On Communication over Unknown Sparse Frequency-Selective Block-Fading Channels

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.