In this article, under a general cost function C, we present a dynamic programming (DP) method to obtain an optimal sequential deterministic quantizer (SDQ) for q-ary input discrete memoryless channel (DMC). The DP method has complexity O(q (N-M)2 M), where N and M are the alphabet sizes of the DMC output and quantizer output, respectively. Then, starting from the quadrangle inequality, two techniques are applied to reduce the DP method's complexity. One technique makes use of the Shor-Moran-Aggarwal-Wilber-Klawe (SMAWK) algorithm and achieves complexity O(q (N-M) M). The other technique is much easier to be implemented and achieves complexity O(q (N2 - M2)). We further derive a sufficient condition under which the optimal SDQ is optimal among all quantizers and the two techniques are applicable. This generalizes the results in the literature for binary-input DMC. Next, we show that the cost function of α-mutual information ( α-MI)-maximizing quantizer belongs to the category of C. We further prove that under a weaker condition than the sufficient condition we derived, the aforementioned two techniques are applicable to the design of α-MI-maximizing quantizer. Finally, we illustrate the particular application of our design method to practical pulse-amplitude modulation systems.

中文翻译：

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离散无记忆通道顺序确定性量化的动态规划


在本文中，在一般成本函数C下，我们提出了一种动态规划（DP）方法来获得q元输入离散无记忆通道（DMC）的最佳顺序确定性量化器（SDQ）。 DP方法的复杂度为O(q(NM)2M)，其中N和M分别是DMC输出和量化器输出的字母大小。然后，从四边形不等式出发，应用两种技术来降低DP方法的复杂度。一种技术利用 Shor-Moran-Aggarwal-Wilber-Klawe (SMAWK) 算法并实现复杂度 O(q (NM) M)。另一种技术更容易实现，复杂度为 O(q (N2 - M2))。我们进一步推导了一个充分条件，在该条件下，最优 SDQ 在所有量化器中都是最优的，并且这两种技术都适用。这概括了二进制输入 DMC 文献中的结果。接下来，我们证明α-互信息（α-MI）最大化量化器的成本函数属于C类别。我们进一步证明，在比我们推导的充分条件更弱的条件下，上述两种技术适用于α-MI最大化量化器的设计。最后，我们说明了我们的设计方法在实际脉冲幅度调制系统中的具体应用。