(N,K)有限存取系統之最佳功率配置準則

Abstract

在本篇論文，我們探討一個有限存取的通訊傳輸系統，該系統由N個彼此獨立的平行通道所組成，且接收端在收到至少K個通道之完整傳輸訊號後才開始進行解碼。因為N個傳輸訊號中，允許僅有K個傳輸訊號到達接收端，此系統因而被稱為(N,K)有限存取通訊系統。對於(N,K)有限存取通訊系統，我們假定系統無法提供接收訊號個數之統計特性，因此該系統之最佳傳輸極限為任意K個通道所對應之交互資訊量(mutual information)之最小值；而找尋最佳功率配置使通道容量最大化便成為了一個最小值最大化的問題(max-min problem)。對於任意通道模型之(N,K)有限存取通訊系統，我們提出了一個系統化的快速搜尋演算法找尋其最佳功率配置，此演算法僅需解至多K個簡化後的最佳化問題，即可得到最佳功率配置，因此可大幅降低找尋最佳功率配置所需要之複雜度。此外，當通道模型為相加雜訊時，此演算法之搜尋步驟可進一步簡化成兩階段注水式功率配置。最後藉由兩階段注水式功率配置之概念，我們進一步探討如何判斷任意通道之『雜訊程度』(degree of noisiness)，並得到『雜訊程度』之定義應為交互資訊量之微分再取反函數所形成之複合函數。

In this dissertation, 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 close-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-distribution 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. Afterwards 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, non-additive and non-Gaussian) limited access channel might be identified by composing the derivative of the mutual information function with its inverse.