## 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. ## Lemma
Let the random variable V have a noncentral chi-square distribution
with degrees of
freedom,
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 . ## ProofThis lemma and a proof for it can be found in (Appendix X, Lemma 10.1)
and in (Appendix A, Lemma A.6)
## Remarks- The probability density function of a noncentral chi-square
distributed random variable with degrees of freedom is
given by
- From this lemma it immediately follows that for
with arbitrary complex constants and a non-zero complex
constant
- Note that a noncentrality parameter
leads to a central chi-square distribution (with an even number of
degrees of freedom) for which the
expected logarithm has been known to be
See (4.352-1.) in **I.S. Gradshteyn, I.M. Ryzhik:***Table of Integrals, Series and Products*, 6th edition, Academic Press, 2000. ISBN: 0-12-294757-6.The most common situation for this special case is where consists only of the squared magnitude of one complex Gaussian random variable (a squared Rayleigh distribution or exponential distribution). In this case the expected logarithm is known to be . - Note that for the random variable is said to
have a squared Rician distribution. In this case the lemma
proves the expected logarithm to be
- Note that can be bounded as follows:
For a proof see Appendix B in **Stefan M. Moser:***The Fading Number of Memoryless Multiple-Input Multiple-Output Fading Channels*, IEEE Transactions on Information Theory, vol. 53, no. 7, pp. 2652–2666, July 2007. (Abstract)*[.pdf]*
## KeywordsChi-square, chi-squared, noncentral chi-square, noncentral chi-squared, expected logarithm, Rayleigh, Rice, Ricean, Rician. ## GeneralizationsThe following lemma has been proven in (Appendix A, Lemma 3)
Lemma: Consider an random matrix
with , where is
deterministic while the entries of are zero-mean
unit-variance IID complex Gaussian. Denoting by the eigenvalues of we have
where is an matrix with entries
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