Minimum Expected Distortion in Gaussian Source Coding with Fading Side Information
📝 Abstract
An encoder, subject to a rate constraint, wishes to describe a Gaussian source under squared error distortion. The decoder, besides receiving the encoder’s description, also observes side information consisting of uncompressed source symbol subject to slow fading and noise. The decoder knows the fading realization but the encoder knows only its distribution. The rate-distortion function that simultaneously satisfies the distortion constraints for all fading states was derived by Heegard and Berger. A layered encoding strategy is considered in which each codeword layer targets a given fading state. When the side-information channel has two discrete fading states, the expected distortion is minimized by optimally allocating the encoding rate between the two codeword layers. For multiple fading states, the minimum expected distortion is formulated as the solution of a convex optimization problem with linearly many variables and constraints. Through a limiting process on the primal and dual solutions, it is shown that single-layer rate allocation is optimal when the fading probability density function is continuous and quasiconcave (e.g., Rayleigh, Rician, Nakagami, and log-normal). In particular, under Rayleigh fading, the optimal single codeword layer targets the least favorable state as if the side information was absent.
💡 Analysis
An encoder, subject to a rate constraint, wishes to describe a Gaussian source under squared error distortion. The decoder, besides receiving the encoder’s description, also observes side information consisting of uncompressed source symbol subject to slow fading and noise. The decoder knows the fading realization but the encoder knows only its distribution. The rate-distortion function that simultaneously satisfies the distortion constraints for all fading states was derived by Heegard and Berger. A layered encoding strategy is considered in which each codeword layer targets a given fading state. When the side-information channel has two discrete fading states, the expected distortion is minimized by optimally allocating the encoding rate between the two codeword layers. For multiple fading states, the minimum expected distortion is formulated as the solution of a convex optimization problem with linearly many variables and constraints. Through a limiting process on the primal and dual solutions, it is shown that single-layer rate allocation is optimal when the fading probability density function is continuous and quasiconcave (e.g., Rayleigh, Rician, Nakagami, and log-normal). In particular, under Rayleigh fading, the optimal single codeword layer targets the least favorable state as if the side information was absent.
📄 Content
I N lossy data compression, side information at the decoder can help reduce the distortion in the reconstruction of the source [1]. The decoder, however, may have access to the side information only through an unreliable channel. For example, in distributed compression over wireless sensor networks, correlated sensor measurements from a neighboring node may be available to the decoder through a fading wireless channel. In this work, we consider a Gaussian source where the encoder is subject to a rate constraint and the distortion metric is the mean squared error of the reconstruction. In addition to the compressed symbol, we assume that the decoder observes the original symbol through a separate analog fading channel. We assume, similar to the approach in [2], that the fading is quasistatic, and that the decoder knows the fading realization but the encoder knows only its distribution. The ratedistortion function that dictates the rate required to satisfy the distortion constraint associated with each fading state is given by Heegard and Berger in [3]. We consider a layered encoding strategy based on the uncertain fading realization in the side-information channel, and optimize the rate allocation among the possible fading states to minimize the expected distortion. In particular, we formulate the distortion minimization as a convex optimization problem, and develop an efficient representation for the Heegard-Berger rate-distortion function under which the optimization problem size is linear in the number of discrete fading states. Furthermore, we identify the conditions under which single-layer rate allocation is expected-distortion-minimizing, and extend these optimality conditions for continuous fading distributions through a limiting process on the primal and dual solutions in the optimization. We show that singlelayer rate allocation is optimal for fading distributions with continuous, quasiconcave probability density functions such as Rayleigh, Rician, Nakagami, and log-normal.
When the side-information channel exhibits no fading, the distortion is given by the Wyner-Ziv ratedistortion function [4]. Rate-distortion is considered in [5], [6] when the side information is also available at the encoder, and in [7] when there is a combination of decoder-only and encoder-and-decoder side information. Successive refinement source coding in the presence of side information is considered in [8], [9]. The side-information scalable rate-distortion region is characterized in [10], in which the user with inferior side information decodes an additional layer of the source-coding codeword. Lossless source coding with an unknown amount of side information at the decoder is considered in [11], in which a fixed data block is broadcast to different users in a variable number of channel uses [12]. In [13], [14], expected distortion is minimized in the transmission of a Gaussian source over a slowly fading channel in the absence of channel state information at the transmitter (CSIT). Broadcast transmission with imperfect CSIT is considered in [15]. Another application of source coding with uncertain side information is in systematic lossy source-channel coding [16] over a fading channel without CSIT. For example, when upgrading legacy communication systems, a digital channel may be added to augment an existing analog channel. In this case the analog reception then plays the role of side information in the decoding of the description from the digital channel. In [17], [18], hybrid digital/analog and digital transmission schemes are considered for Wyner-Ziv coding over broadcast channels. The system model studied in this paper is also related to distributed source coding over multiple links [19], [20] where, besides source coding over a finite-capacity reliable link, noisy versions of the source are described through additional backhaul links with infinite capacity but that are subject to random failure. At the decoder, the realized quality of the side information is determined by the number of backhaul links that are successfully connected. Similar models are considered in [21] for distributed unreliable relay communications.
The remainder of the paper is organized as follows. The system model is described in Section II. Section III derives the minimum expected distortion and presents the convex optimization framework when the side-information channel has discrete fading states. Section IV investigates the optimal rate allocation under different fading distributions in the side-information channel. Section V considers the optimality of single-layer rate allocation under discrete fading states as well as continuous fading distributions. Conclusions are given in Section VI.
Consider the system model shown in Fig. 1. An encoder wishes to describe a real Gaussian source sequence {X} under a rate constraint of R X nats per symbol, where the sequence of random variables are independent identically distributed (i.i.d.) with X ∼
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