Cutoff rate and signal design for the quasi-static Rayleigh-fading space-Time channel

We consider the computational cut-o rate and its implications on signal design for the complex quasistatic Rayleigh at fading spatio-temporal channel under a peak power constraint where neither transmitter nor receiver know the channel matrix. The cut-o rate has an integral representation which is an increasing function of the distance between pairs of complex signal matrices. When the analysis is restricted to nite dimensional sets of signals interesting characterizations of the optimal rate-achieving signal constellation can be obtained. For arbitrary nite dimension, the rate-optimal constellation must admit an equalizer distribution, i.e., a positive set of signal probabilities which equalizes the average distance between signal matrices in the constellation. When the number N of receive antennas is large the distance-optimal constellation is nearly rate-optimal. When the number of matrices in the constellation is less than the ratio of the number of time samples to the number of transmit antennas, the rate-optimal cut-o rate attaining constellation is a set of equiprobable mutually-orthogonal unitary matrices. When the SNR is below a speci ed threshold the matrices in the constellation are rank one and the cut-o rate is achieved by applying all transmit power to a single antenna and using orthogonal signaling. Finally, we derive recursive necessary conditions and suÆcient conditions for a constellation to lie in the feasible set.