Contributions to Reconﬁgurable Filter Banks and...

Linkoping Studies in Science and TechnologyDissertation No. 1344

Contributions to Reconfigurable Filter Banks

and Transmultiplexers

Amir Eghbali

Division of Electronics Systems

Department of Electrical Engineering

Linkoping University, SE–581 83 Linkoping, Sweden

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

E-mail: [email protected]

Linkoping 2010

Contributions to Reconfigurable Filter Banks and Transmultiplexers

c© 2010 Amir Eghbali

Department of Electrical Engineering,Linkoping University,SE–581 83 Linkoping,

Sweden.

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

Printed by LiU-Tryck, Linkoping, Sweden 2010

to my family...

Abstract

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 ofmultimode communications. Here, different bandwidths, from different telecom-munication standards, must be supported. This can be done using multimodetransmultiplexers (TMUXs) which allow different users to share a common chan-nel in a time-varying manner. These TMUXs allow bandwidth-on-demand. Eachuser occupies a specific portion of the channel whose location and width may varywith time.

Another focus among communication engineers is to provide various widebandservices accessible to everybody everywhere. Here, satellites with high-gain spotbeam antennas, on-board signal processing, and switching will be a major comple-mentary part of future digital communication systems. Satellites provide a globalcoverage and customers only need to install a satellite terminal and subscribe to theservice. Efficient utilization of the available limited frequency spectrum, calls foron-board signal processing to perform flexible frequency-band reallocation (FFBR).

In an integrated communication system, TMUXs can operate on-ground whereasFFBR networks can operate on-board. Thus, successful design of dynamic commu-nication systems requires flexible digital signal processing structures. This flexibil-ity (or reconfigurability) must not impose restrictions on the hardware and, ideally,it must come at the expense of simple software modifications. In other words, thesystem is based on a hardware platform whose parameters can be modified withouta need for hardware changes.

This thesis outlines the design and realization of reconfigurable TMUX andFFBR structures which allow dynamic communication scenarios with simple soft-ware reconfigurations. In both structures, the system parameters are determinedin advance. For these parameters, the required filter design problems are solvedonly once. Dynamic communications, with users having different time-varyingbandwidths, are then supported by adjusting some multipliers, commutators, or achannel switch. These adjustments do not require hardware changes and can beperformed online. However, the filter design problem is solved offline. The thesisprovides various illustrative examples and it also discusses possible applicationsof the proposed structures in the context of other communication scenarios, e.g.,cognitive radios.

i

Acknowledgments

I would like to thank my supervisor Professor Hakan Johansson for giving me theopportunity to work as a Ph.D student. However, I should not forget to sincerelythank him for his patience, inspiration, and guidance in helping me deal with myproblems.

I would also like to thank my co-supervisor Docent Per Lowenborg for discussionsand feedback.

Special thanks have to go to all members of my family for their support. Not allproblems can be solved by computers, books, and discussions, etc. One mostly re-quires emotional support and encouragement from beloved ones. God has blessedme with the best of these! I just do not know how to be thankful... I will never beable to do this...

The former and present colleagues at the Division of Electronics Systems, Depart-ment of Electrical Engineering, Linkoping University have created a very friendlyenvironment. They always kindly do their best to help you. You never feel aloneeven 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 inSweden pleasant.

Amir Eghbali

Linkoping, September 2010

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Contents

1 Introduction 11.1 Motivation and Problem Formulation . . . . . . . . . . . . . . . . . . 11.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Basics of Digital Filters 92.1 FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 IIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Note on Stability . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Polyphase Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Special Classes of Filters . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Complementary Filters . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Linear-Phase FIR Filters . . . . . . . . . . . . . . . . . . . . 132.4.3 Nyquist (Mth-band) Filters . . . . . . . . . . . . . . . . . . . 152.4.4 Hilbert Transformers . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 FIR Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Basics of Multirate Signal Processing 213.1 Sampling Rate Conversion: Conventional . . . . . . . . . . . . . . . 21

3.1.1 Noble Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Sampling Rate Conversion: Farrow Structure . . . . . . . . . . . . . 25

v

Contents Contents

3.2.1 Design of the Farrow Structure . . . . . . . . . . . . . . . . . 283.3 General M -Channel FBs . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Filter Design for Modulated FBs . . . . . . . . . . . . . . . . 313.4 General M -Channel TMUXs . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1 Mathematical Representation of TMUXs . . . . . . . . . . . 313.4.2 Duality of FBs and TMUXs . . . . . . . . . . . . . . . . . . . 333.4.3 Approximation of PR in Redundant TMUXs . . . . . . . . . 34

4 Flexible Frequency-Band Reallocation For Real Signals 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Contribution and Relation to Previous Work . . . . . . . . . 384.1.2 Choice of the FFBR Network . . . . . . . . . . . . . . . . . . 384.1.3 MIMO FFBR Network Configuration . . . . . . . . . . . . . 40

4.2 FFBR Network Based on Variable Oversampled Complex Modu-lated FBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Efficient Realization of the FFBR Network . . . . . . . . . . 41

4.3 Alternative I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Complex Versus Real Sampling . . . . . . . . . . . . . . . . . 434.3.2 Arithmetic Complexity: Hilbert Transformer . . . . . . . . . 434.3.3 Arithmetic Complexity: DFT with Complex Inputs . . . . . 444.3.4 Arithmetic Complexity: Complex FFBR Network . . . . . . 45

4.4 Alternative II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.1 Arithmetic Complexity: Real FFBR Network . . . . . . . . . 46

4.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5.1 Arithmetic Complexity: Complex Versus Real FFBR . . . . . 494.5.2 Arithmetic Complexity: Alternative I Versus Alternative II . 504.5.3 Performance: Alternative I Versus Alternative II . . . . . . . 52

4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.6.1 Measure of Complexity . . . . . . . . . . . . . . . . . . . . . 544.6.2 Applicability of Alternatives I and II . . . . . . . . . . . . . . 564.6.3 Filter Bank Design . . . . . . . . . . . . . . . . . . . . . . . . 56

5 A Multimode Transmultiplexer Structure 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Multimode TMUX Structure . . . . . . . . . . . . . . . . . . . . . . 59

5.3.1 Channel Sampling Rates . . . . . . . . . . . . . . . . . . . . . 595.3.2 Sampling Rate Conversion . . . . . . . . . . . . . . . . . . . . 605.3.3 Subcarrier Frequencies . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Implementation and Design Complexity Issues . . . . . . . . . . . . 645.6 TMUX Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.7 Analysis Using Multirate Building Blocks . . . . . . . . . . . . . . . 685.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

vi

Contents Contents

6 A Class of Multimode Transmultiplexers Based on the FarrowStructure 73

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.1.1 Contribution and Relation to Previous Work . . . . . . . . . 74

6.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 74

6.2.2 Some General Issues . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 Proposed Integer SRC Multimode TMUX . . . . . . . . . . . . . . . 76

6.3.1 Variable Integer SRC Using the Farrow Structure . . . . . . . 76

6.3.2 Approximation of Perfect Reconstruction (PR) . . . . . . . . 79

6.3.3 Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3.4 Filter Design Parameters . . . . . . . . . . . . . . . . . . . . 80

6.3.5 Filter Design Criteria . . . . . . . . . . . . . . . . . . . . . . 80

6.4 Proposed Rational SRC Multimode TMUX . . . . . . . . . . . . . . 83

6.4.1 TMUX Illustration . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4.2 Efficient Variable Rational SRC . . . . . . . . . . . . . . . . . 84

6.4.3 Approximation of PR . . . . . . . . . . . . . . . . . . . . . . 87

6.5 TMUX Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5.1 Effects of Bp on the SRC Error . . . . . . . . . . . . . . . . . 90

6.6 Direct Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.6.1 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Reconfigurable Nonuniform Transmultiplexers Using UniformModulated Filter Banks 99

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.1.1 Contribution and Relation to Previous Work . . . . . . . . . 100

7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.3 Nonuniform TMUXs Using Modulated FBs . . . . . . . . . . . . . . 102

7.4 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.4.1 Channel Sampling Periods . . . . . . . . . . . . . . . . . . . . 105

7.4.2 TMUX Illustration . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4.3 Choice of GB . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4.4 Choice of Center Frequency . . . . . . . . . . . . . . . . . . . 105

7.5 Implementation Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.5.1 Choice of M and ρ . . . . . . . . . . . . . . . . . . . . . . . . 110

7.5.2 Filter Design Restrictions . . . . . . . . . . . . . . . . . . . . 112

7.6 Comparison with Existing Multimode TMUXs . . . . . . . . . . . . 112

7.6.1 Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.6.2 Spectrum Efficiency . . . . . . . . . . . . . . . . . . . . . . . 113

7.6.3 Direct or Indirect Design . . . . . . . . . . . . . . . . . . . . 113

7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vii

Contents Contents

8 Applications to Cognitive Radios 1158.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.2 Approach I: Use of DFBR Networks . . . . . . . . . . . . . . . . . . 117

8.2.1 Structure of the DFBR Network . . . . . . . . . . . . . . . . 1188.2.2 User Bandwidth Versus Multiplexing Bandwidth . . . . . . . 1198.2.3 Reconfigurability . . . . . . . . . . . . . . . . . . . . . . . . . 1198.2.4 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.3 Approach II: Use of TMUXs . . . . . . . . . . . . . . . . . . . . . . . 1228.3.1 Structure of the TMUX . . . . . . . . . . . . . . . . . . . . . 1238.3.2 Reconfigurability . . . . . . . . . . . . . . . . . . . . . . . . . 1238.3.3 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.4 Choice of Frequency Shifters . . . . . . . . . . . . . . . . . . . . . . . 1258.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9 Conclusion and Future Work 129

A Derivation of (6.23) 131

B Derivation of (6.35) 135

viii

Acronyms and Abbreviations


ADC Analog to Digital ConverterAFB Analysis Filter BankCLS Constrained Least-SquaresCMFB Cosine Modulated Filter BankDAC Digital to Analog ConverterDFBA Dynamic Frequency-Band AllocationDFBR Dynamic Frequency-Band ReallocationDFT Discrete Fourier TransformESA European Space AgencyEVM Error Vector MagnitudeFB Filter BankFDM Frequency Division MultiplexedFIR Finite-length Impulse ResponseFFBR Flexible Frequency-Band ReallocationFBR Frequency-Band ReallocationGB GuardBandGRB Granularity BandGSM Global System for Mobile communicationsICI Inter-Carrier InterferenceIDFT Inverse Discrete Fourier TransformIIR Infinite-length Impulse ResponseISI Inter-Symbol InterferenceIS-54 Interim Standard-54IS-136 Interim Standard-136LPTV Linear Periodic Time-VaryingLS Least-SquaresLTI Linear Time-InvariantMF/TDMA Multiple Frequency/Time Division Multiple AccessMIMO Multi-Input Multi-OutputMDFT Modified Discrete Fourier TransformMSE Mean Square ErrorNPR Near Perfect ReconstructionPFBR Perfect Frequency-Band ReallocationPR Perfect ReconstructionRF Radio FrequencyQAM Quadrature Amplitude ModulationSFB Synthesis Filter BankSISO Single-Input Single-OutputSRC Sampling Rate ConversionTMUX TransmultiplexerWLAN Wireless Local Area Network

ix


x

1

Introduction

1.1 Motivation and Problem Formulation

Communication engineers aim to design flexible radio systems to handle servicesamong 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 increaseas well. Hence, low-cost multimode1 terminals will be crucial building blocks forfuture 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., interimstandard-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 canuse any standard which suits its requirements on bandwidth, transmission quality,etc. Assume, for example, that a communication channel is shared by three usersA, 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, andaudio. Furthermore, at any time, any user can decide to use any center frequency.

1This is also referred to as multiband, multistandard, universal [7].

1

1. INTRODUCTION

Table 1.1: Bit rate, number of users sharing one channel, and channel spacing indifferent telecommunication standards.

Standard Bit Rate No. of Users Channel Spacing

IS-54/136 48.6 Kbps 3 30 KHzGSM 271 Kbps 8 200 KHzIS-95 1.2288 Kbps 798 1250 KHz

To support multimode communications, we thus need a system which allowsdifferent 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 inmultistandard communications. Multiple access schemes such as code divisionmultiple access, time division multiple access, frequency division multiple access,and orthogonal frequency division multiple access are special cases of a generalTMUX structure [13–15]. To support bandwidth-on-demand, the characteristics ofthe TMUXs must vary with time. Such a communication system has a dynamicallocation of bandwidth. Each user occupies a specific portion of the channel whoselocation 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)Rp

, 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 theuser 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 differentfilters or online filter design. This becomes inefficient when simultaneously consid-ering the increased number of communication scenarios and the desire to supportdynamic communications. Therefore, it is vital to develop low-complexity TMUXswhich dynamically support different communication scenarios with reasonable im-plementation complexity and design effort. One aim of this thesis is to introduceTMUXs which allow different numbers of users, having different bandwidths, toshare 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-boardsignal 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

2The choice of ∆ does not restrict the analysis and design of the TMUX and, hence, throughoutthis thesis we will mostly assume ∆ = 0.

3This is due to the duality of filter banks (FBs) and TMUXs [12].

2

1. INTRODUCTION

D

0 2pwT

X0 X1 X2 XP-1

D

0 2pwT

X0 X1 X2 XP-1 X0

X0

D

0 2pwT

X0 X1 X2 XP-1

D

X0

0 2pwT

X0 X1 X2 XP-1 X0

0 2pwT

X0 X1 X2 XP-1 X0

0 2pwT

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 usersthrough multiple spot beams [37]. Therefore, we need efficient reuse of the limitedavailable frequency spectrum by satellite on-board signal processing [32–57]. Thiscalls for flexible frequency-band reallocation (FFBR) networks [40–50] also referredto as frequency multiplexing and demultiplexing [40, 50–56].

The digital part of the satellite on-board signal processor is a multi-input multi-output system. The number of input signals can differ from that of the output sig-nals. Furthermore, the input/output signals can have different bandwidths. Sucha communication system must support different communication and connectiv-ity scenarios. One such main scenario is based on multiple frequency/time divisionmultiple access (MF/TDMA). Here, the bandwidth of each incoming signal is com-posed of a number of adjacent smaller frequency bands (subbands). Each subbandis 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 channel canbe used by any user. The on-board signal processor reallocates all subbands todifferent output signals and center frequencies.

The principle of this operation is illustrated in Fig. 1.2. Here, different users

3

1. INTRODUCTION

In 1

In 2

FF

BR

Netw

ork

Out 1

Out 2

Out 3

p

Input signal 1

wTin [rad]1 32

p

Input signal 2

wTin [rad]4 65

p

Output signal 1

wTout [rad]13

p

Output signal 2

wTout [rad]45

p

Output signal 3

wTout [rad]2 6

Figure 1.2: Frequency-band reallocation (FBR) for an FFBR network where anysignal in any of the two input signals can be reallocated to any position in any ofthe 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, thebandwidth and center frequency of the users may change in a time-varying man-ner. This necessitates FFBR networks which can dynamically perform reallocationof users with different bandwidths. Consequently, some requirements are imposedon FFBR networks such as flexibility, low complexity, near perfect frequency-bandreallocation, 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 flexibleand low complexity solutions for such FFBR networks. Although the idea of FFBRnetworks stems from satellite-based communications, they are generally applicableto systems which require frequency multiplexing and demultiplexing. This thesiswill 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. Thisflexibility must not restrict the hardware and, ideally, it must come at the expenseof 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 whoseparameters can be modified without hardware changes.

This thesis outlines solutions for the reconfigurable communication scenariosdiscussed above. It is a result of the research performed at the Division of Electron-ics Systems, Department of Electrical Engineering, Linkoping University betweenOctober 2006 and August 2010. The research during this period has resulted inthe following publications [43–46, 63–68]:

1. A. Eghbali, H. Johansson, and P. Lowenborg, “Flexible frequency-bandreallocation MIMO networks for real signals,” in Proc. Int. Symp. ImageSignal Processing Analysis, Istanbul, Turkey, Sept. 2007.

4

1. INTRODUCTION

2. A. Eghbali, H. Johansson, and P. Lowenborg, “Flexible frequency-bandreallocation: complex versus real,” Circuits Syst. Signal Processing, DOI10.1007/s00034-008-9090-3, Jan. 2009.

3. A. Eghbali, H. Johansson, and P. Lowenborg, “An arbitrary bandwidthtransmultiplexer and its application to flexible frequency-band reallocationnetworks,” in Proc. Eur. Conf. Circuit Theory Design, Seville, Spain, Aug.2007.

4. A. Eghbali, H. Johansson, and P. Lowenborg, “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. Lowenborg, “A Farrow-structure-basedmulti-mode transmultiplexer,” in Proc. IEEE Int. Symp. Circuits Syst.,Seattle, Washington, USA, May 2008.

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

7. A. Eghbali, H. Johansson, and P. Lowenborg, “On the filter design for aclass of multimode transmultiplexers,” in Proc. IEEE Int. Symp. CircuitsSyst., Taipei, Taiwan, May. 24-27, 2009.

8. A. Eghbali, H. Johansson, and P. Lowenborg, “Reconfigurable nonuniformtransmultiplexers 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. Lowenborg, “Reconfigurable nonuniformtransmultiplexers based on uniform filter banks,” IEEE Trans. Circuits Syst.I - Regular Papers, accepted for publication.

10. A. Eghbali, H. Johansson, and P. Lowenborg, and H. G. Gockler, “Dy-namic frequency-band reallocation and allocation: From satellite-based com-munication systems to cognitive radios,” J. Signal Processing Syst., DOI10.1007/s11265-009-0348-1, Feb. 2009.

These papers are covered in Chapters 4–8. The following papers were also publishedduring this period but they are not included in this thesis:

1. A. Eghbali, O. Gustafsson, H. Johansson, and P. Lowenborg, “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, andA. Eghbali, “A 403-MHz fully differentialclass-E amplifier in 0.35 um CMOS for ISM band applications,” in Proc.IEEE EWDTS 2008, Lviv, Ukraine, Oct. 9-13, 2008.

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1. INTRODUCTION

3. A. Eghbali, H. Johansson, T. Saramaki, and P. Lowenborg, “On the designof 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-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 andefficient SRC structures are considered. In addition, FBs and TMUXs are studied.The perfect reconstruction is treated and its approximation by redundant TMUXsis considered. Finally, the filter design problem for redundant TMUXs is outlined.

Chapter 4 is based on [43, 45] and it discusses approaches for realizing FFBRnetworks. The chapter introduces two alternatives for processing real signals usingreal input/output and complex input/output FFBR networks. It is shown that thereal case has less overall number of processing units. In addition, the real systemeliminates the need for two Hilbert transformers and is suitable for systems witha large number of users. Finally, issues related to performance and the trend inarithmetic complexity with respect to (i) the prototype filter order, (ii) the numberof FB channels, (iii) the order of the Hilbert transformer, and (iv) the efficiency inFBR 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 fixedinteger SRC, Farrow-based variable rational SRC, and variable frequency shifters.The building blocks, their operation, and the filter design problem along with somedesign examples are considered. It is shown that, by designing the filters only onceoffline, all possible combinations of bandwidths and center frequencies are obtainedonline. This requires simple adjustments of the variable delay parameter of theFarrow-based filters and the variable parameters of the frequency shifters. Usingthe rational SRC equivalent of the Farrow-based filters, the TMUX is described interms of conventional multirate building blocks. The performance and functionalitytests of the FFBR network, discussed in Chapter 4, are also illustrated.

Chapter 6 considers a class of multimode TMUXs proposed by [64–66]. TheTMUXs use the Farrow structure to realize polyphase components of general in-terpolation/decimation filters. This allows integer SRC with different ratios tobe realized using fixed filters and a few variable multipliers. In conjunction withvariable frequency shifters, an integer SRC multimode TMUX is presented and itsfilter design problem, using the minimax and LS methods, is treated. A modelof general rational SRC is then constructed where the same fixed subfilters are

6

1. INTRODUCTION

used to perform rational SRC. Efficient realizations of this rational SRC schemeare presented. Similarly, variable frequency shifters are utilized to derive a generalrational SRC multimode TMUX. By processing quadrature amplitude modulationsignals, the performance of the TMUX is also discussed.

Chapter 7 is based on [67, 68] and it introduces reconfigurable nonuniformTMUXs based on fixed uniform modulated FBs. The proposed TMUXs use cosinemodulated FBs and modified discrete Fourier transform FBs. Users can occupydifferent bandwidths and center frequencies in a time-varying manner. The filterdesign, realization, and the reconstruction error are discussed. Further, the systemparameters and the implementation cost are treated. The chapter also comparesthe proposed TMUXs to those in Chapters 5 and 6.

Chapter 8 is based on [44] and it deals with two approaches for frequencyallocation and reallocation used in the baseband processing of cognitive radios.These approaches can be used depending on the availability of a composite signalcomprising several user signals or the individual user signals. With compositesignals, the FFBR network in Chapter 4 is used. To process individual users, theTMUXs in Chapters 5–7 can be used. Discussions on reconfigurability with respectto cognitive radios are also provided.

Chapter 9 gives some concluding remarks and open issues for future research.

7

1. INTRODUCTION

8

2

Basics of Digital Filters

This chapter reviews some basics of digital filters. First, finite-length impulseresponse (FIR) and infinite-length impulse response (IIR) filters are discussed.Section 2.3 treats the polyphase decomposition. Some classes of filters, viz., powercomplementary, Nyquist, linear-phase FIR, and Hilbert transformers are discussedin Section 2.4. Finally, Section 2.5 outlines the minimax, least-squares (LS), andthe constrained LS (CLS) filter design problems.

2.1 FIR Filters

A causal1 FIR filter of order N has an impulse response with N + 1 coefficientsh(0), h(1), . . . , h(N). The transfer function of an Nth-order FIR filter is [69]

H(z) =

N∑

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∑

k=0

h(k)x(n− k) ⇔ Y (z) = H(z)X(z). (2.2)

1A filter is causal if h(n) = 0, n < 0. A non-causal FIR filter can be made causal by insertionof a proper delay.

9

2. BASICS OF DIGITAL FILTERS

x(n) T TT

h0 h1 h2 hN-1 hN

T

y(n)

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

There are different ways to realize (2.2) and two are shown in Figs. 2.1 and 2.2where the impulse response values are h0, h1, . . . , hN . The FIR filters allow oneto use non-recursive algorithms for their realization thereby eliminating problemswith 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 Ndelay elements.

2.2 IIR Filters

If the length of h(n) is infinite, the filter is called IIR where

H(z) =

∑Nn=0 a(n)z

−n

1−∑Nn=1 b(n)z

−n. (2.3)

With b(n) = 0, n = 0, 1, . . . , N, an IIR filter reduces to an FIR filter. Realizationof IIR filters requires recursive algorithms which may give rise to problems ofinstability. As the poles of IIR filters are not in the origin (as opposed to FIRfilters), their design has extra degrees of freedom. However, care must be taken toplace 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) =+∞∑

n=−∞

h(n)z−n. (2.4)

10


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

|h(n)|≤M1Kn1 , n≥0, (2.5)

|h(n)|≤M2Kn2 , n≤0, (2.6)

then (2.4) converges forK1 < |z| < K2. (2.7)

As z can have a radius r and an angle θ of the form z = rejθ, (2.4) will convergeon every concentric circle with K1 < r < K2. For right-hand (left-hand) sidedsequences, (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 ) =

+∞∑

n=−∞

h(n)e−jnωT . (2.8)

2.3 Polyphase Decomposition

The transfer function in (2.1) can be decomposed as

H(z) =

∞∑

n=−∞

h(nL)z−nL

+z−1∞∑

n=−∞

h(nL+ 1)z−nL (2.9)

. . .

+z−(L−1)∞∑

n=−∞

h(nL+ L− 1)z−nL,

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

H(z) =

L−1∑

i=0

z−iHi(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∑

i=0

z−(L−1−i)Ri(zL), (2.12)

11


where Ri(z) = HL−1−i(z) [12]. The Type I and II polyphase decompositions allowone to efficiently realize the analysis and synthesis filter banks (FBs) of generalFBs, respectively [12].

With polyphase realization, the filters operate at the lowest possible samplingfrequency. Although polyphase decomposition reduces the implementation cost,the total number of multiplications and additions does not change. This costreduction is achieved by operating the adders and multipliers at a lower samplingfrequency. To realize an Nth-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∑

l=0

Yl(zL)z−l =

L−1∑

i=0

Xi(zL)z−i

L−1∑

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), andH(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(z

L) . . . z−LH1(zL)

H1(zL) H0(z

L) . . . z−LH2(zL)

......

. . ....

HL−1(zL) HL−2(z

L) . . . H0(zL)

. (2.15)

2.4 Special Classes of Filters

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

2.4.1 Complementary Filters

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

K∑

k=0

|Hk(ejωT )|2 = c, c > 0. (2.16)

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

K∑

k=0

|Hk(ejωT )|p = c, p ∈ N, c > 0. (2.17)

12


In special cases, the magnitude and power complementary filters satisfy (2.17) forp = 1 and p = 2, respectively. Higher order complementary filters, e.g., p > 2, cangenerate ordinary magnitude and power complementary filters while maintainingsuperior cut-off characteristics [74]. Strictly (or delay) complementary filters arethose who add up to a delay as [12, 69]

K∑

k=0

Hk(ejωT ) = cz−D0 , c 6=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 frequencyresponse of a linear-phase FIR filter can be expressed as

H(ejωT ) = e−j(NωT2 +c)HR(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)

2For Type III linear-phase FIR filters, the number of adders is also reduced.

