Distributed Coding of Correlated Sources and Applications in Sensor Networks

Analog Mappings for Joint Source Channel Coding For 
Single and Multi-Terminal Systems

Introduction

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In this work, joint source channel coding with limited delay is considered. Most information theoretic results assume infinite amount of samples, which is not practically relevant, particularly for sensor networks where limited delay is an important key feature. In this work, optimal joint source-channel coding scheme for a given delay is studied. The point-to-point results are then extended to the main scheme of distributed source-channel coding of correlated sources. The obtained encoding functions are shown to be a continuous relative of, and in fact subsume as a special case, the Wyner-Ziv mappings encountered in digital distributed source coding systems, by mapping multiple source intervals to the same channel interval.

Method : The problem of obtaining the vector transformations (g and h in figure) that optimally map between the m-dimensional source space and the k-dimensional channel space is considered under a given channel power constraint and mean square error distortion measure. Closed form necessary conditions for optimality of the encoder and decoder mappings are derived. The optimal mappings are obtained using an iterative algorithm that updates encoder and decoder mappings according to optimality conditions at each iteration.

Results : Two representative results are presented. Although the proposed method is generally applicable, we present two representative examples based on Gaussian source and channel. The obtained mappings are compared to the prior works (that are based on Archimedian spiral) and as a benchmark, the theoretically optimal curve (OPTA) that can only be achieved with infinite delay. In the first one, two dimensional Gaussian source is mapped to scalar Gaussian channel, and in the second one, scalar Gaussian source is mapped to two dimensional vector channel. The results show substantial improvement of performance of proposed mappings over prior works.

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Distributed case

The proposed method is extended to distributed scenarios including the side information case.

Results : Scalar Gaussian source, channel and side information case is presented, although the method is generally applicable. Fig. 1 below shows the gains of the obtained mapping over linear encoder. The obtained mappings is shown in Fig 2 below.  The obtained mappings is a continous relative of Wyzer-Ziv mappings in the sense that multiple source intervals are mapped to the same channel interval.



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Conditions for Linearity of Optimal Estimation

The analog mapping problem is strongly connected to the problem of optimal estimation. In (asmyptotically high delay) digital communications, "structured" codes, such as linear codes, lattice codes...,  are commonly used. Fortunately, these structured codes can achieve the fundamental limits, i.e., structure comes free. In delay limited mappings, this may not be the case. In this work, we study the conditions  for which the optimal mappings, specifically the estimators (decoders) are structured, i.e., linear. Specifically, we focus on the setting below, looking for conditions for h(Y)=kY for some k.

 

 

Our main result is the necessary and sufficient condition for linearity of optimal estimation, in terms of the characteristic function of the source X, Fx(w) and the noise Z, Fz(w), for a given SNR level, snr:  Fx(w)=(Fz(w))snr . This connection, yields to several important result. One of them is the (asmyptotic) linearity of optimal estimation at high snr, for a Gaussian source, irrespective of the channel. The numerical example below demonstrates how the optimal estimator converges to linear, for a Gaussian source and uniform channel (common setting in source coding),  as snr increases.