On the spectral efficiency of noncoherent doubly selective block-fading channels

In this paper, we consider noncoherent single-antenna communication over doubly selective block-fading channels with discrete block-fading interval N. In our noncoherent setup, neither the transmitter nor the receiver know the channel fading coefficients, though both know the channel statistics. In particular, we consider discrete-time channels whose impulse-response trajectories obey a complex-exponential basis expansion model with uncorrelated coefficients, and we show that such a model holds in the limit N → ∞ for pulse-shaped transmission/reception over certain wide-sense stationary uncorrelated scattering channels. First, we show that, when the inputs are chosen from continuous distributions, the channel's multiplexing gain (i.e., capacity pre-log factor) equals max(0, 1 - N_delay N_Dopp/N), for discrete delay spread N_delay and discrete Doppler spread N_Dopp. Next, for the case of strictly doubly selective fading (i.e., N_Dopp > 1 and N_delay > 1), we establish that, for cyclic-prefixed affine pilot-aided transmission (PAT) schemes designed to minimize the mean-squared error (MSE) attained by pilot-aided minimum-MSE channel estimation, the pre-log factor of the achievable rate is less than the channel's multiplexing gain. We then provide guidelines for the design of PAT schemes whose achievable-rate pre-log factor equals the channel's multiplexing gain and construct an example.