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Advantages of frequency normalized Fourier transform


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"Normalizing" sounds like you divide the fourier transform with some constant in order to interpret the value on the vertical axis, are you sure don't mean something like that? Like, if you run a vector of 1's through your DFT routine, you'll get an output vector with nonzero first element. If you divide your DFT's with this number when plotting you have one common form of normalizing. Problem is that if you normalize you typically won't get back the original time domain vector if you do an inverse DFT so you need to be careful..

 

If you multiply your DFT with frequency you will get some sort of high pass filtering effect, since for [math]f \to 0[/math] the DFT will also approach zero. And if you do an inverse DFT you will get some weird results since multiplication in frequency domain means convolution in time domain.. :blink:

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What are the advantages of normalizing a DFT with the frequency?

E.g.

A plot of DFT*frequency as a function of frequency.

 

A plot of frequency vs frequency is a line of slope 1 so that cannot possibly be what you mean.

 

The "discrete Fourier transform" usually means the Fourier transform on a finite number of points, or in group terms [math] \mathbb Z_n = \dfrac {\mathbb Z}{n \mathbb Z} [/math]. Now the Fourier transform on an Abelian group is taken with respect to the "essentially unique" translation invariant measure (Haar measure) on the group. "Essentially unique" means unique up to a positive real multiple -- which could well be your normalization constant.

 

The natural invariant measure on a finite group is "counting measure" -- the measure of a subset is the number of points in the set. On the other hand in the theory of Fourier transforms on groups one usually chooses the Haar measure of a compact group to be 1 (and finite groups are compact. This is done to make the Fourier inversion theorem simple. So in that case "counting measure is normalized by dividing by the number of elements in the group -- [math]n[/math] in the case of [math]\mathbb Z_n[/math].

 

If this is what you mean by a normalized DFT the normalization merely assures that you are using the same measure for both the Fourier transform and the inverse Fourier transform. It is very convenient, but not essential to adopt such a convention.

 

For a very nice discussion of Fourier analysis on groups, the natural setting, you might want to take a look at Walter Rudin's classic book Fourier Analysis on Groups. The beauty is that all of the different variations on the theme of "Fourier transform" and "Fourier series" that you see in engineering and physics texts are seen as just examples of one overall concept. Oh, yeah, another useful point -- Rudin will not state theorems that are not true, in contrast to some engineering and physics books.

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