How Do Wavelets Compare to Fourier Transforms for Analyzing Non-Uniform Signals?

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Discussion Overview

The discussion revolves around the comparison of wavelets and Fourier transforms for analyzing non-uniform signals. Participants explore the advantages and limitations of both methods, particularly in the context of time-frequency analysis and signal decomposition.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that Fourier transforms struggle with non-uniform signals and suggests using multiple window sizes to capture both high and low frequency components.
  • Another participant introduces the Short-Time Fourier Transform (STFT) and mentions its limitations due to the uncertainty principle, which affects frequency resolution based on window size.
  • A participant expresses confusion about the relationship between wavelets and sampling, questioning how to choose a mother wavelet and the implications of using wavelet transforms for frequency analysis.
  • One participant shares their experience with the Continuous Wavelet Transform (CWT) and notes difficulties in achieving results comparable to STFT.
  • Another participant mentions working on the Non-Uniform Fast Fourier Transform (NUFFT) and seeks assistance in reproducing specific results from existing literature.
  • A participant asks if it is possible to construct time-series data using wavelets in a manner similar to Fourier transforms, indicating interest in the superposition of wavelets.

Areas of Agreement / Disagreement

Participants express various viewpoints on the effectiveness of wavelets versus Fourier transforms, particularly regarding non-uniform signals. There is no consensus on the superiority of one method over the other, and several unresolved questions remain about the practical application of wavelet techniques.

Contextual Notes

Some participants mention specific tools and tutorials, indicating a reliance on external resources for understanding wavelet transforms and their applications. There are also references to unresolved technical challenges and varying experiences with different methods.

Who May Find This Useful

This discussion may be of interest to individuals exploring time-frequency analysis, signal processing, and the application of wavelets and Fourier transforms in various contexts, particularly in non-uniform signal analysis.

likephysics
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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 divide 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 because areas with height 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"

Good luck
 
Last edited by a moderator:
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 anyone interested
 
likephysics said:
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" for example.

I myself am trying to understand the cwt and dwt (there is also wavedec) right know.
Good luck
 
Last edited by a moderator:
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?
 

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