AmirEghbali ContributionstoReconﬁgurableFilterBanksandTransmultiplexers

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Dissertation No. 1344

Contributions to Reconfigurable Filter Banks

and Transmultiplexers

Amir Eghbali

Division of Electronics Systems

Department of Electrical Engineering

Link¨

oping University, SE–581 83 Link¨

oping, Sweden

WWW: http://www.es.isy.liu.se

E-mail: amire@isy.liu.se

Link¨

oping 2010

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c

2010 Amir Eghbali

Department of Electrical Engineering, Link¨oping University,

SE–581 83 Link¨oping, Sweden.

ISBN 978-91-7393-296-7 ISSN 0345-7524

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A current focus among communication engineers is to design flexible radio sys-tems to handle services among different telecommunication standards. Thus, low-cost multimode terminals will be crucial building blocks for future generations of multimode communications. Here, different bandwidths, from different telecom-munication standards, must be supported. This can be done using multimode transmultiplexers (TMUXs) which allow different users to share a common chan-nel 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 comple-mentary 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 commu-nication systems requires flexible digital signal processing structures. This flexibil-ity (or reconfigurabilflexibil-ity) 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 soft-ware 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.

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I would like to thank my supervisor Professor H˚akan 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 L¨owenborg 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 re-quires 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, Depart-ment of Electrical Engineering, Link¨oping 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 Link¨oping, September 2010

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Acronyms and Abbreviations

ADC Analog to Digital Converter

AFB Analysis Filter Bank

CLS Constrained Least-Squares CMFB Cosine Modulated Filter Bank DAC Digital to Analog Converter

DFBA Dynamic Frequency-Band Allocation DFBR Dynamic Frequency-Band Reallocation DFT Discrete Fourier Transform

ESA European Space Agency

EVM Error Vector Magnitude

FB Filter Bank

FDM Frequency Division Multiplexed FIR Finite-length Impulse Response FFBR Flexible Frequency-Band Reallocation FBR Frequency-Band Reallocation

GB GuardBand

GRB Granularity Band

GSM Global System for Mobile communications ICI Inter-Carrier Interference

IDFT Inverse Discrete Fourier Transform IIR Infinite-length Impulse Response ISI Inter-Symbol Interference IS-54 Interim Standard-54 IS-136 Interim Standard-136

LPTV Linear Periodic Time-Varying

LS Least-Squares

LTI Linear Time-Invariant

MF/TDMA Multiple Frequency/Time Division Multiple Access MIMO Multi-Input Multi-Output

MDFT Modified Discrete Fourier Transform

MSE Mean Square Error

NPR Near Perfect Reconstruction

PFBR Perfect Frequency-Band Reallocation

PR Perfect Reconstruction

RF Radio Frequency

QAM Quadrature Amplitude Modulation

SFB Synthesis Filter Bank

SISO Single-Input Single-Output

SRC Sampling Rate Conversion

TMUX Transmultiplexer

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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 multimode1 _{terminals will be crucial building blocks for}

future generations of communication systems. Multistandard communications re-quire to support different bandwidths from different telecommunication standards. Table 1.1 shows the bit rate, number of users sharing one channel, and the chan-nel 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 bandwidth-on-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.

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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, 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. 1.1. Here, the whole frequency spectrum is shared by P users. Each user Xp has a

band-width of π(1+ρ)_R_p , p = 0, 1, . . . , P − 1, and Rp can be an integer or a rational value.

Furthermore, ρ is the roll-off factor and a guardband (GB) of ∆ separates the user signals2. To support such a scenario, we can, in principle, use conventional3 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 consid-ering 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 im-plementation 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 engi-neers also aim to support various wideband services accessible to everybody ev-erywhere [32–39]. Here, satellites with high-gain spot beam antennas, on-board signal processing, and switching will be a major complementary part of future dig-ital 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

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The choice of ∆ does not restrict the analysis and design of the TMUX and, hence, throughout this thesis we will mostly assume ∆ = 0.

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D 0 2p wT X₀ X₁ X₂ XP-1 D 0 2p wT X₀ X₁ X₂ X_P-1 X₀ X₀ D 0 2p wT X₀ X₁ X₂ XP-1 D X₀ 0 2p wT X₀ X1 X2 XP-1 X0 0 2p wT X₀ X₁ X₂ XP-1 X₀ 0 2p wT X₀ X₁ X₂ XP-1 X₀ Case III: D>0 Case I: D<0 Case II: D=0

Figure 1.1: Problem formulation where P users share the frequency spectrum.

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 input multi-output system. The number of input signals can differ from that of the multi-output sig-nals. Furthermore, the input/output signals can have different bandwidths. Such a communication system must support different communication and connectiv-ity scenarios. One such main scenario is based on multiple frequency/time division multiple access (MF/TDMA). Here, the bandwidth of each incoming signal is com-posed 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 chan-nel both in time and frequency [58]. At any time, any portion of the chanchan-nel can be used by any user. The on-board signal processor reallocates all subbands to different output signals and center frequencies.

