1Quantization of Binary-Input Discrete Memoryless Channels, with Applications to LDPC Decoding

Abstract—The quantization of the output of a binary-input discrete memoryless channel to a smaller number of levels is considered. The optimal quantizer, in the sense of maximizing mutual information between the channel input and the quantizer output, may be found by an algorithm with complexity which is quadratic in the number of channel outputs. This is a concave optimization problem, and results from the field of concave optimization are invoked. The quantizer design algorithm is a realization of a dynamic program. Then, this algorithm is applied to the design of message-passing decoders for low-density parity-check codes, over arbitrary discrete memoryless channels. A general, systematic method to find message-passing decoding maps which maximize mutual information at each iteration is given. This may contrasted with existing quantized message-passing algorithms which are heuristically derived. The method finds message-passing decoding maps similar to those given by Richardson and Urbanke’s Algorithm E. Using four bits per message, noise thresholds similar to belief-propagation decoding are obtained. Index Terms—discrete memoryless channel, channel quantiza-tion, mutual information maximization, LDPC decoding I