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

## TL;DR

This paper analyzes the error performance of low-resolution quantized phase modulation systems, deriving optimal detectors, error bounds, and demonstrating that a certain quantization level achieves near-optimal reliability with energy efficiency benefits.

## Contribution

It provides the first comprehensive analysis of optimum detectors and error rates for low-resolution quantized phase modulated wireless systems, including new bounds and asymptotic results.

## Key findings

- Optimal detectors for M-PSK with low-resolution quantization are derived.
- A universal lower bound on symbol error probability is established for insufficient quantization bits.
- Quantization levels of n ≥ log2(M+1) are asymptotically optimal in terms of error decay.

## Abstract

This paper considers a wireless communication system with low-resolution quantizers, in which transmitted signals are corrupted by fading and additive noise. For such wireless systems, a universal lower bound on the average symbol error probability (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 optimum maximum likelihood detector for equi-probable signal points is derived. Utilizing the structure of the derived optimum receiver, a general average 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. Adopting this result for Nakagami-m fading channels, easy-to-evaluate expressions for the average SEP for M-PSK modulation are further derived. It is shown that a transceiver architecture with n-bit quantization is asymptotically optimum in terms of communication reliability if n is greater than or equal to log_2(M +1). That is, the decay exponent for the average SEP is the same and equal to m with infinite-bit and n-bit quantizers for n is greater than or equal to log_2(M+1). On the other hand, it is only equal to half and 0 for n = log_2(M) and n < log_2(M), respectively. An extensive simulation study is performed to illustrate the derived results and energy efficiency gains obtained by means of low-resolution quantizers.