IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, zyxwvuts VOL. 39, NO. zyxwvuts 4, AUGUST 1990 565 Robust Identification of Transfer Functions in the s- and Z-Domains RIK PINTELON AND JOHAN SCHOUKENS zyxwv Abstract-A frequency-domain maximum likelihood estimator (MLE) to estimate the transfer function of linear continuous-time sys- tems has already been developed in [I]. It assumes independent Gauss- ian noise on both the input and the output coefficients. In this paper these results are extended to linear discrete-time systems. It is dem- onstrated that most of the properties of the estimator remain un- changed when it is applied to measured input and output Fourier coef- ficients corrupted with non-Gaussian errors. A robust Gaussian frequency-domainestimator results from this. It is very useful for the practical identificatioil of linear systems. The theoretical results are verified by simulations and experiments. Keywords-Robust identification, maximum likelihood estimation, analog systems, digital systems, transfer function, input-output noise, errors-in-variables model. I. INTRODUCTION HE MAXIMUM likelihood estimator (MLE) devel- T oped in [ l ] to estimate the parameters of the transfer function of a linear analog device, has been built. It as- sumes that the errors in the measured input and output Fourier coefficients are independent zero-mean Gaussian random variables. The literature guarantees, under the standard assumptions of independent, equally distributed errors, asymptotic properties of consistency, asymptotic efficiency and asymptotic normality of the MLE when the amount of measured data (here the number of spectral lines) tends to infinity [2]. This is the only known general theorem concerning maximum likelihood estimation. Un- der nonstandard assumptions there is no a zyxwvuts priori reason why the maximum likelihood estimation should even be useful [3], [4]. Important questions remain to be an- swered concerning what happens if the estimator is ap- plied to data corrupted with noise violating the assump- tions that have been used to derive it [5]-[8]. Here also there is no a priori reason why the estimator should still be useful. For real world processes, the (standard) as- sumptions are not completely met so the practical aptness of the Gaussian frequency-domain estimator given in zyxwvuts [ 13 must be verified. First, the results obtained in [ l ] for lin- ear analog time-invariant systems ( s-domain) are gener- alized to digital systems (2-domain). Then it is demon- strated that most of the properties of the Gaussian Manuscript received May 30, 1989; revised February 13, 1990. This work was supported by the National Fund for Scientific Research, Bel- gium. The authors are with the Department ELEC, Vnje Universiteit Brussel, 1050 Brussel, Belgium. IEEE Log Number 9036463. frequency-domain MLE remain unchanged even when the disturbances of the input and output Fourier coefficients do not satisfy the basic assumptions made to construct it. This explains the good results of the estimation technique obtained in several practical measurement problems such as the real-time equalization of a data acquisition channel [9], the determination of the resonance frequencies and damping coefficients in modal analysis [lo], and the cor- rection of loudspeaker characteristics [ 111. Models where both input and output are disturbed by errors are called errors-in-variable (EV) models [ 121. In an EV model, all variables are treated symmetrically. An- derson [ 131 has demonstrated that it is possible to uniquely identify a system within an EV model only when the power spectra of the errors, or at least their ratios, are known. This poses no practical problem, since the power spectra of the disturbing noise at the input and output of the device under test can easily be measured when no ex- citation signal is present. The stochastic model used in this paper is an EV model in the frequency domain (Fig. l), where X, and Y, are, respectively, the measured input and output spectra, and where X is unknown: Y, = zyx Hx + ay zyx x, = x + a,. (1) The noise sources a, and ay denote the errors in the input and output spectra, respectively. Their variances, or at least their ratio (in case the noise power spectra are flat), are assumed to be known. H is the transfer function of the process zyxwv n r= 1 where for continuous-time systems is the sample period ak = [ f (jwTs) for discrete-time systems where Ts and - - 001 8-9456/90/0800-0565$01 .OO 0 1990 IEEE