  This paper considers the problem of reliable communication over discrete-time
channels whose impulse responses have length $L$ and exactly $S\leq L$ non-zero
coefficients, and whose support and coefficients remain fixed over blocks of
$N>L$ channel uses but change independently from block to block. Here, it is
assumed that the channel's support and coefficient realizations are both
unknown, although their statistics are known. Assuming Gaussian
non-zero-coefficients and noise, and focusing on the high-SNR regime, it is
first shown that the ergodic noncoherent channel capacity has pre-log factor
$1-\frac{S}{N}$ for any $L$. It is then shown that, to communicate with
arbitrarily small error probability at rates in accordance with the capacity
pre-log factor, it suffices to use pilot-aided orthogonal frequency-division
multiplexing (OFDM) with $S$ pilots per fading block, in conjunction with an
appropriate noncoherent decoder. Since the achievability result is proven using
a noncoherent decoder whose complexity grows exponentially in the number of
fading blocks $K$, a simpler decoder, based on $S+1$ pilots, is also proposed.
Its $\epsilon$-achievable rate is shown to have pre-log factor equal to
$1-\frac{S+1}{N}$ with the previously considered channel, while its achievable
rate is shown to have pre-log factor $1-\frac{S+1}{N}$ when the support of the
block-fading channel remains fixed over time.