13


Table 2.1: Typical locations of zeros for linear-phase FIR filters.

Type LocationI ArbitraryII ωT = πIII ωT = 0, πIV ωT = 0

These four types have different expressions for HR(ωT ) as [75]

HR(ωT ) =

h(N2 ) + 2∑

N2n=1 h(

N2 − n) cos(nωT ) Type I

2∑

N−12

n=0 h(N−12 − n) cos(n+1

2 ωT ) Type II

2∑

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

2∑

N−12

n=0 h(N−12 − n) sin(n+1

2 ωT ) Type IV.

(2.25)

Further, [75]

Φ(ωT ) =

−NωT2 Type I,II

−NωT2 + 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 τp(ωT ) is almost constant in thepassband. This makes the delay of all signal components approximately equal. Fornonperiodic signals, τg(ωT ) may be used. For a constant phase delay, β in (2.23) isforced 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 complexconjugate pairs. If the zeros appear off the unit circle, they are mirrored withrespect to the unit circle. This thesis focuses on Types I or II as we deal withlowpass filters. Table 2.1 shows typical locations of the zeros for different linear-phase FIR filters.

3The shape of a periodic bandpass or highpass signal is preserved if β in (2.23) is a multipleof 2π and α is constant [75].

14


2.4.3 Nyquist (Mth-band) Filters

A lowpass non-causal filter h(n) of order N is said to be Mth-band if any of itspolyphase 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) =

1M n = 0

0 n = ±M,±2M, . . .(2.31)

meaning that every Mth sample, except the center tap, is zero. This reducesthe number of multipliers and adders required to realize the filter. If h(n) is anMth-band filter, its delayed version is also an Mth-band filter [12]. In the causalcase, H(z) is an Mth-band filter if the kth polyphase component has the formHk(z) =

1M z−nk . In the time domain, this becomes

h(nM + k) =

1M n = nk

0 otherwise.(2.32)

For an Mth-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 thetransition band contains ωT = π

M . In the context of FBs, ρ can assume any valuesuch 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 Mth-band filter, the sum of M shifted copies of H(z) results in aconstant. In other words,

M∑

k=0

H(zW kM ) = c, WM = e−j 2π

M , c > 0. (2.36)

15


An alternative to (2.36) is obtained from (2.18) with D0 = 0 [12]. Generally, theimpulse 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 designsreal causal linear-phase FIR Nyquist filters. Nyquist filters find applications in, e.g.,transmultiplexers [79], spectrum sensing for cognitive radios [61, 80, 81], samplingrate 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 andH(ejωT ) = H∗(e−jωT ). This results in some redundancy between the portions ofthe spectrum at negative and positive values of ωT [84]. Thus, the information of areal-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 frequenciesand only process the positive part [85]. To preserve the positive frequencies, thereal 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 correspondingIIR non-causal impulse response is

h(n) =

1 n = 02jnπ 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) =

2nπ 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 seethat 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 Hilberttransformer 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(−z2), (2.42)

16


where

G(z) =z−N − E(z2)

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 filterto 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 = 0sin(n)

n n 6= 0(2.45)

and is not realizable. To get a realizable filter, one approximates this ideal transferfunction in the passband(s) and stopband(s) by allowing transition band(s) as wellas 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 δs are, respectively, the passband and stopband ripples with Ωc andΩs being the passband and stopband regions. One can generally have filters withmultiple passband and stopband regions. Then, the specifications must be satisfiedfor 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) issatisfied for desired values of Ωc, Ωs, δc, and δs. A commonly used formula toestimate 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

17


−200

−150

−100

−50

0.2

0.4

0.6

0.8

0

2

4

6

8

δs=δ

c [dB](ω

sT−ω

cT)/π

10

0*

(NB

−N

K)/

NK

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 theorder, 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 practicalalso. As an example, with δc = δs = 0.5, ωsT = 0.3π, and ωcT = 0.2π, we getNB = −5.3059 and NK = −9.5608. Throughout this thesis, the ripples are chosenso that they (i) are practical, and (ii) ensure positive orders. This is achieved if

• δsδc < 0.1 in (2.47).

• δsδc < 10−2620 in (2.48).

Figure 2.3 shows a relative comparison of these positive orders for some typicalvalues 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 inthis thesis, this difference is about 5%. Consequently, the conclusions of the thesisare valid even if (2.48) is used. However, (2.48) slightly changes the fomulations ofcomplexity, etc. Generally and for very small or large ωcT , these formulae sufferfrom estimation inaccuracies. However, there are other methods to estimate the

18


filter order as in, e.g., [90]. As [90] complicates the derivations of the arithmeticcomplexity provided in this thesis, we do not use it.

The filter design problem finds h(n) so as to satisfy a specific criterion. Thiscriterion could be the energy, maximum ripple, or combinations of them leading toLS, 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

∫

ωT∈Ωc

|H(ejωT )− 1|2d(ωT ) +∫

ωT∈Ωs

|H(ejωT )|2W (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 designproblem as


∫

ωT∈Ωc

|H(ejωT )− 1|2d(ωT ) ≤ δ, ωT ∈ Ωc

|H(ejωT )|≤δdes, ωT ∈ Ωs.

Here, δdes is the desired maximum stopband ripple. Further, W (ωT ) is a weightingfunction. A large W (ωT ) results in small (large) stopband approximation errorsfor minimax (LS) designs. This thesis assumes frequency independent weightingfunctions and, thus, W (ωT ) is constant in the frequency range of interest.

19


20

3

Basics of Multirate SignalProcessing

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 andthe Farrow structure. Then, filter banks (FBs) are defined in Section 3.3 wheretheir 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 performedby interpolation (decimation) which increases (decreases) the sampling frequencyof digital signals [12, 69]. An alternative, to perform SRC on digital signals, isto first construct the corresponding analog signal and, then, resample it with thedesired sampling frequency. However, it is more efficient to perform SRC directlyin the digital domain. By changing the sampling frequency, the implementationcost for a given task can be reduced as the adders and multipliers can operateat a lower rate. Interpolation and decimation are two-stage processes comprisinglowpass filters as well as downsamplers and upsamplers. The block diagrams of

21

3. BASICS OF MULTIRATE SIGNAL PROCESSING

Lx(n) y(m)

(a) (b)

x(m) M y(n)

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

x(m) y(n)H(z) M

Figure 3.2: Decimation by M .

Lx(n) y(m)H(z)

Figure 3.3: Interpolation by L.

upsamplers and downsamplers are shown in Fig. 3.1. A downsampler retains everyMth 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∑

k=0

X(z1MW k

M ), (3.2)

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

converting z to z1M ) and shifted (through the terms W k

M ) versions of X(z). Note

that X(z1M ) 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(nL ) 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. Theupsampler 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 filterH(z) limits the bandwidth of x(n) as the original signal can only be preserved ifit is bandlimited to π

M . In Fig. 3.2,

y(n) =+∞∑

k=−∞

x(k)h(nM − k). (3.5)

1This is not necessary to avoid aliasing. For example, if X(ejωT ) is nonzero only atωT∈[ω1T, ω1T + 2π

M] for some ω1T , there is no aliasing [12].

22


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

y(n) =+∞∑

k=−∞

x(k)h(n− kL). (3.6)

For SRC2 by a rational ratio ML , 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 andanti-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 Mfollowed by decimation by L. Generally, G(z) is a lowpass filter with a stopbandedge 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. Formutually 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 multiplier1k where k > 1

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

y(n) =

+∞∑

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 multiratestructure. 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 nobleidentities 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.

2If L > M (L < M), we have interpolation (decimation) by a rational ratio LM

> 1 (ML

> 1).This thesis frequently refers to SRC by a rational ratio Rp > 1.

23


<=> x(m) y(n)M H(z)

Mx(n) y(m)H(z)

x(m) H(zM) y(n)M

x(n) y(m) <=>H(zM)M

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

Mfs

x(m)

fs

y(n)H0(z)

H1(z)

HM-1(z)

z-1

z-1

y(n)

fsMfs

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-1Mfs

x(n)

fs

HM-1(z)

H1(z)

H0(z)

z-1

y(m)

Mfsfs

x(n) H(z)M

M

M

M

H0(z)

x(n)

fs

H1(z)

HM-1(z)

Mfs

y(m)

Figure 3.6: Interpolation with polyphase decomposition and noble identities.

24


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

Nk k Typeeven even Ieven odd IIIodd even IIodd odd IV

x(n)

SL(z) S

2(z) S

1(z)

m

S0(z)

y(n)mm

Figure 3.7: Farrow structure with fixed subfilters Sk(z) and variable fractionaldelay µ.

3.2 Sampling Rate Conversion: Farrow Structure

In conventional SRC and if the SRC ratio changes, new filters are needed. Thisreduces the flexibility in covering different SRC ratios. By utilizing the Farrowstructure [93], shown in Fig. 3.7, this can be solved in an elegant way. The Farrowstructure 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 haveany 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 referredto as the modified Farrow structure [94]. Throughout this thesis, we simply referto it as the Farrow structure. The Farrow structure is efficient for interpolationwhereas, 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]. Thesubfilters can also have even or odd orders Nk. With odd Nk, all Sk(z) are generalfilters whereas for even Nk, the filter S0(z) reduces to a pure delay. The transferfunction of the Farrow structure is

3With infinite-length impulse response (IIR) filters, care must be taken to avoid transients asµ may change for every sample.

25


H(z, µ) =L∑

k=0

Sk(z)µk (3.10)

=

L∑

k=0

Nk∑

n=0

sk(n)z−nµk

=

N∑

n=0

L∑

k=0

sk(n)µkz−n =

N∑

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∑

k=0

sk(n)µk. (3.11)

Further, µ is the fractional delay value4 which defines the time difference betweeneach 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 Tinand Tout to 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 inputsamples, the Farrow structure delays a bandlimited signal by a fixed µ. Figure 3.8shows two delayed versions of a bandlimited signal x(n) = sin(nπ12 ) where µ = 0.25and 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 differentµ. This delay depends on whether one performs decimation or interpolation. Forinterpolation, one can obtain new samples between any two consecutive samples ofx(n). With decimation, one can shift the original samples (or delay them in thetime 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 anexample, 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 inputsample.

4In the modified Farrow structure, 0 < µ < 1.5In the implementation, a group of input samples are present in the delay elements of Sk(z).

For every µ, its corresponding x(n) must be aligned with the point of symmetry of Sk(z).

26


5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

n

Am

pli

tude

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

Am

pli

tude

sin(n1ωT

1)

sin(n2ωT

1/1.75)

sin(n3ωT

1*1.75)

Figure 3.9: Application of the Farrow structure to perform SRC on x(n) = sin(nπ12 ).

27


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 fractional delay µ over the frequency range6 of interest [94, 95, 97–104].The desired causal magnitude and unwrapped phase responses are

Hdes(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 SRCusing only one set of Sk(z) and by simple adjustments of µ. In the non-causal caseand with L subfilters, the Taylor series expansion of (3.13) is [105]

e−jµωT≈L∑

k=0

(−jµωT )kk!

=

L∑

k=0

(−jωT )kk!

µk. (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 methodsto 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 theoriginal 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 outputof a general M -channel FB is

Y (z) =1

P

P−1∑

n=0

X(zWnP )

M−1∑

m=0

Hm(zWnP )Fm(z) (3.17)

where WP is defined as in (2.36). Ideally, the output signal is scaled (by α) anddelayed (by β) version of the input signal, i.e., y(n) = αx(n − β). Such a systemis 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∑

m=0

Hm(z)Fm(z), (3.18)

6The input of the Farrow structure must be bandlimited to this frequency range.

28


x1(m)

xM-1(m)

x0(m)

y(n)x(n)

P P

P

P

P

P

Synthesis FBAnalysis FB

F0(z)

F1(z)

FM-1

(z)

H0(z)

H1(z)

HM-1

(z)

Figure 3.10: General M -channel FB.

whereas the aliasing transfer functions are

Vl(z) =1

P

M−1∑

m=0

Hm(zW lP )Fm(z), l = 1, 2, . . . , P − 1. (3.19)

These FBs are generally linear periodic time-varying (LPTV) systems with a periodM . If there is no aliasing, we have a linear time-invariant (LTI) system [12]. In aPR 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 setXm(z) equals that of X(z). The choice P < M leads to oversampled FBs [12]. IfV0(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 filtersHm(z) and the SFB filters Fm(z). Here,

Y (z) = T0(z)X(z) +M−1∑

l=1

Tl(z)X(zW lM ). (3.22)

The term

T0(z) =1

M

M−1∑

m=0

Fm(z)Hm(z) (3.23)

is the distortion transfer function and

Tl(z) =1

M

M−1∑

m=0

Fm(z)Hm(zW lM ), l = 1, 2, . . . ,M − 1 (3.24)

are the aliasing transfer functions.

29


Analysis FB Synthesis FB

F0(z)M

F1(z)M

FM-1

(z)M

å

H0(z)

H1(z)

HM-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 Nth-order linear-phase FIR prototype filter G(z) =

∑Nn=0 g(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 overlappingfactor [109]) is an integer. For complex modulated FBs,

hm(n) = g(n)W−mnM , (3.27)

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

In the maximally decimated case, we can use modified discrete Fourier transformFBs (MDFT FBs) [110–115]. AnM -channel MDFT FB can equivalently be realizedas (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 andSFBs.

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

hm(n) = amgm(n)e−jπαmMm

(n−Lm−12 ) + a∗mg

∗m(n)e

jπαmMm

(n−Lm−12 ), (3.29)

fm(n) = bmgm(n)e−jπαmMm

(n−Lm−12 ) + a∗mg

∗m(n)e

jπαmMm

(n−Lm−12 ). (3.30)

Here, αm = (Km + 0.5) and gm(n) is the (possibly complex) prototype filter oflength Lm with Mm being the decimation factor in each branch. Each branch has

30


s0(n)

s1(n)

sM-1

(n)

P

P

P

F0(z)

F1(z)

FM-1

(z)

y(n) y(n)^

s0(n)^

s1(n)^

sM-1

(n)^

H0(z) P

PH1(z)

PHM-1

(z)

x0(n)

x1(n)

xM-1

(n)

D(z)

e(n)

Channel

Figure 3.12: General M -channel TMUX.

a center frequency as ±παm

Mmwith Km being an integer where am and bm define the

modulation phase. As opposed to uniform FBs, nonuniform FBs achieve a moregeneral 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 onthese 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) witha 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 = π

2M [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 thepower complementary property.

3.4 General M-Channel TMUXs

A TMUX converts the time multiplexed components of a signal into a frequencymultiplexed version and back [121]. It allows several users to transmit and receiveover 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 generatedby different users or parts of a signal generated by one user. Assume also that wewant to transmit these signals through a channel. As in Fig. 3.12, we can passsk(n) through the transmitter (pulse shaping) filters Fk(z). Then, (3.6) gives

31


pwT

pwT

pwT

F0(z) F1(z) FM-1(z)

(a)

(b)

(c)

FM-1(z)

FM-1(z)

F1(z)

F1(z)

F0(z)

F0(z)

Figure 3.13: M -channel TMUX filters. (a) Overlapping. (b) Marginally overlap-ping. (c) Non-overlapping.

xk(n) =

∞∑

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 complexLTI filter D(z) =

∑LD

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

side, the receiver filters Hk(z) separate the signals and only a downsampling by Pis needed to get the original symbol streams. Ignoring the channel,

Si(z) =

M−1∑

k=0

Sk(z)Tki(zP ), i = 0, 1, . . . ,M − 1 (3.32)

where

Tki(zP ) =

1

P

P−1∑

l=0

Fk(zWlP )Hi(zW

lP ), (3.33)

and WP is defined as in (2.36). Typical characteristics of Fk(z) and Hk(z) areshown in Fig. 3.13. Similar to FBs, TMUXs can be redundant (P > M) or criticallysampled (P =M). To avoid inter-symbol interference (ISI), a level of redundancymay be needed such that P −M≥LD [129]. The output of the TMUX in (3.32) is

Si(z) = Tii(z)Si(z) +P−1∑

k=0,k 6=i

Tki(z)Sk(z) (3.34)

where Tii(z) and Tki(z) represent the ISI and the inter-carrier interference (ICI),respectively [79]. Note that the ISI (ICI) is sometimes also referred to as interband

32


Analysis FBSynthesis FB

F0(z)M

F1(z)M

FM-1

(z)M

å

H0(z)

H1(z)

HM-1

(z)

M

M

M

y(n) y(n)^

X0(z)

X1(z)

XM-1

(z)

^

^

^

X0(z)

X1(z)

XM-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 eachbranch 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 impulseresponse [12]. In this case, designing Fk(z) and Hk(z) so that the decimated (byM) version of Fk(z)Hm(z) becomes a pure delay if k = m and zero otherwise, theTMUX becomes PR. In terms of (3.34), this means

Tii(z) =1

P

P−1∑

l=0

Fi(z1P W l

P )Hi(z1P W l

P ) = αz−β , (3.36)

Tki(z) =1

P

P−1∑

l=0

Fk(z1P W l

P )Hi(z1P W l

P ) = 0. (3.37)

In a PR system, sk(n) = αsk(n − β). The PR properties are independent of thelength and causality of filters, etc., and can be satisfied for both critically sampledand redundant TMUXs. However, for the critically sampled case, there may notalways exist FIR or stable IIR solutions. Therefore, some redundancy makes thesolutions 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 fromthe FB of Fig. 3.11 where

Y (z) =

M−1∑

m=0

Xm(zM )Fm(z). (3.38)

33


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

Xd(z) =

M−1∑

k=0

M−1∑

m=0

Xm(z)Fm(z1/MW kM )Hd(z

1/MW kM ), 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, wehave

ISI =M−1∑

k=0

Fd(z1/MW k

M )Hd(z1/MW k

M ), (3.40)

and

ICI =M−1∑

m=0,m 6=d

M−1∑

k=0

Fm(z1/MW kM )Hd(z

1/MW kM ). (3.41)

The duality of FBs and TMUXs applies to both critically sampled and redundantsystems. It has been shown that if a FB is free from aliasing, the correspondingTMUX 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 thecascade 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 polyphasecomponent of Fk(z)Hm(z). To approximate PR, these ideal conditions must beapproximated as close as desired. Thus, the minimax optimization problem for anNPR 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], k 6=m

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

To simplify (3.42), this thesis uses redundant TMUXs with non-overlappingfilters as shown in Fig. 3.15. The TMUXs are also nonuniform. Consequently, theISI in (3.34) would result from the filters in one branch of the TMUX. In generalnonuniform TMUXs, the ICI in (3.34) becomes time-varying7. However, due to theredundancy, 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 closeas desired. Then, the SFB and AFB filters should be designed such that

7In 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 isdependent on the input at a finite set of frequencies [30].

34


2pwT

F0(z) F1(z) F2(z) FM-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 allpasstransfer function.

Thus, the simplified minimax design problem is


|[Fk(ejωT )Hk(e

jωT )]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 weightingfunctions with Ωs being the stopband region as in (2.46). In the least-squares (LS)sense, (3.43) becomes

min (3.44)

∫

ωT∈[0,π]

|[Fk(ejωT )Hk(e

jωT )]zeroth − 1|2d(ωT )

+

∫

ωT∈Ωs

|Fk(ejωT )|2

W1(ωT )d(ωT )

+

∫

ωT∈Ωs

|Hk(ejωT )|2

W2(ωT )d(ωT ).

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 designreal 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 designmethod. This thesis does not consider the effects of the channel when designingthe TMUXs but some methods can be found in, e.g., [21, 108, 121, 128].

35


36

4

Flexible Frequency-BandReallocation For Real Signals

This chapter discusses a new approach for implementing flexible frequency-bandreallocation (FFBR) networks for bentpipe [38], or transponder [36], satellite pay-loads. We consider two alternatives to process real signals using real and com-plex input/output FFBR networks. After some general and historical introduc-tion in Section 4.1, Section 4.2 briefly reviews the FFBR network. Alternative I,shown in Fig. 4.7, and its arithmetic complexity are treated in Section 4.3. Sec-tion 4.4 introduces Alternative II, shown in Fig. 4.9, and it covers the formulation offrequency-band reallocation (FBR) for real signals, functionality illustration, andthe arithmetic complexity. The comparison of the two alternatives is discussed inSection 4.5. Finally, Section 4.6 gives some concluding remarks.

4.1 Introduction

As discussed in Section 1.1, the European space agency has proposed three majornetwork structures for broadband satellite-based communications in which satellitescommunicate with users through multiple spot beams. This necessitates to reusethe limited available frequency spectrum by satellite on-board signal processing [32–57]. The digital part of the satellite on-board signal processor, or the payload, is amulti-input multi-output (MIMO) system with input/output signals composed ofdifferent users with different bandwidths. The on-board signal processor reallocates

37

4. FLEXIBLE FREQUENCY-BAND REALLOCATION FOR REAL SIGNALS

all users to different output signals and positions in the frequency spectrum. Withbandwidth-on-demand, the bandwidths of different users may vary with time. Thiscan be handled by dividing the input beam into a number of granularity bands(GRBs). At any time, any user can occupy any rational number of GRBs. TheFFBR network operates on multiplexing bandwidths which are integer multiples ofthe GRBs. Each multiplexing bandwidth consists of (i) a user bandwidth which isa rational multiple of the GRB, and (ii) some guardband (GB). There are severalrequirements on FFBR networks as [42]

• Flexibility to handle all FBR scenarios on users with different bandwidths.

• Low complexity to reduce the implementation cost. The amount of improve-ments in system capacity and implementation cost is foreseen to be aboutone or two orders of magnitude [37].

• Near perfect frequency-band reallocation (PFBR) to satisfy any communica-tion performance metric, e.g., error vector magnitude (EVM) [139, 140].

• Simplicity resulting in simple system analysis and design.

4.1.1 Contribution and Relation to Previous Work

There are four types of payloads, viz., bentpipe, full processing, partial process-ing, and hybrid [38]. This chapter focuses on bentpipe payloads whose principleis shown in Fig. 1.2. Here, users with different bandwidths are reallocated to dif-ferent positions in the frequency spectrum. In dynamic communications, thesebandwidths and positions may change in a time-varying manner.

As shown in Fig. 4.1, the FBR based on filter banks (FBs) uses decimation andinterpolation to generate frequency shifts of users [40–42, 48–56]. These approachescan be classified as maximally decimated FBs [40, 55], tree-structured FBs [40,48–54], overlap-save discrete Fourier transform (DFT) and inverse DFT (IDFT)techniques [56], and oversampled complex modulated FBs [41, 42].

In [42], a new class of FFBR networks based on finite-length impulse response(FIR) variable oversampled complex modulated FBs was introduced. The systemin [42] processes complex signals. Thus, the analytic representation of the realuplink satellite signals must be processed by the FFBR network. The frequencymultiplexed results should then be converted to real signals for retransmission. Thisrequires one complex FFBR network and two Hilbert transformers. This chapterrefers to this solution, shown in Fig. 4.7, as Alternative I.

This chapter introduces another alternative to process real signals through a realinput/output FFBR network which has less arithmetic complexity. This solutionis referred to as Alternative II and is shown in Fig. 4.9.

4.1.2 Choice of the FFBR Network

All the FFBR networks in [40–42, 48–56] process complex signals. This chaptershows that if both of the structures in Figs. 4.7 and 4.9 are applicable, the approach

38


0 2p/M wT1

X0 X2

(b)

4p/M 6p/M 2p

XM1

X(ejwT1)

X1

0 2p wT2

(e)

4p 2Mp

X1X1X1 X1

V2(ejwT2)

0 2p/M wT1

(e)

4p/M 2p

X1X1X1 X1

V3(ejwT1)

0 2p/M wT1

(c)

4p/M 2p

H(ejwT1)

0 6p/M wT1

(f)

4p/M 2p

G(ejwT1)

0 2p/M wT1

V1(ejwT1)(d)

4p/M 2p

X1

0 6p/M wT1

Y(ejwT1)(g)

4p/M 2p

X1

H(z) G(z)M Mv1 v2 v3(a)

T1 T1T2

x

T1

y

T1

Decimation Interpolation

Figure 4.1: FBR using decimation and interpolation. Here, only one FB channelis shown but channel combiners can produce outputs from several FB channels.

39


M

M

M

x(n)

Fixed Analysis FB

y0(n)M

M

M

Adjustable Synthesis FB

y1(n)

yq-1(n)C

han

nel C

om

bin

er

y(n)

G0(z)

G1(z)

GM-1

(z)

H0(z)

H1(z)

HN-1

(z)

Figure 4.2: N -channel FFBR network utilizing fixed (adjustable) AFB (SFB).

in Fig. 4.9 is superior to that of Fig. 4.7. By using other FFBR networks, onlythe exact number of operations changes but the superiority of Fig. 4.9 will stillbe preserved. This chapter focuses on the FFBR network in [42] because of thereasons outlined in [43].

4.1.3 MIMO FFBR Network Configuration

The FFBR network, considered here, is an m-input n-output system where m≤n.However, it suffices to only discuss the single-input single-output (SISO) case asthe MIMO case is a duplication of fixed SISO structures (refer to Figs. 17 and 23of [42]) along with some modifications1. Further discussions on the MIMO systemfor both m < n and m = n can be found in [42, 141].

