Effects of the autocorrelation matrix generation method on the model-based sinusoidal parameter estimators

Although the maximum likelihood method gives the optimum solutions for the parameter estimation problem of sinusoids embedded in noise, it is computationally difficult since it generally requires us to solve nonlinear optimization problems. So some model-based parameter estimators with high frequency resolution property are preferred quite often. In order to find these estimates the first step is usually forming the autocorrelation (AC) matrix. In this work the effects of the method utilized in the generation of the AC matrix on the performance of sinusoidal parameter estimators are investigated. One way of forming the AC matrix is to use a Toeplitz structure with either the biased or the unbiased AC lag estimates as the matrix elements. Another way is to use the so-called "covariance method" in the AC matrix generation. In this method the matrix formed is no longer Toeplitz but it is still symmetric. We can think of the Toeplitz AC matrix as a perturbed version of the non-Toeplitz AC matrix. The differences in the performance of the MUSIC spectral estimator with Toeplitz and non-Toeplitz AC matrix usage is related to the perturbation in the AC matrix estimate. For this purpose the 3 × 3 AC matrix is is utilized in the estimation of the frequency of a single sinusoid using the MUSIC frequency estimator. The accuracy of the perturbation analysis is checked with the simulation results. Additionally, the fact that the performance of an estimator with data windowing and Toeplitz AC matrix generation becomes close to the performance of the same estimator with non-Toeplitz AC matrix is shown with simulation studies.