Signal Reconstruction Algorithms for Time-Interleaved...

Linkoping Studies in Science and TechnologyDissertations, No. 1672

Anu Kalidas Muralidharan Pillai

MM UNICATION

YS TEMS Division of Communication Systems

Department of Electrical Engineering (ISY)Linkoping University, SE-581 83 Linkoping, Sweden

www.commsys.isy.liu.se

Linkoping 2015

Signal Reconstruction Algorithms

for Time-Interleaved ADCs

Signal Reconstruction Algorithms for Time-Interleaved ADCs

c© 2015 Anu Kalidas M. Pillai, unless otherwise noted.

ISBN 978-91-7519-062-4ISSN 0345-7524

Printed in Sweden by LiU-Tryck, Linkoping 2015

Abstract

An analog-to-digital converter (ADC) is a key component in many elec-tronic systems. It is used to convert analog signals to the equivalent digitalform. The conversion involves sampling which is the process of convertinga continuous-time signal to a sequence of discrete-time samples, and quan-tization in which each sampled value is represented using a finite numberof bits. The sampling rate and the effective resolution (number of bits) aretwo key ADC performance metrics. Today, ADCs form a major bottleneckin many applications like communication systems since it is difficult to si-multaneously achieve high sampling rate and high resolution. Among thevarious ADC architectures, the time-interleaved analog-to-digital converter(TI-ADC) has emerged as a popular choice for achieving very high samplingrates and resolutions. At the principle level, by interleaving the outputs ofM identical channel ADCs, a TI-ADC could achieve the same resolution asthat of a channel ADC but with M times higher bandwidth. However, inpractice, mismatches between the channel ADCs result in a nonuniformlysampled signal at the output of a TI-ADC which reduces the achievableresolution. Often, in TI-ADC implementations, digital reconstructors areused to recover the uniform-grid samples from the nonuniformly sampledsignal at the output of the TI-ADC. Since such reconstructors operate atthe TI-ADC output rate, reducing the number of computations requiredper corrected output sample helps to reduce the power consumed by theTI-ADC. Also, as the mismatch parameters change occasionally, the recon-structor should support online reconfiguration with minimal or no redesign.Further, it is advantageous to have reconstruction schemes that require fewercoefficient updates during reconfiguration. In this thesis, we focus on reduc-ing the design and implementation complexities of nonrecursive finite-lengthimpulse response (FIR) reconstructors. We propose efficient reconstructionschemes for three classes of nonuniformly sampled signals that can occur atthe output of TI-ADCs.

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Firstly, we consider a class of nonuniformly sampled signals that occur asa result of static timing mismatch errors or due to channel mismatches inTI-ADCs. For this type of nonuniformly sampled signals, we propose threereconstructors which utilize a two-rate approach to derive the correspondingsingle-rate structure. The two-rate based reconstructors move part of thecomplexity to a symmetric filter and also simplifies the reconstruction prob-lem. The complexity reduction stems from the fact that half of the impulseresponse coefficients of the symmetric filter are equal to zero and that, com-pared to the original reconstruction problem, the simplified problem requiresonly a simpler reconstructor.

Next, we consider the class of nonuniformly sampled signals that occur whena TI-ADC is used for sub-Nyquist cyclic nonuniform sampling (CNUS) ofsparse multi-band signals. Sub-Nyquist sampling utilizes the sparsities inthe analog signal to sample the signal at a lower rate. However, the reducedsampling rate comes at the cost of additional digital signal processing thatis needed to reconstruct the uniform-grid sequence from the sub-Nyquistsampled sequence obtained via CNUS. The existing reconstruction schemeis computationally intensive and time consuming and offsets the gains ob-tained from the reduced sampling rate. Also, in applications where the bandlocations of the sparse multi-band signal can change from time to time, thereconstructor should support online reconfigurability. Here, we propose areconstruction scheme that reduces the computational complexity of the re-constructor and at the same time, simplifies the online reconfigurability ofthe reconstructor.

Finally, we consider a class of nonuniformly sampled signals which occur atthe output of TI-ADCs that use some of the input sampling instants forsampling a known calibration signal. The samples corresponding to the cal-ibration signal are used for estimating the channel mismatch parameters.In such TI-ADCs, nonuniform sampling is due to the mismatches betweenthe channel ADCs and due to the missing input samples corresponding tothe sampling instants reserved for the calibration signal. We propose threereconstruction schemes for such nonuniformly sampled signals and show us-ing design examples that, compared to a previous solution, the proposedschemes require substantially lower computational complexity.

Popularvetenskaplig

Sammanfattning

En analog-till-digital-omvandlare (A/D-omvandlare) ar en elektronisk kom-ponent som anvands for att omvandla information (signaler) fran analog tilldigital form. A/D-omvandlare ar bade nyckelkomponenter och flaskhalsari manga sammanhang, t ex i kommunikationssystem som behover hanteramycket information per tidsenhet. Ju mer information som behover omvand-las, desto svarare ar det att praktiskt konstruera en A/D-omvandlare somklarar av att utfora detta utan fel. Prestandan hos A/D-omvandlare matsi form av datatakt (samplingstakt), vilket anger antal sampel (matvarden)per sekund, och den effektiva upplosningen (antal bitar) vilken anger den nu-meriska precisionen hos varje sampel. Det ar framfor allt svart att samtidigterhalla hog datatakt och hog upplosning.

Ett satt att oka datatakten ar att anvanda en sa kallade sammanflatad A/D-omvandlare. I en sadan anvands tva eller flera parallella omvandlare somtar hand om olika sampel. Om M omvandlare anvands parallellt erhalles daen M -fald okning av datatakten. Emellertid finns det alltid analoga match-ningsfel mellan de parallella omvandlarna. Detta ger upphov till sa kalladvikningsdistorsion vilket i sin tur degraderar upplosningen. For att densammanflatade A/D-omvandlaren ska uppna samma upplosning som de in-dividuella omvandlarna behovs darfor digital signalrekonstruktion. Konven-tionella algoritmer for detta problem tenderar att vara berakningsintensiva,dvs de kraver manga aritmetiska operationer (multiplikationer och addi-tioner) for att korrigera varje sampel. Detta innebar en hog effektforbrukn-ing om de ska implementeras i hardvara.

Denna avhandling introducerar nya algoritmer for signalrekonstruktion isammanflatade A/D-omvandlare. De foreslagna algoritmerna kraver bety-dligt farre aritmetiska operationer an befintliga losningar. Darigenom kan

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samma hoga prestanda for den sammanflatade A/D-omvandlaren uppnas,som for befintliga algoritmer, men med lagre beraknings- och implementer-ingskostnad. Avhandlingen behandlar bade konventionella sammanflatadeA/D-omvandlare, som beskrevs ovan, och okonventionella varianter, dar eneller flera av de individuella omvandlarna inte anvands hela tiden eller intealls.

Att inte anvanda omvandlarna hela tiden for den analoga insignalen kan ut-nyttjas for att utfora en robust estimering av mismatch-felen mellan de in-dividuella omvandlarna, vilket behovs for att kunna designa en signalrekon-struktor. Detta sker genom att, i de tidsluckor som da skapas, applicera enkand signal for estimeringen. Att inte alls anvanda en del av de individuellaomvandlarna kan utnyttjas for att dra fordel av strukturer i analoga sig-naler vilket mojliggor sa kallad sub-Nykvist sampling och gles signalbehan-dling. Sub-Nykvist sampling av glesa signaler erbjuder kraftigt reduceraddatatakt. Detta har potential att radikalt reducera effektforbrukningen forA/D-omvandlingen.

Avhandlingen omfattar bade teori och designmetoder for ovan namndatillampningar, samt manga exempel som visar berakningseffektiviteten hosde foreslagna algoritmerna.

Acknowledgments

First and foremost, I would like to express my gratitude to my supervisorProfessor Hakan Johansson for giving me an opportunity to perform the re-search that led to this dissertation. I have immensely benefited from thediscussions with him and from the feedback and numerous insights that heprovided during these four years. His invaluable guidance and encourage-ment has helped me to reach this far.

I would like to acknowledge the support provided by my former co-supervisorDr. J. Jacob Wikner. I would also like to thank him for giving the Ph.D.course “Script-based IC design” which was one of the best Ph.D. coursesthat I have attended. I am very thankful to Dr. Oscar Gustafsson for all thesupport that he had extended to me while I was at the Division of ElectronicsSystems (ES). I also thank him for being the driving-force behind the ESweekly presentations which allowed me to hone my presentation skills as wellas get relevant feedback on my work. I am also grateful to my co-supervisorProf. Erik G. Larsson for granting me the opportunity to join the Divisionof Communication Systems (CommSys) and, for facilitating a stimulatingresearch environment.

I wish to express my warmest thanks to Prakash Harikumar for being a verygood friend and colleague. At times I have sought the help of his retentivememory which has always fascinated me. I also thank him for carefullyproof-reading parts of this dissertation. Special thanks to my friend andcolleague Vishnu Unnikrishnan for always being interested in discussing mywork. I would also like to thank Dr. Hien Quoc Ngo for providing the LATEXtemplate which I have used to prepare this dissertation. Many thanks to allthe former and current colleagues at ES and CommSys who have directly orindirectly helped me in my work. Working with you has been a privilege.

I take this opportunity to thank my wife Roshini for her encouragementand tremendous support which helped me to stay focused on my studies. I

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would also like to thank her parents, Ratnam and Hariprasad, for extendingtheir help when it was needed the most. Their succor during a time of greatpersonal loss allowed me to concentrate on my studies and, for this I amforever indebted to them. I am eternally grateful to my parents, Nirmalaand Muralidharan, for their unconditional love and for supporting me in allthe important decisions that I have taken in life.

It is also acknowledged that parts of the work which resulted in this disser-tation were supported by the Swedish Research Council.

Linkoping, May 2015Anu Kalidas Muralidharan Pillai

Abbreviations

ADC Analog-to-digital converterASIC Application-specific integrated circuitCNUS Cyclic nonuniform samplingDFT Discrete fourier transformDMC Differentiator-multiplier cascadeDSP Digital signal processingFB Filter bankFD Fractional-delayFIR Finite-length impulse responseIDFT Inverse discrete fourier transformPR Perfect reconstructionSFDR Spurious-free dynamic rangeTI-ADC Time-interleaved analog-to-digital converter

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ContentsAbstract iii

Popularvetenskaplig Sammanfattning (in Swedish) v

Acknowledgments vii

Abbreviations ix

I Introduction 1

1 Background 31.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Uniform Sampling . . . . . . . . . . . . . . . . . . . . 61.1.3 FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Decimation . . . . . . . . . . . . . . . . . . . . . . . . 91.1.6 Polyphase Decomposition . . . . . . . . . . . . . . . . 101.1.7 Filter Banks . . . . . . . . . . . . . . . . . . . . . . . 111.1.8 Cosine-Modulated Filter Banks . . . . . . . . . . . . . 121.1.9 Vandermonde Matrices . . . . . . . . . . . . . . . . . . 14

2 Mismatch Error Correction in TI-ADCs 152.1 Time-Interleaved ADCs . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Static Time-Skew Errors in TI-ADCs . . . . . . . . . 172.1.2 Channel Frequency Response Mismatches . . . . . . . 20

2.2 Time-Varying FIR Reconstructors . . . . . . . . . . . . . . . 212.3 Error Metrics and Reconstructor Design . . . . . . . . . . . . 22

2.3.1 Least-Squares Design . . . . . . . . . . . . . . . . . . . 222.3.2 Minimax Design . . . . . . . . . . . . . . . . . . . . . 23

2.4 Reconstructor Complexity . . . . . . . . . . . . . . . . . . . . 242.5 Low-Complexity Reconstruction Schemes . . . . . . . . . . . 25

2.5.1 Two-Rate Based Approach . . . . . . . . . . . . . . . 262.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Sub-Nyquist Sampling of Sparse Multi-Band Signals 313.1 Sub-Nyquist Cyclic Nonuniform Sampling . . . . . . . . . . . 323.2 Reconstruction Using Multi-Level Synthesis Filters . . . . . . 34

3.2.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Reconstruction Using Analysis and Synthesis Filters . . . . . 37

3.3.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Summary and Future Extension . . . . . . . . . . . . . . . . . 40

4 Reconstruction in TI-ADCs with Missing Samples 414.1 TI-ADCs with Missing Samples . . . . . . . . . . . . . . . . . 414.2 Reconstruction Schemes . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Constrained Time-Varying FIR Reconstructor . . . . . 434.2.2 Sub-Band Based Reconstructor . . . . . . . . . . . . . 454.2.3 Pre-Filter Based Reconstructor . . . . . . . . . . . . . 464.2.4 Complexity Comparison . . . . . . . . . . . . . . . . . 47

4.3 Noise Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Summary of Specific Contributions of the Dissertation 535.1 Included Papers . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Not Included papers . . . . . . . . . . . . . . . . . . . . . . . 57

A Alternative Derivation of the Reconstruction Scheme inPaper D 59A.1 Lowpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 60A.2 Conventional Bandpass Filters . . . . . . . . . . . . . . . . . 62A.3 Unconventional Bandpass Filters . . . . . . . . . . . . . . . . 64

B Derivation of the Least-Squares Design in Paper F 67B.1 Constrained Time-Varying FIR Reconstructor . . . . . . . . . 67B.2 Least-Squares Design of Fq(e

jω) and Gq(ejω) in the Sub-band

Based Reconstructor . . . . . . . . . . . . . . . . . . . . . . . 69

II Efficient Reconstruction Schemes for TI-ADCs 81

A Two-Rate Based Low-Complexity Time-Varying Discrete-Time FIR Reconstructors for Two-Periodic NonuniformlySampled Signals 831 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 Nonuniform Sampling and Time-Varying FIR Reconstructors 88

2.1 Reconstructors for Two-Channel TI-ADCs . . . . . . . 89

3 Two-Rate Based Reconstructors . . . . . . . . . . . . . . . . 913.1 Least-Squares Design . . . . . . . . . . . . . . . . . . . 93

4 Polynomial Impulse Response FIR Reconstructors . . . . . . 975 Two-Rate Based Polynomial Impulse Response FIR Recon-

structors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Least-Squares Design . . . . . . . . . . . . . . . . . . . 101

6 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

B Efficient Signal Reconstruction Scheme for M-ChannelTime-Interleaved ADCs 1131 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 M -Channel Two-Rate Based Reconstructors . . . . . . . . . . 121

3.1 Offline Design of F0(z) . . . . . . . . . . . . . . . . . . 123

3.2 Online Design of G(n)0 (z) and G

(n)1 (z) . . . . . . . . . . 126


C Low-Complexity Two-Rate Based Multivariate ImpulseResponse Reconstructor for Time-Skew Error Correctionin M-Channel Time-Interleaved ADCs 1371 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402 M -Periodic Nonuniform Sampling and Reconstruction . . . . 1423 Review of Multivariate Polynomial Impulse Response Recon-

structors and Two-Rate Approach . . . . . . . . . . . . . . . 1433.1 Multivariate Polynomial Impulse Response Recon-

structors . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.2 Two-Rate Approach . . . . . . . . . . . . . . . . . . . 144

4 Proposed Two-Rate Based Multivariate Polynomial ImpulseResponse Reconstructor . . . . . . . . . . . . . . . . . . . . . 1454.1 Reconstructor Design . . . . . . . . . . . . . . . . . . 146


