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Register a new userAnalog Coding of a Source with Erasures
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Added by
Marina Haikin
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Analog coding decouples the tasks of protecting against erasures and noise. For erasure correction, it creates an analog redundancy by means of band-limited discrete Fourier transform (DFT) interpolation, or more generally, by an over-complete expansion based on a frame. We examine the analog coding paradigm for the dual setup of a source with erasure side-information (SI) at the encoder. The excess rate of analog coding above the rate-distortion function (RDF) is associated with the energy of the inverse of submatrices of the frame, where each submatrix corresponds to a possible erasure pattern. We give a partial theoretical as well as numerical evidence that a variety of structured frames, in particular DFT frames with difference-set spectrum and more general equiangular tight frames (ETFs), with a common MANOVA limiting spectrum, minimize the excess rate over all possible frames. However, they do not achieve the RDF even in the limit as the dimension goes to infinity.
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Analog coding is a low-complexity method to combat erasures, based on linear redundancy in the signal space domain. Previous work examined band-limited discrete Fourier transform (DFT) codes for Gaussian channels with erasures or impulses. We extend this concept to source coding with erasure side-information at the encoder and show that the performance of band-limited DFT can be significantly improved using irregular spectrum, and more generally, using equiangular tight frames (ETF). Frames are overcomplete bases and are widely used in mathematics, computer science, engineering, and statistics since they provide a stable and robust decomposition. Design of frames with favorable properties of random subframes is motivated in variety of applications, including code-devision multiple access (CDMA), compressed sensing and analog coding. We present a novel relation between deterministic frames and random matrix theory. We show empirically that the MANOVA ensemble offers a universal description of the spectra of randomly selected subframes with constant aspect ratios, taken from deterministic near-ETFs. Moreover, we derive an analytic framework and bring a formal validation for some of the empirical results, specifically that the asymptotic form for the moments of high orders of subsets of ETF agree with that of MANOVA. Finally, when exploring over-complete bases, the Welch bound is a lower bound on the root mean square cross correlation between vectors. We extend the Welch bound to an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. The lower bound involves moment of the reduced frame, and it is tight for ETFs and asymptotically coincides with the MANOVA moments. This result offers a novel perspective on the superiority of ETFs over other frames.
Over-complete systems of vectors, or in short, frames, play the role of analog codes in many areas of communication and signal processing. To name a few, spreading sequences for code-division multiple access (CDMA), over-complete representations for multiple-description (MD) source coding, space-time codes, sensing matrices for compressed sensing (CS), and more recently, codes for unreliable distributed computation. In this survey paper we observe an information-theoretic random-like behavior of frame subsets. Such sub-frames arise in setups involving erasures (communication), random user activity (multiple access), or sparsity (signal processing), in addition to channel or quantization noise. The goodness of a frame as an analog code is a function of the eigenvalues of a sub-frame, averaged over all sub-frames. For the highly symmetric class of Equiangular Tight Frames (ETF), as well as for other near ETF frames, we show that the empirical eigenvalue distribution of a randomly-selected sub-frame (i) is asymptotically indistinguishable from Wachters MANOVA distribution; and (ii) exhibits a universal convergence rate to this limit that is empirically indistinguishable from that of a matrix sequence drawn from MANOVA (Jacobi) ensembles of corresponding dimensions. Some of these results are shown via careful statistical analysis of empirical evidence, and some are proved analytically using random matrix theory arguments of independent interest. The goodness measures of the MANOVA limit distribution are better, in a concrete formal sense, than those of the Marchenko-Pastur distribution at the same aspect ratio, implying that deterministic analog codes are better than random (i.i.d.) analog codes. We further give evidence that the ETF (and near ETF) family is in fact superior to any other frame family in terms of its typical sub-frame goodness.
Integer-Forcing (IF) is a new framework, based on compute-and-forward, for decoding multiple integer linear combinations from the output of a Gaussian multiple-input multiple-output channel. This work applies the IF approach to arrive at a new low-complexity scheme, IF source coding, for distributed lossy compression of correlated Gaussian sources under a minimum mean squared error distortion measure. All encoders use the same nested lattice codebook. Each encoder quantizes its observation using the fine lattice as a quantizer and reduces the result modulo the coarse lattice, which plays the role of binning. Rather than directly recovering the individual quantized signals, the decoder first recovers a full-rank set of judiciously chosen integer linear combinations of the quantized signals, and then inverts it. In general, the linear combinations have smaller average powers than the original signals. This allows to increase the density of the coarse lattice, which in turn translates to smaller compression rates. We also propose and analyze a one-shot version of IF source coding, that is simple enough to potentially lead to a new design principle for analog-to-digital converters that can exploit spatial correlations between the sampled signals.
The exponential growth in data generation and large-scale data analysis creates an unprecedented need for inexpensive, low-latency, and high-density information storage. This need has motivated significant research into multi-level memory systems that can store multiple bits of information per device. Although both the memory state of these devices and much of the data they store are intrinsically analog-valued, both are quantized for use with digital systems and discrete error correcting codes. Using phase change memory as a prototypical multi-level storage technology, we herein demonstrate that analog-valued devices can achieve higher capacities when paired with analog codes. Further, we find that storing analog signals directly through joint-coding can achieve low distortion with reduced coding complexity. By jointly optimizing for signal statistics, device statistics, and a distortion metric, finite-length analog encodings can perform comparable to digital systems with asymptotically infinite large encodings. These results show that end-to-end analog memory systems have not only the potential to reach higher storage capacities than discrete systems, but also to significantly lower coding complexity, leading to faster and more energy efficient storage.
The Welch Bound is a lower bound on the root mean square cross correlation between $n$ unit-norm vectors $f_1,...,f_n$ in the $m$ dimensional space ($mathbb{R} ^m$ or $mathbb{C} ^m$), for $ngeq m$. Letting $F = [f_1|...|f_n]$ denote the $m$-by-$n$ frame matrix, the Welch bound can be viewed as a lower bound on the second moment of $F$, namely on the trace of the squared Gram matrix $(FF)^2$. We consider an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. We extend the Welch bound to this setting and present the {em erasure Welch bound} on the expected value of the Gram matrix of the reduced frame. Interestingly, this bound generalizes to the $d$-th order moment of $F$. We provide simple, explicit formulae for the generalized bound for $d=2,3,4$, which is the sum of the $d$-th moment of Wachters classical MANOVA distribution and a vanishing term (as $n$ goes to infinity with $frac{m}{n}$ held constant). The bound holds with equality if (and for $d = 4$ only if) $F$ is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on the superiority of ETFs over other frames in a variety of applications, including spread spectrum communications, compressed sensing and analog coding.
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