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 Simulation Example Comparison with conventional methods Conclusions

Introduction to wavelets A wavelet is a waveform of effectively limited duration that has an average value of zero

Wavelet Analysis Comparing wavelet analysis to Fourier analysis Introduction to wavelets

Continues Wavelet Transform (CWT) Wavelet Transform Discrete Wavelet Transform (DWT) Introduction to wavelets `

Continues Wavelet Transform 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. Introduction to wavelets

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

4- Scale (stretch) the wavelet and repeat steps 1 through Repeat steps 1 through 4 for all scales. Introduction to wavelets

Results Introduction 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 wavelets

An Example from Nature: Lunar Surface Introduction to wavelets

Discrete Wavelet Transform Approximations and Details One Stage Filtering Problem: Increasing data volume Introduction to wavelets

Filtering with down sampling Introduction to wavelets

Multi Stage Decomposition Introduction to wavelets

Different Mother wavelets Introduction to wavelets HaarMexican hat PDF’s Derivative Morlet MayerSymletCoifletDaubechies

1) Detecting Discontinuities and Breakdown Points Freqbrk.mat db5 level 5 Introduction to wavelets General Applications for wavelets

2) Detecting Long-Term Evolution Cnoislop.mat db3 level 6 Introduction to wavelets General Applications for wavelets

3) Detecting Self-Similarity vonkoch.mat coif3 continues Introduction to wavelets General Applications for wavelets

4) Identifying Pure Frequencies sumsin.mat db3 level 5 Introduction to wavelets General Applications for wavelets 2 Hz 200 Hz 20 Hz

5) De-Noising Signals noisdopp.mat sym4 level 5 Problem: Loss of Data Introduction to wavelets General Applications for wavelets

Solution: Special Algorithms 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 Introduction to wavelets General 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 wavelets Wavelets in system identification

System Identification Methods: Parametric Non parametric 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. 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 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 Introduction to wavelets Wavelets in system identification

Dyadic Sampling: compression/dilation in the DPWT is by a power of two with Introduction to wavelets Wavelets in system identification

DPWTs are calculated from Analysis equation For orthogonal wavelets An interesting observation Introduction to wavelets Wavelets in system identification

For a source-filter model Introduction to wavelets Wavelets in system identification

Using orthogonality property Introduction to wavelets Wavelets in system identification

It is proved that k=0 is the best choice to prevent aliasing without wasting resources Introduction to wavelets Wavelets in system identification

Discrete time signals Discrete Wavelet Transform (DWT) Introduction to wavelets Wavelets in system identification

System identification using DWT Introduction to wavelets Wavelets in system identification x[n] excitation y[n]=h[n]*x[n] System under test D W T h estimated [n]

i) Choice of excitation System under test: Chebyshev,IIR,10 th order high pass filter with 20db ripple Excitations: Introduction to wavelets Simulation Example

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

Results of changing the coefficients number for Daubeshies Introduction to wavelets Simulation Example

ii) Different Systems wavelet used as excitation and analysing function: Daubechies D4 Introduction to wavelets Simulation Example

System 1: FIR band-stop filter (a) Frequency response (b) Error variation with frequency Introduction to wavelets Simulation Example

System 2: Butterworth IIR, 10 th order Band-stop (a) Frequency response (b) Error variation with frequency Introduction to wavelets Simulation Example

System 3: Chebyshev IIR, 10 th order Band-stop (a) Frequency response (b) Error variation with frequency Introduction to wavelets Simulation Example

System 4: Elliptic IIR, 10 th order Band-stop (a) Frequency response (b) Error variation with frequency Introduction to wavelets Simulation Example

1)Chirp method System under test: Chebyshev high-pass filter Introduction to wavelets Comparison with conventional methods

2) Time domain recursion Introduction to wavelets Comparison with conventional methods System under test: Chebyshev high-pass filter

3) Inverse filtering Introduction to wavelets Comparison with conventional methods System under test: Chebyshev high-pass filter

4) Coherence Introduction 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 wavelets Conclusions

1- R.W.-P. Luk a, R.I. Damper b, “Non-parametric linear time-invariant system identification by discrete wavelet transforms”, Elsevier Inc, M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, “Wavelet Toolbox for use with matlab” Mathworks Inc., کاشانی، حامد، ” کاربرد موجک در شناسايي سيستم“؛ سمينار درس مدلسازی،1383 Introduction to wavelets Refrence

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