A Spectral Matching Technique for ARMA Parameter Estimation

A Spectrai Matching Technique for ARMA Parameter Estimation BENJAMIN FRIEDLANDER

AND

Abstract-Aniterativefrequencydomaintechnique is presentedfor estimatingAR-plus-noiseandautoregessive moving-average(ARMA) parameters. The technique is based on minimizing the error between thesamplepowerspectrumandaspectralmodel.Thevariance of the estimationerror is showntobeclose to the Cramer-Raobound for some requirements computational its

BOAZ PORAT,

MEMBER, IEEE

domain estimation procedure

for ARMA parameters based on

the truncated correlation function {Ro,R^, , * . . ,k,}, in which

all the low order lags are included. The proposed technique involves nonlinear optimizqtion and is more complex than the examples. MYW However, approach. are quite modest as will be shown later. We note that frequency domain techniques for ARMA estimation have also been develI. INTRODUCTION [ 5 ] , Hannan [3] andothers.ThesetechopedbyAnderson techniques for estimatingautoregressiveniquesaretypicallybased on someapproximation of the likeliNmoving-average (ARMA) have been in hood function and use the data directly, rather than in reduced the and engineering literature [11-[51 > [61-[101. form (i,e., thesamplecorrelationfunction).Theircon,putaThis was motivated by the computationa1 of the tionalcomplexity is considerablyhigherthanthat plexity of the maximum likelihood estimator which, in a certain technique presented here. ~~

proposed technique is described in Section for the sense, is the“bestpossible”estimator.ConsiderableattentionThe was devoted in recent years to a class o f ARMAestimation of A~ probesses in noise and in Section 111 for the on the so-called modified Yule-Walker general ARMA case. section Iv techniquesbased results (MYW) equations L61, L1 l1 - u 3 i . The attractive feature Of which indicate that the performance of the spectral matching thesetechniques is thattheyestimatetheARparametersofmethod is veryclose to the Cramer-Rao bound forSome an ARMAprocessbysolving a set of linear equations.These examples. equations involve the sample correlation coefficients {Ji,1 < N , < i < N z ! , where N 1 and N , arefiniteintegers.It is well 11. ESTIMATIONOF ARSIGNALS I N NOISE knownthatARMAestimatorsbased on a finitesetofsample rn this section we considerthe where the observed real correlation coefficients are inefficient, in the statistical sense data y t is a sum of an autoregressive signal X t and a white noise [4] , [ 141 , [ 181 . The loss of efficiency can be made, however, process U t , Thus, very small when the ARMA process is narrowband (i.e., when nearunit the was circle). It y , = X t + ut. (la) are part AR the modes of the shown by Tufts [ I S ] that linear prediction methods based on the MYW equations perform almost as well as the maximum (Ib) sinusoids for method, likelihood in noise. i=l The situation is somewhat different in the case of wideband ( 1 c) var {ut}= u,” , var { u t } = u: processes.The MYW methodmaysuffer a considerable loss of efficiencyduetothefactthatthefirst few correlation co- where ut is also awhitenoiseprocess,uncorrelatedwith ut. efficients {k,, . . . , R N , (orjustthezerothcoefficientThespectraldensityfunction S(w) of the observeddata y , is R o )are excluded from these equations. This effect wasstudied given by in detail in [ 1 4 ] , by looking a t asymptotic the properties of 0 :, -t u; (2) estimators based o n differentsetsofsamplecorrelation cos(0)= ---A(e’W)A(e-iw) efficients. The exclusion of the low lag correlation coefficients where is an inherent feature of the linear prediction formulation for (wideband) ARMA. processes, and can not be easily avoided. A ( z ) = 1 f a 1 z-’ + . . . t a , z - ~ . ( 31 Motivated by the potentfa’ loss of performancewhenthe Thespectralmatchingtechnique involves two steps, [n the MYW technique is used for wideband processes, we sought an first step a nonparametric estimate of the spectral density funcapproach. In this paper we present a frequency tion is obtained, using the correlation method: Let k j be the unbiased estimate of the correlation sequence Manuscriptreccived April 28. 1983; rcviscd October 5. 1983.This work was supported by the Office of Naval Research under Contract NOOO14-82-C-0476. B. Friedlander is with Systems Control Tcchnology Inc.. Palo Alto,

R.“ I -

~

I

T-i

C=

t=i

.v,y-i,

O G K N

FRIEDLANDER AND PORAT: SPECTRAL MATCHING FOR ARMA PARAMETER ESTIMATION N

9(w) = R o t 2

w i R icos i o

(5

i=1

where {wj,1 < i < N } is some window function (a Hamming window was used in our experiments). The choice ofthe number of correlation coefficients N and the window function reflects the desired degree of preliminary smoothing. Typically, N will be larger for narrowband signals and smaller in the wideband case [ 121 . By selecting N = T - 1 and wi = 1, we obtain the unsmoothed correlation method. In the second step, the estimated spectral density function is further smoothed by matching a parametric model (the ARplus-noise model of ( 2 ) , or an ARMA model, to be discussed in Section 111). The number of free parameters in the spectral of thesmoothingdesiredforthe modelreflectstheextent finalresult.Priorinformationregardingthestructureofthe spectral function can be used, when available, to determineth.e model order. The spectral matching procedure consists of computing the set of model parameters that will minimize the squared error s^(w) andthe betweenthenonparametricspectralestimate parametric spectrum S(w). In other words, the AR-plus-noise parameters will be computed by minimizing the following cost function:

4]n

F(al;..,an,u:,u;)=---

J2* [S(w) - s^(w)]2 do.

(6) This function can be considered as an approximate log-likeliis based on a finite hood function (cf. [ 3 ] , [ 5 ] ) . Since s^(o) set of correlation coefficients,all of the information it contains {wj = is given byitsvaluesatadiscretesetoffrequencies (27rZ/N),0 < Z