On the High SNR Capacity of Stationary Gaussian Fading Channels

We present results on the high signal-to-noise ratio (SNR) asymptotic capacity of peak-power limited single-antenna discrete-time stationary complex-Gaussian fading channels with memory, where the transmitter and receiver, while fully cognizant of the fading law, have no access to its realization. Complementing recent results of Lapidoth & Moser about the case where the fading process is regular, we consider here the non-regular case, i.e., the case where the entropy-rate of the fading process is negative infinity. It is demonstrated that while in the former case capacity grows double logarithmically in the SNR with a fading number that is determined by the prediction error in predicting the present fading from noiseless observations of its past, here the asymptotics require a careful analysis of the noisy prediction error, i.e., the asymptotic functional dependence of the prediction error in predicting the present fading from noisy observations of its past. This functional dependence, which can be made explicit in terms of the spectrum of the fading process, may lead to dramatically different asymptotic dependencies of capacity on SNR, e.g., double-logarithmic, logarithmic, or fractional powers of the logarithm of the SNR. The “pre-log”, i.e., the asymptotic ratio of channel capacity to the logarithm of the SNR takes on a particularly simple form: it is the Lebesgue measure of the set of harmonics where the spectral density is zero. The pre-log is unrelated to the “bandwidth” of the process or its “coherence time”. In the light of these results we re-examine some of the models in the literature on fading channels and the asymptotic behaviors associated with them. It is found that what may appear as slight changes in the channel model may lead to dramatically different high-SNR asymptotics.