This paper studies the impact of channel coherence and the role of channel state information (CSI) on the probability of error in wideband multipath fading channels. Inspired by recent ultra wideband channel measurement campaigns, we propose a sparse channel model for time and frequency selective fading in which the number of resolvable channel paths grow sub-linearly with signal space dimensions. As a result, the degrees of freedom (DoF) in the channel scale sub-linearly. With perfect CSI, the DoF reflect the amount of channel diversity whereas with no CSI, the DoF represent the level of channel uncertainty. Based on this model, we investigate the reliability of wideband communication using error exponents. With perfect CSI at the receiver, it is seen that sparse channels are less reliable to communicate over than rich channels in which the paths scale linearly with signaling dimensions. When there is no receiver CSI, we analyze training-based communication schemes aimed at learning the sparse channel and our results reveal a fundamental tradeoff between channel learnability and diversity that affects error performance. We present numerical examples with realistic parameter sets and discuss the conditions under which the two effects can be balanced to obtain the best probability of error in any practical system.
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