III Reconstruction of Sub-Nyquist Sampled Sparse

Multi-Band Signals 153

D Efficient Recovery of Sub-Nyquist Sampled Sparse Multi-Band Signals Using Reconfigurable Multi-Channel Analy-sis and Modulated Synthesis Filter Banks 155

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581.1 Contributions and Outline of the Paper . . . . . . . . 159

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 1622.2 Polyphase Decomposition . . . . . . . . . . . . . . . . 1622.3 Generalized Fractional-Delay Filter . . . . . . . . . . . 163

3 Sub-Nyquist Cyclic Nonuniform Sampling of Sparse Multi-Band Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4 Proposed Reconstruction Using Analysis and Synthesis FBs . 1654.1 Unconventional Bandpass Filters . . . . . . . . . . . . 1654.2 Determining βkmℓ

and αkmℓ. . . . . . . . . . . . . . . 169

5 Proposed Efficient Reconstructor . . . . . . . . . . . . . . . . 1725.1 Synthesis and Analysis FBs . . . . . . . . . . . . . . . 1725.2 Computational Complexity . . . . . . . . . . . . . . . 1735.3 Reconfiguration Complexity . . . . . . . . . . . . . . . 175

6 Design of the Proposed Reconstructor . . . . . . . . . . . . . 1756.1 Prototype Filter Design . . . . . . . . . . . . . . . . . 1766.2 Least-Squares Design of Fℓ(z) and Gℓ(z) . . . . . . . . 1776.3 Design of Reconfigurable Reconstructors . . . . . . . . 1796.4 Design Complexity . . . . . . . . . . . . . . . . . . . . 180

7 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1818 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

IV Reconstruction Schemes for TI-ADCs with Missing

Samples 193

E A Sub-Band Based Reconstructor for M-Channel Time-Interleaved ADCs with Missing Samples 1951 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1982 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . 1993 Proposed Reconstructor . . . . . . . . . . . . . . . . . . . . . 200

3.1 Analysis Filters . . . . . . . . . . . . . . . . . . . . . . 2013.1.1 Reconfigurability . . . . . . . . . . . . . . . . 203

3.2 Synthesis Filters . . . . . . . . . . . . . . . . . . . . . 2043.3 Reconstructor Design . . . . . . . . . . . . . . . . . . 204


F Two Reconstructors for M-Channel Time-InterleavedADCs with Missing Samples 2111 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2153 Constrained Time-Varying FIR Reconstructor . . . . . . . . . 217

3.1 Least-Squares Design . . . . . . . . . . . . . . . . . . . 2184 Sub-Band Based Reconstructor . . . . . . . . . . . . . . . . . 218

4.1 Least-Squares Design . . . . . . . . . . . . . . . . . . . 2215 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . 2226 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

G Prefilter-Based Reconfigurable Reconstructor for Time-Interleaved ADCs with Missing Samples 2291 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2322 Background and Prerequisites . . . . . . . . . . . . . . . . . . 2333 Proposed Reconstructor . . . . . . . . . . . . . . . . . . . . . 235

3.1 Two-Mode Time-Varying Prefilter . . . . . . . . . . . 2353.2 Reconfigurable Part . . . . . . . . . . . . . . . . . . . 2363.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . 2393.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

4 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2415 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Part I

Introduction

1

Chapter 1

Background

Today, digital signal processing (DSP) is extensively used in many appli-cations. Advances in the field of microelectronics have made it possible tocreate efficient and cost-effective DSP hardware. Unlike analog electronics,digital systems are much less susceptible to manufacturing process variationsas well as physical variations such as temperature changes. Moreover, deepsub-micron chip fabrication processes are well suited for digital circuits asthe shrinking transistor sizes offer faster switching speeds and allow us topack more digital logic in a given area. On the other hand, analog circuitdesign has become increasingly challenging especially in the deep sub-micronprocesses. All the above reasons have contributed towards the growing pop-ularity of digital systems. However, many of these digital systems mustinteract with the analog world. For this purpose, analog-to-digital as well asdigital-to-analog interfaces are frequently required at the input and output,respectively, of digital systems.

Typically, an analog-to-digital converter (ADC) samples a continuous-timeanalog signal at a predefined rate (sampling rate) to generate a discrete-timesequence of samples. The analog value of each sample is then representedusing a finite number of bits (resolution). The sampling rate of the ADC isselected depending on the bandwidth of the analog input signal. There existmany ADC architectures that are suitable for different ranges of samplingrates and resolutions [1]. Analog-to-digital converters used in applicationslike communication systems and high-speed digitizers should support veryhigh sampling rates and/or resolutions [2]. In such cases, implementing asingle high-sampling rate ADC is quite challenging and at times infeasible. A

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4 Chapter 1. Background

popular technique to increase the effective sampling rate is to have multipleADCs in a time-interleaved fashion with each ADC operating at a lowersampling rate [3].

In theory, by time-interleaving the outputs of M channel ADCs, a time-interleaved analog-to-digital converter (TI-ADC) can achieve the same res-olution as that of the individual ADCs but with M times higher samplingrate. However, in practice, the channel ADCs suffer from nonidealities suchas gain, offset, and timing errors. These nonidealities manifest mainly due toanalog circuit imperfections caused by variations in manufacturing process,voltage, and temperature [4, 5]. Also, the reduced feature size of transistorsin advanced manufacturing processes make within-die and die-to-die varia-tions more pronounced due to the limited accuracy of the existing lithogra-phy techniques [6]. Due to the random nature of the variations [5–8], eachchannel ADC exhibits different levels of nonidealities which causes channelmismatch errors in TI-ADCs. In a TI-ADC with mismatch errors, the outputis a nonuniformly sampled signal which degrades the achievable resolutionat the output of the TI-ADC [4, 9]. Thus, in order to retain the achievableresolution, TI-ADC implementations must either avoid mismatches betweenthe channel ADCs through careful analog circuit design [7] or use calibrationwherein the mismatch errors are estimated and then compensated for. Theformer approach is extremely challenging especially in newer digital-friendlychip manufacturing processes. Hence, TI-ADC implementations often relyon calibration to mitigate the effects of channel mismatch errors.

The mismatch errors can be broadly classified into linear and nonlinear mis-match errors [9–14]. Linear mismatch errors include gain, offset, timingskew, and frequency response mismatches whereas nonlinear mismatch er-rors occur due to mismatches in the nonlinearity of the channel ADCs. Inthis thesis we consider only linear mismatch errors which typically domi-nate [15]. More specifically, since gain and offset errors can be easily com-pensated, here we deal with only time-skew mismatch errors which occuras a result of nonuniform time skews between the sampling clocks of thechannel ADCs [16–18]. In practice, the time-skew errors can be assumedto be frequency independent only up to a certain output resolution andbandwidth [19]. In high-speed TI-ADCs, the time-skew errors are frequencydependent [20, 21]. Thus, to achieve very high resolutions, each channel ina high-speed TI-ADC is modeled as a general frequency response and thecalibration block should compensate for the frequency-response mismatcherrors.

5

The calibration block utilizes the TI-ADC output to estimate the mismatchparameters which are then used to reconstruct the output so as to compen-sate for the mismatch errors. The reconstruction can be performed eitherin the analog domain [22, 23] or in the digital domain [24]. Analog recon-structors use the mismatch estimates to perform the correction in the analogdomain. Such reconstruction schemes are attractive for low power applica-tions but they require significant design effort. Currently, such techniquesappears only to be effective for low- to moderate-resolution ADCs as thesetechniques are more suited for correcting frequency-independent mismatcherrors. On the other hand, digital reconstructors can be used to achieve ar-bitrarily high resolutions. Such reconstructors use the mismatch estimatesto correct the samples at the output of the TI-ADC. However, these re-constructors require several computations per corrected output sample andhence, consume much power. Also, in TI-ADC implementations, the mis-match parameters can change from time to time. The reconstructor shouldbe easily reconfigurable to cope with changes in the mismatch parameters.Thus, a challenge is to reduce the complexity of the reconstructor as well asto make it easily reconfigurable.

Following this introduction, in Part II of this thesis, we propose efficientsignal reconstruction algorithms for conventional TI-ADCs. Here, we assumethat mismatch parameters are estimated and are available beforehand. It isnoted that efficient reconstruction schemes are also beneficial for estimation,for example, where calibration is performed using simultaneous estimationand compensation through the minimization of an appropriate cost measure.We also propose efficient reconstruction schemes for two unconventional TI-ADCs. In Part III, we consider an unconventional TI-ADC which is usedfor the sub-Nyquist cyclic nonuniform sampling of sparse multi-band signals[25]. In such TI-ADCs, only a few of the channel ADCs are active and,hence, the reconstructor should recover the uniform-grid samples from theavailable samples. In the second type of unconventional TI-ADC consideredin Part IV, some of the ADC sampling instants are reserved for estimatingthe mismatch errors [26]. At these time instances, the input signal will notbe sampled which results in missing samples at the output of the TI-ADC.Thus, the reconstructor used in such TI-ADCs should recover the missingsamples and also correct for the mismatch errors.

Before proceeding to Part II, in Chapters 2, 3, and 4, we introduce somebackground materials and provide a brief overview of the thesis. Finally,Chapter 5 summarizes the specific contributions of the thesis.


1.1 Preliminaries

In this section, after describing the notations, we will briefly review some ofthe basic theory and methods that are used throughout the thesis.

1.1.1 Notations

A continuous-time signal is denoted as x(t) whereas x(n) is used to denotea discrete-time signal. Here, t represents the time axis and n is the timeindex. Bold lowercase letters are used to denote vectors while bold upper-case letters are used to denote matrices. Transpose and conjugate-transposeare represented using (·)T and (·)†, respectively. For a filter with impulseresponse coefficients h(n), we use H(z) to denote its transfer function whichis defined as H(z) =

∑n h(n)z

−n. The frequency response of the filter isdenoted by H(ejωT ) and is obtained from the transfer function by replacingz with ejωT .

1.1.2 Uniform Sampling

Let xa(t) represent a continuous-time signal which is bandlimited to ωc <π/T such that its Fourier transform Xa (jω) defined by

Xa (jω) =1

2π

ˆ ∞

−∞

xa(t)e−jωtdt (1.1)

vanishes outside the interval |ω| ≥ ωc. That is,

Xa (jω) = 0, |ω| ≥ ωc. (1.2)

Then, according to the Nyquist sampling theorem [27–29], xa(t) can be re-constructed from a discrete-time signal x(n) obtained by sampling xa(t) attime instances t = nT where T represents the sampling period. Using Pois-son’s summation formula with xa(t) bandlimited as in (1.2), the Fouriertransform X

(ejωT

)of the uniform-sampling sequence x(n) = xa(nT ) can

be written as [30]

X(ejωT

)=

1

TXa (jω) , ωT ∈ [−π, π]. (1.3)

Uniform sampling can be performed using a single ADC as shown in Fig. 1.1.The analog signal xa(t) is first uniformly sampled at time instants t = nTafter which the sampled value is quantized to a finite number of bits to formthe uniform-sampling sequence x(n).

1.1. Preliminaries 7

Figure 1.1: Uniform sampling using a single ADC.

1.1.3 FIR Filters

In a causal finite-length impulse response (FIR) filter of order N , the impulseresponse coefficients h(n) can be nonzero only for 0 ≤ n ≤ N and are zerofor all other values of n. Thus, its transfer function H(z) is given by

H(z) =N∑

n=0

h(n)z−n. (1.4)

In the case of a linear-phase FIR filter, the impulse response coefficients aresymmetric or anti-symmetric. The frequency response of a linear-phase FIRfilter can be expressed in terms of a real function HR(ωT ) such that

H(ejωT ) = HR(ωT )ejΦ(ωT ) (1.5)

where

Φ(ωT ) = −N

2ωT + c. (1.6)

In (1.6), c = 0 if the filter coefficients h(k) are symmetric and if the coeffi-cients are anti-symmetric, c = π/2. The real function HR(ωT ) is called thezero-phase frequency response of H(ejωT ) [31]. In the case of a noncausallinear-phase FIR filter, centered at n = 0, H(ejωT ) = HR(ωT ).

Let x(n) and y(n) represent the input and output, respectively, of a causalNth-order FIR filter with impulse response h(n). Then, the filtered outputy(n) is obtained by convolving x(n) with h(n) such that

y(n) =

N∑

k=0

x(n− k)h(k). (1.7)

It can be seen from (1.7) that in order to compute each output sample, anNth-order FIR filter requiresN+1 multiplications, N additions, andN delayelements. In a linear-phase FIR filter, due to the symmetry of the impulseresponse coefficients, we require only around N/2 multipliers [31–33].


Figure 1.2: (a) Interpolation by M . (b) Example spectrum of the input,upsampled, and interpolated sequences with interpolation factor M = 3 andan ideal filter H(z). Here, T1 = T/3.

1.1.4 Interpolation

Interpolation is the process of increasing the sampling frequency of a se-quence x(n). As shown in Fig. 1.2(a), interpolation by a factor of M in-volves upsampling by M followed by filtering. The upsampler inserts M − 1zeros between the samples in x(n) such that the upsampled sequence xu(ν)becomes

xu(ν) =

{x(n), ν = nM

0, otherwise.(1.8)

If X(z) represents the z-transform of x(n), then the z-transform of xu(n) isgiven by

Xu(z) = X(zM ). (1.9)

Thus, due to upsampling, M − 1 image terms appear in the frequency bandωT ∈ [0, 2π). These unwanted images are removed using the interpolationfilter H(z) and we obtain the interpolated output y(n). Figure 1.2(b) showsan example spectrum of x(n), xu(ν), and y(ν) when the input x(n) is inter-polated by a factor of three (M = 3) using an ideal interpolation filter H(z).


Figure 1.3: (a) Decimation by M . (b) An example spectrum of x1(n) andthe spectrum of the corresponding decimated sequence y(m) when the dec-imation factor is M = 3. Here, T2 = 3T .

Note that the passband gain of H(z) should be equal to M to preserve thesignal power.

1.1.5 Decimation

Decimation is the process of reducing the sampling rate of a sequence. Deci-mation by a factor of M involves decimation filtering followed by downsam-pling by M as shown in Fig. 1.3(a) [32,33]. The downsampler block shownin Fig. 1.3(a) is used to perform the downsampling operation wherein thedownsampled sequence y(m) contains only every Mth sample in x1(n). Thatis,

y(m) = x1(mM). (1.10)

The z-transform of the downsampled sequence y(m) is given by

Y (z) =1

M

M−1∑

q=0

X1(z1/M e−j2πq/M ). (1.11)

In the summation in (1.11), the terms q = 1, 2, . . . ,M − 1, correspond toM − 1 aliasing terms. Hence, to prevent aliasing into the signal band, adecimation filter H(z) is used to bandlimit x(n) to π/M as shown in Fig.1.3(a). As illustrated for M = 3 in Fig. 1.3(b), with x1(n) bandlimitedto π/M , aliasing terms due to the downsampler will not fall into the signalband.


Figure 1.4: Noble identities.

Figure 1.5: Equivalent representation of the interpolator in Fig. 1.2(a) usingthe M polyphase branches of the filter H(z).

