50
100
150
200
250
position within block
Figure 4.2: Illustration of the magnitude of the auto-correlation matrix Rχ of
a SSB signal.
See algorithm 4.1 on page 20. Darker color illustrates higher
magnitude.
position within block
250
200
150
100
50
50
100
150
200
250
position within block
Figure 4.3: Illustration of the magnitude of the transformed auto-correlation matrix Rχ using Karhunen-Loéve transform of a SSB signal. Darker color illustrates
higher magnitude. The components are arranged by eigenvalue magnitude, all
energy is compacted in the lower left corner. Since there is no energy outside the
diagonal, complete decorrelation is achieved.
24
Compression of signal data
position within block
250
200
150
100
50
50
100
150
200
250
position within block
Figure 4.4: Illustration of the magnitude of the transformed auto-correlation matrix Rχ using Discrete cosine transform of a SSB signal. Darker color illustrates
higher magnitude. The components are arranged by frequency, from low to high.
position within block
250
200
150
100
50
50
100
150
200
250
position within block
Figure 4.5: Illustration of the magnitude of the transformed auto-correlation matrix Rχ using Discrete Fourier transform of a SSB signal. Darker color illustrates
higher magnitude. The components are arranged by frequency, from negative to
positive.
4.2 Quantization
4.2
25
Quantization
When an appropriate transform method has been established the data needs to
be quantized, i.e. truncated to a number of data levels, see figure 4.1 on page 20.
First, 8 bit values are used (see section 5.1.2 on page 38). Then some method
for selecting which data to keep is applied. Both these steps are referred to as
quantization, and is where most of the actual compression occurs. The transform
step will only point out which data to cut in the quantization step.
In this thesis three different approaches are used for quantization; integral
components (section 4.2.1), keeping partial components (section 4.2.2) and timefrequency masking (section 4.2.3). This section aims to describe these approaches.
For example compression ratio calculations, see section 5.1.5 on page 40.
4.2.1
Integral components
First off is the integral components method. The approach for this quantization is
to throw away transform components insignificant to TDOA localization. Components with high energy, containing phase information relevant for estimating the
TDOA, are transmitted uncompressed. Which components to select turns out to
be non-trivial in a noisy environment, as illustrated in section 4.2.1.
The quantization is done by sorting the transform components by the average
energy over all transform blocks and discard components based on the desired
compression ratio.
Since the method decides which components to keep based on the average signal
energy in all blocks it is possible that insignificant samples are kept e.g. during
silent time periods in a speech signal.
The Component Selection Problem
If all three receivers select transform components by signal energy, independently
from each other, they might not select the same components in a noisy environment. This can be mitigated by a protocol allowing the nodes to agree on a set of
transform components to use. Such a protocol is assumed not to be allowed here.
Figure 4.6 on the next page is the result of a simulation with 1000 iterations as
described in algorithm 4.2. The noiseless curve (∞ dB SNR) is not simulated, but
added for clarity.
Algorithm 4.2: DFT component selection on FSS
1. Generate a flat spectrum signal (see 3.3.1 on page 15).
2. Create three received signals by adding uncorrelated noise to get the desired
SNR.
3. Run signal through DFT with block length 128.
4. Choose K components of each transformed signal, according to signal energy.
Compression of signal data
Components found in all three signals
26
120
-10 dB
-6 dB
-2 dB
2 dB
6 dB
10 dB
14 dB
18 dB
22 dB
26 dB
30 dB
∞ dB
100
80
60
40
20
0
0
20
40
60
80
100
120
Components searched
Components found in all three signals
Figure 4.6: Average number of components chosen in three flat spectrum signal
per dB SNR.
20
-10 dB
-6 dB
-2 dB
2 dB
6 dB
10 dB
14 dB
18 dB
22 dB
26 dB
30 dB
∞ dB
15
10
5
0
0
5
10
15
20
Components searched
Figure 4.7: Average number of components chosen in three flat spectrum signal
per dB SNR. Zoomed in at few selected components.
4.2 Quantization
27
5. Find the number of components chosen for all three signals.
As seen in figure 4.7 on the facing page, selecting components by signal energy
independently is not a good approach for WGN-like signals. With a reasonable
amount of components to acquire an adequate compression rate, say 10, we have
very few common components3 for a SNR below 10 dB. Thus, this method seems
useless for flat spectrum signals in white Gaussian noise, which was also confirmed
by simulations (see section 5.2.1 on page 43).
For other types of signals, that has a distribution that differs more from the
background noise, similar methods provide much better results. Figure 4.8 on the
following page is the result of a simulation with 1000 iterations as described in
algorithm 4.3, similar to algorithm 4.2 on page 25 with the notable difference in
the first step.
Algorithm 4.3: DFT component selection on SSB signal
1. Generate a SSB speech signal (see section 3.3.3 on page 17).
2. Continued as algorithm 4.2 on page 25.
As can be seen clearly in figure 4.9 on the next page, selecting components by signal
energy can be done independently in each node with a high degree of certainty that
we will choose the same components in all three nodes, for a reasonable number
of components.
This implies that for signals that differs in distribution from the noise, this is
less of a problem. Similar uncertainty can be found for other quantization schemes,
but is most obvious when keeping or discarding integral components.
3 An interesting note is that the standard deviation is quite small, never above 5 components
for the simulation described.
Compression of signal data
Components found in all three signals
28
120
-10 dB
-6 dB
-2 dB
2 dB
6 dB
10 dB
14 dB
18 dB
22 dB
26 dB
30 dB
∞ dB
100
80
60
40
20
0
0
20
40
60
80
100
120
Components searched
Components found in all three signals
Figure 4.8: Average number of components chosen in three SSB signals per dB
SNR.
20
-10 dB
-6 dB
-2 dB
2 dB
6 dB
10 dB
14 dB
18 dB
22 dB
26 dB
30 dB
∞ dB
15
10
5
0
0
5
10
15
20
Components searched
Figure 4.9: Average number of components chosen in three SSB signals per dB
SNR. Zoomed in at few selected components.
4.2 Quantization
4.2.2
29
Partial Components
It is possible to assign different number of bits to different components in the
transform domain when performing quantization. One approach is to give the
components with higher variance more bits than components with little or no
variance. Assuming that components with higher variance contains more relevant information than components with less variance, this method keeps desired
properties of the signal.
Equation (4.6) optimally assigns bits and minimizes the reconstruction error
of the signal [20, chapter 13.5].
Rk = R +
σ2
1
log2 QLb k
2 1/LB
2
i=1 (σi )
(4.6)
where σk2 is the variance for component k. Rk is the bits assigned to each component k, R is the average number of bits available for assignment and LB is the
total number of components. However, Rk are not guaranteed to be neither integers or positive. Another approach is to use a recursive algorithm 4.4. In this
thesis all signals are assumed to be zero mean signals, hence power and variance
are interchangeable. The algorithm therefore assign bits based on power instead
of variance.
Algorithm 4.4: Zonal sampling, signal power
1. Compute Pk for each component.
2. Set Rk = 0 for all k and set Rb = LB R where Rb is the total number of bits
available for distribution.
3. Sort the energy Pk . Suppose P1 is maximum.
4. If R1 is less than 8 increment it by 1, and divide P1 by 2. If not, proceed
with the next highest energy component until a k such that Rk < 8 is found.
5. Decrement Rb by 1. If Rb = 0 then stop; otherwise go to 3.
This bit allocation scheme is called zonal sampling [20, chapter 13.5]. One
drawback of this method is that it assigns bits based on average values, therefore
it is possible that samples with insignificant information will be assigned bits. E.g.
a speech signal will be assigned bits to transform components with high average
variance even within a time period of silence.
When bits have been assigned quantization is performed. If Rk bits are assigned
the number of levels the bits can represent becomes 2Rk . The phase is quantized
using uniformed quantization, an example of the output of the phase quantization
can be found in figure 4.10 on the following page. The figure shows quantization
with 4 bits, the number of output levels then becomes 24 = 16.
The amplitude data is handled in another way. It is scaled between 0 and
2RMAX − 1, where RMAX is the largest number of bits assigned to a component.
30
Compression of signal data
π
qunatized
non-quantized
Output
π
3
− π3
−π
−π
π
3
− π3
π
Input
Figure 4.10: Phase input and output of a quantifier using 4 bits. The output is
fitted to 24 = 16 levels between −π and π.
Each sample is then rounded to an integer and then set to the minimum of the
rounded value and 2Rk − 1, where Rk is the bits assigned to the samples corresponding transform component. This assures that any symbol corresponding to
a component can be described by the assigned bits. The output of the quantifier
can be found in figure 4.11 on the next page. If the amplitude data were to be
subjected to the same quantization scheme as the phase data, components with
low average energy might be weighted unfairly high in some blocks, distorting the
cross correlation.
4.2.3
Compression Using Time-Frequency Masking
For some non-noise like signals the energy is not distributed evenly in time and
frequency, islands of energy appear. This can be seen in figure 4.12 on page 32,
which shows the energy distribution in the STFT of a SSB speech signal. In other
signals, e.g. a 4-PSK signal, the energy is evenly distributed in the frequency
domain over time. This is shown in figure 4.13 on page 32.
As described in section 2.3 on page 7, the localization system uses short-time
Fourier transform (STFT). The STFT uses a Hamming window to cut out chunks
of the signal. Each chunk overlaps the preceding one with 50%, these chunks is
4.3 Entropy Coding
31
R=8
250
Output
200
150
R=7
100
R=6
50
R=5
R=4
0
0
50
100
150
200
250
Input
Figure 4.11: Output of the amplitude quantifier where RMAX = 8. Each line
corresponds to the output for different assigned bits R.
then Fourier transformed (see (2.11) on page 9). This results in a representation
of the signal in both time and frequency.
By applying a mask to the STFT and cut out the parts with insignificant energy,
the number of samples needed to be sent for estimating TDOA can be greatly
reduced. This is referred to as time-frequency masking (TFM). The compression
ratio can be adjusted by varying the number of elements in the mask.
Compared to choosing the frequency components based on a time average of
the whole signal, e.g. zonal sampling in section 4.2.2 on page 29, this method
gives the possibility to cut out different frequency components in different parts
of the signal. This is a great advantage when localization is done on signals with
an energy distribution in frequency that vary over time, such as the speech signal
shown in figure 4.12 on the next page. However, in signals such as the 4-PSK in
figure 4.13 on the following page the advantage is not as great.
