Low-Resolution Quantization in Phase Modulated Systems: Optimum Detectors and Error Rate Analysis

Abstract

This paper studies optimum detectors and error rate analysis for wireless systems with low-resolution quantizers in the presence of fading and noise. A universal lower bound on the average symbol error probability ( $\mathsf {SEP}$ ), correct for all $M$ -ary modulation schemes, is obtained when the number of quantization bits is not enough to resolve $M$ signal points. In the special case of $M$ -ary phase shift keying ( $M$ -PSK), the maximum likelihood detector is derived. Utilizing the structure of the derived detector, a general average $\mathsf {SEP}$ expression for $M$ -PSK modulation with $n$ -bit quantization is obtained when the wireless channel is subject to fading with a circularly-symmetric distribution. For the Nakagami- $m$ fading, it is shown that a transceiver architecture with $n$ -bit quantization is asymptotically optimum in terms of communication reliability if $n \geq \log _{2}M +1$ . That is, the decay exponent for the average $\mathsf {SEP}$ is the same and equal to $m$ with infinite-bit and $n$ -bit quantizers for $n\geq \log _{2}M+1$ . On the other hand, it is only equal to $\frac {1}2$ and 0 for $n = \log _{2}M$ and $n , respectively. An extensive simulation study is performed to illustrate the accuracy of the derived results, energy efficiency gains obtained by means of low-resolution quantizers, performance comparison of phase modulated systems with independent in-phase and quadrature channel quantization and robustness of the derived results under channel estimation errors.