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In 1 In 2

FFBR Network

Out 1 Out 2 Out 3 p

Input signal 1

wT_in[rad] 1 2 3 p

Input signal 2

wT_in[rad] 4 5 6 p

Output signal 1

wT_out[rad] 1 3 p

Output signal 2

wT_out[rad] 4 5 p

Output signal 3

wT_out[rad] 2 6

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 real-located to different center frequencies. In a dynamic communication system, the bandwidth and center frequency of the users may change in a time-varying man-ner. 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 sub-bands 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 engi-neers 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 reconfigurabil-ity [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 Electron-ics Systems, Department of Electrical Engineering, Link¨oping University between October 2006 and August 2010. The research during this period has resulted in the following publications [43–46, 63–68]:

1. A. Eghbali, H. Johansson, and P. L¨owenborg, “Flexible frequency-band reallocation MIMO networks for real signals,” in Proc. Int. Symp. Image

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2. A. Eghbali, H. Johansson, and P. L¨owenborg, “Flexible frequency-band reallocation: complex versus real,” Circuits Syst. Signal Processing, DOI 10.1007/s00034-008-9090-3, Jan. 2009.

3. A. Eghbali, H. Johansson, and P. L¨owenborg, “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. L¨owenborg, “A multimode transmulti-plexer structure,” IEEE Trans. Circuits Syst. II, vol. 55, no. 3, pp. 279–283, Mar. 2008.

5. A. Eghbali, H. Johansson, and P. L¨owenborg, “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. L¨owenborg, “A class of multimode trans-multiplexers based on the Farrow structure,” Circuits Syst. Signal Processing, 2010, submitted.

7. A. Eghbali, H. Johansson, and P. L¨owenborg, “On the filter design for a class of multimode transmultiplexers,” in Proc. IEEE Int. Symp. Circuits

Syst., Taipei, Taiwan, May. 24-27, 2009.

8. A. Eghbali, H. Johansson, and P. L¨owenborg, “Reconfigurable nonuniform transmultiplexers based on uniform filter banks,” in Proc. IEEE Int. Symp.

Circuits Syst., Paris, France, May 30-June 2, 2010.

9. A. Eghbali, H. Johansson, and P. L¨owenborg, “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. L¨owenborg, and H. G. G¨ockler, “Dy-namic frequency-band reallocation and allocation: From satellite-based com-munication systems to cognitive radios,” J. Signal Processing Syst., DOI 10.1007/s11265-009-0348-1, Feb. 2009.

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. L¨owenborg, “On the com-plexity of multiplierless direct and polyphase FIR filter structures,” in Proc.

Int. Symp. Image Signal Process. Analysis, Istanbul, Turkey, Sept. 2007.

2. 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.

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3. A. Eghbali, H. Johansson, T. Saram¨aki, and P. L¨owenborg, “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, 2010.

1.2 Thesis Outline

The thesis consists of nine chapters where Chapters 2 and 3 deal with the back-ground 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 finite-length impulse response and infinite-finite-length impulse response filters; polyphase de-composition; 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 struc-tures 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 gen-erating 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 in-terpolation/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

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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.

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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 causal1 _{FIR filter of order N has an impulse response with N + 1 coefficients}

h(0), h(1), . . . , h(N ). The transfer function of an N th-order FIR filter is [69] H(z) =

N

X

n=0

h(n)z−n_. _(2.1)

In the time domain and with an input sequence x(n), the output sequence is y(n) = N X k=0 h(k)x(n − k) ⇔ Y (z) = H(z)X(z). (2.2) 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.

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x(n) T T T

h0 h1 h2 hN-1 hN

T

y(n)

Figure 2.1: Direct form realization of an N th-order FIR filter.

x(n) y(n) h0 h1 h2 hN-1 T T T hN 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 h0, h1, . . . , hN. 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

H(z) = PN n=0a(n)z−n 1 −PN n=1b(n)z−n . (2.3)

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.

2.2.1 Note on Stability

The z-transform of h(n) is defined by the Laurent series [69–73]

H(z) =

+∞

X

n=−∞

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This transform exists if h(n) decays to zero as n approaches −∞ and +∞. If [72]

|h(n)|≤M1K1n , n≥0, (2.5)

|h(n)|≤M2K2n , n≤0, (2.6)

then (2.4) converges for

K1< |z| < K2. (2.7)

As z can have a radius r and an angle θ of the form z = rejθ_{, (2.4) will converge}

on every concentric circle with K1 < r < K2. For right-hand (left-hand) sided

sequences, (2.4) will converge on concentric circles exterior (interior) to some ra-dius, say Kc, 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(ejωT) = +∞ X n=−∞ h(n)e−jnωT. (2.8)

2.3 Polyphase Decomposition

The transfer function in (2.1) can be decomposed as

H(z) = ∞ X n=−∞ h(nL)z−nL +z−1 ∞ X n=−∞ h(nL + 1)z−nL (2.9) . . . +z−(L−1) ∞ X n=−∞ h(nL + L − 1)z−nL,

which can be rewritten as [12, 69, 70]

H(z) = L−1 X i=0 z−i_H i(zL). (2.10)

Here, Hi(z) are the polyphase components and

hi(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]

H(z) = L−1 X i=0 z−(L−1−i)_R i(zL), (2.12)

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where Ri(z) = HL−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]

Y (z) = L−1 X l=0 Yl(zL)z−l= L−1 X i=0 Xi(zL)z−i L−1 X j=0 Hj(zL)z−j, (2.13)

where Yl(z), Xi(z), and Hj(z) are the polyphase components of Y (z), X(z), and

H(z), respectively. In a matrix form, (2.13) becomes       Y0(zL) Y1(zL) .. . YL−1(zL)       = H(zL₎       X0(zL) X1(zL) .. . XL−1(zL)       (2.14) where H(zL_{) =}       H0(zL) z−LHL−1(zL) . . . z−LH1(zL) H1(zL) H0(zL) . . . z−LH2(zL) .. . ... . .. ... HL−1(zL) HL−2(zL) . . . H0(zL)       . (2.15)

2.4 Special Classes of Filters

Some classes of digital filters are more suitable for multirate systems. The sequel introduces some of these classes.

