Contributions to Reconfigurable Filter Banks and Transmultiplexers

Transcription

1 Linköping Studies in Science and Technology Dissertation No Contributions to Reconfigurable Filter Banks and Transmultiplexers Amir Eghbali Division of Electronics Systems Department of Electrical Engineering Linköping University, SE Linköping, Sweden WWW: Linköping 2010

2 Contributions to Reconfigurable Filter Banks and Transmultiplexers c 2010 Amir Eghbali Department of Electrical Engineering, Linköping University, SE Linköping, Sweden. ISBN ISSN Printed by LiU-Tryck, Linköping, Sweden 2010

3 to my family...

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5 Abstract A current focus among communication engineers is to design flexible radio systems to handle services among different telecommunication standards. Thus, lowcost multimode terminals will be crucial building blocks for future generations of multimode communications. Here, different bandwidths, from different telecommunication standards, must be supported. This can be done using multimode transmultiplexers (TMUXs) which allow different users to share a common channel in a time-varying manner. These TMUXs allow bandwidth-on-demand. Each user occupies a specific portion of the channel whose location and width 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 customers only need to install a satellite terminal and subscribe to the service. Efficient utilization of the available limited frequency spectrum, calls for on-board signal processing to perform flexible frequency-band reallocation(ffbr). In an integrated communication system, TMUXs can operate on-ground whereas FFBR networks can operate on-board. Thus, successful design of dynamic communication systems requires flexible digital signal processing structures. 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 whose parameters can be modified without a need for hardware changes. This thesis outlines the design and realization of reconfigurable TMUX and FFBR structures which allow dynamic communication scenarios with simple software reconfigurations. 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, commutators, or a channel switch. These adjustments do not require hardware changes and can be performed online. However, the filter design problem is solved offline. The thesis provides various illustrative examples and it also discusses possible applications of the proposed structures in the context of other communication scenarios, e.g., cognitive radios. i

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7 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 guidance in helping me deal with my problems. I would also like to thank my co-supervisor Docent Per Löwenborg for discussions and feedback. Special thanks have to go to all members of my family for their support. 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 being at home! Last but not least, I should thank all my friends whom have made my stay in Sweden pleasant. Amir Eghbali Linköping, September 2010 iii

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9 Contents 1 Introduction Motivation and Problem Formulation Thesis Outline Basics of Digital Filters FIR Filters IIR Filters Note on Stability Polyphase Decomposition Special Classes of Filters Complementary Filters Linear-Phase FIR Filters Nyquist (Mth-band) Filters Hilbert Transformers FIR Filter Design Basics of Multirate Signal Processing Sampling Rate Conversion: Conventional Noble Identity Sampling Rate Conversion: Farrow Structure v

10 Contents Contents Design of the Farrow Structure General M-Channel FBs Filter Design for Modulated FBs General M-Channel TMUXs Mathematical Representation of TMUXs Duality of FBs and TMUXs Approximation of PR in Redundant TMUXs Flexible Frequency-Band Reallocation For Real Signals Introduction Contribution and Relation to Previous Work Choice of the FFBR Network MIMO FFBR Network Configuration FFBR Network Based on Variable Oversampled Complex Modulated FBs Efficient Realization of the FFBR Network Alternative I Complex Versus Real Sampling Arithmetic Complexity: Hilbert Transformer Arithmetic Complexity: DFT with Complex Inputs Arithmetic Complexity: Complex FFBR Network Alternative II Arithmetic Complexity: Real FFBR Network Comparison Arithmetic Complexity: Complex Versus Real FFBR Arithmetic Complexity: Alternative I Versus Alternative II Performance: Alternative I Versus Alternative II Concluding Remarks Measure of Complexity Applicability of Alternatives I and II Filter Bank Design A Multimode Transmultiplexer Structure Introduction Problem Formulation Multimode TMUX Structure Channel Sampling Rates Sampling Rate Conversion Subcarrier Frequencies Filter Design Example Implementation and Design Complexity Issues TMUX Application Analysis Using Multirate Building Blocks Conclusion vi

11 Contents Contents 6 A Class of Multimode Transmultiplexers Based on the Farrow Structure Introduction Contribution and Relation to Previous Work Prerequisites Problem Formulation Some General Issues Proposed Integer SRC Multimode TMUX Variable Integer SRC Using the Farrow Structure Approximation of Perfect Reconstruction (PR) Filter Design Filter Design Parameters Filter Design Criteria Proposed Rational SRC Multimode TMUX TMUX Illustration Efficient Variable Rational SRC Approximation of PR TMUX Performance Effects of B p on the SRC Error Direct Filter Design Design Example Conclusion Reconfigurable Nonuniform Transmultiplexers Using Uniform Modulated Filter Banks Introduction Contribution and Relation to Previous Work Problem Formulation Nonuniform TMUXs Using Modulated FBs System Parameters Channel Sampling Periods TMUX Illustration Choice of GB Choice of Center Frequency Implementation Cost Choice of M and ρ Filter Design Restrictions Comparison with Existing Multimode TMUXs Flexibility Spectrum Efficiency Direct or Indirect Design Conclusion vii

12 Contents Contents 8 Applications to Cognitive Radios Introduction Approach I: Use of DFBR Networks Structure of the DFBR Network User Bandwidth Versus Multiplexing Bandwidth Reconfigurability Modifications Approach II: Use of TMUXs Structure of the TMUX Reconfigurability Modifications Choice of Frequency Shifters Conclusion Conclusion and Future Work 129 A Derivation of (6.23) 131 B Derivation of (6.35) 135 viii

