On the Capacity-Achieving Input of Channels with Phase Quantization

TL;DR

This paper characterizes the capacity-achieving input distributions for channels with multi-bit phase quantization, showing that rotated $2^b$-PSK signals are optimal across various fading scenarios and providing power control strategies.

Contribution

It extends the understanding of optimal inputs to channels with multi-bit phase quantization, proving rotated $2^b$-PSK is optimal and deriving conditions for ergodic capacity with CSI.

Findings

Rotated $2^b$-PSK is capacity-achieving for complex Gaussian channels with $b$-bit phase quantization.

Optimality of rotated $2^b$-PSK holds under noncoherent fast fading Rician channels with LoS.

Power control schemes are proposed for ergodic capacity when CSI is available at both transmitter and receiver.

Abstract

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. However, an exact characterization of the capacity-achieving input distribution for channels with multi-bit phase quantization has not been provided. In this paper, we consider four different channel models with multi-bit phase quantization at the output and identify the optimal input distribution for each channel model. 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 $\frac{2\pi}{2^b}$-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.