DUAL ALGORITHH FOR AR8A SPECIRUH ESII8ATION Y M. Isabel RIBEIRO CAPS, Institclto Superior TQcnico Av.Rovisco Pais, 1 P-1096 Lisboa, PORTUGAL ABSTRACT The present work describes an ANMA estimation algorithm that differs from the known available techniques. It substitutes the autocorrelation estimation sequence by the sequence of estimated reflection coefficients. These are reliably provided by the Burg technique Ell. Then it fits to the process both a sequence of higher order linear predictors (e.g., Levinson algorithm), and a sequence of higher order linear innovations filters (e.g., by recursive inversion). Finally, it obtains the MA coefficients from the linear relations satisfied by the corresponding coefficients of the successive higher order linear predictors, and likewise obtains the AR coefficients from the linear relations satisfied by the corresponding coefficients of the successive higher order innovation filters. We stress that the procedure does not use the sample autocorrelation lags; it uses instead the sequence of sample reflection coefficients, from which it estimates independently of each other and in a dual way, the MA and the AR components of the process. 1. INIRODUCT ION The use of autoregressive moving-average (ARHA) models in spectral estination has received increased atention in the last few years (see e.g. C21-C4lf. Most of the reported techniques involve several steps, the .first of uhich constructs a sample autocovar iance function. Another characteristic of those algorithms is the dependence of the MA component estimation upon the AR component estimation. This work addresses the ARMA estimation problem, presenting an estimation procedure that does not share the two above mentioned common features. On the one hand, it departs from the usual approach of using the sample autocovariance lag sequence, estimating instead from the data the sequence of reflection coefficients. The siniulation results presented here use the Burg technique C11 for the estimation of the reflection coefficients. On the other hand, the AR and MA coefficients are obtained in a dual way. The * The work of the first author was supported by -----_---- INIC (Portugal). Jose H. F. MOURA Ifept. of Electrical and Computer Eng. Carnegie-Mellon University Pittsburgh, PA 15213, USA algorithm constructs from the reflection coefficients two sequences of successively higher order linear filters - one is a sequence of linear predictors, the other is a sequence of innovations filters. The MA part is then obtained by exploring the linear relations that are satisfied by corresponding coefficients of the sequence of linear predictors, while the AR part is obtained similarly but using in turn the coefficients of the ir~creasing order innovations filters. The estimation algorithm dualizes the roles of the AR end MA components, performing the same kind of operations for both components. The estimation algorithm is based on a finite sample of lenght T drawn from the Gaussian, stationary stochastic process Cyn>, with the AHMA signal model where <e,> is a Gaussian, white, zero mean, unit variance noise sequence. The linear, time- invariant representation (1) is assumed to be stable, minimum phase, with no common roots on the numerator and denominator polynomials of its transfer function. The estimation algorithm discussed here assumes apriori knowledge of the number of poles p and zeros q, and that p2.q. We are presently testing an extension of the present procedure that jointly estimates the orders p and q as well a5 the parameters. In section 2 we briefly review the clnderlying theory concerning the estimation algorithm, which is detailed in [GI. Here, the emphasis is on the algorithm aspects of the estimation procedure. Some simulated examples are presented in section 3. Further examples are in [SI. In C71 some preliminary results of this algorithm are compared with those of an estimation procedure based on d-step ahead predictors. A statistical analysis computing analytically the asy~ptotics of the bias and of the error covariance has been carried out and will be presented elsewhere.2. BUbL ESIIiiAIION ALGOBITH# The dual estimation algorithm herein presented is derived from the exact knowledge of the increasing order prediction and innovation filter coefficients associated with the process .Cyrl). We will see that both the AN and the MA components are obtained from the solution of a system of linear equations built up from those coefficients. In the presence of a sample of lenght T of the observed process, those exact vallres are replaced by a suitable estimate. This probles is discussed later, To present the algorithm, we will assume 'that an infinite sample is I)iven, i.e. that we know exactly the above referred filter coefficients. Let structure of (4) allows a recursive implementation of this matrix operation. Hy the Wold decomposition, the elements on line N of matrix W;' constitute, as N goes to infinity, a long AN(N) representation of the AHMA(p,q) process under study. With a dual arcjument, a long MA representation of the process is obtained, for an high value of N, from the Nth order innovation filter, i.e. the elements on line N of the nratrit: wN * Several spectral estimation techniques use long AH or long MA models .35 an intermediate or a final step in the estimation procedure. These approaches correspond to exploring the matrices W N and WM by lines. Un the contrary, the estimation -1 a? i=l,. ..,N, a! = 1, algorithm presented on this section looks upon the columns of those matrices. In fact, both the AH and the MA components of the AHMA process are be the Nth order, one-step ahead prediction error computed from the linear dependencies exhibited by filter coefficients associated with (y,), and the elements on each column of the matrices WN and denote by .1 ' f 2) . -1 cN = aN N For N>p, the elements on each of the first (3) N-¶ columns of the matrix wN are linearly I* variance of the Nth order prediction error. Collecting the