13 Acronyms and Abbreviations Acronyms and Abbreviations ADC AFB CLS CMFB DAC DFBA DFBR DFT ESA EVM FB FDM FIR FFBR FBR GB GRB GSM ICI IDFT IIR ISI IS-54 IS-136 LPTV LS LTI MF/TDMA MIMO MDFT MSE NPR PFBR PR RF QAM SFB SISO SRC TMUX WLAN Analog to Digital Converter Analysis Filter Bank Constrained Least-Squares Cosine Modulated Filter Bank Digital to Analog Converter Dynamic Frequency-Band Allocation Dynamic Frequency-Band Reallocation Discrete Fourier Transform European Space Agency Error Vector Magnitude Filter Bank Frequency Division Multiplexed Finite-length Impulse Response Flexible Frequency-Band Reallocation Frequency-Band Reallocation GuardBand 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 Linear Periodic Time-Varying Least-Squares Linear Time-Invariant Multiple Frequency/Time Division Multiple Access Multi-Input Multi-Output Modified Discrete Fourier Transform Mean Square Error Near Perfect Reconstruction Perfect Frequency-Band Reallocation Perfect Reconstruction Radio Frequency Quadrature Amplitude Modulation Synthesis Filter Bank Single-Input Single-Output Sampling Rate Conversion Transmultiplexer Wireless Local Area Network ix

14 x Acronyms and Abbreviations

15 1 Introduction 1.1 Motivation and Problem Formulation Communication engineers aim to design flexible radio systems to handle services among different telecommunication standards [1 10]. Along with the increase in (i) the number of communication standards (modes), and (ii) the range of services, the requirements on flexibility and cost-efficiency of these radio systems increase as well. Hence, low-cost multimode 1 terminals will be crucial building blocks for future generations of communication systems. Multistandard communications require to support different bandwidths from different telecommunication standards. Table 1.1 shows the bit rate, number of users sharing one channel, and the channel spacing of some popular cellular telecommunication standards, e.g., interim standard-54/136 (IS-54/136), global system for mobile communications (GSM), and IS-95 [11]. To include such standards in a general telecommunication system, one should handle a number of different bandwidths. Consequently, any user can use any standard which suits its requirements on bandwidth, transmission quality, etc. Assume, for example, that a communication channel is shared by three users A, B, and C which respectively transmit video, text, and audio. With bandwidthon-demand, 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 This is also referred to as multiband, multistandard, universal [7]. 1

16 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/ Kbps 3 30 KHz GSM 271 Kbps KHz IS Kbps KHz To support multimode communications, we thus need a system which allows different numbers of users, having different bit rates, to share a common channel. Transmultiplexers (TMUXs) allow different users to share a common channel [12]. Consequently, multimode TMUXs constitute one of the main building blocks in multistandard communications. Multiple access schemes such as code division multiple access, time division multiple access, frequency division multiple access, and orthogonal frequency division multiple access are special cases of a general TMUX structure [13 15]. To support bandwidth-on-demand, the characteristics of the TMUXs must vary with time. Such a communication system has a dynamic allocation of bandwidth. Each user occupies a specific portion of the channel whose location and width may vary with time. The principle of such a communication system is shown in Fig Here, the whole frequency spectrum is shared by P users. Each user X p has a bandwidth of π(1+ρ) R p, p = 0,1,...,P 1, and R p can be an integer or a rational value. Furthermore, ρ is the roll-off factor and a guardband (GB) of separates the user signals 2. To support such a scenario, we can, in principle, use conventional 3 nonuniform TMUXs or FBs, e.g., [16 31]. In a dynamic communication system, these conventional TMUXs and FBs would require either predesign of different filters or online filter design. 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 with reasonable implementation complexity and design effort. One aim of this thesis is to introduce TMUXs which allow different numbers of users, having different bandwidths, to share the whole frequency spectrum in a time-varying manner. As a promise of future digital communication systems, communication engineers also aim to support various wideband services accessible to everybody everywhere [32 39]. 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 [32 37]. Because of the global coverage of satellites, customers only need to install a satellite terminal and subscribe to the service. The European space agency has proposed three major network structures for 2 The choice of does not restrict the analysis and design of the TMUX and, hence, throughout this thesis we will mostly assume = 0. 3 This is due to the duality of filter banks (FBs) and TMUXs [12]. 2

17 1. INTRODUCTION Case I: D<0 X 0 X 1 X 2 X P-1 X 0 0 2p wt X 0 X 1 X 2 X P-1 X 0 0 2p wt Case II: D=0 X 0 X 1 X 2 X P-1 X 0 0 2p wt X 0 X 1 X 2 X P-1 X 0 0 2p wt Case III: D>0 D D X 0 X 1 X 2 X P-1 0 2p wt D D X 0 X 1 X 2 X P-1 0 2p wt Figure 1.1: Problem formulation where P users share the frequency spectrum. X 0 X 0 broadband satellite-based systems in which satellites communicate with the users through multiple spot beams [37]. Therefore, we need efficient reuse of the limited available frequency spectrum by satellite on-board signal processing [32 57]. This calls for flexible frequency-band reallocation (FFBR) networks [40 50] also referred to as frequency multiplexing and demultiplexing [40, 50 56]. The digital part of the satellite on-board signal processor is a multi-input multioutput system. The number of input signals can differ from that of the output signals. Furthermore, the input/output signals can have different bandwidths. Such a communication system must support different communication and connectivity scenarios. One such main scenario is based on multiple frequency/time division multiple access (MF/TDMA). Here, the bandwidth of each incoming signal is composed of a number of adjacent smaller frequency bands (subbands). Each subband is occupied by one (a few) user (users). This MF/TDMA scheme slices the channel both in time and frequency [58]. At any time, any portion of the channel can be used by any user. The on-board signal processor reallocates all subbands to different output signals and center frequencies. The principle of this operation is illustrated in Fig Here, different users 3

