Linköping Studies in Science and Technology Thesis No. 1396 Contributions to Flexible Multirate Digital Signal Processing Structures Amir Eghbali Division of Electronics Systems Department of Electrical Engineering Linköpings universitet, SE–581 83 Linköping, Sweden WWW: http://www.es.isy.liu.se E-mail: [email protected] Linköping 2009 Contributions to Flexible Multirate Digital Signal Processing Structures c 2009 Amir Eghbali Department of Electrical Engineering, Linköpings universitet, SE–581 83 Linköping, Sweden. ISBN 978-91-7393-678-1 ISSN 0280-7971 LIU-TEK-LIC-2009:4 Printed by LiU-Tryck, Linköping, Sweden 2009 to my family... Abstract A current focus among communication engineers is to design flexible radio systems in order to handle services among different telecommunication standards. Efficient support of dynamic interactive communication systems requires flexible and costefficient radio systems. Thus, low-cost multimode terminals will be crucial building blocks for future generations of multimode communication systems. Here, different bandwidths, from different telecommunication standards, must be supported and, thus, there is a need for a system which can handle a number of different bandwidths. This can be done using multimode transmultiplexers (TMUXs) which make it possible for different users to share a common channel in a time-varying manner. These TMUXs allow bandwidth-on-demand so that the resulting communication system has a dynamic allocation of bandwidth to users. Each user occupies a specific portion of the channel where the location and width of this portion may vary with time. Another focus among communication engineers is to provide various wideband services accessible to everybody everywhere. Here, satellites with high-gain spot beam antennas, on-board signal processing, and switching will be a major complementary part of future digital communication systems. Satellites provide a global coverage and if a satellite is in orbit, customers only need to install a satellite terminal and subscribe to the service. Efficient utilization of the available limited frequency spectrum, by these satellites, calls for on-board signal processing to perform flexible frequency-band reallocation (FFBR). Considering these two focuses in one integrated system where the TMUXs operate on-ground and FFBR networks operate on-board, one can conclude that successful design of dynamic communication systems requires high levels of flexibility in digital signal processing structures. In other words, there is a need for flexible digital signal processing structures that can support different telecommunication scenarios and standards. This flexibility (or reconfigurability) must not impose restrictions on the hardware and, ideally, it must come at the expense of simple software modifications. In other words, the system is based on a hardware platform and its parameters can easily be modified without the need for hardware changes. This thesis aims to outline flexible TMUX and FFBR structures which can allow dynamic communication scenarios with simple software reconfigurations on the same hardware platform. In both structures, the system parameters are determined in advance. For these parameters, the required filter design problems are solved only once. Dynamic communications, with users having different time-varying bandwidths, are then supported by adjusting some multipliers of the proposed multimode TMUXs and a simple software programming in the channel switch of the FFBR network. These do not require any hardware changes and can be performed online. However, the filter design problem is solved only once and offline. i Acknowledgments I would like to thank my supervisor Professor Håkan Johansson for giving me the opportunity to work as a Ph.D student. However, I should not forget to sincerely thank him for his patience, inspiration, and wonderful guidance in helping me deal with my research problems. I would also like to thank my co-supervisor Assistant Professor Per Löwenborg for wonderful discussions and feedback. Special thanks have to go to all members of my family for all the support they have provided. Not all problems can be solved by computers, books, and discussions, etc. One mostly requires emotional support and encouragement from beloved ones. God has blessed me with the best of these! I just do not know how to be thankful... I will never be able to do this... The former and present colleagues at the Division of Electronics Systems, Department of Electrical Engineering, Linköping University have created a very friendly environment. They always kindly do their best to help you. You never feel alone even if you come from another country and do not speak fluent Swedish. Actually, you feel it like home! Last but not least, I should thank all my friends whom have made my stay in Sweden pleasant. Amir Eghbali Linköping, January 2009 iii Contents 1 Introduction 1.1 Motivation and Problem Formulation . . . . . . . . . . . . . . . . . . 1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basics of Digital Filters 2.1 FIR Filters . . . . . . . . . . . . . . 2.2 Polyphase Decomposition . . . . . . 2.3 Special Classes of Filters . . . . . . . 2.3.1 Power Complementary Filters 2.3.2 Linear-phase FIR Filters . . . 2.3.3 Nyquist (M th-band) Filters . 2.4 FIR Filter Design . . . . . . . . . . . 1 1 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 8 9 10 10 11 12 3 Basics of Multirate Signal Processing 3.1 Conventional Sampling Rate Conversion . . . . . . . . 3.1.1 Noble Identity . . . . . . . . . . . . . . . . . . 3.2 Sampling Rate Conversion Using the Farrow Structure 3.3 General M -Channel FBs . . . . . . . . . . . . . . . . . 3.4 General M -Channel TMUXs . . . . . . . . . . . . . . 3.4.1 Mathematical Representation of TMUXs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 17 18 22 23 23 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents 3.4.2 Contents Approximation of PR in redundant TMUXs . . . . . . . . . . 25 4 Flexible Frequency-Band Reallocation For Real Signals 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Relation to Previous Work . . . . . . . . . . . . . . . . . . 4.1.2 Remark on the Choice of FFBR Network for Complexity Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 MIMO FFBR Network Configuration . . . . . . . . . . . . 4.2 Review of the FFBR Network Based on Variable Oversampled Complex Modulated FBs . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Efficient Realization of the FFBR Network . . . . . . . . . 4.3 Alternative I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Complex vs. Real Sampling . . . . . . . . . . . . . . . . . . 4.3.2 Arithmetic Complexity of the Hilbert Transformer . . . . . 4.3.3 Arithmetic Complexity of the DFT with Complex Inputs . 4.3.4 Arithmetic Complexity of the Complex FFBR Network . . 4.4 Alternative II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Arithmetic Complexity of the Real FFBR Network . . . . . 4.5 Comparison of Arithmetic Complexity and Performance . . . . . . 4.5.1 Arithmetic Complexity of Complex FFBR vs. Real FFBR . 4.5.2 Arithmetic Complexity of Alternative I vs. Alternative II . 4.5.3 Performance of Alternative I vs. Alternative II . . . . . . . 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Measure of Complexity . . . . . . . . . . . . . . . . . . . . 4.6.2 Applicability of Alternatives I and II . . . . . . . . . . . . . 4.6.3 Filter Bank Design . . . . . . . . . . . . . . . . . . . . . . . 29 . 30 . 30 . 31 . 31 . . . . . . . . . . . . . . . . . 33 33 35 36 36 38 39 40 40 43 43 44 47 49 49 49 49 5 A Multimode Transmultiplexer Structure 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proposed Multimode TMUX Structure . . . . . . . . . . . . . . . . . 5.3.1 Channel Sampling Rates . . . . . . . . . . . . . . . . . . . . . 5.3.2 Sampling Rate Conversion . . . . . . . . . . . . . . . . . . . . 5.3.3 Subcarrier Frequencies . . . . . . . . . . . . . . . . . . . . . . 5.4 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Implementation and Design Complexity Issues . . . . . . . . . . . . 5.6 TMUX Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Analysis of the Nonuniform TMUX Using Multirate Building Blocks 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 52 52 53 55 55 55 56 58 59 61 62 65 6 A Class of Multimode Transmultiplexers Based on the Farrow Structure 67 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 vi Contents 6.3 6.4 6.5 6.6 Contents 6.2.1 Problem Formulation . . . . . . . . . . . . . . . 6.2.2 Some General Issues . . . . . . . . . . . . . . . . Proposed Integer SRC Multimode TMUX . . . . . . . . 6.3.1 Variable Integer SRC Using the Farrow Structure 6.3.2 Efficient Variable Integer SRC . . . . . . . . . . 6.3.3 Arithmetic Complexity . . . . . . . . . . . . . . 6.3.4 Filter Design . . . . . . . . . . . . . . . . . . . . 6.3.5 Filter Design Parameters . . . . . . . . . . . . . 6.3.6 Filter Design Criteria . . . . . . . . . . . . . . . Proposed Rational SRC Multimode TMUX . . . . . . . 6.4.1 Efficient Variable Rational SRC . . . . . . . . . . 6.4.2 Filter Design . . . . . . . . . . . . . . . . . . . . TMUX Performance . . . . . . . . . . . . . . . . . . . . 6.5.1 Effects of Bp on the SRC Error . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 69 70 71 72 73 74 75 75 78 79 80 83 85 86 89 vii Notation ix Notation Notation Acronyms and Abbreviations ADC AFB BER CDMA DFT EDGE ESA EVM FB FDMA FIR FFBR FFT FBR GB GSM ICI IDFT IIR ISI IS-54 IS-136 LS LTI MF/TDMA MIMO OFDM PFBR PR QFT SFB SISO SRC TDMA TMUX WCDMA 3GPP Analog to Digital Converter Analysis Filter Bank Bit Error Rate Code Division Multiple Access Discrete Fourier Transform Enhanced Data Rates for GSM Evolution European Space Agency Error Vector Magnitude Filter Bank Frequency Division Multiple Access Finite-length Impulse Response Flexible Frequency-Band Reallocation Fast Fourier Transform Frequency-Band Reallocation Granularity Band Global System for Mobile communications Inter-Carrier Interference Inverse Discrete Fourier Transform Infinite-length Impulse Response Inter-Symbol Interference Interim Standard-54 Interim Standard-136 Least-Squares Linear Time Invariant Multiple Frequency/Time Division Multiple Access Multi-Input Multi-Output Orthogonal Frequency Division Multiplexing Perfect Frequency-Band Reallocation Perfect Reconstruction Quick Fourier Transform Synthesis Filter Bank Single-Input Single-Output Sampling Rate Conversion Time Division Multiple Access Transmultiplexer Wideband Code Division Multiple Access 3rd Generation Partnership Project x 1 Introduction 1.1 Motivation and Problem Formulation Communication engineers currently aim to design flexible radio systems which can handle services among different telecommunication standards [1]. Along with the increase in (i) the number of communication standards (or modes), and (ii) the range of services provided by the operators, e.g., high bit rate interactive communications, the requirements on flexibility and cost-efficiency of these radio systems increase as well. Hence, low-cost multimode terminals will be crucial building blocks for future generations of communication systems. Multistandard communications require that different bandwidths from different telecommunication standards, are supported. Table 1.1 shows the bit rate, number of users sharing one channel, and the channel spacing of popular cellular telecommunication standards [2]. If such standards are included in a general telecommunication system where any user can, based on its demand, use any standard which suits its requirements (on bandwidth, transmission quality, etc.), there would be a need for a system that can handle a number of different bandwidths. Assume, for example, that a communication channel is shared by three users A, B, and C which respectively transmit video, text, and audio. If bandwidth-on-demand is supported, any user can, at any time, decide to send either of video, text, and audio. Furthermore, at any time, any user can decide to use any center frequency. 1 1. INTRODUCTION Table 1.1: Bit rate, number of users sharing one channel, and channel spacing in different telecommunication standards. Standard Bit Rate No. of Users Channel Spacing IS-54/136 48.6 Kbps 3 30 KHz GSM 271 Kbps 8 200 KHz IS-95 1.2288 Kbps 798 1250 KHz To support multimode communications, there is thus a need for a system which can allow different number of users, having different bit rates, to share a common channel. Transmultiplexers (TMUXs) make it possible for different users to share a common channel and, consequently, multistandard (multimode) TMUXs constitute one of the main building blocks in multistandard communication systems. It is noted that multiple access schemes such as code division multiple access (CDMA), time division multiple access (TDMA), and frequency division multiple access (FDMA) are special cases of a general TMUX theory [3]. To support bandwidth-on-demand such that the users can request any bandwidth at any time, the characteristics of the TMUXs must vary with time. Such a communication system has a dynamic allocation of bandwidth to users so that each user occupies a specific portion of the channel where the location and width of this portion may vary with time. The principle of such a communication system is shown in Fig. 1.1. Here, we assume that the whole frequency spectrum is shared by P users where each user Xp has a bandwidth of π(1+ρ) Rp , p = 0, 1, . . . , P − 1 and Rp can take on integer or rational values. Furthermore, ρ is the roll-off factor and there is a guardband of 2∆ separating the user signals1 . To do this, one can use conventional nonuniform TMUXs, e.g., [4–8], to place different users having different bandwidths at different locations in the frequency spectrum. Assuming a dynamic communication system, these conventional TMUXs would require either predesign of different filters or online design of filters. This becomes inefficient when simultaneously considering the increased number of communication scenarios and the desire to support dynamic communications. Therefore, it is vital to develop low-complexity TMUXs which dynamically support different communication scenarios and require reasonable implementation complexity as well as design effort. One aim of this thesis is to introduce TMUX structures in which different number of users, having different bandwidths, can share the whole frequency band in a time-varying manner. As discussed above, in dynamic communications the bandwidth and number of users sharing the channel may change in a time-varying manner and, thus, the proposed TMUXs must (and will) take this into consideration. As a promise of future digital communication systems, communication engineers also aim to support various wideband services accessible to everybody everywhere [9]. Although the large theoretical bandwidth provided by optical fibers could 1 The choice of ∆ does not restrict the analysis and design of the TMUX and, hence, throughout this thesis we will assume ∆ = 0. 2 1. INTRODUCTION Guard Band (2D) (a) 0D D X0 X1 XP-2 XP-1 w0 w1 wP-2 wP-1 2p wT (b) 0D X0 X1 XP-2 XP-1 w0 w1 wP-2 wP-1 2p wT (c) 0D X0 XP-1 w0 wP-1 2p wT (d) 0D X0 XP-1 w0 wP-1 2p wT Figure 1.1: Formulation of problem for multimode TMUXs where P users share a common channel. make terrestrial networks capable of supporting such services, this bandwidth is hardly available today. Furthermore, there is a gap between the local exchange and the customer which needs to be filled. Thus, it has been concluded that satellites with high-gain spot beam antennas, on-board signal processing, and switching will be a major complementary part of future digital communication systems [9–13]. The reason is that satellites provide a global coverage and if a satellite is in orbit, customers only need to install a satellite terminal and subscribe to the service. Thus, in an integrated operation with terrestrial networks, satellites can have a complementary coverage necessitating a cooperative service delivery [13]. The European space agency (ESA) has proposed three major network structures for broadband satellite-based systems [10] in which satellites communicate with users through multiple spot beams and, therefore, there is a need for efficient reuse of the limited available frequency spectrum by satellite on-board signal processing [9–13]. In technical terms, this calls for flexible frequency-band reallocation (FFBR) networks [14–25] and is also referred to as frequency multiplexing and demultiplexing [14, 15]. The digital part of the satellite on-board signal processor is a multi-input multioutput (MIMO) system and the number of input signals can, in general, differ from that of the output signals. Furthermore, the input/output signals can have different bandwidths and bit rates, e.g., users from different telecommunication standards. The next generation of satellite-based communication systems discussed above must support different communication and connectivity scenarios. One such main scenario is based on multiple frequency/time division multiple access (MF/TDMA) scheme in which the bandwidth of each incoming signal is composed of a number of adjacent smaller frequency bands (or subbands) with each subband being occupied by one (a few) user (users). In other words, the MF/TDMA scheme slices the available capacity of the channel both in time and frequency and at any time, any portion can be used by any user [26]. A main role of the on-board signal processor is to reallocate all subbands to different prespecified output signals and positions in the frequency spectrum. The principle of this operation is illustrated in Fig. 1.2. Here, different users 3 1. INTRODUCTION Output signal 1 In 1 3 1 2 p wTin [rad] Input signal 2 4 5 In 2 6 p FFBR Network Input signal 1 wTin [rad] 3 1 p Out 1 wTout [rad] Output signal 2 Out 2 Out 3 5 4 p wTout [rad] Output signal 3 2 6 p wTout [rad] Figure 1.2: Principle of FBR for an FFBR network where any signal in any of the 2-input signals can be reallocated to any position in any of the 3-output signals. having different bandwidths are present at the input of the FFBR networks and each of these users must be reallocated to different positions in the frequency spectrum. If the communication system is dynamic, the bandwidth and position of the users may change in a time-varying manner. Thus, there will be a need for FFBR networks which can dynamically perform reallocation of users with different bandwidths. Consequently, some requirements are imposed on FFBR networks such as flexibility , low complexity, near perfect frequency-band reallocation (PFBR), simplicity, etc [10]. In practice and similar to Fig. 1.1, there is also a need for guardbands between the subbands so that the network is realizable. It is one aim of this thesis to outline flexible and low complexity solutions for FFBR networks so that different users, present in different composite MF/TDMA input signals, can be reallocated to different positions in different composite MF/TDMA output signals. Furthermore, the solutions must (and will) impose no restrictions on the bandwidth of users or the system operation. To successfully design dynamic communication systems, communication engineers require high levels of flexibility in digital signal processing structures. In other words, there is a need for flexible digital signal processing structures that can be used to support different telecommunication scenarios and standards. This flexibility must not impose restrictions on the hardware and, ideally, it must come at the expense of simple software modifications. This is frequently referred to as reconfigurability [27, 28] meaning that the system is based on a hardware platform and its parameters can easily be modified without the need for hardware changes. This thesis outlines solutions for the flexible communication structures discussed above and is a result of the research performed at the Division of Electronics Systems, Department of Electrical Engineering, Linköping University between October 2006 and December 2008. The research during this period has resulted in the following publications [29–34]: 1. A. Eghbali, H. Johansson, and P. Löwenborg, “Flexible frequency-band reallocation MIMO networks for real signals,” in Proc. Int. Symp. Image Signal Process. Analysis, Istanbul, Turkey, Sept. 2007. 4 1. INTRODUCTION 2. A. Eghbali, H. Johansson, and P. Löwenborg, “Flexible frequency-band reallocation: complex versus real,” Circuits, Syst., and Signal Process., 2008, accepted. 3. A. Eghbali, H. Johansson, and P. Löwenborg, “An arbitrary bandwidth transmultiplexer and its application to flexible frequency-band reallocation networks,” in Proc. Eur. Conf. Circuit Theory Design, Seville, Spain, Aug. 2007. 4. A. Eghbali, H. Johansson, and P. Löwenborg, “A multimode transmultiplexer structure,” IEEE Trans. Circuits Syst. II, vol. 55, no. 3, pp. 279–283, Mar. 2008. 5. A. Eghbali, H. Johansson, and P. Löwenborg, “A Farrow-structure-based multi-mode transmultiplexer,” in Proc. IEEE Int. Symp. Circuits Syst., Seattle, Washington, USA, May 2008. 6. A. Eghbali, H. Johansson, and P. Löwenborg, “A class of multimode transmultiplexers based on the Farrow structure,” IEEE Trans. Circuits Syst. I, 2008, submitted. These papers are covered in Chapters 4-6 of the thesis. The following paper was also published during this period but it is not included in this thesis: 1. A. Eghbali, O. Gustafsson, H. Johansson, and P. Löwenborg, “On the complexity of multiplierless direct and polyphase FIR filter structures,” in Proc. Int. Symp. Image Signal Process. Analysis, Istanbul, Turkey, Sept. 2007. 