The Wavelet Transform ✧ 1 / 35 In­Class Assignments 1. With N being the length of your data, compute fs=morsespace(3,beta,N); for beta=3 followed by psi=morsewave(N,3,beta,fs); and then w=wavetrans(x,{3,beta,fs});. Plot this using wavespecplot(t,x,2*pi./fs,w); where t is the time axis and x is your data. 2. Try different colormaps including colormap lansey. 3. Repeat step #1 with larger value beta=10, or other values of your choice. beta should be greater than or equal to 1/2. 4. The y-axis here is in periods. Make sure that you have it in physical units, e.g. days or hours. 5. Repeat steps #1 and #3 but with the time series being an array of zeros of the same length as your data, with a delta function (that is, a one) in the center. 6. Do the same but for a time series consisting of a cosine of your choice. Mark the period of the cosine as a horizontal line in your wavelet transform. Does it match what you see in the transform? ✧ 2 / 35 Bandpass Filtering Previously we looked at the convolution theorem: ~ ∞ 1 ∞ f (t) = f (τ)g(t − τ)dτ = F(ω)G(ω) eiωtdω. ∫−∞ 2π ∫−∞ If g(t) is smooth, the convolution creates a smoothed version of f (t). g(t) G(ω) If has a Fourier transfor~m is localized in the vicinity of a certain frequency ωo, then f (t) will also be localized around ωo. In this case the convolution becomes a bandpass filter. ✧ 3 / 35 Bandpass Filtering Previously we looked at the convolution theorem: ~ ∞ 1 ∞ f (t) = f (τ)g(t − τ)dτ = F(ω)G(ω) eiωtdω. ∫−∞ 2π ∫−∞ If g(t) is smooth, the convolution creates a smoothed version of f (t). g(t) G(ω) If has a Fourier transfor~m is localized in the vicinity of a certain frequency ωo, then f (t) will also be localized around ωo. In this case the convolution becomes a bandpass filter. What is a good example of a function g(t) whose Fourier transform G(ω) is localized around zero? ✧ 3 / 35 Bandpass Filtering Previously we looked at the convolution theorem: ~ ∞ 1 ∞ f (t) = f (τ)g(t − τ)dτ = F(ω)G(ω) eiωtdω. ∫−∞ 2π ∫−∞ If g(t) is smooth, the convolution creates a smoothed version of f (t). g(t) G(ω) If has a Fourier transfor~m is localized in the vicinity of a certain frequency ωo, then f (t) will also be localized around ωo. In this case the convolution becomes a bandpass filter. What is a good example of a function g(t) whose Fourier transform G(ω) is localized around zero? What is a good example of a function g(t) whose Fourier transform G(ω) is localized around ωo? ✧ 3 / 35 Bandpass Filtering Previously we looked at the convolution theorem: ~ ∞ 1 ∞ f (t) = f (τ)g(t − τ)dτ = F(ω)G(ω) eiωtdω. ∫−∞ 2π ∫−∞ If g(t) is smooth, the convolution creates a smoothed version of f (t). g(t) G(ω) If has a Fourier transfor~m is localized in the vicinity of a certain frequency ωo, then f (t) will also be localized around ωo. In this case the convolution becomes a bandpass filter. What is a good example of a function g(t) whose Fourier transform G(ω) is localized around zero? What is a good example of a function g(t) whose Fourier transform G(ω) is localized around ωo? iω t g(t)e o ⟺ G(ω − ωo) ✧ 3 / 35 A Modulated Gaussian The function g(t)eiωot where g(t) is a Gaussian. ✧ 4 / 35 One­Sided Bandpass If we create a bandpass filter, using the shift theorem, by modulating some function g(t) whose Fourier transform is localized around zero, we obtain the new function ~ iω t g(t) ≡ g(t)e o ⟺ G(ω − ωo). ~ The bandpass filter g(t) will localize in the vicinity of frequency ωo. Thus, if ωo is chosen to be positive (and sufficiently large), we will isolate a band of positive frequencies, and if ωo is chosen to be negative, we will isolate a band of negative frequencies. This is called a one­sided bandpass. An example of a two-sided bandpass would be g(t) cos(ωot) ⟺ [G(ω − ωo) + G(ω + ωo)]/2. Note that a one-sided bandpass necessarily makes a real-valued signal complex. This turns out to be desireable, as it interfaces with an important theory called the analytic signal method. ✧ 5 / 35 A Wavelet Transform ✧ 6 / 35 The Basic Idea The wavelet transform is essentially a stack (or set) of one-sided bandpasses at different frequencies. We bandpass the time series at many different choices of the center of the band, then we look at these all together. We will work with a particular type of wavelet that is strictly one- sided, that is, having no energy at negative frequencies. Such a wavelet is said to be