4.3
Entropy Coding
Entropy coding, such as Huffman coding, is a lossless compression scheme and gives
relatively little compression when compared to the lossy transform-quantization
32
Compression of signal data
1000
Frequency [Hz]
500
0
−500
−1000
0
30
60
90
120
150
180
Time [s]
Figure 4.12: Spectrogram of a SSB speech signal. White is low energy and black
is high energy.
1000
Frequency [Hz]
500
0
−500
−1000
0
30
60
90
120
150
180
Time [s]
Figure 4.13: Spectrogram of a 4-PSK signal. White is low energy and black is
high energy.
4.4 Distortion
33
scheme. Furthermore, since it is lossless, it has no impact on localization performance, which makes it less interesting to simulate and measure.
An otherwise complete compression scheme for localization can be polished by
entropy coding, but this thesis will not give entropy coding a thorough rundown.
4.4
Distortion
This section adds the transform quantization noise to the SNR, and aims to provide
a CRB for an optimal transform. The squared-error distortion,
D = E[(x − x̂)2 ]
(4.7)
is a common measurement of the error introduced by quantization4 . This is also
comparable to the signal power. For a zero mean Gaussian source with variance σ 2 ,
the rate distortion function, the given data rate R (in bits per sample) acheivable
for a given distortion D, is

σ2
1
log2
0 ≤ D ≤ σ2
(4.8)
R(D) = 2
D
0,
D > σ2
which mean that we can express the distortion in terms of the rate as
D(R) = σ 2 2−2R .
(4.9)
for an optimal quantization process5 . For proofs and further details, see [7, chapter
10]. Assuming optimal quantization, for KLT, the decorrelation-optimal transform,
the distortion is [14, chapter 12]
1/N
D = (det Rx )
· 2−2R .
(4.10)
the SNR introduced by the KLT at rate R is
10 log
Ps
Ps
= 10 log
1/N
D
(det Rx )
· 2−2R
(4.11)
The correlation matrix for x = s + ν,
Rx = E[(s + ν)(s + ν)∗ ]
= E[(s + ν)(s∗ + ν ∗ )]
= E[ss∗ + νs∗ + sν ∗ + νν ∗ ]
= E[ss∗ ] + E[νs∗ ] + E[sν ∗ ] + E[νν ∗ ]
.
.
s and ν are independent
= E[ss∗ ] + 0 + 0 + E[νν ∗ ]
= Rs + Rν .
4 This
5 See
(4.12)
distortion is not the only alternative. For more information, see [7, chapter 10].
equation (4.6) on page 29
34
Compression of signal data
Using a FSS signal scaled to let x have a gain SNR of σ 2 , the signal s has power
Ps = σ 2 . The auto-correlation for the signal is then Rs = σ 2 , and for noise
with power Eν = 1, the auto-correlation matrix is Rν = I. The expression then
becomes
Rx = (σ 2 + 1)I
(4.13)
ν
s
+
D
x
+
x̂
Figure 4.14: Gaussian channel with added quantization distortion.
Modelling the transfom as an additive guassian channel depicted in figure 4.14
[20, chapter 9], the added quantization noise is the distortion
1/N
· 2−2R
1/N −2R
= det((σ 2 + 1)I)
·2
1/N
= (σ 2 + 1)N
· 2−2R
D = (det Rx )
= (σ 2 + 1) · 2−2R
so the SNR for the transform coded signal is
σ2
Ps
= 10 log10
.
SNR = 10 log10
Pν + D
1 + (σ 2 + 1) · 2−2R
(4.14)
(4.15)
Reiterating the FIM, previously found as (2.19) on page 11,
J=
N B2
(SNR)2
1
·
· 2 · GGT ,
6
1 + M · (SNR) c
(4.16)
the new SNR gain can be inserted
2
σ2
1+(σ 2 +1)·2−2R
N B2
1
· 2 · GGT
·
J=
2
σ
6
c
1 + M · 1+(σ2 +1)·2
−2R
=
σ4
N B2
1
·
· 2 · GGT
2
2
−2R
2
2
−2R
6
(1 + (σ + 1) · 2
) + M σ (1 + (σ + 1) · 2
) c
(4.17)
Figure 4.15 on the facing page depicts the impact of diffrent data rates per SNR
level.
4.4 Distortion
35
R = ∞
σ, standard deviation of error [m]
105
R = 8
R = 4
R = 2
104
R = 1
103
102
101
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 4.15: Cramér-Rao bound for a 1 s, 1 kHz flat spectrum signal, using different
data rates R. Note that R = 8 and R = ∞ overlaps since 2−2·8 1 and have
little impact on localization performance.
Chapter 5
Evaulation by simulation
5.1
Experiment Setup
h1
s
h2
h3
ν1
+
STFT
ν2
+
STFT
ν3
+
STFT
XC
Sx
q̂
SRP
Figure 5.1: Localization system using TDOA and cross correlation.
The setup described in section 2.3 is used as a test bench for simulations to evaluate
the compression techniques’ impact on localization performance. The test bench
is implemented in Matlab.
Each signal is added with uncorrelated noise with different energy to test the
performance of compression at different levels of SNR, see section 3.1 on page 13.
The test is performed 100 times in order to determine the variance and mean of
the localization error. The data is presented together with the CRB for FSSs (see
section 2.4 on page 10).
Figures show the standard deviation of the error as a function of SNR. The
simulations setup gives an x and y symmetrical error due to the symmetrical node
setup, see figure 5.2 on the next page. Therefore, the figures in this chapter will
show the standard deviation for a combined x and y data set.
The mean error is not shown in the figures, it is low compared to the standard
deviation. Furthermore it will approach zero when the number of iterations is
increased.
37
38
Evaulation by simulation
Figure 5.2: Experiment setup displaying the three receivers as circles and the
transmitter
as a triangle. The distance between the transmitter and receivers are
√
3 · 10 km and the receivers are evenly distributed around the transmitter forming
a circle.
5.1.1
SRP Grid Granularity
To find the SRP peak described in equation 2.14 on page 9, we use a 100 × 100
grid, for positions u, as seen in figure 5.3 on the next page. This grid is constructed to provide a fine grained resolution around the actual transmitter, so
that the localization error due to the grid distance is proportional to the error.
For implementation purposes, both x and y axis symmetry is needed.
There are other methods for finding the maximum SRP value, see e.g.[8], but the
grid solution is easy to implement and good enough for this application. However,
the grid (along with the sensor setup), makes it hard to measure errors of large
magnitudes. The algorithm used will always find a maximum in one of the grid
points, so errors larger than about 104 meters could be omitted, this should be
taken into consideration when interpreting graphs.
5.1.2
Data Rate Reference
Complex data is generated as doubles in Matlab, but both magnitude and phase
data is quantized to 256 levels to be stored in 2 · 8 bits integers. This data is used
to compare with the compressed version and the compression ratio is calculated
using the 8 bit data as reference. As can be seen in figure 5.4 on page 40, this
does not have a significant impact on the localization ability. A reduction to 4
bits does hurt the ability, so 8 bits seems fair to use as a benchmark level. See
also figure 4.15 on page 35.
5.1.3
Block Effects
When analyzing the signal it is divided into blocks using a Hamming window with
50% overlap (as per 4.1.4 on page 21). Doing so doubles the number of blocks
5.1 Experiment Setup
39
×104
3.5
3
y position [m]
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
x position [m]
3
3.5
×104
Figure 5.3: The logarithmic grid used for SRP calculations, with distances in meters. Each dot represent a position u used to calculate the SRP.
needed to be transmitted. The end blocks, containing only half a block of information and zero-padding, could be either kept or discarded, without noticeable
results. In this thesis the end blocks is discarded.
A problem when using zero block overlap is block effects. The characteristics
of the signal can change abruptly from one block to another. E.g in a speech signal
one block may be a period of silence and the following block may be a voiced period.
This will result in severe effects in the block transition and will affect localization.
In the symmetrical simulation system used here, introducing block effects will
have a positive impact on localization performance; the blocks themselves will
correlate at ∆ = 0. Since this is specific to a symmetrical receiver array, this
must be avoided in order to get a fair localization performance measurement of
the compression schemes. Therefore a block overlap is preferred. The drawback of
this is of course the increased data rate needed to transmit all the extra blocks. If
these effects could be reduced or removed without block overlap, the compression
ratio could be cut in half.
The block lengths used are 128 and 512. Different block lengths will affect
40
Evaulation by simulation
CRB
σ, standard deviation of error [m]
104
original
8bit
103
4bit
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.4: Comparison of the standard deviation of localization error at different
SNR levels between CRB, uncompressed, 8 bit quantified, and 4 bit quantified
FSS.
the compression result, but these two are chosen to limit the degrees of freedom.
Longer block lengths will give a more fine grained control over components in each
block, but gives a more computationally complex algorithm and larger overhead.
This is especially true for KLT based compression, but having the same block
length for all algorithms increases comparability. The graphs and table data is
based on block length LB = 512 unless otherwise is stated.
5.1.4
Bandwidth and Signal Length
The bandwidth used is 2 kHz. The signal length is 303616 samples, chosen to give
an approximate CRB of 100 meters at 0 dB SNR given the geometry used. The
signal length is a quotient of the block length and gives L = 4743 blocks when
using 50% overlap.
5.1.5
Compression Ratio
The ratio is calculated using R = 2 · 8 bits per sample as reference. The block
overlap, the type of transform used, and block length is information presumed to
be predefined in the system, and thus unnecessary to transmit.
The total number of bits required by the compression scheme to represent and
reconstruct the data is added together and divided with the total number of bits
5.1 Experiment Setup
41
required to represent the uncompressed data,
ratio =
header + data
.
original
(5.1)
The header part is signal length independent, but the data is signal length dependent. In the sections below the ratio values for different transforms and compressions schemes are described, as well as example calculations.
KLT
The transform base components needed to reconstruct the data is stored. This
gives a larger overhead compared to other transforms used in this thesis. With
integer component quantization the data is paired with a bit vector with the same
length as the block length LB to represent which components that is discarded.
A calculation example can be done with block length LB = 128, K = 8 components kept and signal length L = 128000. Using 50% block overlap this gives a
total of 1999 blocks (throwing away the end blocks).
header = LB + LB · K · 16
data = 1999 · K · 16
original = 16 · L
so
ratio =
16512 + 255872
header + data
=
≈ 0.1330
original
2048000
(5.2)
(5.3)
(5.4)
(5.5)
The same parameters but with K = 4 components kept gives
ratio =
header + data
8320 + 127936
=
≈ 0.0665
original
2048000
(5.6)
DFT and DCT
When using DFT or DCT the transform base components is unnecessary to transmit, since they are known given the block length. With integer component quantization the data is paired with a bit vector with the same length as the block to
represent which components that are discarded. For partial components the data
is paired with a bit vector of size 4 · LB , where LB is the block length. This bit
vector describes the number of bits used for each transform component.