1.1.6 Polyphase Decomposition

Any filter H(z) can generally be expressed in terms of its polyphase compo-nents Hp(z), p = 0, 1, . . . ,M − 1, as [32,34]

H(z) =

M−1∑

p=0

z−pHp(zM ). (1.12)

Polyphase decomposition as in (1.12) along with the noble identities shownin Fig. 1.4 [32], can be used to derive efficient structures for interpolationand decimation. For example, consider the interpolator shown in Fig. 1.2(a).ExpressingH(z) in Fig. 1.2(a) as in (1.12) and then propagating the upsam-pler to the right using the noble identity shown in Fig. 1.4(b), we get thepolyphase structure in Fig. 1.5. It can be seen that, unlike in Fig. 1.2(a),in the polyphase structure the filtering takes place at the lower rate. Inpractice, the upsamplers, delays, and additions in Fig. 1.5 can be replacedwith a commutator [32].


Figure 1.6: M -channel maximally decimated filter bank.

1.1.7 Filter Banks

Consider the M -channel maximally decimated filter bank (FB) shown inFig. 1.6 [32,35]. The z-transform of the output of the analysis filter Bk(z),k ∈ 0, 1, . . . ,M − 1, can be written as

Uk(z) = Bk(z)X(z) (1.13)

where X(z) is the z-transform of the input x(n). Using (1.13), the z-transform of the corresponding decimated output uk(ν) can be written as

Uk(z) =1

M

M−1∑

q=0

Uk

(z1/Me−j2πq/M

)

=1

M

M−1∑

q=0

Bk

(z1/Me−j2πq/M

)X

(z1/Me−j2πq/M

). (1.14)

Since, yk(n) is obtained by upsampling uk(ν) by M , its z-transform Yk(z) isgiven by

Yk(z) = Uk(zM ) =

1

M

M−1∑

q=0

Bk

(ze−j2πq/M

)X

(ze−j2πq/M

). (1.15)

The z-transform of yk(n) can be expressed using (1.15) as

Yk(z) =1

MCk(z)

M−1∑

q=0

Bk

(ze−j2πq/M

)X

(ze−j2πq/M

). (1.16)


Then, the z-transform of the output Y (z) is given by

Y (z) =

M−1∑

k=0

Yk(z) = V0(z)X(z) +

M−1∑

q=1

Vq (z)X(ze−j2πq/M

)(1.17)

where

Vq(z) =1

M

M−1∑

k=0

Ck(z)Bk

(ze−j2πq/M

)(1.18)

for q = 0, 1, . . . ,M − 1. In (1.17), V0(z) is the distortion function and Vq(z),q = 1, 2, . . . ,M − 1, are the M − 1 aliasing terms. It can be seen from (1.17)that with V0(z) = 1 and Vq(z) = 0 for q = 1, 2, . . . ,M − 1, we have perfectreconstruction (PR). That is, y(n) = x(n).

1.1.8 Cosine-Modulated Filter Banks

Approximate PR can be achieved by representing the impulse response co-efficients of the analysis and synthesis filters in terms of a power-symmetriclowpass prototype filter P (z) with cutoff frequency at π/2M , according to

bk(n) = 2p(n) cos

[π

M(k + 0.5)

(n− N

2

)+ (−1)k

π

4

](1.19)

and

ck(n) = 2p(n) cos

[π

M(k + 0.5)

(n− N

2

)− (−1)k

π

4

](1.20)

where p(n) represents the impulse response coefficient of P (z) with filterorder N . Expressing the analysis and synthesis filters as in (1.19) and (1.20)allows us to implement the FB in Fig. 1.6 using the efficient structure in Fig.1.7 [35]. In Fig. 1.7, the Pq(z), q = 0, 1, . . . , 2M − 1, are the 2M polyphasecomponents of P (z) such that

P (z) =

2M−1∑

q=0

z−qPq(z2M ). (1.21)

The complexity of the cosine-modulation block in the analysis FB (synthesisFB) in Fig. 1.7 can be reduced by using a fast-transform algorithm [36, 37]whereas for the prototype filter, only N/M multiplications per input/outputsample are required.


Figure 1.7: Efficient realization of the filter bank in Fig. 1.6 using a cosine-modulated FB.


1.1.9 Vandermonde Matrices

A (K+1)×(K+1) matrixV is a Vandermonde matix if it has the form [38,39]

V =

1 1 · · · 1

x1 x2 · · · xK

x21 x22 · · · x2K

......

. . ....

xK1 xK2 · · · xKK

. (1.22)

The Vandermonde matrix is invertible if and only if the values xk, k =1, 2, . . . ,K − 1, in (1.22) are distinct [32]. The transpose of V is also aVandermonde matrix.

A generalized (K + 1)× (K + 1) Vandermonde matrix has the form [38]

V =

ab11 ab12 · · · ab1K

ab21 ab22 · · · ab2K

......

. . ....

abK1 abK2 · · · abKK

. (1.23)

Chapter 2

Mismatch Error Correction

in TI-ADCs

This chapter provides an overview of Part II of the thesis where we considerreconstruction schemes for conventional TI-ADCs.

2.1 Time-Interleaved ADCs

Analog-to-digital converters supporting very high sampling rates often usetime-interleaving of multiple ADCs to reduce the requirements on the in-dividual ADCs [40]. In an M -channel TI-ADC, the continuous-time signalxa(t) is sampled using M parallel ADCs as shown in Fig. 2.1(a) [3]. Thesampling clocks to the channel ADCs are applied in such a way that, at anygiven time instant, only one channel ADC samples the input. In an idealTI-ADC, the mth channel ADC samples the input xa(t) as shown in Fig.2.1(b).1 In this case, since xm(ν) = xa(νMT + mT ) = x(νM + m), theoutput of the mth channel ADC xm(ν) can be considered as being obtainedfrom the uniform-sampling sequence x(n) as shown in Fig. 2.1(c) wherez = ejωT . Hence, as discussed in Section 1.1.5, the Fourier transform of thedownsampled sequence xm(ν) can be written as

Xm

(ejωT

)=

1

M

M−1∑

k=0

ejωT−2πk

MmX

(ej

ωT−2πkM

). (2.1)

1To simplify the TI-ADC model, in Fig. 2.1 we have ignored the quantizer block whichfollows the sampler in the channel ADC.

15

16 Chapter 2. Mismatch Error Correction in TI-ADCs

Figure 2.1: (a) Ideal M -channel TI-ADC. (b) Block diagram of the mthchannel ADC. (c) Equivalent representation of (b).

Figure 2.2: Equivalent multirate representation of the ideal M -channel TI-ADC in Fig. 2.1(a).

2.1. Time-Interleaved ADCs 17

Now, the interleaved output x(n) in Fig. 2.1(a) can be expressed as

x(n) =M−1∑

m=0

xm(n) (2.2)

where xm(n), m = 0, 1, . . . ,M − 1, are obtained as shown in Fig. 2.2. LetXm

(ejωT

)represent the Fourier transform of xm(n). Then, using (2.1), we

get

Xm

(ejωT

)= e−jωmTXm

(ejωTM

)

=1

M

M−1∑

k=0

e−j 2πkM

mX(ej(ωT− 2πk

M)). (2.3)

Now, taking the Fourier transform on both sides of the equality in (2.2) andusing (2.3), we obtain

X(ejωT

)=

1

M

M−1∑

k=0

X(ej(ωT− 2πk

M))M−1∑

m=0

e−j 2πkM

m = X(ejωT

). (2.4)

The second equality in (2.4) follows from∑M−1

m=0 exp(−j2πkm/M ) = 0 fork 6= 0. Thus, the continuous-time input xa(t) is uniformly sampled by theTI-ADC if the sampling clocks to the channel ADCs are applied as shown inFig. 2.1(b). Hence, the time-skew between the sampling clocks of any twoadjacent ADCs, ADCm and ADCm+1, should be equal to T . However, inpractice, due to mismatches in the channel ADCs and clock routing network,the time-skew between the adjacent channels will not be uniform, resultingin a nonuniform-sampling sequence v(n) at the output of the TI-ADC.

2.1.1 Static Time-Skew Errors in TI-ADCs

At moderate sampling rates and resolutions, the time-skew error can beapproximated as static which means that the time-skews are frequency-independent [19]. In this case, the nonuniform-sampled sequence v(n) canbe written as

v(n) = xa(nT + εnT ) (2.5)

where εn is the percentage deviation of the nth sample from the desiredsampling instant nT . Using (1.2) and (1.3), we can rewrite v(n) using theinverse Fourier transform of x(n) as

v(n) =1

2π

ˆ ωcT

−ωcTejωTεnX

(ejωT

)ejωTnd(ωT ). (2.6)


Figure 2.3: (a) Uniform sampling of xa(t) in an ideal three-channel TI-ADC.(b) Three-periodic nonuniform sampling of xa(t) in a three-channel TI-ADCwith time-skew errors.

Since the output samples in a TI-ADC are formed by interleaving the outputsfrom each channel ADC, the deviations (time-skew errors) are periodic. Inan M -channel TI-ADC, the time-skew errors are M -periodic such that

εn = εn+M . (2.7)

For example, as shown in Fig. 2.3(b), the time-skew errors in a three-channelTI-ADC will be three-periodic with ε3n = ε0, ε3n+1 = ε1, and ε3n+2 = ε2.Figure 2.4 shows the effect of time-skew errors in a three-channel TI-ADC.In Fig. 2.4(a), x(n) is a uniform-sampling sequence at the output of theideal three-channel TI-ADC and v(n) is the corresponding TI-ADC outputsequence when the channel ADCs have static time-skew errors. Figure 2.4(b)shows the amplitude spectrum of x(n), v(n), and a reconstructed sequence.It can be seen that, due to time-skew errors, aliasing terms appear at theoutput of the TI-ADC, degrading the achievable resolution. As illustratedin Fig. 2.4(b), a reconstructor can be used to suppress the aliasing terms atthe output of the TI-ADC. It is noted that, using an appropriate filter orderfor the reconstructor, the aliasing terms can be suppressed even further.Figure 2.5 shows the plot of spurious-free dynamic range (SFDR) versusεmax for a two-channel TI-ADC with time-skew errors ε0,1 = [εmax,−εmax]and where the input is bandlimited to |ωT | ≤ 0.9π. It can be seen that, asthe magnitude of the time-skew errors increases, the achievable resolution atthe output of the TI-ADC decreases.

2.1. Time-Interleaved ADCs 19

0 5 10 15 20 25n

0 5 10 15 20 25n

xa(t) x(n)

xa(t) v(n)

(a) Time-domain signals

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ωT [×π rad]

|X(e

jωT)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ωT [×π rad]

|V(e

jωT)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ωT [×π rad]

|X(e

jωT)|

(b) Amplitude spectrum of the uniform-sampling (top), nonuniform-sampling (middle), and reconstructed (bottom) sequences.

Figure 2.4: Time-domain signals and the corresponding amplitude spectra atthe output of an ideal and a nonideal three-channel TI-ADC. In the nonidealTI-ADC, the channel ADCs have static time-skew errors.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.110

15

20

25

30

35

40

εmax

SFDR

[dB]

Figure 2.5: SFDR vs. εmax in a two-channel TI-ADC with time-skew errorsε0,1 = [εmax,−εmax] and when the input is bandlimited to |ωT | ≤ 0.9π.

2.1.2 Channel Frequency Response Mismatches

In high-speed TI-ADCs supporting very high resolutions, the TI-ADC modelshould be extended to include the channel frequency responses Qm(jω), m =0, 1, . . . ,M − 1, as shown in Fig. 2.6 [11, 20, 41]. Here, assuming that thecontinuous-time input xa(t) is bandlimited as in (1.2), the TI-ADC outputv(n) can be considered as obtained by sampling the output of a time-varyingcontinuous-time system such that

v(n) =1

2π

ˆ ωcT

−ωcTQn(jω)X

(ejωT

)ejωTnd(ωT ) (2.8)

where Qn(jω) = Qn+M (jω), ∀n ∈ Z, and X(ejωT

)represents the Fourier

transform of the uniform-sampling sequence as in (1.3). It can be seenthat with Qn(jω) = ejωTεn , (2.8) reduces to v(n) = xa(nT + εnT ) whichcorresponds to the nonuniform-sampling sequence at the output of a TI-ADC with static time-skew errors.

2.2. Time-Varying FIR Reconstructors 21

Figure 2.6: Model of an M -channel TI-ADC with M different channel fre-quency responses Qm(jω), m = 0, 1, . . . ,M − 1, and a time-varying recon-structor Hn(e

jωT ).

2.2 Time-Varying FIR Reconstructors

In TI-ADC implementations, a digital reconstructor Hn

(ejωT

)can be

used to recover the uniform-sampling sequence x(n) from the nonuniform-sampling sequence v(n) as shown in Fig. 2.6 [41–56]. The reconstructoris a time-varying FIR filter whose impulse-response coefficients hn(k), aredetermined such that the reconstructed output x(n), given by

x(n) =

N/2∑

k=−N/2

v(n− k)hn(k), (2.9)

approximates x(n) [57]. Here, N is the order of the reconstructor. We as-sume noncausal even-order filters to simplify the design and analysis.2 Since,the input of an M -channel TI-ADC with mismatches, is M -periodicallynonuniformly sampled [57–59], the impulse response coefficients of the recon-structor are also M -periodically time-varying. That is, hn(k) = hn+M (k).Further, in practice, the channel mismatch errors are estimated by using oneof the channel ADCs as a reference channel. Due to this, the samples fromthe reference channel require no correction and, hence, hn(k) for the referencechannel is a pure delay. Thus, for an M -channel TI-ADC, the time-varyingFIR reconstructor can be realized using M − 1 separate FIR filters [57]. Itis noted that hn(k) is centered at the sample to be reconstructed.

2 The filters can be easily converted to causal filters by introducing suitable delays.Further, with minor modifications, all the derivations can be applied to the odd-order caseas well.


Substituting (2.8) in (2.9) and rearranging the terms, we get

x(n) =1

2π

ˆ ωcT

−ωcTAn(jω)X

(ejωT

)ejωTnd(ωT ) (2.10)

where

An(jω) =

N/2∑

k=−N/2

hn(k)Qn−k(jω)e−jωTk. (2.11)

Further, if hn(k) perfectly reconstructs the uniform-grid samples, thenx(n) = x(n). Recall that, using the inverse Fourier transform of X(ejωT )[60,61], we have

x(n) =1

2π

ˆ ωcT

−ωcTX

(ejωT

)ejωTnd(ωT ). (2.12)

Thus, comparing (2.10) and (2.12) we see that to obtain perfect reconstruc-tion, we require

An(jω) = 1, ωT ∈ [−ωcT, ωcT ]. (2.13)

However, in practice, perfect reconstruction (PR) is not feasible with realiz-able filters and, moreover, is not a requirement in TI-ADC implementations.Thus, it is sufficient to determine hn(k) so that An(jω) approximates unityin such a way that the reconstruction error e(n) = x(n)− x(n) is minimizedaccording to a specified error metric.

