cop yright© IFAC Identification and System Parameter Estimation. Beijing. PRC 19HH PERFORMANCE ANALYSIS OF THE PISARENKO HARMONIC DECOMPOSITION METHOD P. Stoica* and A. Nehorai** *Department of Automatic Control. Polit echnic Institute of Bucharest, Splaiul Indepemlentei 313. S ectm' 7, R -77 206 Bucharest, Romania **Department of Electrical Engineering. Yale University. New Haven. CT 06520, USA Abstract. A self-contained statistical analysis of the Pisarenko method for estimating sinusoidal frequencies from signal measurements corrupted by white noise is presented. An explicit formula is provided for the asymptotic covariance matrix of the joint estimation errors of the minimum eigenvalue and the minimum eigenfilter coefficients of the data covariance matrix. Our theoretical results extend and reinforce previous results obtained by Sakai [11. A comparison with the performance achieved by the Yule-Walker Method is also reported . Key words. Harmonic analysis, parameter estimation, spectral analysis. I. Introduction Determination of angular frequencies of sinusoidal signals from noisy measurements is a topic of consider- able recent interest (IEEE, 1987). When the measure- ment noise is white, consistent estimates of the sinu- soidal frequencies can be obtained by the so-called Pis- arenko harmonic decomposition method (Pisarenko, 1973; Sakai, 1984) . This method consists of determin- ing the minimum eigenvalue of the data covariance ma- trix and its assoicated eigenfilter. Then the sinusoidal frequencies are determined as the angular positions of the eigenfilter zeros. Recently, Sakai (1984) presented a formula for the asymptotic covariance matrix of the estimation errors of the minimum eigenfilter coefficients. The analysis of (Sakai, 1984) relies on some referenced results, which makes it somewhat difficult to follow. In this paper, we present a self-contained analysis of the statistical performance of the Pisarenko method. Our analysis is more general than that of (Sakai, 1984) since we consider the joint estimation errors of both the min- imum eigenvector and the minimum eigenvalue. The extended result that we obtain exhibits a rather un- usual and interesting feature. While the covariance matrix of the estimated eigenvector depends only on the second order moments of the data, the joint covari- nace matrix of the estimated eigenvector and eigen- value al!jo depends on the fourth order moment of the measurement noise. 'The work of A. Nehorai w"," supported in part by the Air Force Office of Scientific Research under Grant No, AFOSR-88- 0080. 1029 The analysis method of this paper is similar to that in (Stoica and Soderstrom, 1982; Soderstrom and Sto- ica, 1983; Stoica and Soderstrom, 1983; Stoica and co- workers, 1985) and is different from the "periodogram method" used in (Sakai, 1984). We show that Sakai's result may be obtained from our general results, which provides a useful cross-checking of the two methods of analysis used in (Sakai, 1984) and in this paper. We also present a comparison with the performance achieved by the Yule-Walker method . Finally, we note that estimation methods based on eigenanalysis, which are very similar to Pisarenko har- monic decomposition, have been considered in the sys- tem control literature by Levin (1964) and later by oth- ers (Smith and Hilton, 1967; Aoki and Yue, 1970; Fu- ruta and Paquet, 1970). In our opinion, proper credit should be given to these early contributors . 11. Preliminaries Let x(t) denote a signal consisting of m sinusoids m x(t) = L Qk sin (Wkt + .pk) k=l (2.1a) and let y(t) denote the noisy measurement of x(t) y(t) = x(t) + e(t) (2.1b)