1.2 Thesis Outline Chapter 2 reviews the basics of digital filters which will frequently be referred to in the subsequent chapters. It includes the definition of finite-length impulse response (FIR) filters; polyphase decomposition; and some special classes of filters, viz. Nyquist, power complementary, and linear-phase FIR filters. The minimax and least-squares (LS) filter design problems are also treated. In Chapter 3, realization of sampling rate conversion (SRC) using conventional structures and the Farrow structure is discussed. Furthermore, the noble multirate identities are introduced and they are used to derive efficient SRC structures. In addition, the concepts of filter banks (FBs), TMUXs, distortion, cross talk, and aliasing are defined as well. As an important property of multirate systems, perfect reconstruction (PR) is discussed and its approximation by redundant TMUXs is considered. Finally, the filter design problem for redundant TMUXs using the minimax and LS approaches is treated. Chapter 4 is based on a journal and a conference paper [29, 30] and it discusses approaches for realizing FFBR networks. The FFBR network is based on variable oversampled complex-modulated FBs. The chapter introduces two alternatives to 5 1. INTRODUCTION process real signals using real input/output and complex input/output FFBR networks or, simply, real and complex FFBR networks, respectively. Furthermore, the general problem formulation for processing of real signals by the real FFBR network is also discussed. It is shown that the real case has less overall number of processing units, i.e., adders and multipliers, compared to its complex counterpart. In addition, the real system eliminates the need for two Hilbert transformers and is suitable for systems with a large number of users. Finally, issues related to performance and the trend in arithmetic complexity with respect to (i) the prototype filter order, (ii) the number of FB channels, (iii) the order of the Hilbert transformer, and (iv) the efficiency in FBR are also considered. In Chapter 5, which covers a journal and a conference paper [31, 32], a multimode TMUX capable of generating a large set of bandwidths and center frequencies is introduced. The TMUX utilizes fixed integer SRC, Farrow-based variable rational SRC, and variable frequency shifters. The properties of the building blocks as well as the operation of the TMUX are discussed in detail. Furthermore, the filter design problem along with some design examples is considered and it is shown that, by designing the filters only once, all possible combinations of bandwidths and center frequencies are obtained by simple adjustment of the variable delay parameter of the Farrow-based filters as well as the variable parameters of the frequency shifters. Additionally, using the rational SRC equivalent of the Farrow-based filters, the TMUX is described in terms of conventional multirate building blocks and the filter design problem is restated using the blocked transfer function. As an application of the TMUX, the performance and functionality test of the FFBR network discussed in Chapter 4 is illustrated. Chapter 6 discusses a class of multimode TMUXs proposed by a journal and a conference paper [33, 34]. The TMUXs use the Farrow structure to realize polyphase components of general interpolation/decimation filters. In this way, integer SRC with different ratios can be performed using a set of fixed filters, i.e., Farrow subfilters, and variable multipliers. In conjunction with variable frequency shifters, an integer SRC multimode TMUX is presented and its filter design problem, using the minimax and LS methods, is discussed. Furthermore, a model of general rational SRC is constructed where the same fixed subfilters are used to perform rational SRC. Efficient realization of this rational SRC scheme is presented. Similarly, variable frequency shifters are utilized to derive a general rational SRC multimode TMUX which is capable of generating different bandwidths. By processing 16-QAM signals, it is shown that the performance of the TMUX depends on the ripples of the general interpolation/decimation filters. Finally, Chapter 7 outlines some concluding remarks and open issues for future research. 6 2 Basics of Digital Filters In this chapter, some basics of digital filters will be reviewed. These basics are chosen according to their application in subsequent chapters of the thesis. First, the classification of FIR filters is discussed where some straightforward FIR filter realizations are outlined. To derive efficient realizations for the TMUXs and FFBR networks, the polyphase decomposition can be used which will be discussed in Section 2.2. The TMUXs and FFBR networks utilize special classes of filters, viz. power complementary, Nyquist, and linear-phase filters, which will be considered in Section 2.3. Finally, the general formulations of minimax and LS filter design problems are outlined in Section 2.4. 2.1 FIR Filters An FIR filter of order N has an impulse response with finite length and the coefficient values (or impulse response values) in the set h(0), h(1), . . . , h(N ) meaning that there are N + 1 coefficients. The transfer function H(z) of an N th-order 7 2. BASICS OF DIGITAL FILTERS x(n) T h0 T T h1 h2 T hN-1 hN y(n) Figure 2.1: Direct form realization of an N th-order FIR filter. x(n) h0 y(n) h1 T h2 T hN-1 T hN T Figure 2.2: Transposed direct form realization of an N th-order FIR filter. causal1 FIR filter can be written as [35] H(z) = N X h(n)z −n . (2.1) n=0 In the time domain, the output sequence y(n) resulting from an input sequence x(n) can be written as y(n) = N X h(k)x(n − k) ⇔ Y (z) = H(z)X(z). (2.2) k=0 There are different ways to realize the FIR filter in (2.1) and two straightforward realizations are shown in Figs. 2.1 and 2.2 where the set of impulse response values are assumed to be h0 , h1 , . . . , hN . Due to the nature of FIR filters, it is possible to use non-recursive algorithms for their realization and, thereby, problems with instability2 can be eliminated. In this thesis, all the filters are designed to be FIR and, hence, the filters are always stable. In Figs. 2.1 and 2.2, there is a need for N + 1 multiplications, N two-input additions, and N delay elements. In practice, one prefers to use more efficient structures so that the implementation cost can be reduced. Examples could be the polyphase and multiplierless realization [35, 36]. 2.2 Polyphase Decomposition One of the tools to derive efficient structures for digital filters is the polyphase decomposition. The transfer function in (2.1) can be decomposed into its L polyphase 1 A filter h(n) is said to be causal if h(n) = 0 for n < 0. Any non-causal FIR filter can be made causal by insertion of a proper delay. 2 Instability can arise due to poles outside the unit circle. All the poles of an FIR filter are placed at the origin making it unconditionally stable. 8 2. BASICS OF DIGITAL FILTERS components as ∞ X H(z) = +z −1 h(nL)z −nL n=−∞ ∞ X h(nL + 1)z −nL (2.3) n=−∞ ... +z −(L−1) ∞ X h(nL + L − 1)z −nL , n=−∞ which in a compact way becomes H(z) = L−1 X z −i Hi (z L ), (2.4) i=0 where Hi (z) are the polyphase components and hi (n) = h(nL + i), i = 0, 1, . . . , L − 1. (2.5) This decomposition is frequently referred to as Type I polyphase decomposition. On the other hand, the Type II polyphase decomposition of (2.1) is H(z) = L−1 X z −(L−1−i) Ri (z L ), (2.6) i=0 where Ri (z) = HL−1−i (z) [37]. To realize an N th-order FIR filter using the Lpolyphase decomposition, there is a need for L2 subfilters of length NL+1 . Polyphase decomposition makes it possible to have a system where the filters operate at the lowest possible frequency. This is of special interest in the context of FBs and TMUXs as it reduces the implementation cost. The polyphase decomposition brings savings in the implementation cost but the total number of multiplications and additions does not change. However, operating adders and multipliers at a lower rate reduces their implementation cost. 2.3 Special Classes of Filters Some classes of digital filters are more suitable for multirate systems than others. In the next subsections, we will introduce some of these classes to which we will refer later in the thesis. 9 2. BASICS OF DIGITAL FILTERS 2.3.1 Power Complementary Filters A set of filters with frequency responses Hk (ejωT ), k = 0, 1, . . . , K are said to be power complementary if [37] K X |Hk (ejωT )|2 = c, (2.7) k=0 for all ωT and a constant c > 0. In general, a set of filters is said to be complementary of order p if [38] K X |Hk (ejωT )|p = c, p ∈ N. (2.8) k=0 In special cases, the magnitude and power complementary filters are the set which satisfy (2.8) for p = 1 and p = 2, respectively. Higher order complementary filters, i.e., p > 2, can generate ordinary magnitude and power complementary filters while maintaining superior cut-off characteristics [38]. In the filter design problems of this thesis, the power complementary case will frequently be utilized. 2.3.2 Linear-phase FIR Filters An important advantage of FIR filters is that they can be made to have a linear phase. This is done by restricting the impulse response h(n) to be either symmetric or antisymmetric as [35] Symmetric : h(n) Antisymmetric : h(n) = h(N − n), = −h(N − n), n = 0, 1, . . . , N n = 0, 1, . . . , N. (2.9) Specifically, there is only about N2 distinct filter coefficients and, therefore, the number of multiplications for the filter realization can be halved. However, this does not change the number of adders required. The frequency response of a linearphase FIR filter can be expressed as H(ejωT ) = e−j( N ωT 2 +c) HR (ωT ) = ejΘ(ωT ) HR (ωT ), (2.10) where HR (ωT ) is the real zero-phase frequency response with c = 0 and c = π2 for symmetric and antisymmetric h(n), respectively. Furthermore, the phase response Φ(ωT ) is related to Θ(ωT ) and the group delay τg (ωT ) as ( Θ(ωT ) if HR (ωT )≥0 Φ(ωT ) = (2.11) Θ(ωT )±π if HR (ωT ) < 0. and τg (ωT ) = − 10 dΦ(ωT ) , d(ωT ) (2.12) 2. BASICS OF DIGITAL FILTERS In the case of linear-phase FIR filters, the group delay reduces to a constant equal to N2 . Depending on h(n) being symmetric or antisymmetric and N being odd or even, four types of linear-phase FIR filters arise which have different expressions for HR (ωT ) [35]. These four types of linear-phase FIR filters are defined as Type I : h(n) Type II : h(n) Type III : h(n) Type IV : h(n) 2.3.3 = h(N − n), = h(N − n), N even N odd = −h(N − n), = −h(N − n), N even N odd (2.13) Nyquist (M th-band) Filters A lowpass non-causal filter is said to be M th-band if its zeroth polyphase component, i.e., H0 (z) in (2.4), satisfies [39] H0 (z M ) = 1 . M (2.14) Furthermore, the passband and stopband edges are, respectively, given by [40] ωc T = ωs T = π(1 − ρ) M π(1 + ρ) , M (2.15) where ρ is the roll-off factor and 0 < ρ < 1 meaning that the transition band π should always contain ωT = M . In brief, the zeroth polyphase component of an M th-band filter is a constant and the real zero-phase frequency response in (2.10) satisfies 1 π HR (ωT ) = for ωT = . (2.16) 2 M Furthermore, the passband and stopband ripples are related to each other as δs ≤(M − 1)δc . If a filter H(z) is an M th-band filter, the sum of M shifted copies of H(z) results in unity. In other words, M X 2π k H(zWM ) = 1 where WM = e−j M . (2.17) k=0 In the time domain, the impulse response of an M th-band filter satisfies ( 1 if n = 0; M h(n) = 0 if n = ±M, ±2M, . . .. This means that every M th sample, except the center tap, is zero which brings reductions in the number of multipliers and adders required to realize the filter. If h(n) is an M th-band filter, its delayed version is also an M th-band filter. In a 11 2. BASICS OF DIGITAL FILTERS general case, a filter H(z) is said to be an M th-band filter if any of its polyphase components, e.g., Hk (z), has the form Hk (z) = cz −nk . In the time domain, this becomes ( c if n = nk ; h(nM + k) = 0 otherwise. Generally, the impulse response of a Nyquist filter could be causal or noncausal; FIR or infinite-length impulse response (IIR); linear-phase or nonlinear-phase; and real or complex. In this thesis we always design real causal linear-phase FIR Nyquist filters. 2.4 FIR Filter Design The frequency response of an ideal digital filter is equal to one in the passband and zero in the stopband. Furthermore, there is no transition band resulting in a brickwall characteristic. However, such a filter would have an infinite length, i.e., an ideal lowpass sinc function3 , and is not realizable. To get around this, one attempts to approximate this ideal transfer function in the passband and stopband by allowing a transition band as well as some ripples. Thus, the practical specification for a digital filter with frequency response H(ejωT ) is given by4 1 − δc ≤ |H(ejωT )| ≤ 1 + δc , jωT |H(e )| ≤ δs , ωT ∈ Ωc ωT ∈ Ωs (2.18) where δc and δs are, respectively, the passband and stopband ripples with Ωc and Ωs being the passband and stopband regions. As an example, in a lowpass filter, Ωc covers [0, ωc T ] whereas Ωs covers [ωs T, π]. Here, ωc T and ωs T are the passband and stopband edges, respectively. Consequently, after estimating the filter order, the coefficients h(n) must be determined such that (2.18) is satisfied for desired values of Ωc , Ωs , δc , and δs . A commonly used formula to estimate the order N of a linear-phase FIR filter is the Bellanger’s formula given by [35] N≈ − 2 2π log10 (10δs δc ) 3 ωs T − ωc T (2.19) For reasonable filter orders, (2.19) gives a good approximation but in the case of nonlinear-phase FIR filters such formulas do not exist5 and, therefore, a manual search is the only way to find the filter order. The aim of the filter design problem is to find a set of coefficients that satisfy a specific criterion. This criterion could be the energy, maximum ripple, or combinations of them leading to LS, minimax, or constrained LS approaches. In this thesis, 3 Ideally, sin(x) sinc(x) = 1 if x = 0 and sinc(x) = x if x 6= 0. is noted that (2.18) is independent of the filter being FIR or IIR. However, in this section, we consider the design of FIR filters as they are used throughout the thesis. 5 For minimum phase filters, formulas similar to (2.19) can be derived. 4 It 12 2. BASICS OF DIGITAL FILTERS we have used minimax and LS approaches where the minimax design problem can be stated as minimize δ, subject to (2.20) |H(ejωT ) − 1| ≤ δ, |H(ejωT )|≤W (ωT )δ, ωT ∈ Ωc ωT ∈ Ωs . On the other hand, the LS design problem can be stated as minimize Z |H(ejωT ) − 1|2 + ωT ∈Ωc (2.21) |H(ejωT )|2 . ωT ∈Ωs W (ωT ) Z Here, W (ωT ) is a weighting function which weights the approximation error at different frequencies. A large value for W (ωT ) would result in large stopband approximation errors for (2.20) and (2.21). In the examples of this thesis, we have assumed frequency independent weighting functions within each frequency band and, thus, the weightings have constant values in the frequency range of interest. 13 2. BASICS OF DIGITAL FILTERS 14 3 Basics of Multirate Signal Processing This chapter discusses the necessary basics of multirate systems to which we will refer in the remainder of the thesis. In multirate systems, different parts operate at different sampling frequencies which necessitates SRC. In this regards, Sections 3.1 and 3.2 discuss the SRC, i.e., interpolation and decimation, with integer and rational ratios based on the conventional structures as well as the Farrow structure. Using the conventional models for SRC, the idea of FBs is defined in Section 3.3 where their input-output relation as well as the concepts of aliasing and distortion are discussed. Furthermore, the definition and conditions of PR are also considered. As duals of FBs, TMUXs are outlined in Section 3.4 and their input-output relationship, the PR conditions, inter-carrier interference (ICI), and inter-symbol interference (ISI) are described. Finally, the classification of redundant TMUXs with non-overlapping analysis/synthesis filters are discussed. These TMUXs will frequently be used in subsequent chapters of the thesis and, due to this, the filter design problem for these TMUXs is treated thoroughly. 3.1 Conventional Sampling Rate Conversion As the name indicates, in a multirate system, different parts of the system operate at different sampling frequencies and, consequently, there is a need to perform SRC between these parts. This can be performed by interpolation (decimation) which 15 3. BASICS OF MULTIRATE SIGNAL PROCESSING (a) (b) x(n) x(n) y(n) M L y(n) Figure 3.1: (a) M -fold downsampler. (b) L-fold upsampler. increases (decreases) the sampling frequency of digital signals [35, 37]. An alternative way to perform SRC on digital signals is to first construct the corresponding analog signal and, then, resample it with the new sampling frequency. However, it is more efficient to perform SRC directly in the digital domain. By changing the sampling frequency, the implementation cost for a given task can be reduced as one can perform the arithmetic operations, i.e., additions and multiplications, at a lower rate. Both interpolation and decimation are two-stage processes in which lowpass filters as well as downsamplers and upsamplers are involved. The block diagram of upsamplers and downsamplers are shown in Fig. 3.1. A downsampler retains every M th sample of the input signal x(n) and its output sequence can be written as [35, 37] y(n) = x(nM ). (3.1) In the frequency domain, (3.1) becomes Y (z) = M −1 1 1 X k X(z M WM ), M (3.2) k=0 where WM is defined as in (2.17). Specifically, the output signal is a sum of M 1 k ) versions stretched (by converting z to z M ) and shifted (through the terms WM of the input signal. On the other hand, an upsampler adds L − 1 zeros between consecutive samples of the input signal x(n) and, thus, its output becomes [35, 37] y(n) = ( n x( L ) if n = 0, ±L, ±2L, . . . 0 otherwise. (3.3) In the frequency domain, (3.3) can be written as Y (z) = X(z L ). (3.4) This shows that the whole frequency spectrum is compressed by L and, consequently, there are images which must be removed. Unless the input signal is strictly bandlimited, downsampling results in aliasing and, consequently, reducing the sampling rate of a signal by decimation requires an extra filter as shown in Fig. 3.2. This anti-aliasing filter H(z) must limit the bandwidth of the downsampler input as the original content of the signal can only π be preserved if it is bandlimited to M . The time domain expression for the output signal y(n) in Fig. 3.2 can be written as [37] 16 3. BASICS OF MULTIRATE SIGNAL PROCESSING x(n) M H(z) y(n) Figure 3.2: Decimation by a factor of M . x(n) L H(z) y(n) Figure 3.3: Interpolation by a factor of L. +∞ X y(n) = x(k)h(nM − k). (3.5) k=−∞ Similarly, as upsampling causes imaging, increasing the sampling rate of a signal by interpolation would require an interpolation filter as illustrated in Fig. 3.3. This lowpass anti-imaging filter, i.e., H(z), removes the extra images caused by the upsampler. Thus, the time domain expression for the output signal y(n) in Fig. 3.3 can be written as [37] y(n) = +∞ X x(k)h(n − kL). (3.6) k=−∞ To perform SRC1 by a rational ratio M L , interpolation by L must be followed by decimation by M . In other words, Fig. 3.3 must be followed by Fig. 3.2 and, consequently, the cascade of the anti-imaging and anti-aliasing filters would result in one filter, say G(z). Thus, the output sequence y(n) after decimating x(n) by a ratio M L can be written as [37] y(n) = +∞ X x(k)g(nM − kL) (3.7) k=−∞ 3.1.1 Noble Identity A useful identity in multirate digital signal processing is the noble identity which makes it possible to move arithmetic operations inside a multirate structure so that they can be performed at lower frequencies. If the transfer function H(z) is a rational function, i.e., a ratio of polynomials in z or z −1 , the noble identities can be defined as in Fig. 3.4. Using noble identities, a system with a transfer function in terms of z M which is followed by a downsampler or preceded by an upsampler of ratio M , can be moved in a way that the processing is performed at the lower L L > M , we have an interpolation by rational ratio M > 1 in which the sampling frequency M in increased. Otherwise, a decimation by rational L > 1 is performed reducing the sampling frequency. In this thesis, we have frequently referred to SRC by rational ratio Rp > 1. 1 If 17 3. BASICS OF MULTIRATE SIGNAL PROCESSING x(n) x(n) M y(n) <=> x(n) M H(zM) y(n) <=> x(n) H(z) H(zM) M H(z) y(n) M y(n) Figure 3.4: Noble identities which make it possible to move the arithmetic operations to the lower sampling frequency. x(m) M H(z) x(m) Mfs y(n) fs Mfs M H0(z) y(n) H0(z) fs z-1 M x(m) H1(z) H1(z) Mfs y(n) fs z-1 M HM-1(z) HM-1(z) Figure 3.5: Efficient decimation utilizing polyphase decomposition and noble identities. sampling frequency. The combination of these noble identities and the polyphase decomposition in (2.4) enables efficient realizations of multirate structures. Efficient structures for integer decimation and interpolation are, respectively, shown in Figs. 3.5 and 3.6. 3.2 Sampling Rate Conversion Using the Farrow Structure In conventional rational SRC, if it is desired to change the ratio of SRC, there would be a need for new anti-imaging or anti-aliasing filters making it rather difficult when performing a large set of rational SRC ratios. This makes the system less flexible in choosing the set of SRC ratios. However, by utilizing the Farrow structure, shown in Fig. 3.7, this can be solved in an elegant way. The Farrow structure is composed of linear-phase FIR2 subfilters Sk (z), k = 0, 1, . . . , L with either a symmetric (for k even) or antisymmetric (for k odd) impulse response. The subfilters can also have even or odd orders where in the case of odd order, all the subfilters are general filters whereas for the even-order case, S0 (z) reduces to a pure delay. The transfer 2 If IIR variable fractional-delay filters are used, care must be taken to avoid transients as the delay parameter may change for every time index. 