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Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon. Chapter 4: Time frequency and wavelet analysis. Definition Time frequency Shift time fourier transform Winer-ville representations and others Wavelet transform Scalogram Continuous wavelet transform

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Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

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  1. Advanced signal processing Dr. Mohamad KAHLIL Islamic University of Lebanon

  2. Chapter 4: Time frequency and wavelet analysis • Definition • Time frequency • Shift time fourier transform • Winer-ville representations and others • Wavelet transform • Scalogram • Continuous wavelet transform • Discrete wavelet transform: details and approximations • applications

  3. The Story of WaveletsTheory and Engineering Applications • Time frequency representation • Instantaneous frequency and group delay • Short time Fourier transform –Analysis • Short time Fourier transform – Synthesis • Discrete time STFT

  4. Signal processing Signal Processing Freq.-domain techniques Time-domain techniques TF domain techniques Fourier T. Filters Stationary Signals Non/Stationary Signals STFT WAVELET TRANSFORMS Applications Denoising Compression Signal Analysis Disc. Detection BME / NDE Other… 2-D DWT SWT CWT DWT MRA

  5. FT At Work

  6. FT At Work F F F

  7. FT At Work F

  8. Stationary and Non-stationary Signals • FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. Why? • Stationary signals consist of spectral components that do not change in time • all spectral components exist at all times • no need to know any time information • FT works well for stationary signals • However, non-stationary signals consists of time varying spectral components • How do we find out which spectral component appears when? • FT only provides what spectral components exist , not where in time they are located. • Need some other ways to determine time localization of spectral components

  9. Stationary and Non-stationary Signals • Stationary signals’ spectral characteristics do not change with time • Non-stationary signals have time varying spectra Concatenation

  10. Non-stationary Signals 50 Hz 20 Hz 5 Hz Perfect knowledge of what frequencies exist, but no information about where these frequencies are located in time

  11. FT Shortcomings • Complex exponentials stretch out to infinity in time • They analyze the signal globally, not locally • Hence, FT can only tell what frequencies exist in the entire signal, but cannot tell, at what time instances these frequencies occur • In order to obtain time localization of the spectral components, the signal need to be analyzed locally • HOW ?

  12. Short Time Fourier Transform(STFT) • Choose a window function of finite length • Put the window on top of the signal at t=0 • Truncate the signal using this window • Compute the FT of the truncated signal, save. • Slide the window to the right by a small amount • Go to step 3, until window reaches the end of the signal • For each time location where the window is centered, we obtain a different FT • Hence, each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information

  13. STFT

  14. STFT Frequency parameter Time parameter Signal to be analyzed FT Kernel (basis function) STFT of signal x(t): Computed for each window centered at t=t’ Windowing function Windowing function centered at t=t’

  15. STFT at Work Windowed sinusoid allows FT to be computed only through the support of the windowing function 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0 100 200 300 0 100 200 300 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0 100 200 300 0 100 200 300

  16. STFT • STFT provides the time information by computing a different FTs for consecutive time intervals, and then putting them together • Time-Frequency Representation (TFR) • Maps 1-D time domain signals to 2-D time-frequency signals • Consecutive time intervals of the signal are obtained by truncating the signal using a sliding windowing function • How to choose the windowing function? • What shape? Rectangular, Gaussian, Elliptic…? • How wide? • Wider window require less time steps  low time resolution • Also, window should be narrow enough to make sure that the portion of the signal falling within the window is stationary • Can we choose an arbitrarily narrow window…?

  17. Selection of STFT Window Two extreme cases: • W(t) infinitely long:  STFT turns into FT, providing excellent frequency information (good frequency resolution), but no time information • W(t) infinitely short:  STFT then gives the time signal back, with a phase factor. Excellent time information (good time resolution), but no frequency information Wide analysis window poor time resolution, good frequency resolution Narrow analysis windowgood time resolution, poor frequency resolution Once the window is chosen, the resolution is set for both time and frequency.

  18. Heisenberg Principle Frequency resolution: How well two spectral components can be separated from each other in the transform domain Time resolution: How well two spikes in time can be separated from each other in the transform domain Both time and frequency resolutions cannot be arbitrarily high!!! We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals http://engineering.rowan.edu/~polikar/WAVELETS/WTpart2.html

  19. STFT Amplitude ….. ….. time t0 t1 tk tk+1 tn ….. ….. Frequency

  20. The Short Time Fourier Transform • Take FT of segmented consecutive pieces of a signal. • Each FT then provides the spectral content of that time segment only • Spectral content for different time intervals • Time-frequency representation Time parameter Signal to be analyzed FT Kernel (basis function) Frequency parameter STFT of signal x(t): Computed for each window centered at t= (localized spectrum) Windowing function (Analysis window) Windowing function centered at t=

  21. Properties of STFT • Linear • Complex valued • Time invariant • Time shift • Frequency shift • Many other properties of the FT also apply.

