Expectations of a Noncentral Chi-Square Distribution With Application to IID MIMO Gaussian Fading

Abstract

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) (.) and that these families have nice properties (monotonicity, convexity, etc.). Moreover, some tight upper and lower bounds axe derived that are helpful in situations where the closed-form expression of g(m) (.) 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 axe derived for the fading number that describes the asymptotic channel capacity at high signal-to-noise ratios (SNR).