Optimum Performance of Short Block Length Codes Under Multivariate Stationary Rayleigh Fading

The performance of short block length codes in the presence of multi-variate stationary Rayleigh fading coherent channels is studied. The channel model is inspired by multi-symbol OFDM transmissions for ultra-reliable low latency communication (URLLC) services, according to which information is sent in a small number of OFDM symbols to reduce latency. More specifically, the dispersion of the coherent fading channel is generalized from the scalar case, which is well known in the literature, to the multivariate one. The obtained expressions are then particularized to the Rayleigh fading statistics and expressed in closed form as a series expansion. Finally, a high SNR approximation of the ergodic capacity and channel dispersion is derived for this particular fading choice. Results show that the code performance depends on the time-frequency fading correlation through very simple functions of the channel correlation coefficients. In particular, it is shown that the channel dispersion converges to a constant plus the sum of the dilogarithms of the correlation coefficients of the channel process. These formulas provide a very useful tool to design physical layer system parameters from channel correlation information measured at the receiver.