2.4.1 Complementary Filters

The filters Hk(z), k = 0, 1, . . . , K, are power complementary if [12] K

X

k=0

|Hk(ejωT)|2= c, c > 0. (2.16)

In general, Hk(z) are complementary of order p if [74] K

X

k=0

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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

X

k=0

Hk(ejωT) = cz−D0, c6=0. (2.18)

2.4.2 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 N₂ distinct coefficients thereby reducing the number of mul-tipliers. However, this does not change the number2 _{of adders. The frequency}

response of a linear-phase FIR filter can be expressed as H(ejωT) = e−j(N ωT2 +c)H

R(ωT ) = ejΘ(ωT )HR(ωT ), (2.21)

where HR(ωT ) is the real zero-phase frequency response with c = 0 and c = π₂

for symmetric and antisymmetric h(n), respectively. The magnitude response, i.e., |HR(ωT )|, always assumes real positive values whereas HR(ωT ) could be negative.

The phase response is [69, 75]

Φ(ωT ) = (

Θ(ωT ), HR(ωT )≥0

Θ(ωT )−π, HR(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), N even Type II : h(n) = h(N − n), N odd Type III : h(n) = −h(N − n), N even

Type IV : h(n) = −h(N − n), N odd. (2.24)

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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 HR(ωT ) as [75]

HR(ωT ) =                h(N₂) + 2PN2 n=1h(N2 − n) cos(nωT ) Type I 2PN −12 n=0 h(N −12 − n) cos( n+1 2 ωT ) Type II 2PN2−1

n=0 h(N2 − 1 − n) sin((n + 1)ωT ) Type III

2PN −12 n=0 h(N −12 − n) sin( n+1 2 ωT ) Type IV. (2.25) Further, [75] Φ(ωT ) = ( −N ωT2 Type I,II −N ωT2 + π 2 Type III,IV. (2.26)

The group delay τg(ωT ) and the phase delay τp(ωT ) are defined as [69, 75]

τg(ωT ) = − dΦ(ωT ) d(ωT ) , (2.27) and τp(ωT ) = −Φ(ωT ) ωT . (2.28)

The shape of a periodic signal is preserved3 _{if τ}

p(ωT ) is almost constant in the

passband. This makes the delay of all signal components approximately equal. For nonperiodic signals, τg(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 τg(ωT ) = N₂.

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 linear-phase 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].

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2.4.3 Nyquist (M th-band) Filters

A lowpass non-causal filter h(n) of order N is said to be M th-band if any of its polyphase components, i.e., Hk(z), satisfies [12, 69, 76]

Hk(zM) =

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 _Mm. In general and for a non-causal h(n), this gives

h(n) = ( ₁

M n = 0

0 n = ±M, ±2M, . . . (2.31)

meaning that every M th 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 M th-band filter, its delayed version is also an M th-band filter [12]. In the causal case, H(z) is an M th-band filter if the kth polyphase component has the form Hk(z) = _M1z−nk. In the time domain, this becomes

h(nM + k) = ( ₁

M n = nk

0 otherwise. (2.32)

For an M th-band filter, the passband and stopband edges are, respectively, [77] ωcT = π(1 − ρ)

M ωsT = 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 =_Mπ. 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

HR(ωT ) =

1

2, ωT = π

M. (2.34)

Furthermore, the passband and stopband ripples are related to each other as

δs≤(M − 1)δc. (2.35)

If H(z) is an M th-band filter, the sum of M shifted copies of H(z) results in a constant. In other words,

M X k=0 H(zWMk ) = c, WM = e−j 2π M, c > 0. (2.36)

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An alternative to (2.36) is obtained from (2.18) with D0 = 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].

2.4.4 Hilbert Transformers

The spectrum of a real-valued signal is Hermitian symmetric around ωT = 0 and H(ejωT_{) = 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 desir-able 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(ejωT) = (

2 0 < ωT < π

0 −π < ωT < 0. (2.37) From (2.37), we see that there is some ambiguity at ωT = 0 [84]. The corresponding IIR non-causal impulse response is

h(n) =      1 n = 0 2j nπ odd n 0 otherwise. (2.38)

The complex output sequence is then

y(n) = x(n) ∗ h(n) = x(n) + jx(n) ∗ hi(n), (2.39)

where ∗ represents convolution and [84]

hi(n) = ( ₂ nπ odd n 0 even n. (2.40) Further, Hi(ejωT) = ( −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 −12 z−N+ jE(−z2), (2.42)

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where

G(z) = z

−N_{− E(z}2₎

2 . (2.43)

In the FIR case, E(z2_{) has a linear phase with a group delay of N 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(ejωT) = (

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 6= 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 − δc≤ |H(ejωT)| ≤ 1 + δc, ωT ∈ Ωc

|H(ejωT)| ≤ δs, ωT ∈ Ωs. (2.46)

Here, δc and δsare, respectively, the passband and stopband ripples with Ωc and

Ωs 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, Ωc covers [0, ωcT ] whereas Ωs covers [ωsT, π].

