How to implement Continuous Wavelet Transform (CWT) using FFT

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?

On Feb 19, 12:47&#4294967295;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
>
> &#4294967295; 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
>
> &#4294967295;for i = 1:numscales
> &#4294967295; &#4294967295; &#4294967295;% take next scale
> &#4294967295; &#4294967295; &#4294967295;a = scales(i);
>
> &#4294967295; &#4294967295; &#4294967295;% calculate psi the but how?
>
> &#4294967295; &#4294967295; &#4294967295;% convolution theorem
> &#4294967295; &#4294967295; &#4294967295;X(i, :) = ifft(fft(signal) .* fft(psi));
> &#4294967295;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
> &#4294967295;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?

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?

On Feb 20, 1:30&#4294967295;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 in

Have you tried using the psi that Matlab uses?

http://www.mathworks.com/help/toolbox/wavelet/ref/cwtftinfo.html

Dale B, Dalrymple