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

We propose an exact calculation of the probability density function (PDF) and cumulative distribution function (CDF) of mutual information (MI) for a two-user multiple-input multiple-output (MIMO) multiple access channel (MAC) network over block Rayleigh fading channels. This scenario can be found in the uplink channel of MIMO non-orthogonal multiple access (NOMA) system, a promising multiple access technique for 5G networks. 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 investigate the effects of adding antennas in the receiver and its ability to decode the weak user signal. By supposing no channel knowledge at transmitters and successive decoding at receiver, the capacity of the weak user increases and its outage probability decreases with the increment of extra antennas at the receiver end.