Multiple-Description Coding by Dithered Delta-Sigma Quantization

We address the connection between the multiple-description (MD) problem and Delta-Sigma quantization. The inherent redundancy due to oversampling in Delta-Sigma quantization, and the simple linear-additive noise model resulting from dithered lattice quantization, allow us to construct a symmetric and time-invariant MD coding scheme. We show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion. Asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity, the entropy rate of the dithered Delta-Sigma quantization scheme approaches the symmetric two-channel MD rate-distortion function for a memoryless Gaussian source and MSE fidelity criterion, at any side-to-central distortion ratio and any resolution. In the optimal scheme, the infinite-order noise shaping filter must be minimum phase and have a piece-wise flat power spectrum with a single jump discontinuity. An important advantage of the proposed design is that it is symmetric in rate and distortion by construction, so the coding rates of the descriptions are identical and there is therefore no need for source splitting.Comment: Revised, restructured, significantly shortened and minor typos has been fixed. Accepted for publication in the IEEE Transactions on Information Theor

n-Channel Asymmetric Multiple-Description Lattice Vector Quantization

We present analytical expressions for optimal entropy-constrained multiple-description lattice vector quantizers which, under high-resolutions assumptions, minimize the expected distortion for given packet-loss probabilities. We consider the asymmetric case where packet-loss probabilities and side entropies are allowed to be unequal and find optimal quantizers for any number of descriptions in any dimension. We show that the normalized second moments of the side-quantizers are given by that of an $L$-dimensional sphere independent of the choice of lattices. Furthermore, we show that the optimal bit-distribution among the descriptions is not unique. In fact, within certain limits, bits can be arbitrarily distributed.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

Colored-Gaussian Multiple Descriptions: Spectral and Time-Domain Forms

It is well known that Shannon's rate-distortion function (RDF) in the colored quadratic Gaussian (QG) case can be parametrized via a single Lagrangian variable (the "water level" in the reverse water filling solution). In this work, we show that the symmetric colored QG multiple-description (MD) RDF in the case of two descriptions can be parametrized in the spectral domain via two Lagrangian variables, which control the trade-off between the side distortion, the central distortion, and the coding rate. This spectral-domain analysis is complemented by a time-domain scheme-design approach: we show that the symmetric colored QG MD RDF can be achieved by combining ideas of delta-sigma modulation and differential pulse-code modulation. Specifically, two source prediction loops, one for each description, are embedded within a common noise shaping loop, whose parameters are explicitly found from the spectral-domain characterization.Comment: Accepted for publications in the IEEE Transactions on Information Theory. Title have been shortened, abstract clarified, and paper significantly restructure

Packetized Predictive Control for Rate-Limited Networks via Sparse Representation

We study a networked control architecture for linear time-invariant plants in which an unreliable data-rate limited network is placed between the controller and the plant input. The distinguishing aspect of the situation at hand is that an unreliable data-rate limited network is placed between controller and the plant input. To achieve robustness with respect to dropouts, the controller transmits data packets containing plant input predictions, which minimize a finite horizon cost function. In our formulation, we design sparse packets for rate-limited networks, by adopting an an ell-0 optimization, which can be effectively solved by an orthogonal matching pursuit method. Our formulation ensures asymptotic stability of the control loop in the presence of bounded packet dropouts. Simulation results indicate that the proposed controller provides sparse control packets, thereby giving bit-rate reductions for the case of memoryless scalar coding schemes when compared to the use of, more common, quadratic cost functions, as in linear quadratic (LQ) control.Comment: 9 pages, 7 figures. arXiv admin note: text overlap with arXiv:1307.824

