SVD-based criteria for detection of the number of damped / undamped sinusoids in noise & their parameter estimation

Abstract

The thesis addresses two problems: (i) detection of the number of damped/undamped sinusoids in noise, and (ii) their parameter estimation using a nonlinear least squares formulation.

Following the information-theoretic approach to model selection, the thesis develops AIC and MDL criteria for detecting the number of damped/undamped sinusoids. Also, an improved version of the AIC criterion (referred to as the VAIC criterion in the thesis) is motivated. Using the assumptions made by other researchers in the context of similar problems, a unified analytical framework has been developed for analysing the performance of the criteria.

The development of the criteria and the performance analysis make use of certain approximations which become better for large signal-to-noise ratios. The criteria are so well matched to singular value decomposition (SVD)-based methods such as modified forward/backward and forward–backward linear prediction (MFLP/MBLP and MFBLP) methods that the additional computations required, over and above those needed for SVD calculation, are marginal.

The performance of the proposed criteria is compared with that of a recently suggested criterion by Fuchs (for the time-series problem) using simulations. The results show that the MDL and VAIC criteria perform nearly as well as the Fuchs method. However, the Fuchs method is computationally more expensive.

The usefulness of the analysis is verified by comparing the theoretically predicted values of probability of detection with those obtained from simulation, and the results show very good agreement between the two.

The thesis develops the criteria and analysis first for the case of real sinusoids (damped/undamped) and then extends them to the case of complex sinusoids.

The problem of estimating the frequencies, damping factors and amplitudes of superimposed exponential signals in noise is formulated as a nonlinear minimization problem. The alternating projection (AP) algorithm, recently proposed by Ziskind and Wax, is used for solving this minimization numerically.

To apply the Newton–Raphson search technique so as to speed up the convergence of the AP algorithm, the expressions for the gradient and Hessian of the objective function are developed. In this regard, the undamped and damped sinusoids are treated separately because certain analytical difficulties arise in the case of damped sinusoids.

In the thesis, an elegant method is suggested to find the gradient vector and Hessian matrix for the case of damped sinusoids. Simulations are used to compare the performance of the AP algorithm with that of MFBLP (or MBLP), total least squares, and iterative quadratic maximum likelihood methods proposed by Tufts and Kumaresan, Rahman and Yu, and Bresler and Macovski, respectively.