It was widely believed that knowledge of channel state information (CSI) at the receiver imposes a sharp cut-off on the achievability of coherent capacity at large band-widths (or low SNRs). Recent works have shown that by either employing an explicit training-based scheme or an implicit channel-learning and communication scheme, rates intermediate between the coherent and the non-coherent extremes can be achieved. However, to bridge the gap between these two extremes, these works assume that the coherence time of the channel increases as the signaling bandwidth increases, without providing any physical basis that could lead to such a scaling relationship. In this paper, we study the wideband capacity of doubly dispersive underspread wireless channels employing explicit training and communication using short-time Fourier (STF) basis functions, that serve as approximate eigen-functions for such channels. Requirements on coherence time in existing works are naturally replaced with requirements on the time-frequency coherence dimension in STF signaling. Motivated by recent measurement campaigns, we propose a sparse multipath channel model in which the coherence dimension naturally scales with signal-space dimensions. Sparsity in the delay-Doppler domain affords two important benefits that have not be.en recognized thus far: 1) The coherence time requirement necessary to achieve an operational coherence level is dramatically reduced by exploiting sparsity in the delay domain, and 2) Sparsity in the Doppler domain can be used to achieve any operational level of coherence by appropriately scaling the signaling duration as a function of signaling bandwidth.
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