Carrier phase and amplitude estimation for phase shift keying using pilots and data

TL;DR

This paper analyzes least squares estimators for carrier phase and amplitude in M-ary PSK signals with pilots and data, showing phase estimator consistency and asymptotic normality, while amplitude estimation converges to a biased value.

Contribution

It provides a theoretical analysis of the asymptotic properties of least squares estimators for phase and amplitude in PSK signals, including cases without pilots.

Findings

Phase estimator is strongly consistent and asymptotically normal.

Amplitude estimator converges to a biased value depending on noise and constellation.

Monte Carlo simulations confirm theoretical results.

Abstract

We consider least squares estimators of carrier phase and amplitude from a noisy communications signal that contains both pilot signals, known to the receiver, and data signals, unknown to the receiver. We focus on signaling constellations that have symbols evenly distributed on the complex unit circle, i.e., M-ary phase shift keying. We show, under reasonably mild conditions on the distribution of the noise, that the least squares estimator of carrier phase is strongly consistent and asymptotically normally distributed. However, the amplitude estimator is not consistent, but converges to a positive real number that is a function of the true carrier amplitude, the noise distribution and the size of the constellation. Our theoretical results can also be applied to the case where no pilot symbols exist, i.e., noncoherent detection. The results of Monte Carlo simulations are provided and these agree with the theoretical results.