4.2 FFBR Network Based on Variable Oversam-pled Complex Modulated FBs

As shown in Fig. 4.2, the FFBR network uses fixed analysis FB (AFB) filtersHk(z), k = 0, 1, . . . , N − 1, to split x(n) into N subbands. Then, downsam-pling/upsampling by M and the adjustable synthesis FB (SFB) filters Gk(z) per-form FBR. As adjustable SFB filters result in a high implementation cost, theFFBR network is realized using fixed SFB filters and an adjustable channel switch.This requires appropriate choices of system parameters and filter characteristicsbut it reduces the implementation cost.

1For the MIMO case, the channel switch operates between several SISO structures. Further-more, if m < n, some branches at the output of the DFT block in Fig. 4.3 are set to zero.

40


IDFT

a0

a1

aN1

aN

aN

aN

P1(zLWN )

PN1(zLWN )

P0(zLWN )x(n) M

M

M

z1

z1

b0

b1

bN1

Chan

nel S

witch

g0

g1

gN1

DFT

aN1

aN2

a0

aN

aN

aN

PN2(zLWN )

PN1(zLWN )

P0(zLWN ) y(n)

z1

z1

M

M

M

FFBR Network

Figure 4.3: Efficient DFT- and IDFT-based implementation of Fig. 4.2.

4.2.1 Efficient Realization of the FFBR Network

Figure 4.3 shows the architecture of the N -channel FFBR network with complexmultipliers, DFT, IDFT, polyphase components, and input/output commutators.The system assumes that the complex input signal is divided into Q GRBs where

Q =N

A, A > 1, A ∈ N (4.1)

and the GRBs are separated by a GB of 2∆ = 2πǫQ with 0≤ǫ≤1. The choice of

ǫ defines the transition band of the filters. A large transition band would reducethe amount of the frequency spectrum covered by the GRBs and, hence, there is atrade-off. Any user can occupy any rational number of GRBs. To suppress aliasingand to shift the GRBs by all values of 2πq

Q , q = 0, 1, . . . , Q − 1, we should choose

M to be a multiple of Q as2

M = BQ, B ∈ N. (4.2)

To attenuate aliasing by the stopband of the filters, the passbands and transitionbands of the shifted terms should not overlap. This is achieved if

M≤ N

1 + N∆π

< N. (4.3)

Assuming the length-S linear-phase FIR prototype filter

P (z) =

S−1∑

n=0

z−np(n) =

N−1∑

i=0

z−iPi(zN ), (4.4)

the AFB filters are

Hk(z) = βkP (zWk+αN )

= βk

N−1∑

i=0

[z−iαiPi(zNWαN

N )]W−kiN . (4.5)

2Note that N = M AB

= ML where L is the number of FB channels per GRB.

41


0wT

p/N pp p/N

|PR(wT)|

2D

Figure 4.4: Characteristics of P (z).

Here, k = 0, 1, . . . , N − 1, and

WN = e−j 2πN (4.6)

αi = W−αiN (4.7)

βk = W(k+α)(S−1)

2

N . (4.8)

Further,

P (ejωT ) = e−jωT (S−1)

2 PR(ωT ), (4.9)

where PR(ωT ) is the real zero-phase frequency response which should approximatethe magnitude response shown in Fig. 4.4. In addition, α is a real-valued constant toplace the filters at the desired center frequencies. The multipliers βk compensate forthe phase rotations because of the substitution of P (z) with P (zW k+α

N ). Therefore,all the AFB filters become linear-phase FIR filters with the same delay as P (z).The multipliers γk are

γk = βkW−kN , (4.10)

whereas the SFB filters are

Gk(z) = µkrHckr(z), (4.11)

with

ckr = k +Asr (4.12)

µkr = WmrN(S−1)

2M

N (4.13)

mr =

Bsr sr ≥ 0

M +Bsr sr < 0.(4.14)

The parameter sr is the number of GRBs by which the subband r is shifted. It ispositive (negative) if that subband is shifted to the right (left). This informationis required to program the channel switch. Specifically, programming the channelswitch requires knowledge of L, sr, and the number of GRBs that each user oc-cupies. According to (4.13), the constants µkr can be made equal to unity by aproper choice of S and at the cost of some additional delay. This chapter assumesµkr = 1.

42


xa(t)

Anti-Image ADC

Anti-Image ADC

cos(wt)

sin(wt)

I

Q

Figure 4.5: Complex sampling.

xa(t)Digital

ProcessorADC

I

Q

Figure 4.6: Real Sampling.

fsfs/2

fs

xa(t) ya(t)Complex FFBR

22 HilbertHilbertADC DAC

y(n)x(n)

Figure 4.7: Alternative I: complex FFBR network with Hilbert transformer.

4.3 Alternative I

This section discusses Alternative I which consists of a complex FFBR networkand two Hilbert transformers.

4.3.1 Complex Versus Real Sampling

To process the real uplink satellite signal xa(t) by the FFBR network of Fig. 4.3,it must be converted into its analytic representation. This can be done either bycomplex sampling using two 90 degree out-of-phase analog to digital converters(ADCs) or by real sampling which consists of an ADC followed by a digital pro-cessor. The digital processor is composed of a Hilbert transformer followed by adownsampler [57].

In Fig. 4.5, the ADCs operate at fs2 whereas the ADC of Fig. 4.6 operates at

fs. Besides the requirements to attenuate the images arising from the mixers ofFig. 4.5, the high sensitivity to match the two ADCs using analog componentsmakes the real sampling approach more attractive [57, 73]. Hence, the structure inFig. 4.7 can be used to process real signals and its arithmetic complexity will bediscussed in the sequel.

4.3.2 Arithmetic Complexity: Hilbert Transformer

The analytic representation of a real signal x(n) has a zero-valued spectrum for allnegative frequencies [69]. To generate the analytic representation, x(n) is appliedto a complex half-band filter HHB,c(z). As in (2.42), this filter is derived by shiftingthe frequency response of a real lowpass half-band filter HHB,r(z) by π

2 radians3

as [69, 84]HHB,c(z) = jHHB,r(−jz). (4.15)

3A scaling by two may also be required depending on whether HHB,c(z) is used for interpo-lation or decimation (refer to Fig. 4.7). However, this does not affect the filter order.

43


From (2.35) and (2.47), the order of a lowpass real half-band linear-phase FIR4

filter can be approximated as

NH≈−2

3log10(10δs

2)2π

ωsT − ωcT(4.16)

where δs is the stopband ripple with ωsT and ωcT being the stopband and passbandedges, respectively. A real half-band linear-phase FIR filter has an even order ofthe form 4m + 2 where m is an integer [12]. Thus, it is of Type I. To process aspecific GRB by the FFBR network, that GRB should be covered by the passbandof HHB,c(z) which, according to Fig. 4.8, requires

ωsT =π

2+

2∆ + k 2πQ

2,

ωcT =π

2−

2∆ + k 2πQ

2. (4.17)

Here, k∈[

0, 1, . . . , ⌊Q2 − ǫ⌋

]

with ⌊x⌋ being the floor of x. In this way, ωsT < π

and ωcT > 0 and (4.16) becomes

NH≈−2

3log10(10δs

2)Q

ǫ+ k. (4.18)

Having chosen δs, Q, and ǫ, the factor k relates NH to spectrum efficiency η(k) as

η(k) =2π − 2(∆ + kπ

Q )

2π= 1− k + ǫ

Q. (4.19)

The efficiency is the ratio between the part of the frequency spectrum used by theFFBR network and the whole frequency spectrum. In other words, it is the ratiobetween the passband of HHB,c(z) and 2π. Ideally, k = 0 but in systems with alarge N , a small k will result in a large NH . The order of HHB,c(z) equals that ofHHB,r(z) as (4.15) does not alter the filter order.

4.3.3 Arithmetic Complexity: DFT with Complex Inputs

The arithmetic complexity for an N -point complex-input DFT is O(N2) and itrequires N2 complex multiplications and N(N − 1) complex additions [69]. Usingspecialized fast Fourier transform algorithms, the arithmetic complexity for theDFT of a radix-2 length sequence can be reduced to O(N log2N). Any sequencelength can be made radix-2 by zero padding [69].

This chapter focuses on the overall number of real operations. We select thealgorithms in [142–145] which require N(log2N − 3) + 4 real multiplications and3N(log2N−1)+4 real additions. Other DFT realization techniques, e.g., [146–149],may change the exact number of real operations but not the main conclusions.

4At typical stopband attenuations of 60−80dB, the arithmetic complexity of an infinite-lengthimpulse response (IIR) half-band filter is around 60− 70% of that of an half-band FIR filter [99].Hence, IIR half-band filters may result in less arithmetic complexity. The main conclusion of thischapter is independent of the choice of IIR and FIR filters.

44


wTin2pa/Q 2p

2D2D+2p/Q

2D+4p/Q

Hilbert Transformer Characteristics

Figure 4.8: Illustration of the GRBs and the characteristics of HHB,c(z).

4.3.4 Arithmetic Complexity: Complex FFBR Network

As shown in Fig. 4.3, the building blocks of the FFBR network, viz., polyphasecomponents, DFT, IDFT, and complex multipliers, are sandwiched between in-put/output commutators [69]. These blocks produce samples inN parallel branchesbut the output commutator retains one sample out of these branches. Conse-quently, to produce one output sample, the number of operations required by thesebuilding blocks must be divided by N . Both the Alternatives I and II use thearchitecture in Fig. 4.3. Hence, we can ignore this division by N for comparison ofthe arithmetic complexity. The computational workload to produce one complexsample by the N -channel complex FFBR network of Fig. 4.3 is

• 2N real filters of length SN for the AFB and SFB filters operating on complex

data requiring 4S real multiplications and 4(S −N) real additions5.

• Complex multipliers αi, βk, γk resulting in 4N complex multiplications6.

• N -point complex-input IDFT and DFT with 2N(log2N − 3) + 8 real multi-plications and 6N(log2N − 1) + 8 real additions.

To convert complex operations to real operations, [144] assumes that a complexmultiplication requires 3 real multiplications and 5 real additions (3/5) whereas[146] and [69] use (3/3) and (4/2) assumptions, respectively. One complex additionrequires 2 real additions. Thus, using the (4/2) assumption, the overall arithmeticcomplexity becomes 2N log2N+2(4+2S+5N) real multiplications and 6N log2N+2(4+2S−N) real additions whereas (3/3) results in 2N log2N+2(4+2S+3N) realmultiplications and 6N log2N +2(4+2S+N) real additions. These operations runat fs

2M where M < N for an oversampled system. The two Hilbert transformers inFig. 4.7, add the arithmetic complexity of two complex half-band filters running atfs2 if implemented using the polyphase decomposition.

5Depending on S and N , the polyphase components may not have the same lengths. Thearithmetic complexity, derived here, is thus the worst-case scenario. However, if S is chosen suchthat µkr = 1, all the polyphase components will have similar lengths.

6According to (4.7), choosing α = 0, 0.5 further reduces the arithmetic complexity. Thisreduction applies to both the alternatives and is therefore not considered here.

45


fsfs

fs

xa(t) ya(t)RealFFBR

ADC DAC

y(n)x(n)

Figure 4.9: Alternative II: real FFBR network without Hilbert transformer.

4.4 Alternative II

To implement the FFBR network for real signals, Fig. 4.9 suggests to sample thereal signal at fs and feed the sampled data into the real FFBR network at thesame sample rate. Basically, the real FFBR network uses the structure of Fig. 4.3but the polyphase filters operate on real data. Specifically, the polyphase filtersPk(z), k = 0, 1, . . . , N − 1, process real signals. Here, x(n) is a real signal butthe output of the DFT, in the SFB, is a complex signal. As the multipliers αi

are complex, we need to only compute the real part of their output. Then, thepolyphase filters, in the SFB, can process real samples which would reduce theimplementation cost. Similar to Section 4.3 and as in Fig. 4.10, the N -channel realFFBR network assumes that the input signal is divided into Q GRBs separatedby a GB of 2∆. Each GRB is divided into a number of uniform FB channelsconstructed by complex modulating a real linear-phase FIR prototype filter P (z)as in (4.5). Like the complex-input case, FBR is performed by dividing the realinput signal into subbands using the AFB filters Hk; shifting the subbands; andrecombining the shifted subbands using the SFB filters Gk. However, only half ofHk and Gk are involved in the FBR since the spectrum of a real signal spreads in[−π, π] whereas the AFB and SFB filters are defined in [0, 2π].

Figure 4.11(a) shows the frequency spectrum of an input signal consisting ofthree users. This pattern has been generated by the nonuniform TMUX in [46, 63]which we shall discuss in Chapter 5. As discussed in Section 4.1.1, FBR needsfixed AFB and SFB filters and a time-varying channel switch. Figures 4.12 and4.13 show two different channel switches. The corresponding multiplexed outputsignals are shown in Figs. 4.11(b) and (c), respectively.

4.4.1 Arithmetic Complexity: Real FFBR Network

Using earlier discussions, the arithmetic complexity of the real FFBR network is

• 2S real multiplications and 2(S − N) real additions due to real-input AFBand SFB filters Pk(z), k = 0, 1, . . . , N − 1.

• Complex multipliers βk and γk requiring 2N complex multiplications.

• N -point complex-input IDFT and DFT resulting in 2N(log2N − 3) + 8 realmultiplications and 6N(log2N − 1) + 8 real additions.

46


wTin

2p/QGranularity Band

2pa/Q 2p/Q+2pa/Q0

(a)

p

0 pwT

H1H0 H3H2 H5H4 H7H6

(c)

(b)

H0G0+H1G13 3

H6G6+H7G7

6 6

0wT

p

X0 X1 X2

-p

0 pwT

G1G0 G3G2 G5G4 G7G6

(d)

(e)

0wT

p

Y0 Y1 Y2

-p

H0G0+H1G1 H2G2+H3G3+H4G4+H5G5 H6G6+H7G7

0 pwT

G3G2 G5G4 G7G6 G1G0

(f)

(g)

wTp

Y2Y1 Y0

-p

H2G2+H3G3+H4G4+H5G5

6 66 6

-p

-p

-p

-p

X0X1X2

Y0Y1Y2

Y1Y2Y0

0

Guardband

Figure 4.10: FBR by real FFBR network with Q = 8 = N2 . (a) GRBs. (b)–(e)

Recombination of channels. (b), (c), (f), and (g) Combination of channels andreallocation of subbands; Hm

k stands for Hk shifted m GRBs to the right.

47


−60

−40

−20

0

(a)

ωT [rad]

Mag

. [d

B]

−π −0.75π −0.5π −0.25π 0 0.25π 0.5π 0.75π π

X0

X1

X2

−60

−40

−20

0

(a)

ωT [rad]

Mag

. [d

B]

−π −0.75π −0.5π −0.25π 0 0.25π 0.5π 0.75π π

X0

X1

X2

−60

−40

−20

0

(a)

ωT [rad]

Mag

. [d

B]

−π −0.75π −0.5π −0.25π 0 0.25π 0.5π 0.75π π

X0

X1

X2

Figure 4.11: Spectrum of (a) Real input to the FFBR network. (b) and (c)Multiplexed output signals based on the FBR scenarios in Figs. 4.12 and 4.13.Q =M = 10, N = 20, ǫ = 0.125, α = 0.5.

Anal

ysi

s B

ank O

utp

ut

Sy

nth

esis

Ban

k I

nput

Figure 4.12: Scenario I resulting in thevalues of sr in (4.12) to be 3,−2,−2.

Anal

ysi

s B

ank O

utp

ut

Sy

nth

esis

Ban

k I

nput

Figure 4.13: Scenario II resulting in thevalues of sr in (4.12) to be 2, 2,−3.

48


• Complex multiplication of αi on the real outputs of the AFB filters resultingin 2N real multiplications.

• Complex multiplication of the DFT outputs by αi only to compute the realpart requiring 2N real multiplications and N real additions.

With a (4/2) assumption, the overall arithmetic complexity is 2N log2N + 2(4 +S+3N) real multiplications and 6N log2N +2(4+S− 3

2N) real additions whereasa (3/3) assumption results in 2N log2N + 2(4 + S + 4N) real multiplications and6N log2N + 2(4 + S − 1

2N) real additions operating at fsM .

According to Fig. 4.10, the FB channels between [π, 2π] are not involved inthe FBR. Thus, more savings in the arithmetic complexity of the IDFT, DFT,multipliers βk, and γk can be achieved. The amount of these savings depends onthe channel switch position and is thus not considered here7. The main advantageof Alternative II is the elimination of the Hilbert transformers which is independentof the channel switch position.

4.5 Comparison

This section compares the arithmetic complexity of the alternatives in Sections 4.3and 4.4. We consider equal performances and, for this performance, a comparisonof arithmetic complexity is done. First, we will discuss the arithmetic complexi-ties of the real and complex FFBR networks. Second, the structures in Figs. 4.7and 4.9 will be considered. Finally, Alternatives I and II will be compared for a16-quadrature amplitude modulation (QAM) signal and in terms of EVM. Thiscomparison shows that both structures can be designed to have similar perfor-mances but Alternative I would then require additional filters due to the Hilberttransformers. In the sequel and unless otherwise mentioned, the system parametersare those in the caption for Fig. 4.11.

4.5.1 Arithmetic Complexity: Complex Versus Real FFBR

According to Sections 4.3.4 and 4.4, the arithmetic complexity depends on S, thelength of the prototype filter, and N , the number of FB channels. Figure 4.14compares the number of real operations for a 20-channel FFBR network in realand complex cases. The complex FFBR network has more computational workload.Further, it has a larger rate of increase in the number of real operations.

The operating frequency of the real FFBR network is twice that of the complexFFBR network. Thus, the ratio between the number of operations in one time unit,as shown in Fig. 4.15, compares the arithmetic complexity at the same operatingfrequency. For large S, the real FFBR network performs roughly 10 − 15% moreoperations in one time unit. According to Sections 4.3.4 and 4.4, for large S, the

7Using quick Fourier transform [150] and splitting the DFT/IDFT kernel into discrete sineand cosine transforms, more arithmetic complexity savings inside the DFT/IDFT can also beachieved.

49


200 300 400 500 600 700 800

1

2

3

4

5

6

7

Prototype filter length

Rea

l oper

atio

ns

(*1000)

Real FFBR Total

Complex FFBR Total

Real FFBR Add.

Real FFBR Mult.

Complex FFBR Add.

Complex FFBR Mult.

Figure 4.14: Number of real operations in the real and complex FFBR networkswithout considering their frequency of operation.

total arithmetic complexity is dominated by 8S. This can be verified by multiplyingthe total number of real operations in the real FFBR network by two as its operatingfrequency is twice that of its complex counterpart. Consequently, for both real andcomplex FFBR networks, the numbers of real operations in one time unit areasymptotically equal.

4.5.2 Arithmetic Complexity: Alternative I Versus Alterna-tive II

In addition to the individual FFBR networks, the scheme in Fig. 4.7 needs twoHilbert transformers which add the arithmetic complexity of two complex half-band filters running at fs

2 . For large N , the transition band of HHB,c(z) becomessmaller thereby increasing NH . To get around this, one can choose not to use someof the GRBs to widen the transition band (see Fig. 4.8). This reduces both NH andthe spectrum efficiency. Figure 4.16(a) shows the order of HHB,c(z) for differentpercentages of efficiency in a 50-channel complex FFBR network8.

A high efficiency comes at the expense of a large NH . For example, to achievean efficiency of 95% with δc = δs = 0.001, the estimated order of HHB,c(z) isNH = 74. According to Fig. 4.9, this filter operates on a real sequence at the inputside of the FFBR network and on a complex sequence at the output side of theFFBR network. As discussed in Section 2.4.3, the impulse response of a half-bandfilter is symmetric and every other coefficient is zero. Consequently, to process the

8The superiority of Alternative II is more pronounced for systems with large N .

50


200 300 400 500 600 700 800

1.1

1.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26

1.28

1.3

Com

ple

xit

y r

atio

Prototype filter length

Figure 4.15: Number of real operations in the real FFBR over that of the complexFFBR. Here, the number of operations in one time unit are considered.

0 10 20 30 40 50 60 70 80 90 100

100

200

300

400

500

600

Ord

er o

f H

HB

,c(z

)

Efficiency (%)

(a)

0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

Co

mp

lex

ity

rat

io

Efficiency (%)

(b)

250th order

300th order

350th order

400th order

Figure 4.16: Order of HHB,c(z) and the ratio between the total number of realoperations in one time unit for the alternatives of Figs. 4.7 and 4.9 with δs = 0.001.Here, the minimum (maximum) value for the order of HHB,c(z) is 3 (667).

51


real sequence at the input side of Fig. 4.9, we need NH+24 real multiplications and

NH

2 real additions. Similarly, the complex9 sequence at the output side of Fig. 4.9

can be processed at the cost of NH+22 real multiplications and NH real additions.

Example: Assuming NH = 74, we need 34 (3NH + 2) = 168 real operations

which run at M = 502 times higher frequency than the complex FFBR network.

As an example, using a 350th order P (z) and converting all the operations to thesame operating frequency, the arithmetic complexity of Alternative II, in one timeunit, is about 85% of that of Alternative I. Figure 4.16(b) shows the ratio of thenumber of real operations in Alternatives I and II for four different prototype filterorders. With increase in spectrum efficiency, Alternative II results in more savingsin arithmetic complexity10.

Although the real FFBR network performs around 10 − 15% more operationsin one time unit, it eliminates two complex half-band filters. Thus, Alternative IIhas no limitation on the efficiency which means that it has an efficiency of 100%.At this efficiency11 and for practical values of δs and ǫ in systems with a largeN , the arithmetic complexity of Alternative II, in one time unit, is always lessthan 50% of that of Alternative I. Figures 4.17–4.19 show the trend of arithmeticcomplexity savings for some values of N , 10−2≤δs≤10−5, and 0.05≤ǫ≤0.95. Thesefigures show the ratio between the number of real operations in one time unit. Thevalues of S are chosen as S = Nm+ 1, m = 1, 2, . . . , Q. Then, the value of NH isestimated with k = 0 in (4.18). For these values of S and NH , the total numberof real operations, in one time unit, is computed and the average of their ratios isplotted.

4.5.3 Performance: Alternative I Versus Alternative II

To compare the performance of Alternatives I and II in a 20-channel system, a16-QAM signal is processed and the output constellation diagrams are shown inFigs. 4.20 and 4.21. Here, S = 421 and NH = 154. For comparison, the EVM isused which is a metric of transmitter quality in modern communication systems[139, 140]. The EVM is defined as

EVMdB = 20 log

√

√

√

√

∑Ns−1k=0 |e(k)|2

∑Ns−1k=0 |sref (k)|2

. (4.20)

Here, e(k) = s(k) − sref (k) is the complex error sequence with s(k) and sref (k)being the length-Ns measured and reference complex sequences, respectively.

9If the application allows, one can further save the arithmetic complexity by only computingthe real part of the result at the output side of Fig. 4.9.

10For small efficiencies, the complexity of HHB,c(z) is not significant and both the alternativeshave roughly similar complexities. If the complexity of the FFBR network reduces, the superiorityof Alternative II will further be pronounced.

11To get realizable filters, a transition band is needed and, hence, the efficiency of 100%, refersto that achieved by k = 0 in (4.19) which becomes 1− ǫ

Q.

52


24

68

10

x 10−3

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1

δs

N=20

ε

Co

mp

lexity R

atio

Figure 4.17: The ratio between the number of real operations in one time unit forthe alternatives of Figs. 4.7 and 4.9 with different values of δs and ǫ.

Besides the characteristics of P (z), Alternative I uses two complex half-bandfilters whose characteristics further affect the EVM. In this example, the filters weredesigned so that both the alternatives result in roughly equal EVMs. The resultingEVMs for the user X0 in Fig. 4.11, after being processed by Alternatives I andII are −29.60dB and −30.44dB, respectively. The characteristics of the designedP (z), Hk(z) and HHB,c(z) are shown in Fig. 4.22.

If another EVM is desired for Alternative I, both P (z) and HHB,c(z) should beredesigned. On the other hand, Alternative II will only require redesign of P (z).The performance of Alternative I is determined by both HHB,c(z) and P (z). IfP (z) is designed so as to approximate PFBR as close as desired, the ripples ofHHB,c(z) must be roughly of the same12 size as those of P (z). Otherwise, theoverall EVM will be degraded. Assuming S = 421 and NH = 154, the arithmeticcomplexity of Alternative II is about 63% of that of Alternative I. It is desirableto design HHB,c(z) and P (z) jointly so as to achieve the same performance withsmaller orders. Irrespective of the design method, Alternative I will always requireextra arithmetic complexity due to HHB,c(z).

Due to the use of the same filter design method, i.e., minimax, in both alter-natives, it is appropriate to compare these structures in terms of the arithmetic

12This may even require a larger NH than that estimated by (4.18).

53


24

68

10

x 10−3

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

δs

N=50

ε

Co

mp

lexity R

atio


complexity with roughly equal requirements on performance, i.e., EVM. The othercomparison which focuses on performance rather than the arithmetic complexity,is more appropriate if different filter design methods are used.

4.6 Concluding Remarks

The next subsections will discuss some issues that were not treated earlier.

4.6.1 Measure of Complexity

This chapter uses the number of real arithmetic operations and real multiplicationsrather than complex multiplications are used. Multiplications by specific complexnumbers may require less real multiplications than a general complex multiplica-tion. Consequently, the real operations give a more accurate measure of arithmeticcomplexity [151]. When converting complex operations to real operations, eitherof (3/3) or (4/2) assumptions does not alter the total number of real operationsbut the former (latter) results in less multipliers (adders).

