On the Asymptotic Capacity of Fading Channels

We consider a peak-power limited single-antenna flat complexGaussian fading channel where the receiver and transmitter, while fully congnizant of the distribution of the fading process, have no knowledge of its realization. Upper and lower bounds on channel capacity are derived, with special emphasis on tightness in the high SNR regime. Necessary and sufficient conditions (in terms of the autocorrelation of the fading process) are derived for capacity to grow double-logarithmically in the Signal-to-Noise Ratio (SNR). For cases in which capacity increases logarithmically in the SNR, we provide an expression for the “pre-log”, i.e., for the asymptotic ratio between channel capacity and the logarithm of the SNR. This ratio is given by the Lebesgue measure of the set of harmonics where the spectral density of the fading process is zero. We finally demonstrate that the asymptotic dependence of channel capacity on the SNR need not be limited to logarithmic or double-logarithmic behaviors. We exhibit power spectra for which capacity grows as a fractional power of the logarithm of the SNR.