Bounds on the second-order coding rate of the MIMO Rayleigh block-fading channel

We study the second-order coding rate of the multiple-input multiple-output (MIMO) Rayleigh block-fading channel via statistical bounds from information spectrum methods and random matrix theory. Based on an asymptotic analysis of the mutual information density which considers the simultaneous growth of the block length n and the number of transmit and receive antennas K and N, we derive closed-form upper and lower bounds on the optimal average error probability when the code rate is within O(1/√nK) of the asymptotic capacity. A Gaussian approximation is then used to establish an upper bound on the error probability for arbitrary code rates which is shown by simulations to be accurate for small N, K, and n.