in the name of god the compassionate the merciful
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In The Name of God The Compassionate The Merciful. Wavelet Based Methods for System Identification. Nafise Erfanian Saeedi. Presentation Agenda. Introduction to wavelets General applications for wavelets Application of wavelets in system identification - PowerPoint PPT PresentationTRANSCRIPT
Presentation Agenda
Introduction to wavelets General applications for wavelets Application of wavelets in system
identificationSimulation Example Comparison with conventional methodsConclusions
Introduction to wavelets
A wavelet is a waveform of effectively limited duration that has an average value of zero
Continues Wavelet Transform (CWT)
Wavelet Transform
Discrete Wavelet Transform (DWT)
Introduction to waveletsIntroduction to wavelets
`
Five Steps to CWT1- Take a wavelet and compare it to a section at the start of the
original signal.
2- Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. Note that the results will depend on the shape of the wavelet you choose.
Introduction to waveletsIntroduction to wavelets
3- Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.
Introduction to waveletsIntroduction to wavelets
4- Scale (stretch) the wavelet and repeat steps 1 through 3.
5- Repeat steps 1 through 4 for all scales.
Introduction to waveletsIntroduction to wavelets
Results
Introduction to waveletsIntroduction to wavelets
Time
Scale
Small Coefficients
Large Coefficients
Low scale >> Compressed wavelet >> Rapidly changing details >> High frequency
High scale >> Stretched wavelet >> Slowly changing, coarse features >> Low frequency
Introduction to waveletsIntroduction to wavelets
Discrete Wavelet Transform
Approximations and Details
One Stage Filtering
Problem: Increasing data volume
Introduction to waveletsIntroduction to wavelets
Different Mother wavelets
Introduction to waveletsIntroduction to wavelets
Haar Mexican hat PDF’s Derivative Morlet
Mayer Symlet Coiflet Daubechies
1) Detecting Discontinuities and Breakdown Points
Freqbrk.mat
db5 level 5
Introduction to waveletsGeneral Applications for wavelets
2) Detecting Long-Term Evolution
Cnoislop.mat
db3 level 6
Introduction to waveletsGeneral Applications for wavelets
3) Detecting Self-Similarity
vonkoch.mat
coif3 continues
Introduction to waveletsGeneral Applications for wavelets
4) Identifying Pure Frequencies
sumsin.mat
db3 level 5
Introduction to waveletsGeneral Applications for wavelets
2 Hz
200 Hz
20 Hz
5) De-Noising Signals
noisdopp.mat
sym4 level 5
Problem: Loss of Data
Introduction to waveletsGeneral Applications for wavelets
Other Applications:
• Biology for cell membrane recognition, to distinguish the normal from the pathological membranes
• Metallurgy for the characterization of rough surfaces
• Finance (which is more surprising), for detecting the properties of quick variation of values
• Detection of short pathological events as epileptic crises or normal ones as evoked potentials in EEG (medicine)
• Study of short-time phenomena as transient processes
• Automatic target recognition
Introduction to waveletsGeneral Applications for wavelets
Here, we consider wavelet approaches to
analyze signals that are a (linearly) filtered version of some source signal with the purpose
of identifying the characteristics
of the filtering system.
Introduction to waveletsWavelets in system identification
System Identification Methods:
Parametric
Non parametric
Introduction to waveletsWavelets in system identification
Solution one:For a causal system
Problem: Round-off errors accumulate with larger time indices, making this approach impractical for slowly decaying
(i.e., infinite) impulse response functions.
Introduction to waveletsWavelets in system identification
Solution two:Frequency-domain methods for linear systems based on
coherence Analysis
Usually with pseudorandom noise as input
Introduction to waveletsWavelets in system identification
Wavelet representation of signalsFor a finite energy signal:
discrete parameter
wavelet transform (DPWT)
analyzing functions
scale index k
translation index m
Introduction to waveletsWavelets in system identification
Dyadic Sampling:compression/dilation in the DPWT is by a power of two
with
Introduction to waveletsWavelets in system identification
DPWTs are calculated from Analysis equation
For orthogonal wavelets
An interesting observation
Introduction to waveletsWavelets in system identification
It is proved that k=0 is the best choice to prevent aliasing without wasting resources
Introduction to waveletsWavelets in system identification
Discrete time signalsDiscrete Wavelet Transform (DWT)
Introduction to waveletsWavelets in system identification
System identification using DWT
Introduction to waveletsWavelets in system identification
x[n]
excitation
y[n]=h[n]*x[n]System
under testD W T
hestimated[n]
i) Choice of excitation
System under test: Chebyshev,IIR,10th order high pass filter
with 20db ripple
Excitations:
Introduction to waveletsSimulation Example
Results for different excitations
Introduction to waveletsSimulation Example
Haar and Daubechies excitations give very good identification
Results of changing the coefficients number for Daubeshies
Introduction to waveletsSimulation Example
ii) Different Systems
wavelet used as excitation and analysing function:
Daubechies D4
Introduction to waveletsSimulation Example
System 1:FIR band-stop filter
(a) Frequency response
(b) Error variation with
frequency
Introduction to waveletsSimulation Example
System 2:Butterworth IIR,
10th order
Band-stop
(a) Frequency response
(b) Error variation with
frequency
Introduction to waveletsSimulation Example
System 3:Chebyshev IIR,
10th order
Band-stop
(a) Frequency response
(b) Error variation with
frequency
Introduction to waveletsSimulation Example
System 4:Elliptic IIR,
10th order
Band-stop
(a) Frequency response
(b) Error variation with
frequency
Introduction to waveletsSimulation Example
1) Chirp method System under test:
Chebyshev high-pass filter
Introduction to wavelets Comparison with conventional methods
2) Time domain recursionIntroduction to wavelets Comparison with conventional methods
System under test: Chebyshev high-pass filter
3) Inverse filteringIntroduction to wavelets Comparison with conventional methods
System under test: Chebyshev high-pass filter
4) CoherenceIntroduction to wavelets Comparison with conventional methods
System under test: Chebyshev high-pass filter
A new method for non-parametric linear time-invariant system identification based on the discrete wavelet transform (DWT)
is developed.
Identification is achieved using a test excitation to the system under test, that also acts as the analyzing function for the
DWT of the system’s output.
The new wavelet-based method proved to be considerably better than the conventional methods in all cases.
Introduction to waveletsConclusions
1- R.W.-P. Luk a, R.I. Damper b, “Non-parametric linear time-invariant system identification by discrete wavelet transforms”, Elsevier Inc,2005
2- M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, “Wavelet Toolbox for use with matlab” Mathworks Inc., 1996.
حامد،- 3 “کاشانی، ؛” سيستم شناسايي در موجک مدلسازی، کاربرد درس 1383سمينار
Introduction to waveletsRefrence