Exact Capacity-Achieving Inputs of the Noncoherent Block-Rayleigh Channel at Every Coherence Length

Abstract

In this paper, we study the capacity-achieving input of the single-antenna Rayleigh block-fading channel with coherence length T, an average power constraint and no channel state information at either end. Constant-norm signaling, in which every fading block carries the same energy and only its direction is modulated, is the standard design for noncoherent communication. However, to the best of our knowledge, its optimality has been established only asymptotically. We prove that, for every T ≥ 3, there is exactly one threshold signal-to-noise ratio (SNR), such that constant-norm signaling with an isotropic direction is capacity-achieving if and only if the SNR is at or above the threshold, and the capacity is then an explicit one-dimensional integral. Below the threshold, the capacity-achieving input is on-off, i.e., the transmitter is silent with a probability fixed by one scalar equation and otherwise sends a constant-norm block at a higher power. Hence, for every T ≥ 3, the optimal norm law has at most two mass points. Moreover, for every T ≥ 1, the capacity-achieving input is unique up to a per-block phase. The threshold and the on-off law are proved analytically, by counting sign changes of Laplace transforms. Finally, we certify the threshold in brackets narrower than 0.001 dB for every T = 3, …, 200. Over this range, it decreases strictly from about 12.43 dB at T = 3 to about −7.96 dB at T = 200.