On Mutual Information-Based Optimal Quantizer Design

The problem of optimal quantizer design that maximizes the mutual information between the input and the quantized output of a communication channel is considered. It is shown that an optimal quantizer exists which employs convex polytopes as its decision regions in the Euclidean space of likelihood ratios. This complements the previous results in the literature by presenting a new and intuitive proof, establishing a unified treatment of the most general case, extending the solution to continuous-input channels, and providing a characterization for the form of optimal decision rule based on likelihood ratios, whereas the previous results are expressed in terms of the posterior probabilities and depend on the channel input distribution. The result is corroborated with an analytical example employing the transmission of a spherically symmetric random vector source over an additive white Gaussian noise channel.