The Rayleigh product channel model is useful in capturing the performance degradation due to rank deficiency of MIMO channels. In this paper, such a performance degradation is investigated via the distribution of mutual information assuming the block fading channels and the uniform power transmission scheme. Using techniques of free probability theory, the asymptotic variance of mutual information is derived when the dimensions of the channel matrices approach infinity. In this asymptotic regime, the mutual information is rigorously proven to be Gaussian distributed. Using the obtained results, a fundamental tradeoff between multiplexing gain and diversity gain of Rayleigh product channels under the uniform power transmission can be characterized by the closed-form expression at any finite signal-to-noise ratio. Numerical results are provided to compare the outage performance between the Rayleigh product channels and the conventional Rayleigh MIMO channels.

The Diversity-Multiplexing Tradeoff (DMT) is studied for the large Rayleigh product channel at non-asymptotic SNRs. The first result is that, as matrix dimensions growing to infinity, the channel capacity converges to a Gaussian random variable. Based on this, we derive a compact expression for the finite-SNR DMT. From the analytical and numerical results, we gain useful insight into the fundamental tradeoff of the considered channel model in the realistic SNR regime.

In a frequency-selective slow-fading channel in a multiple-input multiple-output (MIMO) system, the channel matrix is of the form of a block matrix. A method is proposed to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. Asymptotic eigenvalue distribution of$HH^*$is also calculated, where the entries of$H$are jointly Gaussian, with a correlation of the form$E[h_{pj}\bar h_{qk}]= \sum_{s=1}^t \Psi^{(s)}_{jk}\mathhat\Psi^{(s)}_{pq}$(where$t$is fixed and does not increase with the size of the matrix). An operator-valued free probability approach is used to achieve this goal. Using this method, a system of equations is derived, which can be solved numerically to compute the desired eigenvalue distribution.