The Expected Logarithm of a Noncentral Chi-Square Random VariableIn the following lemma a closed form expression is given for the expected value of the logarithm of a noncentral chi-square random variable with an even number of degrees of freedom.
Note that all logarithm on this page are natural logarithms.
Let the random variable V have a noncentral chi-square distribution with degrees of freedom, i.e.,
where are IID circularly-symmetric zero-mean unit-variance complex Gaussians and are complex constants. Then
where denotes the noncentrality parameter
and where the function is defined as
for . Here, the function denotes the exponential integral function defined as
and is Euler's psi function given by
where denotes Euler's constant. Note that the functions are continuous, monotonically increasing, and concave in the interval for all . In particular note that are continuous at zero for all .
This lemma and a proof for it can be found in (Appendix X, Lemma 10.1)
and in (Appendix A, Lemma A.6)
Chi-square, chi-squared, noncentral chi-square, noncentral chi-squared, expected logarithm, Rayleigh, Rice, Ricean, Rician.
The following lemma has been proven in (Appendix A, Lemma 3)
where is an matrix with entries
-||- _|_ _|_ / __|__ Stefan M. Moser
Last modified: Wed Sep 9 23:21:00 UTC+8 2015