Here, ωcT and ωsT 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 Ωc, Ωs, δc, and δs. A commonly used formula to

estimate the order of a linear-phase FIR filter is the Bellanger’s formula [88] NB≈ − 2 3log10(10δsδc) 2π ωsT − ωcT . (2.47)

For reasonable orders, (2.47) gives a good approximation. For general nonlinear-phase FIR filters, such formulae do not exist. Then, a manual search is the only

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−200 −150 −100 −50 0.2 0.4 0.6 0.8 0 2 4 6 8 δ s=δc [dB] (ω sT−ωcT)/π 100*(N B −N K )/N K

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

NK≈−20 log10

(√δsδc) − 13

14.6(ωsT − ωcT )/2π

. (2.48)

This thesis uses the Bellanger’s formula. For large values of δc and δs, (2.47) and

(2.48) may result in negative orders but such large ripples may not be practical also. As an example, with δc = δs = 0.5, ωsT = 0.3π, and ωcT = 0.2π, we get

NB= −5.3059 and NK= −9.5608. Throughout this thesis, the ripples are chosen

so that they (i) are practical, and (ii) ensure positive orders. This is achieved if • δsδc< 0.1 in (2.47).

• δsδc< 10−

26

20 in (2.48).

Figure 2.3 shows a relative comparison of these positive orders for some typical values of ωsT − ωcT and δs = δc. As can be seen, there is a maximum of 10%

difference between NB and NK. With the values of δs, δc, ωsT , and ωcT 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 ωcT , these formulae suffer

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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(ejωT) − 1| ≤ δ, ωT ∈ Ωc

|H(ejωT)|≤W (ωT )δ, ωT ∈ Ωs.

On the other hand, the LS design problem is min Z ωT ∈Ωc |H(ejωT) − 1|2d(ωT ) + Z ωT ∈Ωs |H(ejωT_)|2 W (ωT ) d(ωT ). (2.50) Regarding CLS, one could minimize the stopband (passband) energy with con-straints on the passband (stopband) ripples. This thesis formulates the CLS design problem as min δ, subject to (2.51) Z ωT ∈Ωc |H(ejωT) − 1|2d(ωT ) ≤ δ, ωT ∈ Ωc |H(ejωT_)|≤δ des, ωT ∈ Ωs.

Here, δ_desis 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.

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3

Basics of Multirate Signal

Processing

This chapter treats some basics of multirate systems. Sections 3.1 and 3.2 dis-cuss 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 con-sidered. As duals of FBs, transmultiplexers (TMUXs) are outlined in Section 3.4. Finally, redundant TMUXs with non-overlapping filters and their filter design prob-lem 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

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L

x(n)

y(m)

(a)

(b)

x(m)

M

y(n)

Figure 3.1: (a) M -fold downsampler. (b) L-fold upsampler.

x(m)

H(z)

M

y(n)

Figure 3.2: Decimation by M .

L

x(n)

H(z)

y(m)

Figure 3.3: Interpolation by L.

upsamplers and downsamplers are shown in Fig. 3.1. A downsampler retains every M th sample of the input signal as [12, 69]

y(n) = x(nM ). (3.1)

In the frequency domain, (3.1) becomes [12, 69]

Y (z) = 1 M M −1 X k=0 X(zM1Wk M), (3.2)

where WM is defined as in (2.36). The output signal is the sum of M stretched (by

converting z to zM1) and shifted (through the terms Wk

M) versions of X(z). Note

that X(zM1 ) 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(_Ln) if n = 0, ±L, ±2L, . . .

0 otherwise. (3.3)

In the frequency domain, (3.3) becomes [12, 69]

Y (z) = X(zL), (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].

Unless x(n) is lowpass and bandlimited1_{, downsampling results in aliasing.}

Con-sequently, decimation requires an extra filter as in Fig. 3.2. 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) = +∞ X k=−∞ x(k)h(nM − k). (3.5) 1

This is not necessary to avoid aliasing. For example, if X(ejωT_{) is nonzero only at}

ωT∈[ω1T, ω1T+ 2π

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As upsampling causes imaging, interpolation requires a filter as in Fig. 3.3. This lowpass anti-imaging filter H(z) removes the images and [12]

y(n) =

+∞

X

k=−∞

x(k)h(n − kL). (3.6)

For SRC2 _{by a rational ratio}M

L, interpolation by L in Fig. 3.3 must be followed by

decimation by M in Fig. 3.2. 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) =

+∞

X

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] ωsT = 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 co-prime 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

are equal [91]. Note that (3.7) generally fits into the frame work of a linear dual-rate system [92] which can always be represented via a kernel function as

y(n) = +∞ X k=−∞ p(k, n)x(k). (3.9)

3.1.1 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. 3.4. Combination of these noble identities and the polyphase decomposition enables efficient realizations of multi-rate structures. Efficient structures for integer decimation and interpolation are, respectively, shown in Figs. 3.5 and 3.6.