18 3. BASICS OF MULTIRATE SIGNAL PROCESSING x(n) M y(m) H(z) fs Mfs x(n) H0(z) y(m) M fs z-1 H1(z) H0(z) Mfs x(n) M y(m) H1(z) Mfs fs z-1 HM-1(z) HM-1(z) M Figure 3.6: Efficient interpolation utilizing polyphase decomposition and noble identities. x(n) SL(z) S2(z) S1(z) m m S0(z) m y(n) Figure 3.7: Farrow structure with fixed subfilters Sk (z) and variable fractional delay µ. function of the Farrow structure can be written as H(z) = L X Sk (z)µk , |µ| ≤ 0.5 (3.8) k=0 where µ is the fractional delay value. The fractional delay value defines the time difference between each input sample and its corresponding output sample. Assuming Tin and Tout to be the sampling period of the input x(n) and the output y(n), respectively, and considering even/odd order subfilters, µ is defined3 as [32, 33, 41] Even order : Odd order : [nin + µ(nin )]Tin = nout Tout [nin + 0.5 + µ(nin )]Tin = nout Tout (3.9) where nin (nout ) is the input (output) sample index. If the value of µ is a constant for all input samples, the Farrow structure delays all samples of a bandlimited input signal by a fixed value µ. As an example, Fig. 3.8 shows two delayed versions of a bandlimited signal y(n) = sin( nπ 12 ) where µ = 0.25 and µ = 0.45. In both cases, one set of Farrow subfilters, i.e., Sk (z), has been used and it is only the value of µ in 3 In the implementation, a group of input samples are present in the delay elements of the subfilters and, hence, for every value of µ, its corresponding input sample must be aligned with the center tap of the subfilters. 19 3. BASICS OF MULTIRATE SIGNAL PROCESSING 1 y(n) y(n−0.25) y(n−0.45) 0.8 0.6 Amplitude 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 5 10 15 20 n Figure 3.8: Application of the Farrow structure to delay a bandlimited signal y(n) = sin( nπ 12 ). (3.8) which is modified. This modification corresponds to a new set of multipliers µk , k = 0, 1, . . . , L and, hence, all the samples of y(n) are delayed by µ. In general, SRC can be seen as delaying every input sample with a different value. This delay is determined according to whether one performs decimation or interpolation. In the case of interpolation, one can obtain new samples between any two consecutive samples of the original signal. In the case of decimation, one can shift the original samples (or delay them in the time domain) to the positions which would belong to the decimated signal and, hence, some signal samples will be removed but some new samples will be produced. Thus, by controlling the value of µ in (3.9) for every input sample, the Farrow structure can perform SRC. In this case and for decimation by the Farrow structure, Tout > Tin holds where interpolation results in Tout < Tin . As an example, Fig. 3.9 illustrates two versions of a bandlimited signal y(n) = sin( nπ 12 ) where a rational SRC by Rp = 1.75 is performed. In both cases, the same set of filters as those in Fig. 3.8 has been used and it is only the value of µ(nin ) in (3.8) which is modified for every input sample. Generally, the subfilters Sk (z) in Fig. 3.7 can be designed so that H(z) in (3.8) approximates an allpass transfer function having a fractional delay and over the frequency range4 of interest [42, 43]. Furthermore, by utilizing the Farrow 4 The input of the Farrow structure must be bandlimited to this frequency range so that its 20 3. BASICS OF MULTIRATE SIGNAL PROCESSING 1 0.8 sin(n ωT ) 1 1 sin(n ωT /1.75) 0.6 2 1 sin(n ωT *1.75) 3 Amplitude 0.4 1 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 5 10 15 20 Samples 25 30 35 Figure 3.9: Application of the Farrow structure to perform SRC of a bandlimited signal y(n) = sin( nπ 12 ). structure to realize the polyphase components of general interpolation/decimation filters (with the Nyquist filter being a special case), different filters can be obtained through one set of fixed subfilters and several sets of variable multipliers [33, 34, 44]. These two applications of the Farrow structure will be utilized in the TMUX structures proposed in Chapters 5 and 6. Consequently, the main advantage of the Farrow structure is its ability to perform rational SRC using only one set of fixed subfilters and by simple adjustments in the set of variable multipliers which correspond to µ. The transfer function for a pure delay, i.e., z −µ , with z = ejωT can be expanded using the Taylor series as e−jµωT ≈ L X (−jµωT )k k=0 k! = L X (−jωT )k k=0 k! µk . (3.10) Comparing (3.8) and (3.10), it can be seen that one way to obtain a fractional delay filter is to determine the filters Sk (z) so that they approximate Mk th-order differentiators [42]. This way, the Farrow structure approximates an allpass transfer function in the frequency range of interest. Other methods to design the Farrow structure can be found in, e.g., [45–47]. As will be explained in Section 6.3.1, polyphase components of general interpolation/decimation filters with an integer samples can be delayed. 21 3. BASICS OF MULTIRATE SIGNAL PROCESSING Analysis FB H0(z) Synthesis FB x0(m) N N F0(z) N F1(z) N FM1(z) x1(m) x(n) H1(z) N y(n) xM-1(m) HM1(z) N Figure 3.10: General M -channel FB. SRC ratio of Rp , can be realized using the Farrow structure. Consequently, different filters can be obtained through one set of fixed Farrow subfilters and (Rp − 1) sets of multipliers where each set has L variable multipliers [44]. The majority of these multipliers have equal magnitudes which reduces the total amount of variable multipliers required. By considering the transition band of the general interpolation/decimation filters in the filter design problem, it is possible to design approximately Nyquist filters which can be utilized to construct multimode TMUXs. These will be discussed later in Chapter 6. 3.3 General M -Channel FBs An M -Channel FB splits the input signal into M subband signals by the set of analysis filters Hm (z), m = 0, 1, . . . , M − 1. This way, the subbands can go through different subband processing algorithms. To reconstruct the original input signal, there is a need for synthesis filters Fm (z), m = 0, 1, . . . , M − 1. Furthermore, upsamplers and downsamplers by N are also required as shown in Fig. 3.10. The output of a general M -channel FB can be written as N −1 M −1 X 1 X n Y (z) = X(zWN ) Hm (zWNn )Fm (z). N n=0 m=0 (3.11) Hence, the set Xk (z) forms a time-frequency sampled representation of the original signal X(z) and is also referred to as the subband signals. Ideally, the output signal must be a scaled (by α) and delayed (by β) version of the input signal, i.e., y(n) = αx(n − β). Such a system is referred to as PR. If the FB is not PR, there is some aliasing and distortion present and, hence, the value of α is frequency dependent. In this case, the distortion transfer function can be written as V0 (z) = M −1 1 X Hm (z)Fm (z), N m=0 22 (3.12) 3. BASICS OF MULTIRATE SIGNAL PROCESSING where the aliasing transfer function is given by Vn (z) = M −1 1 X Hm (zWNn )Fm (z), N m=0 (3.13) and n = 1, 2, . . . , N − 1. To get a PR system, we must have V0 (ejωT ) Vn (ejωT ) = c, = 0, n = 1, 2, . . . , N − 1, (3.14) for all ωT with c > 0 being a constant. A special class of FBs is achieved by letting N = M which is called a maximally decimated FB in which the number of samples in the set Xk (z) is equal to that of X(z). On the contrary, the choice of N < M leads to the so called oversampled FBs [37]. Depending on how the synthesis and analysis filters are derived from a prototype filter, different classes of FBs, e.g., cosine and discrete Fourier transform (DFT) modulated, are achieved. Furthermore, the choice of synthesis and analysis filters having uniform or nonuniform passbands leads to uniform or nonuniform FBs. Additionally, the synthesis and analysis filters can in general be FIR or IIR. However, the discussion on these issues is not the focus of this thesis and the interested reader is referred to relevant literature such as [37, 39]. In Chapter 4, a variable oversampled complex modulated FB will be discussed and utilized. 3.4 General M -Channel TMUXs By definition, a TMUX converts the time multiplexed components of a signal into a frequency multiplexed version and back [48]. It can also be used for applications such as channel equalization, channel identification, etc. In [49], it was shown that a FB and a TMUX are duals and the transposition of the analysis/synthesis FBs gives the dual TMUX. At the transmitter side, different source signals are multiplexed into one transmit signal by upsamplers and synthesis filters. On the receiver side, the received signal is decomposed into source signals by analysis filters and downsamplers. As it can be predicted, nonideal analysis/synthesis filters result in distortion as well as cross talk between channels. Since analysis/synthesis filters are reversed, the analysis bank removes cross talk introduced by the synthesis bank. 3.4.1 Mathematical Representation of TMUXs Suppose we have a series of symbol streams sk (n), k = 0, 1, . . . , M − 1, either generated by different users or parts of a signal generated by one user, and we want to transmit these signals through a channel5 . As shown in Fig. 3.11, we can pass the signals through a series of transmitter (or pulse shaping) filters Fk (z) which according to (3.6), produce the signals 5 To preserve the generality of the proposed TMUXs, in this thesis, the symbol streams are usually chosen to be wideband such as white Gaussian signals. However, in practice, the symbol streams sk (n) can have limited bandwidths. 23 3. BASICS OF MULTIRATE SIGNAL PROCESSING x0(n) s0(n) P F0(z) s1(n) P F1(z) sM-1(n) P FM-1(z) H0(z) P s^0(n) H1(z) P s^1(n) HM-1(z) P ^s (n) M-1 Channel x1(n) ^ y(n) y(n) D(z) e(n) xM-1(n) Figure 3.11: General M -channel TMUX. xk (n) = ∞ X sk (m)fk (n − mP ). (3.15) m=−∞ The term pulse shaping comes from the fact that the filters Fk (z) take symbols of sk (n) and put pulses fk (n) around them. This is similar to the ideal bandlimited interpolation in which the sum of weighted ideal sinc functions produces the desired signal [50]. Here, we have M users transmitting through one common channel which can be described by a linear time invariant (LTI) filter D(z) followed by the additive noise e(n). At the receiver side, the filters Hk (z) separate the signals and only a downsampling by P is needed to get the original symbol streams. In this system, M signals are multiplexed into one common channel and ignoring the effects of the channel, the input-output relationship can be written as Ŝk (z) = M −1 1 X Sk (z)Tki (z), P (3.16) k=0 where the transfer function Tki (z P ) = P −1 X Fk (zWPl )Hi (zWPl ), (3.17) l=0 relates the output signal ŝi (z) to the input signal sk (z) and WP is defined as in (2.17). Typical characteristics of the filters Fk (z) and Hk (z) are shown in Fig. 3.12. Similar to FBs, TMUXs can be redundant or minimal where the choice of P > M results in a redundant TMUX as opposed to a minimal TMUX in which P = M . The output of the TMUX in (3.16) can also be written as Ŝk (z) = Vkk (z)Sk (z) + P −1 X Vik (z)Si (z) (3.18) i=0,i6=p where Vkk (z) and Vik (z) represent ISI and ICI, respectively [51]. In general, it is desired to have |Vkk (z) − z −ηk | ≤ δISI and |Vik (z)|≤δICI with δISI and δICI being the desired ISI and ICI where ηk is the delay at each branch k of the TMUX. 24 3. BASICS OF MULTIRATE SIGNAL PROCESSING (a) F0(z) F1(z) FM-1(z) wT p (b) F0(z) F1(z) FM-1(z) wT p (c) F0(z) F1(z) FM-1(z) wT p Figure 3.12: M -channel TMUX filters. (a) Overlapping. (b) Marginally overlapping. (c) Non-overlapping. If an LTI filter g(n) is placed between an upsampler and a downsampler of ratio M , the overall system is equivalent to the decimated version of the filter impulse response which becomes g(nM ) [37]. In other words, g(nM ) is the zeroth polyphase component of g(n) as defined in (2.4). In this case, designing the transmit/receive filters so that the decimated version of Fk (z)Hm (z) becomes a pure delay if k = m and zero otherwise, the TMUX becomes a PR system. In a PR system, ŝk (n) = αsk (n − β) which means that the output is a scaled (by α) and delayed (by β) version of the input. The PR properties are independent of filter lengths, causality of filters, etc., and can be satisfied for both minimal and redundant TMUXs. However, for the minimal case, there may not always exist FIR or stable IIR solutions. So, allowing some redundancy will make the solutions feasible. In other words, making the TMUX redundant results in simpler PR conditions. Due to the redundancy, the stopband attenuation of the filters controls the level of cross talk and aliasing. Thus, these terms can be made as small as desired by a proper filter design. 3.4.2 Approximation of PR in redundant TMUXs In this thesis, we will always design systems which approximate PR according to some criterion. The PR condition for a TMUX states that for any two branches k and m, the decimated version of the cascade of synthesis and analysis filters, i.e., [Fk (ejωT )Hm (ejωT )]zeroth , becomes a pure delay if k = m and zero otherwise. In order to approximate PR, the filters should be designed such that they approximate these ideal conditions as close as desired. Thus, assuming the synthesis and analysis filters to be Fk (z) and Hm (z), we have minimize δ, subject to 25 (3.19) 3. BASICS OF MULTIRATE SIGNAL PROCESSING F0(z) F1(z) F2(z) FM-1(z) wT 2p Figure 3.13: Filters of a nonuniform non-overlapping TMUX. |[Fk (ejωT )Hm (ejωT )]zeroth − 1|≤δ, jωT |[Fk (e jωT )Hm (e )]zeroth |≤W (ωT )δ, ωT ∈ [0, π], k = m ωT ∈ [0, π], k6=m where W (ωT ) is the weighting function which controls the approximation error in the filter design. It is well known that increasing the order of synthesis and analysis filters makes it possible to decrease δ and, hence, improve the approximation of PR. To further simplify (3.19), we will use redundant TMUXs with non-overlapping filter responses as shown in Fig. 3.13. It can also be seen that the TMUXs are nonuniform in the sense that the passbands of the filters are different. Consequently, ISI in (3.18) would result from the filters in one branch of the TMUX. In nonuniform TMUXs, the term ICI in (3.18) becomes time-varying. However, due to the redundancy, the stopband attenuation of the filters still controls ICI and, therefore, ICI in (3.18) can be made as small as desired by appropriate stopband attenuation of the filters. To approximate PR as close as desired, the filter Fk (z)Hk (z) should approximate a Nyquist filter as close as desired meaning that the synthesis and analysis filters should be designed such that • They have sufficiently small ripples in their stopbands. • The zeroth polyphase component of Fk (z)Hk (z) approximates an allpass transfer function. Specifically, the filters Fk (z) and Hk (z) are designed so that their cascade approximates a Nyquist filter which in turn necessitates requirements on the transition band of Fk (z) and Hk (z) as well as the passband of the zeroth polyphase component of Fk (z)Hk (z). This results in the simplified minimax design problem as minimize δ, subject to |[Fk (ejωT )Hk (ejωT )]zeroth − 1|≤δ, jωT |Fk (e )|≤W1 (ωT )δ, jωT |Hk (e )|≤W2 (ωT )δ, (3.20) ωT ∈ [0, π] ωT ∈ Ωs ωT ∈ Ωs where k = 0, 1, . . . , M − 1. Furthermore, W1 (ωT ) and W2 (ωT ) are the weighting functions with Ωs defined as in (2.18) representing the stopband of the filters. In the LS sense, (3.20) can be written as minimize 26 (3.21) 3. BASICS OF MULTIRATE SIGNAL PROCESSING Z ωT ∈[0,π] + + |[Fk (ejωT )Hk (ejωT )]zeroth − 1|2 |Fk (ejωT )|2 ωT ∈Ωs W1 (ωT ) Z |Hk (ejωT )|2 . ωT ∈Ωs W2 (ωT ) Z In this thesis, we have frequently used (3.20) and (3.21) with Fk (z) = Hk (z) and W1 (ωT ) = W2 (ωT ) = W (ωT ). In both approaches, a large (small) value for W (ωT ) allows large (small) ripples in the stopband of filters. In this thesis, we will always design real lowpass filters and variable frequency shifters will be utilized to modulate the users into intermediate frequencies. 27 3. BASICS OF MULTIRATE SIGNAL PROCESSING 28 4 Flexible Frequency-Band Reallocation For Real Signals This chapter discusses a new approach for implementing FFBR networks for bentpipe [52] (or transponder [13]) satellite payloads which are based on variable oversampled complex modulated FBs. We consider two alternatives to process real signals using real input/output and complex input/output FFBR networks (or simply, real and complex FFBR networks, respectively). After some general and historical introduction in Section 4.1, Section 4.2 briefly reviews the FFBR network whose real and complex variants are used in both alternatives. Alternative I is discussed in Section 4.3 where the arithmetic complexities of its different building blocks, viz., the Hilbert transformer, DFT, and the complex FFBR network, are treated in detail. Section 4.4 introduces Alternative II and covers the formulation of FBR for real signals, system functionality illustration, and the arithmetic complexity derivation. Comparison of the two alternatives is discussed in Section 4.5 where the number of real operations; growth rate in arithmetic complexity with respect to the prototype filter order and number of FB channels; trend of spectrum efficiency; and the performance of Alternatives I and II based on error vector magnitude (EVM) for a 16-QAM signal are outlined. Finally, Section 4.6 gives some concluding remarks. 29 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 4.1 Introduction As discussed in Section 1.1, ESA has proposed three major network structures for broadband satellite-based communication systems in which satellites communicate with users through multiple spot beams. This gives rise to the necessity of reusing the limited available frequency spectrum by satellite on-board signal processing. The digital part of the satellite on-board signal processor is a MIMO system where input/output signals can be composed of different users with different bandwidths and bit rates. The on-board signal processor reallocates all users to different output signals and positions in the frequency spectrum. In a system supporting bandwidthon-demand, the bandwidths of different users may vary with time which is handled by dividing the input beam into a number of granularity bands (GBs). At any time, any user can occupy any rational number of GBs. There are several requirements on FFBR networks as: • Flexibility to handle all FBR scenarios on users from different telecommunication standards and without restricting the system throughput. • Low complexity to reduce the implementation cost. It is foreseen that the required amount of improvements in system capacity and implementation complexity are about one and two orders of magnitude, respectively [10]. • Near PFBR to satisfy any communication performance metric, e.g., bit error rate (BER), EVM, etc. [53, 54]. • Simplicity resulting in simple system analysis and design. 4.1.1 Relation to Previous Work Generally, there are four types of on-board signal processing architectures (or payloads), viz., bentpipe, full processing, partial processing, and hybrid [52]. This chapter focuses on the application of FBs for bentpipe payloads whose principle is shown in Fig. 4.1. A bentpipe payload reallocates different users with different bandwidths to different positions in the frequency spectrum. To support dynamic communications, the bandwidth and position of the users may change in a timevarying manner necessitating some requirements on FFBR networks. As shown in Fig. 4.2, FB-based FBR [14–25] makes use of decimation and interpolation to generate frequency shifts of users. These approaches can be classified as maximally decimated FBs [15, 16], tree-structured FBs [14, 15, 20–25], overlap-save DFT/inverse DFT (IDFT) techniques [17], and oversampled complex modulated FBs [18, 19]. In [19], a new class of FFBR networks based on FIR variable oversampled complex modulated FBs for bentpipe payloads was introduced and an efficient implementation structure was derived. Furthermore, it was proved that the system can approximate PFBR as close as desired via a proper design. The system in [19] processes complex signals which means that the analytic representation of the real uplink satellite signals must be processed by the FFBR network and the frequency 30 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Output signal 1 In 1 3 1 2 p wTin [rad] Input signal 2 4 5 In 2 6 p FFBR Network Input signal 1 wTin [rad] 3 1 p Out 1 wTout [rad] Output signal 2 Out 2 Out 3 5 4 p wTout [rad] Output signal 3 2 6 p wTout [rad] Figure 4.1: Principle of FBR for an FFBR network where any signal in any of the 2-input signals can be reallocated to any position in any of the 3-output signals. multiplexed results should then be converted to real signals for retransmission. This requires the implementation of one complex FFBR network as well as two Hilbert transformers. Throughout this chapter, we will refer to this solution as Alternative I and it is shown in Fig. 4.8. As the number of FB channels increases, the transition band of the Hilbert transformers becomes smaller resulting in high-order filters and increased arithmetic complexity. In this chapter, we introduce another alternative to process real signals through a real input/output FFBR network1 which in general requires less processing units, i.e., real adders and multipliers, than the complex FFBR network. This solution is referred to as Alternative II and is shown in Fig. 4.10. In addition, the real FFBR network eliminates the need for two Hilbert transformers making it suitable for systems with a large number of FB channels or, equivalently, systems with a large number of users having different bit rates. 4.1.2 Remark on the Choice of FFBR Network for Complexity Comparison All the solutions in [14–25] process complex signals and, thus, regardless of the FFBR network chosen, the main aim of this chapter is to show that the approach in Fig. 4.10 is superior to that of Fig. 4.8. In other words, by using other FB-based FFBR networks, it is only the exact number of operations that changes but the superiority of system in Fig. 4.10 will still be preserved. However, in this chapter we will focus on the FFBR network in [19] due to the reasons outlined in [29]. 