On the Sum-Capacity-Achieving Distributions and Sum-Capacity of 1-Bit ADC MACs in Rayleigh Fading

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This work addresses the sum-capacity-achieving signaling schemes and the sum-capacity of a multiple access channel (MAC) with two mobile users communicating to a base station equipped with 1-bit quantizers. We consider Rayleigh fading channels between the users and the base station where channel state information (CSI) is known only at the base station. Towards this end, we first establish a necessary and sufficient condition refereed to as Kuhn-Tucker condition (KTC) on the input distribution of one user for a given input signal used at another user so that the input/output mutual information (MI) is maximized. By relaxing the power constraint and establishing upper bounds on the MI, we demonstrate that the power constraint of the first user is active. Then using Fubini-Tonelli theorem to exchange the order of integrations between fading and input distributions, and exploiting novel bounds on the output distribution and a related relative entropy, it is shown that the optimal input distribution has a bounded amplitude. Due to the symmetry of the problem, it is then concluded that the sum-capacity-achieving amplitude distributions are bounded, and both users must use full power to achieve the sum-capacity. Next, we exploit the independence of the amplitude and phase of fading gains to show that the optimal inputs are π/2 circular symmetric. Building upon the results on the amplitude and phase, it is then demonstrated that any π/2 circular symmetric input distribution having a constant amplitude is sum-capacity achieving. The sum-capacity is finally obtained in a precise form.