A modified prony algorithm for estimating sinusoidal frequencies

Abstract

We investigate the modified Prony algorithm of Osborne and Smyth (1991, 1993) applied to the estimation of the frequency of oscillation of a sinusoidal signal with added white noise, and compare it with the algorithms usually associated with the names of Prony and Pisarenko. We show that the Pisarenko algorithm returns O(N –1/2) consistent estimates, and that the modified Prony algorithm returns thc least squares estimates which are O(N –1/2) consistent. The transfer function parameter estimates that are returned by these algorithms converge to the true values at the sarne rste as the frequency estimates in each case. A concise expression for the asymptotic covariance matrix of these estimates is given in the simplest case. Small sample properties, starting values and the importance of correctly specifying the number of sinusoids, are investigated in a simulation study.