SPC09-2: ML Estimation of the Frequency and Phase in Noise

The problem of estimating the frequency and carrier phase of a single sinusoid observed in additive, white, Gaussian noise is addressed. Much of the work in the literature considers maximum likelihood (ML) estimation. However, the ML estimator given by the location of the peak of a periodogram in the frequency domain has a very high computational complexity. This paper derives an explicit structure of the ML estimator for data processing in the time domain, assuming only reasonably high signal-to- noise ratio. The result of this approximate ML estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the estimator, and the phase data alone as assumed is not a sufficient statistic. The sample-by-sample iterative processing nature of the estimator enables us to propose a novel, recursive phase-unwrapping algorithm that allows the estimator to be implemented efficiently. To facilitate the performance analysis, an improved, linearized observation model for the instantaneous signal phase that is more accurate than is proposed. This improved model explains physically why the phase data are weighted by the magnitude information in the ML estimator.