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Abstract-Network information theory promises high gains over simple point-to-point communication techniques, at the cost of higher complexity. However, lack of structured coding schemes limited the practical application of these concepts so far. One of the basic elements of a network code is the binning scheme. Wyner and other researchers proposed various forms of coset codes for efficient binning, yet these schemes were applicable only for lossless source (or noiseless channel) network coding. To extend the algebraic binning approach to lossy source (or noisy channel) network coding, recent work proposed the idea of nested codes, or more specifically, nested parity-check codes for the binary case and nested lattices in the continuous case. These ideas connect network information theory with the rich areas of linear codes and lattice codes, and have strong potential for practical applications. We review these recent developments and explore their tight relation to concepts such as combined shaping and precoding, coding for memories with defects, and digital watermarking. We also propose a few novel applications adhering to a unified approach.

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Uniform quantization with dither, or lattice quantization with dither in the vector case, followed by a universal lossless source encoder (entropy coder), is a simple procedure for universal coding with Manuscript

Abstract-The fundamental limits of "systematic" communication are analyzed. In systematic transmission, the decoder has access to a noisy version of the uncoded raw data (analog or digital). The coded version of the data is used to reduce the average reproduced distortion D below that provided by the uncoded systematic link and/or increase the rate of information transmission. Unlike the case of arbitrarily reliable error correction (D ! 0) for symmetric sources/channels, where systematic codes are known to do as well as nonsystematic codes, we demonstrate that the systematic structure may degrade the performance for nonvanishing D: We characterize the achievable average distortion and we find necessary and sufficient conditions under which systematic communication does not incur loss of optimality. The Wyner-Ziv rate distortion theorem plays a fundamental role in our setting. The general result is applied to several scenarios. For a Gaussian bandlimited source and a Gaussian channel, the invariance of the bandwidth-signal-to-nosie ratio (SNR, in decibels) product is established, and the optimality of systematic transmission is demonstrated. Bernoulli sources transmitted over binary-symmetric channels and over certain Gaussian channels are also analyzed. It is shown that if nonnegligible bit-error rate is tolerated, systematic encoding is strictly suboptimal.Index Terms-Gaussian channels and sources, rate-distortion theory, source/channel coding, systematic transmission, uncoded side information, Wyner-Ziv rate distortion.

Lattice coding for signals and networks : a structured coding approach to quantization, modulation, and multiuser information theory / Ram Zamir, Tel Aviv University. pages cm Includes bibliographical references and index. ISBN 978-0-521-76698-2 (hardback) 1. Coding theory. 2. Signal processing-Mathematics. 3. Lattice theory. I. Title. TK5102.92.Z357 2014 003 ′ .54-dc23 2014006008 ISBN 978-0-521-76698-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To my parents Eti and Sasson Zamir Contents Preface page xiii Acknowledgements xv List of notation xviii 4 Dithering and estimation 4.1 Crypto lemma 4.2 Generalized dither viii Contents 4.3 White dither spectrum 4.4 Wiener estimation 4.5 Filtered dithered quantization Summary Problems Historical notes 5 Entropy-coded quantization 5.1 The Shannon entropy 5.2 Quantizer entropy 5.3 Joint and sequential entropy coding* 5.4 Entropy-distortion trade-off 5.5 Redundancy over Shannon 5.6 Optimum test-channel simulation 5.7 Comparison with Lloyd's conditions 5.8 Is random dither really necessary? 5.9 Universal quantization* Summary Problems Historical notes 6 Infinite constellation for modulation 6.1 Rate per unit volume 6.2 ML decoding and error probability 6.3 Gap to capacity 6.4 Non-AWGN and mismatch 6.5 Non-equiprobable signaling 6.6 Maximum a posteriori decoding* Summary Problems Historical notes 7 Asymptotic goodness 7.1 Sphere bounds 7.2 Sphere-Gaussian equivalence 7.3 Good covering and quantization 7.4 Does packing imply modulation? 7.5 The Minkowski-Hlawka theorem 7.6 Good packing 7.7 Good modulation Contents ix 7.8 Non-AWGN 7.9 Simultaneous goodness Summary Problems Historical notes 8 Nested lattices 8.1 Definition and properties 8.2 Cosets and Voronoi codebooks 8.3 Nested linear, lattice and trellis codes 8.4 Dithered codebook 8.5 Good nested lattices Summary Problems Historical notes 9 Lattice shaping 9.1 Voronoi modulation 9.2 Syndrome dilution scheme 9.3 The high SNR case 9.4 Shannon meets Wiener (at medium SNR) 9.5 The mod channel 9.6 Achieving C AWGN for all SNR 9.7 Geometric interpretation 9.8 Noise-matched decoding 9.9 Is the dither really necessary? 9.10 Voronoi quantization Summary Problems Historical notes 10 Side-information problems 10.1 Syndrome coding 10.2 Gaussian multi-terminal problems 10.3 Rate distortion with side information 10.4 Lattice Wyner-Ziv coding 10.5 Channels with side information 10.6 Lattice dirty-paper coding Summary x Contents Problems Historical notes 11 Modulo-lattice modulation 11.1 Separation versus JSCC 11.2 Figures of merit for JSCC 11.3 Joint Wyner-Ziv/dirty-paper coding 11.4 Bandwidth conversion Summary Problems Historical notes 12 Gaussian networks 12.1 The two-help-one problem 12.2 Dirty multiple-access channel 12.3 Lattice network coding 12.4 Interference alignment 12.5 Summary and outlook Summar...