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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.

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