In this paper, we consider a system that consists of $N$ independent parallel channels, where the receiver starts to decode the information being transmitted when it has access to at least $K$ of them. We refer to this system as the $(N,K)$-limited access channel. No prior knowledge for the distribution about which transmissions will be received is assumed. In addition, both the channel inputs and channel disturbances can be arbitrary, except that the mutual information function for each channel is assumed strictly concave with respect to the input power. Hence, the channel capacity below which the code rate is guaranteed to be attainable by a sequence of codes with vanishing error can be determined by the minimum mutual information among any $K$ out of $N$ channels. We then investigate the power allocation that maximizes this minimum mutual information subject to a total power constraint. As a result, the optimal solution can be determined via a systematic algorithmic procedure by performing at most $K$ single-power-sum-constrained maximizations. Based on this result, the closed-form formula of the optimal power allocation for an $(N,K)$ -limited access channel with channel inputs and additive noises, respectively, scaled from two independent and identically distributed random vectors of length $N$ is subsequently established, and is shown to be well interpreted by a two-phase water-filling principle. Specifically, in the first noise-power re- istribution phase, the least $N-K$ noise powers (equivalently, second moments) are first poured (as noise water) into a tank consisting of $K$ interconnected unit-width vessels with solid base heights, respectively, equal to the remaining $K$ largest noise powers. Afterward, those $W$ vessels either with noise water inside or with solid base height equal to the new water surface level are subdivided into $N-K+W$ vessels of rectangular shape with the same heights (as the water surface level) and widths in proportion to their noise powers. In the second signal-power allocation phase, the heights of vessel bases will be first either lifted or lowered according to the total signal power and channel mutual information functions, followed by the usual signal-power water-filling scheme. The two-phase water-filling interpretation then hints that the degree of “noisiness” for a general (possibly, nonadditive and non-Gaussian) limited access channel might be identified by composing the derivative of the mutual information function with its inverse.
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机译：在本文中，我们考虑一个由$ N $个独立并行通道组成的系统，当接收者至少可以访问其中的$ K $个时，接收器便开始对正在传输的信息进行解码。我们将此系统称为$（N，K）$受限访问通道。假设没有关于将要接收传输的分布的先验知识。另外，除了假定每个通道的互信息功能相对于输入功率严格凹入之外，通道输入和通道干扰都可以是任意的。因此，可以由$ N $个信道中的任何$ K $个信道中的最小互信息来确定信道容量，在该信道容量以下，具有消失误差的代码序列可以保证达到码率。然后，我们研究在总功率约束下最大化此最小互信息的功率分配。结果，可以通过执行最多$ K $个单次功率和约束的最大化，通过系统的算法过程确定最佳解决方案。基于此结果，分别从两个独立且分布均匀的长度为$ N的随机矢量按比例缩放具有信道输入和加性噪声的$（N，K）$受限访问信道的最佳功率分配的闭式公式随后建立了$，并通过两相充水原理很好地解释了$。具体而言，在第一个噪声功率重新分配阶段，首先将最小的$ NK $噪声功率（等同于第二时刻）倒入（作为噪声水）到一个由$ K $互连的单位宽度容器与坚实基础组成的罐中高度分别等于剩余的$ K $最大噪声功率。之后，那些内部装有噪声水或基本高度等于新水面高度的$ W $容器被细分为矩形的$ N-K + W $矩形容器，其高度与水面高度相同。宽度与噪声功率成正比。在第二个信号功率分配阶段，首先根据总信号功率和通道互信息功能，升高或降低船基的高度，然后是通常的信号功率注水方案。然后，两阶段注水解释暗示，可以通过将互信息函数的导数与其反函数组合来确定一般（可能是非加性和非高斯）有限访问通道的“嘈杂”程度。
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