2.3 Error Metrics and Reconstructor Design

2.3.1 Least-Squares Design

The impulse response coefficients hn(k), n ∈ [0, 1, . . . ,M − 1], can be deter-mined such that An(jω), n ∈ [0, 1, . . . ,M − 1], approximates unity in theleast-squares sense. Here, the aim is to minimize the energy of the errorbetween the desired and the reconstructed sequences. For this purpose, weuse the error power functions Pn given by [57]

Pn =1

2π

ˆ ωcT

−ωcT|An(jω)− 1|2d(ωT ). (2.14)

2.3. Error Metrics and Reconstructor Design 23

Let hn represent the vector ((N+1)×1 matrix) containing the N+1 impulseresponse coefficients of hn(k) such that

hn = [hn(−N/2) hn(−N/2 + 1) . . . hn(N/2)]T . (2.15)

Then, (2.14) can be written in matrix form as

Pn = hTnSnhn − 2bT

nhn +ωcT

π(2.16)

where Sn is an (N + 1) × (N + 1) matrix with elements Sn,kp, k, p =−N/2,−N/2 + 1, . . . , N/2, given by

Sn,kp =1

2π

ˆ ωcT

−ωcT|Qn−k(jω)| |Qn−p(jω)|

× cos (ω(p− k)T + arg {Qn−k(jω)} − arg {Qn−p(jω)}) d(ωT )(2.17)

andbn = [bn,−N/2 bn,−N/2+1 . . . bn,N/2]

T (2.18)

with

bn,k =1

2π

ˆ ωcT

−ωcT|Qn−k(jω)| cos (ωT − arg {Qn−k(jω)}) d(ωT ) (2.19)

for k = −N/2,−N/2 + 1, . . . , N/2. The value of hn that minimizes Pn in(2.16) is obtained by solving

∂Pn∂hn

= 0 (2.20)

which giveshn = S−1

n bTn . (2.21)

Thus, using a least-squares approach, the coefficients of each Nth-order FIRreconstructor hn(k) can be determined through an (N+1)× (N+1) matrixinversion.

2.3.2 Minimax Design

The SFDR of an ADC is a crucial metric especially in communication ap-plications [62]. In such applications, the reconstructor is required to ensurethat the maximum amplitudes of the spurious frequency components due to


nonuniform sampling, are kept below a specified level. In such cases, hn(k),n = 0, 1, . . . ,M−1, can be determined such that An(jω) approximates unityin the minimax sense. However, a more appropriate measure is obtained byusing a distortion function V0

(ejωT

)and M−1 aliasing functions Vm

(ejωT

),

m = 1, 2, . . . ,M − 1. Using these functions, the input-output relation of thereconstructor in the frequency domain can be written as (see (1.17) in Sec-tion 1.1.7) [57]

X(ejωT

)= V0

(ejωT

)X

(ejωT

)+

M−1∑

m=1

Vm

(ejωT

)X

(ej(ωT−2πm/M)

). (2.22)

Here,

V0

(ejωT

)=

1

M

M−1∑

n=0

An

(ejωT

)(2.23)

and

Vm

(ejωT

)=

1

M

M−1∑

n=0

e−j2πmn/M An

(ej(ωT−2πm/M)

). (2.24)

In (2.23) and (2.24), An

(ejωT

), n ∈ [0, 1, . . . ,M−1], is the 2π-periodic exten-

sion of An(jω) in (2.11). It can be noted from (2.22) that with V0

(ejωT

)= 1

and Vm

(ejωT

)= 0, m = 1, 2, . . . ,M − 1, we attain perfect reconstruction.

However, to achieve a specified SFDR, it is sufficient to ensure that thedistortion and aliasing functions approximate unity and zero, respectively,within the band ωT ∈ [−ωcT, ωcT ]. Thus, the filter coefficients hn(k) canbe determined such that for a maximum specified reconstruction error δ,

|Vm

(ejωT

)− am| ≤ δm, ωT ∈ [−ωcT +

2πm

M,ωcT +

2πm

M]. (2.25)

with δm < δ for m = 1, 2, . . . ,M − 1. Typically, the requirements on thedistortion error δ0 is not the same as the errors in the aliasing terms, δm,m = 1, 2, . . . ,M − 1. In (2.25), a0 = 1 and am = 0 for m = 1, 2, . . . ,M − 1.Unlike the least-squares design which can be carried out online, the minimaxapproach is more suitable for offline design.

2.4 Reconstructor Complexity

In the literature, the complexity of a reconstructor is often measured in termsof the number and type of multipliers required to implement the reconstruc-tor [57, 63]. For example, variable-coefficient multipliers are required if the

2.5. Low-Complexity Reconstruction Schemes 25

filter coefficients in the reconstructor have to be redetermined whenever themismatch parameters change. In such cases, an additional online redesignblock is required to redetermine the coefficients, resulting in increased chiparea and power consumption. As can be seen from (2.21), in an M -channelTI-ADC, online redesign of the time-varying FIR reconstructor would re-quire M − 1 separate (N+1) × (N+1) matrix inversions. Thus, it is stillimportant to have a simpler online redesign block.

Unlike the online redesign block, the reconstructor always runs at the outputof the TI-ADC. Hence, a rough measure of the overall power consumptionis obtained from the number of arithmetic operations per corrected outputsample (computational complexity) as well as the number of delay elementsin the reconstructor. In regular time-varying FIR reconstructors [57], thenumber of adders and delay elements scale proportionally with the numberof multipliers. Hence, the computational complexity of the reconstructor ismeasured in terms of the number of multiplications per corrected outputsample. It can be noted from (2.9) that, in the time-varying FIR recon-structor, each corrected output sample requires N +1 multiplications wherethe value of N depends on the magnitude of the mismatch errors, the band-width supported by the reconstructor, and how small the reconstruction er-ror should be. Further, in reconstructor implementations, variable-coefficientmultipliers are used for filter coefficients whose values are redetermined on-line. However, compared to variable-coefficient multipliers, efficient tech-niques can be used to implement fixed-coefficient multipliers [64,65]. Thus,in order to reduce the area and power consumed by the reconstructor, itis desirable to have a reconstruction scheme with as few variable-coefficientmultipliers as possible.

2.5 Low-Complexity Reconstruction Schemes

In the papers included in Part II of this thesis, we consider the reconstruc-tion of uniform-grid samples from the nonuniformly sampled signal at theoutput of TI-ADCs. There, we assume that the channel ADCs have onlystatic time-skew errors. Among the reconstructors used in such TI-ADCs,the regular time-varying FIR reconstructor [57] requires the minimal numberof multiplications per corrected output sample. However, all the coefficientsin the regular reconstructor have to be redetermined online when the time-skew errors change. Hence, all the filter coefficients in this reconstructor are


implemented using expensive variable-coefficient multipliers. At the otherend of the complexity spectrum is the reconstructor which makes use ofdifferentiator-multiplier cascade (DMC) [18]. For a given specification, theDMC reconstructor requires the least number of variable-coefficient multi-pliers.3 Also, the DMC reconstructor does not require an online redesignblock as it can be reconfigured by directly updating the coefficients of thevariable multipliers with the newly estimated time-skew errors. However,due to a cascaded structure which does not allow the sharing of delay ele-ments, the DMC reconstructor requires more delay elements as well as longerdelays compared to the regular reconstructor. In Part II, we propose threereconstruction schemes that offer trade-offs between online redesign and re-constructor complexities. To reduce the complexity, these reconstructorsutilize a two-rate based approach [66] which earlier has been used only foruniformly sampled signals [67–75]. However, here, the two-rate based ap-proach is extended for the reconstruction of nonuniformly sampled signalswhich requires new design techniques.

2.5.1 Two-Rate Based Approach

The two-rate based approach, on a principle level, is shown in Fig. 2.7(b).Here, the input is first interpolated by a factor of two by using an upsamplerand a half-band filter F (z). The interpolated nonuniformly sampled signalis then reconstructed using the reconstructor Gn(z). Finally, a downsampleris used to make the output rate equal to that of the input. In practice, thetwo-rate based structure in Fig. 2.7(b) is implemented using an equivalentsingle-rate structure as explained in the papers of Part II. Compared to theregular FIR reconstructor in Fig. 2.7(a), the single-rate structure achieveslower complexity. This is because the majority of the multipliers can nowbe implemented using fixed-coefficient multipliers whereas, due to the in-terpolation, Gn(z) requires only a low order reconstructor and hence, fewervariable-coefficient multipliers.

The principle behind the reduction in the complexity of the filter Gn(z) isillustrated in Fig. 2.8 which plots the variation of the reconstructor orderwith its bandwidth ωcT . In Fig. 2.8, the reconstructor is required to recoverthe uniform-grid samples from the output of a four-channel TI-ADC with

3A special case is for M = 2 where the structure in [59] is the more efficient than theDMC reconstructor.

2.5. Low-Complexity Reconstruction Schemes 27

Figure 2.7: (a) Regular time-varying FIR reconstructor. (b) Two-rate basedreconstructor.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

10

20

30

40

50

60

Figure 2.8: Filter order versus reconstructor bandwidth for Hn(z) in a four-channel TI-ADC with time-skew errors ε0,1,2,3 = [−0.02, 0.02,−0.02, 0.02]and with reconstruction error Pn ≤ −80 dB.


channel time-skew errors ε0,1,2,3 = [−0.02, 0.02,−0.02, 0.02]4 such that, afterreconstruction, the error Pn in (2.14) is below −80 dB. It can be seen thatas the bandwidth ωcT increases, the order of the reconstructor increasesroughly by K(π − ωcT )

−1 where K is a constant. As illustrated, if thebandwidth of the reconstructor is ωcT = 0.9π, the regular time-varying FIRreconstructor Hn(z) would require a filter order of NHn = 28. However,due to the interpolation in the two-rate based approach shown in 2.7(b), thebandwidth to be supported by the Gn(z) reconstructor is reduced to 0.45π.Hence, the order required for Gn(z) is lower than that of Hn(z). Also, sincethe filter F (z) is a half-band FIR filter, its every other impulse responsecoefficient is equal to zero [32]. Further, since F (z) is a linear-phase FIRfilter, the nonzero impulse response coefficients are symmetric, and hence,can be implemented with half the number of multipliers.

In the proposed reconstructors in Part II, F (z) is designed offline and itscoefficients are fixed. The coefficients of F (z) are determined such that theycan be used for all εn ∈ [−εmax, εmax]. Even though the design is carriedout offline, it can still be time consuming with design times ranging fromseveral minutes to hours, especially for small reconstruction errors, widerbandwidths and/or for larger M . However, at the cost of a marginal increasein the filter order of Gn(z), the offline design of F (z) can be simplified byusing a standard half-band filter that can be designed straightforwardly.

In Paper A of Part II, we propose a two-rate based reconstructor for two-channel TI-ADCs which is a popular TI-ADC configuration [76–79]. The ba-sic two-rate based approach in Paper A is extended to a general M -channelTI-ADC reconstruction scheme in Paper B. Compared to the regular re-constructor, the reconstructor in Paper B requires fewer variable-coefficientmultipliers and simpler online redesign block. Though the DMC reconstruc-tor [18] requires fewer variable multipliers as well as no online redesign, thereconstructor in Paper B requires significantly fewer delay elements whichalso needs to be taken into account while comparing complexities espe-cially in low-power applications like application-specific integrated circuits(ASICs).

Finally, in Paper C, we use the two-rate based approach to extend the multi-variate impulse response reconstructor, originally proposed in [80]. Thisreconstruction scheme is attractive for M -channel TI-ADCs with a small

4It is noted that, in practice, for TI-ADC implementations, the time-skew error of thereference channel is assumed to be zero.

2.6. Summary 29

maximum possible time-skew error εmax. Unlike the reconstruction scheme inPaper B, the scheme in Paper C requires no online redesign block. However,for a specific design example, it was seen that this reconstructor requiredmore operations per corrected output sample than the other reconstructors.

2.6 Summary

In this chapter we reviewed the effect of channel mismatch errors in TI-ADCs. It was shown that due to mismatches between the channel ADCs,aliasing terms appear at the TI-ADC output which degrades the achievableresolution. We reviewed two different errors metrics and the correspondingdesign for time-varying FIR filters. Further, we discussed that the complex-ity of the reconstructor can be measured in terms of the number of operationsper corrected output sample, the number of fixed and variable-coefficientmultipliers that are required to implement the reconstructor, the number ofdelay elements, and the complexity involved in the online redesign of thereconstructor. The existing reconstruction scheme that gives minimal com-putational complexity, overall delay, and number of delay elements has highredesign complexity. On the other hand, the scheme which has no redesigncomplexity requires more arithmetic operations and higher overall delay andnumber of delay elements. Finally, for correcting static time-skew errors inTI-ADCs, we propose three digital reconstructors which allows the designerto make trade-offs between online redesign and reconstructor complexities.

Chapter 3

Sub-Nyquist Sampling of

Sparse Multi-Band Signals

It is well known that uniform sampling of a signal which is bandlimitedto f < f0, at a sampling frequency of fs ≥ 2f0, results in a uniformlyspaced sequence of samples that can be used to reconstruct the originalsignal. However, in many cases, the signal is sparse in a sense that the actualinformation is contained in a bandwidth much less than f0. One exampleis a frequency-hopping communication system where there are one or morenarrowband carriers (active subbands) that change their center frequencieswithin the band [0, f0) at a certain switching rate. In other words, suchsignals are locally narrowband (in a time frame) but globally wideband (overseveral time frames). In such cases, the traditional approach would require ahigh-speed ADC operating at a rate of fs ≥ 2f0. Hence, within a time frame,the signal is heavily oversampled and the ADC will unnecessarily consumea substantial amount of power.

Sub-Nyquist sampling is becoming increasingly popular in wideband com-munication systems, especially in battery-powered applications where high-speed uniform sampling results in higher power consumption. In such sam-pling schemes, the average sampling rate can be much lower than 2f0 butstill large enough to capture the information content in the signal. In thischapter, we focus on the multi-band (or multi-coset) sampling approachwhere the use of cyclic nonuniform sampling (CNUS) helps to reduce theaverage sampling rate to (in principle) the Landau minimal sampling ratewhich is determined by the frequency occupancy [25]. It is known that,

31

32 Chapter 3. Sub-Nyquist Sampling of Sparse Multi-Band Signals

Figure 3.1: Spectrum of a sparse multi-band signal with M = 32 and fouractive users occupying K = 8 active granularity bands. The top and bottomplots show the occupied granularity bands at two different time frames.

given the sampling pattern for the CNUS approach, the reconstruction canbe carried out via a set of ideal multi-level synthesis filters [81]. However,the straightforward CNUS reconstruction filters have very high design andreconstructor complexity. Also, in spread-spectrum communication systemswhere the active subband locations are different for different time frames,the reconstruction scheme should support online reconfigurability withoutincreasing the complexity.

3.1 Sub-Nyquist Cyclic Nonuniform Sampling

Assume that xa(t) is a real-valued continuous-time signal that carries infor-mation within the frequency band ω ∈ (−2πf0, 2πf0), f0 < 1/(2T ). Uni-form sampling of xa(t) at a sampling frequency of fs = 1/T results in adiscrete-time sequence x(n) = xa(nT ). For the sake of simplicity, hereafterwe assume that T = 1. Now it is assumed that the band ω ∈ [0, π] is dividedinto M granularity bands of equal width π/M . In sparse multi-band signals,at any given time frame, only K of the M granularity bands (K < M) areallocated to users. Here, we use ri ∈ [0, 1, . . . ,M − 1], i = 1, 2, . . . ,K todenote the active granularity bands assigned to users. A user can occupyone or several consecutive granularity bands. Further, to be able to designpractical filters, we assume a certain amount of redundancy (oversampling)which corresponds to transition bands between user bands. Figure 3.1 showsthe principle spectrum of a sparse multi-band signal when M = 32 andK = 8. The top plot in Fig. 3.1 corresponds to the scenario where the ac-tive granularity bands are r1,2,3,4,5,6,7,8 = [6, 7, 14, 15, 22, 23, 24, 25] whereas

3.1. Sub-Nyquist Cyclic Nonuniform Sampling 33

Figure 3.2: (a) Equivalent representation of the available samples uℓ(ν) =x(Mν−mℓ), ℓ = 1, 2, . . . ,K, when the input x(n) is obtained via sub-NyquistCNUS. (b) Reconstruction using multi-level synthesis filters [81].

the bottom plot corresponds to active granularity bands r1,2,3,4,5,6,7,8 =[6, 7, 8, 9, 15, 16, 24, 25].

