A NEW STABLE FEEDBACK LADDER ALGORITHM FOR THE IDENTIFICATION OF MOVING AVERAGE PROCESSESt Caries H. Muravchikt and Martin Marl Durand Bldg. 109 Information Systems Laboratory, Stanford University, Stanford, Ca 94305. Abstract A novel algorithm for the direct identification of the coefficients of a moving average model is presented. The scheme can be represented in a feedback ladder form, recursive in time and order, hence allowing sequential processing of data observed from a process. Potential appli- cations are numerous in different fields such as spectral estimation, automatic control or econometrics. The paper briefly discusses the underlying principles of the method. Results from simulations are included in order to display the general behaviour of the algorithm and its sensitivity to uncertainty in the model order. I. Introduction The problem of identifying the coefficients of a mov- ing average ( MA) model from samples observed from its output is an old and ubiquitious one. Indeed, numerous problems in as different fields as econometrics, com- munications or control make use of MA models, or at least partly MA as in the ARMA case. Specific actions to be performed in such systems require knowledge of the parameters of the model, therefore the necessity for an identification procedure. The difficulty lies in the essen- tial nonlinearity of the problem, unlike the more usual one of estimation of the coefficients of an autoregressive (AR) model. The latter has a nice least squares solution completely similar to regression analysis in statistics - actually, it is a particular case of it -. Recently, a lot of effort was directed towards the efficient solution of the regression equations. In the digital signal processing context the emphasis was on the reduction of the number of computations, improvement of the numerical conditioning, the implementation in real time and the capability of recursively updating the model parameters as new samples became available. Ladder - or lattice - filters emerged in consequence as an important solution to these requirements. In this paper we present a new algorithm of the feedback ladder type that allows the identification of the coefficients of an MA model. Previous work in the prob- lem produced basically three types of solutions: 1) Solve iteratively certain nonlinear equations, for instanoe With a scholarship from CNEA/Senid, Argentina. t This work was supported by the Defense Advanced Research Projects Agency under Contract MDA9O3-80-C-0331 and MDA9O3-82-K-0382. 15A.6 those resulting from the maximum likelihood formula- tion under the assumption of gaussian driving noise. See [4], [5], [17] and [is]. 2) Factor the covariance - or a sample covariance - matrix into its upper-diagonal-lower triangular decomposition to produce asymptotically unbiased estimators. This approach has been pursued in [20] and [12]. 3) Estimate the spectrum and approxi- mate it, as in [9]. Our approach is of the second kind, its particular feature being that it leads to a novel feedback ladder structure that still preserves the advantages mentioned above. This result was to be expected since, roughly, the whitening filter corresponding to this MA problem is an AR or feedback type of filter. Two precedents of the present work, [11] and [12] already anticipated the feed- back structure, although the methods suggested there did not evidently lead to a ladder type solution as the new scheme does. Although no particular comment is made in this paper, the method is applicable to vector processes and complex quantities as well. In section 11 we shall briefly explain the foundations of the factorization of the covariance approach, a prob- lem with close ties to that of spectral factorization. Once this method is established, any scheme producing the above factorization would be adequate, for instance Gaussian elimination, Gram-Schmidt procedure or via Householder transformations. Our technique is based on the so-called Fast Cholesky methods, originally developed in [11], and expanded in [7] and [12]. Also in section II the algorithm and the feedback ladder struc- ture are presented. Simulations were performed to illus- trate certain practical aspects of its use. The results are shown in section III. II. Covariance Factorization and the New Algorithm. As mentioned above, the essence of our identification technique is the factorization of a coven- ance matrix. Assume we are given a - wide sense - sta- tionary, zero mean, stochastic process Ut ' —otoo with covariance RM(t,s) = = R(t—s). If the process has no deterministic component or is regular then - see [1] or [3] and [19] for vector valued processes - by the Wold decomposition theorem the following representation of is possible: lit = .:A(t,s) s (1. a) where is an orthonormal sequence of random vari- ables. The coefficients A(t,s) can be interpreted as the - time varying - impulse response of a filter at time t to an ICASSP 83, BOSTON CH1S41-6/8310000-0683 $1.00 © 1983 IEEE 683