4.1.3 MIMO FFBR Network Configuration The FFBR network considered here is in general an m-input n-output system where m≤n. However, it is sufficient to only discuss the design and properties of the 1 The FFBR network in Fig. 4.4 has complex multipliers and, hence, it is a complex system by nature. However, by real (complex) FFBR, we differentiate between two alternatives of Fig. 4.4 having real (complex) input/output signals. In other words, the complex multipliers are present in both the real and complex FFBR networks. 31 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Decimation (a) x H(z) T1 v1 Interpolation M T1 v2 M T2 v3 T1 G(z) y T1 X(ejwT1) (b) X0 0 X1 2p/M X2 4p/M XM1 6p/M 2p wT1 H(ejwT1) (c) 0 2p/M 4p/M 2p wT1 4p/M 2p wT1 (ejwT1) (d) V1 X1 0 2p/M V2(ejwT2) (e) X1 0 X1 X1 2p X1 4p 2Mp wT2 V3(ejwT1) (e) X1 0 X1 2p/M X1 X1 4p/M 2p wT1 G(ejwT1) (f) 4p/M 0 6p/M 2p wT1 6p/M 2p wT1 Y(ejwT1) (g) X1 0 4p/M Figure 4.2: FBR using decimation and interpolation. Here, only one channel of the FB is shown but in general, channel combiners are needed to produce the outputs from several FB channels. single-input single-output (SISO) case as the MIMO case is a duplication of fixed SISO structures (refer to Figs. 17 and 23 of [19]) along with some modifications2 . Consequently, we focus on the SISO case and further discussions on the MIMO system for both m < n and m = n can be found in [19, 55]. 2 For the MIMO case, the channel switch operates between several SISO structures. Furthermore, if m < n, some branches at the output of the DFT block in Fig. 4.4 are set to zero. 32 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Fixed Analysis FB H0(z) M M G0(z) H1(z) M M G1(z) HN1(z) M M GN1(z) y0(n) Channel Combiner x(n) Adjustable Synthesis FB y1(n) y(n) yq1(n) Flexible Frequency-Band Reallocation Network Figure 4.3: N -channel FFBR network utilizing fixed analysis filters and adjustable synthesis filters. 4.2 Review of the FFBR Network Based on Variable Oversampled Complex Modulated FBs As shown in Fig. 4.3, the FFBR network uses fixed analysis filters Hk (z), k = 0, 1, . . . , N − 1 to split the input signal x(n) into N subbands. Then, downsampling/upsampling by M and the adjustable synthesis filters Gk (z) perform the required frequency shifts of the subbands as well as the recombination of subbands to form the multiplexed output signal y(n). As adjustable synthesis filters result in high implementation cost, the FFBR network is realized using fixed synthesis filters and a channel switch. This requires appropriate choice of system parameters and filter characteristics significantly reducing the arithmetic complexity while having the same functionality as the network in Fig. 4.3. In the next subsection, the efficient realization of the FFBR network with fixed analysis/synthesis filters and a channel switch will be discussed. 4.2.1 Efficient Realization of the FFBR Network Figure 4.4 shows the architecture of the N -channel FFBR network which consists of complex multipliers, DFT, IDFT, polyphase components, and input/output commutators. Briefly, the system assumes that the complex input signal is divided into Q GBs where N (4.1) Q = , A > 1, A ∈ N A and the GBs are separated by a guardband of 2∆ = 2πǫ Q with 0≤ǫ≤1. The choice of ǫ defines the transition band as well as the order of the filters. However, large transition bands would reduce the amount of frequency spectrum covered by the GBs and, hence, there is a tradeoff to be made. Any user can occupy any rational number of GBs meaning that users can have arbitrary variable bandwidths and, through this, bandwidth-on-demand is supported. To suppress aliasing and at the 33 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS aN x(n) M a0 g0 b0 aN1 aN P1(zLWN ) M aN PN1(zLWN ) a1 b1 IDFT z1 aN1 Channel Switch z1 M aN PN1(zLWN ) P0(zLWN ) z1 g1 aN2 aN PN2(zLWN ) DFT M z1 gN1 bN1 M a0 aN P0(zLWN ) M y(n) FFBR Network Figure 4.4: Efficient DFT- and IDFT-based implementation of the N -channel FFBR network in Fig. 4.3. |PR| 2D wT p p/N 0 p/N p Figure 4.5: Characteristics of the prototype filter. same time, to shift the GBs by all values of 2πq Q , the parameter M should be a multiple of Q as3 M = BQ, B ∈ N. (4.2) Furthermore, to attenuate the aliasing terms by the stopband attenuation of the filters, it is required that the passbands and transition bands of the shifted terms do not overlap. This is achieved if M≤ N < N. 1 + Nπ∆ (4.3) Assuming the length-S linear-phase prototype filter with the transfer function P (z) = S−1 X z −n p(n) = n=0 N −1 X z −i Pi (z N ) (4.4) i=0 and the frequency response P (ejωT ) = e−j ωT (S−1) 2 PR (ωT ), (4.5) the analysis filters are 3 It is noted that N = M A = M L where L is the number of FB channels per GB. In the B examples illustrated in this chapter, we assume L = 2. 34 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Hk (z) = βk P (zWNk+α ) = βk N −1 X [z −i αi Pi (z N WNαN )]WN−ki . (4.6) i=0 Here, k = 0, 1, . . . , N − 1 and WN αi βk 2π = e−j N = WN−αi (4.7) (4.8) (k+α)D 2 = WN . (4.9) Furthermore, PR (ωT ) is the real zero-phase frequency response (with the magnitude response shown in Fig. 4.5) and α is a real-valued constant to place the filters at the desired center frequencies. The multipliers βk compensate for the phase rotations due to substitution of P (z) with P (zWNk+α ). Therefore, all the analysis filters become linear-phase filters with the same delay as the prototype filter. The multipliers γk in the synthesis filter bank (SFB) are γk = βk WN−k , (4.10) Gk (z) = µkr Hckr (z), (4.11) whereas the synthesis filters are with ckr µkr mr = k + Asr (4.12) mr N (S−1) 2M = WN ( Bsr = M + Bsr (4.13) sr ≥ 0 sr < 0. (4.14) Each parameter sr is the number of GBs by which subband r is shifted and it is positive (negative) if the subband is shifted to right (left). This information is required to program the channel switch. Specifically, programming the channel switch requires knowledge of L, sr , and the number of GBs each user occupies. A point to note is that the complex constants µkr in (4.13) can be made equal to unity by a proper choice of the prototype filter order and at the cost of some additional delay. Throughout this chapter, we will assume that µkr = 1 and, therefore, as opposed to Fig. 15 of [19], these multipliers are not shown in Fig. 4.4. 4.3 Alternative I In this section, we will discuss Alternative I which consists of a complex FFBR network as well as two Hilbert transformers. 35 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS cos(wt) xa(t) Anti-Image ADC I Anti-Image ADC Q sin(wt) Figure 4.6: Complex sampling. xa(t) ADC Digital Processor I Q Figure 4.7: Real sampling. 4.3.1 Complex vs. Real Sampling As the uplink satellite signal is real, to process it with the FFBR network of Fig. 4.4, the real analog signal xa (t) must be converted into its analytic representation. This can be done either by complex sampling using two 90 degree out-of-phase analog to digital converters (ADCs) as shown in Fig. 4.6, or by real sampling as shown in Fig. 4.7 which consists of an ADC followed by a digital processor. The digital processor is composed of a Hilbert transformer followed by a downsampler [56]. In the complex sampling case, the ADCs operate at a lower frequency f2s compared to the real sampling structure in which the ADCs operate at fs . The accuracy in converting the real signal into its analytic representation limits the system performance and is an important factor in wideband multichannel communications [56]. In addition to the requirements to attenuate the images due to the mixers of Fig. 4.6, the high sensitivity to match two ADCs using analog components4 makes the real sampling approach more attractive [56, 57]. Hence, the system in Fig. 4.8 can be used to process real signals and its arithmetic complexity will be discussed in the next subsections5 . 4.3.2 Arithmetic Complexity of the Hilbert Transformer The complex analytic representation of a real signal has a zero-valued spectrum for all negative frequencies and can be realized by IIR or FIR filter approximations [35]. To generate the analytic representation, the real signal x(n) must be applied to a complex half-band filter HHB,c (z). This filter is derived by shifting the frequency 4 Matching the two ADCs reduces the variations in the gain and phase transfer function between I and Q channels. 5 If the sampling frequency f in Figs. 4.6 and 4.7 is high and due to the limitations of s technology in achieving high sampling frequencies for ADCs, the complex sampling method where fs ADCs operate at 2 may be the only option. However, we assume that the operating frequency of ADCs is at a reasonable range and, hence, the real sampling approach can be utilized. 36 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS x(n) fs xa(t) ADC y(n) fs Complex FFBR 2 Hilbert fs/2 2 Hilbert DAC ya(t) Figure 4.8: Alternative I: complex FFBR network with Hilbert transformer. response of a real lowpass half-band filter HHB,r (z) by π 2 HHB,c (z) = jHHB,r (−jz). radians6 as [35] (4.15) The order of a lowpass half-band linear-phase FIR7 filter can be approximated as [58] NH ≈ 2π −2 log10 (10δs 2 ) 3 ωs T − ωc T (4.16) where δs is the stopband attenuation with ωs T and ωc T being the stopband and passband edges, respectively. To process a specific GB by the FFBR network, it should be covered by the passband of HHB,c (z) which, according to Fig. 4.9, is satisfied if8 2π π 2∆ + k Q + , 2 2 2π π 2∆ + k Q ωc T = − 2 2 ωs T = (4.17) where k = 0, 1, . . . , ⌊ Q 2 − ǫ⌋ with ⌊x⌋ being the floor of x. This way, ωs T < π and ωc T > 0 and the filter order becomes NH ≈ −2 Q log10 (10δs 2 ) . 3 ǫ+k (4.18) 6 A scaling by two may also be required depending on whether the Hilbert transformer is used for interpolation or decimation (refer to Fig. 4.8). However, this does not affect the filter order and is, therefore, not considered here. 7 According to [45], at typical stopband attenuations of 60 −80dB, the arithmetic complexity of an IIR half-band filter is around 60 − 70% of that of an FIR half-band filter. Hence, IIR half-band filters may result in less arithmetic complexity. However, as mentioned in Section 4.1.2, the main conclusion of this chapter is independent of the choice of IIR and FIR filters. 8 Based on the system requirements, the Hilbert transformer characteristics in Fig. 4.9 may not spread over the negative frequencies. In this case, to design Hilbert transformers, a wideband lowpass filter should be shifted in frequency. Hence, the transition band of the filter becomes ∆ + 2kπ which results in higher orders for the Hilbert transformer and further increases the Q superiority of the approach in Section 4.4. However, as opposed to a half-band filter, a general linear-phase FIR filter does not require that δs = δc where δc is the passband ripple. 37 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 2D+4p/Q Hilbert Transformer Characteristics 2D+2p/Q 2p 2pa/Q 2D wTin Figure 4.9: Illustration of the GBs and the characteristics of the Hilbert transformer to control the filter order. In other words, having chosen fixed values for δs , Q, and ǫ, the factor k relates NH to the spectrum efficiency η(k) as9 η(k) = 2π − 2(∆ + 2π kπ Q) =1− k+ǫ . Q (4.19) The efficiency is defined as the ratio between the part of the frequency spectrum used by the complex FFBR network and the whole frequency spectrum. In other words, it is the ratio between the passband of the Hilbert transformer and 2π. Ideally, k should be zero but in systems with a large number of FB channels, small values of k will result in extremely high-order filters that add a large arithmetic complexity as will be discussed in Section 4.5.2. It is noted that the order of HHB,c (z) is equal to that of HHB,r (z) as the frequency shift in (4.15) does not alter the filter order. 4.3.3 Arithmetic Complexity of the DFT with Complex Inputs In general, the order of arithmetic complexity for an N -point complex-input DFT is O(N 2 ) and requires N 2 complex multiplications and N (N − 1) complex additions [35]. However, using specialized fast Fourier transform (FFT) algorithms, the arithmetic complexity for the DFT of a radix-2 length sequence can be reduced to O(N log2 N ). Any sequence length can be made a power of two by zero padding [35] making it possible to utilize these efficient FFT algorithms. In this chapter, we focus on the overall number of real operations and we select the algorithms in [59–62] which require N (log2 N − 3) + 4 real multiplications and 3N (log2 N − 1) + 4 real additions. However, other realizations such as [63–66] can also be used. The aim of this chapter is not to compare different DFT techniques and choosing any other DFT technique will only change the exact number of operations but the main conclusions will still be preserved. 9 The parameters Q and ǫ are imposed by the FFBR network as defined in Section 4.2. Furthermore, according to Fig. 4.9, only the passband of the Hilbert transformer affects the GBs. Hence, δs is chosen to approximate PFBR as close as desired. 38 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 4.3.4 Arithmetic Complexity of the Complex FFBR Network As shown in Fig. 4.4, the building blocks of the FFBR network, viz., polyphase components, DFT, IDFT, and complex multipliers, are sandwiched between input/output commutators [35]. These building blocks produce samples in N parallel branches but the output commutator only retains one sample out of these branches. Consequently, to produce one sample y(n) at the output, the number of operations required by these building blocks must be divided by N . However, both the Alternatives I and II use the architecture in Fig. 4.4 and, hence, in the computation of the arithmetic complexity, we can ignore the division by N as it will be common for both the alternatives. Thus, assuming x(n) to be the analytic representation of a real signal, the computational workload to produce one complex sample by the N -channel complex FFBR network of Fig. 4.4 is S • 2N real filters of length N for the analysis filter bank (AFB) and SFB operating on complex data requiring 4S real multiplications and 4(S − N ) real additions10 . • Complex multipliers αi , βk , γk resulting in 4N complex multiplications11 . • N -point complex-input IDFT and DFT resulting in 2N (log2 N − 3) + 8 real multiplications and 6N (log2 N − 1) + 8 real additions. The input/output commutators and the channel switch do not require any computational workload. To convert complex operations to real operations, it is assumed in [61] that a complex multiplication requires 3 real multiplications and 5 real additions (3/5) whereas [63] and [35] use (3/3) and (4/2) assumptions, respectively. Furthermore, one complex addition requires 2 real additions. Thus, using the (4/2) assumption, the overall arithmetic complexity becomes 2Nlog2 N + 2(4 + 2S + 5N ) real multiplications and 6N log2 N + 2(4 + 2S − N ) real additions whereas (3/3) results in 2N log2 N +2(4+2S +3N ) real multiplications and 6N log2 N +2(4+2S +N ) fs real additions. These operations run at the operating frequency of 2M where M < N for an oversampled system12 . Furthermore, the two Hilbert transformers in Fig. 4.8, add the arithmetic complexity of two complex half-band filters running at the operating frequency of f2s if implemented using the polyphase decomposition. 10 Depending on the prototype filter length and the number of FB channels, the polyphase components may not be of the same length resulting in less arithmetic complexity. Thus, the arithmetic complexity derived here is the worst case scenario. However, if the prototype filter order is chosen such that µkr = 1 which is done in this chapter, then, all the polyphase components will have the same length and, hence, there is a tradeoff to be made. 11 According to (4.8), choosing α = 0, 0.5 leads to further reductions in the arithmetic complexity. However, this reduction applies to both the alternatives discussed in this chapter and is therefore not considered here. 12 According to Section 3.2. in [19], for a fixed value of M and by a proper choice of N , the arithmetic complexity can be minimized. However, this minimization is common for both the Alternatives I and II and, hence, does not affect the overall arithmetic complexity comparison. 39 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS x(n) fs xa(t) ADC fs Real FFBR y(n) fs DAC ya(t) Figure 4.10: Alternative II: real FFBR network without Hilbert transformer. 4.4 Alternative II In this section, we will introduce the Alternative II which utilizes a real FFBR network (see Footnote 1) and eliminates the need for Hilbert transformers. As shown in Fig. 4.10, to implement the FFBR network for real signals, one can sample the real signal at a sampling frequency of fs and feed the sampled data into the real FFBR network at the same sample rate. Basically, the real FFBR network uses the structure of the system in Fig. 4.4 with a difference that the polyphase filters in both AFB and SFB operate on real data and in the SFB, only the real parts of the multiplications by αi need to be computed. Similar to the discussion in Section 4.3 and as shown in Fig. 4.11, the N -channel real FFBR network assumes that the input signal is divided into Q GBs separated by 2∆ and each GB is divided into a number of uniform-band FB channels which are constructed by complex modulating a real linear-phase prototype filter as in (4.6). Like the complex-input case, FBR is performed by dividing the real input signal into subbands by the use of the analysis filters Hk ; shifting the subbands; and recombining the shifted subbands by the use of synthesis filters Gk . However, as can be seen from Fig. 4.11, only half of Hk and Gk are involved in the FBR since the spectrum of a real signal spreads in [−π, π] whereas the filters in AFB and SFB of the FFBR network are defined in [0, 2π]. To illustrate the functionality of the system for real signals, Fig. 4.12(a) shows the frequency spectrum of an input signal consisting of three users with different bandwidths. This pattern has been generated by the nonuniform TMUX in [31, 32]. As discussed in Section 4.1.1, FBR is performed by the use of fixed analysis (synthesis) filters and a channel switch. Figures 4.13 and 4.14 show two different channel switch configurations to perform two scenarios of FBR. The corresponding multiplexed output signals are shown in Figs. 4.12(b) and (c). 4.4.1 Arithmetic Complexity of the Real FFBR Network Considering the discussions in the previous subsection, we have a system with real input and real output with an arithmetic complexity of • 2S real multiplications and 2(S−N ) real additions due to real input polyphase filters. • Complex multipliers βk and γk requiring 2N complex multiplications. 40 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS (a) Granularity Band 2p/Q 0 2pa/Q 2p/Q+2pa/Q (b) Guard Band -p X2 X0 X1 -p X1 X0 p X2 p 0 (c) wTin wT H0 H1 H2 H3 H4 H5 H6 H7 -p p 0 (d) wT G0 G1 G2 G3 G4 G5 G6 G7 -p p 0 (e) H0G0+H1G1 Y1 Y2 -p H2G2+H3G3+H4G4+H5G5 Y1 Y0 Y0 wT H6G6+H7G7 Y2 p 0 (f) wT G2 G3 G4 G5 G6 G7 G0 G1 -p Y0 -p p 0 (g) Y2 6 6 6 H2G2+H3G3+H G +H6 G 4 4 5 5 6 6 H6G6+H7G7 3 3 H0G0+H1G1 Y1 Y2 Y0 Y1 0 p wT wT Figure 4.11: Illustration of FBR by the real FFBR network with Q = 8 and N = 16. (a) Granularity bands. (b)–(e) Recombination of channels. (b),(c),(f), and (g) Combination of channels and reallocation of subbands; Hkm stands for Hk shifted m GBs to the right. • N -point complex-input IDFT and DFT resulting in 2N (log2 N − 3) + 8 real multiplications and 6N (log2 N − 1) + 8 real additions. • Complex multiplication of αi on the real outputs of the analysis filters resulting in 2N real multiplications. • Complex multiplication of the DFT outputs by αi only to compute the real part requiring 2N real multiplications and N real additions. Consequently, using a (4/2) assumption, the overall arithmetic complexity will be 2N log2 N + 2(4 + S + 3N ) real multiplications and 6N log2 N + 2(4 + S − 23 N ) real additions whereas a (3/3) assumption results in 2N log2 N + 2(4 + S + 4N ) real multiplications and 6N log2 N + 2(4 + S − 21 N ) real additions operating at fs the operating frequency of M . In this approach, there is no need for the Hilbert 41 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Mag. [dB] (a) 0 −20 −40 −60 −π X0 −0.75π −0.5π −0.25π 0 ω [rad] X1 0.25π X2 0.5π π 0.75π Mag. [dB] (a) 0 −20 −40 −60 −π X X 1 −0.75π −0.5π −0.25π 0 ω [rad] X 2 0.25π 0 0.5π π 0.75π Mag. [dB] (a) 0 −20 −40 −60 −π X2 −0.75π −0.5π −0.25π 0 ω [rad] 0.25π X0 0.5π X1 0.75π π Synthesis Bank Input Analysis Bank Output Figure 4.12: Spectrum of (a) Real input to the FFBR network. (b) and (c) Multiplexed output signals based on the FBR scenarios in Fig. 4.13 and Fig. 4.14. Q = M = 10, N = 20, ǫ = 0.125, α = 0.5. Figure 4.13: Scenario I resulting in the values of sr in (4.12) to be {3, −2, −2}. transformers. This brings considerable savings in the arithmetic complexity as the operating frequency of the Hilbert transformers in Fig. 4.8, is M times that of the complex FFBR network. According to the problem formulation in Fig. 4.11, the FB channels between [π, 2π] are not involved in the reallocation of the subbands. Thus, more savings in the arithmetic complexity of the IDFT, DFT, multipliers βk , and γk can be achieved. However, the amount of these savings depends on the channel switch position for each scenario of FBR and is thus not considered here13 . The main advantage of the approach utilizing the real FFBR network is the elimination of 13 Using the concept of quick Fourier transform (QFT) defined in [67] and splitting DFT/IDFT kernel into discrete sine and cosine transforms, more arithmetic complexity savings inside the DFT/IDFT blocks can also be achieved. However, further discussion on these savings is outside the scope of this chapter. 