18 1. INTRODUCTION Input signal p Input signal p In 1 wt in [rad] In 2 wt in [rad] FFBR Network Out 1 Out 2 Out 3 Output signal p p wt out [rad] Output signal wt out [rad] Output signal wt p out [rad] Figure 1.2: Frequency-band reallocation (FBR) for an FFBR network where any signal in any of the two input signals can be reallocated to any position in any of the three output signals. are present at the input of the FFBR networks and each of them must be reallocated to different center frequencies. In a dynamic communication system, the bandwidth and center frequency of the users may change in a time-varying manner. This necessitates 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, simplicity, etc. [37]. In practice, one may need GBs between the subbands so that the network is realizable. It is one aim of this thesis to outline flexible and low complexity solutions for such FFBR networks. Although the idea of FFBR networks stems from satellite-based communications, they are generally applicable to systems which require frequency multiplexing and demultiplexing. This thesis will also outline some of these applications in the context of cognitive radios. To successfully design dynamic communication systems, communication engineers require high levels of flexibility in digital signal processing structures. This flexibility must not restrict the hardware and, ideally, it must come at the expense of simple software modifications. This is frequently referred to as reconfigurability [4, 6, 59 62] meaning that the system is based on a hardware platform whose parameters can be modified without hardware changes. This thesis outlines solutions for the reconfigurable communication scenarios discussed above. It is a result of the research performed at the Division of Electronics Systems, Department of Electrical Engineering, Linköping University between October 2006 and August The research during this period has resulted in the following publications [43 46, 63 68]: 1. A. Eghbali, H. Johansson, and P. Löwenborg, Flexible frequency-band reallocation MIMO networks for real signals, in Proc. Int. Symp. Image Signal Processing Analysis, Istanbul, Turkey, Sept

19 1. INTRODUCTION 2. A. Eghbali, H. Johansson, and P. Löwenborg, Flexible frequency-band reallocation: complex versus real, Circuits Syst. Signal Processing, DOI /s , Jan 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 A. Eghbali, H. Johansson, and P. Löwenborg, A multimode transmultiplexer structure, IEEE Trans. Circuits Syst. II, vol. 55, no. 3, pp , Mar 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 A. Eghbali, H. Johansson, and P. Löwenborg, A class of multimode transmultiplexers based on the Farrow structure, Circuits Syst. Signal Processing, 2010, submitted. 7. A. Eghbali, H. Johansson, and P. Löwenborg, On the filter design for a class of multimode transmultiplexers, in Proc. IEEE Int. Symp. Circuits Syst., Taipei, Taiwan, May , A. Eghbali, H. Johansson, and P. Löwenborg, Reconfigurable nonuniform transmultiplexers based on uniform filter banks, in Proc. IEEE Int. Symp. Circuits Syst., Paris, France, May 30-June 2, A. Eghbali, H. Johansson, and P. Löwenborg, Reconfigurable nonuniform transmultiplexers based on uniform filter banks, IEEE Trans. Circuits Syst. I - Regular Papers, accepted for publication. 10. A. Eghbali, H. Johansson, and P. Löwenborg, and H. G. Göckler, Dynamic frequency-band reallocation and allocation: From satellite-based communication systems to cognitive radios, J. Signal Processing Syst., DOI /s , Feb These papers are covered in Chapters 4 8. The following papers were also published during this period but they are 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 G. Mehdi, N. Ahsan, A. Altaf, and A. Eghbali, A 403-MHz fully differential class-e amplifier in 0.35 um CMOS for ISM band applications, in Proc. IEEE EWDTS 2008, Lviv, Ukraine, Oct. 9-13,

20 1. INTRODUCTION 3. A. Eghbali, H. Johansson, T. Saramäki, and P. Löwenborg, On the design of adjustable fractional delay FIR filters using digital differentiators, in Proc. IEEE Int. Conf. Green Circuits Syst., Shanghai, China, June 21-23, Thesis Outline The thesis consists of nine chapters where Chapters 2 and 3 deal with the background material. The main contributions of the thesis appear in Chapters 4 8. Chapter 2 reviews the basics of digital filters. It includes the definition of finitelength impulse response and infinite-length impulse response filters; polyphase decomposition; and some special classes of filters. The minimax, least-squares (LS), and the constrained LS filter design problems are also treated. Chapter 3 discusses sampling rate conversion (SRC) using conventional structures and the Farrow structure. Furthermore, the noble multirate identities and efficient SRC structures are considered. In addition, FBs and TMUXs are studied. The perfect reconstruction is treated and its approximation by redundant TMUXs is considered. Finally, the filter design problem for redundant TMUXs is outlined. Chapter 4 is based on [43, 45] and it discusses approaches for realizing FFBR networks. The chapter introduces two alternatives for processing real signals using real input/output and complex input/output FFBR networks. It is shown that the real case has less overall number of processing units. 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. Chapter 5 covers [46, 63] and it introduces a multimode TMUX capable of generating a large set of bandwidths and center frequencies. The TMUX utilizes fixed integer SRC, Farrow-based variable rational SRC, and variable frequency shifters. The building blocks, their operation, and the filter design problem along with some design examples are considered. It is shown that, by designing the filters only once offline, all possible combinations of bandwidths and center frequencies are obtained online. This requires simple adjustments of the variable delay parameter of the Farrow-based filters and the variable parameters of the frequency shifters. Using the rational SRC equivalent of the Farrow-based filters, the TMUX is described in terms of conventional multirate building blocks. The performance and functionality tests of the FFBR network, discussed in Chapter 4, are also illustrated. Chapter 6 considers a class of multimode TMUXs proposed by [64 66]. The TMUXs use the Farrow structure to realize polyphase components of general interpolation/decimation filters. This allows integer SRC with different ratios to be realized using fixed filters and a few 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 treated. A model of general rational SRC is then constructed where the same fixed subfilters are 6

