In The Name of God The Compassionate The Merciful

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In The Name of God The Compassionate
The Merciful
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Wavelet Based Methodsfor System Identification

• Nafise Erfanian Saeedi

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Presentation Agenda

• Introduction to wavelets
• General applications for wavelets
• Application of wavelets in system identification
• Simulation Example
• Comparison with conventional methods
• Conclusions

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Introduction to wavelets

• A wavelet is a waveform of effectively limited
  duration that has an average value of zero

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Introduction to wavelets

• Wavelet Analysis
• Comparing wavelet analysis to Fourier analysis

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Introduction to wavelets
Introduction to wavelets

• Continues Wavelet Transform (CWT)
• Wavelet Transform
• Discrete Wavelet Transform (DWT)

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Introduction to wavelets
Introduction to wavelets

• Continues Wavelet Transform

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Introduction to wavelets
Introduction to wavelets

• Five Steps to CWT
• 1- 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.

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Introduction to wavelets
Introduction to wavelets

• 3- Shift the wavelet to the right and repeat
  steps 1 and 2 until you've covered the whole
  signal.

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Introduction to wavelets
Introduction to wavelets

• 4- Scale (stretch) the wavelet and repeat steps 1
  through 3.
• 5- Repeat steps 1 through 4 for all scales.

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Introduction to wavelets
Introduction to wavelets

• Results

Large Coefficients
Scale
Small Coefficients
Time
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Introduction to wavelets
Introduction to wavelets

• Low scale gtgt Compressed wavelet gtgt Rapidly
  changing details gtgt High frequency
• High scale gtgt Stretched wavelet gtgt Slowly
  changing, coarse features gtgt Low frequency

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Introduction to wavelets
Introduction to wavelets

• An Example from Nature Lunar Surface

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Introduction to wavelets
Introduction to wavelets

• Discrete Wavelet Transform
• Approximations and Details
• One Stage Filtering
• Problem Increasing data volume

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Introduction to wavelets
Introduction to wavelets

• Filtering with down sampling

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Introduction to wavelets
Introduction to wavelets

• Multi Stage Decomposition

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Introduction to wavelets
Introduction to wavelets

• Different Mother wavelets

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Introduction to wavelets
General Applications for wavelets

• 1) Detecting Discontinuities and Breakdown Points
• Freqbrk.mat
• db5 level 5

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Introduction to wavelets
General Applications for wavelets

• 2) Detecting Long-Term Evolution
• Cnoislop.mat
• db3 level 6

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Introduction to wavelets
General Applications for wavelets

• 3) Detecting Self-Similarity
• vonkoch.mat
• coif3 continues

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Introduction to wavelets
General Applications for wavelets

• 4) Identifying Pure Frequencies
• sumsin.mat
• db3 level 5

2 Hz
20 Hz
200 Hz
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Introduction to wavelets
General Applications for wavelets

• 5) De-Noising Signals
• noisdopp.mat
• sym4 level 5
• Problem Loss of Data

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Introduction to wavelets
General Applications for wavelets

• Solution Special Algorithms

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Introduction to wavelets
General 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

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Introduction to wavelets
Wavelets in system identification

• 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.

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Introduction to wavelets
Wavelets in system identification

• System Identification Methods
• Parametric
• Non parametric

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Introduction to wavelets
Wavelets 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.

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Introduction to wavelets
Wavelets in system identification

• Solution two
• Frequency-domain methods for linear systems based
  on coherence Analysis
• Usually with pseudorandom noise as input

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Introduction to wavelets
Wavelets in system identification

• Wavelet representation of signals
• For a finite energy signal
• discrete parameter
• wavelet transform (DPWT)
• analyzing functions
• scale index k
• translation index m

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Introduction to wavelets
Wavelets in system identification

• Dyadic Sampling
• compression/dilation in the DPWT is by a power of
  two
• with

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Introduction to wavelets
Wavelets in system identification

• DPWTs are calculated from Analysis equation
• For orthogonal wavelets
• An interesting observation

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Introduction to wavelets
Wavelets in system identification

• For a source-filter model

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Introduction to wavelets
Wavelets in system identification

• Using orthogonality property

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Introduction to wavelets
Wavelets in system identification

• It is proved that k0 is the best choice to
  prevent aliasing without wasting resources

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Introduction to wavelets
Wavelets in system identification

• Discrete time signals
• Discrete Wavelet Transform (DWT)

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Introduction to wavelets
Wavelets in system identification

• System identification using DWT

ynhnxn
hestimatedn
xn excitation
System under test
D W T
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Introduction to wavelets
Simulation Example

• i) Choice of excitation
• System under test
• Chebyshev,IIR,10th order high pass filter
• with 20db ripple
• Excitations

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Introduction to wavelets
Simulation Example

• Results for different excitations

Haar and Daubechies excitations give very good
identification
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Introduction to wavelets
Simulation Example

• Results of changing the coefficients number for
  Daubeshies

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Introduction to wavelets
Simulation Example

• ii) Different Systems
• wavelet used as excitation and analysing
  function
• Daubechies D4

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Introduction to wavelets
Simulation Example

• System 1
• FIR band-stop filter
• (a) Frequency response
• (b) Error variation with
• frequency

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Introduction to wavelets
Simulation Example

• System 2
• Butterworth IIR,
• 10th order
• Band-stop
• (a) Frequency response
• (b) Error variation with
• frequency

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Introduction to wavelets
Simulation Example

• System 3
• Chebyshev IIR,
• 10th order
• Band-stop
• (a) Frequency response
• (b) Error variation with
• frequency

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Introduction to wavelets
Simulation Example

• System 4
• Elliptic IIR,
• 10th order
• Band-stop
• (a) Frequency response
• (b) Error variation with
• frequency

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Introduction to wavelets
Comparison with conventional methods

• Chirp method
• System under test
• Chebyshev
• high-pass filter

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Introduction to wavelets
Comparison with conventional methods

• 2) Time domain recursion

System under test Chebyshev
high-pass filter
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Introduction to wavelets
Comparison with conventional methods

• 3) Inverse filtering

System under test Chebyshev
high-pass filter
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Introduction to wavelets
Comparison with conventional methods

• 4) Coherence

System under test Chebyshev
high-pass filter
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Introduction to wavelets
Conclusions

• 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
  systems output.
• The new wavelet-based method proved to be
  considerably better than the conventional methods
  in all cases.

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Introduction to wavelets
Refrence

• 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.
• 3- ??????? ????? ?????? ???? ?? ??????? ??????
  ?????? ??? ????????1383

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Thank youforYour Kind Attention