Some expectations of a non-central chi-square distribution with an even number of degrees of freedom

Abstract

The non-central chi-square distribution plays an important role in communications, for example in the analysis of mobile and wireless communication systems. It not only includes the important cases of a squared Rayleigh distribution and a squared Rice distribution, but also the generalizations to a sum of independent squared Gaussian random variables of identical variance with or without mean, i.e., a "squared MIMO Rayleigh" and "squared MIMO Rice" distribution. In this paper closed-form expressions are derived for the expectation of the logarithm and for the expectation of the n-th power of the reciprocal value of a non-central chi-square random variable. It is shown that these expectations can be expressed by a family of continuous functions g(m)(.) and that these families have nice properties (monotonicity, convexity, etc.). Moreover, some tight upper and lower bounds are derived that are helpful in situations where the closed-form expression of g(m)(.) is too complex for further analysis.