On the Exact Distribution of Mutual Information of Two-user MIMO MAC Based on Quotient Distribution of Wishart Matrices

We propose the exact calculation of the probability density function (PDF) and cumulative distribution function (CDF) of mutual information (MI) for a two-user MIMO MAC network over block Rayleigh fading channels. So far the PDF and CDF have been numerically evaluated since MI depends on the quotient of two Wishart matrices, and no closed-form for this quotient was available. We derive exact results for the PDF and CDF of extreme (the smallest/the largest) eigenvalues. Based on the results of quotient ensemble the exact calculation for PDF and CDF of mutual information is presented via Laplace transform approach and by direct integration of joint PDF of quotient ensemble’s eigenvalues. Furthermore, our derivations also provide the parameters to apply the Gaussian approximation method, which is comparatively easier to implement. We show that approximation matches the exact results remarkably well for outage probability, i.e. CDF, above 10%. However, the approximation could also be used for 1% outage probability with a relatively small error. We apply the derived expressions to analyze the effects of adding receiving antennas on the receiver’s performance. By supposing no channel knowledge at transmitters and successive decoding at receiver, the capacity of the first user increases and outage probability decreases with extra antennas, as expected. Index Terms Multiple access channel, multiple-input multiple-output, mutual information, outage probability, Rayleigh fading, Wishart matrices, quotient ensemble, extreme eigenvalues, gap probabilities. G. Pivaro is at Rice University (USA), as part of his PhD in a joint agreement between Unicamp (Brazil) and Rice University (USA) (email: [email protected]). Santosh Kumar is with the Department of Physics, Shiv Nadar University, India (email: [email protected]; [email protected]). G. Fraidenraich is with the Department of Communications, State University of Campinas (Unicamp), 13083-852, Campinas, SP, Brazil (email:[email protected]). G. Pivaro and G. Fraidenraich were supported in part by Capes and CNPq, Brazil.