A Note on Functional Averages over Gaussian Ensembles

Random matrix theory was introduced to the theoretical physics community byWigner in his work on nuclear physics in the 1950s [1, 2]. Since that time, the subject is an important and active research area in mathematics, and it finds applications in fields as diverse as the Riemann conjecture, physics, chaotic systems, multivariate statistics, wireless communications, signal processing, compressed sensing, and information theory. In the last decades, a considerable amount of work has emerged in the communications and information theory on the fundamental limits of communication channels that make use of results in random matrix theory [3–5]. For this reason, computing averages over certain matrix ensembles becomes extremely important in many situations. To be more specific, consider the well-known case of the single user MIMO channel with multiple transmitting and receiving antennas. Denoting the number of transmitting antennas by t and the number of receiving antennas by r, the channel model is