Low-precision analog-to-digital conversion and mutual information in channels with memory

The discrete-time channel with memory and additive Gaussian noise under output analog-to-digital (A/D) conversion (quantization) is considered. Traditionally, the design of the components performing the quantization step targets to reproduce the received waveform as close as possible, often using the squared error as the metric. Instead, we focus on mutual information as a cost function for the design of A/D converters for channels with memory. It is shown that 1-bit/sample quantization is sufficient to achieve the full information rate of one bit per channel use, for all channels and BPSK modulation at infinite signal-to-noise ratio; these quantizers, however, are not given by Lloyd-Max quantizers for all channels. Based on that result, we design scalar and multi-dimensional A/D converters adopting mutual information as a design criterion. These A/D converters differ from conventional converters in that the binary indices of each quantization region do not correspond to a real value representing the received sample; rather, those indices represent the likelihoods of the corresponding channel input given the received sample, leading to the notion of analog-to-likelihood converters. Simulation results are presented to illustrate the effectiveness of the proposed design.