Using integer component quantization a calculation example can be done using
LB = 128 block length, K = 8 components kept and signal length L = 128000.
Using 50% block overlap this gives a total of 1999 blocks (throwing away the end
blocks).
header = LB
data = 1999 · K · 16
original = 16 · L
(5.7)
(5.8)
(5.9)
42
Evaulation by simulation
so
128 + 255872
header + data
=
= 0.1250
original
2048000
The same parameters but with K = 4 components kept gives
ratio =
ratio =
header + data
128 + 127936
=
≈ 0.0625
original
2048000
(5.10)
(5.11)
Using partial components quantization a calculation example can be done using
LB = 128 block length, R = 2 · 4 bits per sample on average and signal length
L = 128000. Using 50% block overlap this gives a total of 1999 blocks (throwing
away the end blocks).
header = 4 · LB
data = 1999 · LB · R
original = 16 · L
(5.12)
(5.13)
(5.14)
so
512 + 2046976
header + data
=
≈ 0.9975
(5.15)
original
2048000
The same parameters but with R = 2 · 2 bits per sample on average gives
ratio =
ratio =
header + data
512 + 1023488
=
≈ 0.5
original
2048000
(5.16)
Time-Frequency Masking
When using time-frequency masking each block is paired with a vector with the
same length as the block. The vector describes which components that is discarded
and is necessary to reconstruct the signal. Since this information is paired with
each block, no signal header is needed.
A calculation example can be done using LB = 128 block length, signal length
128000, and 90% of the energy thrown away. For a FSS this gives approximately
10000 samples kept. Using 50% block overlap this gives a total of 1999 blocks
(throwing away the end blocks).
header = 0
(5.17)
data = block header + actual data = 1999 · LB + 10000 · 16
original = 16 · L
so
ratio =
header + data
0 + (255872 + 160000)
=
≈ 0.203
original
2048000
(5.18)
(5.19)
(5.20)
The same parameters but with 95% of the energy thrown gives approximately 5000
samples kept.
ratio =
header + data
0 + (255872 + 80000)
=
≈ 0.164
original
2048000
(5.21)
5.2 Evaluation of Compression Impact
5.1.6
43
Localization Ability Adjusted Compression Ratio
Another interesting measure of compression performance is to multiply the ratio
with the (probably) decreased localization ability ratio. The motivation for this
is found in the time proportional CRB variance; twice the standard deviation can
be compensated by four times the signal length (see section 2.4 on page 10). This
gives a localization ability adjusted ratio of
2
header + data · σcompressed σoriginal
.
adjusted ratio =
original
(5.22)
This is only precise for signals that does not vary in distribution over time, but is
rather fair for signals that do not change dramatically.
An adjusted ratio value of above 1 means that the compression scheme reduced
the localization ability more than it reduced the data. A value below 1 is a
successful compression.
5.2
Evaluation of Compression Impact
The CRB used is for FSSs of similar bandwidth, regardless of the other signal types
illustrated.
The signals used (FSS, 8-PSK, and SSB signals) are described in section 3.3 on
page 14. Signals of each type with bandwidth 2 kHz were generated and three
different WGN signals were added. The simulation was done 100 times per signal
and SNR level. The PSK signals have deterministic maximum length sequence contents, the FSS are randomized each time and the SSB signals comes from different
time chunks from the same audio book.
The only transform types used in simulations are KLT and DFT. The DCT is
omitted since the signals are complex valued, and the DCT is optimized for real
signals. Some early simulations were done using DCT, but they did not seem to
diverge from the DFT case enough to merit further studies.
KLT is only used on the integral component quantization scheme, since it would
carry a large overhead for other types of quantization.
The ratio value tables comes in pairs, the ratio achieved for a given simulation
(an average over the iterations) and the localization ability adjusted ratio described
in section 5.1.6. A ratio below 1 is a compression, a value above 1 is a data
expansion, i.e., a lower value is better.
5.2.1
Compression Using Integral DFT Components
Here, we keep K components out of 512 in each block, based on the average signal
energy in each block.
With the limitations of the grid mentioned in section 5.1.1 on page 38, effectively capping the possible error, figure 5.5 on page 45 shows that keeping integral
components is rather useless, since the nodes have a hard time separating the signal from the noise. Table 5.1 on the following page further clarifies this, with ratio
44
Evaulation by simulation
values over 1 for all SNR and numbers of components kept. See also section 4.2.1
on page 25 and 7.1.1 on page 85 for further discussion of this phenomenon.
For this type of compression, FSS and PSK behaves similarly, which can be seen
in figure 5.6 on the next page and table 5.2 on page 46. The same conclusion is
valid here, keeping integral components seems useless for noise-like signals, when
no communication between the nodes are allowed.
SNR
all
8bit
1.00
K=1
0.00
K=2
0.01
K=4
0.02
K=8
0.03
K = 16
0.06
K = 32
0.12
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
0.90
4.12
19.51
47.76
166.42
328.14
646.80
379.27
1190.14
633.86
328.07
K=2
1.55
10.19
41.31
112.43
204.54
256.03
152.31
76.32
140.57
53.28
25.08
K=4
3.98
18.69
85.32
232.49
185.65
59.18
4.32
2.31
1.86
1.66
1.83
K=8
13.55
44.48
192.33
163.24
53.76
3.89
2.32
1.69
1.42
1.65
1.66
K = 16
23.04
81.08
192.61
51.15
4.80
2.67
2.12
1.54
1.48
1.38
1.56
K = 32
29.68
37.46
15.46
5.51
2.83
2.34
1.63
1.45
1.36
1.59
1.23
(b) Adjusted ratio.
Table 5.1: Ratio calculations when keeping K integral DFT components for flat
spectrum signals.
5.2 Evaluation of Compression Impact
45
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.5: The standard deviation of the localization error using flat spectrum
signals and keeping K integral DFT components.
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.6: The standard deviation of the localization error using 8-PSK signals
and keeping K integral DFT components.
46
Evaulation by simulation
SNR
all
8bit
1.00
K=1
0.00
K=2
0.01
K=4
0.02
K=8
0.03
K = 16
0.06
K = 32
0.12
K=8
12.98
50.13
176.58
234.56
58.62
2.69
1.72
1.41
1.12
1.16
1.18
K = 16
26.59
63.66
88.00
8.41
4.13
1.69
1.29
1.21
0.99
0.97
0.99
K = 32
23.43
21.82
13.96
7.54
4.51
3.47
2.25
1.52
1.21
1.70
1.35
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
1.01
5.67
24.02
59.60
167.60
556.28
829.72
613.29
291.07
4.03
6.35
K=2
2.26
11.33
50.66
124.12
252.68
366.76
435.61
2.14
1.27
1.11
1.10
K=4
4.75
22.72
109.92
232.94
273.84
3.20
16.29
1.26
0.98
0.95
1.30
(b) Adjusted ratio.
Table 5.2: Ratio calculations when keeping K integral DFT components for 8-PSK
signals.
For SSB, which differs from the noise, more interesting things appear. As seen
in figure 5.7 on the next page, the localization ability follows that of the 8bits
signal. Combining this with the ratio values seen in table 5.3a on page 48, we
get the localization ability adjusted ratio values of table 5.3b on page 48. With
numbers below 1 for SNRs below ∼ 20 dB, indicating a data compression. This
is a positive result. This holds for most of the numbers of components kept, but
K = 2 or K = 4 delivers best results.
5.2 Evaluation of Compression Impact
47
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.7: The standard deviation of the localization error using SSB signals and
keeping K integral DFT components.
σ, standard deviation of error
10
CRB
8bit
K =
K =
K =
K =
K =
K =
8
6
1
2
4
8
16
32
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.8: The quotient of the standard deviation of the localization error for
SSB signals using 8 bit as reference. Each line corresponds to keeping K integral
DFT components. If the y axis value is 2, the σ is two times that of the 8 bit
signal.
48
Evaulation by simulation
SNR
all
8bit
1.00
K=1
0.00
K=2
0.01
K=4
0.02
K=8
0.03
K = 16
0.06
K = 32
0.12
K=8
0.04
0.09
0.13
0.17
0.25
0.31
0.34
0.48
0.66
0.77
1.03
K = 16
0.05
0.11
0.20
0.26
0.23
0.34
0.44
0.63
0.85
0.87
1.29
K = 32
0.08
0.15
0.29
0.37
0.44
0.55
0.50
0.74
1.07
1.28
1.38
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
0.06
0.05
0.28
0.20
0.14
0.16
0.23
0.31
0.39
0.53
0.81
K=2
0.04
0.08
0.08
0.13
0.17
0.19
0.24
0.29
0.50
0.69
1.03
K=4
0.04
0.07
0.11
0.15
0.19
0.27
0.28
0.31
0.50
0.65
0.78
(b) Adjusted ratio.
Table 5.3: Ratio calculations when keeping K integral DFT components for single
side band signals.
5.2.2
Compression Using Integral KLT Components
Switching transform method to the KLT and comparing figure 5.9 on the facing
page for KLT and figure 5.5 on page 45 for DFT, there are noticeable difference.
The KLT performs more evenly along the SNR values, beating the DFT for weak
signals and the opposite for strong signals.
The ratio calculations differs, table 5.4a on the facing page indicates about 50%
larger overhead than its DFT counterpart, resulting in worse ratio values for those
K-SNR combinations with lower ratio values. Where DFT performs abhorrently
bad1 , the KLT performs better, but fails to get below 1.
8-PSK tells the same story; KLT performs more evenly, but still bad.
For SSB signals, the KLT captures the signal properties well, giving an adjusted
ratio under 1 for all SNR levels and all K numbers2 of components saved.
1 Ratio values of over 100, indicating a required data rate of above 100 times the original rate
for the same localization ability. This can not be called compression at all.
