On the Design and Analysis of Shannon-Kotel’nikov Mappings for Joint-Source-Channel Coding
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In this dissertation, we explore the possibility of transmitting discrete-time, continuous amplitude sources over discrete-time, continuous-amplitude channels by using non-linear direct source-channel mappings. This is a joint source-channel coding technique where there is no real distinction between the source coding part and the channel coding part. In contrast to traditional digital communication systems, these techniques are only suited for transmitting sources which tolerate a certain amount of distortion, e.g. images and sound. The reason for this restriction is that the channel noise is part of the total distortion, thus making error-free transmission impossible. However, the analog nature of the scheme provides both high spectral efficiency and robustness when the mappings are properly designed. As there are no known non-linear maps which actually achieves optimality, we investigate the mechanisms which degrade the performance of source-channel coding systems, hoping to obtain some pointers on how todesign systems with as little performance loss as possible. We identify severalloss factors causing performance degradations; among them we mention mismatched channel symbol distribution and mismatched error-sequence distribution. Given an additive white Gaussian noise channel with an average power constraint, and a mean-squared error distortion measure, it is shownthat both the loss from having non-Gaussian distributed channel symbols, and the loss from having non-Gaussian reconstruction error/noise, equals the relative entropy of the actual distribution and the capacity-achievingGaussian. A class of joint source-channel coding schemes which we have called Shannon-Kotel’nikov mappings is shown to provide both robustness to unknown channel conditions, and high spectral efficiency in the sense that themappings operate close to the theoretical performance bounds. A mappingconsist of a non-linear curve, or function, which maps a source point directly into the channel space. By having different dimensions of the source and channel spaces, both bandwidth reduction (compression) and bandwidthexpansion (error control) can be achieved. The optimization of a 2:1 bandwidth-reducing system is shown, for both Gaussian and Laplacian sources over additive white Gaussian noise channels. These are shown to perform quite well relative to digital systems. Furthermore, a 4:1 bandwidthreducingsystem consisting of a cascade of two 2:1 mappings is tried out as away to avoid the complexity increase associated with higher dimensions, but this shows worse performance than known channel-optimized vector quantizers. A problem with the analog nature of the Shannon-Kotel’nikov mappings is that they do not automatically interface with existing digital communicationsystems. Whether one wants to store the received mappings, or transmitthem further through a digital transport network, a digitization step is necessaryin order to obtain a bit-representation. We propose some very simple transcoding schemes which digitize the received channel symbols directly, instead of first decoding to native representation and re-encoding with a digital source coder. This proves to be simple, yet effective for interfacing the Shannon-Kotel’nikov mappings with a digital system. Moreover, the transcoding technique which produces multi-level channel symbols proves to be able to suppress most of the channel noise, enabling serial multi-hop communication without accumulation of channel noise.