  In contrast to the prevalent assumption of rich multipath in information
theoretic analysis of wireless channels, physical channels exhibit sparse
multipath, especially at large bandwidths. We propose a model for sparse
multipath fading channels and present results on the impact of sparsity on
non-coherent capacity and reliability in the wideband regime. A key implication
of sparsity is that the statistically independent degrees of freedom in the
channel, that represent the delay-Doppler diversity afforded by multipath,
scale at a sub-linear rate with the signal space dimension (time-bandwidth
product). Our analysis is based on a training-based communication scheme that
uses short-time Fourier (STF) signaling waveforms. Sparsity in delay-Doppler
manifests itself as time-frequency coherence in the STF domain. From a capacity
perspective, sparse channels are asymptotically coherent: the gap between
coherent and non-coherent extremes vanishes in the limit of large signal space
dimension without the need for peaky signaling. From a reliability viewpoint,
there is a fundamental tradeoff between channel diversity and learnability that
can be optimized to maximize the error exponent at any rate by appropriately
choosing the signaling duration as a function of bandwidth.
