Digital Communications

Appendix E: The Wishart Distribution

The Wishart distribution may be considered as a matrix generalization of the chi‐squared distribution, or, in the case of non‐integer degrees of freedom, of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defined over symmetric, positive‐definite matrix‐valued random variables.

E.1 Introduction to Wishart Distribution

Chi‐square distribution is widely encountered in telecommunication applications where the powers/SNRs of multiple Gaussian signals are summed for receive diversity with MRC or transmit diversity with MRT (see Appendix F and Chapter 12). If denote real‐valued independent Gaussian random variables with identical variances but different means , then has a chi‐square distribution with n degrees of freedom. If , the chi‐square distribution is said to be central, otherwise non‐central. The corresponding pdf’s are given by (F.111) and (F.120), respectively.

Now consider n random vectors each with q components. Each component a_ij