2

If L > M (L < M ), we have interpolation (decimation) by a rational ratio L M >1 (

M L >1).

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<=>

x(m)

M

H(z)

y(n)

M

x(n)

H(z)

y(m)

x(m)

H(z

M

₎

_M

_y(n)

x(n)

_M

H(z

M

₎

y(m)

_<=>

Figure 3.4: Noble identities which allow us to move the arithmetic operations to the lower sampling frequency.

Mfs x(m) fs y(n) H0(z) H₁(z) HM-1(z) z-1 z-1 y(n) fs Mfs x(m) H(z) M M M M HM-1(z) fs y(n) H1(z) H₀(z) Mfs x(m)

Figure 3.5: Decimation with polyphase decomposition and noble identities.

y(m) z-1 Mfs x(n) fs HM-1(z) H1(z) H0(z) z-1 y(m) Mfs fs x(n) M H(z) M M M H₀(z) x(n) fs H₁(z) HM-1(z) Mfs y(m)

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Table 3.1: Types of the linear-phase FIR filters Sk(z).

Nk k Type

even even I even odd III

odd even II odd odd IV x(n) S L(z) S2(z) S1(z) m S₀(z) y(n) m m

Figure 3.7: Farrow structure with fixed subfilters Sk(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

subfil-ters Sk(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 2.4.2.

When Sk(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 Nk. With odd Nk, all Sk(z) are general

filters whereas for even Nk, the filter S0(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 µmay change for every sample.

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H(z, µ) = L X k=0 Sk(z)µk (3.10) = L X k=0 Nk X n=0 sk(n)z−nµk = N X n=0 L X k=0 sk(n)µkz−n= N X n=0 h(n, µ)z−n_.

Here, |µ| ≤ 0.5 and N is the order of the overall impulse response

h(n, µ) =

L

X

k=0

sk(n)µk. (3.11)

Further, µ is the fractional delay value4 _{which defines the time difference between}

each input sample and its corresponding output sample. In the rest of the thesis, we use h(n) and H(z) instead of h(n, µ) and H(z, µ), respectively. Assuming Tin

and Toutto be the sampling period of x(n) and y(n), respectively, µ is5[63, 65, 96]

Even Nk : [nin+ µ(nin)]Tin= noutTout

Odd Nk : [nin+ 0.5 + µ(nin)]Tin= noutTout (3.12)

where nin(nout) is the input (output) sample index. If µ is constant for all input

samples, the Farrow structure delays a bandlimited signal by a fixed µ. Figure 3.8 shows two delayed versions of a bandlimited signal x(n) = sin(nπ

12) where µ = 0.25

and µ = 0.45. In both cases, one set of Sk(z) has been used and only µ is modified.

In general, SRC can be seen as delaying every input sample with a different µ. This delay depends on whether one performs decimation or interpolation. For interpolation, one can obtain new samples between any two consecutive samples of x(n). With decimation, one can shift the original samples (or delay them in the time domain) to the positions which would belong to the decimated signal. Hence, some signal samples will be removed but some new samples will be produced. Thus, by controlling µ for every input sample, the Farrow structure performs SRC. For decimation, Tout > Tin whereas interpolation results in Tout < Tin. As an

example, Fig. 3.9 illustrates two versions of a bandlimited signal x(n) = sin(nπ₁₂) where a rational SRC by Rp = 1.75 is performed. In both cases, the same Sk(z)

as those in Fig. 3.8 have been used and only µ(nin) is modified for every input

sample.

4

In the modified Farrow structure, 0 < µ < 1.

5

In the implementation, a group of input samples are present in the delay elements of Sk(z).

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5 10 15 20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 n Amplitude x(n) x(n−0.25) x(n−0.45)

Figure 3.8: Application of the Farrow structure to delay x(n) = sin(nπ₁₂).

5 10 15 20 25 30 35 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Samples Amplitude sin(n 1ωT1) sin(n 2ωT1/1.75) sin(n 3ωT1*1.75)

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3.2.1 Design of the Farrow Structure

Generally, Sk(z) are designed so that H(z) approximates an allpass transfer

func-tion with a fracfunc-tional delay µ over the frequency range6_{of interest [94, 95, 97–104].}

The desired causal magnitude and unwrapped phase responses are

H_des(ejωT) = e−j(∆+µ)ωT, (3.13)

Φdes(ωT ) = −(∆ + µ)ωT, (3.14)

where

∆ = maxk(Nk)

2 . (3.15)

The main advantage of the Farrow structure is its ability to perform rational SRC using only one set of Sk(z) and by simple adjustments of µ. In the non-causal case

and with L subfilters, the Taylor series expansion of (3.13) is [105]

e−jµωT≈ L X k=0 (−jµωT )k k! = L X k=0 (−jωT )k k! µ k_. _(3.16)

Comparing (3.10) and (3.16), one way to obtain a fractional delay filter is to deter-mine Sk(z) so that they approximate kth-order differentiators [102]. Other methods

to design the Farrow structure can be found in, e.g., [94, 95, 97–104].