21 1. INTRODUCTION used to perform rational SRC. Efficient realizations of this rational SRC scheme are presented. Similarly, variable frequency shifters are utilized to derive a general rational SRC multimode TMUX. By processing quadrature amplitude modulation signals, the performance of the TMUX is also discussed. Chapter 7 is based on [67, 68] and it introduces reconfigurable nonuniform TMUXs based on fixed uniform modulated FBs. The proposed TMUXs use cosine modulated FBs and modified discrete Fourier transform FBs. Users can occupy different bandwidths and center frequencies in a time-varying manner. The filter design, realization, and the reconstruction error are discussed. Further, the system parameters and the implementation cost are treated. The chapter also compares the proposed TMUXs to those in Chapters 5 and 6. Chapter 8 is based on [44] and it deals with two approaches for frequency allocation and reallocation used in the baseband processing of cognitive radios. These approaches can be used depending on the availability of a composite signal comprising several user signals or the individual user signals. With composite signals, the FFBR network in Chapter 4 is used. To process individual users, the TMUXs in Chapters 5 7 can be used. Discussions on reconfigurability with respect to cognitive radios are also provided. Chapter 9 gives some concluding remarks and open issues for future research. 7

22 8 1. INTRODUCTION

23 2 Basics of Digital Filters This chapter reviews some basics of digital filters. First, finite-length impulse response (FIR) and infinite-length impulse response (IIR) filters are discussed. Section 2.3 treats the polyphase decomposition. Some classes of filters, viz., power complementary, Nyquist, linear-phase FIR, and Hilbert transformers are discussed in Section 2.4. Finally, Section 2.5 outlines the minimax, least-squares (LS), and the constrained LS (CLS) filter design problems. 2.1 FIR Filters A causal 1 FIR filter of order N has an impulse response with N + 1 coefficients h(0),h(1),...,h(n). The transfer function of an Nth-order FIR filter is [69] N H(z) = h(n)z n. (2.1) n=0 In the time domain and with an input sequence x(n), the output sequence is N y(n) = h(k)x(n k) Y(z) = H(z)X(z). (2.2) k=0 1 A filter is causal if h(n) = 0, n < 0. A non-causal FIR filter can be made causal by insertion of a proper delay. 9

24 2. BASICS OF DIGITAL FILTERS x(n) T T T T h 0 h 1 h 2 h N-1 h N y(n) x(n) Figure 2.1: Direct form realization of an N th-order FIR filter. h 0 h 1 h 2 h N-1 h N y(n) T T T T Figure 2.2: Transposed direct form realization of an N th-order FIR filter. There are different ways to realize (2.2) and two are shown in Figs. 2.1 and 2.2 where the impulse response values are h 0,h 1,...,h N. The FIR filters allow one to use non-recursive algorithms for their realization thereby eliminating problems with instability. This thesis always deals with non-recursive stable FIR filters. Figures 2.1 and 2.2 need N + 1 multiplications, N two-input additions, and N delay elements. 2.2 IIR Filters If the length of h(n) is infinite, the filter is called IIR where N n=0 H(z) = a(n)z n 1 (2.3) N n=1b(n)z n. With b(n) = 0, n = 0,1,...,N, an IIR filter reduces to an FIR filter. Realization of IIR filters requires recursive algorithms which may give rise to problems of instability. As the poles of IIR filters are not in the origin (as opposed to FIR filters), their design has extra degrees of freedom. However, care must be taken to place the poles inside the unit circle to ensure stability Note on Stability The z-transform of h(n) is defined by the Laurent series [69 73] H(z) = + n= 10 h(n)z n. (2.4)

25 2. BASICS OF DIGITAL FILTERS This transform exists if h(n) decays to zero as n approaches and +. If [72] then (2.4) converges for h(n) M 1 K n 1, n 0, (2.5) h(n) M 2 K n 2, n 0, (2.6) K 1 < z < K 2. (2.7) As z can have a radius r and an angle θ of the form z = re jθ, (2.4) will converge on every concentric circle with K 1 < r < K 2. For right-hand (left-hand) sided sequences, (2.4) will converge on concentric circles exterior (interior) to some radius, say K c, determined by the radius of the largest (smallest) pole [72]. If (2.4) converges for r = 1, the Fourier transform of h(n) exists and it is defined as [69, 71] H(e jωt ) = + n= 2.3 Polyphase Decomposition The transfer function in (2.1) can be decomposed as H(z) = n= +z 1... n= +z (L 1) n= which can be rewritten as [12, 69, 70] h(n)e jnωt. (2.8) h(nl)z nl h(nl+1)z nl (2.9) h(nl+l 1)z nl, L 1 H(z) = z i H i (z L ). (2.10) i=0 Here, H i (z) are the polyphase components and h i (n) = h(nl+i), i = 0,1,...,L 1. (2.11) This decomposition is frequently referred to as the Type I polyphase decomposition. The Type II polyphase decomposition of (2.1) is [12] L 1 H(z) = z (L 1 i) R i (z L ), (2.12) i=0 11

