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On Binary Quantizer For Maximizing Mutual Information
IEEE Transactions on Communications ( IF 8.3 ) Pub Date : 2020-09-01 , DOI: 10.1109/tcomm.2020.3002910
Thuan Duc Nguyen , Thinh Nguyen

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 $r(y) = \frac {P(Y=y|X=0)}{P(Y = y|X=1)}$ is a strictly increasing or 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 $r^{*}>0$ . In addition, we also characterize necessary conditions using fixed point theorem for the optimality and uniqueness of a quantizer. Based on these conditions, we propose an efficient procedure for determining all locally optimal quantizers, and thus, a globally optimal quantizer can be found. Our results also confirm some previous results using alternative elementary proofs.

中文翻译:

用于最大化互信息的二进制量化器

我们考虑一个带有二进制输入的通道 $X$ 被连续值噪声破坏,导致连续值输出 $Y$ . 最佳二进制量化器用于量化连续值输出 $Y$ 到最终的二进制输出 $Z$ 最大化互信息 $I(X; Z)$ . 我们表明,当信道条件密度的比率 $r(y) = \frac {P(Y=y|X=0)}{P(Y = y|X=1)}$ 是一个严格的递增或递减函数 $y$ ,那么具有单个阈值的量化器可以最大化互信息。此外,我们证明了最佳量化器(可能具有多个阈值)是具有阈值向量的量化器,其元素都是 $r(y)=r^{*}$ 对于一些常数 $r^{*}>0$ . 此外,我们还使用不动点定理来表征量化器的最优性和唯一性的必要条件。基于这些条件,我们提出了一种确定所有局部最优量化器的有效程序,从而可以找到全局最优量化器。我们的结果还使用替代的基本证明证实了之前的一些结果。
更新日期:2020-09-01
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