Optimal Signaling Scheme and Capacity of Non-Coherent Rician Fading Channels with 1-Bit Output Quantization

Low resolution analog-to-digital converter (ADC) has been considered as a promising solution to save power and cost in communication systems using high bandwidth and/or multiple RF chains. The goal of this work is to address the design of optimal signaling schemes and establish the capacity limit of Rician fading channels with 1- bit output quantization. This fading channel can be used to accurately model a wide range of wireless channels with LOS components, including emerging mmWave communications. The focus is on non-coherent fast fading channels where neither the transmitter nor the receiver knows the channel state information (CSI). By first examining the continuity of the input-output mutual information, the existence of the optimal input signal is validated. Moreover, the optimal input is shown to be $\pi/2$ circularly symmetric. A necessary and sufficient condition for an input signal to be optimal, which is referred to as the Kuhn-Tucker condition (KTC), and Lagrangian optimization problem are then established. By exploiting novel log-quadratic bounds on the Gaussian $Q$-function, it is then demonstrated that for a given mass point's amplitude, the corresponding rotated mass points through the phase of LOS component must form a square grid centered at zero. Furthermore, by establishing an upper bound on the KTC coefficient, we show that there are exact four mass points in the optimal distribution. As a result, the capacity-achieving input is a rotated QPSK constellation, and the rotation angle is the phase of the LOS coefficient. By using this input, the channel capacity is finally established in closed-form.