Joint ML/MAP Estimation of the Frequency and Phase of a Single Sinusoid With Wiener Carrier Phase Noise

Abstract

We address here the issue of jointly estimating the angle parameters of a single sinusoid with Wiener carrier phase noise and observed in additive, white, Gaussian noise (AWGN). We develop the theoretical foundation for time-domain, phase-based, joint maximum likelihood (ML) estimation of the unknown carrier frequency and the initial carrier phase, with simultaneous maximum a posteriori probability (MAP) estimation of the time-varying carrier phase noise. The derivation is based on the amplitude and phase-form of the noisy received signal model together with the use of the best, linearized, additive observation phase noise model due to AWGN. Our newly derived estimators are closed-form expressions, consisting of both the phase and the magnitude of all the received signal samples. More importantly, they all have a low-complexity, sample-by-sample iterative processing structure, which can be implemented iteratively in real-time. As a basis for comparison, the Cramer-Rao lower bound (CRLB) for the ML estimators and the Bayesian CRLB (BCRLB) for the MAP estimator are derived in the presence of carrier phase noise, and the results simply depend on the signal-to-noise ratio (SNR), the observation length and the phase noise variance. It is theoretically shown that the estimates obtained are unbiased, and the mean-square error (MSE) of the estimators attain the CRLB/BCRLB at high SNR. The MSE performance as a function of the SNR, the observation length and the phase noise variance is verified using Monte Carlo simulation, which shows a remarkable improvement in estimation accuracy in large phase noise.

Using conditional averaging of delayed signals to measure phase shift angle

“…The theoretical basis for phase-based estimation of the unknown carrier frequency in the time domain using the ML method has been discussed with taking into account the initial carrier phase, with simultaneous a posteriori maximum likelihood (MAP) estimation of the time-varying carrier phase noise. It was theoretically demonstrated that the obtained estimates are unbiased, and the performance of the mean squared error as a function of SNR, observation length, and phase noise variance was verified using Monte Carlo simulations [9].…”

“…The problem of joint estimation of angular parameters of a single sinusoid with Wiener phase noise of the carrier and white Gaussian noise observed in the additive has been presented in [9]. The theoretical basis for phase-based estimation of the unknown carrier frequency in the time domain using the ML method has been discussed with taking into account the initial carrier phase, with simultaneous a posteriori maximum likelihood (MAP) estimation of the time-varying carrier phase noise.…”