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