42 Synthesis Bank Input Analysis Bank Output 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS Figure 4.14: Scenario II resulting in the values of sr in (4.12) to be {2, 2, −3} the Hilbert transformers which is independent of the channel switch operation. Furthermore, the unused FB channels in the real FFBR network can be shared in a MIMO configuration specially in the case of an m-input n-output system where m < n. 4.5 Comparison of Arithmetic Complexity and Performance This section compares the arithmetic complexity and performance of the alternatives introduced in Sections 4.3 and 4.4 based on the overall number of real operations. We consider that both alternatives have roughly equal performances and for this performance, a comparison of the arithmetic complexity is performed. First, we will discuss the arithmetic complexities of the individual real and complex FFBR networks as two variants of Fig. 4.4. Second, the alternatives of Figs. 4.8 and 4.10 will be considered. Finally, based on EVM as a performance metric, we will compare the performance of Alternatives I and II for a 16-QAM signal. The main aim of the performance comparison is to show that both structures can be designed to have similar performances but Alternative I would then require additional filters (arithmetic complexity) due to the presence of the Hilbert transformers. In later discussions and unless otherwise mentioned, the system parameters are those used in Fig. 4.12. 4.5.1 Arithmetic Complexity of Complex FFBR vs. Real FFBR According to the arithmetic complexity relations derived in Sections 4.3.4 and 4.4, the arithmetic complexity depends on the prototype filter length S and the number of FB channels N . Under these assumptions, Fig. 4.15 compares the number of real operations for a 20-channel FFBR network in real and complex cases. As can be seen, the complex FFBR network has more computational workload than its real counterpart. In addition, the rate of increase in the number of real operations is larger in the case of the complex FFBR network. 43 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 7 Real FFBR Total Complex FFBR Total Real FFBR Add. Real FFBR Mult. Complex FFBR Add. Complex FFBR Mult. Real operations (*1000) 6 5 4 3 2 1 200 300 400 500 600 Prototype filter length 700 800 Figure 4.15: Number of real operations in the real and complex FFBR networks. However, the operating frequency of the real FFBR network is twice that of the complex FFBR network. This means that in one time unit, every adder or multiplier of the complex FFBR network can perform two computations whereas the adders and multipliers of the real FFBR network perform only one computation. Thus, the ratio between the number of operations in one time unit, as shown in Fig. 4.16, compares the arithmetic complexity at the same operating frequency. It can be seen that for high prototype filter orders, the real FFBR network performs roughly 10 − 15% more operations in one time unit. According to the arithmetic complexity relations derived in Sections 4.3.4 and 4.4, for high prototype filter orders, the total arithmetic complexity is dominated by S. Consequently, at high prototype filter orders and for both real and complex FFBR networks, the number of real operations in one time unit are asymptotically equal14 . However, as will be discussed in the next subsection, the real FFBR network eliminates the need for two Hilbert transformers resulting in less overall arithmetic complexity and higher efficiency. 4.5.2 Arithmetic Complexity of Alternative I vs. Alternative II In addition to the individual FFBR networks, as shown in Fig. 4.8, two Hilbert transformers are needed along with the complex FFBR network. These Hilbert transformers add the arithmetic complexity of two complex half-band filters run14 For large values of S and after multiplying the total number of real operations in the real FFBR network by two (as its operating frequency is twice that of the complex FFBR network), the total number of real operations in both real and complex FFBR networks is dominated by 8S. 44 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 1.3 1.28 1.26 Complexity ratio 1.24 1.22 1.2 1.18 1.16 1.14 1.12 1.1 200 300 400 500 600 Prototype filter length 700 800 Figure 4.16: Number of real operations in the real FFBR over that of the complex FFBR. Here, the number of operations in one time unit are considered. ning at the operating frequency of f2s . Recalling from Section 4.3.2, in systems with large number of FB channels, the transition band of the complex half-band filter HHB,c (z) in (4.15) becomes smaller which in turn increases the filter order and the additional arithmetic complexity. To get around this, one can choose not to use some of the GBs to relax the requirements on the filter transition band (see Fig. 4.9). This reduces the additional arithmetic complexity but on the other hand, it reduces the efficiency of the system. To illustrate this, Fig. 4.17(a) shows the order of HHB,c (z) for different percentages of efficiency in a 50-channel complex FFBR network15 . As can be seen, a high efficiency comes at the expense of high-order filters which further shows the superiority of Alternative II. For example, to achieve an efficiency of 95%, the approximate order of the complex half-band filter is estimated to be NH = 74. According to Fig. 4.10, this filter operates on a real sequence at the input side of the FFBR network and on a complex sequence at the output side of the FFBR network. As discussed in Section 2.3.3, the impulse response of a half-band filter is symmetric and every other coefficient is zero. Consequently, to process the real sequence, there is need for NH4+2 real multiplications and N2H real additions. Similarly, the complex sequence can be processed at the cost of NH2+2 real multiplications and NH real additions. Assuming NH = 74, the arithmetic complexity relations derived above may not result in large values but note that these operations run at M = 50 2 times higher frequency than the complex FFBR network. As an example, using a 350th order prototype filter and converting all the operations to the same operating frequency, 15 The superiority of Alternative II is more pronounced for systems with large number of FB channels. 45 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS (a) Order of HHB,c(z) 600 500 400 300 200 100 0 10 20 30 40 50 60 70 80 90 100 70 80 90 100 Efficiency (%) (b) Complexity ratio 1.4 1.2 250th order 300th order 350th order 400th order 1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 60 Efficiency (%) Figure 4.17: Order of the Hilbert transformer and the ratio between the total number of real operations in one time unit for the alternatives of Figs. 4.10 and 4.8 and with δs = 0.001. the overall number of real operations performed by Alternative II in one time unit, is about 85% of that of Alternative I. Figure 4.17(b) shows the ratio of the number of operations in Alternatives I and II at the same operating frequency for four different prototype filter orders. As can be seen, with increase in spectrum efficiency, Alternative II results in more savings in arithmetic complexity16 . To achieve an efficiency17 of 100% in systems with large number of users, the number of real operations performed by Alternative II, in one time unit, will always be less than 50% of that of Alternative I. In conclusion, although the real FFBR network performs around 10 − 15% more operations in one time unit than the complex FFBR network, it eliminates the need for two complex half-band filters. Thus, the approach utilizing the real FFBR network, i.e., Alternative II, has no limitation on the efficiency which means 16 For small efficiencies, the complexity of the Hilbert transformer is not significant and, hence, both the alternatives have roughly similar complexities which is in accordance with Fig. 4.16 as well. If the complexity of the FFBR network reduces, the superiority of Alternative II will further be pronounced. 17 To get realizable filters, a transition band is needed and, hence, the efficiency of 100%, refers ǫ to that achieved by k = 0 in (4.19) which becomes 1 − Q = 99.5% for this example. 46 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 1 Quadrature 0.5 0 −0.5 −1 −1 −0.5 0 0.5 In−Phase 1 Figure 4.18: Alternative I. that it has an efficiency of 100%. At this efficiency and for systems with large number of users, the arithmetic complexity of Alternative II is always less than 50% of that of Alternative I. 4.5.3 Performance of Alternative I vs. Alternative II To compare the performance of Alternatives I and II in a 20-channel system, a 16-QAM signal is processed by these alternatives and the output constellation diagrams are shown in Figs. 4.18 and 4.19. The order of the prototype filter in both cases is 420 and, furthermore, in Alternative I, a 154th order Hilbert transformer is used. To numerically compare the two alternatives, EVM is used which is a metric of transmitter quality in modern communication systems such as IS-54/IS-136, GSM/EDGE, 3GPP-WCDMA, and 802.11a/b/g [53, 54]. The EVM is a statistical estimate of the magnitude of the error vector normalized by the ideal signal and is defined as v u PNs −1 u |e(k)|2 EV M dB = 20 log(t PNsk=0 ) (4.20) −1 2 k=0 |sref (k)| where e(k) = s(k) − sref (k) is the complex error sequence with s(k) and sref (k) being the length-Ns measured and reference complex sequences, respectively18 . Besides the characteristics of the prototype filter which is common for both alternatives, Alternative I uses two complex half-band filters whose characteristics further affect EVM. In this example, the filters were designed so that both the 18 In the rest of the thesis, EVM will be used to compare different systems and filter design methods. 47 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 1 Quadrature 0.5 0 −0.5 −1 −1 −0.5 0 0.5 In−Phase 1 Figure 4.19: Alternative II. alternatives result in roughly equal EVMs. The resulting EVMs for the user X0 in Fig. 4.12, after being processed by Alternatives I and II are −29.60dB and −30.44dB, respectively. If a smaller EVM is desired for Alternative I, both the prototype filter and the Hilbert transformer should be redesigned. On the other hand, Alternative II will only require redesign of the prototype filter. It is noted that the performance of Alternative I is determined by the passband ripple of the Hilbert transformer and the stopband attenuation of the prototype filter. If the prototype filter is designed such that the FFBR network can approximate PFBR as close as desired, the passband ripple of the Hilbert transformer must be at the same order as the stopband ripple of the prototype filter19 . Otherwise, the overall EVM value will be degraded. Assuming the orders of the filters in this example with N = 20, which result in roughly the same EVMs, the arithmetic complexity of Alternative II is about 63% of that of Alternative I. The main purpose of this illustrative example is to show that if both alternatives are designed to have roughly equal EVMs, Alternative I would require two extra Hilbert transformers resulting in a higher complexity compared to Alternative II. Due to the use of the same filter design method, i.e., minimax, in both alternatives, it is appropriate to compare the structures in terms of complexity with roughly equal requirements on performance, i.e., EVM. The other comparison which focuses on performance rather than complexity, is more appropriate for cases where different filter design methods are involved. An example could be to compare the performance of design methods such as LS, minimax, or constrained LS with filters having the same complexity. However, this type of comparison is not relevant in this chapter but such a comparison will later be used in Chapter 6. 19 This may even require higher filter orders than that estimated by (4.18) which would add extra arithmetic complexity. 48 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 4.6 Concluding Remarks The next subsections will discuss some issues that have not been treated in the previous sections. 4.6.1 Measure of Complexity The measure of arithmetic complexity discussed in this chapter is the number of operations and real multiplications rather than complex multiplications are used. The reason is that multiplications by specific complex numbers may require less real multiplications than a general complex multiplication. Consequently, comparing the real operations gives a more accurate measure of arithmetic complexity [68]. Furthermore, in converting complex operations to real operations, using the (3/3) or (4/2) assumption (see Section 4.3.4) does not alter the total number of real operations shown in Fig. 4.15(a) but the former (latter) results in less multipliers (adders). Hence, using a (3/3) assumption, the arithmetic complexity values will be different from those of Fig. 4.15(b) but the trend in arithmetic complexity will still be the same. 4.6.2 Applicability of Alternatives I and II In hybrid, full processing, and partial processing payloads, there is a need for complex data in subsequent stages of the on-board processing. In these cases, the only choice will be to use Alternative I so that the intermediate complex signals can be used for necessary processing, e.g., modulation/demodulation, coding/decoding, etc. Alternative II can thus be used in systems where the need for intermediate complex signals can be eliminated. 4.6.3 Filter Bank Design Basically, to design the FB for the real FFBR network, the method outlined in [19] can be used. However, in contrast to Alternative I, Alternative II uses half of the spectrum of a complex signal. Due to this, a worst-case degradation of 6dB in the values of EVM may be introduced which can simply be overcome by reducing the approximation error specification (by a factor of two) in the filter design procedure. 49 4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS 50 5 A Multimode Transmultiplexer Structure This chapter introduces a multimode TMUX structure which can generate a large set of bandwidths and center frequencies. The structure utilizes fixed integer SRC, Farrow-based variable rational SRC, and variable frequency shifters. A main advantage of this TMUX is that it only needs one filter design beforehand. Specifically, the filters in the fixed integer SRC blocks and the subfilters of the Farrow structure are designed only once. Then, all possible combinations of bandwidths and center frequencies are obtained by properly adjusting the variable delay parameter of the Farrow-based filters and the variable parameters of the frequency shifters. Following the general and historical introduction in Section 5.1, Section 5.2 outlines the problem for the TMUX. Then, Section 5.3 discusses the building blocks and the operation of the proposed multimode TMUX whereas Section 5.4 deals with the filter design problem. After a discussion on implementation complexity in Section 5.5, two applications of the proposed TMUX are covered in Section 5.6. In Section 5.7, the TMUX is described in terms of conventional multirate building blocks making it possible to use techniques based on the blocked transfer function to design the filters. Finally, Section 5.8 concludes the chapter. 51 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE 5.1 Introduction As discussed in Section 1.1, a current focus in the communications area is to develop flexible radio systems which aim to seamlessly support services across several radio standards [1]. A major part of the research in this area is to cost-efficiently implement multimode transceivers. The simplest approach to cope with multimode problems is to use a custom device for each communication mode. However, with the growing number of standards, communication modes, and the demand for more functionality, this approach is becoming increasingly unacceptable both in terms of manufacturing cost and energy consumption. Thus, it is vital to develop new low-cost multimode terminals. TMUXs allow several signals (users) to share a common channel and, thus, they constitute fundamental building blocks in communication systems. Multimode communication systems require multimode TMUXs that support different bandwidths which may vary with time. In other words, the users can request any bandwidth at any time. For example, a communication channel can be shared by three users that simultaneously transmit video, text, and voice. Here, each user occupies a specific portion of the channel and this portion may vary with time. As discussed in Section 3.4, a TMUX is the dual of a FB and is composed of a SFB followed by an AFB. The SFB (AFB) is constructed as a parallel connection of a number of branches with each branch being realized by digital bandpass interpolators (decimators). Multimode TMUXs thus require interpolators and decimators with variable parameters. These blocks can be constructed using variable upsamplers (downsamplers) and bandpass filters which have variable center frequencies and bandwidths. However, when the number of modes increases, the degree of variability grows which implies that the implementation complexity of such an approach may become intolerably high. 5.2 Problem Formulation In a similar manner to Fig. 1.1, we assume that the whole frequency band is occupied by a number of users having different bandwidths and center frequencies. According to Fig. 5.1, one way of dealing with such a scenario is to divide the whole frequency band into Q GBs separated by a guardband of 2∆ and any user p can occupy any rational Rp GBs. In this chapter, we model the input patterns on which the FFBR network in Chapter 4 is operating. To do so and in accordance 2π with [19], the guardband and GB values are chosen as 2ǫπ Q and Q − 2∆ where 0≤ǫ≤1. In general, one can select any value for the guardband and GBs according to the requirements of the system to be modeled. 52 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE (a) Granularity Band (BGB) 2p/Q Guard Band 1BGB 0 (b) 2pa/Q 2p/Q+2pa/Q 1.5BGB 2p wT 2p-2p/Q +2pa/Q 1BGB 2.2BGB 1.75BGB 2p wT 0 (c) 3.2BGB 2.7BGB 2p wT 0 (d) 5.9BGB 1BGB 2p wT 0 Figure 5.1: Formulation of the GBs and examples of different user signals having arbitrary bandwidths. Synthesis FB Analysis FB ^ n -jw 0 ejw0n x0(n0) L F(z) H0(z) L F(z) ^ n -jw 1 jw1n e x1(n1) e H1(z) ^ y(n) y(n) L F(z) ^ F(z) L x^0(n0) H1(z) ^ F(z) L ^x (n1) 1 ^ F(z) L ^x (nP-1) P-1 e ^ e-jwP-1n ejwP-1n xP-1(nP-1) H0(z) HP-1(z) HP-1(z) Figure 5.2: Proposed multimode TMUX consisting of fixed integer SRC, variable rational SRC, and variable frequency shifters. Here, Hp↓ (z) and Hp↑ (z) represent Farrow-based filter for decimation and interpolation. 5.3 Proposed Multimode TMUX Structure In this section, we will introduce a multimode TMUX which can generate arbitrary bandwidths and center frequencies1 . We define a GB2 as the minimum bandwidth a user can occupy and assume that users are separated by guardbands which means that the TMUX is slightly redundant. As discussed in Section 3.4.2, the proposed multimode TMUX requires a small redundancy (oversampling) so that it can generate all possible modes without channel interference and using only one set of fixed filters. Without such an assumption, one would require redesign of filters for each mode which is cumbersome. In addition, it is well known that redundancy is needed anyhow in communication systems to ensure a high-performance transmission [37]. Specifically, redundancy may be of particular interest in cases where 1 The bandwidths and center frequencies are in practice limited to rational numbers due to finite precision (wordlength). However, at the expense of additional implementation cost, any desired precision can be achieved. 2 In multistandard communications, a GB can be chosen such that all bandwidths are rational multiples of the GB. 53 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Mag. [dB] Mag. [dB] 0 −20 −40 −60 Mag. [dB] 0 −20 −40 −60 Mag. [dB] 0 −20 −40 −60 Mag. [dB] (a) 0 −20 −40 −60 0 −20 −40 −60 0 0.25π 0.5π 0.75π π ω [rad] 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π (b) 0 0.25π 0.5π 0.75π π ω [rad] (c) 0 0.25π 0.5π 0.75π π ω [rad] (d) 0 0.25π 0.5π 0.75π π ω [rad] (e) 0 0.25π 0.5π 0.75π π ω [rad] Figure 5.3: (a) Output of integer interpolator. (b) Output of Farrow-based decimator. (c) and (d) Outputs of frequency shifters. (e) Output of Farrow-based interpolator. L = 12, Rp = 2.5, ωp = 0.3π, ω̂p = 0.3π. there exists severe channel distortion in some frequency bands [69]. We also assume that any user p can occupy rational Rp (t) number of GBs3 where 1 ≤ Rp (t) ≤ Q with Q being the total number of GBs in the whole frequency range. As shown in Fig. 5.2, the TMUX generates a GB through upsampling by L followed by a lowpass filter. As users can have bandwidths which are rational multiples of the GB, the Farrow-based filter (refer to Section 3.2) performs decimation by rational ratios Rp . To place the users in appropriate positions in the frequency spectrum, variable frequency shifters are utilized. Finally, all users are summed for transmission in the channel. In the AFB, the received signal is first frequency shifted such that the desired signal can be processed in the baseband. Then, a Farrow-based interpolator (by ratio Rp ) followed by decimation by L is used to obtain the desired signal. It is also noted that, like, e.g., orthogonal frequency division multiplexing-based (OFDM-based) TMUXs, the output of the TMUX is a complex signal. Figure 5.3 illustrates the principle of the structure by plotting the frequency spectrum at the output of the lowpass filter, Farrow-based filters, and the frequency shifters with a uniformly distributed random input. 3 The value of R (t) is constant during the time frame in which user signal p is transmitted. p In later discussions, the time index t will be omitted. 54 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE 5.3.1 Channel Sampling Rates As the proposed TMUX is aimed for a multimode communication system, the users X0 , X1 , . . . , XP −1 can generally have different sampling (bit) rates. This means that in one time frame, i.e., the time during which a signal is transmitted, the number of samples (and, hence, the time index np ) processed in each branch of the TMUX of Fig. 5.2 can be different from other branches. Mathematically, the sampling periods of the TMUX inputs, i.e., T0 , T1 , . . . , TP −1 , must satisfy T0 R0 = T1 R1 = . . . = TP −1 RP −1 = LTy , (5.1) where Ty is the sampling period of y(n). 