Approximate distribution of log 2 ( A + χ 2 ) and its applications

This study concerns the approximate distribution of the random variable log 2 ( A + χ 2 ) and its applications in the performance analysis of communication systems where A is 0 or 1 and χ 2 a chi‐square distributed random variable. The authors prove that log 2 ( A + χ 2 ) is approximately Gaussian distributed and derive the expressions of the mean and variance of the approximate distribution. The approximate Gaussian distribution provides a new way to simplify the derivation of performance metrics and obtain analytical results. Then the authors utilise the approximate results in two applications, which are the approximation of the Gaussian Q ‐function and capacity analysis, respectively. For one thing, the approximate distribution can be explored to deduce a new approximate equivalent expression of the Q ‐function on the basis of which an accurate approximation of the Gaussian Q ‐function can be developed. For another, the approximate Gaussian distribution provides an intuitive approach to analyse the channel capacity of Rayleigh‐fading multiple antenna systems. The approximate statistical distributions of the channel capacities of orthogonal space‐time block codes, single‐input multiple‐output, multiple‐input single‐output, transmit antenna selection with maximal‐ratio combining and multiple‐input multiple‐output systems are presented. The applicability of the approximate distribution can be demonstrated through simulation results in both applications.