The constrained total least squares with regularization and its use in ill-conditioned signal restoration

The recovery or restoration of an input signal from the impulse response and output signal of a linear, time-invariant system involves solving a set of linear equations in which both the data matrix and the observation vector are corrupted by noise Since the 1980s, the total least squares technique has been a powerful tool for solving a noise perturbed system of linear equations. However, the total least squares method is based on the assumptions that the noise perturbations contained in both the data matrix and the observation vector are statistically independent of each other and have equal values of variances. Violation of these assumptions degrades the performance of this technique or even results in its failure. To combat the correlated noise entries in the data matrix and the observation vector, a modified version of the total least squares method, called the constrained total least squares has been developed. The constrained total least squares technique attempts to find the input signal estimate by minimizing the Frobenius norm of noise entries of the data matrix and the observation vector. It has been shown that the constrained total least squares has the maximum likelihood estimator properties. However, the noise content affects the minimization process, and causes numerical ill-conditioning, both of which degrade its performance.;In this dissertation, a new approach, a modified version of the constrained total least squares technique, called the regularized constrained total least squares technique, is developed for the restoration of an input signal from noisy data. Based on the constrained total least squares, the proposed technique estimates the input signal by minimizing the Frobenius norm of noise entries in the data matrix and the observation vector under the linear equality constraints on the input signal vector. In view of the linear algebraic relations among the noise entries in the data matrix and the observation vector, the presented technique transforms the problem of minimization of Frobenius norm with linear equality constraints into an unconstrained minimization problem. Therefore, numerical optimization techniques can be applied. A perturbation analysis is performed to ascertain the applicability of the foregoing technique. It indicates that variance and the overall mean square error of the solution obtained by using the presented technique is reduced while increasing the stability. To illustrate the validity of the presented technique, both simulated data with additive white Gaussian noise and actual data with unknown noise characteristics are considered. The results show that, when the regularization parameter is appropriately chosen, the new technique is superior to the constrained total least squares technique.