The Expected Logarithm of a Noncentral Chi-Square Random Variable

In 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 2m degrees of freedom, i.e.,

v=sum of gauss magnitude squared

where u_1^m are IID circularly-symmetric zero-mean unit-variance complex Gaussians complex normal and mu_1^m are complex constants. Then

expected log of v

where s^2 denotes the noncentrality parameter

noncentrality parameter s^2

and where the function g_m is defined as

definition of g_m

for m positive integer. Here, the function ei denotes the exponential integral function defined as

definition of ei

and psi(m) is Euler's psi function given by

euler's psi function

where gamma approximately 0.577 denotes Euler's constant. Note that the functions g_m are continuous, monotonically increasing, and concave in the interval from 0 to infinity for all m positive integer. In particular note that g_m are continuous at zero for all m positive integer.

Proof

This lemma and a proof for it can be found in (Appendix X, Lemma 10.1)

Amos Lapidoth, Stefan M. Moser: Capacity Bounds via Duality with Applications to Multiple-Antenna Systems on Flat Fading Channels, IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2426-2467, October 2003. (Abstract) [Download]

and in (Appendix A, Lemma A.6)

Stefan M. Moser: Duality-Based Bounds on Channel Capacity, Ph.D. thesis, Swiss Federal Institute of Technology (ETH), Switzerland, October 2004. Under the supervision of Prof. Dr. Amos Lapidoth. Diss. ETH No. 15769. Hartung-Gorre Verlag Konstanz, January 2005, ISBN: 3-89649-956-4. (Abstract, Download)

Remarks

  • The probability density function of a noncentral chi-square distributed random variable with n degrees of freedom is given by

    prob. density of 
  noncentral chi-square

    where i_nu(.) denotes the modified Bessel function of the first kind of order nu in reals. So basically the lemma gives an analytic expression for the following integral:

    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 1/2.
  • From this lemma it immediately follows that for u_j^m iid complex gaussian with arbitrary complex constants mu_1^m and a non-zero complex constant lambda

    exp. log of
  noncentral chi-square

  • Note that a noncentrality parameter s^2=0 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

    g_m(0)=psi(m)

    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 m=1 where v 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 g_1(0)=psi(1)=-gamma.

  • Note that for m=1 the random variable v is said to have a squared Rician distribution. In this case the lemma proves the expected logarithm to be

    g_1(s^2)=log(s^2)-ei(-s^2)

  • Note that g_m can be bounded as follows:

    log(xi)-ei(-xi) le g_m(xi) le log(m+xi)

    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 m times n random matrix h=hbar+w with m less or equal n, where hbar is deterministic while the entries of w are zero-mean unit-variance IID complex Gaussian. Denoting by phi_j (j=1,...m) the eigenvalues of hbar hermi(hbar) we have

e[log det hbar hermi(hbar)]=...

where xi_i (i=1,...,m) is an m times m matrix with entries

(xi_i)_{k,l}=...


-||-   _|_ _|_     /    __|__   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: Wed Sep 9 23:21:00 UTC+8 2015