In the case of such sparse multi-band signals, uniform sampling will generatemore samples than what is required to prevent information loss. The numberof samples that is generated during the sampling process can be reduced byusing CNUS which only uses a subset x(Mn − mℓ), ℓ = 1, 2, . . . ,K withmℓ ∈ [0, 1, . . . ,M − 1] of the uniform samples x(n). The available samplesuℓ(ν) = x(Mν − mℓ) can be considered as obtained from the uniform-gridsamples x(n) as shown in Fig. 3.2(a). A practical implementation of theCNUS is anM -channel TI-ADC where only a subset of the channels are used.By properly selecting the sampling instants mℓ [82–84], a reconstructor canbe used to recover the uniformly sampled sequence x(n) from x(Mn −mℓ)for a given set of K granularity bands.


3.2 Reconstruction Using Multi-Level Synthesis

Filters

A reconstruction scheme using a set of K multi-level synthesis filters Aℓ(z),ℓ = 1, 2, . . . ,K, as shown in Fig. 3.2(b), was proposed in [81]. Below, weshow that using ideal multi-level synthesis filters Aℓ(z) in Fig. 3.2(b) we can,in principle, perfectly recover x(n) from the available samples uℓ(ν) [81].

To show this, we start with the z-transforms of the sequences xℓ(n), ℓ =1, 2, . . . ,K in Fig. 3.2(b) which can be expressed as

Xℓ(z) = z−mℓX(z). (3.1)

Since the sequences uℓ(ν) in Fig. 3.2(b) are obtained by decimating xℓ(n),their z-transforms can be written as

Uℓ(z) =1

M

M−1∑

q=0

Xℓ

(z1/Me−j2πq/M

). (3.2)

Using (3.1) and (3.2), the z-transforms of yℓ(n) can be written as

Yℓ(z) = Uℓ(zM ) =

1

M

M−1∑

q=0

Xℓ

(ze−j2πq/M

)

=1

M

M−1∑

q=0

z−mℓej2πqmℓ/MX(ze−j2πq/M

). (3.3)

Now, the z-transforms of xℓ(n) are given by

Xℓ(z) = zmℓAℓ(z)Yℓ(z) =1

MAℓ(z)

M−1∑

q=0

ej2πqmℓ/MX(ze−j2πq/M

). (3.4)

Then, the Fourier transform of the output X(z) can be written as

X(z) =

K∑

ℓ=1

Xℓ(z) =1

ME0(z)X (z) +

1

M

M−1∑

q=1

Eq(z)X(ze−j2πq/M

)(3.5)

where

Eq(z) =1

M

K∑

ℓ=1

ej2πqmℓ/MAℓ(z) (3.6)

3.2. Reconstruction Using Multi-Level Synthesis Filters 35

for q = 0, 1, . . . ,M−1. In (3.5), the termsX(ze−j2πq/M

), q = 1, 2, . . . ,M−1

correspond to the M − 1 aliasing terms of X(z). Since, for perfect recon-struction, X(z) = X(z), it follows from (3.5) that we must have E0(z) = Mand Eq(z) = 0 for q = 1, 2, . . . ,M − 1. Now, assuming that the magnituderesponses of the ideal synthesis filters Aℓ(z) are zero in the M − K non-active subbands, the filters Aℓ(z) are to be designed such that they cancelthe aliasing terms that fall into the K active subbands. Since the Nyquistband is divided into M non-overlapping subbands, the signal in each ac-tive subband |ω| ∈ [rkπ/M, (rk + 1)π/M ], rk ∈ [0, 1, . . . ,M − 1], fulfills thesampling theorem for bandpass decimation. Hence, the shifted copies of anactive subband will not overlap with itself. Further, from each of the re-maining K − 1 active bands, exactly one shifted copy will also fall into theabove active subband. Due to this, in any active subband, K − 1 aliasingterms will be present. Thus, the frequency response of the K synthesis filtersAℓ(e

jω) in the active subband ω ∈ [rkπ/M, (rk +1)π/M ] can be determinedby solving the systems of K equations

1 1 · · · 1

ej2πq

(k)1 m1M ej

2πq(k)1 m2M · · · ej

2πq(k)1 mKM

......

. . ....

ej2πq

(k)K

m1M ej

2πq(k)K

m2M · · · ej

2πq(k)K

mKM

︸︷︷︸E(k)

A1(ejω)

A2(ejω)...

AK(ejω)

=

M0...0

(3.7)

where q(k)i ∈ [1, 2, . . . ,M − 1], i = 1, 2, . . . ,K, correspond to the 2πq

(k)i /M -

shifted copies (aliasing terms) of X(z) that fall into the kth subband. Asolution for (3.7) exists if the matrix E(k) is invertible. For a given set ofactive subband locations, this can be ensured through a proper selection ofthe sampling instants mℓ [81–84].

Next, we illustrate the design of the multi-level synthesis filters using anexample. Here, assume that M = 8, K = 3, and the active subbands bandsare r1,2,3 = {1, 4, 6}. That is, the spectrum of the uniformly sampled signalX(ejω) is as shown in Fig. 3.3. For the sampling points m1,2,3 = {0, 3, 5}, wewill have three synthesis filters A1(z), A2(z), and A3(z). In the following,for simplicity, we assume that the reconstruction is performed using ideal

synthesis filters. As can be seen from Fig. 3.3, the aliasing terms q(1)1,2 =

{3, 4} q(2)1,2 = {3, 7}, and q

(3)1,2 = {1, 4} fall into the subbands r1, r2, and r3,


Figure 3.3: Spectrum of a uniformly sampled sparse multi-band signal andits aliasing terms. Here, M = 8 and K = 3. The shaded terms inX(ej(ω−2πq/M)), q = 1, 2, . . . , 7, are the aliasing terms that fall in the ac-tive subbands.

3.3. Reconstruction Using Analysis and Synthesis Filters 37

respectively. Thus, in the subband r1, (3.7) becomes

1 1 1

ej2π3·0

8 ej2π3·3

8 ej2π3·5

8

ej2π4·0

8 ej2π4·3

8 ej2π4·5

8

A1(ejω)

A2(ejω)

A3(ejω)

=

M00

. (3.8)

Solving (3.8), we get the response of synthesis filters in subband r1 asA1(e

jω) = 4, A2(ejω) = 5.2263e−j1.1781, and A3(e

jω) = 5.2263ej1.1781.Similarly, we get A1(e

jω) = 0, A2(ejω) = 5.6569e−j0.7854, and A3(e

jω) =5.6569ej0.7854 in subband r2, and A1(e

jω) = 4, A2(ejω) = 2.1648e−j0.3927,

and A3(ejω) = 2.1648ej0.3927. The responses of A1(e

jω), A2(ejω), and

A2(ejω) in the negative frequency bands are obtained by taking the complex

conjugate of the responses of A1(ejω), A2(e

jω), and A2(ejω), respectively, in

the corresponding positive frequency bands.

3.2.1 Complexity

In practice, the coefficients of the synthesis filters can be determined straight-forwardly, assuming no a priori relations between the filters. If N is theorder of Aℓ(z), then with polyphase implementation, the reconstructor hasa computational complexity of roughly NK/M multiplications per correctedoutput sample. As M increases, K also increases such that the ratio K/Mremains the same. However, as M increases, the subbands become narrowerand as a result, N increases. Due to this, the reconstructor can becomeintolerably costly in real-time applications. Also, all the synthesis filtershave to be be redesigned if the location of the active subbands change ata later time frame. Since regular filter design is cumbersome to be carriedout online when the filter order is high, the online design complexity of thereconstructor is high.

3.3 Reconstruction Using Analysis and Synthesis

Filters

In Paper D of Part III, we derive an efficient reconstruction scheme, shownhere in Fig. 3.4, by describing the reconstruction in terms of both analysisand synthesis filters as shown in Fig. 3.5. Each analysis filter Bk(z) extractsone of the K active granularity bands. The filtering by Bk(z) is followed by


Figure 3.4: Efficient reconfigurable reconstructor proposed in Paper D.

3.3. Reconstruction Using Analysis and Synthesis Filters 39

Figure 3.5: Reconstruction of sub-Nyquist sampled sparse multi-band signalusing analysis and synthesis filters.

downsampling by M so as to have the extracted active granularity band atthe lower sampling rate fs/M . The low-rate signal is then placed at theoriginal active granularity band location at the higher rate fs via upsam-pling by M followed by bandpass filtering via Ck(z). The synthesis filtersCk(z) thus provide a bank of K different conventional bandpass filters whichcan be implemented using an efficient cosine-modulated FB consisting of aprototype filter and a transform block (see Section 1.1.8). Each analysis fil-ter Bk(z) is an unconventional bandpass decimation filter with K non-zeropolyphase branches. In Paper D, we show that the multi-level synthesisfilters in [81] can be expressed in terms of analysis and synthesis filters asshown in Fig. 3.5. However, to get further insights and understanding ofthe efficient reconfigurable reconstruction scheme described in Paper D, wederive the scheme in Fig. 3.5 using a different approach in Appendix A.

3.3.1 Complexity

In Paper D, we show that expressing the reconstruction problem in termsof analysis and synthesis FB as shown in Fig. 3.5, allows us to derive theefficient reconfigurable reconstructor shown in Fig. 3.4. There, we showthat compared to the reconstructor that uses only a synthesis FB [81], theproposed reconstructor can deliver an order-of-magnitude reduction in com-putational complexity. In addition to this, the proposed reconstructor can bereconfigured online with a single K×K matrix inversion. Table 1 illustratesthe complexity savings of the proposed reconstructor when the informa-tion containing frequency bands are {[3.2–3.8], [7.2–7.8], [11.2–12.8]}×π/16,M = 128, K = 18, and the reconstructor is designed to keep the aliasingterms below −40 dB (for details, see Example 2 in Section 7 of Paper D).


Table 1: Reconstructor complexity comparison.

Reconstructor Complexity5

C N Reconfiguration

Only synthesis FB [81] 164 20934 18 [1163 × 1163]

Proposed 24 648 One [18× 18]

3.4 Summary and Future Extension

In this chapter we reviewed the concept of using CNUS for the sub-Nyquistsampling of sparse multi-band signals. It was shown that a TI-ADC withonly few active channel ADCs can be used to straightforwardly implementthe sub-Nyquist CNUS scheme. We also reviewed the existing reconstructionscheme which uses only a synthesis FB. Here, we introduced a reconfigurablereconstruction scheme that offers substantial reduction in complexity. Theproposed reconstruction scheme is derived by describing the reconstructionin terms of both the analysis and synthesis FB. Since the polyphase com-ponents of the filters in the analysis FB are generalized FD filters, all theanalysis filters are expressed using a common set of fixed subfilters which aredesigned offline. The different analysis filters are then realized using differ-ent sets of multipliers. When the reconstructor is reconfigured online, onlythe coefficients of these multipliers have to be redetermined, thus loweringthe complexity of the online redesign block.

In our discussions, we assumed that the channel ADCs have no mismatcherrors. As seen in Chapter 2, the achievable resolution of a TI-ADC canbe degraded by channel mismatch errors. Using an approach similar to thesub-band based reconstructor in Section 4.2.2 of Chapter 4, the proposedreconstruction scheme can be modified to compensate for static time-skewerrors in the active channel ADCs. As in the case of the sub-band basedreconstructor, here also the reconstructor can be reconfigured easily when-ever the time-skew errors change. However, the above scheme cannot bestraightforwardly used for correcting frequency-response mismatches whichcan be a limiting factor in applications requiring very high sampling rates.Thus, an interesting future extension is to derive a structure that can beused to correct frequency-dependent mismatches as well.

5C and N represent the number of multiplications per corrected output sample and thenumber of multipliers to be updated during reconfiguration, respectively. The reconfigu-ration complexity is the number of online matrix inversions.

Chapter 4

Reconstruction in TI-ADCs

with Missing Samples

4.1 TI-ADCs with Missing Samples

In Chapter 2, we showed that due to mismatches between the channel ADCs,the output of a conventional TI-ADC will be a nonuniformly sampled versionof the bandlimited input signal xa(t). In particular, it was shown that time-skew mismatches between the channel ADCs result in a nonunifom-samplingsequence v(n) given by

v(n) = xa(nT + εnT ) (4.1)

where T is the sampling period and εnT represents, for the nth sample, thedeviation of the actual sampling instant from the uniform sampling instantnT . In the rest of this chapter, we assume T = 1 for simplicity. It wasalso shown in Chapter 2 that digital reconstructors can be used at the out-put of TI-ADCs to compensate for these mismatch errors and approximatelyrecover the uniformly sampled signal x(n). In practice, the mismatch param-eters have to be estimated first before they can be used by the reconstructor.An iterative background estimation scheme that achieves robust estimationcompared to other background estimation schemes was proposed in [26]. Inthis scheme, a known calibration signal is injected to the TI-ADC input atpredefined sampling instants t = rMc, r ∈ Z, as shown in Fig. 4.1. Thus,every Mcth output sample from the TI-ADC corresponds to a nonuniformly

41

42 Chapter 4. Reconstruction in TI-ADCs with Missing Samples

Figure 4.1: Block digram of the background estimation scheme in [26] wherea known calibration signal is used to estimate the mismatch parameters.

sampled version of ca(t). The nonuniformly sampled calibration sequencec(r) is fed to an estimator which compares c(r) with a known reference se-quence cref(r) = ca(rMc), and thereby estimates the mismatch between thechannel ADCs. In order to ensure that c(r) is composed of samples fromall the channel ADCs, in an M -channel TI-ADC, Mc is chosen such thatM and Mc are co-prime. At the input sampling instants reserved for thecalibration signal, the input signal xa(t) will not be sampled by the TI-ADCresulting in missing samples at the TI-ADC output. Thus, the input to thereconstructor, y(n), is given by

y(n) =

{0, n = rMc

v(n), otherwise.(4.2)

Figure 4.2 illustrates this for a three-channel TI-ADC (M = 3) with time-skew errors where every fourth sampling instant (Mc = 4) is reserved for thecalibration signal. Thus, the reconstructor used in such TI-ADCs shouldrecover these missing samples in addition to compensating for the mismatcherrors.

4.2 Reconstruction Schemes

In [26], the reconstructor was realized using an iterative scheme with goodconvergence rate. However, this scheme has a high computational complex-ity which is measured in terms of the number of operations required percorrected output sample. Further, the reconstructor makes use of a recur-sive structure which, in addition to limiting the maximum data rate [85],can also lead to stability problems. In this section, we provide an overviewof the three nonrecursive FIR reconstructors proposed in Part IV.

4.2. Reconstruction Schemes 43

Figure 4.2: (a) Illustration of the sampling instants in a three-channel TI-ADC (M = 3) where every fourth sampling instant (Mc = 4) is reserved forthe calibration signal. (b) Corresponding nonuniformly sampled sequence atthe output of the three-channel TI-ADC.

