This paper studies reliability of communication over wideband
multipath channels in the finite energy regime. Motivated by
measurement results and physical arguments, our analysis focuses on
sparse multipath scenarios, where the degrees of freedom (DoF) in
the channel scale sub-linearly with the signaling dimension
(time-bandwidth product). This is in contrast to the implicit
assumption of rich multipath in most existing works, where the DoF
scale linearly. We consider signaling using orthogonal short-time
Fourier (STF) waveforms that serve as approximate eigenfunctions for
underspread channels and relate multipath sparsity in the
delay-Doppler domain to channel coherence in time and frequency. Our
focus is on the non-coherent scenario, where we employ a
training-based STF communication scheme and investigate its
reliability under a finite energy constraint using random coding
error exponents. Our analysis reveals a fundamental tradeoff between
channel learnability and diversity: optimizing the tradeoff at any
given packet payload and energy yields the largest error exponent
and the smallest probability of error. For channels that are
asymmetrically sparse in delay and Doppler, we show that the minimum
error probability is achievable at all code lengths (signaling
dimension) by appropriately adapting the STF packet configuration
(signaling duration and bandwidth) to the level of sparsity in delay
and Doppler. For symmetrically sparse channels, we show the
existence of an optimal code length at which the minimum error
probability is achieved, regardless of the packet configuration.
Numerical results are provided to illustrate the implications of the
results.