3.3 General M -Channel FBs

An M -Channel FB splits the input signal into the M subbands Xm(z), m =

0, 1, . . . , M − 1, using the analysis FB (AFB) filters Hm(z). To reconstruct the

original input signal, we need the synthesis FB (SFB) filters Fm(z). Furthermore,

upsamplers and downsamplers by P are also required as in Fig. 3.10. The output of a general M -channel FB is Y (z) = 1 P P −1 X n=0 X(zWPn) M −1 X m=0 Hm(zWPn)Fm(z) (3.17)

where WP is defined as in (2.36). Ideally, the output signal is 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 a FB is near PR (NPR), some aliasing and distortion exist. Therefore, α is frequency dependent and the distortion transfer function is

V0(z) = 1 P M −1 X m=0 Hm(z)Fm(z), (3.18) 6

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x1(m) x_M-1(m) x0(m)

y(n)

x(n)

P P P P P P

Synthesis FB

Analysis FB

F₀(z) F₁(z) F_M-1(z) H₀(z) H₁(z) H_M-1(z)

Figure 3.10: General M -channel FB.

whereas the aliasing transfer functions are

Vl(z) = 1 P M −1 X m=0 Hm(zWPl)Fm(z), l = 1, 2, . . . , P − 1. (3.19)

These FBs are generally linear periodic time-varying (LPTV) systems with a period M . If there is no aliasing, we have a linear time-invariant (LTI) system [12]. In a PR FB,

V0(ejωT) = c, c > 0 (3.20)

Vl(ejωT) = 0, l = 1, 2, . . . , P − 1. (3.21)

If P = M , the FB is maximally decimated and the number of samples in the set Xm(z) equals that of X(z). The choice P < M leads to oversampled FBs [12]. If

V0(z) is allpass (has linear-phase), we have no amplitude (phase) distortion.

Figure 3.11 shows an M -channel maximally decimated FB with the AFB filters Hm(z) and the SFB filters Fm(z). Here,

Y (z) = T0(z)X(z) + M −1 X l=1 Tl(z)X(zWMl ). (3.22) The term T0(z) = 1 M M −1 X m=0 Fm(z)Hm(z) (3.23)

is the distortion transfer function and Tl(z) = 1 M M −1 X m=0 Fm(z)Hm(zWMl ), l = 1, 2, . . . , M − 1 (3.24)

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Analysis FB

Synthesis FB

F₀(z) M F₁(z) M F_M-1(z) M

å

H₀(z) H₁(z) H_M-1(z) M M M y(n) x(n)

Figure 3.11: M -channel maximally decimated FB.

To obtain the AFB and SFB filters, one can modulate a single N th-order linear-phase FIR prototype filter G(z) =PN

n=0g(n)z−n. With cosine modulation [106–

108], hm(n) = 2g(n) cos[(m + 0.5) π M(N − n + M + 1 2 )], (3.25) fm(n) = 2g(n) cos[(m + 0.5) π M(n + M + 1 2 )] = hm(N − n). (3.26) In a PR cosine modulated FB (CMFB), N = 2KM − 1 and K (the overlapping factor [109]) is an integer. For complex modulated FBs,

hm(n) = g(n)WM−mn, (3.27)

fm(n) = hm(n). (3.28)

In the maximally decimated case, we can use modified discrete Fourier transform FBs (MDFT FBs) [110–115]. An M -channel MDFT FB can equivalently be realized as (see Figs. 7.11 and 7.12 of [76])

• Two SRC stages with ratios M2 and 2 while adding some phase offset between

these stages.

• Two separate FBs where the phase offset is applied outside the AFBs and SFBs.

If an MDFT FB is PR, N is an integer as KM + s where 0≤s < M. The choice of AFB and SFB filters, having uniform or nonuniform passbands, leads to uniform or nonuniform FBs [12] which can also be obtained by modulation as [23]

hm(n) = amgm(n)e− jπαm Mm (n−Lm−12 )+ a∗ mgm∗(n)e jπαm Mm (n−Lm−12 ), (3.29) fm(n) = bmgm(n)e− jπαm Mm (n−Lm−12 )_{+ a}∗ mg∗m(n)e jπαm Mm (n−Lm−12 )_. _(3.30) Here, αm = (Km+ 0.5) and gm(n) is the (possibly complex) prototype filter of

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s₀(n) s₁(n) s M-1(n) P P P F₀(z) F₁(z) F_M-1(z) y(n) y(n)^ s₀(n) ^ s 1(n) ^ s M-1(n) ^ H₀(z) P P H₁(z) P H_M-1(z) x₀(n) x 1(n) x M-1(n) D(z) e(n) Channel

Figure 3.12: General M -channel TMUX.

a center frequency as ±παm

Mm with Kmbeing an integer where amand bmdefine the modulation phase. As opposed to uniform FBs, nonuniform FBs achieve a more general time and frequency tiling [92]. Note that sine modulated FBs (SMFBs) can be obtained similar to (3.25) and (3.26). The exponentially modulated FBs (with complex filters) are a combination of SMFBs and CMFBs [107, 108].

For any FB, the AFB and SFB filters can be FIR or IIR. Further discussion on these issues is not the focus of this thesis and the interested reader is referred to, e.g., [12, 69, 76].