4.2.1 Constrained Time-Varying FIR Reconstructor

As seen from Section 2.1.1, due to mismatches, the output of a TI-ADCwithout missing samples is an M -periodically nonuniformly sampled versionof the input xa(t). In Section 2.2, it was shown that the uniform-grid sam-ples can be recovered from the M -periodically nonuniformly sampled signalusing an M -periodically time-varying FIR reconstructor, hn(k). Here, dueto the missing samples, the TI-ADC output y(n) is an MMc-periodicallynonuniformly sampled version of xa(t). That is, hn(k) = hn+Mt(k) whereMt = MMc. Hence, we could use an Mt-periodically time-varying FIR re-constructor to recover the uniform-grid samples from y(n). However, unlikethe time-varying reconstructor in [57], here, the missing samples in y(n)restrict some of the impulse response coefficients to be zero. Specifically,

hn(k) = 0, ∀ k = −n+ rMc, r ∈ Z. (4.3)

Assuming that, for a given n, Rn denotes the number of non-zero impulseresponse coefficients in hn(k), the reconstructed output x(n) can be writtenas

x(n) =

Rn∑

i=1

y(n− ki)hn(ki) (4.4)


where ki ∈ [−Nhn/2,−Nhn

/2 + 1, . . . , Nhn/2], i = 1, 2, . . . , R, represent the

indices of the non-zero impulse response coefficients in hn(k). In (4.4), forsimplicity, it is assumed that hn(k) is a noncausal filter (see Footnote 2) witheven order Nhn

.

It was seen in Section 2.2 that in the case of the time-varying reconstructor,how well x(n) approximates the uniform-grid samples x(n) depends on theapproximation error between An(jω) in (2.11) and unity. This is also ap-plicable for the constrained time-varying reconstructor except, here, due tothe missing samples, An(jω) is given by

An(jω) =

Rn∑

i=1

hn(ki)Qn−ki(jω)e−jωki (4.5)

where Qn(jω) represents the channel frequency response. In Paper F, we usea least-squares approach to determine the coefficients of each hn(ki) througha matrix inversion of size Rn × Rn. There, we assume that the TI-ADCsuffers only from static time-skew errors which corresponds to Qn(jω) =ejωεn . In Appendix B.1, using the least-squares approach, we derive a closed-form expression for hn(ki) that can be used to compensate general channelfrequency response mismatches [20].

The constrained time-varying reconstructor requiresMMc FIR filters6 hn(k),n ∈ [0, 1, . . . ,MMc− 1], where the order of each filter, Nhn

, depends on howclose the sample to be reconstructed is to the uniform-grid sample. Thus, thefilter corresponding to the missing sample requires the highest order whichcan be seen from the design example in Paper F. For a given set of time-skewerrors and Mc, the computational complexity of the reconstructor measuredin terms of the number of multiplications per corrected output sample, isminimal for this reconstructor and is equal to

∑MMc−1n=0 Rn/MMc. However,

if the time-skew errors change, the coefficients of all the MMc filters need tobe redetermined through MMc separate matrix inversion resulting in highonline redesign complexity.

6 In TI-ADC implementations, the mismatch parameters are estimated and recon-structed with respect to a reference channel. Thus, in a period of MMc samples, Mc − 1samples from the reference channel are available and require no correction. Hence, inpractice, for the constrained time-varying reconstructor, we require only MMc −Mc + 1separate FIR filters. However, in the design example in Section 4.2.4 as well as in Paper Fof Part IV, we use all the MMc FIR filters to be able to compare with the correspondingexample in [26].


4.2.2 Sub-Band Based Reconstructor

The problem of reconstructing the uniform-grid samples from the output aTI-ADC with missing samples is similar to that of recovering the missingsamples from a sub-Nyquist sampled sparse multi-band signal discussed inChapter 3. Utilizing this observation, in Paper E, it is shown that the Mt-periodically nonuniformly sampled signal at the output of an M -channel TI-ADC with missing samples can be performed using a set of K-channel analy-sis and synthesis FBs whereK = Mt−M . In the sub-band based reconstruc-tor, the Nyquist band is divided into Mt sub-bands of equal length π/Mt andthe firstK bands are assumed to be active. Due to the missing samples, onlythe inputs to the K polyphase branches of the analysis filters Bk(e

jω) arenon-zero. Similar to the reconstructor in the sub-Nyquist CNUS case, here,the K non-zero polyphase components Bkmℓ

(ejω), mℓ ∈ [0, 1, . . . ,Mt − 1],ℓ = 1, 2, . . . ,K, approximate generalized fractional-delay filters so that

Bkmℓ(ejω) ≈ βkmℓ

ej(ω(mℓ+εℓ)/Mt+αkmℓsgn(ω)), ω ∈ (−π, π) (4.6)

where βkmℓand αkmℓ

are the modulus and angle, respectively, of a corre-sponding complex constant ckmℓ

which is determined using a matrix inver-sion, and εℓ is the time-skew error corresponding to the sampling instantmℓ.

The complexity of the online redesign block is reduced by expressing thepolyphase components of all the analysis filters using a common set of L+1fixed subfilters Fq(e

jω) and Gq(ejω), q = 0, 1, . . . , L, such that [86]

Bkmℓ(ejω) = γkmℓ

L∑

q=0

(dℓMt

)q

Fq(ejω) + ζkmℓ

L∑

q=0

(dℓMt

)q

Gq(ejω) (4.7)

where γkmℓ= βkmℓ

cos(θkmℓ), ζkmℓ

= βkmℓsin(θkmℓ

), θkmℓ= αkmℓ

+ π/4,dℓ = mℓ + εℓ, and Mt = MMc. Thus, the different polyphase branches canbe obtained via different sets of values for γkmℓ

, ζkmℓ, and dℓ. When the

time-skew errors change, only the general multipliers corresponding to γkmℓ,

ζkmℓ, and dℓ need to be updated with the corresponding new values. Also,

the new values of the constants γkmℓand ζkmℓ

can be determined through asingle K×K matrix inversion. The filters Fq(e

jω) and Gq(ejω) are designed

offline and are then fixed. In Paper F, we propose a least-squares approachfor determining the coefficients of these filters.7 Further, like in the sub-Nyquist CNUS reconstructor, the bandpass filters in the synthesis FB are

7Appendix B.2 contains the detailed equations for the least-squares design of Fq(ejω)

and Gq(ejω) filters.


implemented using a cosine-modulated FB. The impulse response coefficientsof the prototype filter for the cosine-modulated FB are determined offlineand are then fixed.

The computational complexity of the sub-band based reconstructor can beevaluated as

Cm ≈ NP

Mt+ log2(Mt) +

2(L+ 1)NF

Mt+

2LK

Mt+

2K2

Mt(4.8)

where NP is the order of the prototype filter for the cosine-modulated FBand NF is the order of the fixed subfilters in the analysis FB. The first twoterms in the expression for Cm in (4.8), correspond to the computational com-plexity of the cosine-modulated synthesis FB assuming that the 2Mt ×Mt

transform block is implemented using a fast-transform algorithm [36]. Thethird, fourth, and fifth terms are the computational complexities of the2(L + 1) fixed subfilters Fq(e

jω) and Gq(ejω), the 2L multipliers (dℓ/Mt)

q,q = 1, 2, . . . , L, and the 2K2 multipliers corresponding to the scaled cos(·)and sin(·) terms in the analysis FB, respectively. As can be seen from thedesign example in Paper F, compared to the constrained time-varying recon-structor, the sub-band based reconstructor has higher computational com-plexity. However, in contrast to the time-varying reconstructor, the onlineredesign of the reconstructor requires only a single K ×K matrix inversion.

4.2.3 Pre-Filter Based Reconstructor

As noted in Section 4.2.1, even though the constrained time-varying recon-structor has minimal computational complexity, the online redesign com-plexity is high due to the MMc matrix inversions that need to be performedonline (see Footnote 6). Further, all the multipliers in the time-varying re-constructor should be implemented using expensive variable-coefficient mul-tipliers as the coefficients of all the filters in the reconstructor can changeduring the online redesign.

In paper G of Part IV, we propose a pre-filter based reconstruction schemeshown here in Fig. 4.3. In this scheme, the overall implementation com-plexity is reduced by approximately recovering the missing samples using apre-filter and then correcting the remaining mismatch errors using an itera-tive compensation structure. Since the pre-filter only approximately recoversthe missing samples, its filter coefficients depend only on the location of the


Figure 4.3: Prefilter-based reconstruction scheme.

Table 1: Complexity comparison for the different reconstruction schemes.

Reconstructor Order Complexity9

Cf Cv Cm Redesign10

Recursive [26] 396 543 11 554 None

Nonrecursive constrained 68 0 31 31 28

Nonrecursive sub-band 960 27 48 75 1

Nonrecursive pre-filter 154 28 3 31 None

missing samples and hence, can be determined offline and then fixed. Theiterative structure consists of fixed subfilters and variable multipliers, likein [56]. Whenever the mismatch parameters change, the reconstructor canbe reconfigured directly by updating the variable multipliers. As the co-efficients of the pre-filter and subfilters in the compensation structure aredetermined offline and then fixed, they can be implemented using cheaperfixed-coefficient multipliers.

4.2.4 Complexity Comparison

Table 1 compares the complexities of the different reconstruction schemeswhen they are used to reconstruct the output of a four-channel TI-ADC(M = 4) in which every seventh sample is used to estimate the mismatchparameters (Mc = 7).8 It is assumed that the timing mismatches in thechannel ADCs are ε0,1,2,3 = [0.01,−0.05, 0.04,−0.03] and the reconstructorbandwidth is ωc = 0.8π. Further, it is required that the reconstructor shouldkeep the aliasing terms below −50 dB. Here, to be able to compare with the

8Refer to Paper G for details on how the values in Table 1 were obtained.9Cf and Cv represent the number of fixed-coefficient and variable-coefficient multiplica-

tions per corrected output sample, respectively.10Number of online matrix inversions.


corresponding example in [26], the time-skew error of the reference chan-nel, ε0, is not assumed to be zero. It can be seen from Table 1 that thepre-filter based reconstructor requires the same number of multiplicationsper corrected output sample as the constrained time-varying reconstructorbut with only fewer variable-coefficient multiplications per corrected outputsample.

4.3 Noise Gain

All reconstruction schemes for TI-ADCs with missing samples amplify thenoise at the input of the reconstructor. This noise gain is due to the presenceof missing samples which restricts some of the impulse response coefficients ofthe reconstructor to be equal to zero. Consider, for example, the constrainedtime-varying FIR reconstructor used in the four-channel TI-ADC case inSection 4.2.4. However, here, for simplicity, we assume that the TI-ADConly contains missing samples and has no time-skew error.11 When there areno time-skew errors, the filters H6(e

jω), H13(ejω), H20(e

jω), and H27(ejω),

which recover the missing samples can be linear-phase FIR filters. Thisalso the case for the prefilter in the prefilter-based reconstructor where thenoise gain is due to the linear-phase FIR filter G(z) (see Fig. 4.3) whichrecovers the missing sample. Further, the reconstructor Hn(z) in Fig. 4.3only compensates for the mismatch errors and, hence, will not introduce anyextra noise gain.

Assuming that H6(ejω), H13(e

jω), H20(ejω), and H27(e

jω) are noncausalfilters, we have, Hn(e

jω) = Hn,R(ω) where Hn,R(ω), n ∈ [6, 13, 20, 27], is thecorresponding real-valued zero-phase frequency response (see Section 1.1.3).Also, the mid-tap impulse response coefficient hn(0), n ∈ [6, 13, 20, 27] (andsome other taps depending on n and the order of hn(k)), is equal to zero.The mid-tap value hn(0) can be represented in terms of its Fourier transformas

hn(0) =1

2π

ˆ π

−πHn,R(ω) dω. (4.9)

Since Hn,R(ω) = 1 for ω ∈ [−ωc, ωc], for the term on the right side of theequality in (4.9) to be equal to zero, Hn,R(ω) must have negative values for

11The extra small time-skew errors will only have a very small effect on the noise gainanalysis and conclusions.

4.3. Noise Gain 49

|ω| ∈ [ωc, π]. That is,ˆ π

ωc

Hn,R(ω) dω = A (4.10)

where A = −ωc/π. Now, if σ2e represents the noise power at the input of the

filter hn(k), the noise power at the output of the filter is given by

σ2xe

= σ2e

1

2π

ˆ π

−πH2

n,R(ω) dω. (4.11)

Assuming ideal filters with Hn,R(ω) = 1 in |ω| ∈ [0, ωc], the minimum valueof Hn,R(ω) in ω ∈ (ωc, π] can be determined by solving the problem:

minimize

ˆ π

ωc

H2n,R(ω) dω subject to

ˆ π

ωc

Hn,R(ω) dω = A. (4.12)

By dividing the frequency band ω ∈ (ωc, π] into W bands of width ∆ω, theintegrals in (4.12) can be approximated using summations. Thus, (4.12) canbe approximated as

minimize ∆ω

W−1∑

q=0

(H

(q)n,R(ω)

)2subject to ∆ω

W−1∑

q=0

H(q)n,R(ω) = A (4.13)

where H(q)n,R(ω) is the value of Hn,R(ω) in the qth band.

Now, since ∆ω is a constant, (4.13) can be restated as

minimize

W−1∑

q=0

(H

(q)n,R(ω)

)2subject to

W−1∑

q=0

H(q)n,R(ω) =

A

∆ω. (4.14)

Since Hn,R(ω) is real-valued, the function to be minimized can be rewrittenas [87]

W−1∑

q=0

(H

(q)n,R(ω)

)2=

1

W

W−1∑

q=0

H(q)n,R(ω)

2

+

1

W

W−2∑

q=0

W−1∑

r=q+1

(H

(q)n,R(ω)−H

(r)n,R(ω)

)2

=1

W

(A

∆ω

)2

+1

W

W−2∑

q=0

W−1∑

r=q+1

(H

(q)n,R(ω)−H

(r)n,R(ω)

)2

(4.15)


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

ω [×π rad]

H6,R(ω

)

Figure 4.4: Frequency response of H6,R(ω) for the constrained time-varyingreconstructor designed for the four-channel TI-ADC specification given inSection 4.2.4 where ωc = 0.8π.

where the second equality in (4.15) is obtained by using the constraint in(4.14). It can be seen that the solution to (4.14) is obtained when the secondterm on the right side of the equality in (4.15) is equal to zero. For this,

the value of H(q)n,R(ω) in the W frequency bands in ω ∈ (ωc, π] should all be

equal to a constant, say A1. Thus,

Hn,R(ω) =

{1, |ω| ∈ [0, ωc]

A1, |ω| ∈ (ωc, π].(4.16)

Substituting (4.16) in (4.9) and solving for hn(0) = 0, we get

A1 = − ωc

π − ωc. (4.17)

This is also verified by Fig. 4.4 which plots the spectrum of H6,R(ω) for theconstrained time-varying FIR reconstructor designed for the four-channelTI-ADC specification in Section 4.2.4 where ωc = 0.8π. Using (4.16) in(4.11), we get the noise gain as

σ2xe

σ2e

=ωc

π+

ω2c

π − ωc. (4.18)

Figure 4.5 illustrates how the noise gain at the output of the reconstructionfilter Hn(e

jω), used to recover the missing sample, increases with recon-structor bandwidth ωc. It can be seen that the noise gain is not an issue innarrowband reconstructors but as ωc approaches π, the noise gain increasesrapidly. For example, at ωc = 0.8π, the noise gain at the output of the FIR

4.4. Summary 51

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−15

−10

−5

0

5

10

15

20

25

ωc [×π rad]

Noise

gain

[dB]

Figure 4.5: Noise gain at the output of the filterHn(ejω) versus reconstructor

bandwidth ωc.

filter recovering the missing sample correspond to approximately 1.8 bitsdegradation in the resolution. However, the above noise gain affects only ev-ery Mcth output sample and, hence, the average degradation at the outputof the reconstructor will be less than 1.8 bits. Thus, if Mc = 7, the noisegain will result in an average degradation of around 0.7 bits at the outputof the reconstructor.

