On fixed input distributions for noncoherent communication over high-SNR Rayleigh-fading channels

It is well known that independent and identically distributed Gaussian inputs, scaled appropriately based on the signal-to-noise ratio (SNR), achieve capacity on the additive white Gaussian noise (AWGN) channel at all values of SNR. In this correspondence, we consider the question of whether such good input distributions exist for frequency-nonselective Rayleigh-fading channels, assuming that neither the transmitter nor the receiver has a priori knowledge of the fading coefficients. In this noncoherent regime, for a Gauss-Markov model of the fading channel, we obtain explicit mutual information bounds for the Gaussian input distribution. The fact that Gaussian input generates bounded mutual information motivates the search for better choices of fixed input distributions for high-rate transmission over rapidly varying channels. Necessary and sufficient conditions are derived for characterizing such distributions for the worst case scenario of memoryless fading, using the criterion that the mutual information is unbounded as the SNR gets large. Examples of both discrete and continuous distributions that satisfy these conditions are given. A family of fixed input distributions with mutual information growth rate of O((log log SNR)(1-u)), u > 0 are constructed. It is also proved that there does not exist a single fixed-input distribution that achieves the optimal mutual information growth rate of log log SNR.