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

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(ldr) 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(ldr) is too complex for further analysis.