4.4 Summary

In this chapter, we introduced three reconstruction schemes that can be usedto recover the uniform-grid samples from the nonuniformly sampled signalat the output of an unconventional TI-ADC. In the TI-ADC under consid-eration, some of the input sampling instants are reserved for estimating thechannel mismatch parameters. Thus, in such TI-ADCs, nonuniform sam-pling is a result of the mismatch errors as well as the missing samples. Usinga design example, the complexities of the three new schemes were compared


with the existing iterative reconstruction scheme that uses recursive struc-tures. It was seen that the proposed schemes provide around an order-of-magnitude lower complexity compared to the existing iterative scheme. Wealso showed that, for reasonable bandwidths, the reconstruction schemes forTI-ADCs with missing samples have a noise gain greater than unity andthat the magnitude of the noise gain is dependent on the bandwidth of thereconstructor.

Chapter 5

Summary of Specific

Contributions of the

Dissertation

The contributions of this dissertation are divided into three parts: In Part II,we propose efficient signal reconstruction schemes for static time-skew errorcorrection in TI-ADCs. The proposed schemes offer trade-offs between onlineredesign and reconstructor complexities. Part III of the dissertation dealswith the reconstruction of sub-Nyquist sampled sparse multi-band signalswhich are obtained via CNUS. There, we propose an efficient reconfigurablereconstruction scheme that can potentially deliver an order-of-magnitudereduction in complexity compared to the existing scheme. In Part IV, wepropose three reconstruction schemes for TI-ADCs in which some of thesampling instants are reserved for estimating the mismatch parameters. Inaddition to correcting the channel mismatch errors, the reconstructors usedin such TI-ADCs should also recover the missing samples corresponding tothe time instants reserved for estimation.

In the above parts, we provide the steps for designing the reconstructors.Design examples are used to compare the complexity of the proposed schemeswith that of the existing schemes.

53

54 Chapter 5. Contributions of the Dissertation

5.1 Included Papers

Brief summaries of the papers included in this dissertation are as follows:

Paper A: Two-Rate Based Low-Complexity Time-VaryingDiscrete-Time FIR Reconstructors for Two-Periodic Nonuni-formly Sampled Signals

Authored by Anu Kalidas M. Pillai, and Hakan Johansson.

Published in Sampling Theory in Signal and Image Processing, 2013. Partsof this work have been presented at two conferences [88,89].

This paper deals with time-varying finite-length impulse response (FIR) fil-ters used for reconstruction of two-periodic nonuniformly sampled signals.The complexity of such reconstructors increases as their bandwidth ap-proaches the whole Nyquist band. Reconstructor design that yields min-imum reconstructor order requires expensive online redesign while thosemethods that simplify online redesign result in higher reconstructor complex-ity. This paper utilizes a two-rate approach to derive a single-rate structurewhere part of the complexity of the reconstructor is moved to a symmet-ric filter so as to reduce the number of multipliers. The symmetric filter isdesigned such that it can be used for all time-skew errors within a certainrange, thereby reducing the number of coefficients that need online redesign.The basic two-rate based reconstructor is further extended to completelyremove the need for online redesign at the cost of a slight increase in thetotal number of multipliers.

Paper B: Efficient Signal Reconstruction Scheme for M-ChannelTime-Interleaved ADCs


Published in the Analog Integrated Circuits and Signal Processing, 2013.

In time-interleaved analog-to-digital converters (TI-ADCs), the timing mis-matches between the channels result in a periodically nonuniformly sam-pled sequence at the output. Such nonuniformly sampled output limits theachievable resolution of the TI-ADC. In order to correct the errors due totiming mismatches, the output of the TI-ADC is passed through a digi-tal time-varying finite-length impulse response (FIR) reconstructor. Such

5.1. Included Papers 55

reconstructors convert the nonuniformly sampled output sequence to a uni-formly spaced output. Since the reconstructor runs at the output rate ofthe TI-ADC, it is beneficial to reduce the number of coefficient multipliersin the reconstructor. Also, it is advantageous to have as few coefficient up-dates as possible when the timing errors change. Reconstructors that reducethe number of multipliers to be updated online do so at a cost of increasednumber of multiplications per corrected output sample. This paper proposesa technique which can be used to reduce the number of reconstructor coef-ficients that need to be updated online without increasing the number ofmultiplications per corrected output sample.

Paper C: Low-Complexity Two-Rate Based Multivariate ImpulseResponse Reconstructor for Time-Skew Error Correction in M-Channel Time-Interleaved ADCs


Published in the proceedings of IEEE International Symposium on Circuitsand Systems, 2013.

Nonuniform sampling occurs in time-interleaved analog-to-digital convert-ers (TI-ADC) due to timing mismatches between the individual channelanalog-to-digital converters (ADCs). Such nonuniformly sampled outputwill degrade the achievable resolution in a TI-ADC. To restore the degradedperformance, digital time-varying reconstructors can be used at the outputof the TI-ADC, which in principle, converts the nonuniformly sampled out-put sequence to a uniformly sampled output. As the bandwidth of thesereconstructors increases, their complexity also increases rapidly. Also, sincethe timing errors change occasionally, it is important to have a reconstructorarchitecture that requires fewer coefficient updates when the value of the tim-ing error changes. Multivariate polynomial impulse response reconstructoris an attractive option for an M -channel reconstructor. If the channel tim-ing error varies within a certain limit, these reconstructors do not need anyonline redesign of their impulse response coefficients. This paper proposes atechnique that can be applied to multivariate polynomial impulse responsereconstructors in order to further reduce the number of fixed-coefficient mul-tipliers, and thereby reduce the implementation complexity.

Paper D: Efficient Recovery of Sub-Nyquist Sparse Multi-bandSignals Using Reconfigurable Multi-Channel Analysis and Modu-lated Synthesis Filter Banks



Submitted to the IEEE Transactions on Signal Processing, 2015. This is anextension of a conference publication [90].

Sub-Nyquist cyclic nonuniform sampling (CNUS) of a sparse multi-bandsignal generates a nonuniformly sampled signal. In order to recover themissing uniform-grid samples, the sequence obtained via CNUS is passedthrough a reconstructor. At present, these reconstructors have very highdesign and implementation complexity that offsets the gains obtained dueto sub-Nyquist sampling. In this paper, we propose a scheme that reducesthe design and implementation complexity of the reconstructor. In contrastto the existing reconstructors which use only synthesis filter bank (FB), theproposed reconstructor utilizes both analysis and synthesis FBs which makesit feasible to achieve an order-of-magnitude reduction of the complexity. Theanalysis filters are implemented using polyphase networks whose branchesare allpass filters with distinct fractional delays and phase shifts. In order toreduce both the design and the implementation complexity of the synthesisFB, the synthesis filters are implemented using a cosine-modulated FB. Inaddition to the reduced complexity of the reconstructor, the proposed recov-ery scheme also supports online reconfigurability which is required in flexible(multi-mode) systems where the user subband locations vary with time.

Paper E: A Sub-Band Based Reconstructor for M-Channel Time-Interleaved ADCs with Missing Samples


Published in the proceedings of IEEE International Conference on Acoustics,Speech and Signal Processing, 2014.

This paper proposes a scheme for the recovery of a uniformly sampled se-quence from the output of a time-interleaved analog-to-digital converter (TI-ADC) with static time-skew errors and missing samples. Nonuniform sam-pling occurs due to timing mismatches between the individual channel ADCsand also due to missing input samples as some of the sampling instants arereserved for estimating the mismatches in the TI-ADC. In addition to us-ing a non-recursive structure, the proposed reconstruction scheme supportsonline reconfigurability and reduces the computational complexity of thereconstructor as compared to a previous solution.

5.2. Not Included papers 57

Paper F: Two Reconstructors for M-Channel Time InterleavedADCs with Missing Samples


Published in the proceedings of IEEE International New Circuits and Sys-tems Conference, 2014.

In this paper, we explore two nonrecursive reconstructors which recover theuniform-grid samples from the output of a time-interleaved analog-to-digitalconverter (TI-ADC) that uses some of the sampling instants for estimatingthe mismatches in the TI-ADC. Nonuniform sampling occurs due to timingmismatches between the individual channel ADCs and also due to missing in-put samples. Compared to a previous solution, the reconstructors presentedhere offer substantially lower computational complexity.

Paper G: Prefilter-Based Reconfigurable Reconstructor for Time-Interleaved ADCs with Missing Samples


Published in the IEEE Transactions on Circuits and Systems: II - ExpressBreifs, 2015.

This brief proposes a reconstruction scheme for the compensation offrequency-response mismatch errors at the output of a time-interleavedanalog-to-digital converter (TI-ADC) with missing samples. The missingsamples are due to sampling instants reserved for estimating the channelmismatch errors in the TI-ADC. Compared to previous solutions, the pro-posed scheme offers a substantially lower computational complexity.

5.2 Not Included papers

The following publications by the author are not included because they wereearlier versions of the journal publications included in the dissertation.

[P1] A. K. M. Pillai and H. Johansson, “Efficient signal reconstructionscheme for time-interleaved ADCs,” Proc. IEEE Int. New CircuitsSyst. Conf., Montreal, Canada, Jun. 17–20, 2012, pp. 357–360.


[P2] A. K. M. Pillai and H. Johansson, “Time-skew error correction in two-channel time-interleaved ADCs based on a two-rate approach and poly-nomial impulse responses,” Proc. IEEE Int. Midwest Symp. CircuitsSyst., Boise, ID, USA, Aug. 5–8, 2012, pp. 1136–1139.

[P3] A. K. M. Pillai and H. Johansson, “Efficient reconfigurable scheme forthe recovery of sub-Nyquist sampled sparse multi-band signals,” Proc.IEEE Global Conf. Signal Information Process., Austin, TX, USA,Dec. 3–5, 2013, pp. 1294–1297.

Appendix A

Alternative Derivation of the

Reconstruction Scheme in

Paper D

Starting with the multi-level synthesis filters in [81], Paper D of Part IIIshows that the reconstruction can be described in terms of the analysis andsynthesis filters. There, we also show that the non-zero polyphase com-ponents of the unconventional decimation filter Bk(z) can be expressed asan allpass filter with a distinct fractional delay and a phase shift. In thisappendix, however, we show this using a different approach which gives fur-ther insight and understanding of the efficient reconfigurable reconstructionscheme described in Paper D. For this purpose, we first consider conven-tional lowpass and bandpass filters used in resampling (interpolation anddecimation). This is because the unconventional bandpass filters Bk(z) canbe viewed as extensions of the conventional bandpass filters whereas thepolyphase components of the conventional bandpass filters in turn can beconsidered as a generalization of the polyphase components of lowpass filters.It is noted that lowpass filtering using polyphase networks was consideredin [91], but here we use a different derivation which is suitable for the CNUSrecovery problem. It will be shown that the conventional bandpass filter canbe expressed in terms of a polyphase network whose branches are allpassfilters with distinct fractional delays and phase shifts. Here, the phase shiftvalues are directly given by one of the columns of a discrete Fourier transform(DFT) matrix. Extending the results derived for the conventional bandpass

59

60 Appendix A. Alternative Derivation of the Scheme in Paper D

filter, we will then show that, for the unconventional bandpass filters Bk(z),the gains of the allpass filters are generally different and the values of thephase shifts are not given by simple expressions but are obtained through amatrix inversion.

A.1 Lowpass Filters

Both interpolation and decimation require an (approximately) Mth-bandfilter with cutoff frequency at π/M . Recall that if h(n) denotes the im-pulse response of the Mth-band lowpass filter, its transfer function, H(z),is defined as

H(z) =∞∑

n=−∞

h(n)z−n. (A.1)

The M -fold polyphase representation of H(z) is given by [32]

H(z) =

M−1∑

m=0

z−mHm(zM ) (A.2)

where

Hm(z) =∞∑

n=−∞

h(nM +m)z−n (A.3)

denotes the mth polyphase component of H(z). In the lowpass decimationcase, the (noncausal) filter’s frequency response H(ejω) is ideally unity inthe passband |ω| ∈ [0, π/M ] and zero in the stopband |ω| ∈ (π/M, π].

Next, we will show that the Mth-band lowpass filter H(ejω) is obtainedif its polyphase components are fractional-delay (FD) filters with distinctFD values [91]. It is noted that, here we show only the “if” part. The“only if” part can be shown as in [92]. Here, we divide the frequency range[−π/M, 2π− π/M ] into M adjacent regions of equal width 2π/M as shownin Fig. A.1(a). Thus, region p, p ∈ [0, 1, . . . ,M−1], covers the frequencies in[−π/M +2πp/M, −π/M +2π(p+1)/M ]. Next, it is assumed that Hm(ejω)are FD filters according to

Hm(ejω) =1

Mejωm/M , ω ∈ [−π, π] (A.4)

and Hm(ejω) are 2π-periodic with respect to ω. The latter feature impliesthat the frequency response of the polyphase component for any ω ∈ [−π +

A.1. Lowpass Filters 61

Figure A.1: (a) Spectrum of a lowpass filter with cutoff frequency at π/M .The frequency range [−π/M, 2π− π/M ] is divided into M adjacent regionsof equal width 2π/M . (b) Spectrum of a bandpass filter with passband inthe frequency range [rkπ/M, (rk + 1)π/M ].

2πp, −π + 2π(p + 1)] and all integers p should be equal to Hm(ejω) in ω ∈[−π, π]. Hence, Hm(ejω) = ej(ω−2πp)m/M/M for ω ∈ [−π+2πp, −π+2π(p+1)] and all integers p. It is further noted that Hm(ejωM ) are compressed(by M) versions of the corresponding frequency responses Hm(ejω). Thismeans that Hm(ejωM ) for ω ∈ [−π/M + 2πp/M, −π/M + 2π(p + 1)/M ]equals Hm(ejω) for ω ∈ [−π + 2πp, −π + 2π(p + 1)]. Thus, Hm(ejωM ) =ej(ωM−2πp)m/M/M for ω ∈ [−π/M +2πp/M, −π/M +2π(p+1)/M ] and allintegers p.

For a real lowpass filter with a cutoff frequency at π/M and a passbandgain of unity, we require that H(ejω) = 1 for p = 0 and H(ejω) = 0 forp = 1, 2, . . . ,M − 1. Since H(ejω) is 2π-periodic, it suffices to considerp = 0, 1, . . . ,M − 1 as this corresponds to a frequency range of 2π. UsingHm(ejωM ) = ej(ωm−2πpm/M)/M in (A.2), we obtain

H(ejω) =1

M

M−1∑

m=0

e−jωmej(ωm−2πpm/M)

=

{1, p = 0

0, p = 1, 2, . . . ,M − 1.(A.5)

Thus, the FD polyphase components in (A.4) results in an overall lowpassfilter with cutoff frequency at π/M .


