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The Expected Logarithm of a Noncentral Chi-Square Random Variable

Lemma

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 .

Proof

This 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

  

  where denotes the modified Bessel function of the first kind of order . So basically the lemma gives an analytic expression for the following integral:

  

  Note that the real and imaginary parts of a circularly-symmetric zero-mean complex Gaussian random variable are independent zero-mean real Gaussian random variables of variance .
• 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, to appear in IEEE Transactions on Information Theory, vol. 53, no. 7, July 2007. (Abstract) [Download]

Keywords

Chi-square, chi-squared, noncentral chi-square, noncentral chi-squared, expected logarithm, Rayleigh, Rice, Ricean, Rician.

Generalizations

The following lemma has been proven in (Appendix A, Lemma 3)

	Angel Lozano, Antonia M. Tulino, Sergio Verdú: High-SNR Power Offset in Multiantenna Communication, IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4134-4151, December 2005.

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

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Associate Professor at National Chiao
_|_     -- --|-    _     /  /   Tung University (NCTU), Hsinchu, Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser.cm.nctu.edu.tw/