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Lecture 19 The Wavelet Transform:

Lecture 19 The Wavelet Transform

Some signals obviously have spectral characteristics that vary with time:

Some signals obviously have spectral characteristics that vary with time Motivation

Criticism of Fourier Spectrum:

Criticism of Fourier Spectrum It’s giving you the spectrum of the ‘whole time-series’ Which is OK if the time-series is stationary But what if its not? We need a technique that can “march along” a timeseries and that is capable of: Analyzing spectral content in different places Detecting sharp changes in spectral character

Fourier Analysis is based on an indefinitely long cosine wave of a specific frequency:

Fourier Analysis is based on an indefinitely long cosine wave of a specific frequency Wavelet Analysis is based on an short duration wavelet of a specific center frequency time, t time, t

Wavelet Transform:

Wavelet Transform Inverse Wavelet Transform All wavelet derived from mother wavelet

Inverse Wavelet Transform:

Inverse Wavelet Transform wavelet with scale, s and time, t time-series coefficients of wavelets build up a time-series as sum of wavelets of different scales, s, and positions, t

Wavelet Transform:

Wavelet Transform complex conjugate of wavelet with scale, s and time, t time-series coefficient of wavelet with scale, s and time, t I’m going to ignore the complex conjugate from now on, assuming that we’re using real wavelets

Wavelet:

Wavelet change in scale: big s means long wavelength normalization wavelet with scale, s and time, t shift in time Mother wavelet

Shannon Wavelet Y(t) = 2 sinc(2t) – sinc(t):

Shannon Wavelet Y (t) = 2 sinc(2t) – sinc(t) mother wavelet t =5, s=2 time

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Fourier spectrum of Shannon Wavelet frequency, w Spectrum of higher scale wavelets w

Thus determining the wavelet coefficients at a fixed scale, s can be thought of as a filtering operation g(s,t) =  f(t) Y[(t-t)/s] dt = f(t) * Y(-t/s) where the filter Y(-t/s) is has a band-limited spectrum, so the filtering operation is a bandpass filter:

Thus determining the wavelet coefficients at a fixed scale, s can be thought of as a filtering operation g (s, t ) =  f(t) Y [(t- t )/s] dt = f( t ) * Y (- t /s) where the filter Y (- t /s) is has a band-limited spectrum, so the filtering operation is a bandpass filter

not any function, Y(t) will work as a wavelet:

not any function, Y (t) will work as a wavelet admissibility condition: Implies that Y ( w ) 0 both as w 0 and w  , so Y ( w ) must be band-limited

a desirable property is g(s,t)0 as s0 :

a desirable property is g (s, t ) 0 as s0 p-th moment of Y (t) Suppose the first n moments are zero (called the approximation order of the wavelet), then it can be shown that g (s, t )  s n+2 . So some effort has been put into finding wavelets with high approximation order.

Discrete wavelets: choice of scale and sampling in time:

Discrete wavelets: choice of scale and sampling in time s j =2 j and t j , k = 2 j k D t Then g ( s j ,t j,k ) = g jk where j = 1, 2, …  k = -… -2, -1, 0, 1, 2, … Scale changes by factors of 2 Sampling widens by factor of 2 for each successive scale

dyadic grid:

dyadic grid

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The factor of two scaling means that the spectra of the wavelets divide up the frequency scale into octaves (frequency doubling intervals) w ny w ½ w ny ¼ w ny 1 / 8 w ny

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As we showed previously, the coefficients of Y 1 is just the band-passes filtered time-series, where Y 1 is the wavelet, now viewed as a bandpass filter. This suggests a recursion. Replace: w ny w ½ w ny ¼ w ny 1 / 8 w ny w ny w ½ w ny with low-pass filter

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And then repeat the processes, recursively …

Chosing the low-pass filter:

Chosing the low-pass filter It turns out that its easy to pick the low-pass filter, f lp (w). It must match wavelet filter, Y ( w ). A reasonable requirement is: |f lp ( w )| 2 + | Y ( w )| 2 = 1 That is, the spectra of the two filters add up to unity. A pair of such filters are called Quadature Mirror Filters . They are known to have filter coefficients that satisfy the relationship: Y N-1-k = (-1) k f lp k Furthermore, it’s known that these filters allows perfect reconstruction of a time-series by summing its low-pass and high-pass versions

To implement the ever-widening time sampling tj,k = 2jkDt we merely subsample the time-series by a factor of two after each filtering operation:

To implement the ever-widening time sampling t j , k = 2 j k D t we merely subsample the time-series by a factor of two after each filtering operation

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time-series of length N HP LP  2  2 HP LP  2  2 HP LP  2  2 … g (s 1 ,t) g (s 2 ,t) g (s 3 ,t) Recursion for wavelet coefficients g (s 1 ,t): N/2 coefficients g (s 2 ,t): N/4 coefficients g (s 2 ,t): N/8 coefficients Total: N coefficients

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Coiflet low pass filter From http://en.wikipedia.org/wiki/Coiflet Coiflet high-pass filter time, t time, t

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Spectrum of low pass filter frequency, w Spectrum of wavelet frequency, w

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stage 1 - hi time-series stage 1 - lo

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stage 2 - hi Stage 1 lo stage 2 - lo

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stage 3 - hi Stage 2 lo stage 3 - lo

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stage 4 - hi Stage 3 lo stage 4 - lo

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stage 5 - hi Stage 4 lo stage 6 - lo

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stage 5 - hi Stage 4 lo stage 6 - lo Had enough?

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Putting it all together … time, t scale long wavelengths short wavelengths | g (s j ,t)| 2

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stage 1 - hi LGA Temperature time-series stage 1 - lo

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time, t scale long wavelengths short wavelengths

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