A FREQUENCY RESPONSE ESTIMATION METHOD BASED ON SMOOTHING AND THRESHOLDING.pdf

TECHNICAL NOTEA FREQUENCY RESPONSE ESTIMATION METHODBASED ON SMOOTHING AND THRESHOLDINGP. BODIN AND B. WAHLBERGS3-Automatic Control, The Royal Institute of Technology, 100 44 Stockholm, SwedenSUMMARYA standard approach for estimating the frequency function of a linear dynamical system is to usespectral estimation. Traditionally, this is done by smoothing the noisy frequency data using linearlters. The method has proved to be successful in most cases and is widely used.However, if the frequency response has ne details appearing only locally in frequency, the lossof resolution caused by smoothing might result in unacceptable errors. In this paper, a dierentmethod for frequency response estimation is suggested. The method utilizes recently proposedwavelet based denoising schemes combined with traditional smoothing techniques. The wavelettransform is applied in the frequency domain in order to provide a suitable frequency window.Tested through simulations, this approach provides an alternative when traditional methods fail.KEY WORDS frequency response estimation; smoothing; wavelets; thresholding1. INTRODUCTIONA commonly used technique in system identication is to estimate the frequency responseof a time-invariant linear dynamical system. Methods for this are described in moststandard books in the eld1;10;13;17. Over the past ve years, much attention has been paidto the possibility of using wavelets in approximation theory and for recovering functionscorrupted by noise. It was suggested6;7 that thresholding of the wavelet coecients shouldbe used for smoothing of noise corrupted data. These methods are based on removingsmall wavelet coecients where the choice of threshold level is based on an asymptoticstatistical result. The major advantages with the methods are that wavelets can ecientlyNovember 24, 1997. To appear in International Journal of Adaptive Control and Signal Processing. approximate a large class of functions that would be dicult to approximate f