Hi all, I'm having some troubles implementing the Continuous Wavelet Transform (CWT) using the Fast Fourier Transform (FFT). I understand that the idea to do this is to first define psi~(a, t) = (1 / sqrt(a)) * conj(psi(-t / a)) where conj denotes a complex conjugate and psi is the original wavelet function. Now the CWT can be expressed as the convolution of the original signal and psi~. Then I can calculate the convolution using the convolution theorem and FFT. So the algorithm should like for i = 1:numscales % take next scale a = scales(i); % calculate psi the but how? % convolution theorem X(i, :) = ifft(fft(signal) .* fft(psi)); end; The problem I'm having is how to calculate the values for the wavelet function psi. Suppose I want to use the Morlet wavelet, psi = exp(-t^2/2) * cos(5*t), which is defined in the interval -4 <= t <= 4. The first step is to replace the t with (-t /a) as suggested by the definition of psi~, so the equation becomes psi = exp(-(-t/a)^2/2) * cos(5*(-t/a)). This is as far as I get. If I do something like t = linspace(-4, 4, 4096) to calculate values of psi I'm getting the wrong results. How do I calculate the psi to get the code working?

# How to implement Continuous Wavelet Transform (CWT) using FFT

Started by ●February 19, 2012

Reply by ●February 20, 20122012-02-20

On Feb 19, 12:47�pm, "tt-" <tuomas.p@n_o_s_p_a_m.gmail.com> wrote:> Hi all, > > I'm having some troubles implementing the Continuous Wavelet Transform > (CWT) > using the Fast Fourier Transform (FFT). > > I understand that the idea to do this is to first define > > � psi~(a, t) = (1 / sqrt(a)) * conj(psi(-t / a)) > > where conj denotes a complex conjugate and psi is the original wavelet > function. Now the CWT can be expressed as the convolution of the original > signal and psi~. Then I can calculate the convolution using the convolution > theorem and FFT. > > So the algorithm should like > > �for i = 1:numscales > � � �% take next scale > � � �a = scales(i); > > � � �% calculate psi the but how? > > � � �% convolution theorem > � � �X(i, :) = ifft(fft(signal) .* fft(psi)); > �end; > > The problem I'm having is how to calculate the values for the wavelet > function psi. > > Suppose I want to use the Morlet wavelet, psi = exp(-t^2/2) * cos(5*t), > which is defined in the interval -4 <= t <= 4. > > The first step is to replace the t with (-t /a) as suggested by the > definition of psi~, so the equation becomes psi = exp(-(-t/a)^2/2) * > cos(5*(-t/a)). > > This is as far as I get. If I do something like > �t = linspace(-4, 4, 4096) > to calculate values of psi I'm getting the wrong results. > > How do I calculate the psi to get the code working?Are you sure the convolution integral and the continuous wavelet integral are of the same form?

Reply by ●February 20, 20122012-02-20

Yes, I'm quite sure. I used the material available at http://www.mathworks.se/help/toolbox/wavelet/gs/f3-1000759.html#bsu1pdf for reference. So this is probably the same way it is implemented in Matlab.>Are you sure the convolution integral and the continuous wavelet >integral are of the same form?

Reply by ●February 20, 20122012-02-20

On Feb 20, 1:30�pm, "tt-" <tuomas.p@n_o_s_p_a_m.gmail.com> wrote:> Yes, I'm quite sure. I used the material available at > http://www.mathworks.se/help/toolbox/wavelet/gs/f3-1000759.html#bsu1pdf > for reference. So this is probably the same way it is implemented inHave you tried using the psi that Matlab uses? http://www.mathworks.com/help/toolbox/wavelet/ref/cwtftinfo.html Dale B, Dalrymple