A.2 Conventional Bandpass Filters

Next, we will focus on the bandpass filter case where the passband ofthe desired real bandpass filter Bk(e

jω) is assumed to cover the bandω ∈ [rkπ/M, (rk + 1)π/M ], rk ∈ [0, 1, . . . ,M − 1], and thus also ω ∈[2π − (rk + 1)π/M, 2π − rkπ/M ] as shown in Fig. A.1(b). Regarding thetransfer function, the point of departure is again the polyphase representa-tion in (A.2). However, to obtain a bandpass filter, we cannot use the FDpolyphase component in (A.4) since this choice automatically results in alowpass filter as shown in Appendix A.1. As will be shown below, a band-pass filter is obtained by adding phase offsets to the FD filters in (A.4). Inother words, we replace (A.4) with

Bkm(ejω) =1

Mej(ωm/M+αkm sgn(ω)), ω ∈ [−π, π] (A.6)

where sgn(ω) is the sign of ω. Again, we assume that the frequency range[−π/M, 2π−π/M ] is divided into M adjacent regions of equal width 2π/Mas shown in Fig. A.1(b). Thus, if Bk(e

jω) occupies the left (right) half ofthe region p, p ∈ [0, 1, . . . ,M − 1], it will also occupy the right (left) half ofthe M − pth region as shown in Fig. A.1(b). Comparing Figs. A.1(a) andA.1(b), it can be seen that, unlike in the lowpass filter case where H(ejω)equals unity in the whole region 0, here, Bk(e

jω) should not be equal tounity in the whole region p. Instead, as the passband of a real bandpassdecimation filter cover only half of such a region, we get two requirementsper region p. As will be shown below, if we satisfy the two requirements forregion p, we will also automatically satisfy the two requirements for regionM−p. Hence, in total, we will end up with a system of M equations with Munknowns αkm. Also, it will be shown that the values of αkm are obtainedas one of the columns in a scaled DFT matrix.

The difference from the lowpass case is that, due to the sgn(ω) in (A.6),Bkm(ejωM ) = ej(ωm−2πpm/M−αkm)/M in the left part of region p whereasBkm(ejωM ) = ej(ωm−2πpm/M+αkm)/M in the right part of the same region.Using these expressions in (A.2), we get

Bk(e−jω) =

1

M

M−1∑

m=0

ej2πpm/Mejαkm (A.7)

for ω ∈ [−π/M + 2πp/M, 2πp/M ] (left part of region p) and

Bk(ejω) =

1

M

M−1∑

m=0

e−j2πpm/Mejαkm (A.8)

A.2. Conventional Bandpass Filters 63

for ω ∈ [2πp/M, −π/M + 2π(p + 1)/M ] (right part of region p). Above,we used Bk(e

−jω) in (A.7) since real filters are assumed. For real filters,Bk(e

jω) = 1 (Bk(ejω) = 0) in the passband (stopband) region implies

Bk(e−jω) = 1 (Bk(e

−jω) = 0) as well.

It can be seen that (A.7) and (A.8) also correspond to the right and the lefthalf, respectively, of region M − p. Hence, the two requirements on Bk(e

jω)in region p equals the two requirements in region M − p. Consequently,it suffices to solve a system of M equations for M values of q = ±p thatcorrespond to the frequency band ω ∈ [0, π]. If M is even, these values areq = 0,±1,±2, . . . ,M/2 and if M is odd, we have q = 0,±1,±2, . . . ,±(M −1)/2. For convenience, we will however restate the requirements in terms ofpositive values of q which is possible because e−j2πpm/M = ej2π(M−p)m/M .Then, q = 0, 1, 2, . . . ,M − 1, and they correspond to the left half of the Mregions in Fig. A.1(b). Thus, we obtain the system of equations

Dvk = bk (A.9)

where D is an M ×M matrix with the elements dqm = 1M ej2πqm/M , q,m =

0, 1, . . . ,M −1, and vk is a vector (M ×1 matrix) with M unknowns υk,m =ejαkm, m = 0, 1, . . . ,M − 1. Further, bk is a vector (M × 1 matrix) withelements ♭k,m, m = 0, 1, . . . ,M−1, that correspond to the required frequencyresponse in the left half of the M subbands which will result in an overallBk(e

jω) shown in Fig. A.1(b). That is, bk contains M − 1 zeros and unityfor the position corresponding to the passband ω ∈ [rkπ/M, (rk + 1)π/M ],rk ∈ [0, 1, . . . ,M−1]. Assuming that ♭k,w(rk) = 1, the value of w(rk) is givenby

w(rk) =

rk+12 , odd rk

M − rk2 , even rk 6= 0

0, rk = 0

. (A.10)

It is noted that D is the inverse discrete Fourier transform (IDFT) matrix.Therefore, the inverse of D always exists and equals the DFT matrix. Since♭k,w(rk) = 1 and ♭k,m = 0 for m 6= w(rk), υk,m are obtained as

υk,m = e−j2πw(rk)m/M , m = 0, 1, . . . ,M − 1. (A.11)

Consequently, the phase offsets αkm in (A.6) become

αkm =−2πw(rk)m

M, m = 0, 1, . . . ,M − 1. (A.12)

Thus, with αkm as in (A.12), the polyphase components in (A.6) result inan overall bandpass filter with passband in the region ω ∈ [rkπ/M, (rk +1)π/M ].


Figure A.2: (a) Bandpass decimator in the kth branch of the analysis FB inthe proposed reconstructor. (b) Polyphase representation of the bandpassdecimator in (a) when x(n) is obtained via sub-Nyquist CNUS.

A.3 Unconventional Bandpass Filters

We now extend the above results to bandpass filters used in the analysisFB of the proposed reconstructor (Fig. 3.5). Figure A.2(a) shows the kthbranch of the analysis FB in the proposed reconstructor. Recall that sincethe input signal is sub-Nyquist sampled as explained in Section 3.1, it canbe considered that the available input samples uk(ν) = x(Mν − mℓ), ℓ =1, 2, . . . ,K, are obtained from the uniform-grid samples x(n) as shown inFig. 3.2(a). Thus, due to missing samples, the inputs to M −K polyphasebranches of the bandpass filter Bk(z) in Fig. A.2(a) will be equal to zero.This implies that

Bk(z) =

K∑

ℓ=1

z−mℓBkmℓ(zM ) (A.13)

wheremℓ ∈ [0, 1, . . . ,M−1], ℓ = 1, 2, . . . ,K, are theK sampling instants andBkmℓ

(ejω) are the K non-zero polyphase components of Bk(z). Hence, thebandpass decimator in Fig. A.2(a) can be redrawn as shown in Fig. A.2(b).Similar to the bandpass filter in Appendix A.2, the polyphase components ofthe bandpass filter Bk(z) are generalized FD filters but, here, the non-zeropolyphase components Bkmℓ

(ejω) can be written as

Bkmℓ(ejω) ≈ βkmℓ

Mej(ωmℓ/M+αkmℓ

sgn(ω)), ω ∈ (−π, π). (A.14)

In (A.14), βkmℓand αkmℓ

are the modulus and angle, respectively, of a cor-responding complex constant υkmℓ

. Moreover, proceeding in the same way

A.3. Unconventional Bandpass Filters 65

as in Appendix A.2, the vector vk, containing all the K complex constantsυkmℓ

, mℓ ∈ [0, 1, . . . ,M−1], ℓ = 1, 2, . . . ,K, can be determined using matrixinversion as

vk = D−1bk. (A.15)

Here, D is a K × K generalized Vandermonde matrix whose elements aredetermined by the K sampling points mℓ and the K regions qi, whose lefthalves are occupied by the active subbands ri, such that

D =1

M

ej2πq1m1/M ej2πq1m2/M · · · ej2πq1mK/M

ej2πq2m1/M ej2πq2m2/M · · · ej2πq2mK/M

......

. . ....

ej2πqKm1/M ej2πqKm2/M · · · ej2πqKmK/M

(A.16)

where qi,mℓ ∈ [0, 1, . . . ,M−1], i, ℓ = 1, 2, . . . ,K. It can be seen from (A.16)that there is always at least one set of sampling instants that correspondsto an invertible matrix, namely mℓ = 0, 1, . . . ,K, since for these samplingpoints the generalized Vandermonde matrix D reduces to a Vandermondematrix. However, these sampling instants may not guarantee that the matrixD is well conditioned. In order to ensure that D is well conditioned, optimalsampling instants can be selected depending on the active subband locationsas outlined in [82,84]. Further, bk is a vector (K×1 matrix) containing K−1zeros and unity for the position k.

Appendix B

Derivation of the

Least-Squares Design in

Paper F

In Paper F of Part IV, we propose least-squares design for the two reconstruc-tors used in TI-ADCs with missing samples. In this appendix, we providethe derivation of these least-squares designs since the derivations were notincluded in Paper F due to space limitations.

B.1 Constrained Time-Varying FIR

Reconstructor

In Paper F, the least-squares design proposed for the constrained time-varying reconstructor assumes that the channel ADCs have only static time-skew errors. However, here, we consider the more general case where eachchannel ADC has a frequency response Qn(jω). The expression for the statictime-skew error case considered in Paper F can be obtained by substitutingQn(jω) = ejωεn .

Let hn represent the vector (Rn × 1 matrix) containing the Rn non-zeroimpulse response coefficients in hn(k) such that

hn = [hn(k1) hn(k2) . . . hn(kRn)]T . (B.1)

67

68 Appendix B. Least-Squares Design in Paper F

Now, hn, the vector (Nhn+1×1 matrix) containing all the impulse response

coefficients of hn(k) can be written as

hn = Dnhn (B.2)

where Dn is a Nhn+ 1×Rn matrix given by

Dn = [dn,1 dn,2 . . . dn,Rn ], (B.3)

with

dn,i =[d(i)n,−Nhn/2

d(i)n,−Nhn/2+1 . . . d

(i)n,Nhn/2

]T(B.4)

and

d(i)n,k =

{1, k = ki

0, otherwise(B.5)

for i = 1, 2, . . . , Rn, k = −Nhn/2,−Nhn

/2 + 1, . . . , Nhn/2. Then, assuming

that xa(t) is bandlimited to |ω| ≤ ω0 < π, the error power function

Pn =1

2π

ˆ ω0

−ω0

|An(jω) − 1|2 dω (B.6)

can be expressed in terms of the non-zero impulse response coefficients hn

as

Pn = hTnSnhn − 2bT

n hn +ω0

π

= hTnD

TnSnDnhn − 2bT

nDnhn +ω0

π(B.7)

where Sn is an Nhn+ 1 × Nhn

+ 1 matrix with elements Sn,kp, k, p =−Nhn

/2,−Nhn/2 + 1, . . . , Nhn

/2, given by

Sn,kp =1

2π

ˆ ω0

−ω0

|Qn−k(jω)| |Qn−p(jω)|

× cos (ω(p − k) + arg {Qn−k(jω)} − arg {Qn−k(jω)}) dω (B.8)

andbn = [bn,−Nhn/2

bn,−Nhn/2+1 . . . bn,Nhn/2]T (B.9)

with

bn,k =1

2π

ˆ ω0

−ω0

|Qn−k(jω)| cos (ω − arg {Qn−k(jω)}) dω (B.10)

B.2. Least-Squares Design of Fq(ejω) and Gq(e

jω) 69

for k = −Nhn/2,−Nhn

/2 + 1, . . . , Nhn/2. The value of hn that minimizes

Pn in (B.7) is obtained by solving

∂Pn∂hn

= 0 (B.11)

which giveshn = (DnSn)

−1 bTn . (B.12)

B.2 Least-Squares Design of Fq(ejω) and Gq(e

jω) in

the Sub-band Based Reconstructor

Here, we derive the closed-form expression for determining the impulse re-sponse coefficients of the subfilters Fq(e

jω) and Gq(ejω) in the analysis FB

of the sub-band based reconstructor in Paper F. The closed-form expressionis derived using the least-squares design outlined in Section 4.1 of Paper F.

Let NF denote the order of the 2(L + 1) subfilters Fq(ejω) and Gq(e

jω),q = 0, 1, . . . , L, in the analysis FB and NC the order of the synthesis filtersCk(e

jω). In the following equations, for simplicity, we assume that the filtersare noncausal with even order. Then, the vectors (NF + 1 × 1 matrices) fqand fq containing the impulse response coefficients of Fq(e

jω) and Gq(ejω),

respectively, are given by

fq = [fq(−NF /2) fq(−NF /2 + 1) . . . fq(NF /2)]T (B.13)

andgq = [gq(−NF /2) gq(−NF /2 + 1) . . . gq(NF /2)]

T (B.14)

where fq(n) and gq(n), n = −NF/2,−NF /2 + 1, . . . , NF /2, are the impulseresponse coefficients of Fq(e

jω) and Gq(ejω), respectively. Similarly, the

vector (NC + 1 × 1 matrix) ck containing the impulse response coefficientsof Ck(e

jω) is given by

ck = [ck(−NC/2) ck(−NC/2 + 1) . . . ck(NC/2)]T (B.15)

where ck(n), n = −NC/2,−NC/2 + 1, . . . , NC/2, are the impulse responsecoefficients of Ck(e

jω). Then, the distortion function (p = 0) and aliasing


functions (p = 1, 2, . . . , N − 1), Vp(ejω) in (10) of paper F can be expressed

in matrix form as12

Vp(ejω) =

1

Ne(ω,NC)CU(ω, p)h (B.16)

where

e(ω,NC) =[e−jω(−NC/2) e−jω(−NC/2+1) . . . e−jωNC/2

], (B.17)

C = [c1 c2 . . . cK ] , (B.18)

U(ω, p) =[uT1 uT

2 . . . uTK

]T, (B.19)

andh =

[fT0 gT0 fT1 gT1 . . . fTL gT

L

]T. (B.20)

In (B.19)

uk(ω, p) =

K∑

ℓ=1

dkℓ(ω, p) (B.21)

for k = 1, 2, . . . ,K, with

dkℓ(ω, p) =[a(0)kℓ (ω, p) b

(0)kℓ (ω, p) . . . a

(K)kℓ (ω, p) b

(K)kℓ (ω, p)

], (B.22)

a(q)kℓ (ω, p) = akℓ(ω, p)e(ωN,NF )

(dℓN

)q

, (B.23)

b(q)kℓ (ω, p) = bkℓ(ω, p)e(ωN,NF )

(dℓN

)q

, (B.24)

for q = 0, 1, . . . , L,

akℓ(ω, p) = γkmℓe−j(ω−2πp/N)dℓ , (B.25)

andbkℓ(ω, p) = ζkmℓ

e−j(ω−2πp/N)dℓ , (B.26)

where γkmℓand ζkmℓ

are as in Section 4 of Paper F. The vector e(ωN,NF )in (B.23) and (B.24) is obtained by replacing ω and NC in (B.17) with ωNand NF , respectively. With Vp(e

jω) as in (B.16), from (12) in Paper F, theerror power function P0 can be written as

P0 =1

N2hTS0h− 2

N2r0h+

1

N2

ω0

π(B.27)

12Here, N corresponds to Mt in Section 4.2.

B.2. Least-Squares Design of Fq(ejω) and Gq(e

jω) 71

and Pp, p = 1, 2, . . . , N − 1, become

Pp =1

N2hTSph (B.28)

where

Sp =1

2π

ˆ ω0

ω0

U†(ω, p)CTe†(ω,NC)e(ω,NC)CU(ω, p) dω (B.29)

for p = 0, 1, . . . , N − 1, and

r0 =1

2π

ˆ ω0

ω0

Re {e(ω,NC)CU(ω, 0)} dω. (B.30)

Then, the h that minimizes P in (11) of Paper F can be found by solving∂P/∂h and is given by

h =

N−1∑

p=0

Sp

−1

r0. (B.31)

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Part II

Efficient Reconstruction

Schemes for TI-ADCs

81

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