As in Section 4.3.4, this chapter does not consider the division by N in thearithmetic complexity of the FFBR networks. Doing so, Alternative II becomes

54


24

68

10

x 10−3

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

δs

N=100

ε

Co

mp

lexity R

atio


−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Qu

ad

ratu

re

In−Phase

Figure 4.20: Alternative I.

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Qu

ad

ratu

re

In−Phase

Figure 4.21: Alternative II.

55


−40

−20

0

(a)

ωT [rad]

Mag

. [d

B]

0 0.2π 0.4π 0.6π 0.8π π

−40

−20

0

(b)

ωT [rad]

Mag

. [d

B]

0 0.2π 0.4π 0.6π 0.8π π 1.2π 1.4π 1.6π 1.8π 2π

−40

−20

0

(c)

ωT [rad]

Mag

. [d

B]

−π −0.8π −0.6π −0.4π −0.2π 0 0.2π 0.4π 0.6π 0.8π π

Figure 4.22: Characteristics of (a) P (z). (b) Hk(z). (c) HHB,c(z).

more superior and the upper bound in Fig. 4.16(b) hovers around 0.5 instead of1.4.

4.6.2 Applicability of Alternatives I and II

The hybrid, full processing, and partial processing payloads need complex data inthe subsequent stages of the on-board signal processing. Then, the only choice willbe Alternative I so that the intermediate complex signals can be used for necessaryprocessing, e.g., modulation/demodulation, coding/decoding, etc. Alternative IIcan thus be used if the need for intermediate complex signals can be eliminated.

4.6.3 Filter Bank Design

To design the FB for the real FFBR network, the method of [42] can be used.In contrast to Alternative I, Alternative II uses half of the spectrum of a complexsignal. Due to this, a worst-case degradation of 6dB in the EVM may be introducedwhich can be overcome by halving the approximation error.

56

5

A Multimode TransmultiplexerStructure

This chapter introduces a multimode transmultiplexer (TMUX) which can generatea large set of bandwidths and center frequencies. It utilizes fixed integer samplingrate conversion (SRC), Farrow-based variable rational SRC, and variable frequencyshifters. Following an introduction in Section 5.1, Section 5.2 outlines the problemfor the TMUX. Then, Section 5.3 discusses the building blocks and the TMUXoperation whereas Section 5.4 deals with the filter design. After a discussion onimplementation complexity in Section 5.5, two applications of the proposed TMUXare covered in Section 5.6. In Section 5.7, the TMUX is analyzed using conventionalmultirate building blocks. Finally, Section 5.8 concludes the chapter.

5.1 Introduction

As discussed in Section 1.1, a current focus in the communications area is to de-velop flexible radio systems which seamlessly support services across several radiostandards [1–10]. A major research topic in this area is to cost-efficiently imple-ment multimode transceivers. The simplest approach for multimode problems is touse a custom device for each communication mode [8]. With the growing numberof communication standards, this approach is becoming inefficient. Thus, it is vitalto develop new low-cost multimode terminals.

A TMUX allows several users to share a common channel and multimode

57

5. A MULTIMODE TRANSMULTIPLEXER STRUCTURE

0

(b)

2p wT

1.5BGRB 1.75BGRB 2.2BGRB 1BGRB

0

(d)

2p wT

5.9BGRB 1BGRB

0

(c)

2p wT

3.2BGRB 2.7BGRB

2p/QGranularity Band (BGRB) Guardband (GB)

2pa/Q 2p/Q+2pa/Q 2p-2p/Q+2pa/Q

0 2p

(a)

wT

1BGRB

Figure 5.1: Formulation of the GRBs and examples of different user signals havingarbitrary bandwidths.

communications require multimode TMUXs that support different (time-varying)bandwidths. In other words, the users can request any bandwidth at any time. Asdiscussed in Section 3.4, a TMUX is the dual of a filter bank (FB) whose synthesisFB (SFB) and analysis FB (AFB) consist of a parallel connection of a number ofbranches. Each branch is realized by digital bandpass interpolators and decima-tors. Multimode TMUXs thus require interpolators and decimators with variableparameters. These blocks can be constructed using variable upsamplers (downsam-plers) and bandpass filters with variable center frequencies and bandwidths. Withconventional interpolators and decimators discussed in Section 3.1, the implemen-tation cost of such multimode TMUXs grows proportional to the level of flexibilityin center frequencies and bandwidths.

5.2 Problem Formulation

Similar to Fig. 1.1, we assume that the whole frequency spectrum is occupied bya number of users having different bandwidths and center frequencies. Accordingto Fig. 5.1, one can divide the whole frequency band into Q granularity bands(GRBs) separated by a guardband (GB) of 2πǫ

Q where α is as in (4.5). Any user pcan occupy any rational number of GRBs. This chapter models the input patternson which the flexible frequency-band reallocation (FFBR) network in Chapter 4 isoperating. For this purpose and in accordance with [42], the values of the GB andthe GRB are chosen as 2πǫ

Q and 2πQ − 2πǫ

Q where 0≤ǫ≤1. One can generally selectany value for the GBs and the GRBs according to the system requirements.

58


H0(z)

H1(z)

HP-1

(z)

x0(n0)

x1(n1)

xP-1

(nP-1)

L

L

L

F(z)

F(z)

F(z)

ejw

0n

ejw

1n

ejw

P-1n

y(n) y(n)^

x0(n0)^

x1(n1)^

xP-1

(nP-1)^

H0(z) F(z) L

ê

-jw0n

ê

-jw1n

H1(z) LF(z)

HP-1

(z) L

ê

-jwP-1

n

F(z)

Synthesis FB Analysis FB

Figure 5.2: Proposed multimode TMUX consisting of fixed integer SRC, variablerational SRC, and variable frequency shifters. Here, H↓

p (z) and H↑p (z) represent

Farrow-based filters for decimation and interpolation, respectively.

5.3 Multimode TMUX Structure

This section introduces a multimode TMUX, shown in Fig. 5.2, which can generatearbitrary1 bandwidths and center frequencies. A GRB is the minimum bandwidtha user can occupy. The users are separated by GBs of ∆≥0 so that the TMUXis slightly redundant. We mostly consider ∆ = 0. As discussed in Section 3.4.3,the proposed multimode TMUX requires a small redundancy so as to generate allpossible modes without channel interference and filter redesign. Redundancy maybe of particular interest if there exists severe channel distortion in some frequencybands [20]. We also assume that any user p can occupy a rational Rp(t) number ofGRBs2 where 1 ≤ Rp(t) ≤ Q.

The TMUX generates a GRB through interpolation by an integer L. To getbandwidths which are rational multiples of the GRB, the Farrow-based filter per-forms decimation by rational ratios Rp. As discussed in Section 3.2, for interpola-tion (decimation) one can use the Farrow (transposed Farrow) structure. To placethe users in appropriate positions in the frequency spectrum, variable frequencyshifters are utilized. In the AFB, the received signal is first frequency shifted suchthat the desired signal can be processed in the baseband. Then, a Farrow-based in-terpolator (by ratio Rp) followed by decimation by an integer L obtains the desiredsignal. Figure 5.3 plots the frequency spectrum at the output of different blockswith a uniformly distributed random input.

5.3.1 Channel Sampling Rates

In multimode systems, the users X0, X1, . . . , XP−1, can generally have differentdata rates. This means that in one time frame, the number of processed samples

1The bandwidths and center frequencies are in practice limited to rational numbers due tofinite precision. At the expense of additional implementation cost, any precision can be achieved.

2The value of Rp(t) is constant during the time frame in which the user signal p is transmitted.In the sequel, the time index t will be omitted.

59


−60−40−20

020

ωT1 [rad]

(a)M

ag.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−60−40−20

020

ωT2 [rad]

(b)

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−60−40−20

020

ωT2 [rad]

(c)

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−60−40−20

020

ωT2 [rad]

(d)

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−60−40−20

020

ωT1 [rad]

(e)

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

Figure 5.3: (a) Output of F (z). (b) Output of H↓p (z). (c) and (d) Outputs of

frequency shifters. (e) Output of H↑p (z). L = 12, Rp = 2.5, ωp = 0.3π, ωp = 0.3π,

and T2 = RpT1.

in each branch of Fig. 5.2 can be different from the others. Mathematically, thesampling periods of the TMUX inputs, i.e., T0, T1, . . . , TP−1, must satisfy

T0R0 = T1R1 = . . . = TP−1RP−1 = LTy, (5.1)

where Ty is the sampling period of y(n). In the proposed redundant TMUX,∑P−1

p=0Rp

L < 1 which is in contrast to a critically sampled TMUX with∑P−1

p=0Rp

L =1.

5.3.2 Sampling Rate Conversion

As shown in Fig. 5.2, integer interpolation and decimation by L require lowpassfilters F (z) and F (z), respectively. The stopband and passband edges of these filters

are defined as in (2.33). This also sets the value of the GRB as 2π(1+ρ)L . Further,

SRC by a rational value Rp is performed by the Farrow-based filters resulting in

the set of bandwidths 2π(1+ρ)L Rp, p = 0, 1, . . . , P − 1. Each of the systems H↓

p (z)

60


and H↑p (z) employs a filter with a transfer function given by (3.10) and it performs

SRC by a variable rational ratio Rp. The values of µ are given by (3.12). Furtherdetails can be found in Section 3.2.

5.3.3 Subcarrier Frequencies

According to Section 5.2, the input signal is divided into a number of GRBs sepa-rated by a GB of 2πǫ

Q . Having generated the user signals, they must be modulatedinto specific locations in the frequency spectrum. To avoid the inter-carrier inter-ference (ICI), the user signals should not overlap. Consequently, for user k

ωk =

χ0

2 if p = 0∑k−1

p=0 χp +χk

2 if p 6=0.(5.2)

Here, χp = ⌈Rp⌉ 2πQ , p = 0, 1, . . . , k, and ⌈x⌉ is the ceiling of x. The ceiling

operation ensures that users do not share3 a GRB. In general, the bandwidthsare time-varying and only the GRB is fixed. With no frequency-band reallocation(FBR), we have ωk = ωk. In case of FBR, ωk becomes

ωk = ωk + Γ(t, ωk) (5.3)

where Γ(t, ωk) is a time-varying function expressing the FBR or frequency multi-plexing or any frequency offset.

5.4 Filter Design

Due to redundancy and according to Section 3.4.3, the ICI is determined by thestopband attenuation of F (z) and F (z). In each branch and ignoring H↓

p (z) and

H↑p (z), the transfer function from xp(np) to xp(np) is the zeroth polyphase com-

ponent of F (z)F (z) [12]. This polyphase component controls the inter-symbolinterference (ISI). To make this polyphase component unity4, F (z)F (z) must bean Lth-band filter. Thus, F (z) and F (z) are the spectral factors of an Lth-bandfilter (refer to Section 2.4.3). This also holds when H↓

p (z) and H↑p (z) are present

provided that they approximate a fractional delay filter with a delay µ throughouttheir respective frequency bands. For H↓

p (z), only the GRB needs to be coveredwhereas in the AFB, the whole band except for a small band near π must be cov-ered. The reason is that the output of F (z) is bandlimited to the GRB. However,in the AFB, y(n) is processed by H↑

p (z). Consequently, the complexity of H↓p (z)

will be less than that of H↑p (z). We thus need to determine F (z) and F (z) such

3This is specific for the FFBR network in Chapter 4.4For causal filters, one of the polyphase components should be a pure delay.

61


0

0.2

0.4

0.6

0.8

1

0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π πωT [rad]

Mag

nit

ude

−0.01

−0.005

0

0.005

0.01

0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9πωT [rad]

Err

or

Figure 5.4: Magnitude response and approximation error of the Farrow structuredesigned by (5.5) with δ4 = 0.01 and ω2T = 0.9π. Here, 6 subfilters with orders24, 24, 8, 20, 6, 8 have been used.

that

|[F (ejωT )F (ejωT )]zeroth − 1| ≤ δ1, ωT ∈ [0, π],

|F (ejωT )| ≤ δ2, ωT ∈ [ω1T, π],

|F (ejωT )| ≤ δ3, ωT ∈ [ω1T, π], (5.4)

where ω1T = π(1+ρ)L , and [F (ejωT )F (ejωT )]zeroth denotes the zeroth polyphase

component of F (z)F (z). The Farrow-based filter H(z, µ) should be designed suchthat

|H(ejωT , µ)− e−jωTµ| ≤ δ4, ωT ∈ [0, ω2T ], µ∈[−0.5, 0.5]. (5.5)

Additionally, ω2T = ω1T for the SFB. In the AFB, the spectral width of y(n), inFig. 5.2, determines ω2T . For example, at a typical spectrum utilization percentageof 90%, we have ω2T = 0.9π.

All δi, i = 1, 2, 3, 4, in (5.4) and (5.5) can be reduced to any desired level bysimply increasing the filter order. To design the TMUX, one should

• solve (5.4) to get F (z) and F (z).

• solve (5.5) to get the Farrow subfilters in the SFB and AFB.

62


0

0.2

0.4

0.6

0.8

1

0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9π πωT [rad]

Mag

nit

ude

−1

−0.5

0

0.5

1x 10

−3

0 0.1π 0.2π 0.3π 0.4π 0.5π 0.6π 0.7π 0.8π 0.9πωT [rad]

Err

or

Figure 5.5: Magnitude response and approximation error of the Farrow structuredesigned by (5.5) with δ4 = 0.001 and ω2T = 0.9π. Here, 7 subfilters with orders38, 38, 18, 32, 14, 20, 6 have been used.

Having solved these problems once offline, only µ and the variable frequency shifterschange online. The filter pair F (z) and F (z) can be designed as outlined in, e.g.,[77, 130–138], whereas the Farrow-based filters may be designed as described in,e.g., [94, 95, 97–104]. The proposed multimode TMUX can thus be designed toapproximate perfect reconstruction (PR) as close as desired for all possible modesby separately solving three conventional filter design problems offline.

5.4.1 Example

A series of filters with δ1 = δ4 = 0.01, 0.001, ω1T = 0.0875π, ω2T = 0.9π, andL = 12 are assumed. As the stopband attenuation of F (z) and F (z) suppresses theICI, they have been designed with different values of δ2 = δ3. With fixed δ1, thereare similar constraints on [F (ejωT )F (ejωT )]zeroth and H(ejωT , µ). The stopband

attenuation of F (z) and F (z) is then the only parameter which changes. Figures 5.4and 5.5 show the magnitude response and the approximation error of the Farrowstructure for µ = 0, 0.05, 0.1, 0.15, . . . , 0.5. These structures have been designedto approximate allpass transfer functions in the frequency band ωT∈[0, 0.9π] andthe resulting values for δ4 are, respectively, 0.01 and 0.001.

The magnitude responses at the stopband of F (z) = F (z) and the passband of[F (ejωT )F (ejωT )]zeroth for some of the designed filters are also shown in Figs. 5.6

63


−70

−60

−50

−40

−30

0.09π 0.3π 0.5π 0.7π 0.9π πωT [rad]

Mag

nit

ude

[dB

]

−0.01

−0.005

0

0.005

0.01

0.2π 0.4π 0.6π 0.8π πωT [rad]

Mag

nit

ude

Figure 5.6: Magnitude response in the stopband (top) of F (z) = F (z) and thepassband (bottom) of [F (ejωT )F (ejωT )]zeroth for δ1 = 0.01 and L = 12.

and 5.7. For all of these filters, the values of δ1 are, respectively, 0.01 and 0.001but they have different stopband attenuations. Figure 5.8 shows the average errorvector magnitude (EVM), discussed in Section 4.5.3, for three multimode setupsin a 16-quadrature amplitude modulation (QAM) signal. The values of δ1 = δ4 seta lower bound on the EVM. However, the EVM can be decreased to any level andfor all possible modes by decreasing δi, i = 1, 2, 3, 4. Figure 5.9 plots the frequencyspectrum of y(n) for these multimode setups. As can be seen, ∆ = 0.

5.5 Implementation and Design Complexity Issues

The previous section showed that the proposed TMUX can be designed to have assmall errors as desired for all possible modes through three separate offline filterdesigns. This is attractive compared to solutions that require either one set offilters for each mode or online filter design. There is still room for complexityreductions by modifying the proposed structure. This section points out some ofthese.

A motivation to using integer interpolation to generate a GRB is that regularinteger interpolation structures are more efficient than Farrow-based structureswhen implementing an interpolator with a relatively large conversion ratio L [152].

64


−70

−60

−50

−40

−30

0.09π 0.3π 0.5π 0.7π 0.9π πωT [rad]

Mag

nit

ude

[dB

]

−1

−0.5

0

0.5

1x 10

−3

0.2π 0.4π 0.6π 0.8π πωT [rad]

Mag

nit

ude

Figure 5.7: Magnitude response in the stopband (top) of F (z) = F (z) and thepassband (bottom) of [F (ejωT )F (ejωT )]zeroth for δ1 = 0.001 and L = 12.

This is true if multi-stage structures [77] are utilized which should indeed be appliedfor large L. If the bandwidth of the users often matches the GRB, this option(and the dual in the AFB) appears as the most natural choice. If the users oftenoccupy wider bandwidths than the GRB, it may then be worth to use a smallerL. The Farrow-based filters in the SFB and AFB can then both work either as aninterpolator or as a decimator. This allows us to find the best trade-off betweenthe complexities of the integer and rational SRC parts. Some results are availablefor interpolators and decimators [152] but the problem is more complex here as wedeal with TMUXs. The overall optimum will depend on how often the users takeon narrow or wide bandwidths.

Another issue is the filter design. The previous section outlined the separatefilter design which is attractive as known techniques can be adopted. Although thisgives a good suboptimum overall solution, it is slightly overdesigned. To reducethe complexity, one can design all filters simultaneously which can, in principle,be done using standard nonlinear optimization techniques. This has successfullybeen used for fixed FBs and TMUXs [79] but the problem is much more complexhere as we deal with multimode TMUXs. This implies that the requirements mustbe satisfied for all possible modes. Consequently, the number of constraints growswith the number of modes. Simultaneous optimization may therefore be practicallyfeasible only for problems that have a few modes.

65


40 50 60 70 80 90

−40

−35

−30

Stopband attenuation [dB]

EV

M [

dB

]

(a) δ1=δ

4=0.01

Setup 1: Rp=1.3,2.5,3.7

Setup 2: Rp=1.1,1.9,4.1

Setup 3: Rp=1.75,2.7,5.1

40 45 50 55 60 65 70 75 80 85

−50

−40

−30

Stopband attenuation [dB]

EV

M [

dB

]

(a) δ1=δ

4=0.001

Figure 5.8: The resulting EVM for some multimode setups at different stopband at-tenuations of F (z)F (z) with fixed errors in [F (ejωT )F (ejωT )]zeroth and H(ejωT , µ).

5.6 TMUX Application

This section considers two applications of the proposed TMUX. Having designedthe TMUX for an EVM of −100dB, achieved with δi = 10−5, i = 1, 2, 3, 4, thestructure in Fig. 5.10 can be used for functionality/performance test of the FFBRnetwork in Chapter 4. The values for the GRB and GB are chosen as 2π

Q − 2πǫQ

and 2πǫQ where 0≤ǫ≤1. To verify the functionality, four different user signals

X0, X1, X2, X3 with Rp = 1.75, 1.25, 2, 3.5 and ωp = 0.2π, 0.6π, π, 1.6π areassumed. The frequency spectrum of the input and the multiplexed output of theFFBR network with Q =

∑

p ⌈Rp⌉ = 10 GRBs are shown in Figs. 5.12(a) and5.12(b). The scenario of FBR in Fig. 5.12(b), results in ωp = π, 1.8π, 1.4π, 0.4π.To illustrate the performance, Fig. 5.12(c) shows the values of EVM for differentstopband attenuations of P (z) in (4.4) and with a 16-QAM signal. The stopbandattenuation of P (z) in (4.4) mainly determines the error of the FFBR network [42].

66


−60

−40

−20

0

ωT [rad]

Mag

. [d

B]

(a)

0 0.25π 0.5π 0.75π π 1.25π

−60

−40

−20

0

ωT [rad]

Mag

. [d

B]

(b)

0 0.25π 0.5π 0.75π π 1.25π

−60

−40

−20

0

ωT [rad]

Mag

. [d

B]

(c)

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π

Figure 5.9: Spectrum of y(n) for multimode setups of Fig. 5.8. (a) Setup 1. (b)Setup 2. (c) Setup 3.

FFBRNetwork

SynthesisFB

x0(n0)

x1(n1)

xP-1

(nP-1)

AnalysisFB

x0(n0)

x1(n1)

xP-1

(nP-1)

^

^

^

Figure 5.10: Setup for functionality/performance test.

2p/QGranularity Band (BGRB)=2p(1+r)/L Guardband

2pa/Q 2p/Q+2pa/Q 2p-2p/Q+2pa/Q

0 2p wT

1BGRB

L-th Band Filter

Figure 5.11: The GB, the GRB, and the L-th band filter. As 2πQ − 2πǫ

Q = 2π(1+ρ)L ,

we can have ρ≤L(1−ǫ)Q − 1.

67


−80−60−40−20

0

ωT [rad]

Mag

.

(a)

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

X3

−80−60−40−20

0

ωT [rad]

Mag

.

(b)

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

X3

40 45 50 55 60 65 70

−40

−20

Stopband attenutation of P(z) [dB]

EV

M [

dB

]

(c)

Figure 5.12: (a) and (b) Functionality test of an FFBR network. (c) Performancetest of an FFBR network. L = 12, Q = 10, ǫ = 0.125.

Lxm

(nm) Am

Ap D

p(z) B

p^Fp(z) L x

p(np)^F

m(z) C

m(z)B

m

Farrow Decimator Farrow Interpolator

e jwmn

e-jwpn^

Figure 5.13: Equivalent path between xm(nm) and xp(np).

5.7 Analysis Using Multirate Building Blocks

This section describes the proposed TMUX in terms of conventional multiratebuilding blocks [18]. This is done by utilizing the rational SRC equivalent of theFarrow-based filter [153]. In each branch, the Farrow-based filter for interpolation

by Rp =Ap

Bp> 1 can be modeled as the cascade of upsampling by Ap, the finite-

length impulse response (FIR) filter Dp(z), and downsampling by Bp. Similarly,a cascade of upsampling by Bp, the FIR filter Cp(z), and downsampling by Ap

can be used to model decimation by Rp =Ap

Bp> 1. Consequently, each branch

can be modeled as in Fig. 5.13. Considering the frequency shifters, Fig. 5.13 canbe redrawn as in Fig. 5.14 where zm = ejωmT and zp = ejωpT . This structure issimilar to [19] with some differences as:

1. The nonuniform TMUX in [19] does not utilize frequency shifters and the

68


Lxm

(nm) Am

Ap D

p(z) B

p^Fp(z) L x

p(np)^F

m(z) C

m(z)B

m

Farrow Decimator Farrow Interpolator

^z (zp/zm)z

Figure 5.14: Equivalent path between xm(np) and xm(np) considering the effectsof frequency shifters in the frequency domain.

LBm

xm

(nm) ^ xm

(nm)^LBm

F(zBm)Cm

(z) Am F(zBm)D

m(z)A

m

Figure 5.15: Simplified equivalent path between xm(nm) and xm(nm).

termzpzmz does not appear in the formulations. As the filters of Fig. 5.2 are

lowpass and the frequency shifts of a filter do not alter the characteristicsof its baseband equivalent, the term

zpzmz does not affect the analysis of the

TMUX. Therefore, similar analysis as that in [19] can be used for the presentTMUX.

2. Instead of single SFB and AFB filters in [19], the present TMUX uses thecascade of a periodic filter, i.e., Fm(zBm) or Fp(z

Bp), and the FIR equivalentof the Farrow structure, i.e, Cm(z) or Dp(z).

Note also that instead of designing bandpass filters, the proposed TMUX designslowpass filters. Then, frequency shifters modulate the user signals. This not onlygives bandpass characteristics but also increases the reconfigurability regarding thecenter frequencies. To clarify the second difference, Fig. 5.14 is redrawn in Fig. 5.15where p = m and Bp = Bm = B. Figure 5.16 shows the transfer function of the

two cascaded filters, i.e., Fm(zB)Cm(z) or Dm(z)Fm(zB). This cascaded filter islowpass with passband and stopband edges as

ωcT =π(1− ρ)

LB, (5.6)

ωsT =2π

B− π(1 + ρ)

LB. (5.7)

From [19], the blocked transfer function of Fig. 5.2 is T (z) = Φ(z)Ψ(z) where

Φ(z) =[

φ0(z) φ1(z) . . . φP−1(z)]T,

Ψ(z) =[

ψ0(z) ψ1(z) . . . ψP−1(z)]

. (5.8)

For the proposed TMUX,

φp(z) = Dp(z)Fp(zBp),

ψp(z) = Fm((zpzm

z)Bm)Cm(zpzm

z). (5.9)

69


(1-r)p/(LB)

(1+r)p/(LB)

2p/B-(1+r)p/(LB)2p/B

2p/B-(1-r)p/(LB)

wT

2p/B+(1+r)p/(LB)

2p/B+(1-r)p/(LB)

4p/B-(1+r)p/(LB)

4p/B

4p/B-(1-r)p/(LB)

2p2p-(1+r)p/(LB)

2p-(1-r)p/(LB)

Figure 5.16: The cascade of a periodic filter and the FIR equivalent of the Farrowstructure. The dashed line is the FIR equivalent of the Farrow structure, i.e., Cm(z)or Dm(z), whereas the solid line is the periodic filter, i.e., Fm(zB) or Fm(zB).

