Asymptotic analysis of the least squares estimate of 2-D exponentials in colored noise

Abstract

This paper considers the problem of estimating the parameters of complex-valued sinusoidal signals observed in colored noise. This problem is a special case of the general problem of estimating the parameters of a complex-valued homogeneous random field with mixed spectral distribution from a single observed realization of it. The large sample properties of the least squares estimator of the exponentials' parameters are derived, making no assumptions as to the probability distribution of the observed field. It is shown that the least squares estimator is asymptotically unbiased. A simple expression for the estimator asymptotic covariance matrix is derived. The derivation shows that, asymptotically, the least squares estimation of the parameters of each exponential is decoupled from the estimation of the parameters of the other exponentials. Assuming the observed field is a realization of a Gaussian random field, it is further demonstrated that the asymptotic error covariance matrix of the least squares estimate attains the Cramer-Rao bound, even for modest dimensions of the observed field and low signal-to-noise ratios.