2 Of those simulated.
5.2 Evaluation of Compression Impact
SNR
all
8bit
1.00
K=1
0.01
K=2
0.01
K=4
0.02
49
K=8
0.04
K = 16
0.09
K = 32
0.18
K=8
7.46
9.06
12.87
9.88
13.82
20.68
20.27
22.16
31.73
32.18
33.68
K = 16
3.24
3.41
6.30
7.65
9.89
12.13
12.84
17.41
21.59
23.05
24.01
K = 32
1.97
1.93
3.00
3.79
4.88
6.33
8.42
9.40
11.78
15.90
17.04
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
2.57
11.51
51.03
94.13
156.37
141.76
85.82
53.38
52.80
51.43
42.22
K=2
4.77
18.86
60.20
58.00
44.98
46.93
44.52
45.61
54.96
57.11
54.42
K=4
7.95
21.32
36.68
28.85
28.75
26.07
29.07
26.94
37.91
45.47
45.76
(b) Adjusted ratio.
Table 5.4: Ratio calculations when keeping K integral KLT components for flat
spectrum signals.
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.9: The standard deviation of the localization error using flat spectrum
signals and keeping K integral KLT components.
50
Evaulation by simulation
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.10: The standard deviation of the localization error using 8-PSK signals
and keeping K integral KLT components.
SNR
all
8bit
1.00
K=1
0.01
K=2
0.01
K=4
0.02
K=8
0.04
K = 16
0.09
K = 32
0.18
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
3.25
16.01
61.82
125.48
166.60
239.48
157.16
88.85
68.99
73.37
77.60
K=2
6.03
22.53
77.21
87.62
74.25
57.24
60.51
111.87
234.65
436.85
1007.08
K=4
8.55
23.94
33.21
29.80
26.33
26.90
25.96
24.08
25.30
21.00
21.86
K=8
8.74
7.14
12.69
13.71
15.38
17.77
18.14
28.74
22.43
31.33
29.36
K = 16
3.93
4.17
7.24
7.35
8.80
12.69
14.64
16.18
22.16
24.35
30.28
K = 32
2.05
2.41
2.70
3.58
4.83
7.19
8.39
11.18
11.59
16.47
18.57
(b) Adjusted ratio.
Table 5.5: Ratio calculations when keeping K integral KLT components for 8-PSK
signals.
5.2 Evaluation of Compression Impact
SNR
all
8bit
1.00
K=1
0.01
K=2
0.01
K=4
0.02
51
K=8
0.04
K = 16
0.09
K = 32
0.18
K=8
0.14
0.13
0.19
0.20
0.27
0.48
0.59
0.64
0.54
0.42
0.40
K = 16
0.23
0.23
0.27
0.29
0.49
0.52
0.51
0.58
0.70
0.53
0.50
K = 32
0.37
0.33
0.34
0.37
0.51
0.60
0.79
0.79
0.96
0.78
0.96
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
K=1
0.09
0.11
0.10
0.14
0.15
0.18
0.16
0.20
0.21
0.18
0.17
K=2
0.05
0.07
0.09
0.11
0.15
0.15
0.17
0.23
0.22
0.16
0.15
K=4
0.07
0.10
0.12
0.12
0.17
0.24
0.24
0.18
0.23
0.19
0.18
(b) Adjusted ratio.
Table 5.6: Ratio calculations when keeping K integral KLT components for single
side band signals.
52
Evaulation by simulation
σ, standard deviation of error [m]
104
CRB
8bit
K =
K =
K =
K =
K =
K =
103
1
2
4
8
16
32
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.11: The standard deviation of the localization error using SSB signals
and keeping K integral KLT components.
σ, standard deviation of error
10
CRB
8bit
K =
K =
K =
K =
K =
K =
8
6
1
2
4
8
16
32
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.12: The quotient of the standard deviation of the localization error for
SSB signals using 8 bit as reference. Each line corresponds to keeping K integral
KLT components. If the y axis value is 2, the σ is two times that of the 8 bit
signal.
5.2 Evaluation of Compression Impact
5.2.3
53
Compression Using Partial Components
For figures showing the partial component scheme, “R = 4” should be interpreted
as R = 2 · 4 bits per component on average. This is consistent with the “8bit”
notation for the reference signals. Here, the same number of bits are assigned to
the magnitude and the phase data for each component.
The first interesting result here is that the figures are strikingly similar for both
FSS, 8-PSK and SSB signals, see figures 5.13 to 5.15 on pages 55–56. The second
promising finding is that for 2 bits on average, the localization ability adjusted
ratio is below 1 for all signals with 2 dB SNR or lower, see tables 5.7b, 5.9b, and
5.8b. The specific scheme we use, described in section 4.2.2 on page 29, fails for
lower number of bits allocated than 2. This is thought to be due to the poor phase
granularity at 1 bit components. If further data reduction is wanted, there should
probably be a redistribution of bits from the amplitude data to the phase data.
SNR
all
8bit
1.00
R=2
0.50
R=4
1.00
R=6
1.50
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
R=2
0.65
0.66
0.92
0.56
0.87
1.07
1.36
1.53
2.50
4.30
5.58
R=4
0.85
0.98
1.14
0.98
1.13
1.29
1.67
2.34
2.69
4.53
7.13
R=6
1.24
1.59
1.53
1.14
1.54
1.67
1.77
1.99
2.08
2.22
2.84
(b) Adjusted ratio.
Table 5.7: Ratio calculations using partial components for flat spectrum signals.
54
Evaulation by simulation
SNR
all
8bit
1.00
R=2
0.50
R=4
1.00
R=6
1.50
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
R=2
0.74
0.76
0.97
0.63
0.72
1.45
1.67
2.48
3.13
5.11
8.83
R=4
0.86
1.05
1.24
1.00
1.11
1.44
1.22
2.10
2.97
3.76
6.70
R=6
1.30
1.58
1.96
1.59
1.72
1.90
1.50
1.92
1.71
1.90
2.85
(b) Adjusted ratio.
Table 5.8: Ratio calculations using partial components for 8-PSK signals.
SNR
all
8bit
1.00
R=2
0.50
R=4
1.00
R=6
1.50
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
R=2
0.92
0.91
0.82
0.78
1.02
1.07
1.43
1.71
1.77
3.44
4.19
R=4
1.08
1.10
1.19
1.16
1.21
1.13
1.38
1.72
2.28
2.19
2.92
R=6
1.52
2.30
1.56
1.43
1.40
1.61
1.67
1.72
2.16
2.64
3.12
(b) Adjusted ratio.
Table 5.9: Ratio calculations using partial components for single side band signals.
5.2 Evaluation of Compression Impact
55
CRB
σ, standard deviation of error [m]
104
8bit
R=2
R=4
103
R=6
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.13: The standard deviation of the localization error using flat spectrum
signals and keeping partial DFT components. Each line corresponds to different
number of bits assigned.
CRB
σ, standard deviation of error
10
8bit
R=2
8
R=4
R=6
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.14: The quotient of the standard deviation of the localization error for
flat spectrum signals and keeping partial DFT components. The figure use 8 bit
as reference. Each line corresponds to different number of bits assigned. If the y
axis value is 2, the σ is two times that of the 8 bit signal.
56
Evaulation by simulation
CRB
σ, standard deviation of error [m]
104
8bit
R=2
R=4
103
R=6
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.15: The standard deviation of the localization error using 8-PSK signals
and keeping partial DFT components. Each line corresponds to different number
of bits assigned.
CRB
σ, standard deviation of error
10
8bit
R=2
8
R=4
R=6
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.16: The quotient of the standard deviation of the localization error for
8-PSK signals and keeping partial DFT components using 8 bit as reference. Each
line corresponds to different number of bits assigned. If the y axis value is 2, the
σ is two times that of the 8 bit signal.
5.2 Evaluation of Compression Impact
57
CRB
σ, standard deviation of error [m]
104
8bit
R=2
R=4
103
R=6
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.17: The standard deviation of the localization error using single side band
signals and keeping partial DFT components. Each line corresponds to different
number of bits assigned.
CRB
σ, standard deviation of error
10
8bit
R=2
8
R=4
R=6
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.18: The quotient of the standard deviation of the localization error for
single side band signals and keeping partial DFT components using 8 bit as reference. Each line corresponds to different number of bits assigned. If the y axis
value is 2, the σ is two times that of the 8 bit signal.
58
Evaulation by simulation
5.2.4
Compression Using Time-Frequency Masking
For TFM, “70%” the interpretation should be “70% of the signal energy is cut”. The
trend for TFM on FSS follows the one for the uncompressed version (figure 5.19
on page 60), but not closely enough (figure 5.20 on page 60) for it to achieve
compression when the ratio is adjusted for the localization ability regression when
a lot of the signal energy is cut. A slight compression is achieved for high SNR
levels when cutting only a small part of the signal energy. (see table 5.10b).
Time-frequency DFT masking for 8-PSK behaves almost exactly as for FSS,
compare the values of table 5.11 on the next page to those of table 5.10.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
95%
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
90%
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
0.17
80%
0.22
0.22
0.22
0.22
0.22
0.23
0.23
0.23
0.23
0.23
0.23
70%
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
0.30
60%
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
0.39
50%
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
40%
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
30%
0.79
0.79
0.79
0.79
0.79
0.79
0.79
0.79
0.79
0.79
0.79
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
76.84
293.50
181.11
44.01
21.33
15.03
10.70
7.68
8.44
11.71
11.01
95%
30.33
21.37
14.37
4.70
3.43
3.01
2.59
2.24
1.83
2.18
2.07
90%
9.41
6.22
4.72
2.35
1.88
1.73
1.26
1.24
1.09
1.42
1.34
80%
2.45
2.17
2.24
1.40
1.28
1.09
0.90
1.00
0.74
1.09
0.92
70%
1.73
1.49
1.60
1.23
0.95
0.99
0.84
0.80
0.78
0.72
0.93
60%
1.33
1.26
1.38
1.11
1.06
0.92
0.78
0.89
0.57
0.66
0.85
50%
0.94
1.26
1.09
1.11
0.86
0.94
0.95
1.03
0.90
0.84
0.86
40%
0.99
1.11
1.09
1.07
1.01
0.98
0.88
1.04
0.94
0.90
0.91
30%
1.07
1.00
1.26
1.05
1.03
1.19
1.34
0.96
0.99
0.94
0.84
(b) Adjusted ratio.
Table 5.10: Ratio calculations using TFM for flat spectrum signals. The percentages corresponds to the amount of energy thrown.