26 2. BASICS OF DIGITAL FILTERS where R i (z) = H L 1 i (z) [12]. The Type I and II polyphase decompositions allow one to efficiently realize the analysis and synthesis filter banks (FBs) of general FBs, respectively [12]. With polyphase realization, the filters operate at the lowest possible sampling frequency. Although polyphase decomposition reduces the implementation cost, the total number of multiplications and additions does not change. This cost reduction is achieved by operating the adders and multipliers at a lower sampling frequency. To realize an N th-order FIR filter using the L-polyphase decomposition, we need L subfilters of length roughly N+1 L. To do so, (2.2) is rewritten as [70] L 1 L 1 L 1 Y(z) = Y l (z L )z l = X i (z L )z i H j (z L )z j, (2.13) l=0 i=0 where Y l (z), X i (z), and H j (z) are the polyphase components of Y(z), X(z), and H(z), respectively. In a matrix form, (2.13) becomes Y 0 (z L ) X 0 (z L ) Y 1 (z L ). = X 1 (z L ) H(zL ) (2.14). where H(z L ) = Y L 1 (z L ) j=0 X L 1 (z L ) H 0 (z L ) z L H L 1 (z L )... z L H 1 (z L ) H 1 (z L ) H 0 (z L )... z L H 2 (z L ) H L 1 (z L ) H L 2 (z L )... H 0 (z L ) 2.4 Special Classes of Filters. (2.15) Some classes of digital filters are more suitable for multirate systems. The sequel introduces some of these classes Complementary Filters The filters H k (z), k = 0,1,...,K, are power complementary if [12] K H k (e jωt ) 2 = c, c > 0. (2.16) k=0 In general, H k (z) are complementary of order p if [74] K H k (e jωt ) p = c, p N, c > 0. (2.17) k=0 12

27 2. BASICS OF DIGITAL FILTERS In special cases, the magnitude and power complementary filters satisfy (2.17) for p = 1 and p = 2, respectively. Higher order complementary filters, e.g., p > 2, can generate ordinary magnitude and power complementary filters while maintaining superior cut-off characteristics [74]. Strictly (or delay) complementary filters are those who add up to a delay as [12, 69] K H k (e jωt ) = cz D0, c 0. (2.18) k= Linear-Phase FIR Filters The FIR filters can have a linear phase so as to preserve the shape of the signals. This requires h(n) to be either symmetric or antisymmetric as [69] Symmetric : h(n) = h(n n), n = 0,1,...,N (2.19) Antisymmetric : h(n) = h(n n), n = 0,1,...,N. (2.20) Then, we have about N 2 distinct coefficients thereby reducing the number of multipliers. However, this does not change the number 2 of adders. The frequency response of a linear-phase FIR filter can be expressed as H(e jωt ) = e j(nωt 2 +c) H R (ωt) = e jθ(ωt) H R (ωt), (2.21) where H R (ωt) is the real zero-phase frequency response with c = 0 and c = π 2 for symmetric and antisymmetric h(n), respectively. The magnitude response, i.e., H R (ωt), always assumes real positive values whereas H R (ωt) could be negative. The phase response is [69, 75] Φ(ωT) = { Θ(ωT), HR (ωt) 0 Θ(ωT) π, H R (ωt) < 0. (2.22) In general, the linear-phase response can be of the form [75] Φ(ωT) = αωt +β. (2.23) Depending on h(n) being symmetric or antisymmetric and N being odd or even, four types of linear-phase FIR filters are defined as [69, 75] Type I : h(n) = h(n n), Type II : h(n) = h(n n), N even N odd Type III : h(n) = h(n n), N even Type IV : h(n) = h(n n), N odd. (2.24) 2 For Type III linear-phase FIR filters, the number of adders is also reduced. 13

28 2. BASICS OF DIGITAL FILTERS Table 2.1: Typical locations of zeros for linear-phase FIR filters. Type Location I Arbitrary II ωt = π III ωt = 0,π IV ωt = 0 These four types have different expressions for H R (ωt) as [75] H R (ωt) = h( N 2 )+2 N 2 n=1 h(n 2 2 N 1 2 n=0 h(n 1 2 N 2 1 n=0 h(n 2 2 N 1 2 n=0 h(n 1 n)cos(nωt) Type I 2 n)cos( n+1 2 ωt) Type II 1 n)sin((n+1)ωt) Type III 2 n)sin( n+1 2 ωt) Type IV. (2.25) Further, [75] Φ(ωT) = { NωT 2 Type I,II NωT 2 + π 2 Type III,IV. (2.26) The group delay τ g (ωt) and the phase delay τ p (ωt) are defined as [69, 75] τ g (ωt) = dφ(ωt) d(ωt), (2.27) and τ p (ωt) = Φ(ωT) ωt. (2.28) The shape of a periodic signal is preserved 3 if τ p (ωt) is almost constant in the passband. This makes the delay of all signal components approximately equal. For nonperiodic signals, τ g (ωt) may be used. For a constant phase delay, β in (2.23) is forced to be zero whereas for a constant group delay, β in (2.23) can be arbitrary. Linear-phase FIR filters have a constant group delay of τ g (ωt) = N 2. The zeros of a real-valued linear-phase FIR filter are either real or as complex conjugate pairs. If the zeros appear off the unit circle, they are mirrored with respect to the unit circle. This thesis focuses on Types I or II as we deal with lowpass filters. Table 2.1 shows typical locations of the zeros for different linearphase FIR filters. 3 The shape of a periodic bandpass or highpass signal is preserved if β in (2.23) is a multiple of 2π and α is constant [75]. 14

