Asymptotic capacity of underspread and overspread doubly selective MIMO channels

In this paper, we consider stationary time- and frequency-selective MIMO 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 one of the system parameters, the number of transmit antennas and channel parameters as delay spread, Doppler bandwidth and channel spread factor (the product of the previous two parameters). For critically spread channels (channel spread factor of 1), it is widely believed that the dominant term of high-SNR expansion of the capacity is log(log(SNR)) or in other words, the pre-log (the coefficient of log(SNR)) is zero. We provide a very simple scheme showing that for critically spread and mildly overspread channels a non-zero pre-log exists under certain conditions. We specify these conditions in terms of the Doppler bandwidth and the delay spread. We reason that for nearly critically spread channels, MIMO systems exhibit same degrees of freedom as that of a SISO system. At higher channel spread factor (overspread case), the log(SNR) term vanishes and log(log(SNR)) term becomes the dominant capacity term. We specify the range of existence for log(SNR) regime.