Discrete-Time Wavelets: Time-Frequency Localization, Shift-Invariance, Modeling Of Linear Time-Invariant Systems

14 It is shown further how the discrete wavelet transform can be shift invariant in the sense that the coecients of the transform at a scale for a shifted sequence can be computed from the coecients at the same scale from the original sequence. We derived conditions for the analyzing lters in order to obtain that kind of shift invariance. The transfer functions of the lters must not overlap with their translations for in frequency. The reconstruction lter for the co-ecients of shifted sequences is just a linear time-invariant all pass lter. Finally, we found for the linear time-invariant systems modeled by discrete-time wavelets the minimum mean square error parameter estimates and the value of minimum mean square error. The following quantities which aect the estimates and which are related to discrete-time wavelets are pointed out or dened: the orthogonality, wavelet time-crosscorrelation, inner product of wavelet time-crosscorrela-tion and input signal autocorrelation, time-frequency local-ization of wavelets. Due to the localized main energy support of wavelets in the time-frequency domain a clear physical interpretation of the reduced modeling error can be given. An on-line least mean square parameter identica-tion adaptive algorithm is proposed and it is shown how its behavior is aected by the wavelet related quantities mentioned before. The orthogonality and good time-frequency localization can enhance the convergence speed and numerical properties of the algorithm. REFERENCES [ 1] A. Grossmann and J. Morlet, \Decomposition of Hardy functions into square integrable wavelets of constant shape", SIAM J. When the input is not white noise, for orthogonal wavelets we have that IJ (l) is zero in the points which correspond to the shifts needed to obtain from I (k) new wavelets at the same scale. Also, if I (k) and J (k) are on the same scale the maximum of 3 JI (l) is out of origin for J 6 = I. When we assume u (l) is concentrated around the origin, we can again expect (4.10) be small for I 6 = J. The multiresolution analysis based on wavelets can help to physically interpret the error induced by improper modeling, i.e., the eects of reduced order modeling. The characterization of those eects is an involved problem in the case of standard modeling methods. For instance, this is the case if we are using the rational model for the transfer function of some linear time-invariant system [13], [14]. However, when the wavelets are …