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.


Non-central chi-square distribution, Rayleigh, Rice, expected
logarithm, expected reciprocal value.

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Senior Researcher & Lecturer, ETH Zurich, Switzerland
_|_     -- --|-    _     /  /   Adj. Professor, National Chiao Tung University (NCTU), Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser-isi.ethz.ch/

Last modified: Fri Apr 27 10:29:57 UTC+8 2007