Optimal Memoryless Relays With Noncoherent Modulation

We derive optimal memoryless relays for the case of noncoherent modulation over additive white Gaussian noise (AWGN) channels with or without fading. The derivation is flexible, as it can be applied to any binary hypothesis observed at the relay. We investigate several channels, including random phase and fading, and apply different modulation schemes, namely on-off-keying (OOK) and orthogonal frequency-shift-keying (FSK). We find that at low signal-to-noise ratios (SNR) the relay censors its observation, as it only transmits at nonzero energy if the observations seem reliable. Compared to the known results that optimal memoryless relays for the case of coherent binary-phase-shift-keying (BPSK) are combinations of soft-information and a hard-limiter, the noncoherent relays have considerably less emphasis on soft-information and converge much faster to the hard-limiter.