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.
Highlights
Estimating the parameters, e.g., the carrier frequency and phase of a sine wave with noise is a classic and important issue in communications [1]–[3], biomedical engineering [4]– [6], radar/sonar applications [7], [8], and other areas of signal processing, e.g., power-quality monitoring in the power grid [9], [10]
The other popular alternative is called phase-based, timedomain estimation, using the received signal phases as the observation data samples to be fed into the estimator, where the received phase is expressed as the sum of the transmitted signal phase and an additive observation phase noise (AOPN) due to the AWGN [13], [14]
The derivation is based on the fact that using the received signal phases with the instantaneous received amplitude information incorporated in the AOPN model leads to the same maximum likelihood (ML)/maximum a posteriori probability (MAP) estimates of the angle parameters as using the in-phase and quadrature components of the received signals [31]
Summary
Estimating the parameters, e.g., the carrier frequency and phase of a sine wave with noise is a classic and important issue in communications [1]–[3], biomedical engineering [4]– [6], radar/sonar applications [7], [8], and other areas of signal processing, e.g., power-quality monitoring in the power grid [9], [10]. The derivation is based on the fact that using the received signal phases with the instantaneous received amplitude information incorporated in the AOPN model leads to the same ML/MAP estimates of the angle parameters as using the in-phase and quadrature components of the received signals [31] They all have a low-complexity, sample-by-sample iterative processing structure, which avoids the operation of matrix inversion and can be implemented in real-time. 4) The linear minimum-MSE (LMMSE) implementation of the weighted phase averager (WPA) estimator is discussed as an alternative for the frequency estimation in carrier phase noise Even though it performs worse than the ML estimator, it is easier to implement in practice, since it makes use of the phase differences between the contiguous noisy received signal samples and can avoid the phase unwrapping in most cases.
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