29 2. BASICS OF DIGITAL FILTERS Nyquist (M th-band) Filters A lowpass non-causal filter h(n) of order N is said to be Mth-band if any of its polyphase components, i.e., H k (z), satisfies [12, 69, 76] H k (z M ) = 1 M. (2.29) Here, N = KM m with K and m being integers. Then, k = M m mod M (2.30) where m mod M represents the remainder of m M. In general and for a non-causal h(n), this gives { 1 M h(n) = n = 0 (2.31) 0 n = ±M,±2M,... meaning that every Mth sample, except the center tap, is zero. This reduces the number of multipliers and adders required to realize the filter. If h(n) is an Mth-band filter, its delayed version is also an Mth-band filter [12]. In the causal case, H(z) is an Mth-band filter if the kth polyphase component has the form H k (z) = 1 M z n k. In the time domain, this becomes h(nm +k) = { 1 M n = n k 0 otherwise. (2.32) For an M th-band filter, the passband and stopband edges are, respectively, [77] ω c T = π(1 ρ) M ω s T = π(1+ρ) M, (2.33) where ρ is the roll-off factor (excess bandwidth [75]) and 0 < ρ < 1 so that the transition band contains ωt = π M. In the context of FBs, ρ can assume any value such that ρ > 0 [78]. In brief, H(z) has a real zero-phase frequency response where H R (ωt) = 1 2, ωt = π M. (2.34) Furthermore, the passband and stopband ripples are related to each other as δ s (M 1)δ c. (2.35) If H(z) is an Mth-band filter, the sum of M shifted copies of H(z) results in a constant. In other words, M H(zWM) k = c, W M = e j 2π M, c > 0. (2.36) k=0 15

30 2. BASICS OF DIGITAL FILTERS An alternative to (2.36) is obtained from (2.18) with D 0 = 0 [12]. Generally, the impulse response of a Nyquist filter could be causal or non-causal; FIR or IIR; linear-phase or nonlinear-phase; and real or complex. This thesis always designs real causal linear-phase FIR Nyquist filters. Nyquist filters find applications in, e.g., transmultiplexers [79], spectrum sensing for cognitive radios [61, 80, 81], sampling rate conversion [12, 69], and pulse shaping in communications [82, 83] Hilbert Transformers The spectrum of a real-valued signal is Hermitian symmetric around ωt = 0 and H(e jωt ) = H (e jωt ). This results in some redundancy between the portions of the spectrum at negative and positive values of ωt [84]. Thus, the information of a real-valued signal can be obtained from its spectrum for ωt [0,π]. It is also desirable for, e.g., single sideband communications, to discard the negative frequencies and only process the positive part [85]. To preserve the positive frequencies, the real signal x(n) is passed through a complex linear-phase filter [84] H(e jωt ) = { 2 0 < ωt < π 0 π < ωt < 0. (2.37) From(2.37), weseethatthereissomeambiguityatωt = 0[84]. Thecorresponding IIR non-causal impulse response is 1 n = 0 2j h(n) = nπ odd n (2.38) 0 otherwise. The complex output sequence is then where represents convolution and [84] y(n) = x(n) h(n) = x(n)+jx(n) h i (n), (2.39) h i (n) = { 2 nπ odd n 0 even n. (2.40) Further, H i (e jωt ) = { j 0 < ωt < π j π < ωt < 0. (2.41) In the literature, (2.41) is also referred to as the Hilbert transformer [12, 86, 87]. This thesis uses the term Hilbert transformer for (2.37). From (2.39), we can see that the real and imaginary parts of y(n) are related by a Hilbert transform, i.e., a phase shift of π 2 at all frequencies as in (2.41). One way to design a Hilbert transformer is to shift a real lowpass half-band filter G(z) of length 2N as [69, 84] H(z) = j2g( jz) = ( 1) N 1 2 z N +je( z 2 ), (2.42) 16

31 2. BASICS OF DIGITAL FILTERS where G(z) = z N E(z 2 ). (2.43) 2 IntheFIRcase, E(z 2 )hasalinear phasewithagroupdelayofn samples. Further, E(z) is a wideband lowpass filter. This thesis shifts a real lowpass half-band filter to obtain a Hilbert transformer. Thus, we have causal linear-phase FIR filters. 2.5 FIR Filter Design The frequency response of an ideal digital filter is equal to unity in the passband(s) and zero in the stopband(s). In other words, H(e jωt ) = { 1 in passband(s) 0 in stopband(s). (2.44) Furthermore, there are no transition band(s) resulting in a brick-wall characteristic. Such a filter has an infinite length, e.g., an ideal lowpass sinc function, as h(n) = { 1 n = 0 sin(n) n n 0 (2.45) and is not realizable. To get a realizable filter, one approximates this ideal transfer function in the passband(s) and stopband(s) by allowing transition band(s) as well as some ripples. Thus, the practical specification for a digital filter is 1 δ c H(e jωt ) 1+δ c, ωt Ω c H(e jωt ) δ s, ωt Ω s. (2.46) Here, δ c and δ s are, respectively, the passband and stopband ripples with Ω c and Ω s being the passband and stopband regions. One can generally have filters with multiple passband and stopband regions. Then, the specifications must be satisfied for all of these regions. Further, one can allow different ripples in these regions. As an example, in a lowpass filter, Ω c covers [0,ω c T] whereas Ω s covers [ω s T,π]. Here, ω c T and ω s T are the passband and stopband edges, respectively. After estimating the filter order, h(n) must be determined such that (2.46) is satisfied for desired values of Ω c, Ω s, δ c, and δ s. A commonly used formula to estimate the order of a linear-phase FIR filter is the Bellanger s formula [88] N B 2 3 log 2π 10(10δ s δ c ) ω s T ω c T. (2.47) For reasonable orders, (2.47) gives a good approximation. For general nonlinearphase FIR filters, such formulae do not exist. Then, a manual search is the only 17

