On the Uniqueness of Binary Quantizers for Maximizing Mutual Information
This provides theoretical insights for signal processing and communication systems, but it is incremental as it confirms and extends previous results with alternative proofs.
The paper tackles the problem of designing an optimal binary quantizer for a channel with binary input and continuous output to maximize mutual information, showing that under certain monotonicity conditions, a single-threshold quantizer is optimal and the optimal threshold is unique, enabling efficient algorithms like bisection.
We consider a channel with a binary input X being corrupted by a continuous-valued noise that results in a continuous-valued output Y. An optimal binary quantizer is used to quantize the continuous-valued output Y to the final binary output Z to maximize the mutual information I(X; Z). We show that when the ratio of the channel conditional density r(y) = P(Y=y|X=0)/ P(Y =y|X=1) is a strictly increasing/decreasing function of y, then a quantizer having a single threshold can maximize mutual information. Furthermore, we show that an optimal quantizer (possibly with multiple thresholds) is the one with the thresholding vector whose elements are all the solutions of r(y) = r* for some constant r* > 0. Interestingly, the optimal constant r* is unique. This uniqueness property allows for fast algorithmic implementation such as a bisection algorithm to find the optimal quantizer. Our results also confirm some previous results using alternative elementary proofs. We show some numerical examples of applying our results to channels with additive Gaussian noises.