Sparse Packetized Predictive Control for Networked Control over Erasure Channels

We study feedback control over erasure channels with packet-dropouts. To achieve robustness with respect to packet-dropouts, the controller transmits data packets containing plant input predictions, which minimize a finite horizon cost function. To reduce the data size of packets, we propose to adopt sparsity-promoting optimizations, namely, ell-1-ell-2 and ell-2-constrained ell-0 optimizations, for which efficient algorithms exist. We derive sufficient conditions on design parameters, which guarantee (practical) stability of the resulting feedback control systems when the number of consecutive packet-dropouts is bounded.Comment: IEEE Transactions on Automatic Control, Volume 59 (2014), Issue 7 (July) (to appear

Sparsely-Packetized Predictive Control by Orthogonal Matching Pursuit

We study packetized predictive control, known to be robust against packet dropouts in networked systems. To obtain sparse packets for rate-limited networks, we design control packets via an L0 optimization, which can be effectively solved by orthogonal matching pursuit. Our formulation ensures asymptotic stability of the control loop in the presence of bounded packet dropouts.Comment: 3-page extended abstract for MTNS 2012 with 3 figure

Zero-Delay Rate Distortion via Filtering for Vector-Valued Gaussian Sources

We deal with zero-delay source coding of a vector-valued Gauss-Markov source subject to a mean-squared error (MSE) fidelity criterion characterized by the operational zero-delay vector-valued Gaussian rate distortion function (RDF). We address this problem by considering the nonanticipative RDF (NRDF) which is a lower bound to the causal optimal performance theoretically attainable (OPTA) function and operational zero-delay RDF. We recall the realization that corresponds to the optimal "test-channel" of the Gaussian NRDF, when considering a vector Gauss-Markov source subject to a MSE distortion in the finite time horizon. Then, we introduce sufficient conditions to show existence of solution for this problem in the infinite time horizon. For the asymptotic regime, we use the asymptotic characterization of the Gaussian NRDF to provide a new equivalent realization scheme with feedback which is characterized by a resource allocation (reverse-waterfilling) problem across the dimension of the vector source. We leverage the new realization to derive a predictive coding scheme via lattice quantization with subtractive dither and joint memoryless entropy coding. This coding scheme offers an upper bound to the operational zero-delay vector-valued Gaussian RDF. When we use scalar quantization, then for "r" active dimensions of the vector Gauss-Markov source the gap between the obtained lower and theoretical upper bounds is less than or equal to 0.254r + 1 bits/vector. We further show that it is possible when we use vector quantization, and assume infinite dimensional Gauss-Markov sources to make the previous gap to be negligible, i.e., Gaussian NRDF approximates the operational zero-delay Gaussian RDF. We also extend our results to vector-valued Gaussian sources of any finite memory under mild conditions. Our theoretical framework is demonstrated with illustrative numerical experiments.Comment: 32 pages, 9 figures, published in IEEE Journal of Selected Topics in Signal Processin

An Upper Bound to Zero-Delay Rate Distortion via Kalman Filtering for Vector Gaussian Sources

We deal with zero-delay source coding of a vector Gaussian autoregressive (AR) source subject to an average mean squared error (MSE) fidelity criterion. Toward this end, we consider the nonanticipative rate distortion function (NRDF) which is a lower bound to the causal and zero-delay rate distortion function (RDF). We use the realization scheme with feedback proposed in [1] to model the corresponding optimal "test-channel" of the NRDF, when considering vector Gaussian AR(1) sources subject to an average MSE distortion. We give conditions on the vector Gaussian AR(1) source to ensure asymptotic stationarity of the realization scheme (bounded performance). Then, we encode the vector innovations due to Kalman filtering via lattice quantization with subtractive dither and memoryless entropy coding. This coding scheme provides a tight upper bound to the zero-delay Gaussian RDF. We extend this result to vector Gaussian AR sources of any finite order. Further, we show that for infinite dimensional vector Gaussian AR sources of any finite order, the NRDF coincides with the zero-delay RDF. Our theoretical framework is corroborated with a simulation example.Comment: 7 pages, 6 figures, accepted for publication in IEEE Information Theory Workshop (ITW