32 2. BASICS OF DIGITAL FILTERS 8 100*(N B N K )/N K (ω s T ω c T)/π δ s =δ c [db] Figure 2.3: Relative comparison of the orders estimated by (2.47) and (2.48). way to find the filter order. Note that there exist other formulae to estimate the order, e.g., Kaiser [89], as N K 20log 10( δ s δ c ) (ω s T ω c T)/2π. (2.48) This thesis uses the Bellanger s formula. For large values of δ c and δ s, (2.47) and (2.48) may result in negative orders but such large ripples may not be practical also. As an example, with δ c = δ s = 0.5, ω s T = 0.3π, and ω c T = 0.2π, we get N B = and N K = Throughout this thesis, the ripples are chosen so that they (i) are practical, and (ii) ensure positive orders. This is achieved if δ s δ c < 0.1 in (2.47). δ s δ c < in (2.48). Figure 2.3 shows a relative comparison of these positive orders for some typical values of ω s T ω c T and δ s = δ c. As can be seen, there is a maximum of 10% difference between N B and N K. With the values of δ s, δ c, ω s T, and ω c T used in this thesis, this difference is about 5%. Consequently, the conclusions of the thesis are valid even if (2.48) is used. However, (2.48) slightly changes the fomulations of complexity, etc. Generally and for very small or large ω c T, these formulae suffer from estimation inaccuracies. However, there are other methods to estimate the 18

33 2. BASICS OF DIGITAL FILTERS filter order as in, e.g., [90]. As [90] complicates the derivations of the arithmetic complexity provided in this thesis, we do not use it. The filter design problem finds h(n) so as to satisfy a specific criterion. This criterion could be the energy, maximum ripple, or combinations of them leading to LS, minimax, or CLS approaches. The general minimax design problem is min δ, subject to (2.49) H(e jωt ) 1 δ, ωt Ω c H(e jωt ) W(ωT)δ, ωt Ω s. On the other hand, the LS design problem is min H(e jωt ) 1 2 H(e d(ωt)+ ωt Ω c ωt Ωs jωt ) 2 d(ωt). (2.50) W(ωT) Regarding CLS, one could minimize the stopband (passband) energy with constraints on the passband (stopband) ripples. This thesis formulates the CLS design problem as min δ, subject to (2.51) ωt Ω c H(e jωt ) 1 2 d(ωt) δ, ωt Ω c H(e jωt ) δ des, ωt Ω s. Here, δ des is the desired maximum stopband ripple. Further, W(ωT) is a weighting function. A large W(ωT) results in small (large) stopband approximation errors for minimax (LS) designs. This thesis assumes frequency independent weighting functions and, thus, W(ωT) is constant in the frequency range of interest. 19

34 20 2. BASICS OF DIGITAL FILTERS

35 3 Basics of Multirate Signal Processing This chapter treats some basics of multirate systems. Sections 3.1 and 3.2 discuss the sampling rate conversion (SRC) based on the conventional structures and the Farrow structure. Then, filter banks (FBs) are defined in Section 3.3 where their input-output relation and the perfect reconstruction (PR) conditions are considered. As duals of FBs, transmultiplexers (TMUXs) are outlined in Section 3.4. Finally, redundant TMUXs with non-overlapping filters and their filter design problem are treated. 3.1 Sampling Rate Conversion: Conventional Different parts of a multirate system operate at different sampling frequencies. Consequently, there is a need for SRC between these parts. This can be performed by interpolation (decimation) which increases (decreases) the sampling frequency of digital signals [12, 69]. An alternative, to perform SRC on digital signals, is to first construct the corresponding analog signal and, then, resample it with the desired 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 the adders and multipliers can operate at a lower rate. Interpolation and decimation are two-stage processes comprising lowpass filters as well as downsamplers and upsamplers. The block diagrams of 21

