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Register a new userRemote Source Coding under Gaussian Noise : Dueling Roles of Power and Entropy Power
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Added by
Michael Gastpar
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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The distributed remote source coding (so-called CEO) problem is studied in the case where the underlying source, not necessarily Gaussian, has finite differential entropy and the observation noise is Gaussian. The main result is a new lower bound for the sum-rate-distortion function under arbitrary distortion measures. When specialized to the case of mean-squared error, it is shown that the bound exactly mirrors a corresponding upper bound, except that the upper bound has the source power (variance) whereas the lower bound has the source entropy power. Bounds exhibiting this pleasing duality of power and entropy power have been well known for direct and centralized source coding since Shannons work. While the bounds hold generally, their value is most pronounced when interpreted as a function of the number of agents in the CEO problem.
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An extension of the entropy power inequality to the form $N_r^alpha(X+Y) geq N_r^alpha(X) + N_r^alpha(Y)$ with arbitrary independent summands $X$ and $Y$ in $mathbb{R}^n$ is obtained for the Renyi entropy and powers $alpha geq (r+1)/2$.
This paper gives improved R{e}nyi entropy power inequalities (R-EPIs). Consider a sum $S_n = sum_{k=1}^n X_k$ of $n$ independent continuous random vectors taking values on $mathbb{R}^d$, and let $alpha in [1, infty]$. An R-EPI provides a lower bound on the order-$alpha$ Renyi entropy power of $S_n$ that, up to a multiplicative constant (which may depend in general on $n, alpha, d$), is equal to the sum of the order-$alpha$ Renyi entropy powers of the $n$ random vectors ${X_k}_{k=1}^n$. For $alpha=1$, the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov which relies on the sharpened Youngs inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.
Source-channel coding for an energy limited wireless sensor node is investigated. The sensor node observes independent Gaussian source samples with variances changing over time slots and transmits to a destination over a flat fading channel. The fading is constant during each time slot. The compressed samples are stored in a finite size data buffer and need to be delivered in at most $d$ time slots. The objective is to design optimal transmission policies, namely, optimal power and distortion allocation, over the time slots such that the average distortion at destination is minimized. In particular, optimal transmission policies with various energy constraints are studied. First, a battery operated system in which sensor node has a finite amount of energy at the beginning of transmission is investigated. Then, the impact of energy harvesting, energy cost of processing and sampling are considered. For each energy constraint, a convex optimization problem is formulated, and the properties of optimal transmission policies are identified. For the strict delay case, $d=1$, $2D$ waterfilling interpretation is provided. Numerical results are presented to illustrate the structure of the optimal transmission policy, to analyze the effect of delay constraints, data buffer size, energy harvesting, processing and sampling costs.
This note contributes to the understanding of generalized entropy power inequalities. Our main goal is to construct a counter-example regarding monotonicity and entropy comparison of weighted sums of independent identically distributed log-concave random variables. We also present a complex analogue of a recent dependent entropy power inequality of Hao and Jog, and give a very simple proof.
We determine the rate region of the vector Gaussian one-helper source-coding problem under a covariance matrix distortion constraint. The rate region is achieved by a simple scheme that separates the lossy vector quantization from the lossless spatial compression. The converse is established by extending and combining three analysis techniques that have been employed in the past to obtain partial results for the problem.
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