On the Accuracy of Phase Extraction from a Known-Frequency Noisy Sinusoidal Signal

TL;DR

This paper analyzes the accuracy of phase estimation from known-frequency sinusoidal signals under synchronous sampling, considering additive and phase noise, and provides a comprehensive model validated by simulations.

Contribution

It introduces a closed-form asymptotic PDF for phase estimators in known-frequency signals with noise, and characterizes RMSE behavior across different noise and sample conditions.

Findings

Estimator converges to CRLB at high SNR

RMSE saturates at low SNR and high noise levels

Three distinct RMSE regimes identified based on SNR and sample size

Abstract

Accurate phase extraction from sinusoidal signals is a crucial task in various signal processing applications. While prior research predominantly addresses the case of asynchronous sampling with unknown signal frequency, this study focuses on the more specific situation where synchronous sampling is possible, and the signal's frequency is known. In this framework, a comprehensive analysis of phase estimation accuracy in the presence of both additive and phase noises is presented. A closed-form expression for the asymptotic Probability Density Function (PDF) of the resulting phase estimator is validated by simulations depicting Root Mean Square Error (RMSE) trends in different noise scenarios. This estimator is asymptotically efficient, converging rapidly to its Cram\'er-Rao Lower Bound (CRLB). Three distinct RMSE behaviours were identified based on SNR, sample count (N), and noise level: (i) saturation towards a random guess at low Signal to Noise Ratio (SNR), (ii) linear decrease with the square roots of N and SNR at moderate noise levels, and (iii) saturation at high SNR towards a noise floor dependent on the phase noise level. By quantifying the impact of sample count, additive noise, and phase noise on phase estimation accuracy, this work provides valuable insights for designing systems requiring precise phase extraction, such as phase-based fluorescence assays or system identification.