A Refined Wiener–Levinson Method in Frequency Analysis

Abstract

This paper is concerned with the problem of determining unknown frequencies $\omega _1 , \ldots ,\omega _I $, using the first N observed values of a discrete-time signal $\{ {x(m)} \}_{m = 0}^{N - 1} $ arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well-defined frequencies $\omega _j $, $j = 1,2, \ldots ,I$. In [K. Pan and E. B. Saff, J. Approx. Theory, 71 (1992), pp. 239–251 (see also [W. B. Jones, O. Njastad, W. J. Thron, and H. Waadeland, J. Comput. Appl. Math., 46 (1993), pp. 217–228]), we proved that unknown frequencies $\omega _j $, $j = 1,2, \ldots ,I$, in a periodic discrete-time signal can be determined by zeros of Szego polynomials with respect to some distribution function by using the first N samples with a rate of convergence of ${1 / N}$. We introduce a refined way to obtain a rate of convergence of ${1 / N}^p $ by using about $p^N $ samples of the signals, where p is any given positive integer.