On statistical analysis of Pisarenko tone frequency estimator

doi:10.1016/0165-1684(93)90092-O

Signal Processing

Volume 31, Issue 3, April 1993, Pages 349-353

https://doi.org/10.1016/0165-1684(93)90092-O Get rights and content

Abstract

In a recent paper by Anarim and Istefanopulos, a statistical analysis of the Pisarenko Harmonic Decomposition (PHD) for frequency estimation was carried out. The expression for the variance of the frequency estimate from Anarim and Istefanopulos differs significantly from the expression for the variance of the PHD frequency estimate obtained in earlier studies, e.g. by Sakai, Stoica and Nehorai. In this short communication, we reproduce the analysis from Anarim and Istefanopulos, and obtain an expression for the variance which is identical to the expression derived by Sakai, Stoica and Nehorai.

Zusammenfassung

In einer kürzlich erschienenen Arbeit von Anarim und Istefanopulos wurde eine statistische Analyse der harmonischen Zerlegung nach Pisarenko (PHD) zur Frequenzschätzung durchgeführt. Der Ausdruck für die Varianz der geschätzten Frequenz unterscheidet sich erheblich von dem Ausdruck für dieselbe Varianz, wie er in früheren Untersuchungen (z.B. von Sakai, Stoica und Nehorai) gefunden wurde. In diesem Kurzbeitrag führen wir die Analyse von Anarim und Istefanopulos erneut durch und erhalten einen Ausdruck für die Varianz, der mit dem durch Sakai, Stoica und Nehorai hergeleiteten Ausdruck übereinstimmt.

Résumé

Dans un article récent d'Anarim et Istefanopulos, une analyse statistique de la décomposition harmonique de Pisarenko (PHD), utilisée en estimation de fréquence, a été réalisée. L'expression de la variance de l'estimée de fréquence d'Anarim et Istefanopulos diffère de manière significative d l'expression de la variance de l'estimée de fréquence PHD obtenue dans des études antérieures telles que par Sakai, Stoica et Nehorai. Nour reproduisons dans cette correspondance l'analyse d'Anarim et Istefanopulos, et obtenons une expression de la variance qui est identique à celle dérivée par Sakai, Stoica et Nehorai.

References (8)

E. Anarim et al.
Statistical analysis of Pisarenko type tone frequency estimator
Signal Processing
(September 1991)
P. Stoica et al.
Overdetermined Yule-Walker estimation of the frequencies of multiple sinusoids: Accuracy aspects
Signal Processing
(February 1989)
J. Ängeby et al.
Statistical analysis of autoregressive frequency estimates
S.M. Kay
Modern Spectral Estimation, Theory and Application
(1988)

There are more references available in the full text version of this article.

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Citation Excerpt :
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Among these, the Pisarenko harmonic decomposer (PHD) [9] is of historical interest, since it was the first to exploit the eigenstructure of the covariance matrix of the observations, and its performance has been extensively studied [10,11]. Interestingly, for the case of a real-valued single sinusoid the PHD estimator can be implemented in a very simple way [12]. Some attempts to improve the performance of the PHD estimator can be found in the literature.
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A two-stage autocorrelation approach is proposed for single-tone frequency estimation of a real sinusoid in white noise. In the first stage, we transform the received data to another noisy sinusoidal sequence of same frequency via an autocorrelation procedure. Autocorrelation functions of the converted sequence are then employed for frequency estimation in the second stage. A simple frequency estimator is first derived, followed by a generalized algorithm. Computer simulations are included to illustrate the accuracy of our proposed approach via comparison with Cramér–Rao lower bound and several conventional frequency estimators.
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A number of statistical analyses of the Pisarenko harmonic decomposition (PHD) method for single-tone frequency estimation were carried out in the literature. Based on asymptotic derivations, however, none of them can provide an accurate frequency variance expression for arbitrary data record lengths and/or signal-to-noise ratios (SNRs). With the use of a simple variance analysis technique, a new closed form expression of the PHD variance is derived in this paper. An approximate PHD variance formula for sufficiently large data lengths and SNRs, as well as the upper and lower performance bounds are also included.
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^☆: This work has been supported by the Swedish Research Council for Engineering Sciences under contract 91-676.

¹: On leave from the Department of Automatic Control, Polytechnic Institute of Bucharest, Splaiul Independentei 313, R-77206 Bucharest, Roumania

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Short communicationOn statistical analysis of Pisarenko tone frequency estimator☆

Abstract

Zusammenfassung

Résumé

Signal Processing

Signal Processing

Statistical analysis of autoregressive frequency estimates

Modern Spectral Estimation, Theory and Application

Short communication
On statistical analysis of Pisarenko tone frequency estimator☆