Optimum soft decision decoding with graceful degradation

We demonstrate that the complex decision variables eminating directly from the channel receiver may be efficiently used for optimum decoding. The data is protected by a linear block designed for the minimum mean-square error between the numerical representations of the input and output words. This criterion affords a natural graceful degradation property in the sense that if the channel noise randomly exceeds the performance capabilities of the code, any decoding errors are confined to the least significant positions. The channel model is described by a conditional probability density function which obeys a realistic additive assumption.A general optimum design procedure for selecting the encoding and decoding rules is presented. It involves the use of generalized Fourier transform coefficients of the channel density function. We examine the structure of the decoder and emphasize the special case where the channel is memoryless. In that case the optimum decoder uses a state space mechanization resembling the Viterbi decoder. Practical examples of coherent phase and frequency digital modulation formats are discussed. The general results are further extended to the situation where optimum symbol-by-symbol rules are employed