To further simplify this, assume Fm(z) = F (z) and Fm(z) = F (z) for m =0, 1, . . . , P − 1. If p = m, then zp = zm and (5.9) becomes

φp(z) = Dp(z)F (zBp) (5.10)

ψp(z) = F (zBp)Cp(z). (5.11)

The values of the ICI and ISI are mainly determined by the expressions in (5.8)

which themselves depend on the ratiozpzm

in (5.9). For the desired signal Xd(z),the relation m = p = d holds. Then, ωd = ωd = ωd also holds and the systemcan approximate PR as close as desired via a proper design of the SFB and AFBfilters. If m 6= p (or equivalently, ωm 6= ωp), the signals are considered as undesiredand will be attenuated by the AFB filters. The reason is that ωk in (5.2) ensuresno overlap of the user signals. Therefore, the undesired signals which have passedthrough the SFB filter Fm((

zpzmz)Bm)Cm(

zpzmz), will fall in the stopband of the

AFB filter Dp(z)Fp(zBp) and be attenuated. The amount of the ICI and ISI is

determined by the cascade of these filters.In conclusion, if the TMUX is designed for a worst-case error δw, it can produce

arbitrary bandwidths while approximating PR with smaller errors than δw. Then,we need a fixed set of filters and the only parameter to support arbitrary band-widths is the fractional delay of the Farrow structure. Although the same analysismethods as for the existing TMUXs can be used here, the implementation is differ-ent. The conventional rational SRC blocks (upsamplers, downsamplers, and filters)are only used for modeling whereas the TMUX implements these blocks implicitlyusing integer and rational SRC.

5.8 Conclusion

This chapter introduced a multimode TMUX capable of generating a large set ofbandwidths and center frequencies. The TMUX utilizes fixed integer SRC, Farrow-based variable rational SRC, and variable frequency shifters. It needs only oneoffline filter design beforehand. Then, all possible combinations of bandwidths andcenter frequencies are easily obtained by online adjustment of (i) the variable delay

70


parameters of the Farrow-based filters, and (ii) the variable parameters of the fre-quency shifters. Design examples were provided for illustration. Furthermore, theTMUX was described in terms of conventional multirate building blocks allowingone to use the design techniques based on the blocked transfer function.

71


72

6

A Class of MultimodeTransmultiplexers Based on the

Farrow Structure

This chapter introduces a class of multimode transmultiplexers (TMUXs) in whichthe Farrow structure realizes the polyphase components of general lowpass filters.After a brief introduction, Section 6.2 gives some prerequisites whereas Section 6.3considers the design of approximately Nyquist filters. Further, a TMUX capableof performing integer sampling rate conversion (SRC) is proposed. Extending theinteger SRC to rational SRC, Section 6.4 introduces a multimode TMUX whichcan generate arbitrary bandwidths. The TMUX performance is investigated inSection 6.5 and some concluding remarks are given in Section 6.7.

6.1 Introduction

As discussed in Section 1.1, multimode communications require to support differentbandwidths, resulting from various telecommunication standards, e.g., global sys-tem for mobile communications (GSM), interim standard-54/136 (IS-54/36), andIS-95 [11]. To include these standards in a general telecommunication system, weshould handle a number of different bandwidths. Thus, multimode communica-tions require that several users, with different bit rates, share a common channel.A TMUX allows different users to share a common channel and multimode TMUXsthus constitute one of the main building blocks in multistandard communications[121]. With bandwidth-on-demand, the TMUX characteristics must vary with

73

6. A CLASS OF MULTIMODE TRANSMULTIPLEXERS BASED ON THEFARROW STRUCTURE

time. This supports a dynamic allocation of bandwidth so that each user occupiesa time-varying portion of the channel.


TMUXs, as duals of filter banks (FBs), are composed of a synthesis FB (SFB)followed by an analysis FB (AFB). The SFB and AFB consist of a parallel con-nection of a number of branches [12]. Each branch is realized by digital inter-polators/decimators. In uniform TMUXs, the bandwidths and center frequenciesof these interpolators/decimators are fixed. Multimode TMUXs require interpo-lators/decimators with variable parameters. In other words, we need nonuniformTMUXs and FBs which can modify their characteristics in a time-varying manner.With fixed traditional TMUXs and FBs, e.g., [16–31], a set of filters, designed for aFB, can be adopted for a TMUX [12]. This does not apply to multimode TMUXsthough. The use of conventional TMUXs, in multimode communications, requireseither predesign of different filters or online filter design. This becomes complicatedwhen supporting dynamic communications.

Chapter 5 introduced a multimode TMUX consisting of fixed integer SRC andFarrow-based variable rational SRC. It needs three sets of filters and the Farrowstructure in the AFB has a higher arithmetic complexity. This chapter introducesan alternative method to design approximately Nyquist filters where the Farrowstructure realizes the polyphase components of general lowpass filters. The zerothpolyphase component is a Type I linear-phase finite-length impulse response (FIR)filter whereas the remaining polyphase components are realized by the Farrowstructure. Specific constraints are imposed on these lowpass filters such that theyare approximately power complementary. The power complementary property al-lows us to obtain integer SRC multimode TMUXs. Incorporating the integer SRC,in a general rational SRC structure, a rational SRC multimode TMUX is thenconstructed. Different filter design techniques are considered and compared. Incontrast to the TMUX in Chapter 5, the Farrow structures in the SFB and AFBof the present chapter have equal complexities. The main advantage of the pro-posed TMUXs is to support dynamic communications with reasonable design effortand an offline filter design. The proposed TMUXs are near perfect reconstruction(NPR). We do not cancel the inter-carrier interference (ICI) and inter-symbol in-terference (ISI). Instead, we suppress them by a proper filter design.

6.2 Prerequisites

This section discusses some prerequisites and general issues.

6.2.1 Problem Formulation

As in Fig. 1.1, we assume the whole frequency spectrum to be shared by P users.

Each user has a bandwidth of π(1+ρ)Rp

, p = 0, 1, . . . , P−1, and Rp =Ap

Bpwith integers

74


Ap and Bp. Furthermore, ρ is the roll-off factor and a guardband (GB) of ∆≥0separates the users. This chapter assumes ∆ = 0 to allow a minimum GB betweenthe users. The choice of ∆ does not affect the filter design problem. Generally, Pcan vary with time. For a discussion independent of time, the sequel assumes thatin each time frame, P is time-invariant. Therefore, in each time frame, the TMUXoperates as a fixed nonuniform TMUX.

Assume α time frames (modes) and the delay of the TMUX being Nd. Whenchanging P , a new packet of data, containingNs samples, is sent. Due to transients,the receiver first receives Nd unreliable samples. The ratio between the number ofreliable samples and the total number of samples gives the efficiency as

η =αNs

αNs + (α− 1)Nd=

1

1 + α−1α

Nd

Ns

. (6.1)

Here, the term α−1α is asymptotically equal to unity. Then, choosing Ns > Nd

increases η and the effect of Ns is more pronounced than that of α. Thus, for everyNd, we can choose a proper Ns so as not to degrade η. Note that the proposedTMUXs have linear-phase FIR filters and Nd is known [152].

6.2.2 Some General Issues

Some general issues need to be outlined. First, to avoid ICI, the users do notoverlap and the TMUXs are slightly redundant. Redundancy simplifies the filterdesign as discussed in Section 3.4.3. It also enables one to avoid infeasible cases[20, 22, 24, 26, 27].

Second, we consider packets of data separated by time slots. This is similarto time division multiple access systems like, e.g., GSM. The TMUX parameterschange only when switching from one mode to another. For each mode, the TMUXis fixed and it works as any regular fixed TMUX. In other words, the proposedTMUX realizes a (large) number of fixed TMUXs. The switching between them isdone by changing a few multipliers (in the Farrow structure) and the SRC ratios.Thus, for each and every mode, we should ensure that the ISI and ICI are small,as in any regular fixed TMUX. Transients are only present when switching fromone mode to another. An alternative to using the proposed TMUXs is to use a setof different and separately designed fixed TMUXs. When changing from one modeto another, we then change one fixed TMUX to another, and we are facing exactlythe same problem regarding transients. Note that [67, 68] solves the problem withtransients but it does not provide a full flexibility as compared to the proposedTMUXs. A thorough discussion on the differences and similarities of [67, 68] andthe proposed TMUXs can be found in [68] and Chapter 7.

Third, this chapter uses the Farrow structure as outlined in Section 3.2. Here,Sk(z) are designed to obtain a fractional delay filter, i.e., H(z) = z−µ. Although wehave linear-phase FIR filters Sk(z), we use the term the Farrow structure insteadof the modified Farrow structure [94]. The Farrow structure and the transposedFarrow structure [3, 95] are used, respectively, for interpolation and decimation, asseen later in Figs. 6.4, 6.5, 6.14, and 6.15.

75


G0(z)

G1(z)

GP-1

(z)

x0(n0)

x1(n1)

xP-1

(nP-1)

ejw

0n

ejw

1n

ejw

P-1n

y(n) y(n)^

G0(z) x

0(n0)^

x1(n1)^

xP-1

(nP-1)^

ê

-jw0n

ê

-jw1n

G1(z)

GP-1

(z)

ê

-jwP-1

n


^

^

^ RP-1

R1

R0R0

R1

RP-1

Figure 6.1: Integer SRC multimode TMUX composed of variable integer SRCs andadjustable frequency shifters. The actual realization of SRC with an integer ratioRp is performed by the structures in Figs. 6.4 and 6.5. The SRC structures in thisfigure are only used for analysis purposes.

6.3 Proposed Integer SRC Multimode TMUX

With general lowpass filters, we can construct an integer SRC multimode TMUXas in Fig. 6.1. It consists of upsamplers/downsamplers Rp, p = 0, 1, . . . , P − 1;

lowpass filters Gp(z) and Gp(z); and adjustable frequency shifters ωp and ωp. IfTp is the sampling period in branch p, then

T0R0

=T1R1

= . . . = Ty, (6.2)

where Ty is the sampling period of y(n). If∑P−1

p=01Rp

< 1, the TMUX is redundant

whereas∑P−1

p=01Rp

= 1 gives a critically sampled TMUX.

6.3.1 Variable Integer SRC Using the Farrow Structure

The Type I polyphase decomposition of a filter Gp(z) is [12, 69, 70]

Gp(z) =

Rp−1∑

m=0

z−mGp,m(zRp). (6.3)

If Gp(z) is a general causal ideal lowpass filter of order N , we have

Gp(z) =

z−N2 in passband

0 in stopband.(6.4)

Comparing (6.3) and (6.4) gives

Gp,m(z) =

z−

N2

−m

Rp in passband

0 in stopband.(6.5)

76


x(n)

SQ

(z) S5(z) S

3(z) S

1(z)

mp,m1mp,m

3mp,m5mp,m

Q

yp,m(z)

yyp,m

(n)

Figure 6.2: Realization of Ψp,m(z).

x(n)

yFp,m

(n)

SP(z) S

4(z) S

2(z) S

0(z)

mp,m2mp,m

4mp,mP

Fp,m(z)

Figure 6.3: Realization of Φp,m(z).

Therefore, a general interpolation/decimation filter can be designed by choosingGp,0(z) to be an N0-th order Type I linear-phase FIR filter and utilizing the Farrowstructure to realize Gp,m(z), m = 1, 2, . . . , Rp − 1, of odd1 order N1 as

N0 =N

Rp= N1 + 1, (6.6)

where

Gp,m(z) =L∑

k=0

Sk(z)µkp,m, (6.7)

with

µp,m =−mRp

+1

2⇒ µp,m = −µp,Rp−m. (6.8)

Note that (6.8) relates the SRC ratio Rp to the values of µ in Fig. 3.7. For everynew Rp, we should only compute Rp new values of µp,m whereas Sk(z) need notchange. Specifically, (6.8) can be computed recursively as

µp,m+1 =−1

Rp+ µp,m, µp,0 =

1

2. (6.9)

Efficient Variable Integer SRC

Using (6.7) and (6.8), Gp,m(z) and Gp,Rp−m(z) can be written as

Gp,m(z) = Φp,m(z) + Ψp,m(z), (6.10)

Gp,Rp−m(z) = Φp,m(z)−Ψp,m(z). (6.11)

According to Figs. 6.2 and 6.3 with Q = 2⌊L+12 ⌋ − 1 and P = 2⌊L

2 ⌋, we have

Φp,m(z) =

⌊L2 ⌋

∑

k=0

Gp,2k(z)µ2kp,m, (6.12)

Ψp,m(z) =

⌊L+12 ⌋

∑

k=1

Gp,2k−1(z)µ2k−1p,m . (6.13)

1With proper modifications, even-order filters can also be designed [152].

77


Sk(z)

Gp,0(z)

Rp fs

y(m)

01

Rp-1k = 0, 1, ..., L

fs

x(n)

Fixed Variable

mp,m k

Figure 6.4: Efficient interpolation by a variable integer ratio Rp using fixed subfil-ters, variable multipliers, and commutators.

Sk(z)

Gp,0(z)

mp,mfs/Rp

y(m)01

Rp-1k = 0, 1, ..., L

kx(n)

fs

Variable Fixed

Figure 6.5: Efficient decimation by a variable integer ratio Rp using fixed subfilters,variable multipliers, and commutators.

In this way, two polyphase components can be realized at the cost of one [152].Incorporating commutators, as in Figs. 6.4 and 6.5, SRC by a variable integer ratioRp requires fixed filters Sk(z) and Gp,0(z); variable multipliers µk

p,m; and commu-

tators. Due to (6.8), only the distinct values of µkp,m must be considered. Thus,

variable integer SRC needs either a set of precomputed values or some variablemultipliers.

Arithmetic Complexity

The arithmetic complexity of the fixed parts, in Figs. 6.4 and 6.5, results from theN0-th order filterGp,0(z); the Farrow structure composed of L+1 subfilters ofN1-th

order; and L additional structural adders2. Then, we roughly require 3N1(L+2)2 +

3L+52 fixed arithmetic operations. For each extra coefficient, in Sk(z), we need

3(L+2)2 additional fixed arithmetic operations. The structures in Figs. 6.4 and 6.5

require some variable multipliers as well. The Farrow-based SRC is generally moreefficient than regular SRC, except for an Rp which is either small or factorable intointeger parts [154]. This is true even if one does not consider the reconfigurabilityin SRC which is obtained by using the Farrow structure.

2The decimator of Fig. 6.5 has an additional multi-input structural adder.

78


6.3.2 Approximation of Perfect Reconstruction (PR)

Generally, TMUXs have two sources of interference as in (3.35). The filters betweenXp(z) and Xp(z) cause ISI, whereas the filters between Xi(z) and Xp(z), give rise

to ICI. In Fig. 6.1, a desired signal Xd(z), can be expressed as

Xd(z) =

Rd−1∑

m=0

P−1∑

p=0

Xp(zRpRdW

mRp

Rd(zdzp

)Rp)Gp(z1

RdWmRd

zdzp

)Gd(z1

RdWmRd

) (6.14)

where zp = ejωpT , zd = ejωdT , and Wα = e−j 2πα . If p 6=d (p = d), then (6.14)

represents the effect of ICI (ISI) on Xd(z). Generally, the ICI has P − 1 terms andthe complete ICI is represented by their summation. In the proposed redundantTMUX, the users do not overlap in the transition bands and passbands. Thus, allICI terms fall in the stopband of Gd(z). With brickwall lowpass filters,

Gp(z) =

1 in passband

0 in stopband.(6.15)

Then, δICI = 0 and δISI = 0 in (3.35). Thus, a redundant PR TMUX requiresbrickwall lowpass filters. Such a PR TMUX is unrealizable. This chapter deals withrealizable NPR TMUXs. Therefore, the ICI (ISI) is controlled by the stopband(passband) ripples of Gp(z) and Gp(z). By decreasing these ripples, arbitrarilygood NPR TMUXs can be obtained.

In maximally decimated nonuniform FBs, the AFB and SFB filters should havespecific center frequencies. Otherwise, infeasible cases occur [20, 22, 24, 26, 27] andNPR FBs may not even be achievable. To avoid this, we can shift some AFB andSFB filters to specific center frequencies [27]. Duality of FBs and TMUXs givesrise to similar problems for critically sampled nonuniform TMUXs. The proposedredundant TMUX does not have infeasible cases as the users do not overlap. Thefrequency shifters are chosen so as to ensure feasibility. Feasibility is achieved if(with ∆ = 0)

ωp =

π(1+ρ)R0

if p = 0

π(1 + ρ)[ 1Rp

+∑p−1

k=02Rk

] if p 6=0.(6.16)

Note that these frequency shifters are similar to those in [27].

6.3.3 Filter Design

With p = d, zp = zd, and Rp = Rd in (6.14), we have Xp(z) = Xp(z)Fp(z) where

Fp(z) =

Rp−1∑

m=0

Gp(z1

RpWmRp

)Gp(z1

RpWmRp

). (6.17)

This is the zeroth polyphase component of Gp(z)Gp(z). Therefore, Sk(z) andGp,0(z) must be determined such that

79


• Gp(z) and Gp(z) have small stopband ripples.

• Fp(z) approximates an allpass function.

In other words, Gp(z)Gp(z) approximates an Rp-th band filter as defined in Sec-tion 2.4.3.

6.3.4 Filter Design Parameters

The free design parameters are the coefficients of Sk(z) and Gp,0(z). The valuesof N1, N0, L, ρ, µ

kp,m, and Rp are fixed during the optimization. Note that the

filter design problem is solved only once and offline. After determining Sk(z) andGp,0(z) only once, a large set of SRC ratios can be supported.

6.3.5 Filter Design Criteria

To design the filters3, we can use both least-squares (LS) or minimax methodsor their combinations. However, the application mainly determines the designmethod. For example, one could be interested in minimizing (i) the ripples, or (ii)the energy of the filter frequency response in specific bands. In some applications,combinations of energy and ripples may also be useful. The sequel considers theminimax and LS approaches which are the most commonly used methods [75].

Minimax Design

The minimax filter design problem is

min∀Rp

δ subject to

|Fp(ejωT )− 1| ≤ δ, ωT ∈ [0, π]

|Gp(ejωT )|≤W (ωT )δ, ωT ∈ [ωsT, π], (6.18)

where δ and W (ωT ) are, respectively, the ripples and the weighting function with

ωsT =π(1 + ρ)

Rp. (6.19)

This chapter uses flat weighting functions reducing W (ωT ) to a constant W .The filter specifications are satisfied for every mode. Compared to conventionalTMUXs, this filter design is more complicated but solved only once and offline.

Figure 6.6 shows the filters resulting from (6.18) and Rp = 2, 3, . . . , 30. Thevalues of ρ, W , L, N1, and N0 are 0.2, 1, 5, 17, and 18, respectively. Furthermore,δ = 2.56×10−3. The characteristics of the filters with W = 0.2 and W = 5 are,respectively, shown in Figs. 6.7 and 6.8. Here, δ = 1.025×10−2 and δ = 9.79×10−4,respectively.

3To avoid having two sets of Sk(z), we assume that Gp(z) = Gp(z) but the TMUX discussion

is treated in a general case where Gp(z) may differ from Gp(z).

80


−3

−2

−1

0

1

2

3x 10

−3

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

Figure 6.6: Approximate Rp-th band filters designed with W = 1 in (6.18).

−0.03

−0.02

−0.01

0

0.01

0.02

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

Figure 6.7: Approximate Rp-th band filters designed with W = 0.2 in (6.18).

81


−3

−2

−1

0

1

2

3x 10

−3

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π


LS Design

One way to formulate the LS filter design problem is

min∀Rp

∫ π

0

|Fp(ejωT )− 1|2d(ωT ) + 1

W

∫ π

ωsT

|Gp(ejωT )|2d(ωT ). (6.20)

The designed LS filters are shown in Figs. 6.9–6.11. Here, the same parameters asthose in Figs. 6.6–6.8 have been used.

A Note on Filter Order

For the same ρ and passband/stopband ripples, we can compare N in (6.6) withthat estimated by the well-known formulae, e.g., Bellanger [88], Kaiser [89], asdiscussed in (2.47) and (2.48). This comparison shows that N in (6.6) is about20% − 30% larger than those estimated by [88, 89]. However, [88, 89] do notaccount for the power complementary property. This property requires an excessorder of about 20% − 30% [68] to those estimated by [88, 89]. Consequently, theproposed design method does not necessarily increase the arithmetic complexityas compared to regular filter design methods. Further, the filter order (and thenumber of delay elements) of the Farrow-based and the regular SRC structuresdiffer by a maximum of 10% [154].

82


−3

−2

−1

0

1

2

3x 10

−3

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π


6.4 Proposed Rational SRC Multimode TMUX

Section 6.3.1 revealed that the Farrow structure can implement variable integerSRC. Additional downsamplers/upsamplers allow one to construct a rational SRCmultimode TMUX. The TMUX, in Fig. 6.12, consists of upsamplers/downsamplersAp, Bp, p = 0, 1, . . . , P − 1; lowpass filters Gp(z) and Gp(z); and adjustable fre-quency shifters ωp and ωp. If Tp is the sampling period in branch p, then

T0B0

A0= T1

B1

A1= . . . = Ty, (6.21)

where Ty is the sampling period of y(n). The proposed redundant TMUX assumes∑P−1

p=0Bp

Ap< 1 whereas

∑P−1p=0

Bp

Ap= 1 gives a critically sampled TMUX.

6.4.1 TMUX Illustration

To illustrate the TMUX, we can study one of its branches. Redundancy allows us toadd signals from other branches and to construct a composite signal. Figure 6.13plots the outputs of different blocks with a uniformly distributed random inputwhere Rp = 19

9 . In Fig. 6.13(e), a lowpass filter can remove the images allowingone to retrieve the desired symbols by a downsampler.

83


−3

−2

−1

0

1

2

3x 10

−3

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

Figure 6.10: Approximate Rp-th band filters designed with W = 0.2 in (6.20).

6.4.2 Efficient Variable Rational SRC

With conventional upsamplers, downsamplers, and filters, SRC by a ratioAp

Bpre-

quires [12, 69]

• Downsampling by Bp after interpolation by Ap.

• Upsampling by Bp before decimation by Ap.

For a fixed Ap, changing the values of Bp gives a number of rational SRC ra-tios. According to Section 6.3.1, integer SRC by Ap needs a set of fixed filters.

Thus, any rational ratioAp

Bpcan be handled using fixed filters and different upsam-

plers/downsamplers Bp. To construct structures for variable rational SRC, we canreplace SRC by Ap in Fig. 6.12, with its equivalent structure from Figs. 6.4 and6.5.

In Fig. 6.14, some outputs of the Farrow-based interpolation are not needed.The commutator retains every Bp-th sample. In Fig. 6.15, some inputs of theFarrow-based decimation are zero. This saves the arithmetic complexity as thesesamples need not be processed. This is illustrated in Fig. 6.16 for a group of inputsamples x0, x1, . . . , xAp−1. For other groups, e.g., xAp

, xAp+1, . . . , x2Ap−1, thelocation of these zero-valued samples changes but the savings in the arithmetic

84


−3

−2

−1

0

1

2

3x 10

−3

ωT [rad]

Fp(e

jωT)−

1

0 0.2π 0.4π 0.6π 0.8π π

−60

−40

−20

0

ωT [rad]

Gp(e

jωT)

[dB

]

0 0.2π 0.4π 0.6π 0.8π π


G0(z)

G1(z)

GP-1

(z)

x0(n0)

x1(n1)

xP-1

(nP-1)

ejw

0n

ejw

1n

ejw

P-1n

y(n) y(n)^

G0(z) x

0(n0)^

x1(n1)^

xP-1

(nP-1)^

ê

-jw0n

ê

-jw1n

G1(z)

GP-1

(z)

ê

-jwP-1

n


^

^

^ AP-1

A1

A0A0

A1

AP-1

B0

B1

BP-1

B0

B1

BP-1

Figure 6.12: Rational SRC multimode TMUX composed of variable rational SRCsand adjustable frequency shifters. The actual realization of rational SRC with aratio Rp is performed by the structures in Figs. 6.14 and 6.15. The SRC structuresin this figure are only used for analysis purposes.

85


−40−20

020

ωT1 [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(a)

−40−20

020

ωT2 [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(b)

−40−20

020

ωT2 [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(c)

−40−20

020

ωT2 [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(d)

−40−20

020

ωT3 [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(e)

Figure 6.13: Output spectrum. (a) Interpolation by Ap = 19. (b) Downsamplingby Bp = 9. (c) and (d) Frequency shifters ωp = ωp = 0.5684π. (e) Upsampling byBp = 9. This illustration uses the filters in Fig. 6.6 where T1, T2, and T3 representdifferent sampling periods at the input/output of the SRC blocks.

Sk(z)

Gp,0(z)

mp,m

Rp fs

y(m)

01

Ap-1k = 0, 1, ..., L

kfs

x(n) Bp:1

Fixed Variable

Figure 6.14: Efficient interpolation by a variable rational ratio Rp =Ap

Bpusing the

Farrow structure.

86


Sk(z)

Gp,0(z)

mp,mRp fs

y(m)01

Ap-1k = 0, 1, ..., L

kBpx(n)

fs

Variable Fixed

Figure 6.15: Efficient decimation by a variable rational ratio Rp =Ap

Bpusing the

Farrow structure.

