On spectral and root forms of sinusoidal frequency estimators

Abstract

Several frequency estimation methods consist of two stages. In the first stage, the coefficients of a polynomial A( z) are consistently estimated from the available measurements of the sinusoids-in-noise signal; let A ̂ (z) denote the estimated A-polynomial. In the second stage, the sinusoidal frequencies are determined either from the locations of the peaks of the ‘spectrum’ | A ̂ ( e iω )| −2, or from the angular positions of the roofs of A ̂ (z) situated nearest the unit circle. The two aforementioned possibilities in the second stage lead to the spectral, respectively the root forms of various sinusoidal frequency estimators. It is shown here that the frequency errors corresponding to these two forms have the same large-sample variances for any (consistent) estimation method used in the first stage. In the finite-sample case, however, it is shown by means of Monte-Carlo simulations that the root form may outperform the spectral form.