3.3.1 Filter Design for Modulated FBs

To design the prototype filter G(z), we can use any standard filter design technique, e.g., [12, 69, 76, 78, 106, 113–119]. The MDFT FB has a typical lowpass G(z) with a stopband edge as ωsT = 2π_M [76]. The CMFB has a typical lowpass G(z) with a

stopband edge as ωsT =π(1+ρ)_2M and a 3-dB cutoff frequency at ωT = _2Mπ [117, 120].

If 0 < ρ≤1, only the adjacent branches overlap. With 1 < ρ≤2 (or ρ > 2), two (or at least three) adjacent branches overlap [78]. In both FBs, G(z) satisfies the power complementary property.

3.4 General M -Channel TMUXs

A TMUX converts the time multiplexed components of a signal into a frequency multiplexed version and back [121]. It allows several users to transmit and receive over a common channel. A TMUX, e.g., [12, 13, 15–20, 24, 30, 46, 63–69, 76, 79, 116, 117, 119–124], is also referred to as a FB transceiver, e.g., [21, 125–128].

3.4.1 Mathematical Representation of TMUXs

Assume 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. Assume also that we want to transmit these signals through a channel. As in Fig. 3.12, we can pass sk(n) through the transmitter (pulse shaping) filters Fk(z). Then, (3.6) gives

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p wT p wT p wT

F

₀

(z) F

₁

(z)

F

_M-1

(z)

(a) (b) (c)

F

_M-1

(z)

F

_M-1

(z)

F

₁

(z)

F

₁

(z)

F

₀

(z)

F

₀

(z)

Figure 3.13: M -channel TMUX filters. (a) Overlapping. (b) Marginally overlap-ping. (c) Non-overlapoverlap-ping. xk(n) = ∞ X m=−∞ sk(m)fk(n − mP ). (3.31)

The filters Fk(z) take symbols of sk(n) and put pulses fk(n) around them. Here,

M users transmit through one common channel described by a possibly complex LTI filter D(z) =PLD

n=0d(n)z−nfollowed by an additive noise e(n). At the receiver

side, the receiver filters Hk(z) separate the signals and only a downsampling by P

is needed to get the original symbol streams. Ignoring the channel, ˆ Si(z) = M −1 X k=0 Sk(z)Tki(zP), i = 0, 1, . . . , M − 1 (3.32) where Tki(zP) = 1 P P −1 X l=0 Fk(zWPl)Hi(zWPl), (3.33)

and WP is defined as in (2.36). Typical characteristics of Fk(z) and Hk(z) are

shown in Fig. 3.13. Similar to FBs, TMUXs can be redundant (P > M ) or critically sampled (P = M ). To avoid inter-symbol interference (ISI), a level of redundancy may be needed such that P − M≥LD[129]. The output of the TMUX in (3.32) is

ˆ Si(z) = Tii(z)Si(z) + P −1 X k=0,k6=i Tki(z)Sk(z) (3.34)

where Tii(z) and Tki(z) represent the ISI and the inter-carrier interference (ICI),

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Analysis FB

Synthesis FB

F₀(z) M F₁(z) M F_M-1(z) M

å

H₀(z) H₁(z) H_M-1(z) M M M y(n) y(n)^ X₀(z) X₁(z) X_M-1(z) ^ ^ ^ X₀(z) X₁(z) X_M-1(z)

Figure 3.14: M -channel critically sampled TMUX.

(cross-band) ISI [21]. It is desired to have

|Tii(z) − z−ηi| ≤ δISI

|Tki(z)| ≤ δICI (3.35)

with δISI and δICI being the desired ISI and ICI where ηi is the delay in each

branch i of the TMUX.

If an LTI filter is placed between an upsampler and a downsampler of ratio M , the overall system is equivalent to the decimated (by M ) version of its impulse response [12]. In this case, designing Fk(z) and Hk(z) so that the decimated (by

M ) version of Fk(z)Hm(z) becomes a pure delay if k = m and zero otherwise, the

TMUX becomes PR. In terms of (3.34), this means

Tii(z) = 1 P P −1 X l=0 Fi(z 1 PWl P)Hi(z 1 PWl P) = αz−β, (3.36) Tki(z) = 1 P P −1 X l=0 Fk(z 1 PWl P)Hi(z 1 PWl P) = 0. (3.37)

In a PR system, ˆsk(n) = αsk(n − β). The PR properties are independent of the

length and causality of filters, etc., and can be satisfied for both critically sampled and redundant TMUXs. However, for the critically sampled case, there may not always exist FIR or stable IIR solutions. Therefore, some redundancy makes the solutions feasible [20, 22, 24, 26, 27] and it also simplifies the PR conditions.

3.4.2 Duality of FBs and TMUXs

Duality [12] of TMUXs and FBs allows one to obtain the TMUX of Fig. 3.14 from the FB of Fig. 3.11 where

Y (z) =

M −1

X

m=0

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This signal is then transmitted over a common channel. With ˆy(n) = y(n) and ignoring the scaling factors,

ˆ Xd(z) = M −1 X k=0 M −1 X m=0 Xm(z)Fm(z1/MWMk )Hd(z1/MWMk), d = 0, 1, . . . , M − 1. (3.39)

Like aliasing and distortion in FBs, we can define two error sources for TMUXs, i.e., ISI and ICI. In terms of (3.35)–(3.37) and for a critically sampled TMUX, we have ISI = M −1 X k=0 Fd(z1/MWMk)Hd(z1/MWMk), (3.40) and ICI = M −1 X m=0,m6=d M −1 X k=0 Fm(z1/MWMk)Hd(z1/MWMk ). (3.41)

The duality of FBs and TMUXs applies to both critically sampled and redundant systems. It has been shown that if a FB is free from aliasing, the corresponding TMUX is free from ICI [76].

