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 Gaussian tools from random matrix theory. Based on an asymptotic analysis of the information density which considers the simultaneous growth of the block length n and the number of transmit and receive antennas K and N, respectively, we derive closed-form upper and lower bounds on the optimal average error probability when the code rate is within O(1/sqrt{ 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. A comparison to practical low-density paritycheck (LDPC) codes reveals a striking similarity between the empirical and theoretical slopes of the error-probability curve, seen as functions of n or the signal-to-noise ratio (SNR). This allows one to predict in practice by how much n or the SNR must be increased to realize a desired error probability improvement.