A DFT based alternating projection algorithm for parameter estimation of superimposed complex sinusoids

Abstract

In this paper, we consider the alternating projection algorithm suggested by Ziskind and Wax [13] for parameter estimation of superimposed complex sinusoids (cisoids) embedded in noise, and show how the objective function being maximized at each step can be expressed as a non-linear function of the Fourier transforms of the observed data and discrete cisoids. We obtain elegant expressions for the objective functions for the case when the number of cisoids is two and develop a discrete Fourier transform (DFT) based algorithm for this case. The expressions for the case of three or more cisoids are quite cumbersome, but have a nice recursive structure that is illustrated by taking the case of three cisoids as an example. The details of the recursion for the general case are given in the appendix. We discuss some of the approximations and simplifications that lead to substantial reduction in computation. Simulation results are presented to show how the developed algorithm performs in comparison with the other techniques.