1. In The Name of God The Compassionate The Merciful

2. Wavelet Based Methods for System Identification Nafise Erfanian Saeedi

3. Presentation Agenda Introduction to wavelets General applications for wavelets Application of wavelets in system identification Simulation Example Comparison with conventional methods Conclusions

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

5. Wavelet Analysis Comparing wavelet analysis to Fourier analysis Introduction to wavelets

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

7. Continues Wavelet Transform Introduction to wavelets

8. 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

9. 3- Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal. Introduction to wavelets

10. 4- Scale (stretch) the wavelet and repeat steps 1 through 3. 5- Repeat steps 1 through 4 for all scales. Introduction to wavelets

11. Results Introduction to wavelets Time Scale Small Coefficients Large Coefficients

12. Low scale >> Compressed wavelet >> Rapidly changing details >> High frequency High scale >> Stretched wavelet >> Slowly changing, coarse features >> Low frequency Introduction to wavelets

13. An Example from Nature: Lunar Surface Introduction to wavelets

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

15. Filtering with down sampling Introduction to wavelets

16. Multi Stage Decomposition Introduction to wavelets

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

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

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

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

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

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

23. Solution: Special Algorithms Introduction to wavelets General Applications for wavelets

24. 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

25. 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

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

27. 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

28. 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

29. 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

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

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

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

33. Using orthogonality property Introduction to wavelets Wavelets in system identification

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

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

36. 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]

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

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

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

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

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

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

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

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

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

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

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

48. 4) Coherence Introduction to wavelets Comparison with conventional methods System under test: Chebyshev high-pass filter

49.  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

50. 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- کاشانی، حامد، ” کاربرد موجک در شناسايي سيستم“؛ سمينار درس مدلسازی،1383 Introduction to wavelets Refrence

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