Sk(z)

Gp,0(z)

mp,mRp fs

y(m)0

Ap-1 k = 0, 1, ..., L

kBp-1 zeros

Bp-1 zerosBp+1

2Bp+1

Figure 6.16: Efficient decimation by Rp =Ap

Bpusing the Farrow structure and by

incorporating the effect of the upsampling by Bp into Fig. 6.15.

complexity are preserved. Using (6.3), (6.7), and (6.8) with Rp = Ap, we have

Gp(z) = Gp,0(zAp) +

Ap−1∑

m=1

z−mL∑

k=0

Sk(zAp)(

−mAp

+1

2)k. (6.22)

If some inputs of the Ap branches are zero, we can discard their correspond-ing polyphase components in (6.22). Thus, only a subset of the values in m =1, 2, . . . , Ap − 1, will be used.

6.4.3 Approximation of PR

Similar to (6.14), a desired Xd(z) in Fig. 6.12 can be written as (see Appendix A)

Xd(z) =

P−1∑

p=0

Ad−1∑

m=0

Bp−1∑

k=0

Xp(zApBdAdBpW

mApBdBp

Ad(zdzp

)ApBpW

kAp

Bp)×

Gp(zBd

AdBpWmBdBp

Ad(zdzp

)1

BpW kBp

)Gd(z1

AdWmAd

). (6.23)

Note that (6.14) is a special case of (6.23) in which Ap = Rp and Bp = 1. Similar

to (6.17) and with p = d, (6.23) gives Xp(z) = Xp(z)Fp(z) where

Fp(z) =

Ap−1∑

m=0

Bp−1∑

k=0

Gp(z1

ApWmApW k

Bp)Gp(z

1ApWm

Ap). (6.24)

87


Like Section 6.3.2, the choice of zdzp

allows one to control the ISI and ICI. For

SRC byAp

Bp, we can design the integer SRC by Ap and, then, perform the integer

upsampling/downsampling by Bp. In this case, (4.17) becomes [69]

ωsT =π(1 + ρ)

max(Ap, Bp). (6.25)

With fixed Ap in Fig. 6.12, aliasing is avoided if

Ap

1 + ρ≥Bp≥2. (6.26)

Equations (6.25) and (6.26) give

ωsT =π(1 + ρ)

Ap. (6.27)

Figure 6.17 shows the magnitude of Xd(z) for 12 multimode setups consisting ofP = 2, 3, 4, 5, 6 users. Here, xp(n) = δ(n) and each multimode setup has anumber of SRC ratios as in Table 6.1. Although the filters are designed for sets ofAp, the choice of zd

zpenables one to control the ISI and ICI. We can directly use

(6.23) in the filter design but this approach is not favorable as it complicates thedesign. Instead, we can control ωp by (6.16) and use the simpler design problemsof (6.18) and (6.20).

The proposed TMUX can also be analyzed using the method in [19]. Thiswas earlier performed in Chapter 5. The blocked transfer function of Fig. 6.12 isT (z) = Φ(z)Ψ(z) where

Φ(z) =[

φ0(z) φ1(z) . . . φP−1(z)]T, (6.28)

Ψ(z) =[

ψ0(z) ψ1(z) . . . ψP−1(z)]

. (6.29)

and

φp(z) = Gp(z), (6.30)

ψp(z) = Gp(zzdzp

). (6.31)

6.5 TMUX Performance

To illustrate the TMUX performance, the error vector magnitude (EVM) [139, 140]and the multimode setups of Table 6.1 with 16-quadrature amplitude modulation(QAM) data are considered. Each multimode setup corresponds to one specific timeframe as in Section 6.2.1. According to Fig. 6.18, the TMUX provides bandwidth-on-demand and the whole frequency spectrum is shared by any number of users4.Figure 6.19 shows the average EVM for these multimode setups.

88


−5

0

5

x 10−3

ωT [rad]

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(a) δISI

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

0

0.005

0.01

ωT [rad]

Mag

.

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(b) δICI

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

Figure 6.17: The ISI and ICI in Fig. 6.12 with xp(n) = δ(n) for the multimodesetups of Table 6.1 and the filters used in Fig. 6.6.

Table 6.1: SRC ratios for the multimode setups of Fig. 6.19. As an example, forthe first setup, R0 = 29

10 and R1 = 2311 .

Setup A B P1 [29, 23] [10, 11] 22 [15, 27, 23, 6] [4, 5, 4, 1] 43 [29, 17, 27, 19] [4, 4, 8, 3] 44 [20, 19, 17, 9, 19, 17] [3, 2, 3, 1, 2, 3] 65 [25, 27, 29] [7, 8, 7] 36 [13, 24, 13] [2, 7, 5] 37 [10, 9, 7] [1, 2, 3] 38 [28, 23] [11, 10] 29 [9, 4, 3] [2, 1, 1] 310 [11, 7, 14] [2, 3, 3] 311 [29, 23, 9] [5, 10, 2] 312 [30, 21, 17, 28, 18] [1, 2, 3, 5, 5] 5

89


−40

−20

0

20

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−40

−20

0

20

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−40

−20

0

20

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−40

−20

0

20

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

Figure 6.18: Transmitted signal for the first four multimode setups of Table 6.1.

The LS filters have a smaller EVM and the filters designed with W = 1 aresuperior to those with W = 5 and W = 0.2. Thus, the EVM is determined by the(i) stopband of Gp(z), and (ii) passband of Fp(z). Although the LS approach issuperior according to EVM, some systems may constrain the ripples. Then, themore appropriate option would be to use the minimax or the constrained LS (CLS)approaches.

Irrespective of the design technique, this TMUX has an indirect filter design.In other words, the filters are designed only for sets of Ap. Consequently, theconstraints in the filter design are only satisfied for the values of Ap. Then, rational

SRC by a ratioAp

Bpis realized by choosing the sets of Bp as in (6.26). However,

as Fig. 6.19 shows, the TMUX can satisfy any desired ICI and ISI by a properfilter design. Without additional constraints, due to Bp, the designed filters aresuboptimal and they have some overdesign as we shall see in the next subsection.

6.5.1 Effects of Bp on the SRC Error

A comparison between (6.17) and (6.24) reveals that downsampling by Bp resultsin a sum of Bp terms. Thus, a larger Bp would result in a larger error. However,

ensuring (6.26) allows one to attenuate these terms by the stopband of Gp(z).

4For illustration purposes, the values of Ap and Bp are chosen so that y(n), in Fig. 6.12,occupies between 90− 99% of the frequency range [0, 2π].

90


1 2 3 4 5 6 7 8 9 10 11 12

−50

−48

−46

−44

−42

−40

Multimode setup

Aver

age

EV

M [

dB

]

LS, W=1

LS, W=5

LS, W=0.2

Minimax, W=1

Minimax, W=0.2

Minimax, W=5

Figure 6.19: Average EVM of 16-QAM signals in multimode setups of Table 6.1for the TMUX in Fig. 6.12 and with the filters in Figs. 6.6–6.11.

To illustrate this, four values of Ap are chosen randomly and the values of Bp

are determined using (6.26). This gives a set of Rp =Ap

Bp. Further, a cascade of

interpolation by Rp and decimation by Rp, with the filters in Fig. 6.6, is performed.Figure 6.20(a) shows the resulting EVM. As can be seen, a larger Bp increases theEVM. Figure 6.20(b) shows the EVM for all 232 possible unique values of Rp

achieved by Ap = 2, 3, . . . , 30 and (6.26) with ρ = 0.2. There is an upper bound

on the EVM mainly determined by the stopband of Gp(z). Thus, the stopbandattenuation can be increased to compensate for the additional error due to Bp.

As Fig. 6.20(b) shows, the EVM varies in a range of about 10 dB. Hence,increasing the stopband attenuation, by 10 dB, would decrease the highest EVMto a desired level. According to [88], increasing the stopband attenuation by 10dB, would increase the order of linear-phase FIR filters by about 10%. On theother hand, direct optimization would require an increase of 232

29 = 800% in theoptimization complexity. This shows that it may indeed be preferable to have aslight overdesign and avoid direct optimization. However, direct optimization canbe performed for limited sets of Ap and Bp as we shall see in the sequel.

91


2 4 6 8 10 12 14 16 18 20 22

−56

−54

−52

−50

−48

Bp

EV

M [

dB

]

(a)

Ap = 6

Ap = 13

Ap = 18

Ap = 27

5 10 15 20 25 30

−56

−54

−52

−50

−48

−46

(b)

Rp

EV

M [

dB

]

Figure 6.20: Effects of Bp on the error for the cascade of interpolation by Rp =Ap

Bp

and decimation by Rp =Ap

Bp. (a) Increase of EVM with the increase in Bp for

a fixed Ap. (b) Upper bound of EVM for all possible values of Rp achieved byAp = 2, 3, . . . , 30 and (6.26) with ρ = 0.2.

hn(k) y(n)x(n)

Figure 6.21: Equivalent model for each branch in the TMUX of Fig. 6.12.

6.6 Direct Filter Design

As discussed before, one can design the TMUX filters for the sets of Ap. Specifically,the specifications are satisfied only for the sets of Ap. Then, one can choose proper

values of Bp so as to perform rational SRC by a ratio Rp =Ap

Bp. This indirect

design method results in (i) suboptimal filters, and (ii) overdesign. To overcomethese, we can include additional filter design constraints arising from the sets ofBp. Then, the specifications are satisfied for all sets of Ap and Bp. This makes thedesign direct.

In one branch of the TMUX, i.e., a cascade of interpolation by Ad

Bdand decima-

92


tion by Ad

Bd, the input-output relation is

Y (z) =

Ad−1∑

m=0

Bd−1∑

l=0

G(z1

AdWmAd

)G(z1

AdW lBdWm

Ad)X(zW lAd

Bd)

=

Bd−1∑

l=0

Tl(z)X(zW lAd

Bd), (6.32)

where

Tl(z) =

Ad−1∑

m=0

G(z1

AdWmAd

)G(z1

AdW lBdWm

Ad), l = 0, 1, . . . , Bd − 1. (6.33)

Here, (6.32) is obtained by setting Ap = Ad, Bp = Bd, and zp = zd in (6.23).Consequently, T0(z) reduces to (6.17). In the time-variant dual rate system [92]represented by (6.32), the output signal is the result of operating (see Footnote 7in Chapter 3)

• T0(z) on X(z).

• Tl(z), l = 1, . . . , Bd − 1, on the frequency shifted (by 2lπAd

Bd) versions of the

input, i.e., X(zW lAd

Bd).

Ideally,

T0(z) = 1

Tl(z) = 0, l = 1, . . . , Bd − 1. (6.34)

Then,∑Bd−1

l=0 Tl(z) = 1. As shown in Fig. 6.21, the system can also be modeled asthe operation of a time-varying periodic filter hn(k) on the input signal as [155]

y(n) =∑

k

x(k)∑

m

g(mBd −Adk)g(nAd −mBd)

=∑

k

x(n− k)∑

m

g(mBd − nAd + kAd)g(nAd −mBd),

=∑

k

x(n− k)hn(k). (6.35)

whereHn(z) =

∑

k

hn(k)z−k. (6.36)

Thus, we have Bd impulse responses (see Appendix B)

hn(k) =∑

m

g(mBd − nAd + kAd)g(nAd −mBd), n = 0, 1, . . . , Bd − 1. (6.37)

93


One can determine hn(k) by feeding δ(n − m), m = 0, 1, . . . , to the system andcomputing the corresponding outputs ym(n). The impulse responses hn(k) arethen given as hn(n−m) = ym(n).

In a PR system, the output signal is a delayed version of the input signal.Ignoring the delay and defining the error e(n) = y(n)− x(n), we have

e(n) =1

2π

∫ π

−π

(Hn(ejωT )− 1)X(ejωT )ejnωT d(ωT ). (6.38)

Ideally, e(n) = 0 and, then, Hn(ejωT ) = 1. In practice, Hn(e

jωT ) can generallyonly approximate unity in the frequency range of interest. Consequently, the filterdesign problem should determine hn(k) so that e(n) is minimized according tosome criterion. As e(n) depends on both Hn(e

jωT ) and X(ejωT ), one generallyrequires knowledge about the spectrum of the input signal to determine optimumfilters. In practice, there is no complete knowledge about X(ejωT ) and one acceptsa suboptimum solution instead. In this regard, it can be convenient to make useof the Lp-norm of a general signal S(ejωT ) given by [69]

‖S(ejωT )‖p = p

√

1

2π

∫ π

−π

|S(ejωT )|pd(ωT ). (6.39)

Using the triangle inequality of integrals in (6.38), we have

|e(n)|≤ 1

2π

∫ π

−π

|Hn(ejωT )− 1||X(ejωT )|d(ωT ) (6.40)

which in terms of L∞-norm gives

|e(n)|≤‖Hn(ejωT )− 1‖∞‖X(ejωT )‖∞. (6.41)

As‖S(ejωT )− 1‖∞ = max

ωT|S(ejωT )− 1|, (6.42)

minimizing the maximum of |Hn(ejωT ) − 1| corresponds to minimizing the max-

imum of e(n) resulting from that particular hn(k). From earlier discussions, theindirect minimax filter design problem can be rewritten as

Case I: min∀Ad

δ subject to

|Fd(ejωT )− 1| ≤ δ, ωT ∈ [0, π]

|Gd(ejωT )|≤δs, ωT ∈ [ωsT, π]. (6.43)

Here, δs is the desired stopband attenuation and ωsT is given by (6.27). Further,Fd(z) is given by (6.17). The indirect CLS filter design problem is

Case II : min∀Ad

δ subject to

94


Table 6.2: The SRC ratios for the multimode setups used in Fig. 6.22.

Setup Ap Bp P1 [15, 18, 13] [4, 7, 2] 32 [17, 19, 9] [4, 7, 2] 33 [20, 19, 13, 17] [3, 3, 4, 3] 44 [10, 3, 5] [1, 1, 2] 35 [9, 4, 3] [2, 1, 1] 36 [11, 7, 14] [2, 3, 3] 3

∫ π

0

|Fd(ejωT )− 1|2d(ωT ) ≤ δ, ωT ∈ [0, π]

|Gd(ejωT )|≤δs, ωT ∈ [ωsT, π]. (6.44)

One can express the direct minimax filter design problem as

Case III : min∀Ad,Bd

δ subject to

1

Bd

Bd−1∑

n=0

|Hn(ejωT )− 1| ≤ δ, ωT ∈ [0, π]

|Gd(ejωT )|≤δs, ωT ∈ [ωsT, π] (6.45)

where ωsT is given by (6.25). Similarly, (6.40) can also be written in terms of theL2-norm leading to the direct CLS filter design problem as

Case IV : min∀Ad,Bd

δ subject to

1

Bd

Bd−1∑

n=0

∫ π

0

|Hn(ejωT )− 1|2d(ωT ) ≤ δ, ωT ∈ [0, π]

|Gd(ejωT )|≤δs, ωT ∈ [ωsT, π]. (6.46)

6.6.1 Design Example

This section considers some design examples for the direct and indirect filter de-signs. Here, the values of L, N1, and δs are 5, 17, and 0.01 respectively. Further-more, (6.43), (6.44), (6.45), and (6.46) result, respectively, in the values of δ tobe 6.21×10−4, 1.28×10−7, 2.70×10−4, and 2.49×10−8. Note that in all of thesefilter design problems, (6.26) holds and their difference lies in the fact whether theyinclude Bd in the optimization or not.

Figure 6.22 shows the average EVM for 16-QAM signals according to the mul-timode setups of Table 6.2. Generally, the CLS approach results in a smaller EVM

95


−48

−46

−44

−42

−40

−38

Multimode setup

Aver

age

EV

M [

dB

]

1 2 3 4 5 6

Case I

Case II

Case III

Case IV

Figure 6.22: The EVM values for the multimode setups in Table 6.2.

than the minimax method. Furthermore, the direct filter design in Case III (IV)reduces the EVM compared to the indirect filter design of Case I (II). Comparedto the direct minimax filter design, the direct CLS method brings a larger improve-ment in system performance.

With δs = 0.01, the EVM values hover around −40dB. In other words, theICI is controlled by δs and the four filter design problems decrease the ISI eitherdirectly or indirectly. The overall EVM is determined by both ISI and ICI. Thevalues of δs and δ are correlated. For the same filter orders, decreasing δs wouldincrease δ. For example, with the filter orders used in Fig. 6.22, having δs = 0.001results in about 4dB larger EVM than that depicted in Fig. 6.22. In this case,the EVM is mainly determined by δ and the difference between these filter designproblems is less pronounced. A desired EVM can be achieved by choosing propervalues of L and N1.

Although the direct filter design has a smaller EVM, it requires to solve amore complex design problem. This increased design complexity is a result ofthe increased number of constraints and is proportional to the number of setsAp, Bp. Consequently, the convergence time will increase and issues regardingmemory problems may also arise. However, the memory problems can partially bealleviated by careful choice of the number of grid points for ωT , etc. As this TMUXrequires offline filter design, the memory problems are generally more importantthan the convergence time.

96


6.7 Conclusion

A class of multimode TMUXs was introduced in which the Farrow structure real-izes general interpolation/decimation filters. These TMUXs support variable SRCratios using fixed filters and variable multipliers. Efficient realization structuresare derived and different filter design techniques are compared.

This TMUX does not need online filter design. This comes at the expense ofa more complicated filter design problem but it suffices to solve it only once andoffline. Then, we need to simply adjust some multipliers online. In terms of EVM,the LS approach is better than the minimax method but some applications maynecessitate the use of the minimax method.

Different filter design formulations result in different performances. However,the direct filter design has a better control over the TMUX noise but it has a largerdesign complexity.

97


98

7

Reconfigurable NonuniformTransmultiplexers Using

Uniform Modulated Filter Banks

This chapter introduces reconfigurable nonuniform transmultiplexers (TMUXs)based on uniform modulated filter banks (FBs). After a brief introduction inSection 7.1, Section 7.2 outlines the problem formulation. The proposed TMUXis introduced in Section 7.3 where its filter design, realization, and reconstructionerror are discussed. Section 7.4 discusses the system parameters and topics such aschannel sampling periods, guardbands (GBs), and center frequencies. Section 7.5treats the implementation cost and the parameter selection whereas Section 7.6compares the proposed TMUX to the existing multimode TMUXs. Some conclud-ing remarks are given in Section 7.7.

7.1 Introduction

A TMUX enables one to transmit different data streams through a single chan-nel [12]. As discussed in Section 3.4, TMUXs consist of a synthesis FB (SFB)followed by an analysis FB (AFB) which are a parallel connection of a number ofbranches. Each branch is realized by digital interpolators/decimators. The SFB fil-ters cover different regions of the frequency spectrum thereby packing independentdata streams into adjacent frequency bands. This leads to uniform or nonuniformTMUXs where the passbands of the SFB filters determine the frequency bands.

Nonuniform TMUXs need different interpolators and decimators necessitating

99

7. RECONFIGURABLE NONUNIFORM TRANSMULTIPLEXERS USINGUNIFORM MODULATED FILTER BANKS

different bandpass filters. To freely determine the bandwidths of each data stream,we should either (i) design a large set of filters, or (ii) design the filters online.In other words, if the frequency division multiplexed (FDM) scenario (multimodesetup) changes, new sets of filters are required. This becomes involved if the FDMscenarios change in a time-varying manner. We consequently need TMUXs which,in a time-varying manner, cover different FDM scenarios.

The duality of FBs and TMUXs allows us to use the conventional filter designmethods in the case of fixed TMUXs. As the characteristics of multimode TMUXschange in a time-varying manner, they do not generally allow one to use theseconventional filter design techniques. Consequently, multimode TMUXs requirespecial filter design techniques [46, 63–66]. However, it is desired to obtain mul-timode TMUXs based on conventional FBs so as to use the existing design andrealization techniques.


Chapters 5 and 6 used the Farrow structure [93] to obtain redundant near perfectreconstruction (NPR) multimode TMUXs. The filters are designed only once andoffline. Then, the TMUXs are reconfigured by adjusting (i) the fractional delay ofthe Farrow structure, and (ii) some frequency shifters.

Nonuniform FBs and TMUXs can also be obtained by combining some of theAFB and SFB filters and assigning1 them to each user. Although this gives anonuniform system, it may result in infeasible cases [20, 22, 24, 26, 27]. Otheralternatives are, to name a few, nonuniform modulated FBs [23]; tree structuredFBs [25]; semi-infinite programming [16, 31]; dividing nonuniform FBs into uniformFBs [28]; FBs with complex filters [29]; and the recombination structures [17, 24,26]. These may have less problems associated with infeasible cases but they have alow degree of reconfigurability. They obtain nonuniform filters through dedicatedadders, modulators, or structures. To change the AFB and SFB filters, theseadders, modulators, or structures must change.

The cosine modulated FBs (CMFBs) and modified discrete Fourier transformFBs (MDFT FBs) are well-studied. They are generally uniform and we can obtainuniform TMUXs using the duality of TMUXs and FBs [116]. This chapter outlinesreconfigurable nonuniform TMUXs based on fixed uniform modulated FBs. TheTMUXs use parallel (polyphase) processing and any user is processed by a numberof TMUX branches. One branch represents one granularity band (GRB) and anyuser occupies integer multiples of the GRB. This only requires some adjustablecommutators with no additional arithmetic complexity. Consequently, the band-width of each user can change by simple adjustments of these commutators. Theproposed TMUXs can be either critically sampled or redundant.

With the proposed TMUXs, it suffices to design the filters only once offline. Asthe TMUXs use standard modulated FBs, e.g., CMFBs and MDFT FBs, we canobtain perfect reconstruction (PR) or NPR TMUXs. This chapter shows how

1This is also referred to as the split-and-add method [20].

100


• To use fixed uniform modulated FBs to obtain reconfigurable nonuniformTMUXs.

• To determine the system parameters so as to minimize the arithmetic com-plexity.

In the proposed TMUXs, each polyphase component is transmitted by a feasi-ble uniform TMUX. Thus, there are no infeasible cases. The data rates for allpolyphase components are equal. As a number of these components belong to onespecific user, different users will have different data rates.

Note that [21] uses the concept in [22] to convert TMUXs with a rational sam-pling rate conversion (SRC) into TMUXs with an integer SRC. Then, similar tothe proposed TMUX, polyphase components of input signals are processed in dif-ferent branches. However, each branch of [21] has a specific SRC factor therebydecreasing the realization regularity.

7.2 Problem Formulation

As in Fig. 1.1, we assume that P users share the whole frequency spectrum. Eachuser occupies a time-varying portion of the frequency spectrum. By adjusting thisportion at any point, any user can have any center frequency. Here, ∆ determinesthe amount of overlap or GB. Any scenario can be obtained by an appropriate ∆.Generally, we can have

• Case I with ∆ < 0 in which case different user spectra overlap and we do nothave a GB.

• Case II with minimal GB where ∆ = 0 and we have neither overlap nor extraGB.

• Case III with ∆ > 0 resulting in an extra GB.

Depending on ∆, we can have critically sampled (Case I) or redundant (Case II andIII) TMUXs. For example, the European space agency has outlined three networktopologies for broadband communication systems using satellites [37]. In thesesatellites, users are multiplexed between (and within) different satellite beams [39]which necessitates ∆ > 0 [43, 44]. Without intermediate processing, e.g., multi-plexing as discussed above, one may choose Case I so as to increase the spectrumefficiency. For practical reasons, one may anyhow allow ∆ > 0 relaxing the sub-sequent signal processing, e.g., band limitation or digital to analog conversion.Redundant TMUXs may be necessary when the channel distortion is significant insome bands [20].

101


X0,0

(z)

Analysis FBSynthesis FB

F0(z)M

X0,1

(z) F1(z)M

X0,M

0-1

(z) FM

0-1

(z)M

X1,0

(z) FM

0+1

(z)M

X1,1

(z) FM

0+2

(z)M

X1,M

1-1

(z) FM

0+M

1

(z)M

XP-1,0

(z) FM-M

P

(z)M

XP-1,1

(z) FM-M

P+1

(z)M

XP-1,M

P-1

(z) FM-1

(z)M

å

H0(z)

H1(z)

HM

0-1

(z)

HM

0+1

(z)

HM

0+2

(z)

HM

0+M

1

(z)

HM-M

P

(z)

HM-M

P+1

(z)

HM-1

(z)

M

M

M

M

M

M

M

M

M

y(n) y(n)^

X0,0

(z)

X0,1

(z)

X0,M

0-1

(z)

X1,0

(z)

X1,1

(z)

X1,M

1-1

(z)

XP-1,0

(z)

XP-1,1

(z)

XP-1,M

P-1

(z)

^

^

^

^

^

^

^

^

^

Figure 7.1: Proposed nonuniform TMUX with M channels and P≤M users.

7.3 Nonuniform TMUXs Using Modulated FBs

We can treat polyphase components of different users as independent data streamsand transmit them in any TMUX branch2. Consequently, different users havedifferent bandwidths and a nonuniform TMUX is obtained.

Assume the structure in Fig. 7.1 and

Xp(z) =

Mp−1∑

l=0

z−lXp,l(zMp), p = 0, 1, . . . , P − 1. (7.1)

2This is similar to multicarrier modulation applied to multiple users.

102


Then,

Y (z) =

P−1∑

p=0

Mp−1∑

l=0

Xp,l(zM )Fαp,l

(z). (7.2)

The integer Mp is the number of branches that the user Xp(z) occupies. It corre-sponds to the bandwidth assigned to Xp(z).

