Time – frequency resolution problem Concepts of scale and translation

1. 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

2. 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?

3. 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

4. 1:44,500,000 1:2,500,000 1:375,500 1:62,500

5. 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

6. 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

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

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

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

10. 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

11. 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.

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

13. 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)

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

15. Inverse CWT provided that

16. Properties of Continuous Wavelet Transform • Linearity • Translation • Scaling • Wavelet shifting • Wavelet scaling • Linear combination of wavelets

17. Example

18. Example

19. Example