Lecture 19 The Wavelet Transform
Lecture 19
The Wavelet Transform
Some signals obviously have spectral characteristics that vary with timeMotivation
Criticism of Fourier SpectrumIts giving you the spectrum of thewhole time-series
Which is OK if the time-series is stationaryBut what if its not?
We need a technique that can march along a timeseries and that is capable of:
Analyzing spectral content in different placesDetecting sharp changes in spectral character
Fourier Analysis is based on an indefinitely long cosine wave of a specific frequencyWavelet Analysis is based on an short duration wavelet of a specific center frequencytime, ttime, t
Wavelet TransformInverse Wavelet TransformAll wavelet derived from mother wavelet
Inverse Wavelet Transformwavelet withscale, s and time, ttime-seriescoefficientsof waveletsbuild up a time-series as sum of wavelets of different scales, s, and positions, t
Wavelet Transformcomplex conjugate of wavelet withscale, s and time, ttime-seriescoefficient of wavelet withscale, s and time, tIm going to ignore the complex conjugate from now on, assuming that were using real wavelets
Waveletchange in scale:big s means long wavelengthnormalizationwavelet withscale, s and time, tshift in timeMother wavelet
Shannon Wavelet
Y(t) = 2 sinc(2t) sinc(t)mother wavelett=5, s=2time
Fourier spectrum of Shannon Waveletfrequency, wSpectrum of higher scale waveletsw
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 workas a waveletadmissibility condition:Implies that Y(w)0 both as w0 and w, so Y(w) must be band-limited
a desirable property is g(s,t)0 as s0 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)sn+2. So some effort has been put into finding wavelets with high approximation order.
Discrete wavelets:choice of scale and sampling in timesj=2j
and
tj,k = 2jkDt
Then g(sj,tj,k) = gjk
where j = 1, 2, k = - -2, -1, 0, 1, 2, Scale changes by factors of 2Sampling widens by factor of 2 for each successive scale
dyadic grid
The factor of two scaling means that the spectra of the wavelets divide up the frequency scale into octaves (frequency doubling intervals)wnywwnywny1/8wny
As we showed previously, the coefficients of Y1 is just the band-passes filtered time-series, where Y1 is the wavelet, now viewed as a bandpass filter.
This suggests a recursion. Replace:
wnywwnywith
low-pass filter
And then repeat the processes, recursively
Chosing the low-pass filterIt turns out that its easy to pick the low-pass filter, flp(w). It must match wavelet filter, Y(w). A reasonable requirement is:
|flp(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:
YN-1-k = (-1)k flpk
Furthermore, its 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
time-series of length NHPLP22HPLP22HPLP22g(s1,t)g(s2,t)g(s3,t)Recursion for wavelet coefficientsg(s1,t): N/2 coefficientsg(s2,t): N/4 coefficientsg(s2,t): N/8 coefficientsTotal: N coefficients
Coiflet low pass filterFrom http://en.wikipedia.org/wiki/CoifletCoiflet high-pass filtertime, ttime, t
Spectrum of low pass filterfrequency, wSpectrum of waveletfrequency, w
stage 1 - hitime-seriesstage 1 - lo
stage 2 - hiStage 1 lostage 2 - lo
stage 3 - hiStage 2 lostage 3 - lo
stage 4 - hiStage 3 lostage 4 - lo
stage 5 - hiStage 4 lostage 6 - lo
stage 5 - hiStage 4 lostage 6 - loHad enough?
Putting it all together time, tscalelongwavelengthsshortwavelengths|g(sj,t)|2
stage 1 - hiLGA Temperature time-seriesstage 1 - lo
time, tscalelongwavelengthsshortwavelengths