To incorporate ∆, we should use the SFB and AFB filters with proper centerfrequencies. One can allow a GB by not using some TMUX branches. To exemplify,assume M0 = 1 and M1 = 2. Then, X0(z) uses one branch corresponding to F0(z)while X1(z) uses two branches corresponding to F2(z) and F3(z). In this way, onebranch, corresponding to F1(z), acts as a GB. Thus, the index to the proper filteris

αp,l =

∑p−1m=0Mm + l + pDg if p 6= 0

l if p = 0,(7.3)

where l = 0, 1, . . . ,Mp−1. Furthermore, Dg is the (generally time-varying) numberof branches acting as GBs. In Fig. 7.1, we assume Dg = 1. In the AFB,

Xp,l(z) =

M−1∑

m=0

Y (z1/MWmM )Hαp,l

(z1/MWmM ). (7.4)

Setting y(n) = y(n) and after some manipulations, (7.4) gives

Xp,l(z) =P−1∑

p=0

Mp−1∑

l=0

Xp,l(z)M−1∑

m=0

Fαp,l(z1/MWm

M )Hαp,l(z1/MWm

M ). (7.5)

With p = l = d in (7.5), the inter-symbol interference (ISI) can be represented as

Xd,d(z) = Xd,d(z)

M−1∑

m=0

Fαd,d(z1/MWm

M )Hαd,d(z1/MWm

M ). (7.6)

As 0≤αp,l≤M − 1, (7.6) can be simplified and some manipulations allow one toexpress the ISI similar to (3.40). To compute the inter-carrier interference (ICI),we have

Xpd,ld(z) =P−1∑

p=0,p 6=pd

Mp−1∑

l=0,l 6=ld

Xp,l(z)M−1∑

m=0

Fαp,l(z1/MWm

M )Hαpd,ld(z1/MWm

M ) (7.7)

which can be simplified so that the ICI is computed similar to (3.41). If an NPRTMUX has a maximum mean square error (MSE) of ǫmax in each branch, the MSEfor Xp,l(z) is

ǫp,l =1

Np,l

Np,l−1∑

n=0

[xp,l(n)− xp,l(n)]2, Np,l =

Np

Mp(7.8)

103


Xp(z)

Xp,0

(z)

Xp,1

(z)

Xp,Mp-1

(z)

Xp(z)

Figure 7.2: Time-varying commutators in the proposed TMUX whereMp can varywith time.

where ǫp,l≤ǫmax and Np is the length of Xp(z). Here, the total MSE for Xp(z)depends on that of its polyphase components. For Xp(z), the MSE is (ǫmax = 0for a PR TMUX)

ǫp =1

Np

Np−1∑

n=0

[xp(n)− xp(n)]2 (7.9)

=1

MpNp,l

Mp−1∑

l=0

Np,l−1∑

n=0

[xp,l(n)− xp,l(n)]2

=1

Mp

Mp−1∑

l=0

ǫp,l≤1

MpMpǫmax = ǫmax.

Thus, independent of Mp, the user Xp(z) has a maximum MSE of ǫmax. In otherwords, even if the polyphase components of each user are processed inMp branches,the overall reconstruction error for that specific user is only determined by the ap-proximation error of the corresponding FB and for the case withMp = 1. Therefore,we need to design the prototype filter for the corresponding FB. Then, a recon-figurable nonuniform TMUX is obtained by (i) interchanging the AFB and SFB,and (ii) adding some time-varying commutators. Figure 7.2 shows the architectureof these time-varying commutators. These adjustable commutators do not requireextra arithmetic complexity but they do give some control hardware overhead, i.e.,multiplexing of the polyphase components. Specifically, (7.3) describes this con-trol mechanism but it does not actually require any arithmetic complexity. Such acontrol mechanism is generally common for any polyphase realization.

7.4 System Parameters

The TMUX requires to design the prototype filter, say G(z), of the correspond-ing FB as discussed in Section 3.3.1. This design requires the passband andstopband ripples δc and δs; roll-off ρ; and the number of TMUX channels M .These are the basic system parameters but the application can also constrainthem. One such constraint arises from the multimode setups represented by Mp =M0,M1, . . . ,MP−1. With these parameters, some characteristics such as thechannel sampling periods Tp, center frequencies Ωp, and the GB ∆ are determined.

104


7.4.1 Channel Sampling Periods

The proposed TMUX assigns Mp branches to the user Xp(z) which has a sampling

period of Tp. Then, TpMp

M = Ty where Ty is the sampling period of y(n) in Fig. 7.1.

Note that with∑P−1

p=0Mp

M < 1 we have a redundant TMUX whereas∑P−1

p=0Mp

M = 1gives a critically sampled TMUX.

7.4.2 TMUX Illustration

Figure 7.3 shows the frequency spectrum of y(n) for three cases where P = 2, 3, 4.These figures are obtained by an MDFT-based TMUX with M = 20. We onlyadjust the commutators to choose the correct polyphase component. The corre-sponding AFB filters are shown in Fig. 7.5(a). For these filters, ρ = 0.2 and onlythe adjacent filters overlap. Further, Dg = 1 and we have offset αp,l (by 1) so that3

the user X0 has all its spectrum in ωT≥0. In Fig. 7.3(a), Mp = 11, 7 whereasin Figs. 7.3(b) and (c), we have Mp = 9, 5, 3 and Mp = 7, 3, 5, 1, respectively.We can define different multimode setups by the vectors Mp. A multimode setupis the configuration of a number of users occupying the whole frequency spectrum.If Mp changes, the multimode setup changes also but the filter coefficients neednot change.

Figure 7.4 illustrates a CMFB-based TMUX with the AFB filters of Fig. 7.5(b).As ρ = 0.5, the adjacent filters overlap. Furthermore, Dg = 1 and we have alsooffset αp,l (by 1). In Fig. 7.4(a), Mp = 10, 7 whereas in Figs. 7.4(b) and (c), wehave Mp = 8, 5, 3 and Mp = 7, 3, 4, 1, respectively.

7.4.3 Choice of GB

As the proposed TMUX uses maximally decimated FBs, the user spectra can over-lap as in Case I. In the TMUX illustration, both of the Cases II and III are coveredto show that the proposed TMUX can be reconfigured for any ∆. The GBs affectthe spectrum efficiency but not the reconstruction error or the filter design.

A specific ρ allows Case II with a minimal GB. With Dg = 1, the MDFT-basedTMUX achieves Case II if ωsT = 2π

M whereas the CMFB-based TMUX allows aminimal GB if ωsT = π

M . This can be seen in Figs. 7.3 and 7.4. Figure 7.3 has noextra GB as for this specific4 MDFT FB, ωsT = 2π

M . In contrast, Fig. 7.4 followsCase III as ρ = 0.5 gives ωsT < 2π

M .

7.4.4 Choice of Center Frequency

Some applications, e.g., cognitive radios [44, 59–62, 156–158], may require time-varying center frequencies. The proposed TMUX allows this by simple adjustmentsof the commutators. Assume, for example, Mp = 2, 3, 4, Dg = 1, and an offset

3This is for illustration purposes only. The spectrum of the user X0, in Fig. 1.1, is partiallycentered on ωT = 0 and ωT = 2π.

4The design method for this FB is outlined in [76] where ωsT = 2πM

.

105


−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(a)

X0

X1

−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(b)

X0

X1

X2

−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

(c)

X0

X1

X2

X3

Figure 7.3: Illustration of the MDFT-based TMUX where 2, 3, and 4 users haveoccupied the frequency spectrum ωT∈[0, 2π]. Here, ∆ = 0.

(by 1) in αp,l. Then, X0(z) uses two branches corresponding to F1(z) and F2(z)where α0,0 = 1 and α0,1 = 2. On the other hand, X1(z) uses three branches F4(z)−F6(z). Further, X2(z) occupies four frequency bands covered by F8(z) − F11(z).In an M -channel MDFT FB, the center frequency for Fm(z), m = 0, 1, . . . ,M − 1,is 2πm

M . Therefore, the center frequency for X0(z) is3πM whereas X1(z) and X2(z)

have center frequencies 10πM and 19π

M , respectively. If Xp(z) occupies Mp branches,its center frequency Ωp is5

Ωp =2π

M

1

Mp + 1

Mp−1∑

l=0

αp,l. (7.10)

7.5 Implementation Cost

Any FB realization can be adopted for the proposed TMUX. The main aim is thusto determine the system parameters so as to minimize the arithmetic complexity.The TMUX may also require some considerations with respect to the number ofmultimode setups. This section treats the problem of determining the system pa-rameters. In Scenario I, we assume that all system parameters are adjustable. We

5For a CMFB, the term 2πM

must be modified.

106


−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

(a)

X0

X1

−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

(b)

X0

X1

X2

−40

−20

0

ωT [rad]

Ma

g.

[dB

]

0 0.2π 0.4π 0.6π 0.8π π

(c)

X0

X1

X2

X3

Figure 7.4: Illustration of the CMFB-based TMUX where 2, 3, and 4 users haveoccupied the frequency spectrum ωT∈[0, π]. Here, ∆ > 0.

outline a general optimization problem to minimize the overall per-sample arith-metic complexity. The Scenario II assumes a fixed transition band for G(z) and adifferent optimization problem is discussed. We consider the MDFT-based TMUXbut the discussion can be extended to CMFBs after appropriate modifications.

Scenario I

To process M complex-valued input samples, the M -channel MDFT FB requiresM(4K + log2M − 3) + 4 real multiplications and M(4K + 3 log2M − 1) − 4 realadditions [109]. Thus, the total per-sample arithmetic complexity is 8K+4 log2M−4. The order of the prototype filter G(z), in an MDFT FB, can be obtained as [88]

NMDFT ≈ KM (7.11)

≈ −2

3log10(10δcδs)

2π2πM − π(1−ρ)

M

≈ −2

3log10(10δcδs)

2M

1 + ρ.

Consequently, we define the total per-sample arithmetic complexity as

C(M,ρ, δc, δs) =−32 log10(10δcδs)

3(1 + ρ)+ 4 log2M − 4. (7.12)

107


−80

−60

−40

−20

0

ωT [rad]

Mag

. [d

B]

(a)

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

−50

−40

−30

−20

−10

0

ωT [rad]

Mag

. [d

B]

0 0.25π 0.5π 0.75π π

(b)

Figure 7.5: The AFB filters for the TMUXs illustrated in Figs. 7.3 and 7.4 whereM = 20. (a) MDFT FB with ρ = 0.2. (b) CMFB with ρ = 0.5.

As shown in Fig. 7.6, for a fixed ρ, a larger M increases C(M,ρ, δc, δs). Further,for a fixedM , decreasing ρ will increase C(M,ρ, δc, δs). Thus, we can determineMso that C(M,ρ, δc, δs) does not increase unnecessarily. Note also the discontinuityof the curves due to the integer values of K.

We can use other formulae, e.g., Kaiser [89], to obtain NMDFT . This slightlychanges (7.12) but not the overall conclusions. In addition, (7.11) does not accountfor the power complementary property of G(z). We found experimentally that ifNMDFT is estimated by (7.11) and G(z) is to satisfy the power complementaryproperty with an error of δc, we need to increase NMDFT by 20%− 30%.

Scenario II

If the user spectra overlap, we can fix the transition band of G(z). For example, wecan increaseM and allow the overlap of several user spectra. Then, (7.11) becomes

NMDFT = KM =Φ

Ψ=

− 4π3 log10(10δcδs)

Ψ(7.13)

which would give

K =Φ

MΨ. (7.14)

108


1020304050600

0.5

1

20

40

ρ

(a)

M

C(M

,ρ,δ

c,δs)

1020304050600

0.5

1

40

60

80

ρ

(b)

M

C(M

,ρ,δ

c,δs)

Figure 7.6: Arithmetic complexity of the M -channel MDFT FB using (7.12). (a)δs = δc = 0.01. (b) δs = δc = 0.0001.

Here, Ψ represents the transition band of G(z). With constant δc and δs, the valueof Φ becomes fixed. Then, Φ

Ψ can be adjusted and (7.12) becomes

C(M,ρ, δc, δs) =8Φ

MΨ+ 4 log2M − 4. (7.15)

To minimize C(M,ρ, δc, δs), we should have

∂C(M,ρ, δc, δs)

∂M=

4

M(−2Φ

MΨ+ log2 e) = 0 (7.16)

and the optimum number of channels is

Mopt =2Φ

Ψ log2 e. (7.17)

As shown in Fig. 7.7, C(M,ρ, δc, δs) indeed decreases by increasing M . Accordingto Section 3.3, K is selected to be an integer. With an integer K, the curves inFig. 7.7 will be sparser but the general trend will still be preserved. Then, Mopt

may not be valid but a valid (integer) value of Mopt can easily be obtained byadjusting Φ

Ψ .Figure 7.7 suggests to choose M to be larger than that required by the applica-

tion. This not only decreases the arithmetic complexity, it increases the flexibilityin terms of Tp, Ωp, and Mp. IfM is close to Mopt, the value of C(M,ρ, δc, δs) doesnot change significantly and we can also avoid a (possibly) large Mopt by choosing

109


20 40 60 80 100 120 140 160 180 200

30

40

50

60

70

80

90

M

C(M

,ρ,δ

c,δs)

Φ/Ψ=105

Φ/Ψ=70

Φ/Ψ=52

Φ/Ψ=42

Φ/Ψ=35

Figure 7.7: Arithmetic complexity of the M -channel MDFT FB using (7.15) andδs = δc = 0.001.

a moderate M < Mopt. Some systems may anyhow favor a large M from thechannel equalization point of view. Then, the channel is divided into very smallfrequency bands in which it has a flat frequency response [120, 159].

Although a large M is advantageous due to the minimization of C(M,ρ, δc, δs),this may not be useful from another point of view. In hardware realization, theimplementation complexity could be affected by the number of connections, etc. Inthe case of the proposed TMUX, these connections are directly proportional to M .Consequently, one should bear this in mind so as not to increase the complexity ofthese connections [92].

7.5.1 Choice of M and ρ

Consider different time instants with P (t) users. Assume also that each user occu-pies Mp(t), p = 0, 1, . . . , P (t) − 1, branches and Dg(t) branches are used as GBs.Thus,

(P (t)− 1)Dg(t) +

P (t)−1∑

p=0

Mp(t) ≤M (7.18)

meaning that Mp(t), P (t), and Dg(t) determine M . Then, an appropriate ρ canbe chosen to meet the criterion on C(M,ρ, δc, δs). With fixed ripples δc and δs, we

110


0 0.2 0.4 0.6 0.8 1

35

40

45

50

55

60

65

ρ

C(M

,ρ,δ

c,δs)

Figure 7.8: Trend of C(M,ρ, δc, δs), in (7.12), for different values of ρ where P = 3,M = 10, and δc = δs = 0.001.

can solve an optimization problem as

minM,ρ,t

C(M,ρ, δc, δs) (7.19)

subject to (P (t)− 1)Dg(t) +

P (t)−1∑

p=0

Mp(t) ≤M.

With a maximum value for P (t), the flexibility can be defined. To exemplify, as-sume P = 3, M = 10, and Dg = 1. Thus, we can obtain all possible combinations

ofMp so that (7.18) holds. In other words,∑2

p=0Mp≤8 gives the number of multi-mode setups defined by the vectors Mp. Here, between one and three users occupythe whole frequency spectrum. This gives 41 multimode setups. Letting Dg = 0

results in∑2

p=0Mp≤10 and it allows 67 multimode setups. Here, Mp = 2, 3, 5has been considered equivalent to Mp = 3, 2, 5 or Mp = 3, 5, 2. However,these three multimode setups assign different center frequencies Ωp to each user.

Figure 7.8 shows the trend of C(M,ρ, δc, δs) in an MDFT-based TMUX withP = 3, M = 10, and δc = δs = 0.001. As can be seen, we can choose a ρ to satisfythe constraints on C(M,ρ, δc, δs). The discussion above considers Scenario I. WithScenario II, ρ is fixed giving a rather different problem. However, both problemsminimize C(M,ρ, δc, δs) subject to the constraints in (7.18).

Even though the proposed TMUX allows a large number of multimode setups,an standardized communication system may anyhow require fewer setups. ThisTMUX is still advantageous in an standardized system with fewer modes. As anexample, the long term evolution requires 6 scalable bandwidths as 1.4, 3, 5, 10,

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15, and 20 MHz [160]. For these bandwidths, it may still be advantageous to usethe proposed TMUX rather than using a number, e.g., 6, of separately designedTMUXs. Generally, this TMUX becomes more advantageous if the number ofmultimode setups is large.

7.5.2 Filter Design Restrictions

According to Section 3.3.1, G(z) has a passband edge which is proportional to1M . For moderate M , the well-known estimation formulae, viz., Hermann, Kaiser,and Bellanger, can be used [69]. At very small (or large) passband edges, theseestimation formulae suffer from inaccuracies. To overcome this, we must considerother order estimation formulae, e.g., [90]. Irrespective of the order estimation,the numerical challenges to design very sharp G(z) will always exist ifM increases.However, we can also use efficient design techniques, e.g., frequency response mask-ing [79], to obtain very narrow band filters.

7.6 Comparison with Existing Multimode TMUXs

This section compares the proposed TMUX to the existing multimode TMUXsdiscussed in Chapters 5 and 6. Issues such as flexibility, realization regularity,spectrum efficiency, and filter design are considered.

7.6.1 Flexibility

In the proposed TMUX, the bandwidth of each user is an integer multiple of theGRB. This GRB is fixed and it is equal to the passband width of G(z). This issimilar to [64–66] as they also allow bandwidths equal to integer Bp multiples of

the GRB with a spectral width of 2π(1+ρ)Ap

. In contrast to the fixed GRB in the

proposed TMUX, the width of the GRB in [64–66] can have different values. The

TMUXs in [46, 63] start from a fixed GRB with a spectral width of 2π(1+ρ)L . Then,

they allow each user to occupy bandwidths which are rational Rp multiples of theGRB.

From the flexibility point of view, [46, 63] are superior to the proposed TMUXas well as [64–66]. However, the flexibilities of the proposed TMUX and [64–66]may be comparable if we do not consider the flexibility in the center frequencies.One can increase the flexibility of the proposed TMUXs by increasing the numberof branches but it then brings challenges with filter design. Note that any of theseTMUXs can be configured to support a given multimode setup. To do so, propervalues of L, ρ,

Ap

Bp, M , and Mp need to be determined so that

2π(1 + ρ)Rp

L=

2π(Mp + 1)

M=

2π(1 + ρ)Bp

Ap. (7.20)

Table 7.1 compares these parameters for some of the multimode setups in Sec-tion 7.4.2. Here, the values for ρ and L are chosen in accordance with those out-

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Table 7.1: Comparison of parameters to support the same multimode setup.

Ref. Parameters

Proposed M = 20, Mp = 11, 9, 7, 5[46, 63] L = 12, ρ = 0.05, Rp = 6.86, 5.71, 4.57, 3.43

[64–66] ρ = 0.2,Ap

Bp= 2, 2.4, 3, 4

lined in earlier chapters. We use the MDFT-based TMUX as [46, 63–66] transmita complex composite signal. Furthermore, we do not compare the reconstructionerrors and the complexity as the proposed TMUX can be PR whereas those in[46, 63–66] are NPR. Further, the frequency shifters in [46, 63–66] would requireadditional hardware. On the other hand, the commutators in the proposed TMUXmay also give some control hardware overhead. Therefore, a detailed complexitycomparison is rather subtle.

7.6.2 Spectrum Efficiency

The proposed TMUX is critically sampled and the user spectra can overlap. Thisis in contrast to the redundant TMUXs of [46, 63–66]. Thus, the proposed TMUXhas a better spectrum efficiency. According to Section 7.2, the application mainlydetermines the amount of the GB. If the application allows it, the proposed TMUXcan support scenarios with spectra overlap. This increases the spectrum efficiency.

7.6.3 Direct or Indirect Design

The proposed TMUX allows one to use the filter design techniques for generalmodulated FBs. These techniques are direct as they include ISI and ICI in theirfilter design [118, 119]. However, [46, 63–65] indirectly design the filters by utilizingtheir redundancy. This redundancy simplifies the filter design but it has someoverdesign thereby increasing the implementation cost. Therefore, there is a trade-off between filter design complexity and implementation cost.

Generally and without the knowledge of multimode setups, the indirect designmay be the only option. However, if all Mp (or the most common ones) are known,we can use direct design [66]. Consequently, the proposed TMUX is superior tothose in [46, 63–65] as it always designs the filters directly.

7.7 Conclusion

In this chapter, a reconfigurable nonuniform TMUX based on uniform modulatedFBs was introduced. It uses polyphase processing and each user is processed by anumber of TMUX branches. Any user can occupy different bandwidths and centerfrequencies. This comes at the expense of some adjustable commutators and re-quires no additional arithmetic complexity. The chapter considered both CMFBsand MDFT FBs to obtain reconfigurable TMUXs. Issues related to filter design,

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realization, implementation cost, and parameter selection were discussed. Further-more, the proposed TMUX was compared to some existing multimode TMUXs.

114

8

Applications to Cognitive Radios

This chapter discusses two approaches for frequency allocation and reallocationused in baseband processing of cognitive radios. These approaches can be useddepending on the availability of a composite signal comprising several user signalsor the individual user signals. After an introduction in Section 8.1, Section 8.2outlines Approach I which is based on flexible frequency-band reallocation (FFBR)networks. Then, Section 8.3 discusses Approach II based on transmultiplexers(TMUXs). Discussions on reconfigurability with respect to cognitive radios arealso provided. Section 8.4 treats the issues regarding the choice of the centerfrequencies. Finally, some concluding remarks are given in Section 8.5.

8.1 Introduction

One perspective in the design of communication systems is to increase the spec-trum utilization using cognitive radios. A cognitive radio is a network of intelligentco-existing radios which senses1 the environment to find available frequency slots,white spaces, or spectrum holes [59, 61]. Then, it modifies its transmission char-

1Spectrum sensing is mostly done by digital baseband processing. As these algorithms requirea long time to detect an available channel, one can combine radio frequency (RF) and analogcircuits for a faster spectrum sensing [161].

115

8. APPLICATIONS TO COGNITIVE RADIOS

0 f

Primary UserOverlay Transmission

Pow

er S

pec

trum

Den

sity

Frequency

Figure 8.1: Overlay approach for spectrum sharing in a cognitive radio.

ADC andDAC

Reconfiguration

RadioFrequency

BasebandProcessing

From receiver

To transmitter

To user

From user

Figure 8.2: General block diagram of a cognitive radio composed of digital to analogconverters (DACs), analog to digital converters (ADCs), baseband processing, andRF part.

acteristics to use that particular frequency slot. Figure 8.1 illustrates the overlay2

spectrum sharing [164] or the opportunistic spectrum access [157, 162] or the dy-namic spectrum access [60]. Here, secondary users occupy the frequency slots notused3 by the primary users. One of the main tasks in a cognitive radio is conse-quently the spectrum mobility [61, 156] or the dynamic frequency allocation [59]or the dynamic spectrum allocation [62, 157]. This chapter uses the term dynamicfrequency-band allocation (DFBA). Being dynamic means that the transmissionparameters, e.g., bandwidth, center frequency, transmission power, communicationstandard, etc., may vary with time [156]. One should at least be able to change thecenter frequency and bandwidth although other parameters may also change [60].This is also referred to as the reconfigurability [4, 6, 59–62]. The general blockdiagram of a cognitive radio is shown in Fig. 8.2.

Another perspective in the design of communication systems calls for satellitesto play a complementary role supporting various wideband services accessible toeverybody everywhere [32–39]. For this purpose, the European space agency hasproposed three major network structures for broadband satellite-based communi-cation systems [37]. This requires an efficient use of the limited available frequency

2The underlay spectrum sharing or the ultra wideband [162], exploits spread spectrum. Userstransmit at certain portions of spectrum regarded as noise by the primary (licensed) users [163].

3Under certain conditions, the secondary users need not wait for a vacant channel. This allowsa simultaneous transmission over the same time or frequency [165].

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spectrum by satellite on-board signal processing [32–57].

Similar to satellite-based communication systems which require both on-groundDFBA and on-board dynamic frequency-band reallocation (DFBR), the ad hoc-or infrastructure-based cognitive radios can also utilize these tasks. In the adhoc-based networks, individual users can utilize DFBA while DFBR can be per-formed by the base stations of an infrastructure-based network. However, DFBAcan also be deployed by the individual users in an infrastructure-based network.Furthermore, DFBR is applicable to composite signals comprising several userswith different but fixed bandwidths. On the other hand, DFBA is applicable toindividual users enabling them to change their bandwidths as well.

Both DFBA and DFBR can be realized using interpolation/decimation withvariable parameters. For large sets of variable conversion factors, the implementa-tion complexity of this approach increases. Complexity reduction can be achievedby reconfigurable structures. These solutions must perform various tasks by simplemodifications and without hardware changes. These modifications are applied to(i) the values of some multipliers, or (ii) the operation of some commutators andchannel switches. Further, the filter coefficients do not change which enables us tosolve the filter design problem only once and offline. Specifically, one must be ableto reprogram the same hardware.

As both DFBR and DFBA process baseband digital signals, this chapter focuseson the frequency allocation and reallocation for baseband processing of cognitiveradios. We consider operations related to changing the center frequency and band-width of the user signals. Similar to bentpipe satellite payloads [33, 38], these donot require any modulation/demodulation, coding/decoding, etc.

This chapter discusses the reconfiguration and parameter selection issues whenadopting the DFBA and DFBR for cognitive radios. Two basically different ap-proaches, referred to as Approach I and II, are discussed. They are appropriatedepending on the availability of (i) a composite signal comprising several user sig-nals, or (ii) the individual user signals. An alternative to deal with compositesignals is to first divide, using a filter bank (FB), the composite signal into its cor-responding user signals and, then, use Approach II on each user signal. However,it is more efficient to directly use Approach I. Consequently, we will not discussthis alternative. Combinations of Approaches I and II provide an increased degreeof freedom to allocate and reallocate the user signals.

