Capacity of Sparse Multipath Channels in the Ultra-Wideband Regime

This paper studies the ergodic capacity of time-and frequency-selective multipath fading channels in the ultrawide-band (UWB) regime when training signals are used for channel estimation at the receiver. Motivated by recent measurement results on UWB channels, we propose a model for sparse multipath channels. A key implication of sparsity is that the independent degrees of freedom in the channel scale sublinearly with the signal space dimension (product of signaling duration and bandwidth). Sparsity is captured by the number of resolvable paths in delay and Doppler. Our analysis is based on a training and communication scheme that employs signaling over orthogonal short-time Fourier (STF) basis functions. STF signaling naturally relates sparsity in delay-Doppler to coherence in time and frequency. We study the impact of multipath sparsity on two fundamental metrics of spectral efficiency in the wideband/low-SNR limit introduced by Verdu: first- and second-order optimality conditions. Recent results by Zheng et al. have underscored the large gap in spectral efficiency between coherent and noncoherent extremes and the importance of channel learning in bridging the gap. Building on these results, our results lead to the following implications of multipath sparsity: (1) the coherence requirements are shared in both time and frequency, thereby significantly relaxing the required scaling in coherence time with SNR; (2) sparse multipath channels are asymptotically coherent - for a given but large bandwidth, the channel can be learned perfectly and the coherence requirements for first- and second-order optimality met through sufficiently large signaling duration; and (3) the requirement of peaky signals in attaining capacity is eliminated or relaxed in sparse environments.