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تسجيل مستخدم جديدAnalog Coding Frame-work
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Marina Haikin
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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.
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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.
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