Analysis Of The Constrained Total Least Squares Method And Its Relation To Harmonic Superresolution

Abstract

The Constrained Total least Squares (CTLS) method is a generalized least squares technique used to solve an overdetermined set of linear equations whose coefficients are noisy. The CTLS method is a natural extension of the Total Least Squares method to the case where the noise components of the coefficients are algebraically related. This paper presents a number of analytical properties of the CTLS method and solution, and sets forth their application to harmonic superresolution. the CTLS solution as formulated is derived as a constrained minimization problem. It is shown that the CTLS problem can be reduced to an unconstrained minimization problem over a smaller set of variables. A perturbation analysis of the CTLS solution valid for small noise levels, is derived, and from it the root mean square error of the CTLS solution is obtained in closed form. It is shown that the CTLS problem is equivalent to a constrained parameter maximum-likelihood problem. A complex version of the Newton method for finding the minimum of a real function of several complex variables is derived and applied to find the CTLS solution. Finally, the CTLS technique is applied to frequency estimation of sinusoids and direction of arrival estimation of wavefronts impinging on a linear uniform array. Simulation results show that the CTLS method is more accurate than the Modified Forward Backward Linear Prediction technique of Tufts and Kumaresan. Also, the CTLS, MUSIC, and ROOT-MUSIC techniques are compared for angle-of-arrival estimation.