Parameter Estimation of a Single Chirp in the Presence of Wiener Phase Noise With Unknown Variance

This study addresses the problem of estimating the parameters of a single chirp signal affected by Wiener phase noise with an unknown variance and observed in an additive white Gaussian noise (AWGN) environment. We derive the time-domain joint maximum likelihood (ML) estimators for the three phase coefficients (initial phase, initial frequency, frequency rate), along with the maximum a posteriori probability (MAP) estimator for the phase noise. As a critical input for the joint ML and MAP estimators, we further derive the ML estimation of the unknown phase noise variance. All derived estimators are closed-form expressions based on the phases and magnitudes of the received signal samples. To establish benchmarks for comparison in scenarios with phase noise, we concurrently derive the Cramer-Rao lower bounds (CRLBs) for the ML estimators of the phase coefficients and the phase noise variance, along with the Bayesian CRLB (BCRLB) for the MAP estimator. The joint ML estimators for the phase coefficients and the MAP estimator for the phase noise are unbiased, and their mean-square errors (MSEs) asymptotically achieve the CRLBs and BCRLB at high signal-to-noise ratio (SNR). The MSE performance of the joint ML and MAP estimators is validated using Monte Carlo simulations considering both exact and estimated phase noise variances. In scenarios with large phase noise variance, the MSE performance of the ML estimators for the phase coefficients demonstrates a significant improvement compared with the most current estimators.