5.2 Evaluation of Compression Impact
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
95%
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
90%
0.17
0.17
0.17
0.16
0.16
0.16
0.16
0.16
0.16
0.16
0.16
80%
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
59
70%
0.30
0.30
0.30
0.29
0.29
0.29
0.29
0.29
0.29
0.29
0.29
60%
0.39
0.39
0.39
0.38
0.38
0.38
0.38
0.38
0.37
0.37
0.37
50%
0.50
0.50
0.50
0.49
0.49
0.48
0.48
0.48
0.48
0.48
0.48
40%
0.63
0.63
0.63
0.62
0.61
0.61
0.61
0.60
0.60
0.60
0.60
30%
0.79
0.79
0.79
0.78
0.77
0.77
0.76
0.76
0.76
0.76
0.76
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
89.19
327.29
148.63
35.67
18.90
12.97
10.09
8.89
6.56
9.19
11.74
95%
40.91
16.50
12.67
4.79
3.81
3.26
2.28
1.87
1.90
2.23
2.66
90%
6.47
7.42
4.02
2.17
1.46
1.70
1.25
1.19
0.98
0.98
1.45
80%
2.84
2.53
2.17
1.24
1.22
1.07
0.94
0.81
0.70
0.89
0.98
70%
1.48
1.90
1.82
1.13
1.08
1.08
0.84
0.69
0.64
0.70
0.82
60%
1.36
1.36
1.50
1.07
1.00
0.81
0.73
0.85
0.76
0.74
0.94
50%
1.06
1.42
1.38
0.85
0.91
0.97
0.86
0.73
0.75
0.75
0.90
40%
1.03
1.41
1.61
1.06
1.00
1.07
0.85
0.96
0.69
0.86
1.05
30%
1.04
1.48
1.34
1.03
1.01
1.09
1.03
0.82
0.80
1.10
1.02
(b) Adjusted ratio.
Table 5.11: Ratio calculations using TFM for 8-PSK signals. The percentage
corresponds to the amount of energy thrown.
60
Evaulation by simulation
σ, standard deviation of error [m]
104
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
103
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.19: The standard deviation of the localization error using flat spectrum
signals and time-frequency masking. The percentages corresponds to the amount
of energy thrown.
10
σ, standard deviation of error
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
8
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.20: The quotient of the standard deviation of the localization error for
flat spectrum signals and time-frequency masking, using 8 bit as reference. The
percentages corresponds to the amount of energy thrown. If the y axis value is 2,
the σ is two times that of the 8 bit signal.
5.2 Evaluation of Compression Impact
61
σ, standard deviation of error [m]
104
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
103
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.21: The standard deviation of the localization error using 8-PSK signals
and time-frequency masking The percentages cores-ponds to the amount of energy
thrown.
10
σ, standard deviation of error
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
8
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.22: The quotient of the standard deviation of the localization error for 8PSK signals and time-frequency masking, using 8 bit as reference. The percentage
corresponds to the amount of energy thrown. If the y axis value is 2, the σ is two
times that of the 8 bit signal.
62
Evaulation by simulation
For low SNR levels, TFM performs rather well for SSB signals. Here, the scheme
removes a lot of noise, improving localization ability. Illustration in figures 5.23
to 5.24 on the next page. In table 5.12b, one can see that, interestingly enough,
one should cut relatively little energy from the signal. Figures 5.23 to 5.24 on the
next page show the performance of this scheme on SSB speech signals.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
0.13
0.13
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
95%
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
90%
0.15
0.14
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
80%
0.21
0.17
0.14
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13
70%
0.28
0.23
0.16
0.14
0.13
0.13
0.13
0.13
0.13
0.13
0.13
60%
0.37
0.32
0.22
0.15
0.14
0.13
0.13
0.13
0.13
0.13
0.13
50%
0.47
0.42
0.31
0.18
0.15
0.14
0.14
0.14
0.14
0.14
0.14
40%
0.60
0.55
0.43
0.26
0.17
0.15
0.15
0.15
0.15
0.15
0.15
30%
0.77
0.71
0.60
0.39
0.22
0.18
0.17
0.16
0.16
0.16
0.16
(a) Ratio.
SNR
−10
−6
−2
2
6
10
14
18
22
26
30
8bit
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
99%
0.28
1.29
4.30
7.28
17.36
27.74
36.21
59.24
100.35
156.10
222.69
95%
0.07
0.26
0.94
1.33
2.82
4.00
6.24
8.83
17.22
18.73
32.83
90%
0.06
0.14
0.53
0.78
1.58
2.32
2.91
4.48
8.24
10.42
13.43
80%
0.09
0.12
0.22
0.47
0.67
1.18
1.15
2.04
3.28
3.93
5.65
70%
0.10
0.13
0.14
0.27
0.53
0.59
0.95
0.99
2.05
2.00
3.68
60%
0.18
0.18
0.16
0.24
0.48
0.50
0.62
0.76
1.37
1.30
2.03
50%
0.25
0.23
0.19
0.20
0.34
0.48
0.43
0.65
0.77
0.79
1.28
40%
0.38
0.39
0.34
0.19
0.27
0.29
0.41
0.45
0.72
0.72
1.08
30%
0.53
0.56
0.47
0.28
0.27
0.34
0.36
0.43
0.51
0.57
0.76
(b) Adjusted ratio.
Table 5.12: Ratio calculations using time-frequency masking for single side band
signals. The percentages corresponds to the amount of energy thrown.
5.2 Evaluation of Compression Impact
63
σ, standard deviation of error [m]
104
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
103
102
101
100
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.23: The standard deviation of the localization error using single side band
signals and time-frequency masking. The percentage correspond to the amount of
energy thrown.
10
σ, standard deviation of error
CRB
8bit
99%
95%
90%
80%
70%
60%
50%
40%
30%
8
6
4
2
0
−10
−5
0
5
10
SNR
15
20
25
30
[dB]
Figure 5.24: The quotient of the standard deviation of the localization error for
single side band signals and time-frequency masking. The figure uses 8 bit as
reference. The percentages corresponds to the amount of energy thrown. If the y
axis value is 2, the σ is two times that of the 8 bit signal.
Chapter 6
Recorded field data for
evaluation
6.1
Introduction
To try to validate the simulations a field experiment was performed. Data was
collected using four receiver positions and one transmitter position. The geometry
was chosen to be similar to the one used in the simulations. However, the distance
between transmitter and receiver was 5 km due to time and resource constraints.
The geometry is illustrated in figure 6.1 on the following page.
The forth position was not part of the original simulation array, this one was
used for redundancy if some part of the field experiment failed. This was not
necessary and to ensure comparability the signals recorded at this position was
not used.
Due to lack of equipment only two receivers was used. One was moved between
three locations, the second receiver was stationary at the forth position. The transmitter and receivers were synchronized using a time reference from Global Positioning System (GPS), so that the localization system will perceive the recordings
as simultaneous.
6.2
Signals Used
The signals used was 8-PSK, SSB, and WGN at both 2 kHz and 10 kHz bandwidth.
Each signal was generated at 4 different energy levels separated by 10 dB, for a
total of 24 signals. The different energy levels was combined by modulating each
signal to different frequencies. These frequencies can be found in table 6.1 on
page 68. The 10 kHz and 2 kHz signals was sent sequentially since they use the
same frequency band. The bandwidth of the combined signals were 50 kHz and
250 kHz, respectively. The frequency spectrum of the signals, in baseband, can be
found in figure 6.2 on page 67.
65
66
Recorded field data for evaluation
o
o
Δ
ø
o
Figure 6.1: Map illustrating the transmitter position (∆), the receiver positions
(o), and the stationary receiver (ø). The location is Linköping, Sweden. The map
is acquired from www.openstreetmap.org.
6.3
Transmitting and Receiving
When receiving the signal the transmitter power was adjusted so that the received
SNR at the mobile receiver was approximately 20 dB for the highest energy level
signal. However, the transmitting power was limited to 5 W, some of the data
recordings therefore had a lower SNR. The SNR at the stationary receiver was
approximately 30 dB for the highest energy level signals.
At each position 24 recordings were performed for the 10 kHz signals and 5
recordings for the 2 kHz signals. The 10 kHz signals had a duration of 15.9 seconds,
and the 2 kHz signals had a duration of 79.5 seconds. The signal length is chosen
by the signal generator; it only supports signals smaller than 4 million samples.
Given the total bandwidths of the signals sent, this introduces a limit of 16 and
80 seconds, respectively. A period of silence was introduced as a start-of-signal
indicator.
The transmitter repetitively transmitted the signal, always starting at integer
seconds using the GPS time reference. Recording was done for a period of 34.0 and
162.0 seconds respectively. This was done to assure that a complete signal would
be received.
Four dipole antennas were used, each one connected to separate tuner channels
6.3 Transmitting and Receiving
67
−40
dB
−60
−80
−100
−120
−22 −18 −14 −10 −6
−2 0 2
6
10
14
18
22
90
110
kHz
(a) Frequency spectrum for 2 kHz signals sent.
−40
dB
−60
−80
−100
−120
−110 −90 −70 −50 −30 −10 0 10
30
50
70
kHz
(b) Frequency spectrum for 10 kHz signals sent.
Figure 6.2: Frequency spectra for sent signals. The first four signals from left are
FSS at different energy levels, the next four are PSK and the last four are SSB.
The frequency values are relative to the center frequency of 306 MHz.
68
Recorded field data for evaluation
Signal
FSS
FSS
FSS
FSS
SSB
SSB
SSB
SSB
8-PSK
8-PSK
8-PSK
8-PSK
SNR
20
10
0
-10
20
10
0
-10
20
10
0
-10
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
Frequency
305.978 MHz
305.982 MHz
305.986 MHz
305.990 MHz
305.994 MHz
305.998 MHz
306.002 MHz
306.006 MHz
306.010 MHz
306.014 MHz
306.018 MHz
306.022 MHz
(a) 2 kHz signals.
Signal
FSS
FSS
FSS
FSS
SSB
SSB
SSB
SSB
8-PSK
8-PSK
8-PSK
8-PSK
SNR
20
10
0
-10
20
10
0
-10
20
10
0
-10
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
Frequency
305.890 MHz
305.910 MHz
305.930 MHz
305.950 MHz
306.050 MHz
306.070 MHz
306.090 MHz
306.110 MHz
305.970 MHz
305.990 MHz
306.010 MHz
306.030 MHz
(b) 10 kHz signals.
Table 6.1: Center frequencies used for each transmitted signal. SNR levels are
desired levels at the mobile receiver.
on the receiver side. A computer with a data acquisition card was used to record
the received signals. The tuner shifted the signals center frequency to an intermediate frequency of 10.4 MHz. This intermediate channel was connected to the
data acquisition card and sampled at 160 MHz. The sampling card demodulated
and decimated the signal to a sampling frequency of 156.250 kHz and 312.500
kHz for the 2 kHz and 10 kHz bandwidth signals respectively, thus fulfilling the
Nyquist–Shannon sampling theorem.
