Generalized bandpass filters for decoding block codes

Abstract

We show that the optimum decoder for a linear code over a finite field can be implemented by a bank of bandpass filters that are defined in a generalized frequency domain. This decoder has a fixed signal processing structure and uses complex-valued arithmetic operations instead of the usual finite field ones, This structure closely resembles one of the usual digital filter mechanizations. The emphasis in this paper is on general additive channels. The maximum likelihood decoder configuration for a rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/n</tex> code is a bank of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> bandpass filters whose outputs are the decoded information digits. The parameters that define the filters are directly related to the channel statistics, and each of the band-pass filters is composed of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> branch filters where the order of the field is the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> th power of a prime. The processing requirements can further be reduced by detuning each branch filter and following it by a real roundoff operation. These possibilities have not been fully exploited and remain an open question. A procedure for determining the filter weights needed in the branch filters that uses fast transform techniques is demonstrated. The probability of correct decoding can be computed using functions defined over the transform domain. Finally, we show that suboptimal decoders, which may also include detection capabilities, can be derived from this work. A hierarchy of suboptimal decoding levels is proposed so as to reduce the average computational requirements of the decoder.