Wavelets and Fourier transform

Main Question or Discussion Point

I am new to wavelets.
I was reading about wavelets and Fourier transforms. So the main disadvantage of Fourier Transform is that you cannot use it on a non-uniform signal.
Even though you use it you have to use a window and select your region of interest.
If the window is small enough you can see the high frequency components, but not the low frequency components. But if the window is large, then you see the LF components but not the HF components.
[Small window:good time resolution, bad freq res. Large window: bad time resolution, good freq resolution.]
So why not use both windows on the same signal. Maybe even use a lot of windows from narrow to wide.
Wouldn't this give us all the LF,HF information about the signal?

In wavelets, how do you decide on a mother wavelet?
If I have understood correctly, wavelet transform is similar to sampling a signal. In sampling you multiply the signal with a delta function. In WT, you multiply the signal with a mother wavelet. Instead of doing it once, you change the frequency of(expand/compress) the mother wavelet and multiply it with the signal each time. correct?
This will give us several results with HF only, LF only components of the original signal. So what after this?
Isn't this similar to sampling a signal(say well beyond Nyquist rate) and then decimating it in steps to get different frequency components?

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Wow lots of questions.

Look, in general, you start off with fourier -(gives you all frequencies in your signal)
Then, you go to STFT (short time fourier transform) in other words devide your time/distance into blocks and each time calculate the frequancy components in each block.
Heisenberg (Hope I spelt his name right) sais that looking at finite blocks is going to smear your freaquancies (-Uncertainty principle) depending on the size of your block=window.
apparently, STFT is not very good becuase areas with hight freq. and areas with low freq. will demand different sized windows. Here comes in the Wavelet principle.

From your questions it is clear you know some stu but it is all muddled up in your head. So in this case I suggest you read the tutorial of a man named Dr. Polikar Robi which will start right from the beginig up to DWT (Discrete Wavelet Transform).

http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html" [Broken]

Good luck

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John, thanks for the reply. A day after posting I read the Robi Polikar's tutorial. Really neat.
Cleared up most of my doubts. I could not understand the basys vectors and the reason wavelets are better for de-noising compared to a filter.
Now I am trying to analyse a non-uniform signal to see all the different frequencies. STFT gives good results, but when I apply CWT I don't see anything close to what I see in STFT.
I am trying wavelet packets, but still no luck.

Well, I am working on the Non uniform fast fourier transform, which can be use on non uniform signal now days, and i am stuck some where....I am trying to produce the same results as from
A.J.W.Duijndam and M.A.Schonewillie but i am not able to produce my time signal back any one interested

John, thanks for the reply. A day after posting I read the Robi Polikar's tutorial. Really neat.
Cleared up most of my doubts. I could not understand the basys vectors and the reason wavelets are better for de-noising compared to a filter.
Now I am trying to analyse a non-uniform signal to see all the different frequencies. STFT gives good results, but when I apply CWT I don't see anything close to what I see in STFT.
I am trying wavelet packets, but still no luck.
I don't know what you are working with, I work with Matlab, but I can give you a few tutorials on wavelets in matlab http://visl.technion.ac.il/documents/wavelet_ug.pdf" [Broken] for example.

I myself am trying to understand the cwt and dwt (there is also wavedec) right know.
Good luck

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Thanks. I am using matlab too. I followed the matlab example from their help docs. Didn't help much. I'll go thru your examples. What is a wavedec?

Does anyone know if i can construct-compose time-series consist of wavlets? e.g. if i have as data a characteristic wave height and a wave length i can produce time-series with Fourier transform which equals of a superposition of sinus waves. i want to do the same but with wavelets. possible?