The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
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DOI：
10.1109/tit.2019.2948636
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发表时间：
2020-04-01
影响因子：
2.5
通讯作者：
Shamai, Shlomo (Shitz)
中科院分区：
文献类型：
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作者：
Dytso, Alex;Yagli, Semih;Shamai, Shlomo (Shitz)
研究了一种具有峰值功率约束输入的n维加性高斯噪声信道。众所周知，在这种情况下，当n = 1时，实现能力的输入分布是具有有限多个质量点的离散分布，当n = 1时，实现能力的输入分布被支撑在有限多个同心壳上。然而，由于之前的证明技术，甚至没有一个关于质量点/壳的确切数量的界限。本文在给出质量点和壳数的第一个确定界限的同时，提供了另一种证明能力实现输入分布质量点/壳数有限的方法，为解决许多这类问题铺平了另一条道路。本文的第一个主要结果是一个阶紧隐界，该隐界表明，与向下移位的能达输出概率密度函数的零点数相比，能达输入分布中的质量点数在两个因子之内。接下来，利用该隐式边界提供最优输入分布支持大小的第一个确定上限，即O（A2）上限，其中a表示对输入振幅的约束。本文的第二个主要结果将第一个结果推广到> 1的情况，表明对于n - 1的每一个维度，最优输入分布包含的壳数为O（A2）。最后，本文的第三个主要结果用额外的平均功率约束重新考虑了n = 1的情况，证明了一个类似的O（A2）界。
This paper studies an n-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when n = 1 the capacity-achieving input distribution is discrete with finitely many mass points, and when n > 1 the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity-achieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an O(A2) upper bound where A denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case whenn > 1, showing that, for each and every dimension n - 1, the number of shells that the optimal input distribution contains is O(A2). Finally, the third main result of this paper reconsiders the case n = 1 with an additional average power constraint, demonstrating a similar O(A2) bound.