3.4.3 Approximation of PR in Redundant TMUXs

In a PR TMUX and for any two branches k and m, the decimated version of the cascade of the SFB and AFB filters is a pure delay if k = m and zero otherwise [12]. In this regard, we use [Fk(z)Hm(z)]zeroth to represent the zeroth polyphase component of Fk(z)Hm(z). To approximate PR, these ideal conditions must be

approximated as close as desired. Thus, the minimax optimization problem for an NPR TMUX is

|[Fk(ejωT)Hm(ejωT)]zeroth− 1|≤δ, ωT ∈ [0, π], k = m |[Fk(ejωT)Hm(ejωT)]zeroth|≤W (ωT )δ, ωT ∈ [0, π], k6=m

where W (ωT ) is the weighting function. Note that (3.42) considers non-causal filters. It is well known that increasing the order of the SFB and AFB filters allows one to decrease δ and, hence, improve the approximation of PR.

To simplify (3.42), this thesis uses redundant TMUXs with non-overlapping filters as shown in Fig. 3.15. The TMUXs are also nonuniform. Consequently, the ISI in (3.34) would result from the filters in one branch of the TMUX. In general nonuniform TMUXs, the ICI in (3.34) becomes time-varying7_{. However, due to the}

redundancy, the stopband attenuation of the filters still controls the ICI. Therefore, the ICI can be made as small as desired by increasing this stopband attenuation. To meet NPR conditions, Fk(z)Hk(z) should approximate a Nyquist filter as close

as desired. Then, the SFB and AFB filters should be designed such that

7

In LTI systems, the output at any frequency only depends on the input at the same frequency. These nonuniform TMUXs are LPTV systems. Then, the output at any given frequency is dependent on the input at a finite set of frequencies [30].

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2p wT

F

₀

(z) F

₁

(z)

F

₂

(z)

F

_M-1

(z)

Figure 3.15: Filters of a nonuniform non-overlapping TMUX.

• They have sufficiently small ripples in their stopbands.

• The zeroth polyphase component of Fk(z)Hk(z) approximates an allpass

transfer function.

Thus, the simplified minimax design problem is

|[Fk(ejωT)Hk(ejωT)]zeroth− 1|≤δ, ωT ∈ [0, π] |Fk(ejωT)|≤W1(ωT )δ, ωT ∈ Ωs

|Hk(ejωT)|≤W2(ωT )δ, ωT ∈ Ωs

where k = 0, 1, . . . , M − 1. Furthermore, W1(ωT ) and W2(ωT ) are the weighting

functions with Ωsbeing the stopband region as in (2.46). In the least-squares (LS)

This thesis frequently uses Fk(z) = Hk(z) and W1(ωT ) = W2(ωT ) = W (ωT ).

Then, Fk(z) and Hk(z) are the spectral factors of a Nyquist filter [130–138].

Specif-ically, with linear-phase FIR filters and Fk(z) = Hk(z), the resulting Nyquist filter,

i.e., Fk(z)Hk(z), has double zeros in the z-plane. Further, we will always design

real lowpass filters and variable frequency shifters will modulate the users into in-termediate frequencies. Similar to (2.51), we will also use the constrained LS design method. This thesis does not consider the effects of the channel when designing the TMUXs but some methods can be found in, e.g., [21, 108, 121, 128].

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AmirEghbali ContributionstoReconﬁgurableFilterBanksandTransmultiplexers

Contributions to Reconfigurable Filter Banks

and Transmultiplexers

Amir Eghbali

Division of Electronics Systems

Department of Electrical Engineering

Link¨

oping University, SE–581 83 Link¨

oping, Sweden

WWW: http://www.es.isy.liu.se

E-mail: amire@isy.liu.se

Link¨

oping 2010

Contents

Acronyms and Abbreviations

1

Introduction

1.1

Motivation and Problem Formulation

FFBR Network

Input signal 1

Input signal 2

Output signal 1

Output signal 2

Output signal 3

1.2

Thesis Outline

2

Basics of Digital Filters

2.1

FIR Filters

2.2

IIR Filters

2.2.1

Note on Stability

2.3

Polyphase Decomposition

2.4

Special Classes of Filters

2.4.1

Complementary Filters

2.4.2

Linear-Phase FIR Filters

2.4.3

Nyquist (M th-band) Filters

2.4.4

Hilbert Transformers

2.5

FIR Filter Design

3

Basics of Multirate Signal

Processing

3.1

Sampling Rate Conversion: Conventional

L

x(n)

y(m)

(a)

(b)

x(m)

M

y(n)

x(m)

H(z)

M

y(n)

L

x(n)

H(z)

y(m)

3.1.1

Noble Identity

<=>

x(m)

M

H(z)

y(n)

M

x(n)

H(z)

₎

_M

_y(n)

_M

₎

_<=>