WAVELET AND IDENTIFICATION WAVELET AND IDENTIFICATION Hamed Kashani

Fourier Analysis Base functions: Sine and Cosine. Transfer signal from time domain to frequency domain. Useful if the frequency content is of great importance. Drawback: time information is lost.(drift, trends, abrupt changes, and beginnings and ends of events)

Short-time Fourier Analysis Maps a signal into a two-dimensional function of time and frequency. Drawback: Window is the same for all frequencies.

Continues Wavelet Transform Wavelet coefficient ScaleShift Mother wavelet

Scale and Position Scaling Shifting Large scale Low frequency Small scale High frequency

Some Mother Wavelets HaarMexican hat PDF’s Derivative Morlet MayerSymletCoifletDiubechies Selection is usually based on input signal and application

Wavelet’s properties Admissibility condition is necessary for reconstruction

Discrete wavelet Dyadic discretisation

Reconstruction Continuous: Discrete: Partly: Is useful for signal decomposition

Example Input Signal Haar matrix

Example (decomposition)

Example (reconstruction)

LTV System Identification Some of important LTV systems: Air conditioning and refrigeration systems Power systems under fluctuating load. Aircraft with TV angle of attack. Wireless communication channels in cellular phone applications. LTV System Identification Selection of model structureAlgorithm and performance analysis

Selection of Model Structure Representation of impulse response in transform domain Impulse response matrix Time shift matrix Input vector Output vector

Representation of impulse response in transform domain Based on input transformation (t is fixed) Based on input-output transformation Based on output transformation (  is fixed) All relations can be represented in matrix form Basis function Parameters

Identification Using the Input-Side Transformation System Representation Structure selection: Output vector Formation : Input matrix Formation : Biorthogonal approximation vector selection: Calculate N(t) = U(t)  Calculate estimated coefficients: Repeat above procedure for t+1

Some Definitions

Exp: Input-Side Transformation Based Ident. The time-domain representation of the time-varying system to be identified System changes linearly expect at 11 th step, where a step change which represents an abrupt occurs

Exp (following): Basis Selection Magnitude response of  1,  2,  3  1 : db4.  2 : Result of convolution of db4 LP and HP (up sampled by 2 ).  3 : Result of convolution of 2 db4 LP (up sampled by 2 ).

Exp (following): Results The input-side transformation based identification: U: the time-domain representation of the identification result L: the identification error

References M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, “Wavelet Toolbox for use with matlab” Mathworks Inc., D. B. Percival, A. T. Walden, “Wavelet Mathematics for Time Series Analysis”, Cambridge University Press, ‌H. Zaho, J. Bentsman, “Block Diagram Reduction of the Interconnected Linear Time-Varying Systems in the Time-Frequency Domain” Accepted for publication by multidimensional systems and signal processing, H. Rotstein, S. Raz, “Gobar Transformation of Time-Varying systems: exact representation and approximation”, IEEE Trans. Autom. Control, 44 No.4, Apr. pp , H. Zaho, J. Bentsman, “Bi-orthogonal Wavelet Based Identification of Fast Linear Time-Varying Systems part I: system representations”, Journal of Dynamic systems, Measurement and control, Vol.123, pp , H. Zaho, J. Bentsman, “Bi-orthogonal Wavelet Based Identification of Fast Linear Time-Varying Systems part II: algorithms and performance analysis”, Journal of Dynamic systems, Measurement and control, Vol.123, pp , 2001.