Parameter estimation for exponential signals in colored noise using the pseudo-autoregressive (PAR) model

Abstract

Most modem techniques for high resolution processing of closely spaced signals assume either uncorrelated noise or require knowledge of the noise covariance matrix. These assumptions are often invalid in practice. Here we propose a Pseudo-Autoregressive (PAR (M, <i>p</i>) model for estimation of an arbitrary number of signals M in the presence of a <i>p</i>-th order autoregressive (AR) noise environment. We derive the Cramer-Rao Lower Bound (CRLB) for the parameters of damped exponential signals in the colored noise case. A closed-form expression for the Cramer-Rao Lower Bound for the Pseudo-Autoregressive (PAR(<i>M,p</i>) model is obtained. Some special cases are investigated, for example, the PAR(<i>M,p</i>) model for <i>p</i> = 0, i.e., the white noise case, where our results agree with previous research results. We then evaluate the Cramer-Rao Lower Bound for two possibly closely spaced signals in a colored noise environment, showing that the colored noise assumption can lead to a much lower variance bound for the exponential parameters than under the white noise assumption.

An algorithmic procedure is presented for the identification of the parameters of exponential signals, measured in colored noise. Previous papers on identifying sinusoids in noise have concentrated mainly on white noise disturbances. In a practical environment however, the disturbance is usually colored; sea-clutter in a radar context, is a lowpass type noise for example. When least squares type estimates are used in the colored noise environment, this usually leads to an unacceptable bias in the estimates. We propose an identification method, named Singular Value Decomposition Bias Elimination (SVDBE), in which it is assumed that the noise can be represented well by an AR process. The parameters of this noise model are then iteratively estimated along with the exponential signal parameters, via Singular Value Decomposition (SVD) based least squares. The iteration process starts with the white noise assumption, and improves on that by allowing the parameters in the noise model to vary away from the white noise case. A high order modal decomposition is found, and the best subset of the identified modes is selected. Simulations assess the merits of the introduced SVDBE algorithm, by comparison of the estimation results with the derived Cramer-Rao Lower Bound.