5.3.2 Sampling Rate Conversion As shown in Fig. 5.2, integer interpolation and decimation by L requires lowpass filters F (z) and F̂ (z), respectively. The stopband edges of these filters are defined as the edges of an Lth-band filter given by (2.15) with M = L. This also sets the value of the GB as 2π(1+ρ) . Further, SRC by the rational value Rp is performed L by the Farrow-based filters resulting in the set4 of bandwidths 2π(1+ρ) Rp , p = L 0, 1, . . . , P −1. In the TMUX of Fig. 5.2, each of systems Hp↓ (z) and Hp ↑(z) employs a filter with a transfer function given by (3.8) and performs SRC by variable rational ratio Rp . The fractional delay values for decimation and interpolation are given by (3.9). 5.3.3 Subcarrier Frequencies According to the problem formulation discussed in Section 5.2, the input signal is divided into a number of GBs which are separated by a guardband of 2∆. This means that, having generated the user signals through the SRC structures, they must be modulated into specific locations in the frequency spectrum for transmission. At this stage and to avoid cross talk, it must be ensured that the user signals do not overlap. Consequently, the subcarrier ωk for user k must be computed as ωk = k−1 X χp + p=0 χk 2 (5.2) where χp = ⌈Rp ⌉ 2π Q , p = 0, 1, . . . , k with ⌈x⌉ being the ceiling of x. In other words, using the ceiling operation, we ensure that the user signals do not even share a GB5 . In general, the bandwidths are time-varying and only the value of guardband 4 The narrow-band output of F (z), as in Fig. 5.3(a), can be expanded by any rational ratio R p through Hp↓ (z), as in Fig. 5.3(b). Thus, for fixed values of L and ρ, the values of Rp determine the set of P bandwidths. 5 This is specific for the FFBR network in Chapter 4 and may be ignored or modified for other system formulations. 55 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Magnitude 1 0.8 0.6 0.4 0.2 0 0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π π ωT [rad] 0.01 Error 0.005 0 −0.005 −0.01 0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π ωT [rad] Figure 5.4: Magnitude response and approximation error of the Farrow structure designed by (5.5) with δ4 = 0.01 and ω2 = 0.9π. Here, 6 subfilters each having a maximum order of 24 have been used. is fixed. If there is no FBR of user signals, we have ω̂k = ωk . However, in case of FBR, the subcarrier ω̂k becomes ω̂k = ωk + f (t, ωk ) (5.3) where f (t, ωk ) is a time-varying function expressing the FBR (or any frequency multiplexing) scenario. In general, f (t, ωk ) can be defined by the system which processes y(n). 5.4 Filter Design The filters F (z) and F̂ (z), respectively, suppress the channel cross talk and make the overall transfer functions, between xp (np ) and x̂p (np ), approximate unity. As the TMUX is slightly redundant, the level of cross talk and aliasing resulting from the rational SRC is determined by the stopband attenuation of these filters and can thus easily be suppressed to any desired level. Further, ignoring the rational SRCs, it is well known that the transfer function from xp (np ) to x̂p (np ) is the zeroth polyphase component of F (z)F̂ (z) [37]. To make this polyphase component unity6 , F (z)F̂ (z) must be an Lth-band filter. The filters F (z) and F̂ (z) should thus be designed so that (i) the zeroth polyphase component of F (z)F̂ (z) approximates unity, and (ii) the stopband attenuations of F (z) and F̂ (z) are high enough. This 6 For causal filters, the polyphase component should be a pure delay. 56 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Magnitude 1 0.8 0.6 0.4 0.2 0 0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π π ωT [rad] −3 1 x 10 Error 0.5 0 −0.5 −1 0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π ωT [rad] Figure 5.5: Magnitude response and approximation error of the Farrow structure designed by (5.5) with δ4 = 0.001 and ω2 = 0.9π. Here, 6 subfilters each having a maximum order of 38 have been used. also holds when the rational SRCs are present provided that the Farrow-based filter in (3.8) approximates a fractional delay filter with a delay µ throughout its respective frequency band. For the Farrow-based filter in the SFB, only the GB needs to be covered whereas in the AFB, the whole band except for a small band near π must be covered. The reason is that the output of each integer interpolator (in the SFB) is bandlimited to the GB. However, in the AFB, the sum of user signals, i.e., y(n), is processed by the Farrow-based filter and, therefore, this sum determines the frequency band. Consequently, the complexity of the Farrow-based filter in the SFB will be less than that of the AFB. The discussion above reveals that the proposed TMUX can be designed by determining F (z) and F̂ (z) such that |[F (ejω )F̂ (ejω )]zeroth − 1| ≤ δ1 |F (ejω )| ≤ δ2 ω ∈ [0, π], ω ∈ [0, ω1 ], |F̂ (ejω )| ≤ δ3 ω ∈ [0, ω1 ], (5.4) where ω1 = π(1+ρ) , and [F (ejω )F̂ (ejω )]zeroth denotes the zeroth polyphase compoL nent of F (z)F̂ (z). In addition, the Farrow-based filter in the SFB and AFB, i.e., H(z, µ), should be designed such that |H(ejω , µ) − e−jωµ | ≤ δ4 for ω ∈ [0, ω2 ] (5.5) for all µ ∈ [−0.5, 0.5]. Additionally, ω2 = ω1 for the SFB whereas in the AFB, it is the width of the spectrum of y(n), in Fig. 5.2, which determines ω2 . For example, at a typical spectrum utilization percentage of 90%, we have ω2 = 0.9π. 57 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Magnitude [dB] −30 −40 −50 −60 −70 0.09π 0.3π 0.5π 0.7π 0.9π π ωT [rad] Magnitude 0.01 0.005 0 −0.005 −0.01 0.2π 0.4π 0.6π 0.8π π ωT [rad] Figure 5.6: Magnitude response at the stopband of F (z) = F̂ (z) and the passband of [F (ejω )F̂ (ejω )]zeroth for δ1 = 0.01 and L = 12. It is well known that all δi , i = 1, 2, 3, 4, in (5.4) and (5.5) can be reduced to any desired levels by simply increasing the filter order. To design the TMUX, there is a need to solve (i) one filter design problem to get the filter pair F (z) and F̂ (z) as in (5.4), and (ii) two filter design problems to get the subfilters of the Farrow structures in the SFB and AFB as in (5.5). Having solved these problems only once, it is only the values of the fractional delays, i.e., µ, and the parameters of the variable frequency shifters that change for every new configuration of standards. The filter pair F (z) and F̂ (z) can, for example, be designed as outlined in [40] whereas the Farrow-based filters may be designed as described in [42]. In other words, the proposed multimode TMUX can be designed to approximate PR as close as desired for all possible modes by separately solving three conventional filter design problems. 5.4.1 Example As discussed in the previous subsection, the proposed multimode TMUX can approximate PR as close as desired by proper design of the filters in the fixed and integer SRC blocks. To illustrate this, a series of filters with fixed δ1 = δ4 = {0.01, 0.001}, ω1 = 0.0875π, ω2 = 0.9π, and L = 12 are assumed. As the stopband attenuation of F (z) and F̂ (z) suppresses channel cross talk, they have been designed with different values of δ2 = δ3 . Thus, there are similar constraints on [F (ejω )F̂ (ejω )]zeroth and H(ejω , µ) with the stopband attenuation of F (z) and F̂ (z) being the only parameter which changes. Figures 5.4 and 5.5 show the mag58 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Magnitude [dB] −30 −40 −50 −60 −70 0.09π 0.3π 0.5π 0.7π 0.9π π ωT [rad] −3 Magnitude 1 x 10 0.5 0 −0.5 −1 0.2π 0.4π 0.6π 0.8π π ωT [rad] Figure 5.7: Magnitude response at the stopband of F (z) = F̂ (z) and the passband of [F (ejω )F̂ (ejω )]zeroth for δ1 = 0.001 and L = 12. nitude response and the approximation error of the Farrow structures for some values of µ, i.e., µ = 0, 0.05, 0.1, 0.15, . . . , 0.5. These structures have been designed to approximate allpass transfer functions in the frequency band [0, 0.9π] and the resulting values for δ4 are, respectively, 0.01 and 0.001. The magnitude response at the stopband of F (z) = F̂ (z) and the passband of [F (ejω )F̂ (ejω )]zeroth for some of the designed filters are also shown in Figs. 5.6 and 5.7. For all these filters, the values of δ1 are, respectively, 0.01 and 0.001 but they have different stopband attenuations. Using these filters, Fig. 5.8 shows the average EVM (refer to Section 4.5.3) values for three multimode setups in a 16-QAM signal. It illustrates the fact that EVM resulting from the TMUX can be made as small as desired for all possible modes by decreasing δi , i = 1, 2, 3, 4. It can be seen that the values of δ1 = δ4 set a lower bound on the EVM. However, this lower bound can be decreased to any level by increasing the filter order and decreasing δi , i = 1, 2, 3, 4. 5.5 Implementation and Design Complexity Issues In the previous section, it was shown and demonstrated that the proposed TMUX can be designed to have as small errors as desired for all possible modes through three separate filter designs. This is attractive compared to solutions that require either one set of filters for each mode or online filter design whenever a new mode is desired. However, there is still room for complexity reductions by modifying the proposed structure. Details are beyond the scope of this chapter which aims 59 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE (a) δ1=δ4=0.01 Setup 1: R ={1.3,2.5,3.7} p EVM [dB] −30 Setup 2: R ={1.1,1.9,4.1} p Setup 3: R ={1.75,2.7,5.1} p −35 −40 40 50 60 70 80 90 Stopband attenuation [dB] (a) δ1=δ4=0.001 −25 EVM [dB] −30 −35 −40 −45 −50 40 45 50 55 60 65 70 75 80 85 Stopband attenuation [dB] Figure 5.8: EVM at different stopband attenuations of F (z)F̂ (z) with fixed errors in [F (ejω )F̂ (ejω )]zeroth and H(ejω , µ). to outline the main course to follow when implementing multimode TMUXs, but we will in this section point out some possible ways to reduce the complexity and issues for future research. A motivation to using integer interpolation to generate signals with the spectral width of a GB is that regular integer interpolation structures are more efficient than Farrow-based structures when it comes to implementing an interpolator with a relatively large conversion ratio L [44]. This is true if multi-stage interpolation structures [40] are utilized which should be done for large values of L. If the bandwidth of the users often matches the GB, this option (and the dual in the AFB) appears the most natural choice. On the other hand, if the users often occupy wider bandwidths than the GB, then it may be worth to use a smaller L in the integer SRC stages. The Farrow-based filter in the SFB and AFB can then both work either as interpolator or decimator. In this way, one can find the best trade-off between the complexity of the integer SRC part and the rational SRC part to reduce the overall complexity. Some results are available for interpolators and decimators [44] but the problem is more complex here as we deal with TMUXs. It is thus of interest to extend the results of [44] to multimode TMUXs. It is noted that, as the overall optimum will depend on how often the users take on narrow or wide bandwidths, it is not a trivial task to derive it mathematically. Another issue is the filter design. In the previous section, we outlined the separate filter design which is attractive as known techniques can be adopted. Although this gives us a good suboptimum overall solution, it is slightly overdesigned and has a somewhat higher complexity than necessary. To reduce the complexity, one 60 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE x0(n0) x1(n1) Synthesis FB FFBR Network Analysis FB ^x (n0) 0 ^x (n1) 1 x^P-1(nP-1) xP-1(nP-1) Figure 5.9: Setup for functionality/performance test. can design all filters simultaneously which can, in principle, be done using standard nonlinear optimization techniques. This has successfully been used for fixed FBs and TMUXs [51] but the problem is much more complex here as we deal with multimode TMUXs. This implies that, in the optimization, the prespecified requirements must be satisfied for all possible modes. Consequently, the number of constraints grows with the number of modes. Simultaneous optimization may therefore be practically feasible only for problems that have a few modes. However, it is interesting to investigate how many modes one can handle using simultaneous optimization. 5.6 TMUX Application In this section, two applications of the proposed TMUX are considered. Having designed the TMUX for a specific EVM7 (e.g., −100dB achieved with δi = 10−5 , i = 1, 2, 3, 4), the setup in Fig. 5.9 can be used for functionality/performance test of the FFBR network defined in [19]. According to [31] and as shown in Fig. 5.10, 2πǫ the values for the GB and guardband8 are chosen as 2π Q − 2∆ and Q where 0≤ǫ≤1. To verify the functionality of the FFBR network, four different user signals {X0 , X1 , X2 , X3 } with Rp = {1.75, 1.25, 2, 3.5} resulting in subcarrier frequencies ωp = {0.2π, 0.6π, π, 1.6π} are assumed. The frequency spectrum of the input and P the multiplexed output of the FFBR network with Q = p ⌈Rp ⌉ = 10 GBs are shown in Fig. 5.11(a). The scenario of FBR shown in Fig. 5.11(b) , results in ω̂p = {π, 1.8π, 1.4π, 0.4π}. To illustrate the noise behavior of the FFBR network, Fig. 5.11(c) shows the values of EVM for different prototype filter stopband attenuations in a 16-QAM signal. As can be seen, the stopband attenuation of the FFBR network’s prototype filter is the main source of aliasing suppression [19]. 7 If the TMUX is designed to have a very small EVM, it can detect larger errors that arise due to the system under test. 8 According to Fig. 5.10, small values of ρ would result in high orders for F (z) and F̂ (z). Using the ceiling operation, (5.2) allows some additional guardband which can be utilized to relax the requirements on ρ and, hence, reduce the order of F (z) and F̂ (z). 61 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE L-th Band Filter 2p/Q Granularity Band (BGB)=2p(1+r)/L Guard Band 1BGB 2p-2p/Q 2p wT +2pa/Q 2p/Q+2pa/Q 2pa/Q 0 Figure 5.10: Guardband, GB, and the L-th band filter. As can have ρ≤ L(1−ǫ) Q 2π Q − 2πǫ Q = 2π(1+ρ) , L we − 1. Mag. [dB] (a) 0 −20 −40 −60 X0 Mag. 0 0.25π 0 −20 −40 −60 0.5π X2 0.75π X3 0 EVM [dB] X1 0.25π π ω [rad] (b) X3 1.25π X0 0.5π 0.75π π ω [rad] (c) 1.5π X2 1.25π 1.75π 2π X1 1.5π 1.75π 2π −20 −40 40 45 50 55 60 65 70 Stopband attenutation [dB] Figure 5.11: (a) and (b) Functionality test of a FFBR network using the proposed TMUX. (c) Performance test of a FFBR network using the proposed TMUX. L = 12, Q = 10, ǫ = 0.125. 5.7 Analysis of the Nonuniform TMUX Using Multirate Building Blocks This section shows that the proposed TMUX can alternatively be described in terms of conventional multirate building blocks which may be useful in further analysis of the overall system. This is done by utilizing the rational SRC equivalent of the Farrow-based filter [70]. In each branch of the TMUX, the Farrow-based filter for A interpolation by Rp = Bpp > 1 can be modeled as the cascade of upsampling by Ap , FIR filter Dp (z), and downsampling by Bp . Similarly, a cascade of upsampling by Bp , FIR filter Cp (z), and downsampling by Ap can be used to model decimation A by Rp = Bpp > 1. Consequently, each branch of the TMUX between Xm (z) and X̂p (z) can be drawn as shown in Fig. 5.12. Considering the effect of the frequency shifters in the frequency domain, Fig. 5.12 can be redrawn as in Fig. 5.13 where 62 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE xm(nm) Bm Cm(z) L Fm(z) ^ n -jw p e jwmn e Ap D (z) Am p Farrow Decimator Bp F^p(z) L x^p(np) Farrow Interpolator Figure 5.12: Equivalent path between xm (nm ) and x̂p (np ). xm(nm) L Fm(z) Bm Cm(z) Am z (z^p/zm)z Farrow Decimator Ap D (z) p F^p(z) Bp L x^p(np) Farrow Interpolator Figure 5.13: Equivalent path between xm (np ) and x̂m (np ) considering the effects of frequency shifters in the frequency domain. zm = ejωm and ẑp = ej ω̂p . This structure is similar to the TMUX discussed in [4]. However, there are some differences as : 1. The nonuniform TMUX in [4] does not utilize frequency shifters and, thus, ẑ the term zmp z does not appear in the formulations. However, as the filters of the TMUX in Fig. 5.2 are lowpass and the frequency shifts of a filter transfer function do not alter the characteristics of its baseband equivalent, the term ẑp zm z does not affect the mathematical analysis of the TMUX. Therefore, similar analysis as that in [4] can be used to analyze the present TMUX. 2. Instead of single synthesis and analysis filters in the nonuniform TMUX of [4], the present TMUX uses the cascade of a periodic filter, i.e., Fm (z Bm ) or F̂p (z Bp ), and the FIR equivalent of the Farrow structure, i.e, Cm (z) or Dp (z). To clarify the second difference, the simplified structure of Fig. 5.13 is shown in Fig. 5.14 where p = m and Bp = Bm = B. Consequently, Fig. 5.15 shows the transfer function of the two cascaded filters, i.e., Fm (z B )Cm (z) or Dm (z)F̂m (z B ). This cascaded filter has a lowpass characteristic with passband and stopband edges at π(1−ρ) π(1+ρ) and 2π LB B − LB , respectively. Hence, using the analysis in [4], the blocked transfer function of the TMUX in Fig. 5.2 can be written as T (z) = Φ(z)Ψ(z) where Φ(z) Ψ(z) = = φ0 (z) φ1 (z) ψ0 (z) ψ1 (z) ... ... φP −1 (z) ψP −1 (z) For the proposed TMUX, we have = Dp (z)F̂p (z Bp ), ẑp ẑp ψp (z) = Fm (( z)Bm )Cm ( z). zm zm T . , (5.6) φp (z) 63 (5.7) 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE xm(nm) LBm F(zBm)Cm(z) ^ Bm Am F(z )Dm(z) Am LBm x^m(nm) Figure 5.14: Simplified equivalent path between xm (nm ) and x̂m (nm ). (1+r)p/(LB) 2p/B-(1-r)p/(LB) (1-r)p/(LB) 2p/B-(1+r)p/(LB) 2p/B+(1-r)p/(LB) 2p/B 2p/B+(1+r)p/(LB) 4p/B-(1-r)p/(LB) 2p-(1-r)p/(LB) 4p/B 2p-(1+r)p/(LB) 2p 4p/B-(1+r)p/(LB) wT Figure 5.15: The cascade of a periodic filter and the FIR equivalent of the Farrow structure. The dashed filter is the FIR equivalent of the Farrow structure, i.e., Cm (z) or Dm (z), whereas the periodic filter, i.e., Fm (z B ) or F̂m (z B ), is shown with the solid line. To further simplify this, one can assume Fm (z) = F (z) and F̂m (z) = F̂ (z) for m = 0, 1, . . . , P − 1. If p = m, then ẑp = zm and (5.7) becomes φp (z) = Dp (z)F̂ (z Bp ), ψp (z) = F (z Bp )Cp (z). (5.8) Similar to the discussion in Section 3.4.1 and ignoring the Farrow-based Filters, for the desired user Xd (z), the TMUX output can be written as X̂d (z) = Vdd (z)Xd (z)+ PP −1 i=0,i6=d Vid (z)Xi (z) where Vdd (z) and Vid (z) represent ISI and ICI, respectively. In general, it is desired to have |Vdd (z) − z −ηd | ≤ δISI and |Vid (z)| ≤ δICI with δISI and δICI being the allowed ISI and ICI. However, the values of ICI and ISI are mainly determined by (5.6) which itself ẑ depends on the ratio zmp in (5.7). In other words, for the desired signal X̂d (z), the relation m = p = d and, hence, ωd = ω̂d = ωd holds. Consequently, the system can approximate PR as close as desired via proper design of synthesis and analysis filters. On the other hand, if m 6= p (or equivalently, ωm 6= ω̂p ), the signals are considered as undesired and will be attenuated by the analysis filters. This is due to the fact that the values ωk in (5.2) ensure that the user signals will not overlap. Therefore, according to (5.7), the undesired signals which have passed through the ẑ ẑ synthesis filter Fm (( zmp z)Bm )Cm ( zmp z), will fall in the stopband of the analysis filter Dp (z)F̂p (z Bp ) and be attenuated. A point to note is that, the amount of ICI and ISI is defined by the cascade of the filters on the analysis side. In other words, the system performance depends on the stopband attenuation of the Lth-band filter as well as the errors caused by the Farrow structure. In conclusion, if the TMUX is designed for a worst case error δw , then, it will be able to produce arbitrary bandwidths while approximating PR with smaller errors than δw . This means that the system can have a fixed set of filters and the only parameter to produce arbitrary bandwidths will be the fractional delay values defined by (3.9). 64 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE Although the same analysis methods as for existing TMUXs can be used here, the implementation is different. In other words, the conventional rational SRC blocks (upsamplers, downsamplers, and frequency selective filters) are only used for clarification whereas the TMUX implements these blocks implicitly using integerconversion blocks and Farrow-based rational-conversion blocks. 5.8 Conclusion In this chapter, a multimode TMUX capable of generating a large set of bandwidths and center frequencies was introduced. The structure utilizes fixed integer SRC, Farrow-based variable rational SRC, and variable frequency shifters. The TMUX needs only one filter design beforehand and having designed the filters only once, all possible combinations of bandwidths and center frequencies are easily obtained. To do so, one only requires to adjust the variable delay parameter of the Farrow-based filters and the variable parameters of the frequency shifters. Design examples are provided to illustrate the functionality and performance of the proposed TMUX. Furthermore, the TMUX is described in terms of conventional multirate building blocks which allows the application of techniques based on the blocked transfer function to design the filters. 