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
C scale, position 


f t  scale, position, t dt

1
C a, b  
a
Wavelet Scale
coefficient
Shift



t b 
f t  
dt
 a 
Mother
wavelet
Scale and Position
Large scale
Low frequency
Small scale
High frequency
Scaling
Shifting
Some Mother Wavelets
Haar
Mayer
Mexican hat
PDF’s Derivative
Symlet
Coiflet
Morlet
Diubechies
Selection is usually based on input signal and application
Wavelet’s properties


 u du  0



 2 u du  1

C 



  

d
   0  0
Admissibility condition is necessary for reconstruction
Discrete wavelet
Dyadic discretisation
 j ,k t   2  2 t  k 

j
2
j
j, k  
Reconstruction
Continuous:
Discrete:
1
f t  
k
f t  





jZ
Partly:

D j t  
1  t  b  da.db
C a, b 

 2
a  a  a
C  j, k  j ,k t 
kZ

C  j, k  j ,k t 
kZ
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 structure
Algorithm and performance analysis
Selection of Model Structure
Representation of impulse response in transform domain
Input vector
Impulse response matrix
Output vector
Time shift matrix
Representation of impulse response in transform
domain
Based on input transformation (t is fixed)
Basis function
Parameters
Based on output transformation (t is fixed)
Based on input-output transformation
All relations can be represented in matrix form
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 11th 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., 1996.
D. B. Percival, A. T. Walden, “Wavelet Mathematics for Time Series
Analysis”, Cambridge University Press, 2000.
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.729-741, 1999.
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
585-592, 2001.
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 593-600, 2001.