8.2 Approach I: Use of DFBR Networks

For DFBR, we assume that signals from several users, e.g., mobile handsets in acellular network or computers in a wireless local area network (WLAN), have beenadded into a composite signal at a main station, e.g., a base station in a cellularnetwork or an access point in a WLAN. This main station finds available frequencyslots and reallocates each user to one of them. In a dynamic communication system,users can occupy any bandwidth at any time. Such a main station is similar to abentpipe satellite payload [33, 38] with its idea of operation shown in Fig. 8.3. The

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In 1

In 2

DF

BR

Netw

ork

Out 1

Out 2

Out 3

2p

Input signal 1

wTin [rad]1 32

2p

Input signal 2

wTin [rad]4 65

2p

Output signal 1

wTout [rad]13

2p

Output signal 2

wTout [rad]45

2p

Output signal 3

wTout [rad]2 6

Figure 8.3: Approach I: DFBR networks process composite signals to reallocateusers from one composite input signal to another composite output signal.

composite signals are processed by the DFBR network and the users are reallocatedto new frequency slots. These slots could be different antenna beams of a satellitepayload or different cells in a cellular network. Multiple antennas of a satellitepayload perform signal filtering in spatial rather than frequency domain. This issimilar to the techniques utilizing multiple antennas for cognitive radios [166]. TheDFBR networks could also be useful for the centralized4 cooperative5 cognitiveradios [167]. They can also be considered as secondary base stations in licensedband cognitive radios [168]. In licensed band networks, the DFBR can coexist withthe primary networks and opportunistically operate in an overlay transmission.

The DFBR network can be a mutli-input multi-output system as it can havea number of composite input and output signals. The dynamic nature of theDFBR networks allows the users to occupy any suitable6 frequency slot in a time-varying manner. Each user can be sent in contiguous or separate frequency bandsrequiring contiguous or fragmented DFBR [62]. The separate frequency bandscan be considered as a multi-spectrum transmission. Specifically, as white spacesare mostly fragmented [169], the user signals can be transmitted in several non-contiguous frequency bands.

8.2.1 Structure of the DFBR Network

This chapter uses the term DFBR which is essentially similar to FFBR in Chapter 4.Consequently, the DFBR network has the structure of the FFBR network as in,e.g., Fig. 4.3. One can in principle use any of the FFBR networks outlined inSections 4.1.1 and 4.1.2.

4A centralized entity, e.g., the DFBR network, controls the spectrum allocation and access.5Non-cooperative networks do not share the interference measurements of each user with the

others. Generally, cooperative networks are more accurate in sensing the spectrum [61].6The frequency slot depends on spatial and temporal parameters such as the number of avail-

able slots, user movement, and activity of primary users, etc. [156]. The operation of the DFBRnetwork is independent of these parameters.

118


Granularity Band (BGRB) Guardband (GB)

0 2p

(a)

wT1BGRB

0 2p

(b)

wT3.5BGRB1.75BGRB

Additional GBUser Bandwidth

Multiplexing Bandwidth

0 2p

(c)

wT3BGRB2BGRB

Figure 8.4: User bandwidth versus multiplexing bandwidth.

8.2.2 User Bandwidth Versus Multiplexing Bandwidth

The DFBR networks divide the user signals into a number of granularity bands(GRBs) on which the frequency shifts are performed. As the DFBR networksutilize FBs, the multiplexing bandwidth must be an integer multiple of the GRB.The DFBR networks perform frequency shifts on users whose bandwidths are,in general, rational multiples of the GRB. In this way, bandwidth-on-demand issupported. An important issue is to ensure that the users do not share a GRB whichcan be achieved by allowing some extra guardband (GB). However, the extra GBsaffect the spectrum efficiency resulting in a trade-off. As in Fig. 8.4, a multiplexingbandwidth contains a user bandwidth and some extra GB.

8.2.3 Reconfigurability

A cognitive radio should adjust its operating parameters without hardware mod-ifications [4]. It is built on the platform for a software defined radio with theprocessing mainly in the digital domain [162]. There are several reconfigurableparameters such as operating frequency, modulation method, transmission power,communication standard, etc. In the context of adaptable operating frequency, orflexible frequency carrier tuning [62], a cognitive radio changes its operating fre-quency. However, this should not restrict the system throughput and hardware.

The DFBR networks can perform any frequency shift of any user having anybandwidth, using a channel switch. This switch seamlessly directs different FBchannels to their desired outputs and requires no arithmetic complexity. In addi-tion, the system parameters are determined and fixed only once offline. Then, thereconfigurable operation is performed by reconfiguring the channel switch online.

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−60

−40

−20

0

(a)

ωT [πrad]

|X(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

X3

−60

−40

−20

0

(b)

ωT [πrad]

|Y1(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

X3

−60

−40

−20

0

(c)

ωT [πrad]

|Y2(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

X3

Figure 8.5: Input pattern and the reallocated outputs using the channel switchconfigurations in Figs. 8.7 and 8.8.

Here, the user bandwidths are predetermined but can be arbitrary. The DFBRnetwork makes a hand off by changing the operating frequency [61].

Figures 8.5 and 8.6 show two cases where, respectively, four and three users haveoccupied the whole frequency band between [0, 2π]. To generate these user signals,the multimode TMUX of Chapter 5 has been used. In Fig. 8.5(a), the user signalsX0, X1, X2, X3 occupy, respectively, user bandwidths of 1, 2.9, 3.6, 1.9 GRBs.Each GRB has a width of 2π

Q − 2ǫπQ with Q = 10 and ǫ = 0.125. In Fig. 8.6(a),

the user signals X0, X1, X2 occupy 1, 6.9, 1.9 GRBs, respectively. As can beseen, the user signals can occupy any rational number of GRBs. To ensure thatthe users do not share a GRB, one can add some extra GB. This difference in theamount of the GBs between different users can be recognized from Figs. 8.5 and8.6.

These examples assume the DFBR network to operate on the same antennabeam. By having several DFBR networks, the users can be reallocated betweendifferent antenna beams as well. This requires a duplication of DFBR networksand a channel switch capable of directing the user signals between different DFBRnetworks. Each branch of the channel switches in Figs. 8.7–8.10 represents theoperation of two FB channels as each GRB contains two FB channels. Specifically,the values of N , M , and L, in Fig. 4.3, are 20, 10, and 2 respectively.

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−60

−40

−20

0

(a)

ωT [πrad]

|X(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

−60

−40

−20

0

(b)

ωT [πrad]

|Y1(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

−60

−40

−20

0

(c)

ωT [πrad]

|Y2(e

jωT)|

[d

B]

0 0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π

X0

X1

X2

Figure 8.6: Input pattern and the reallocated outputs using the channel switchconfigurations in Figs. 8.9 and 8.10.

An

aly

sis

Ban

k O

utp

ut

Sy

nth

esis

Ban

k I

np

ut

Figure 8.7: Scenario I.

An

aly

sis

Ban

k O

utp

ut

Sy

nth

esis

Ban

k I

np

ut

Figure 8.8: Scenario II.

An

aly

sis

Ban

k O

utp

ut

Sy

nth

esis

Ban

k I

np

ut

Figure 8.9: Scenario I.

An

aly

sis

Ban

k O

utp

ut

Sy

nth

esis

Ban

k I

np

ut

Figure 8.10: Scenario II.

121


TM

UX Out

2p

User 1 at t1,B1,f1

wT

2p

User 1 at t2,B2,f2

wT

2p

User 1 at t3,B3,f3

wT

t,B,f

Figure 8.11: Approach II: TMUXs used to perform DFBA. At any time tk, eachuser can decide its bandwidth Bk and operating frequency fk.

8.2.4 Modifications

The use of DFBR networks in cognitive radios needs some modifications whichare mainly related to the choice of the system parameters. For different systemparameters, the implementation complexity may be different. However, once theparameters are chosen, the implementation complexity remains constant and thesystem can be easily reconfigured on the same hardware platform.

For the DFBR networks, the width of a GRB must be proportional to that of

the spectrum holes. Thus, one requires to choose a value for the BGRB = 2π(1−ǫ)Q ,

in Fig. 8.4, such that the bandwidth of any spectrum hole can be represented as arational multiple of BGRB .

8.3 Approach II: Use of TMUXs

Using a TMUX-based solution, each user terminal can adjust its operating fre-quency and bandwidth. The basic idea is depicted in Fig. 8.11 where different band-widths and center frequencies can be generated using multirate signal processingtechniques. These TMUXs can also be regarded as the time-spectrum blocks [158]which can transmit any amount of data at any time interval and on any portion ofthe frequency spectrum. This finds application if licensed users choose frequency di-vision multiple access and/or time division multiple access as their spectrum accessmode. Then, the spectrum holes are identified in the time/frequency plane [170].As shown in Fig. 8.12, the interpolation part represents the transmitter where avariable filter places the desired user signal at the required center frequency. Thereceiver, i.e., the decimation part, is designed accordingly so that the input signalis recovered with reasonable and controllable levels of error.

Similar to straightforward DFBR solutions, one can use conventional nonuni-form TMUXs to place users with different bandwidths at different center frequen-cies. This becomes inefficient when simultaneously considering the increased num-

122


G(z)Mv1(a)

T1

x1 y

Interpolation

H(z) Mv2

T2

Decimation

0 2p/MwT1

X1

(b)

4p/M 6p/M 2p

0 2p/M

(c)

4p/M 2p

G(ejwT2)

0 2p/M

V2(ejwT2)(d)

4p/M 2p

X1

X1(ejwT1)

wT2

wT2

Figure 8.12: Principle of TMUXs using multirate building blocks.

ber of communication scenarios and the desire to support dynamic communications.

In this context, TMUX structures of the general form shown in Fig. 8.13 areintroduced. In the synthesis FB, the system Cp performs interpolation by a rational

ratio Rp whereas the system Cp in the analysis FB performs decimation by arational ratio Rp. These blocks enable one to transmit and receive baseband signals,having arbitrary bandwidths, through a common channel.

8.3.1 Structure of the TMUX

Any of the TMUXs in Chapters 5–7 can be used here. Note that the TMUXin Chapter 7 has a rather different structure. Instead of variable lowpass filtersand frequency shifters as in Chapters 5–6, it performs bandpass rational SRCusing flexible commutators and fixed bandpass filters. However, one can generallydescribe it in terms of Fig. 8.13.

8.3.2 Reconfigurability

The cognitive radio must adjust its operating frequency and bandwidth withouthardware modifications. The DFBR networks partially provide this capability but

123


C0

C1

CP-1

x0(n0)

x1(n1)

xP-1

(nP-1)

ejw

0n

ejw

1n

ejw

P-1n

y(n) y(n)^

C0

x0(n0)^

x1(n1)^

xP-1

(nP-1)^

ê

-jw0n

ê

-jw1n

C1

CP-1

ê

-jwP-1

n


^

^

^

Figure 8.13: General structure of a multimode TMUX where systems Cp and Cp

perform rational sampling rate conversion (SRC).

they have no control over the user bandwidth. In contrast, the TMUX-basedapproaches add reconfigurability to the user bandwidth as well. Furthermore, thesesolutions bring flexible receiver signal filtering [62] by changing the transmitter andreceiver filters.

As can be seen from Figs. 8.5 and 8.6, the TMUX allows different numbersof users, e.g., four and three, with different user bandwidths to occupy the wholefrequency band between [0, 2π]. These TMUXs provide this full reconfigurabilitywithout any hardware changes.

8.3.3 Modifications

Similar to the DFBR networks, one requires certain system parameters to eliminatethe need for any hardware change while having a simple reconfigurability.

Regarding DFBA, there are different ways to perform SRC which could beuseful in different scenarios. The TMUX in [46, 63] generates a GRB through

integer interpolation by, e.g., W , resulting in BGRB = 2π(1+ρ)W where ρ is the

roll-off. Then, rational Rp multiples of BGRB can be created using the Farrowstructure. One can determine BGRB according to the bandwidth of the spectrumholes. Then, any user may occupy any rational number of spectrum holes.

The TMUX in [64–66] assumes no GRBs and it allows the users to occupy anyportion of the spectrum. It utilizes the Farrow structure to perform general rationalSRC by, e.g., Rp =

Ap

Bp. This allows one to cover a large set of user bandwidths.

Here, one can also assume a GRB of size BGRB = 2π(1+ρ)Ap

. Then, users can have

bandwidths which are integer Bp multiples of BGRB .

Although [67, 68] propose a slightly different TMUX, one can also assume

BGRB = 2π(1+ρ)M . Then, users have bandwidths which are integer Mp multiples

of BGRB . Note that this applies to the case with modified discrete Fourier trans-

124


form FBs. For cosine modulated FBs, similar formulae can be derived.

8.4 Choice of Frequency Shifters

To perform a hand off without loss of information, the DFBR network requiresthe users not to share a GRB. Consequently, a lossless reallocation requires to (i)generate appropriate frequency division multiplexed (FDM) input patterns, and(ii) determine proper parameters for the DFBR networks. To generate the inputpatterns, the reconfigurability of the TMUXs in Fig. 8.13 can be used. Specifically,after generating the user signals with desired bandwidths, the frequency shiftersωp, p = 0, 1 . . . , P −1, can be computed to allow some extra GB. Here, an exampleusing the TMUX in Chapter 5 is provided. Assuming bandwidths that are rational,e.g., Rp, multiples of BGRB , the subcarrier ωp for user p is

ωp =

F0

2 if p = 0∑p−1

k=0 Fk +Fp

2 if p 6=0.(8.1)

where Fp = ⌈Rp⌉ 2πQ , p = 0, 1, . . . , k, with ⌈x⌉ being the ceiling of x. Here, Fp is

the multiplexing bandwidth and the ceiling operation ensures that the users do notshare a GRB. This formulation applies to the case where DFBA and DFBR aresimultaneously used. Otherwise, formulae similar to (6.16) can be used instead. Inpractice, one may anyhow require some extra GB due to the design margins.

In Figs. 8.5 and 8.6, the users occupy Rp = 2.9, 3.6, 1.9, 6.9 GRBs. This ne-cessitates an extra GB which is Ep = 0.1, 0.4, 0.1, 0.1 multiples of 2π

Q . Therefore,the spectrum efficiency decreases as some parts of the spectrum are not used. Fora set of values Rp, p = 0, 1, . . . , P − 1, about

ηdec =

2πQ

∑P−1p=0 (⌈Rp⌉ −Rp)

2π=

∑P−1p=0 ⌈Rp⌉ −Rp

Q(8.2)

percent of the spectrum in [0, 2π] is not used by the DFBR network. In the examplesof Figs. 8.5 and 8.6, about 6% and 2% of the total spectrum is not used due to theextra GBs.

To decrease these percentages, one can increase Q by, e.g., K times, whichwould, in turn, decrease ηdec. In this case, (8.2) becomes

ηdec =

∑P−1p=0 ⌈KRp⌉ −KRp

KQ. (8.3)

However, increasing Q would increase the order of the prototype filter P (z) in (4.4).For each K, the prototype filter of the DFBR network would have a transition bandof 2πǫ

KQ [42, 43]. As the order of a linear-phase finite-length impulse response filter

is inversely proportional to the width of its transition band [69], there is a trade-offbetween the spectrum efficiency and the arithmetic complexity.

125


0 5 10 15

310

320

330

340

350

ηdec.

(%)

Rea

l oper

atio

ns

(a)

Rp = 1,2.9,3.6,1.9

Rp = 1,6.9,1.9

Rp = 1.75,1.25,2,3.5

1 2 3 4 5 6 7 8 9 100

5

10

15

K

ηd

ec. (

%)

(b)

Figure 8.14: Trade-off between spectrum efficiency and arithmetic complexity. (a)Decrease in spectrum efficiency versus per-sample arithmetic complexity. (b) Trendof spectrum efficiency versus different K in (8.3).

With a K-fold increase in Q, the length of the prototype filter and the numberof FB channels increase proportional to K. Figure 8.14 shows the trend in spec-trum efficiency with respect to the per-sample arithmetic complexity of the DFBRnetwork discussed in Section 4.2. Here, the examples of Figs. 8.5 and 8.6 as wellas that of [63] with Rp = 1.75, 1.25, 2, 3.5 are considered and K = 1, 2, . . . , 10.As can be seen, a larger K increases the per-sample arithmetic complexity but itdecreases ηdec. The values of Rp mainly determine the maximum and minimumamounts of ηdec. Hence, for every set of Rp, one can determine a K such that ηdecand the per-sample arithmetic complexity are within the acceptable ranges.

8.5 Conclusion

This chapter discussed two approaches for the baseband processing in cognitiveradios based on DFBR and DFBA. They can support different bandwidths andcenter frequencies for a large set of users and are easily reconfigurable.

In DFBR networks, composite FDM signals comprising several users are pro-cessed and the users are reallocated to new center frequencies. They are applicableto cognitive radios with multiple antennas [166]; centralized cooperative cognitive

126


radios [167]; and secondary base stations in licensed band cognitive radios [168]. InDFBA networks, each user controls its operating frequency and bandwidth. Thesenetworks can be regarded as the time-spectrum blocks [158].

The reconfigurability of DFBA and DFBR is performed either by a channelswitch, in DFBR, or by variable multipliers/commutators, in DFBA. The examplesin Figs. 8.5 and 8.6 show the increased flexibility to allocate and reallocate any userto any center frequency by simultaneous utilization of DFBA and DFBR. In thiscase, the individual users can occupy any available frequency slot and be reallocatedby the base station.

Basically, utilizing any of Approaches I and II in cognitive radios only requiresmodifications imposed by the special choice of the system parameters. After choos-ing these parameters once, we must design the filters to satisfy any desired level oferror. Then, the same hardware can be reconfigured in a simple manner.

127


128

9

Conclusion and Future Work

This thesis introduced nonuniform transmultiplexers (TMUXs) which enable dif-ferent numbers of users, having different bandwidths, to share the whole frequencyband in a time-varying manner. The TMUXs support dynamic communicationsand are reconfigurable. The TMUX in Chapter 5 uses a two-stage sampling rateconversion (SRC) composed of integer and rational SRC schemes. Adjustable fre-quency shifters, along with SRC of an arbitrary rational ratio, allow the users tooccupy any bandwidth and center frequency. On the other hand, Chapter 6 intro-duced a TMUX where SRC is performed in the context of a conventional rationalSRC scheme. Here, integer SRC along with integer upsamplers/downsamplers andadjustable frequency shifters are used. Both of these TMUXs use the Farrow struc-ture. Further, Chapter 7 proposed TMUXs based on modulated filter banks (FBs).Here, adjustable commutators are used enabling the users to occupy the desiredbandwidths and center frequencies. All of these TMUXs require neither filter re-design nor hardware changes. Specifically, the filters are designed only once offline.Then, the TMUXs are reconfigured by simple modifications without restricting thesystem operation. These modifications apply to some multipliers, i.e., the frac-tional delay of the Farrow structure and the variable parameters of the frequencyshifters, and some commutators.

The thesis also outlined solutions for flexible frequency-band reallocation (FFBR)networks. The FFBR network in Chapter 4 enables different users, in differentcomposite multiple frequency/time division multiple access (MF/TDMA) input

129

9. CONCLUSION AND FUTURE WORK

signals, to be reallocated to different positions in different composite MF/TDMAoutput signals. Although Chapter 4 considers real signals, the FFBR network cangenerally be used for complex signals. The reallocation is done using simple modifi-cations in the channel switch and it does not require any filter redesign or hardwarechanges. The FFBR solutions do not restrict the bandwidth of the users or thesystem operation.

In all of these chapters, the problems of filter design were discussed and illustra-tive examples were provided. The thesis also outlined possible applications of theproposed TMUXs and FFBR networks in the context of cognitive radios. Theseapplications are mainly related to spectrum mobility thereby allowing the usersto dynamically occupy any bandwidth and center frequency. As topics of futureresearch, the following issues are identified:

1. Application of cosine modulated FBs (CMFBs) to derive FFBR networks forreal signals. It would specifically be interesting to compare the flexibility infrequency-band reallocation between complex modulated FBs and CMFBs.

2. Development of efficient methods to produce the complex periodic sequencesresulting from the variable frequency shifters. This is also referred to as directdigital frequency synthesis. These methods should be reconfigurable as thevalues of the frequency shifters, in the proposed TMUXs, can change online.

3. Realization of the proposed solutions in hardware for some emerging stan-dards, e.g., long term evolution.

4. Derivation of integer SRC using multi-stage structures to reduce the arith-metic complexity. Specifically, it would be interesting to propose efficientrealization structures for the TMUX in Chapter 5. Then, one can allow asmaller granularity band thereby increasing the reconfigurability.

5. Development of methods to design the Farrow structure which further reducethe implementation cost. A paper has already been published which outlinesan idea on how to reduce the orders of Farrow subfilters. This idea can bedirectly applied to the TMUX in Chapter 5. Generalizations of this idea forapplication to the TMUX in Chapter 6 would also be possible.

130

A

Derivation of (6.23)

The sequel derives (6.23). Figure A.1 shows the structure of the transmultiplexerwith the desired signal Xd(z) in the analysis filter bank. Using (3.4), we have

Xp(z) = Xp(zAp)Gp(z), p = 0, 1, . . . , P − 1 (A-1)

G0(z)

G1(z)

GP-1

(z)

x0(n0)

x1(n1)

xP-1

(nP-1)

ejw

0n

ejw

1n

ejw

P-1n

y(n)x

d(nd)^

^e

-jwdn

Gd(z)^ Ad

A0

A1

AP-1

B0

B1

BP-1

Bd

x0

-

x1

-

-xP-1

x0

~

x1

~

~xP-1

y0(n)

y1(n)

yP-1

(n)

xd-x

d

g

Figure A.1: The synthesis filter bank and the desired signal for the transmultiplexerin Chapter 6.

131

APPENDIX A. DERIVATION OF (6.23)

where (3.2) gives

Xp(z) =1

Bp

Bp−1∑

k=0

Xp(z1

BpW kBp

) (A-2)

=1

Bp

Bp−1∑

k=0

Xp([z1

BpW kBp

]Ap)Gp(z1

BpW kBp

).

With frequency shifts by ωp and −ωd and assuming zp = ejωpT and zd = ejωdT ,we get

Yp(z) = Xp(z

zp), (A-3)

and

Y (z) =

P−1∑

p=0

Yp(z). (A-4)

In the analysis filter bank,

Xd(z) = Y (zzd) =

P−1∑

p=0

Xp(zzdzp

), (A-5)

which using (A-2) becomes

Xd(z) =1

Bp

P−1∑

p=0

Bp−1∑

k=0

Xp([(zzdzp

)1

BpW kBp

]Ap)Gp((zzdzp

)1

BpW kBp

). (A-6)

Further,

Xd(z) = Xd(zBd)Gd(z) (A-7)

=1

Bp

P−1∑

p=0

Bp−1∑

k=0

Xp([(zBdzdzp

)1

BpW kBp

]Ap)Gp((zBdzdzp

)1

BpW kBp

)Gd(z)

=1

Bp

P−1∑

p=0

Bp−1∑

k=0

Xp(zApBdBp (

zdzp

)ApBpW

kAp

Bp)Gp(z

BdBp (

zdzp

)1

BpW kBp

)Gd(z).

Further, the downsampler by Ad gives

Xd(z) =1

Ad

Ad−1∑

m=0

Xd(z1

AdWmAd

). (A-8)

Ignoring the scaling factors, (A-8) and (A-7) give

Xd(z) =

P−1∑

p=0

Bp−1∑

k=0

Ad−1∑

m=0

Xp([z1

AdWmAd

]ApBdBp (

zdzp

)ApBpW

kAp

Bp)×

Gp([z1

AdWmAd

]BdBp (

zdzp

)1

BpW kBp

)Gd(z1

AdWmAd

). (A-9)

132


After some manipulations, we have

Xd(z) =

P−1∑

p=0

Ad−1∑

m=0

Bp−1∑

k=0

Xp(zApBdAdBpW

mApBdBp

Ad(zdzp

)ApBpW

kAp

Bp)×

Gp(zBd

AdBpWmBdBp

Ad(zdzp

)1

BpW kBp

)Gd(z1

AdWmAd

) (A-10)

which is (6.23).

133


134

B

Derivation of (6.35)

The sequel derives (6.35) using Fig. B.1. For interpolation byAp

Bp, we have

y3(n) =∑

k

x(k)g(nBp − kAp). (B-1)

Asy5(n) =

∑

m

y3(m)g(n−mBp) (B-2)

andy(n) = y5(nAp), (B-3)

we get

y5(n) =∑

m

∑

k

x(k)g(mBp − kAp)g(n−mBp). (B-4)

Then,

y(n) =∑

m

∑

k

x(k)g(mBp − kAp)g(nAp −mBp). (B-5)

Finally

y(n) =∑

k

x(k)∑

m

g(mBp − kAp)g(nAp −mBp). (B-6)

Now, replace k with n− k and (6.35) follows.

135

APPENDIX B. DERIVATION OF (6.35)

G(z) ApBpG(z) BpApx(n)y1(n)

^y2(n) y3(n) y4(n) y5(n)

y(n)

Figure B.1: Cascade of interpolation and decimation byAp

Bpfor the transmultiplexer

in Chapter 6.

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