65 5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE 66 6 A Class of Multimode Transmultiplexers Based on the Farrow Structure This chapter introduces a class of multimode TMUXs which uses the Farrow structure to realize the polyphase components of general lowpass integer interpolation/decimation filters. In this way, integer SRC with different ratios can be performed using a set of fixed filters, i.e., Farrow subfilters, and variable multipliers. This does not require any redesign of filters as various filters with different passband/stopband edges are achieved through one set of common Farrow subfilters and some variable multipliers. Following a historical background in Section 6.1, Section 6.2 gives some prerequisites such as the problem where different users share a common channel and some general issues. In Section 6.3, the design of approximately Nyquist filters using the Farrow structure is discussed and a TMUX capable of performing integer SRC is proposed. By extending the integer SRC to general rational SRC, a multimode TMUX capable of generating arbitrary bandwidths is introduced in Section 6.4. The performance of the proposed TMUX is investigated in Section 6.5 and, finally, some concluding remarks are given in Section 6.6. 6.1 Introduction In Chapter 5, a multimode TMUX consisting of fixed integer SRC, Farrow-based variable rational SRC, and adjustable frequency shifters with a capability to generate arbitrary bandwidths was introduced. It is composed of different filters for 67 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE the integer and rational SRC blocks and, thus, there is a need to design three sets of filters, i.e., integer SRC filters, Farrow subfilters in the AFB, and the Farrow subfilters in the SFB. Furthermore, the frequency bands (and the complexity) of the Farrow-based filters in AFB and SFB are different. In other words, the Farrow structure in the AFB has a wider frequency band resulting in a higher complexity. This chapter introduces an alternative method to design approximately Nyquist filters where the Farrow structure realizes the polyphase components of general lowpass integer interpolation/decimation filters. Furthermore, specific constraints are imposed on the transition band of these filters such that by appropriate design of the Farrow subfilters, the resulting interpolation/decimation filters are power complementary. Satisfying the power complementary property makes it possible to obtain integer SRC multimode TMUXs capable of transmitting and receiving signals with controllable levels of distortion and cross talk. Incorporating the integer SRC blocks in a general rational SRC model, a multimode TMUX which can generate arbitrary bandwidths is then constructed. Different design techniques, to obtain approximately Nyquist filters, are considered and compared through the performance of the TMUX. In contrast to the TMUX in Chapter 5, the Farrow structures in the AFB and SFB of the present chapter have equal complexities and, furthermore, there is no need for integer SRC filters. Due to this, we only require to design one set of filters as the Farrow structures in the AFB and SFB are chosen to be equal. Specifically, there is no need to redesign the filters as various filters with different passband/stopband edges can be achieved by one set of common Farrow subfilters. Up to now, there exists little work (almost none to the knowledge of the author) about TMUX structures with high levels of flexibility in SRC and bandwidths. Hence, it is not adequate (and fair) to compare the TMUXs proposed in this chapter as well as Chapter 5, with other TMUX structures. To do a comparison, one needs to limit the flexibility of the proposed TMUX structures and doing so, their main advantages are not considered. The main focus (and advantage) of the TMUXs proposed in this chapter, is to support dynamic communications with reasonable design effort and implementation complexity. In other words, the filters of the TMUX are designed only once and offline. Then, a large set of bandwidths and communication scenarios are dynamically supported. 6.2 Prerequisites In this section, some prerequisites and general issues will briefly be discussed. 6.2.1 Problem Formulation According to Fig. 1.1, we assume that the whole frequency spectrum is shared by P users where each user has a bandwidth of π(1+ρ) Rp , p = 0, 1, . . . , P − 1 and A Rp can have integer or rational vales as Rp = Bpp with Ap and Bp being integers. Furthermore, ρ is the roll-off factor and there is a guardband of 2∆ separating the 68 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Analysis FB Synthesis FB ^ jw0n e-jw0n jw1n ^ n -jw 1 e x (n0) 0 C e x (n1) 1 ^ y(n) y(n) C P-1 C^ ^x (n1) 0 1 1 ^ n -jw P-1 e e (nP-1) x^ (n0) e 1 jwP-1n x C^ 0 0 C^ C P-1 P-1 ^x (nP-1) P-1 Figure 6.1: General model of a multimode TMUX where systems Cp and Ĉp perform SRC. users. Throughout this chapter, we assume ∆ = 0 as the choice of ∆ does not restrict the filter design problem. Generally, the value of P can be different at each time instant meaning that the number of users, sharing a common channel, can vary with time. To make the discussion independent of time, in the rest of the chapter we will assume that at each time frame for which we consider the operation of the TMUX, the value of P is time invariant. As will be illustrated in Section 6.5, the TMUXs introduced in this chapter can handle cases where the value of P may change. In other words, if the multimode setup changes, the value of P may change as well. 6.2.2 Some General Issues This chapter discusses multimode TMUXs of the form shown in Fig. 6.1. In the SFB, the system Cp performs interpolation by the ratio Rp whereas the system A Ĉp in the AFB performs decimation by the ratio Rp . The value of Rp = Bpp can generally be rational with the integer value being a special case achieved by Bp = 1. These blocks make it possible to transmit and receive signals having arbitrary bandwidths according to the scheme in Fig. 1.1. In order to share a common channel by several users having different bandwidths, adjustable frequency shifters are required. Some general issues need to be outlined. First, to avoid cross talk and aliasing, the users do not overlap and, hence, the TMUXs are slightly redundant. Due to the redundancy, the level of cross talk and aliasing is determined by the stopband attenuation of the interpolation/decimation filters and, thus, these levels can be made as small as desired by proper design of the filters. Furthermore, the redundancy simplifies the analysis of the TMUX as it suffices just to focus on one branch and, therefore, ignore the frequency shifters. Second, in the present chapter we do not consider any guardbands between users and it is only ensured that the cutoff frequencies of different filters in the SFB do not overlap. This is taken care of through a set of variable frequency shifters. Guardbands can account for design margins and mismatches, and they 69 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Synthesis FB Analysis FB ^ n -jw 0 jw0n e x (n0) 0 R0 0 1 R1 ^ y(n) y(n) G (z) (nP-1) RP-1 G P-1 R0 x^ (n0) ^ (z) G R1 ^x (n1) 0 1 1 ^ n -jw P-1 e e P-1 0 e 1 jwP-1n x ^ (z) G ^ n -jw 1 jw1n e x (n1) e G (z) ^ G (z) (z) P-1 RP-1 ^xP-1(nP-1) Figure 6.2: First variant of the multimode TMUX in Fig. 6.1 which is composed of variable integer SRC and adjustable frequency shifters. Actual realization of SRC with integer ratio Rp is performed by the structures in Figs. 6.6 and 6.7. The SRC model, shown here, is only used for illustration and analysis purposes. can also ease the frequency synthesis problem by choosing proper multiplications which arise due to the frequency shifter blocks. Guardbands do not restrict the filter design problem as it is only defined by the stopband/passband edges as well as the ripples of the filter frequency response. As discussed in Section 5.3.3 and based on the application, these frequency shifters can be computed to consider a guardband as well. Third, like OFDM-based TMUXs, the output signals which are transmitted over the channel, are complex. Fourth, due to the nature of multimode communication systems, different users can request any sampling (bit) rates and, hence, in one time frame during which a signal is transmitted, the number of samples (or the time index np ) in each branch of the TMUXs can be different from others. Finally, note that the systems Cp and Ĉp perform SRC by integer (rational) ratios and are efficiently realized by the structures in Figs. 6.16 and 6.17. However, for clarity and analysis purposes, the TMUX structures in Figs. 6.2 and 6.14 include the conventional models of integer and rational SRC discussed in Chapter 3. 6.3 Proposed Integer SRC Multimode TMUX As discussed in Section 3.2, the Farrow structure can be used to obtain general lowpass interpolation/decimation filters through fixed subfilters and variable multipliers. These general filters can then be utilized to construct an integer multimode TMUX which is shown in Fig. 6.2. This TMUX consists of upsampling/downsampling by variable integer ratio Rp , p = 0, 1, . . . , P − 1; lowpass interpolation/decimation filters, i.e., Gp (z) for interpolation and Ĝp (z) for decimation; and adjustable frequency shifters, i.e., frequency shifts by ωp and ω̂p . Assuming Tp to be the sampling period at branch p of the TMUX, we have T1 T0 = = . . . = Ty , R0 R1 70 (6.1) 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE where Ty is the sampling period of y(n) in Fig. 6.2. As the TMUX is aimed for a multimode communication system, it must (at any time) be capable of supporting different bandwidths. To do so, the required bandwidths are generated through interpolation by Rp . Using modulation by variable frequency shifters, the users are placed at any appropriate position in the frequency spectrum and, finally, all users are summed to form y(n) for transmission. To recover a specific user, the received signal ŷ(n) is demodulated and passed through baseband decimation by the use of a lowpass filter Ĝp (z) followed by downsampling by Rp . 6.3.1 Variable Integer SRC Using the Farrow Structure Each polyphase branch of general interpolation/decimation filters can be realized by a Farrow structure having a distinct fractional delay value [34, 44]. Consequently, integer SRC blocks can be implemented using fixed subfilters and variable multipliers. In other words, if the SRC ratio is to be changed, there is only a need to modify the set of multipliers µ which correspond to the set of fractional delays computed by (3.9). The Type I polyphase decomposition of a filter Gp (z), used for SRC by ratio Rp , can be written as [37] Rp −1 Gp (z) = X z −m Gp,m (z Rp ), (6.2) m=0 where Gp,m (z) denote the polyphase components. If Gp (z) is a general interpolation/decimation filter of order N , it approximates z −N/2 in the passband and zero in the stopband. To be more specific, the filter design problem is to approximate an ideal brick-wall filter whose frequency response has a magnitude of unity in the passband (with a delay corresponding to z −N/2 ) and zero in the stopband. Thus, in the passband, each term z −m Gp,m (z Rp ) in (6.2) should have a delay of z −N/2 meaning that Gp,m (z) should approximate an allpass transfer function with a fractional delay of ( N2 − m)/Rp [34, 44]. To conclude, a general interpolation/decimation filter can be designed by choosing its zeroth polyphase component, i.e., Gp,0 (z), to be a Type I linear-phase FIR filter of even order N0 and utilizing the Farrow structure to realize the polyphase components Gp,m (z), m = 1, 2, . . . , Rp − 1 so that they have an odd order1 of N1 such that N N0 = = N1 + 1, (6.3) Rp and Gp,m (z) = L X Sk (z)µkp,m , (6.4) k=0 where µp,m = 1 With −m 1 + ⇒ µp,m = −µp,Rp −m . Rp 2 proper modifications, even-order filters can also be designed [44]. 71 (6.5) 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE - yp,m(z) yRpm(n) x(n) Fp,m(z) ym(n) Figure 6.3: Realization of polyphase components Gp,m (z) and Gp,Rp −m (z) utilizing the symmetry of µp,m . yp,m(z) yyp,m(n) mQ p,m 5 mp,m S5(z) SQ(z) 3 mp,m S3(z) 1 mp,m S1(z) x(n) Figure 6.4: Realization of Ψp,m (z). Q = 2⌊ L+1 2 ⌋ − 1 according to (6.7). 6.3.2 Efficient Variable Integer SRC Considering the antisymmetry of µp,m in (6.5) and as shown in Fig. 6.3, the polyphase components Gp,m (z) and Gp,Rp −m (z) can be written as Gp,m (z) = Φp,m (z) + Ψp,m (z), Gp,Rp −m (z) = Φp,m (z) − Ψp,m (z), (6.6) where Φp,m (z) and Ψp,m (z), shown in Figs. 6.4 and 6.5, are defined as ⌊L 2⌋ Φp,m (z) = X Gp,2k (z)µ2k p,m , k=0 ⌊ L+1 2 ⌋ Ψp,m (z) = X Gp,2k−1 (z)µ2k−1 p,m . (6.7) k=1 Consequently, different lowpass filters can be achieved by a set of fixed filters, i.e., the Farrow subfilters, and some variable multipliers, i.e., µkp,m . Besides lowpass filters, the efficient polyphase realization of integer SRC requires commutators as well [35, 37]. Thus, as shown in Figs. 6.6 and 6.7, SRC by variable integer ratio Rp can be performed using • A set of fixed filters, i.e., the zeroth polyphase component Gp,0 (z) and the Farrow subfilters Sk (z). • Variable multipliers due to the fractional delays µp,m . • Commutators [35, 37]. 72 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Fp,m(z) yFp,m(n) P mp,m 4 mp,m S4(z) SP(z) 2 mp,m S2(z) S0(z) x(n) Figure 6.5: Realization of Φp,m (z). P = 2⌊ L2 ⌋ according to (6.7). Fixed x(n) fs Variable Gp,0(z) k mp,m Sk(z) 0 1 Rp-1 k = 0, 1, ..., L y(m) Rp fs Figure 6.6: Efficient interpolation by integer ratio Rp using fixed subfilters, variable multipliers, and commutators. The structures in Figs. 6.6 and 6.7 are composed of fixed and variable parts. The fixed part refers to Gp,0 (z) and Sk (z) where the variable part accounts for the multipliers related to fractional delays µkp,m . Due to the antisymmetry of µp,m , only the distinct values of µkp,m must be taken into consideration for a realization2 . Consequently, to perform variable integer SRC there would be a need for either a set of precomputed values µkp,m or some variable multipliers which can cover these distinct multiplications. Assuming Rp = 2, 3, . . . , 30 with L = 5 and N1 = 17, realizing the block of µkp,m requires 2174 nonzero multipliers of which 981 have distinct nonzero magnitudes. Specifically, to perform variable integer SRC by 29 ratios in the range Rp = 2, 3, . . . , 30, there is a need for either 982 precomputed values or some variable multipliers which can cover these distinct multiplications. These 981 distinct nonzero multiplications account for 29 linear-phase FIR filters. 6.3.3 Arithmetic Complexity The arithmetic complexity of the fixed parts in the structures of Figs. 6.6 and 6.7 results from the N0 -th order zeroth polyphase component Gp,0 (z) and a Farrow structure composed of L + 1 subfilters of N1 -th order and L adders3 . Here, the L multipliers corresponding to µ in Fig. 3.7 are included in the variable part. Thus, considering the symmetry or antisymmetry of Sk (z) and Gp,0 (z), we roughly require 3N1 (L+2) + 3L+5 fixed arithmetic operations4 . Hence, for each extra coefficient in 2 2 2 About 55% of the total variable multipliers µkp,m have distinct magnitude values. structure for decimation, in Fig. 6.7, has an additional multi-input adder as well. 4 For antisymmetric S (z), the center tap is zero which brings some additional (but small) k reduction in arithmetic complexity. 3 The 73 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Variable 0 1 x(n) fs Fixed Gp,0(z) k mp,m y(m) fs/Rp Sk(z) k = 0, 1, ..., L Rp-1 Figure 6.7: Efficient decimation by integer ratio Rp using fixed subfilters, variable multipliers, and commutators. Sk (z), we require 3(L+2) additional fixed arithmetic operations. According to (6.3), 2 an increase of one coefficient in Sk (z) amounts indirectly to Rp extra coefficients in the overall filter. Assuming N1 = 17 and L = 5, we roughly require 190 fixed arithmetic operations. In total, to realize 29 linear-phase FIR filters, there is a need for about 1170 arithmetic operations. 6.3.4 Filter Design In general, there are two important source of interference in TMUXs. The filters in each branch of the TMUX, between Xp (z) and X̂p (z), cause ISI whereas the contribution of signals from other branches, i.e., between Xi (z) and X̂p (z), gives rise to ICI. Assuming Vpp and Vip to represent ISI and ICI, respectively, it is desired to have |Vpp − z −ηp | ≤ δISI and |Vip |≤δICI with δISI and δICI being the allowed ISI and ICI where ηp is the delay at each branch p of the TMUX in Fig. 6.2. Considering the redundancy of the TMUX and the fact that user signals do not overlap, the level of ICI is here controlled by the stopband attenuation of Gp (z) and Ĝp (z). Furthermore, ISI can be controlled by appropriate choice of Gp (z) and Ĝp (z) so that the zeroth polyphase component of Gp (z)Ĝp (z) approximates an allpass transfer function. In a general redundant nonuniform TMUX, different branches have different values of Rp and, thus, the term Vip becomes time-varying. However, according to the discussion above, the important issue is to control the stopband attenuation of Gp (z) and Ĝp (z) as well as the zeroth polyphase component of Gp (z)Ĝp (z). To approximate PR as close as desired, the filter Gp (z)Ĝp (z) should approximate an Rp th-band filter as close as desired [37]. According to the discussion in the previous subsection, in the Farrow-structure-based design of the interpolation/decimation filters, only the passband and the stopband have been considered. However, to utilize these filters in a TMUX, there must be requirements on the transition band of the filters as well. In other words, the subfilters of the Farrow structure must be determined such that the resulting interpolation/decimation filters satisfy the power complementary property. Consequently, the filters Gp (z) and Ĝp (z) should be designed such that • They have sufficiently small ripples in their stopbands. • The zeroth polyphase component of Gp (z)Ĝp (z) approximates an allpass 74 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE transfer function. Assuming one set of fixed subfilters, resulting in Gp (z) = Ĝp (z) for the TMUX of Fig. 6.2, the zeroth polyphase component of the filter Gp (z)Ĝp (z) can be written as5 Fp (ejωT ) = [Gp (ejωT )Ĝp (ejωT )ej Rp −1 = X j(ωT − 2πn Rp ) [Gp (e N ωT 2 ]zeroth N j(ωT − 2πn Rp ) 2 2 )e ] . (6.8) n=0 Thus, the subfilters Sk (z) must be determined such that Gp (ejωT ) is a lowpass filter and Fp (ejωT ) has an allpass transfer function. 6.3.5 Filter Design Parameters To design approximately Nyquist filters by the Farrow structure, the free optimization parameters are the coefficients of the Farrow subfilters, i.e., Sk (z). In other words, the values of N1 , N0 , L, ρ, Rp , and µkp,m are fixed during the optimization and the filter design procedure only determines the coefficients of Sk (z). A crucial point is that the filter design problem is solved only once and offline. After determining the Farrow subfilters Sk (z) only once, a large set of SRC ratios (and bandwidths) can be supported with any desired minimum ISI and ICI. Specifically, implementation of different bandwidths (and SRC ratios) is obtained by choosing proper values of µp,m and using appropriate number of polyphase components, i.e., Rp . For example, in each of the Figs. 6.8-6.13, all the 29 filters are achieved by one common set of impulse responses Sk (z) and their difference lies only in the multiplier values µp,m and the number of polyphase components Rp . 6.3.6 Filter Design Criteria To design the filters of the TMUX6 , both LS and minimax methods (or combinations of them) can be used but it is the application which determines the design method. In the following subsections, we will consider alternatives of the minimax and LS approaches where the filters designed using these methods will be compared later. Minimax Design The filter design problem in the minimax sense can be formulated as over all Rp , minimize δ subject to N ωT 5 For convenience in the filter design, adding the term ej 2 results in a noncausal filter in order to include the center tap in the optimization. 6 To avoid having two sets of subfilters, we assume that G (z) = Ĝ (z). However, the discussion p p of the TMUX is treated in a general case with different filters in the AFB and SFB. 75 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −3 3 x 10 1 0 p F (ejωT)−1 2 −1 −2 −3 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.8: Approximate Rp -th band filters designed with W = 1 in (6.9). |Fp (ejωT ) − 1| ≤ δ, |Gp (ejωT )|≤W (ωT )δ, ωT ∈ [0, π] ωT ∈ [ωs T, π], (6.9) where δ and W (ωT ) are, respectively, the desired maximum ripples and the weighting function with ωs T defined as in (2.15) with M = Rp . Throughout this chapter, we assume a flat weighting function and, consequently, the term W (ωT ) reduces to a constant value represented by W . According to (6.9), a larger (smaller) value for W allows the stopband ripples of filters to be larger (smaller). The deviation of Fp (ejωT ) from an allpass transfer function (or a pure delay) controls the ISI whereas the stopband attenuation of Gp (ejωT ) controls the ICI through Vip . Consequently, there is a need for a simultaneous optimization to minimize δ over all Rp values of interest. In this way, the requirements on the filter characteristics are satisfied for every mode resulting in a more complicated filter design problem compared to conventional uniform and fixed nonuniform TMUXs. However, as discussed in Section 6.3.5, the filter design problem needs to be solved only once and offline. Figure 6.8 shows the characteristics of the approximate Rp th-band filters and their corresponding zeroth polyphase components resulting from a simultaneous optimization of (6.9) for Rp = {2, 3, . . . , 30} where the values for ρ, W , L, N1 , and N0 are chosen to be 0.2, 1, 5, 17, and 18 respectively. Furthermore, the simultaneous optimization results in δ = 2.56×10−3 . The characteristics of the filters resulting from W = 0.2 and W = 5 in (6.9) are shown in Figs. 6.9 and 6.10. Here, the simultaneous optimization results in δ = 1.025×10−2 and δ = 9.79×10−4 , respectively. According to (6.3), the choice of N0 and N1 depends both on the overall desired filter order N and the SRC ratio Rp . However, based on the desired levels of ISI and ICI, one can choose optimal values for L and N1 [44]. 