On the Capacity-Achieving Input of Channels with Phase Quantization
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Several information-theoretic studies on channels with output quantization have identified the capacity-achieving input distributions for different fading channels with 1-bit in-phase and quadrature (I/Q) output quantization. But can analytical results on the capacity-achieving input also be obtained for multi-bit quantization? We answer the question in the affirmative by considering multi-bit phase quantization. We first consider a complex Gaussian channel with b-bit phase-quantized output and prove that the capacity-achieving distribution is a rotated 2^b-phase shift keying (PSK). The analysis is then extended to multiple fading scenarios. We show that the optimality of rotated 2^b-PSK continues to hold under noncoherent fast fading Rician channels with b-bit phase quantization when line-of-sight (LoS) is present. When channel state information (CSI) is available at the receiver, we identify 2π/2^b-symmetry and constant amplitude as the necessary and sufficient conditions for the ergodic capacity-achieving input distribution; which a 2^b-PSK satisfies. Finally, an optimum power control scheme is presented which achieves ergodic capacity when CSI is also available at the transmitter.
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