The Dispersion of Mismatched Joint Source-Channel Coding for Arbitrary Sources and Additive Channels

Abstract

We consider a joint source channel coding (JSCC) problem in which we desire to transmit an arbitrary memoryless source over an arbitrary additive channel. We propose a mismatched coding architecture that consists of Gaussian codebooks for both the source reproduction sequences and channel codewords. The natural nearest neighbor encoder and decoder, however, need to be judiciously modified to obtain the highest communication rates at finite blocklength. In particular, we consider a unequal error protection (UEP) scheme in which all sources are partitioned into disjoint power type classes. We also regularize the nearest neighbor decoder so that an appropriate measure of the size of each power type class is taken into account in the decoding strategy. For such an architecture, we derive ensemble-tight second-order and moderate deviations results. Our first-order (optimal bandwidth expansion ratio) result generalizes the seminal results by Lapidoth (1996, 1997). The dispersion of our JSCC scheme is a linear combination of the mismatched dispersions for the channel coding saddle-point problem by Scarlett, Tan and Durisi (2017) and the rate-distortion saddle-point problem by the present authors, thus also generalizing these results. A. Main Contributions and Related WorksOur main contributions are summarized as follows: (i) We propose a JSCC architecture using Gaussian codebooks, with modified minimum distance encoding and decoding, to transmit an arbitrary memoryless source over an arbitrary additive memoryless channel. We argue in Section II-C that this architecture generalizes and unifies works by Lapidoth [2], [3]. While the Gaussian codebooks are similar to those in [2], [3], our encoding and decoding schemes differ somewhat. To capture the JSCC nature of the problem, we draw inspiration from works by Csiszár [6] and Wang, Ingber and Kochman [7] who respectively established the error exponent and second-order asymptotics for sending a DMS over a DMC. The authors employed the method of types and an unequal error protection (UEP) scheme (cf. Shkel, Tan and Draper [8]). In our work, we introduce a natural partition of the source sequences into types; however, the notion of types has to be defined carefully since the source need not be discrete. We also regularize the nearest neighbor decoder [2] so that an appropriate measure of the size of each type class is carefully taken into account in the decoding strategy. Our architecture (which is shown in Figure 1) and subsequent analyses allow us to show that the maximum attainable rate is the ratio between the Gaussian capacity and Gaussian rate-distortion function. (ii) The main contribution, however, is the derivation of ensemble-tight second-order coding rates and moderate deviations constants for the architecture so described. By allowing a non-vanishing ensemble excess-distortion probability, we shed light on the backoff from the maximum attainable rate at finite blocklengths. This complements the results of Kostina and Verdú [9] who also derived the dispersion of tr...

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