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I am currently designing an alogirthm that is either based on Fourier Transform approach, or the Wavelet Transform Approach, or the combination of the two. Since Wavelet is new to me, I am having difficulty to decide which approach to take.

The system could be modelled as follows: between each time slot, we will be receving incoming data point, and we are supposed to analyze the frequency component from these incoming data point to see whether a problem has occur. There are two approach of this problem: Fourier Transform or Wavelet Transform.

The difference of two is explained by here :Difference between Fourier transform and Wavelets

It seems to me that, the only advantage Wavelet offers is that inaddition to having the frequency spectrum, it is also showing the time component such that it allows you to see at which time does this frequency peaks.

My question is: if we decide to aggregate the incoming data point for a certain time (say $[t_0, t_1]$), and then perform a FFT on those aggregated data. And because we already had the information that the frequency spectrum produced from FFt is for signal between $[t_0,t_1]$, given that we don't care at exactly what $t$ the problem occurs, does the Wavelet approach still offer any advantage over the Fourier?

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As you mention, the pure Fourier Transform will not give you any time resolution (time information is hidden in the phase, which is usually not visualized). If you just take a certain time interval $[t_0, t_1]$, that is a subset of your full signal domain you will have some implicit time localization. The wavelet transform will not give you any further information...

If you take a short time interval that you want to analyze with the Fourier Transform you might also want to have a look at windowing. In formulas I would suggest you to cut out the interval of interest by a relatively smooth window function $w$ that is supported on $[t_0, t_1]$ and then perform the Fourier Transform on $f \cdot w$, which is also supported on the interval above. That will give you better results, as the Fourier Transform is quite sensitive to discontinuities.

Hope this helps...

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  • $\begingroup$ Yes, the thing I am worrying is the discontinuities. How will that disrupt the frequency spectrum? $\endgroup$ – kuku Oct 2 '14 at 6:30
  • $\begingroup$ I will try to give you a rather non-mathematical explanation: If you perform the Fourier Transform on a signal $f$, that is only defined on the interval $[t_0, t_1]$ you will have to make sure that you do not have discontinuities along the border, i.e. $f(t_0) = f(t_1)$, as the FT will interpret your signal as periodic. Any discontinuity will result in a blurred FT, mathematically speaking you can characterize it as something sharp convolved with something rather dislocalized. $\endgroup$ – Chris Oct 2 '14 at 6:35

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