36 3. BASICS OF MULTIRATE SIGNAL PROCESSING (a) x(m) M y(n) (b) x(n) L y(m) Figure 3.1: (a) M-fold downsampler. (b) L-fold upsampler. x(m) H(z) M y(n) x(n) L H(z) y(m) Figure 3.2: Decimation by M. Figure 3.3: Interpolation by L. upsamplers and downsamplers are shown in Fig A downsampler retains every Mth sample of the input signal as [12, 69] In the frequency domain, (3.1) becomes [12, 69] Y(z) = 1 M y(n) = x(nm). (3.1) M 1 k=0 X(z 1 M W k M ), (3.2) where W M is defined as in (2.36). The output signal is the sum of M stretched (by converting z to z M) 1 and shifted (through the terms WM k ) versions of X(z). Note that X(z M) 1 is not periodic by 2π. Adding the shifted versions gives a signal with a period of 2π so that the Fourier transform can be defined. An upsampler adds L 1 zeros between consecutive samples of x(n) and [12, 69] y(n) = { x( n L ) if n = 0,±L,±2L,... 0 otherwise. (3.3) In the frequency domain, (3.3) becomes [12, 69] Y(z) = X(z L ), (3.4) and the whole frequency spectrum is compressed by L giving rise to images. The upsampler and downsampler are linear time-varying systems [12]. Unlessx(n)islowpassandbandlimited 1, downsamplingresultsinaliasing. Consequently, decimation requires an extra filter as in Fig This anti-aliasing filter H(z) limits the bandwidth of x(n) as the original signal can only be preserved if it is bandlimited to π M. In Fig. 3.2, y(n) = + k= x(k)h(nm k). (3.5) 1 This is not necessary to avoid aliasing. For example, if X(e jωt ) is nonzero only at ωt [ω 1 T,ω 1 T + 2π M ] for some ω 1T, there is no aliasing [12]. 22

37 3. BASICS OF MULTIRATE SIGNAL PROCESSING As upsampling causes imaging, interpolation requires a filter as in Fig This lowpass anti-imaging filter H(z) removes the images and [12] y(n) = + k= x(k)h(n kl). (3.6) For SRC 2 by a rational ratio M L, interpolation by L in Fig. 3.3 must be followed by decimation by M in Fig Consequently, the cascade of the anti-imaging and anti-aliasing filters results in one filter, say G(z). Thus, the output is [12] y(n) = + k= x(k)g(nm kl). (3.7) This thesis will frequently use this cascade and its dual, i.e., interpolation by M followed by decimation by L. Generally, G(z) is a lowpass filter with a stopband edge at [12, 69] ω s T = min( π M, π L ) = π max(m,l). (3.8) In practice, there is a roll-off factor as in (2.33). If M and L are mutually coprime numbers, a decimator can be obtained by transposing the interpolator. For mutually coprime M and L, the following three systems 1. Upsampling by M followed by downsampling by L 2. Downsampling by L followed by upsampling by M 3. Upsampling by km followed by downsampling by kl followed by multiplier 1 k where k > 1 areequal[91]. Notethat(3.7)generallyfitsintotheframeworkofalineardual-rate system [92] which can always be represented via a kernel function as y(n) = + k= p(k, n)x(k). (3.9) Noble Identity The noble identity allows one to move the filtering operations inside a multirate structure. If 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 Combination of these noble identities and the polyphase decomposition enables efficient realizations of multirate structures. Efficient structures for integer decimation and interpolation are, respectively, shown in Figs. 3.5 and If L > M (L < M), we have interpolation (decimation) by a rational ratio L M > 1 (M L > 1). This thesis frequently refers to SRC by a rational ratio R p > 1. 23

38 3. BASICS OF MULTIRATE SIGNAL PROCESSING x(m) H(z M ) y(n) M <=> x(m) y(n) M H(z) x(n) M H(z M ) y(m) <=> x(n) H(z) M y(m) Figure 3.4: Noble identities which allow us to move the arithmetic operations to the lower sampling frequency. x(m) H(z) M Mf s y(n) f s x(m) Mf s z -1 M H 0 (z) y(n) f s H 0 (z) M H 1 (z) x(m) Mf s H 1 (z) y(n) f s z -1 M H M-1 (z) H M-1 (z) Figure 3.5: Decimation with polyphase decomposition and noble identities. x(n) M H(z) y(m) f s Mf s x(n) f s H 0 (z) M z -1 y(m) Mf s H 0 (z) H 1 (z) M x(n) H 1 (z) y(m) f s Mf s H M-1 (z) M z -1 H M-1 (z) Figure 3.6: Interpolation with polyphase decomposition and noble identities. 24

39 3. BASICS OF MULTIRATE SIGNAL PROCESSING Table 3.1: Types of the linear-phase FIR filters S k (z). N k k Type even even I even odd III odd even II odd odd IV x(n) S L (z) S 2 (z) S 1 (z) S 0 (z) m m m y(n) Figure 3.7: Farrow structure with fixed subfilters S k (z) and variable fractional delay µ. 3.2 Sampling Rate Conversion: Farrow Structure In conventional SRC and if the SRC ratio changes, new filters are needed. This reduces the flexibility in covering different SRC ratios. By utilizing the Farrow structure [93], shown in Fig. 3.7, this can be solved in an elegant way. The Farrow structure is composed of linear-phase finite-length impulse response (FIR) 3 subfilters S k (z), k = 0,1,...,L, with either a symmetric (for k even) or antisymmetric (for k odd) impulse response. According to Table 3.1, these subfilters could have any of the four types of the linear-phase FIR filters discussed in Section When S k (z) are linear-phase FIR filters, the Farrow structure is often referred to as the modified Farrow structure [94]. Throughout this thesis, we simply refer to it as the Farrow structure. The Farrow structure is efficient for interpolation whereas, for decimation, it is better to use the transposed Farrow structure [3, 95] so as to avoid aliasing. This chapter only considers integer and rational SRC ratios. Then, the decimators are obtained by transposing the corresponding interpolators [12]. This is in contrast to the irrational case which is more subtle [3, 95]. The subfilters can also have even or odd orders N k. With odd N k, all S k (z) are general filters whereas for even N k, the filter S 0 (z) reduces to a pure delay. The transfer function of the Farrow structure is 3 With infinite-length impulse response (IIR) filters, care must be taken to avoid transients as µ may change for every sample. 25