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Smyth, G. K. (2000). Employing symmetry constraints for improved frequency estimation by eigenanalysis methods. Technometrics 42, 277-289.

# Employing Symmetry Constraints for Improved Frequency Estimation by Eigenanalysis Methods

Gordon K. Smyth, Department of Mathematics, University of Queensland

## Abstract

The problem of extracting sinusoid signals from noisy observations made at equally spaced times is considered. Eigenanalysis methods, such as Pisarenko's method and the extended Prony method, find the eigenvector with minimum eigenvalue of a suitably chosen matrix, and then obtain the complex sinusoids as the roots of the polynomial which has the components of the eigenvector as coefficients. For the sinusoids to be undamped, it is necessary that the roots lie on the unit circle, and hence that the eigenvector be conjugate symmetric. It is shown how this symmetry constraint can be incorporated into eigenanalysis estimation methods in a routine way. The practical importance of the constraint is investigated by Cramér-Rao variance bound calculations and by simulation. Thee data examples are included. The following conclusions are made: (i) The symmetry constraint is straightforward to implement, and reduces the amount of computation required. (iii) The relative reduction in variance of the constrained over the unconstrained frequency estimators is arbitrarily large for frequencies close together or near a multiple of p. However there are also frequency values for which the symmetry constraint gives no reduction in variance at low noise-to-signal ratios. (iv) The relative reduction in variance converges to zero for large sample sizes. (v) The symmetry constraint increases the breakdown noise-to-signal ratios above which the various methods fail to give useful results.

Keywords: Prony's method; Pisarenko's harmonic decomposition; sinusoid signals; complex exponentials; least squares; discrete spectra.