MAP harmonic retrieval in Gaussian colored noise with prior information

Abstract

In this paper we deal with the MAP estimation of the parameters of a single complex sinusoid in Gaussian white and colored noise. We analyze the effects of modeling the frequency and amplitude as random variables, deriving the expressions of the MAP estimators, and we analytically compute the CRB's for a tone with Gaussian frequency and amplitude distributions. Moreover, we explicitly derive closed-form expressions of the optimal estimators in the case of Gaussian and Rayleigh distributed amplitude. In particular, we show that, in the colored noise case, the amplitude information can also help improving the performance of the frequency estimator, and vice versa. We propose an efficient FFT-based implementation structure of the optimal estimators, which can be further simplified when the noise can be modeled as an AR process; besides, when the estimators cannot be expressed in closed form, we propose a simple univariate optimization algorithm in order to compute the estimates. Simulation results are reported, which validate the theoretical results and witness the significant performance improvement that can be achieved embodying the parameter priors into the frequency and amplitude estimators.