  22. Alternate Representation of STFT STFT : The inverse FT of the windowed spectrum, with a phase factor

  23. Filter Interpretation of STFT X(t) is passed through a bandpass filter with a center frequency of Note that (f) itself is a lowpass filter.

  24. x(t) X Filter Interpretation of STFT X x(t)

  25. Resolution Issues All signal attributes located within the local window interval around “t” will appear at “t” in the STFT Amplitude time n k Frequency

  26. Time-Frequency Resolution • Closely related to the choice of analysis window • Narrow window  good time resolution • Wide window (narrow band)  good frequency resolution • Two extreme cases: • (T)=(t) excellent time resolution, no frequency resolution • (T)=1 excellent freq. resolution (FT), no time info!!! • How to choose the window length? • Window length defines the time and frequency resolutions • Heisenberg’s inequality • Cannot have arbitrarily good time and frequency resolutions. One must trade one for the other. Their product is bounded from below.

  27. Time-Frequency Resolution Frequency Time

  28. Time Frequency Signal Expansion and STFT Synthesis Basis functions Coefficients (weights) Synthesis window Synthesized signal • Each (2D) point on the STFT plane shows how strongly a time • frequency point (t,f) contributes to the signal. • Typically, analysis and synthesis windows are chosen to be identical.

  29. STFT Example 300 Hz 200 Hz 100Hz 50Hz

  30. STFT Example

  31. STFT Example a=0.01

  32. STFT Example a=0.001

  33. STFT Example a=0.0001

  34. STFT Example a=0.00001

  35. Discrete Time Stft

  36. The Story of WaveletsTheory and Engineering Applications • Time – frequency resolution problem • Concepts of scale and translation • The mother of all oscillatory little basis functions… • The continuous wavelet transform • Filter interpretation of wavelet transform • Constant Q filters

  37. Time – Frequency Resolution • Time – frequency resolution problem with STFT • Analysis window dictates both time and frequency resolutions, once and for all • Narrow window  Good time resolution • Narrow band (wide window)  Good frequency resolution • When do we need good time resolution, when do we need good frequency resolution?

  38. Scale & Translation • Translation  time shift • f(t) f(a.t) a>0 • If 0<a<1 dilation, expansion  lower frequency • If a>1  contraction  higher frequency • f(t)f(t/a) a>0 • If 0<a<1  contraction  low scale (high frequency) • If a>1  dilation, expansion  large scale (lower frequency) • Scaling  Similar meaning of scale in maps • Large scale: Overall view, long term behavior • Small scale: Detail view, local behavior

  39. The Mother of All Oscillatory Little Basis Functions • The kernel functions used in Wavelet transform are all obtained from one prototype function, by scaling and translating the prototype function. • This prototype is called the mother wavelet Translation parameter Scale parameter Normalization factor to ensure that allwavelets have the same energy

  40. Continuous Wavelet Transform translation Mother wavelet Normalization factor Scaling: Changes the support of the wavelet based on the scale (frequency) CWT of x(t) at scale a and translation b Note: low scale  high frequency

  41. Computation of CWT Amplitude Amplitude bN b0 time b0 bN time Amplitude Amplitude bN b0 b0 bN time time

  42. WT at Work High frequency (small scale) Low frequency (large scale)

  43. Why Wavelet? • We require that the wavelet functions, at a minimum, satisfy the following: Wave… …let

  44. The CWT as a Correlation • Recall that in the L2 space an inner product is defined as then Cross correlation: then

  45. The CWT as a Correlation • Meaning of life: W(a,b) is the cross correlation of the signal x(t) with the mother wavelet at scale a, at the lag of b. If x(t) is similar to the mother wavelet at this scale and lag, then W(a,b) will be large. wavelets

  46. Filtering Interpretation of Wavelet Transform • Recall that for a given system h[n], y[n]=x[n]*h[n] • Observe that • Interpretation:For any given scale a (frequency ~ 1/a), the CWT W(a,b) is the output of the filter with the impulse response to the input x(b), i.e., we have a continuum of filters, parameterized by the scale factor a.

  47. What do Wavelets Look Like??? • Mexican Hat Wavelet • Haar Wavelet • Morlet Wavelet

  48. Constant Q Filtering • A special property of the filters defined by the mother wavelet is that they are –so called – constant Q filters. • Q Factor: • We observe that the filters defined by the mother wavelet increase their bandwidth, as the scale is reduced (center frequency is increased) w (rad/s)

  49. B B B B B B B Constant Q STFT f0 2f0 3f0 4f0 5f0 6f0 2B 4B 8B CWT f0 2f0 4f0 8f0

  50. Inverse CWT provided that

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