Instantaneous Frequency (IF) estimation based on time-varying autoregressive (TVAR) modeling has been shown to perform well in practical scenarios when the IF variation is rapid and/or non-linear and only short data records are available for modeling. A challenging aspect of implementing IF estimation based on TVAR modeling is the efficient computation of the time-varying coefficients by solving a set of linear equations referred to as the generalized covariance equations. Conventional approaches such as Gaussian elimination or direct matrix inversion are computationally inefficient for solving such a system of equations especially when the covariance matrix has a high order. We implement two recursive algorithms for efficiently inverting the covariance matrix. First, we implement the Akaike algorithm which exploits the block-Toeplitz structure of the covariance matrix for its recursive inversion. In the second approach, we implement the Wax-Kailath algorithm that achieves a factor of 2 reduction over the Akaike algorithm in the number of recursions involved and the computational effort required to form the inverse matrix. Although a TVAR model works well for IF estimation of frequency modulated (FM) components in white noise, when the model is applied to a signal containing a finitely correlated signal in addition to the white noise, estimation performance degrades; especially when the correlated signal is not weak relative to the FM components. We propose a decorrelating TVAR (DTVAR) model based IF estimation and a DTVAR model based linear prediction error filter for FM interference rejection in a finitely correlated environment. Simulations show notable performance gains for a DTVAR model over the TVAR model for moderate to high SIRs. / Master of Science

This research develops efficient solution methods for linear systems with scalar and multi-level Toeplitz structure. Toeplitz systems are common in one-dimensional signal-processing applications, and typically correspond to temporal- or spatial-invariance in the underlying physical phenomenon. Over time, a number of algorithms have been developed to solve these systems economically by exploiting their structure. These developments began with the Levinson-Durbin recursion, a classical fast method for solving Toeplitz systems that has become a standard algorithm in signal processing. Over time, more advanced routines known as superfast algorithms were introduced that are capable of solving Toeplitz systems with even lower asymptotic complexity. For multi-dimensional signals, temporally- and spatially-invariant systems have linear-algebraic descriptions characterized by multi-level Toeplitz matrices, which exhibit Toeplitz structure on multiple levels. These matrices lack the same algebraic properties and structural simplicity of their scalar analogs. As a result, it has proven exceedingly difficult to extend the existing scalar Toeplitz algorithms for their treatment. This research presents algorithms to solve scalar and two-level Toeplitz systems through a constructive approach, using methods devised for specialized cases to build more general solution methods. These methods extend known scalar Toeplitz inversion results to more general scalar least-squares problems and to multi-level Toeplitz problems. The resulting algorithms have the potential to provide substantial computational gains for a large class of problems in signal processing, such as image deconvolution, non-uniform resampling, and the reconstruction of spatial volumes from non-uniform Fourier samples.