Consider a channel whose the input alphabet set X={x1,x2,…,xK}\mathbb {X}=\{x_{1},x_{2}, {\dots },x_{K}\} contains KK discrete symbols modeled as a discrete random variable XX having a probability mass function p(x)=[p(x1),p(x2),…,p(xK)]\mathbf {p}(\mathbf {x}) = [p(x_{1}), p(x_{2}), {\dots }, p(x_{K})] and the received signal YY being a continuous random variable. YY is a distorted version of XX caused by a channel distortion, characterized by the conditional densities p(y|xi)=ϕi(y)p(y|x_{i})=\phi _{i}(y) , i=1,2,…,Ki=1,2, {\dots },K . To recover XX , a quantizer QQ is used to quantize YY back to a discrete output Z={z1,z2,…,zN}\mathbb {Z} =\{z_{1}, z_{2}, {\dots }, z_{N}\} corresponding to a random variable ZZ with a probability mass function p(z)=[p(z1),p(z2),…,p(zN)]\mathbf {p}(\mathbf {z}) = [p(z_{1}), p(z_{2}), {\dots }, p(z_{N})] such that the mutual information I(X;Z)I(X;Z) is maximized subject to an arbitrary constraint on p(z)\mathbf {p}(\mathbf {z}) . Formally, we are interested in designing an optimal quantizer Q∗Q^{*} that maximizes βI(X;Z)−C(Z)\beta I(X;Z) - C(Z) where β\beta is a positive number that controls the trade-off between maximizing I(X;Z)I(X;Z) and minimizing an arbitrary cost function C(Z)C(Z) . Let p(x|y)=[p(x1|y),p(x2|y),…,p(xK|y)]\mathbf {p}(\mathbf {x}|y)=[p(x_{1}|y),p(x_{2}|y), {\dots },p(x_{K}|y)] be the posterior distribution of XX for a given value of yy , we show that for any arbitrary cost function C(.)C(.) , the optimal quantizer Q∗Q^{*} separates the vectors p(x|y)\mathbf {p}(\mathbf {x}|y) into convex regions. Using this result, a method is proposed to determine an upper bound on the number of thresholds (decision variables on yy ) which is used to speed up the algorithm for finding an optimal quantizer. Numerical results are presented to validate the findings.

中文翻译：

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约束下互信息最大化的最优量化器结构


考虑一个通道，其输入字母集 X={x1,x2,…,xK}\mathbb {X}=\{x_{1},x_{2}, {\dots },x_{K}\} 包含 KK离散符号建模为具有概率质量函数 p(x)=[p(x1),p(x2),…,p(xK)]\mathbf {p}(\mathbf {x}) = 的离散随机变量 XX [p(x_{1}), p(x_{2}), {\dots }, p(x_{K})] 和接收信号 YY 是连续随机变量。 YY 是由通道失真引起的 XX 的失真版本，其特征为条件密度 p(y|xi)=phii(y)p(y|x_{i})=\phi _{i}(y) , i =1,2,…,Ki=1,2, {\dots },K 。为了恢复 XX ，使用量化器 QQ 将 YY 量化回离散输出 Z={z1,z2,…,zN}\mathbb {Z} =\{z_{1}, z_{2}, {\dots } , z_{N}\} 对应于概率质量函数 p(z)=[p(z1),p(z2),…,p(zN)]\mathbf {p}(\mathbf { z}) = [p(z_{1}), p(z_{2}), {\dots }, p(z_{N})] 使得互​​信息 I(X;Z)I(X;Z ) 在 p(z)\mathbf {p}(\mathbf {z}) 的任意约束下最大化。形式上，我们感兴趣的是设计一个最大化 βI(X;Z)−C(Z)\beta I(X;Z) - C(Z) 的最优量化器 Q*Q^{*}，其中 β\beta 是正数控制最大化 I(X;Z)I(X;Z) 和最小化任意成本函数 C(Z)C(Z) 之间的权衡的数字。令 p(x|y)=[p(x1|y),p(x2|y),…,p(xK|y)]\mathbf {p}(\mathbf {x}|y)=[p( x_{1}|y),p(x_{2}|y), {\dots },p(x_{K}|y)] 是 XX 对于给定值 yy 的后验分布，我们证明对于任何任意成本函数 C(.)C(.) ，最优量化器 Q∗Q^{*} 将向量 p(x|y)\mathbf {p}(\mathbf {x}|y) 分成凸区域。使用该结果，提出了一种确定阈值数量上限（yy 上的决策变量）的方法，该方法用于加速寻找最佳量化器的算法。 给出了数值结果来验证研究结果。