The frequency spectrum for the demodulated received signals can be found in
figure 6.3 on the next page. A peak is introduced around 0 kHz for the 2 kHz
signals and more peaks are introduced for the 10 kHz signals. Little time was
spent on figuring out why these peaks occurred since they are positioned between
the signal bands and can be handled at the post processing discussed in the section
below. However, one theory for the 0 kHz peak is that it is introduced because of
drifting in the local oscillator in the tuner.
The peaks at approximately −20 and 24 kHz are believed to be other signals.
Other concurrent measurements of the same frequency spectrum done by other
Swedish Defence Research Agency (FOI) projects indicates that there were other
signals present. If such signals were to be present in some of the signal bands,
they would likely not correlate between recordings and have little impact on uncompressed localization ability.
6.3 Transmitting and Receiving
69
−10
dB
−20
−30
−40
−50
−22 −18 −14 −10 −6
−2 0 2
6
10
14
18
22
90
110
kHz
(a) Frequency spectrum for the received 2 kHz signals.
−10
dB
−20
−30
−40
−50
−110 −90 −70 −50 −30 −10 0 10
30
50
70
kHz
(b) Frequency spectrum for the received 10 kHz signals.
Figure 6.3: Example of frequency spectra for the received signals. The first four
signals from left are FSS at different energy levels, the next four are PSK and the
last four are SSB.
70
Recorded field data for evaluation
6.4
Post Processing
The signals were processed in Matlab to cut out the correct parts of the signals.
The transmitting and recording always started and ended at integer seconds using
the GPS reference. Since each signal sampled was twice the length of the original
signal sent there were silent periods in the received signals. These silent parts
were used to find where the signal started and ended. The cuts were placed at
integer seconds to ensure that no time delay affecting TDOA was introduced in the
processing.
The signals were then demodulated and low-pass filtered to cut out the 3
different signals at the 4 different energy levels. The peaks discussed in section 6.3
on page 66 were removed by the low-pass filters. Given that four channels were
used and 24 recordings were done for the 10 kHz signal a total number of 96
signals were extracted for each position, signal type and SNR level. Using the
same calculation for the 2 kHz signals a total number of 20 signals were gathered
for each position, signal type and SNR level.
However, for the 10 kHz bandwidth signals, one channel at each position had a
SNR below 10 dB for the highest energy flat spectrum signal, these channels were
discarded. A total number of 72 useful signals were therefore gathered with 10
kHz bandwidth. For the 2 kHz signals one channel was bad at two of the positions
and at one of the positions the SNR was over 30 dB for one of the channels. The
data from these channels where also discarded to get matching SNR levels. A total
number of 15 useful signals were therefore gathered with 2 kHz bandwidth.
SNR levels were calculated using an empty frequency band between the modulated signals as noise reference. This noise band was chosen between the two flat
spectrum signal with the highest energy where no peak was detected.
6.5
Localization Using Recorded Data
s1
STFT
s2
STFT
s3
STFT
XC
Sx (ω)
×
STx (ω)
q̂
SRP
1 Sx (0)
Figure 6.4: Modified localization system estimating the position from phase shift
compensated cross correlation on recorded data.
The localization system used for simulation used an impulse response to filter
the signal and added white Gaussian noise. A modified system was used for the
field data. Since the signals have passed the actual channel no impulse response
was used and no noise was added. The geometry was different so a modified grid
was used to calculate the steered response power. The original grid was translated
and scaled to fit the new geometry.
6.5 Localization Using Recorded Data
71
The output of the localization system was not as expected, the localization
bias was heavy. The output, using the high SNR 10 kHz bandwidth flat spectrum
signals, can be found in figure 6.5a on the next page. The bad localization performance was found to be due to that the local oscillators in the receivers was not
locked in phase to each other between the recordings. This introduced a phase
shift in the cross spectral density in 2.10 on page 7 which now has to be modified
to
Sm n (ω) = Ss (ω)ejω∆m,n (q) ej(θm −θn ) + Sνm νn (ω),
(6.1)
where ej(θm −θn ) is the result of the unlocked phases. Under ideal conditions with
no noise the cross-correlation simplifies to
Sm n (ω) = Ss (ω)ejω∆m,n (q) ej(θm −θn ) .
(6.2)
By inserting ωk = 0 the expression becomes
Sm n (0) = Ss (0)ej(θm −θn ) .
(6.3)
It is now possible to compensate for the phase shift by dividing the cross-correlation
function with Sm n (0), the new compensated auto-correlation function then becomes
Ss (ω)ejω∆m,n (q)
Sm n (ωk )
=
.
(6.4)
STm n (ωk ) =
Sm n (0)
Ss (0)
By doing this the localization performance was significantly increased.
This phase compensation is only possible for signals with high SNR since it
assumes that there is an ideal noise free channel at Ss (0). Because of this the
phase shift was estimated using the high SNR signals from each measurement and
applied on all the lower energy signals as well. This is not possible in real scenarios
when localization is done on low SNR signals; there exists other methods to mitigate
this but they are outside the scope of this thesis. The new modified localization
system using phase error compensation is found in figure 6.4 on the preceding
page.
The output of the modified localization system with phase shift compensation,
using the same set of signals as earlier, can be found in figure 6.5b on the following
page.
When using the system on the 2 kHz signals the localization result was poor.
This can be seen in figure 6.6 on page 73. This can be explained by the CRB, the
variance for the 2 kHz signals becomes 25 times larger given the geometry, SNR
and the length of the signals used. The same poor results was achieved using the
SSB speech signal with both 2 and 10 kHz bandwidth, however this is not as easy
to explain and little time was spent on doing so. The PSK behaved similar to FSS.
72
Recorded field data for evaluation
(a) Output of localization system without com-(b) Output of localization system with compenpensation for phase error
sation for phase error
Figure 6.5: Output of localization system before and after phase error compensation for high SNR 10 kHz bandwidth FSS. The receivers are illustrated with small
circles, the transmitter as a triangle and the large dashed circle has its center in
the bias and its radius is the standard deviation. Each dot corresponds to an
estimated position.
6.6 Evaluation of Compression Impact
73
Figure 6.6: The localization output using 2 kHz bandwidth high SNR FSS. The
receivers are illustrated with small circles, the transmitter as a triangle and the
large dashed circle has its center in the bias and its radius is the standard deviation.
Each dot corresponds to an estimated position.
6.6
Evaluation of Compression Impact
This section will only look at the effect of compression when localizing 10 kHz
flat spectrum signals due to the similar behavior using PSK and poor localization
results using the recorded SSB data. The presentation is to be regarded as a
complement to chapter 5 on page 37.
The mean error e is calculated using the norm of the localization error along
the x and y axis,
q
e=
µ2x + µ2y .
(6.5)
The standard deviation of the error, ς, is calculated by taking the square root of
the mean variance along the x and y axis.
r
σx2 + σy2
ς=
.
(6.6)
2
The localization performance for the uncompressed flat spectrum signals is
found in table 6.2 on the next page.
6.6.1
Compression Using DFT Components
The output of integral component quantization using DFT on the high SNR flat
spectrum signals is found in figure 6.7 on page 75. The localization performance
was fairly good when using 32 and 16 components compared to the uncompressed
version found in figure 6.5b on the preceding page. Table 6.3 shows the standard
74
Recorded field data for evaluation
SNR
−10
0
10
20
dB
dB
dB
dB
Uncompressed
4548
2546
1982
1664
(a) Standard deviation of error
SNR
−10
0
10
20
dB
dB
dB
dB
Uncompressed
1027
2003
1394
1286
(b) Mean error
Table 6.2: Tables of the standard deviation and mean of localization error using
uncompressed flat spectrum signals.
deviation and mean of the localization error. Table 6.4 on page 76 shows the
calculated ratios, all adjusted ratios is well below 1.
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
2487
2562
3669
3499
K=2
3285
3508
4322
3551
K=4
4004
3800
3876
3550
K=8
4691
4550
3463
3333
K = 16
4850
4480
3096
3191
K = 32
5279
4506
3024
2865
(a) Standard deviation of error
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
5701
5521
3812
2784
K=2
5415
4007
1364
1346
K=4
5257
2721
2215
1201
K=8
3016
1150
2822
1327
K = 16
1040
2650
3163
1400
K = 32
1109
2568
3141
1160
(b) Mean error
Table 6.3: Tables of the standard deviation and mean of localization error using
integral component quantization with DFT on flat spectrum signals.
6.6 Evaluation of Compression Impact
75
(a) Keeping 4 components using DFT
(b) Keeping 8 components using DFT
(c) Keeping 16 components using DFT
(d) Keeping 32 components using DFT
Figure 6.7: Illustrating output of localization using compression scheme with integral component quantization and DFT on high SNR flat spectrum signals. The
receivers are illustrated with small circles, the transmitter as a triangle and the
large dashed circle has its center in the bias and its radius is the standard deviation.
Each dot corresponds to an estimated position.
76
Recorded field data for evaluation
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
0.00
0.00
0.00
0.00
K=2
0.01
0.01
0.01
0.01
K=4
0.02
0.02
0.02
0.02
K=8
0.03
0.03
0.03
0.03
K = 16
0.06
0.06
0.06
0.06
K = 32
0.12
0.12
0.12
0.12
K=8
0.03
0.10
0.09
0.12
K = 16
0.07
0.19
0.15
0.23
K = 32
0.17
0.39
0.29
0.37
(a) Ratio.
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
0.00
0.00
0.01
0.02
K=2
0.00
0.01
0.03
0.03
K=4
0.01
0.03
0.06
0.07
(b) Adjusted ratio.
Table 6.4: Ratio calculations using integral component quantization with DFT
on flat spectrum signals.
6.6.2
Compression Using KLT Components
The output of integral quantization using KLT on the high SNR flat spectrum
signals is found in figure 6.8 on the next page. The localization performance was
fairly good when using 32, 16 and 8 components compared to the uncompressed
version found in figure 6.5b on page 72. Table 6.5 show the standard deviation and
mean of the localization error. Table 6.6 on page 78 shows the calculated ratios,
all adjusted ratios is well below 1.
