Optimum symbol-by-symbol mean-square error channel coding

Abstract

An <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k)</tex> linear block code is used to convey <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> information digits, each having a numerical value attached, in such a way as to minimize the mean-square error of the individual information digits regardless of the other <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)</tex> positions. This criterion is used because it provides a graceful degradation in the coding performance when the channel temporarily becomes less reliable. Unambiguous encoding is employed, and the channel is assumed to obey a realistic additive property. An optimum pair of symbol encoding and decoding functions are derived, and it is shown that the optimum symbol encoding can be performed in a linear fashion. The related decoding role is the conditional mean estimator. Fourier transforms are central to the optimal design of both rules. The optimum symbol rules tire identical with the properly isolated parts of the optimum mean-square error rules when the code is used to carry the k Information digits considered as a single complete numerical entity, The same channel coding design is therefore optimum for a wide variety of Information configurations. Techniques for implementing the optimum symbol decoders are examined with special emphasis on memoryless channels where the storage requirements for the necessary weighting coefficients are drastically reduced.