76 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Fp(ejωT)−1 0.02 0.01 0 −0.01 −0.02 −0.03 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.9: Approximate Rp -th band filters designed with W = 0.2 in (6.9). Least Squares Design The filter design problem in the LS sense can be written as over all Rp , minimize Z π 1 π |Gp (ejωT )|2 . |Fp (ejωT ) − 1|2 + W ωs T 0 Z (6.10) Similar to the minimax formulation and according to (6.10), a larger (smaller) W makes the stopband energy of filters larger (smaller). The characteristics of simultaneously designed approximate Rp th-band filters and their corresponding zeroth polyphase components resulting from (6.10) are shown in Figs. 6.11-6.13. Here, the same parameters as those in Figs. 6.8-6.10 have been used. Choosing either of these design techniques depends on the application as well as the measure of performance considered. In this chapter, we use the energy of the noise introduced by the TMUX and, consequently, the LS method becomes superior. The reason is that the LS approach optimizes the filters in the energy sense and, thus, it results in a smaller noise energy. However, by comparing Figs. 6.8-6.13, it can be seen that Fp (ejωT ) of the minimax filters have larger ripples than those achieved through the LS method. It is also well known that the maximum stopband ripple of the LS filters may be larger than that of the minimax filters. However, the stopband ripples of the LS filters decay in the stopband and, consequently, the average ripple of the minimax filters will be larger. As will be seen later, the performance of the TMUX is equally dependent on both the ripples at the (i) stopband of Gp (ejωT ), and (ii) passband of Fp (ejωT ). These ripples are correlated and increasing one of them (through the weighting functions) makes the other ripple smaller. 77 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −3 3 x 10 1 0 p F (ejωT)−1 2 −1 −2 −3 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.10: Approximate Rp -th band filters designed with W = 5 in (6.9). 6.4 Proposed Rational SRC Multimode TMUX The discussion in Section 6.3.1 reveals that the Farrow structure can be used to implement interpolators/decimators with variable integer SRC ratios. To model rational SRC, one can combine additional downsamplers/upsamplers with integer interpolators/decimators and, consequently, construct a variable rational SRC multimode TMUX which is shown in Fig. 6.14. The P -channel TMUX consists of upsamplers/downsamplers Ap , Bp , p = 0, 1, . . . , P − 1; lowpass interpolation/decimation filters, i.e., Gp (z) for interpolation and Ĝp (z) for decimation; and adjustable frequency shifters, i.e., frequency shifts by ωp and ω̂p . Assuming the sampling period at branch p of the TMUX to be Tp , we have T0 B0 B1 = T1 = . . . = Ty , A0 A1 (6.11) where Ty is the sampling period of y(n) in Fig. 6.14. In the SFB, the TMUX generates the required bandwidths through interpolation by Ap followed by downsampling by Bp . Then, variable frequency shifters place the users at appropriate positions in the frequency spectrum where they are summed to form y(n) for transmission. In the AFB, the received signal ŷ(n) is first frequency shifted to process the desired user signal in the baseband. Then, a lowpass filter Ĝp (z) removes the cross talk and images which arise due to upsampling by Bp . Finally, a downsampling by Ap is used to obtain the desired signal. Figure 6.15 illustrates the principle of operation by plotting the outputs of different blocks in one branch and with a uniformly distributed random input, for an SRC ratio of Rp = 19 9 . As can be seen, a lowpass filter can remove the images in Fig. 6.15(e) and, thus, make it possible to retrieve the desired symbols. 78 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −3 3 x 10 1 0 p F (ejωT)−1 2 −1 −2 −3 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.11: Approximate Rp -th band filters designed with W = 1 in (6.10). 6.4.1 Efficient Variable Rational SRC Considering conventional multirate building blocks, i.e., upsamplers, downsamA plers, and filters, rational SRC by ratio Bpp can be performed using [35, 37] • Downsampling by Bp after interpolation by Ap . • Upsampling by Bp before decimation by Ap . For a fixed Ap , one can change the values of Bp resulting in a number of rational SRC ratios. However, to avoid aliasing, care must be taken to ensure that the inputs to the downsamplers are bandlimited7 . According to the discussion in Section 6.3.1, integer SRC with ratio Ap can be performed using a set of fixed subfilters A and, thus, any rational ratio Bpp can be handled using a set of fixed subfilters and different upsamplers/downsamplers. Consequently, we can replace the SRC by integer value Ap in Fig. 6.14, with their equivalent structures from Figs. 6.6 and 6.7 and, hence, construct structures for variable rational SRC using fixed subfilters, variable multipliers, and commutators. In case a downsampler is added, as shown in Fig. 6.16, some values at the output of the Farrow-based interpolation are not needed and the commutator will retain every Bp th sample. On the other hand, if an upsampler is added, as shown in Fig. 6.17, some branches of the commutator which feeds the Farrow-based decimation, are set to zero. Therefore, savings in the arithmetic complexity are achieved as there would be no need for these samples to be processed. This is further illustrated in Fig. 6.18 where the first group of input samples or the first 7 This fundamental limit is imposed due to the nature of upsamplers and downsampler so that downsampling produces no aliasing. In other words, this limit is not imposed by the TMUX. 79 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −3 3 x 10 1 0 p F (ejωT)−1 2 −1 −2 −3 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.12: Approximate Rp -th band filters designed with W = 0.2 in (6.10). step in realization, consisting of Ap samples {x0 , x1 , . . . , xAp −1 }, is considered. For other batches of input samples, e.g., {xAp , xAp +1 , . . . , x2Ap −1 }, the location of the groups of zero-valued samples, which are fed into the polyphase branches, will change. However, the savings in the arithmetic complexity will still be preserved. Substituting (6.4) and (6.5) in (6.2) with Rp = Ap , the transfer function of a general interpolation/decimation filter Gp (z), composed of Ap polyphase components, can be written as Ap −1 Gp (z) = Gp,0 (z Ap ) + X z −m m=1 L X k=0 Sk (z Ap )( −m 1 k + ) . Ap 2 (6.12) If some of the inputs to the Ap branches are zero, their corresponding polyphase components in (6.12) can be discarded in the realization and, hence, only a subset of the values in the range m = 1, 2, . . . , Ap − 1 will be used. This brings savings in the arithmetic complexity. 6.4.2 Filter Design A As discussed before, rational interpolation (decimation) by ratio Bpp , Ap > Bp requires that integer interpolation (decimation) by Ap is performed before (after) A downsampling (upsampling) by Bp . Thus, to get an SRC by Bpp , it is required to design the integer SRC by Ap and, then, perform the integer SRC by Bp through either a commutator, in case of interpolation in Fig. 6.16, or an upsampler, in case of decimation in Fig. 6.17. In other words, if the filters for the integer SRC are designed so that they can perform integer SRC with small levels of aliasing and 80 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −3 3 x 10 1 0 p F (ejωT)−1 2 −1 −2 −3 0 0.2π 0.4π 0.6π 0.8π π 0.6π 0.8π π ωT [rad] Gp(ejωT) [dB] 0 −20 −40 −60 0 0.2π 0.4π ωT [rad] Figure 6.13: Approximate Rp -th band filters designed with W = 5 in (6.10). Synthesis FB x0(n0) A0 G0(z) B0 G1(z) B1 Analysis FB e-jw0n jw1n ^ n -jw 1 e x1(n1) A1 ^ ejw0n ^ y(n) y(n) GP-1(z) A0 x^0(n0) B1 ^ (z) G 1 A1 ^x (n1) 1 BP-1 ^ (z) G P-1 AP-1 e e e AP-1 ^ (z) G 0 ^ n -jw P-1 jwP-1n xP-1(nP-1) B0 BP-1 ^x (nP-1) P-1 Figure 6.14: Second variant of the multimode TMUX in Fig. 6.1 which is composed of variable rational SRC and adjustable frequency shifters. Actual realization of rational SRC with Rp is performed by the structures in Figs. 6.16 and 6.17. The SRC model, shown here, is only used for illustration and analysis purposes. images, the same filters can be utilized to achieve rational SRC as well. Considering fixed values of Ap in Fig. 6.14, the values of Bp must be chosen such that the output of the integer downsampling by Bp is bandlimited to [0, π] and, consequently, no aliasing occurs during decimation by Ap in Fig. 6.14. In other words, the values for Ap , Bp , and ρ should be chosen such that Ap ≥Bp ≥2, {Ap , Bp }∈N. 1+ρ (6.13) However, in practical implementations and due to finite wordlength effects, the A 19 ratio Bpp cannot take on arbitrary values such as 19 3 = 3.6666 and 9 = 2.1111 but as we shall see later (and also illustrated in Fig. 6.15), the designed filters can handle such cases as well. Additionally, in practice, the set of values for Ap and Bp 81 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Mag. [dB] Mag. [dB] (a) 20 0 −20 −40 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT1 [rad] 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π 1.25π 1.5π 1.75π 2π (b) 0 0.25π 0.5π 0.75π π ωT [rad] 2 Mag. [dB] Mag. [dB] (c) 20 0 −20 −40 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT2 [rad] (d) 0 0.25π 0.5π 0.75π π ωT [rad] Mag. [dB] 2 (e) 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT3 [rad] Figure 6.15: Spectrum at the output of (a) Interpolation by Ap = 19. (b) Downsampling by Bp = 9. (c) and (d) Frequency shifters with ωp = ω̂p = 0.5684π. (e) Upsampling by Bp = 9. For this illustration, the filters in Fig. 6.8 have been used where T1 , T2 , and T3 represent different sampling periods at the input/output of the SRC blocks. Fixed x(n) fs Gp,0(z) Variable k mp,m Sk(z) k = 0, 1, ..., L 0 1 Ap-1 Bp:1 y(m) Rp fs Figure 6.16: Efficient interpolation by rational ratio Rp = structure. Ap Bp using the Farrow are limited and known which simplifies the filter design problem. Consequently, one can achieve smaller values for δ without the need to increase N1 , N0 , and L. In other words, the filter design problem reduces to simplified versions of (6.9) and (6.10) in which a sparser grid of Rp are considered. 82 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Variable 0 1 x(n) fs Bp Fixed Gp,0(z) k mp,m y(m) Rp fs Sk(z) k = 0, 1, ..., L Ap-1 Figure 6.17: Efficient decimation by rational ratio Rp = structure. 0 Bp-1 zeros Bp+1 Bp-1 zeros 2Bp+1 Ap-1 Gp,0(z) k mp,m Ap Bp using the Farrow y(m) Rp fs Sk(z) k = 0, 1, ..., L A Figure 6.18: Efficient decimation by Rp = Bpp using the Farrow structure and by incorporating the effect of the upsampling by Bp into Fig. 6.17. 6.5 TMUX Performance To illustrate the performance of the proposed TMUX, the average EVM (refer to Section 4.5.3) is used and to verify the functionality of the proposed TMUX, 12 multimode setups consisting of P = 2, 3, 4, 5, 6 users are considered. Each setup consists of a number of SRC ratios as shown in Table 6.1. According to Fig. 6.19, by having different number of users, the TMUX can provide bandwidthon-demand and the whole frequency spectrum can be shared by any number of users8 . To reconstruct any user signal X̂p (z) in any of the multimode setups, in the AFB, y(n) is multiplied by an appropriate term e−j ω̂p n so that the desired user signal is in the baseband. Then, an upsampling by Bp is performed. This upsampling compresses the frequency axis and generates some images. However, due to the redundancy and the fact that user signals do not overlap, the stopband attenuation of the lowpass filter Ĝp (z) removes all the images9 . Consequently, only the baseband content of the desired signal remains and the original signal can be reconstructed through a simple downsampling by Ap . Thus, each signal can be reconstructed with controllable levels of ICI and ISI. Note that ICI arises due to the filters in one branch of the TMUX. Figure 6.20 shows the average values of EVM for these multimode setups. As can be seen, the filters designed through the LS approach in (6.10) result in a smaller EVM compared to the minimax technique. Furthermore, in both LS 8 For illustration purposes, the values of A and B are chosen so that y(n), in Fig. 6.14, p p occupies between 90 − 99% of the frequency range [0, 2π]. 9 These images would contribute to ICI and, thus, they are attenuated by the stopband attenuation of the lowpass filters. 83 Mag. [dB] Mag. [dB] Mag. [dB] Mag. [dB] 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT [rad] 1.25π 1.5π 1.75π 2π 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT [rad] 1.25π 1.5π 1.75π 2π 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT [rad] 1.25π 1.5π 1.75π 2π 20 0 −20 −40 0 0.25π 0.5π 0.75π π ωT [rad] 1.25π 1.5π 1.75π 2π Figure 6.19: Transmitted signal y(n) for the first 4 multimode setups of Table 6.1. and minimax cases, the filters designed with W = 1 are superior to those designed through W = 5, 0.2. This shows that EVM is equally determined by the ripples at (i) the stopband of Gp (ejωT ), and (ii) the passband of Fp (ejωT ). Consequently, PR can be approximated as close as desired by decreasing these ripples. Finally, the effect of different weightings is more distinct in the minimax approach but it can be seen that in both cases, the choice of W = 1 results in a smaller EVM. The distinction in EVM can further be explained by noting the more distinct difference in the ripples of the filters in Figs. 6.8-6.10. Due to the presence of upsampling by Bp in the AFB, there are images present at the input of the decimator by Ap . These images must be removed by the lowpass filter Ĝp (z) and by increasing the stopband attenuation of this filter, it is ensured that all the undesired images are attenuated and, hence, the level of cross talk is small enough. Although the LS approach shows a superiority according to the performance measure considered in this chapter, in some systems there may be a restriction on the maximum allowable ripples. In such systems, the more appropriate option would be to use the minimax approach to ensure that the individual Rp -th band filters have the desired ripples. However, irrespective of the design technique used and in contrast to the ordinary TMUXs, the filter design problem for the proposed TMUX is indirect as it must simultaneously be capable of handling a large set of rational SRC ratios with a reasonable ISI and ICI. In other words, the filters are designed for sets of Ap and, then, by choosing the sets of Bp , the TMUX satisfies some minimum ICI and ISI constraints. If the sets of Ap and Bp are known, one can include additional constraints making the optimization more direct. To keep the filter design complexity at a reasonable level, it may however be preferred to slightly overdesign the 84 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE Table 6.1: SRC ratios for the multimode setups of Fig. 6.20. 29 the first setup, R0 = 10 and R1 = 23 11 . Setup A B 1 [29, 23] [10, 11] 2 [15, 27, 23, 6] [4, 5, 4, 1] 3 [29, 17, 27, 19] [4, 4, 8, 3] 4 [20, 19, 17, 9, 19, 17] [3, 2, 3, 1, 2, 3] 5 [25, 27, 29] [7, 8, 7] 6 [13, 24, 13] [2, 7, 5] 7 [10, 9, 7] [1, 2, 3] 8 [28, 23] [11, 10] 9 [9, 4, 3] [2, 1, 1] 10 [11, 7, 14] [2, 3, 3] 11 [29, 23, 9] [5, 10, 2] 12 [30, 21, 17, 28, 18] [1, 2, 3, 5, 5] As an example, for P 2 4 4 6 3 3 3 2 3 3 3 5 filters and, hence, not include any additional constraints. Even in an offline filter design, there is still a limit on the size of the optimization problem which can be handled. In other words, it may not practically be possible to solve a very large direct filter design problem and, thus, slight overdesign may be a better option. However, direct optimization can be possible for limited sets of Ap and Bp . 6.5.1 Effects of Bp on the SRC Error According to the input-output relation of a downsampler in (3.2), the presence of downsampling by Bp results in a sum of Bp stretched and shifted signals. Thus, one could expect that a larger Bp would result in a larger error due to the increased number of added signals. However, by ensuring (6.13), these signals can be attenuated by the stopband attenuation of Ĝp (ejωT ) and, consequently, the level of noise can be controlled. To illustrate this, four values of 2≤Ap ≤30 are chosen randomly and for these values, their appropriate values of Bp are determined according to (6.13). This gives a set of values for Rp . Furthermore, a cascade of interpolation by Rp and decimation by Rp with the filters in Fig. 6.8 is performed. Figure 6.21(a) shows the values of EVM for this cascade and as can be seen, by increasing the values of Bp , the error of the SRC increases as well. However, Fig. 6.21(b) shows the values of EVM for all 232 possible unique values of Rp which are achieved by Ap = {2, 3, . . . , 30} and (6.13). It can be seen that there is an upper bound on the error which is mainly determined by the stopband attenuation of Ĝp (ejωT ). However, the ripples in the passband of Fp (ejωT ) also play a roll in the error but the main source of attenuating the images is the stopband attenuation of Ĝp (ejωT ). Thus, the stopband attenuation of the filters can be reduced to compensate for the additional noise 85 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE −40 Average EVM [dB] −42 −44 −46 −48 LS, W=1 LS, W=5 LS, W=0.2 Minimax, W=1 Minimax, W=0.2 Minimax, W=5 −50 1 2 3 4 5 6 7 Multimode setup 8 9 10 11 12 Figure 6.20: Average EVM of 16-QAM signals in multimode setups of Table 6.1 for the TMUX in Fig. 6.14 with the filters in Figs. 6.8-6.13. arising from Bp . Figure 6.21(b) shows that the EVM varies in a range of about 10 dB. In other words, a decrease of about 10 dB in the stopband attenuation, would decrease the highest EVM to a desired level. According to Bellanger’s formula to estimate the order of FIR linear-phase filters as in (2.19), a decrease of 10 dB in the stopband attenuation, would increase the filter order by about 10%. On the other hand, to make the optimization direct, an increase of 232 29 = 800% in the optimization complexity would be required. This obviously shows that it may indeed be preferable to have a slight overdesign and, hence, avoid the direct optimization. Even considering the slight overdesign, this TMUX is still superior to other nonuniform TMUXs from the reconfigurability and flexibility points of view. The reason is that it requires to design the filters only once and, having done this, a large set of rational SRC ratios can easily be implemented by simple reconfigurations. Note that the presence of cross talk from other channels makes the EVM of the overall TMUX, shown in Fig. 6.20, larger than that achieved for a simple cascade of interpolation and decimation by Rp which is shown in Fig. 6.21(b). 6.6 Conclusion A class of multimode TMUXs has been introduced in which the Farrow structure is used to obtain general interpolation/decimation filters. These TMUXs support variable SRC ratios using fixed Farrow subfilters and variable multipliers. Efficient realization structures for both integer and rational SRC are derived and, 86 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE (a) EVM [dB] −48 −50 A =6 −52 p Ap = 13 −54 Ap = 18 Ap = 27 −56 2 4 6 8 10 12 Bp 14 16 18 20 22 (b) −46 EVM [dB] −48 −50 −52 −54 −56 5 10 15 R 20 25 30 p Figure 6.21: Effects of Bp on the error for the cascade of interpolation by Rp = Ap Bp A and decimation by Rp = Bpp . (a) Increase of EVM with the increase in Bp for a fixed Ap . (b) Upper bound of EVM for all possible values of Rp achieved by Ap = {2, 3, . . . , 30} and (6.13). furthermore, different filter design techniques to obtain the Farrow subfilters, are considered and compared. The focus of this chapter is on the structure of the TMUX and the channel noise has not been included as it is assumed to have a flat frequency response. As an example, the effect of a channel with a nonflat frequency response can be compensated as discussed in [48]. Doing so, there would be a need for two sets of Farrow subfilters as the filters in the SFB and AFB would be different. As a result of the filter design considered in this chapter, there is no need for online design of filters. This comes at the expense of a more complicated filter design problem but it suffices to solve it only once and offline. Having done this, there is only a need to adjust some parameters, i.e., multipliers due to the distinct fractional delays. Furthermore, in terms of EVM, the LS approach is better than the minimax method but in some applications, it may be more appropriate to utilize the minimax approach. 87 6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THE FARROW STRUCTURE 88 7 Conclusion and Future Work In this thesis, nonuniform flexible TMUXs are introduced in which different number of users having different bandwidths can share the whole frequency band in a time-varying manner. These TMUXs can support multimode dynamic communication scenarios and are easily reconfigurable. Furthermore, they require neither redesign of filters nor hardware changes. Specifically, the TMUX filters are designed only once and, then, the operation of the TMUX is easily reconfigured by simple modifications and there is no restriction on the system operation. In addition, the thesis outlines flexible and low complexity solutions for FFBR networks in which different users, present in different composite MF/TDMA input signals, can be reallocated to different positions in different composite MF/TDMA output signals. This reallocation comes at the expense of simple modifications in the channel switch and does not require any filter redesign or hardware changes. Furthermore, the FFBR solutions impose no restrictions on the bandwidth of users or the system operation. As topics of future research, the following issues are identified: 1. Application of cosine modulated FBs to derive FFBR networks to process real signals and detailed comparison of these with the real FFBR in Chapter 4. It would specifically be interesting to compare the flexibility in FBR between complex and cosine modulated FBs. 2. Application of direct filter design for multimode TMUXs for cases where the 89 7. CONCLUSION AND FUTURE WORK set of bandwidths are known. 3. 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