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
5177
5371
5469
5121
K=2
5427
5186
4599
4231
K=4
5245
5304
4502
3789
K=8
5082
5112
3107
2542
K = 16
5067
4438
2851
2563
K = 32
5170
4101
2755
2543
(a) Standard deviation of error
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
818
716
522
731
K=2
668
1660
449
401
K=4
770
1444
1110
1082
K=8
435
439
2202
937
K = 16
396
701
1625
1035
K = 32
924
1488
1325
927
(b) Mean error
Table 6.5: Tables of the standard deviation and mean of localization error using
integral component quantization with KLT on flat spectrum signals.
6.6 Evaluation of Compression Impact
77
(a) Keeping 4 components using KLT
(b) Keeping 8 components using KLT
(c) Keeping 16 components using KLT
(d) Keeping 32 components using KLT
Figure 6.8: Illustrating output of localization using compression scheme with integral component quantization and KLT on high SNR flat spectrum signals. The
receivers are illustrated with small circles, the transmitter as a triangle and the
large dashed circle has its center in the bias and its radius is the standard deviation.
Each dot corresponds to an estimated position.
78
Recorded field data for evaluation
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
0.01
0.01
0.01
0.01
K=2
0.01
0.01
0.01
0.01
K=4
0.03
0.03
0.03
0.03
K=8
0.06
0.06
0.06
0.06
K = 16
0.11
0.11
0.11
0.11
K = 32
0.23
0.23
0.23
0.23
K=8
0.06
0.15
0.10
0.10
K = 16
0.13
0.24
0.18
0.20
K = 32
0.26
0.43
0.34
0.39
(a) Ratio.
SNR
−10
0
10
20
dB
dB
dB
dB
K=1
0.01
0.02
0.03
0.04
K=2
0.02
0.04
0.05
0.06
K=4
0.03
0.08
0.09
0.09
(b) Adjusted ratio.
Table 6.6: Ratio calculations using integral component quantization with KLT
on flat spectrum signals.
6.6.3
Compression Using Partial Components
The partial components worked good down to 2 bits on average per component.
The localization output for the high SNR flat spectrum signals can be found in
figure 6.9 on the next page, for comparison the uncompressed version is found
in 6.5b on page 72. Table 6.7 shows the standard deviation and mean of the
localization error. Table 6.8 on page 80 shows the calculated ratios, the localization
performance adjusted ratios is slightly lower than the original ratios.
SNR
−10
0
10
20
dB
dB
dB
dB
2 bits
4538
2741
1842
1720
4 bits
4455
2753
1800
1564
6 bits
4478
2592
1797
1559
(a) Standard deviation of error
SNR
−10
0
10
20
dB
dB
dB
dB
2 bits
1028
2316
1301
1026
4 bits
753
2196
945
1195
6 bits
740
2097
1039
1185
(b) Mean error
Table 6.7: Tables of the standard deviation and mean of localization error using
partial component quantization with DFT.
6.6 Evaluation of Compression Impact
79
(a) Assigning 6 bits using partial components (b) Assigning 4 bits using partial components
(c) Assigning 2 bits using partial components
Figure 6.9: Illustrating output of localization using compression scheme with partial component quantization and DFT. The receivers are illustrated with small
circles, the transmitter as a triangle and the large dashed circle has its center in
the bias and its radius is the standard deviation. Each dot corresponds to an
estimated position.
80
Recorded field data for evaluation
SNR
−10
0
10
20
dB
dB
dB
dB
2 bits
0.50
0.50
0.50
0.50
4 bits
1.00
1.00
1.00
1.00
6 bits
1.50
1.50
1.50
1.50
(a) Ratio.
SNR
−10
0
10
20
dB
dB
dB
dB
2 bits
0.50
0.58
0.43
0.53
4 bits
0.96
1.17
0.82
0.88
6 bits
1.45
1.56
1.23
1.32
(b) Adjusted ratio.
Table 6.8: Ratio calculations using partial component quantization with DFT on
flat spectrum signals.
6.6.4
Compression Using Time-Frequency Masking
For time-frequency masking, “70%” should be interpreted as “70% of the energy is
cut”. The compression behaves well when cutting smaller amounts of data, this is
consistent with the simulations. For the high SNR signal the output is illustrated
in figure 6.10 to 6.11 on pages 81–82. Table 6.9 shows the standard deviation and
mean of the localization error. The standard deviation is significantly less when
cutting 30% than cutting 99% for the high SNR signals. Table 6.10 on page 83
shows the calculated ratios, almost all adjusted ratios are below 1.
SNR
−10
0
10
20
dB
dB
dB
dB
30%
4799
2946
2092
1820
40%
4679
3058
2169
1866
60%
4683
3393
2522
1979
70%
5013
4123
2665
2023
80%
5008
4431
3109
2585
90%
5045
4960
3608
3087
95%
5296
5405
4176
3349
99%
4926
4713
5034
4569
95%
1122
710
1384
753
99%
1539
1326
707
1016
(a) Standard deviation of error
SNR
−10
0
10
20
dB
dB
dB
dB
30%
553
2043
1469
1252
40%
489
2294
1509
1283
60%
1053
2350
1520
1270
70%
1423
1751
1440
1235
80%
1107
1540
1060
1052
90%
1447
1635
1349
839
(b) Mean error
Table 6.9: Tables of the standard deviation and mean of localization error using
time-frequency masking on flat spectrum signals.
6.6 Evaluation of Compression Impact
81
(a) 99% energy thrown
(b) 95% energy thrown
(c) 90% energy thrown
(d) 80% energy thrown
Figure 6.10: Illustrating output of localization using compression scheme with
time-frequency masking for high SNR flat spectrum signals. The receivers are
illustrated with small circles, the transmitter as a triangle and the large dashed
circle has its center in the bias and its radius is the standard deviation. Each dot
corresponds to an estimated position.
82
Recorded field data for evaluation
(a) 70% energy thrown
(b) 60% energy thrown
(c) 40% energy thrown
(d) 30% energy thrown
Figure 6.11: Illustrating output of localization using compression scheme with
time-frequency masking on high SNR flat spectrum signals. The receivers are
illustrated with small circles, the transmitter as a triangle and the large dashed
circle has its center in the bias and its radius is the standard deviation. Each dot
corresponds to an estimated position.
6.6 Evaluation of Compression Impact
SNR
−10
0
10
20
dB
dB
dB
dB
30%
0.76
0.75
0.73
0.70
40%
0.60
0.59
0.57
0.55
60%
0.37
0.36
0.35
0.34
70%
0.28
0.28
0.27
0.27
83
80%
0.21
0.21
0.21
0.20
90%
0.16
0.16
0.16
0.16
95%
0.14
0.14
0.14
0.14
99%
0.13
0.13
0.13
0.13
80%
0.26
0.64
0.51
0.49
90%
0.19
0.60
0.52
0.54
95%
0.19
0.62
0.61
0.56
99%
0.15
0.44
0.82
0.96
(a) Ratio.
SNR
−10
0
10
20
dB
dB
dB
dB
30%
0.85
1.01
0.81
0.84
40%
0.64
0.86
0.68
0.70
60%
0.39
0.65
0.57
0.48
70%
0.34
0.73
0.49
0.39
(b) Adjusted ratio.
Table 6.10:
signals.
Ratio calculations using time-frequency masking on flat spectrum
Chapter 7
Conclusion and discussion
7.1
Comments on Compression and Noise
There are several aspects of noise needed to be considered when applying data
compression in a TDOA system. This section will discuss a few of them.
7.1.1
Separate Signal from Noise
As mentioned in section 3.2 on page 14, different signals pose different difficulties
when separating them from the noise. For weak (low SNR) noise-like signals,
such as FSS with white Gaussian noise, it is hard to distinguish relevant signal
information from noise in a single receiver. However, as the signal length increases,
cross correlation will be able to find the signal anyway since the noise is assumed
to be uncorrelated; TDOA works for negative SNR [9]. This means that information
that is impossible, or at least very hard, to label as important in a single node
might be useful in the correlation system.
7.1.2
Noise Reduction
For radar signals, it has been shown that a compression ratio of about 1:100 can
be achieved using Singular Value Decomposition (SVD) with little impact on TDOA
localization [12]. This is in part because compression can actually increase the SNR,
since transform components with more energy often contains a higher percentage
of signal information than the components with less energy do. This is why some
of the compression schemes used performs better at lower SNR levels in terms of
adjusted ratio. This denoising effect is seen in figure 4.15 on page 35. All curves
show the same trend at low SNR but those with lower data rate performes worse at
high SNR. Thus, the performance adjusted ratio will be worse for high SNR signals
at low data rates.
85
86
7.2
Conclusion and discussion
Comments on the Field Recordings
The behavior of the compressed flat spectrum signals from the recorded field data
was similar to the ones used in the simulations. This supports the theory that
compression can be applied in actual TDOA localization systems.
When using the single side band signals together with the localization system
the output was not as expected. Why this is so was not thoroughly investigated.
This neither prove or disprove the results of the simulations, further studies are
needed.
The drifting in the local oscillator mentioned in section 6.3 on page 66 could be
of significance, especially when localization is done on narrow bandwidth signals.
While studying the frequency spectrum of the received signal, using a high number of frequency bins, the drift was estimated to be approximately 2 Hz. When
estimating the TDOA 512 frequency bins was used, this gives approximately 4 Hz
width of each bin for the 2 kHz signal. By decreasing the number of frequency
bins the impact of drifting could be reduced. Another method could be using a
automatic frequency control circuit in the tuner.
The adjusted ratios was good when using the compression schemes on the
recorded data, the compression had little impact on the standard deviation of the
localization error. However, the adjusted ratios and standard deviations calculated
using the recorded data has low precision due to the granularity in the steered
response power grid. This will have to be considered when interpreting the data.
When simulating, the estimated positions were gathered at the dense part of the
grid. Since the error were higher for the recorded data the precision was reduced.
A more fine grained grid would shine some more light on the localization precision.
7.3
Proposed Use
Data compression in a localization system is a non-trivial problem, but not a
useless one. In the introduction chapter (page 1), the motivation behind it was
presented. This thesis work has not found a good catch-all compression scheme,
but shines some light on possible solutions if the sought after signal is of a known
type. The compression scheme using partial components gives interesting results,
the signal independent trend of the 2 bits partial component adjusted ratio values
is promising. This compression scheme should be fine tuned before actual use,
but seems to be able to successfully compress the data. A combination of partial
components and time-frequency masking could improve the result. For SSB signals,
several of the tested compression schemes gave good results.
7.4
Localization Ability Reduction
If lower localization accuracy is tolerated, compression can do more. A lot of the
adjusted compression ratio numbers presented in chapter 5 are well above 1 due to
compensation for reduced accuracy. In general, there is a trade off between data
rate and localization ability, see section 4.4 on page 33.
