828 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 4, AUGUST 1998 Nonparametric Identification Assuming Two Noise Sources: A Deconvolution Approach Tam´ as Dab´ oczi Abstract— Nonparametric identification of linear systems is investigated in this paper. Nonparametric identification is the estimation of the time record of the impulse response of the system. It is a deconvolution problem, i.e., inverse operation of the convolution of the impulse response and the excitation signal. The problem is ill posed, i.e., deconvolution amplifies the measurement noise to a great extent. The noise has to be suppressed with the price of a bias in the estimate. A tradeoff has to be found between the noisy and biased estimates. Because of the need for repeatability and to reduce the subjectivity, the level of noise reduction has to be set algorithmically. This paper introduces a method that optimizes the parameter(s) of deconvo- lution filters and, thus, controls the level of noise reduction. The proposed method assumes observation noise sources for both the measurement of the excitation signal and the system output. Index Terms—Deconvolutions, identification, inverse problem, nonparametric identification, signal reconstruction, signal restoration. I. INTRODUCTION I DENTIFICATION of linear systems is the estimation of some characteristic function of the system (e.g., impulse response, transfer function, step response etc.). Parametric identification consists of two steps. First, an appropriate model should be set up for the structure of the system; then the parameters of the system have to be estimated. The result of nonparametric methods is the time or other domain waveform of a characteristic function of the system (impulse response, transfer function etc.), without any explicit information about the structure of the system. As a consequence, nonparamet- ric identification is preferred if the structural information is lacking. The output signal of the investigated system is the con- volution of the excitation signal with the impulse response. Thus, nonparametric identification is a deconvolution problem (inverse operation of convolution). The problem is ill-posed, which means that a small uncertainty in the measurement leads Manuscript received May 18, 1997; revised December 21, 1998. This work was supported by the Hungarian Fund for Scientific Research under Grants OTKA F16457, F26136, and OTKA T021 003. This work was also supported in part by the National Institute of Standards and Technology (NIST), Gaithersburg, MD, under Grant 43NANB614 883 and by the Electronic Instrumentation and Metrology Group (led by B. A. Bell). The author is with the Department of Measurement and Information Systems, Technical University of Budapest, Budapest, H-1521 Hungary. Publisher Item Identifier S 0018-9456(98)09943-4. to large deviations in the estimation [1], [2]. In other words, the deconvolution process amplifies the measurement noise. This noise has to be suppressed [3]. However, suppression of the noise leads to bias of the useful signal. A tradeoff has to be found in the level of the noise reduction to keep a balance between the variance and the bias. The level of noise reduction is usually controlled with one or a few parameters of the deconvolution filter. In order to ensure repeatability and reduce the level of subjectivity, the level of noise reduction should be determined algorithmically, which is the aim of this paper. It will be shown that the extension of a previous work [4], [13] improves the capabilities of the algorithm signifi- cantly, by taking both input and output observation noises into account. The results can be used also for signal reconstruction problems. In this case, the second noise source, in addition to the output noise, is the uncertainty of the estimate of the transfer function. II. PREVIOUS WORKS One of the earliest results is reported in [5], developed by Guillaume and Nahman in the early 1980’s. The estimated impulse response is calculated in the frequency domain. The measured signal is transformed into the discrete Fourier trans- form (DFT) domain, filtered, and transformed to the discrete time domain. The originally real signal becomes complex because of the computational errors. The standard deviation of the imaginary part of the estimated signal is minimized to obtain the optimal filter parameter. The optimization criterion is quite heuristic; there is no proof for optimality yet. More- over, the errors of the imaginary part depend on the precision and the number representation of the computer and on the implementation of the DFT and inverse DFT (IDFT) routines. A more systematic approach is reported by Parruck and Riad [6], [7]. They investigate the tail part of the estimated step response and define the conditions of the good reconstruction based on the empirical mean and standard deviation of the tail part. The optimal reconstruction satisfies the conditions; however, the solution is not unique. One has to choose the right parameter of the inverse filter from an interval. Moreover, the borders of the interval are not exactly defined, only with “much smaller” type conditions. 0018–9456/98$10.00  1998 IEEE