analytic. It is built up entirely from contributions due to positively-rotating Fourier components. To isolate the negatively-rotating variability we use the complex conjugate of the analytic wavelet. This filter has no support at positive frequencies as is said to be anti­analytic. ✧ 7 / 35 The Wavelet Transform For analytic wavelets, the wavelet transform can be written as ∞ 1 τ − t w(t, s) = ψ ( ) z(τ)dτ. ∫−∞ s s This describes the act of convolving the data with a filter ψ(t) that is rescaled by a scaling parameter s. This states that when we choose s > 1, thus making ψ broader in time, we also proportionately decrease its magnitude. ✧ 8 / 35 The Wavelet Transform For analytic wavelets, the wavelet transform can be written as ∞ 1 τ − t w(t, s) = ψ ( ) z(τ)dτ. ∫−∞ s s This describes the act of convolving the data with a filter ψ(t) that is rescaled by a scaling parameter s. This states that when we choose s > 1, thus making ψ broader in time, we also proportionately decrease its magnitude. Compare the wavelet transform definition with the convolution ∞ (f ∗ g)(t) ≡ f (τ)g(t − τ)dτ. ∫−∞ What are the differences? ✧ 8 / 35 Wavelet vs. Convolution Comparison with the convolution shows that we are actually convolving with the time­reversed wavelet, leading the time argument of the convolution is reversed from its usual appearance. The reason for this is that the wavelet transform is usually expressed in terms of a very close relative of the convolution, the cross­correlation. The cross-correleation theorem, a variant of the convolution theorem, is ∞ 1 ∞ h(t) = f (τ)g(τ − t)dτ = F(ω)G∗(ω) eiωtdω ∫−∞ 2π ∫−∞ which involves a conjugation in the frequency domain. These are not substantially different, but unlike the convolution operator, the cross-correleation operator does depend on the order. Note that the term ‘cross-correlation’ also has other meanings! ✧ 9 / 35 The Wavelet Transform Q: What is happening in the frequency domain? ✧ 10 / 35 The Wavelet Transform Q: What is happening in the frequency domain? A convolution in time is a multiplication in frequency. And, we have also learned the scaling thereom that tells us how things scale in one domain when you stretch or compress them in the other. Putting these together we have: ∞ ∞ 1 τ − t 1 ∗ iωt w(t, s) = ψ ( ) z(τ)dτ = Ψ (sω)Z(ω)e dω ∫−∞ s s 2π ∫−∞ for the frequency-domain form of the wavelet transform. Thus, if Ψ(ω) is localized around a particular frequency, then Ψ(sω) is localized around a different frequency. For this reason we can think of the wavelet transform as a set of bandpasses. For each s value, the wavelet transform w(t, s) will be localized around a different positive frequency. ✧ 10 / 35 The Role of Scale s ✧ 1 The time-domain wavelet s ψ(t/s) for different choices of s. 11 / 35 The Role of Scale s The frequency-domain wavelet Ψ(sω) for different choices of s. ✧ 12 / 35 Examples of Wavelets A broad family of wavelets in the time domain. ✧ 13 / 35 Examples of Wavelets The same family of wavelets in the frequency domain. ✧ 14 / 35 That's Pretty Much It So the wavelet transform is simply a stack of one-sided bandpasses. This means the magnitude-squared of the wavelets transform, 2 |w(t, s)| , describes the variability in the vicinity of a certain time t and also a certain period of oscillation, set by the scale s. It is convenient to convert the scale s into an equivalent frequency, the scale frequency, which ends up being proportional to 1/s. The wavelet can be thought of as a definition of an oscillatory signal, and the magnitude of the wavelet transform as the magnitude of the local projection onto time and frequency shifted versions of that test signal. ✧ 15 / 35 Which Wavelet to Use? One stumbling block used to be determining which wavelet to use. “A practical question is which wavelet to use for a particular application. In addition to the popular but only approximately analytic Morlet wavelet, a variety of analytic wavelets have been proposed, including the Cauchy-Klauder-Morse-Paul, Derivative of Gaussian, lognormal or log Gabor, Shannon, and Bessel wavelets. However, this profusion of wavelet types serves as a barrier to acquiring the knowledge necessary to carry out practical applications. In the absence of a suitable unifying theory for wavelet behaviors, the choice of a particular wavelet for a particular problem may even appear arbitrary.