7.5 Future
7.5
87
Future
This section discusses areas where further research can be done.
7.5.1
Amplitude Data
It may be possible to cut the data needed to be transmitted from all but one node
in half by simply discarding the magnitude data and only keeping the phase data,
and estimating the amplitude from the one node that did not discard its amplitude
data. This is more convenient if one of the nodes is part of the system which does
the correlation and localization; this is often the case. Using one of the listening nodes as base node would also partly solve the component selection problem
presented in section 4.2.1 on page 25, one less signal need to be compressed.
7.5.2
Phase-amplitude Data Optimization
Further effort should be spent on combining bit allocation schemes with an amplitudephase data consideration and combine this with a TFM scheme. Most of the TDOA
relevant information exists in the phase data when the receivers are less than a
wave length away from the transmitter. Allocating schemes that takes this into
consideration could improve the results. An optimization problem approach to the
trade off between amplitude and phase data might be a useful way forward.
7.5.3
Block Length and Ratio
The block length chosen has an impact on the compression results, both in quality
and ratio. Exactly how depends on compression scheme and signal type, and to
what end the data is used for. For noise-like signals, the DFT block length should be
more thoroughly studied, one idea is that there are signal type dependent optimal
block lengths.
7.5.4
Impact on Node-Base Transmission Redundancy
Like in most compression schemes, redundancy between data points is reduced.
If the transmission between a system node (that uses compression) and the node
responsible for the correlation introduces bit errors, the results may be worse off
than it would have been without compression. However, the compression schemes
use block overlap and introduce redundancy to the transmitted signals.
These effects, and whether there will be graceful degradation1 or not, are out
of scope.
1 “Graceful degradation: Degradation of a system in such a manner that it continues to
operate, but provides a reduced level of service rather than failing completely,” [4]
88
7.5.5
Conclusion and discussion
Other Areas to Look at
It would be interesting to investigate adaptive compression schemes, to see if systems that use higher order cumulants and moments to fit signal to known distribution types can achieve better compression rates.
Looking into the signals received, e.g. phase shift in constellation points for
PSK, might give very good compression rates for high SNR levels.
Lists
List of Figures
1.1
A common system setup used for localization in electronic warfare scenarios.
1
2.1
2.2
2.3
2.4
6
7
8
2.6
ˆ
Two receivers, each hyperbola corresponds to an estimated time delay ∆.
Three receivers locating one transmitter using TDOA. . . . . . . . . . . .
Localization system using TDOA and cross correlation (XC). . . . . . . .
Experiment setup displaying the three receivers as circles and the transmitter as a triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The magnitude of a SRP scaled between 0.0 and 1.0 illustrated with contour lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CRB for FSS, 1 kHz signal bandwidth at different signal lengths . . . . .
10
12
3.1
3.2
3.3
3.4
Gaussian channel. . . . . . . . . . . . .
Frequency energy distributions for WGN
Phase-shift keying signal constellations .
SSB signal creation. . . . . . . . . . . . .
.
.
.
.
13
15
16
17
4.1
4.2
Transform coding compression. . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the magnitude of the auto-correlation matrix Rχ of a SSB
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the magnitude of the transformed auto-correlation matrix
Rχ using Karhunen-Loéve transform of a SSB signal. . . . . . . . . . . . .
Illustration of the magnitude of the transformed auto-correlation matrix
Rχ using Discrete cosine transform of a SSB signal. . . . . . . . . . . . . .
Illustration of the magnitude of the transformed auto-correlation matrix
Rχ using Discrete Fourier transform of a SSB signal. . . . . . . . . . . . .
Average number of components chosen in three flat spectrum signal . . . .
Average number of components chosen in three flat spectrum signal . . . .
Average number of components chosen in three SSB signals . . . . . . . .
Average number of components chosen in three SSB signals . . . . . . . .
Phase input and output of a quantifier using 4 bits. . . . . . . . . . . . . .
Output of the amplitude quantifier . . . . . . . . . . . . . . . . . . . . . .
Spectrogram of a SSB speech signal. . . . . . . . . . . . . . . . . . . . . .
Spectrogram of a 4-PSK signal. . . . . . . . . . . . . . . . . . . . . . . . .
Gaussian channel with added quantization distortion. . . . . . . . . . . .
20
2.5
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
89
. . . . . . . . .
and bandwidth
. . . . . . . . .
. . . . . . . . .
. . . . . . . .
limited FSS.
. . . . . . . .
. . . . . . . .
.
.
.
.
8
23
23
24
24
26
26
28
28
30
31
32
32
34
90
Conclusion and discussion
4.15 Cramér-Rao bound for a 1 s, 1 kHz flat spectrum signal, using different
data rates R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
Localization system using TDOA and cross correlation. . . . . . . . . . .
Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The logarithmic grid used for SRP calculations, with distances in meters. .
Comparison of the standard deviation of localization error at different SNR
levels between CRB, uncompressed, 8 bit quantified, and 4 bit quantified
FSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using flat spectrum signals
and keeping K integral DFT components. . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using 8-PSK signals and
keeping K integral DFT components. . . . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using SSB signals and
keeping K integral DFT components. . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for SSB .
The standard deviation of the localization error using flat spectrum signals
and keeping K integral KLT components. . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using 8-PSK signals and
keeping K integral KLT components. . . . . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using SSB signals and
keeping K integral KLT components. . . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for SSB
signals using 8 bit as reference. . . . . . . . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using flat spectrum signals
and keeping partial DFT components. . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for flat
spectrum signals and keeping partial DFT components. . . . . . . . . . .
The standard deviation of the localization error using 8-PSK signals and
keeping partial DFT components. . . . . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for 8-PSK
signals and keeping partial DFT components using 8 bit as reference. . . .
The standard deviation of the localization error using single side band
signals and keeping partial DFT components. . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for single
side band signals and keeping partial DFT components using 8 bit as
reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The standard deviation of the localization error using flat spectrum signals
and time-frequency masking. . . . . . . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for flat
spectrum signals and time-frequency masking, using 8 bit as reference. . .
The standard deviation of the localization error using 8-PSK signals and
time-frequency masking . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for 8-PSK
signals and time-frequency masking, using 8 bit as reference. . . . . . . .
The standard deviation of the localization error using single side band
signals and time-frequency masking. The percentage correspond to the
amount of energy thrown. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The quotient of the standard deviation of the localization error for single
side band signals and time-frequency masking. . . . . . . . . . . . . . . .
35
37
38
39
40
45
45
47
47
49
50
52
52
55
55
56
56
57
57
60
60
61
61
63
63
7.5 Future
Map illustrating the transmitter position (∆), the receiver positions (o),
and the stationary receiver (ø). . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Frequency spectra for sent signals . . . . . . . . . . . . . . . . . . . . . . .
6.3 Example of frequency spectra for the received signals . . . . . . . . . . . .
6.4 Modified localization system estimating the position from phase shift compensated cross correlation on recorded data. . . . . . . . . . . . . . . . . .
6.5 Output of localization system before and after phase error compensation
for high SNR 10 kHz bandwidth FSS. . . . . . . . . . . . . . . . . . . . .
6.6 The localization output using 2 kHz bandwidth high SNR FSS. . . . . . .
6.7 Illustrating output of localization using compression scheme with integral
component quantization and DFT on high SNR flat spectrum signals. . .
6.8 Illustrating output of localization using compression scheme with integral
component quantization and KLT on high SNR flat spectrum signals. . .
6.9 Illustrating output of localization using compression scheme with partial
component quantization and DFT. . . . . . . . . . . . . . . . . . . . . . .
6.10 Illustrating output of localization using compression scheme with timefrequency masking on high SNR flat spectrum signals. . . . . . . . . . . .
6.11 Illustrating output of localization using compression scheme with timefrequency masking on high SNR flat spectrum signals. . . . . . . . . . . .
91
6.1
66
67
69
70
72
73
75
77
79
81
82
List of Tables
1.1
Transmission rates in different media used in localization systems. . . . .
2
3.1
The number of samples needed for flat spectrum signals at different bandwidths and SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Ratio calculations when keeping K integral DFT components for flat spectrum signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Ratio calculations when keeping K integral DFT components for 8-PSK
signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Ratio calculations when keeping K integral DFT components for single
side band signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Ratio calculations when keeping K integral KLT components for flat spectrum signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Ratio calculations when keeping K integral KLT components for 8-PSK
signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Ratio calculations when keeping K integral KLT components for single
side band signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Ratio calculations using partial components for flat spectrum signals. . . .
5.8 Ratio calculations using partial components for 8-PSK signals. . . . . . .
5.9 Ratio calculations using partial components for single side band signals. .
5.10 Ratio calculations using TFM for flat spectrum signals. The percentages
corresponds to the amount of energy thrown. . . . . . . . . . . . . . . . .
5.11 Ratio calculations using TFM for 8-PSK signals. The percentage corresponds to the amount of energy thrown. . . . . . . . . . . . . . . . . . . .
5.12 Ratio calculations using time-frequency masking for single side band signals. The percentages corresponds to the amount of energy thrown. . . .
5.1
44
46
48
49
50
51
53
54
54
58
59
62
92
Conclusion and discussion
6.1
6.2
Center frequencies used for each transmitted signal . . . . . . . . . . . . .
Tables of the standard deviation and mean of localization error using uncompressed flat spectrum signals. . . . . . . . . . . . . . . . . . . . . . . .
6.3 Tables of the standard deviation and mean of localization error using integral component quantization with DFT on flat spectrum signals. . . . .
6.4 Ratio calculations using integral component quantization with DFT on
flat spectrum signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Tables of the standard deviation and mean of localization error using integral component quantization with KLT on flat spectrum signals. . . . .
6.6 Ratio calculations using integral component quantization with KLT on flat
spectrum signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Tables of the standard deviation and mean of localization error using partial component quantization with DFT. . . . . . . . . . . . . . . . . . . .
6.8 Ratio calculations using partial component quantization with DFT on flat
spectrum signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Tables of the standard deviation and mean of localization error using timefrequency masking on flat spectrum signals. . . . . . . . . . . . . . . . . .
6.10 Ratio calculations using time-frequency masking on flat spectrum signals.
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List of Algorithms
4.1
4.2
4.3
4.4
Transformation of Rx . . . . . . . . .
component selection on FSS . . .
component selection on SSB signal
Zonal sampling, signal power . . . . .
DFT
DFT
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