Asymptotic capacity of underspread and overspread stationary time- and frequency-selective channels

In this paper, we consider stationary time- and frequency-selective channels. No channel knowledge neither at the transmitter nor at the receiver is assumed to be available. We investigate the capacity behavior of these doubly selective channels as a function of the channel parameters delay spread, Doppler bandwidth and channel spread factor (the product of the delay spread and the Doppler bandwidth). We shed light on different capacity regimes at high values of signal to noise ratio (SNR) in which the dominant capacity term is either of order log(SNR) or log(log(SNR)), depending on the channel conditions (delay spread, Doppler Bandwidth and channel spread factor). For critically spread channels (channel spread factor of 1), it is widely believed that the dominant term of the high-SNR expansion of the capacity is of order log (log(SNR)) or in other words, that the pre-log (the coefficient of log(SNR)) is zero. We provide a very simple scheme that shows that even for critically spread channels a non-zero pre-log might exist under certain conditions. We also specify these conditions in terms of Doppler bandwidth and delay spread. We also show that a nonzero pre-log might exist even for over-spread channels (channel spread factor greater than 1). We specify the channel conditions which govern the range of existence of the log(SNR) regime. At higher channel spread factor, the log(SNR) term vanishes and a log(log(SNR)) term becomes the dominant capacity term. We specify the range of this log(log(SNR)) regime and also provide bounds for the coefficient of this log(log(SNR)) term (the pre-loglog).