Expectations of a noncentral chi-square distribution with application to IID MIMO Gaussian fading

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 (inverse moments) of a noncentral chi-square random variable of even degree of freedom. It is shown that these expectations can be expressed by a family of continuous functions g_m(middot) 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(middot) is too complex for further analysis. As an example of the applicability of these results, in the second part of this paper an independent and identically distributed (IID) Gaussian multiple-input-multiple-output (MIMO) fading channel with a scalar line-of-sight component is analyzed. Some new expressions are derived for the fading number that describes the asymptotic channel capacity at high signal-to-noise ratios (SNR).