We consider a channel with discrete input ${X}$, a continuous noise that corrupts the input ${X}$ to produce the continuous-valued output ${U}$. A thresholding quantizer is then used to quantize the continuous-valued output ${U}$ to the final discrete output ${V}$. The goal is to jointly design a thresholding quantizer that maximizes the mutual information between input and quantized output ${I}$(${X}$;${V}$) while minimizing a pre-specified function of the quantized output $F(p_{V})$. A general dynamic programming algorithm is proposed having the time complexity $O(KNM^{2})$ where N, M and ${K}$ are the sizes of input ${X}$, output ${U}$ and quantized output ${V}$, respectively. Moreover, we show that if $F(p_{V})=\displaystyle \sum _{i=1}^{K}g_{i}(p_{v_{i}})$ where $g_{i}(.)$ is a convex function, $p_{v_{i}} \in p_{V}=\{p_{v}1,\cdots ~p_{v_{K}}\}$ is the probability mass function of output $v_{i}\in V$ and the channel conditional density $p(u|x)$ satisfies the dominated condition (often true in practice), then the existing SMAWK algorithm can be applied to reduce the time complexity of the dynamic programming algorithm from $O(KNM^{2}$) to ${O}$(KNM). Both theoretical and numerical results are provided to verify our contributions.
Thresholding Quantizer Design for Mutual Information Maximization Under Output Constraint
2020-05-01
148867 byte
Aufsatz (Konferenz)
Elektronische Ressource
Englisch
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