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New Mathematica user here. I was trying to use FourierTransform today and getting perplexing minus signs in places I didn't expect them. After perusing the documentation, it seems to be due to the fact that what Mathematica calls FourierTransform is actually an inverse Fourier transform by the usual definition.

Questions:

Why is it defined like this?

There are of course many different conventions for Fourier transforms (angular vs. regular frequency, normalizations, etc.). Is there somewhere I can find a summary of Mathematica conventions?

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Yly
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1 Answers1

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The first line in definition can be a little misleading. You need to look down more

Mathematica graphics

When comparing it to the more standard one with minus sign

https://en.wikipedia.org/wiki/Fourier_transform

Mathematica graphics

But that is because , down in the same help page we see

enter image description here

And the above is the one you want to use, to change the definition to the way you want it.

I normally use FourierParameters->{1,-1} which gives standard one used in signal processing with the minus sign as expected. I have these FourierParameters->{1,-1} written on a yellow stick note attached to the corner of computer screen so I do not forget them because there are others.

So using FourierParameters->{1,-1} in the above gives

$$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} \,dt $$

which is what our text in DSP uses. Hence your code should be like this

 FourierTransform[f[t], t, w, FourierParameters -> {1, -1}]

And not just

 FourierTransform[f[t], t, w]

This way there will be no surprises.

Nasser
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  • Thanks. Do you have any idea why the default is the way it is? I have never seen a definition of Fourier transform without a minus in the exponent anywhere else. This seems like a really insidious gotcha. – Yly May 19 '20 at 23:45
  • @Yly I do not know. Mathematica is almost 30 years old. May be at the time they wrote this back in the days, the other definition was more popular? I really do not know., But I always use FourierParameters this way, I always know which definition I am using. – Nasser May 19 '20 at 23:47
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    @Yly - that's the definition for physics. – ciao May 20 '20 at 00:04
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    that's the definition for physics. @Yly well, this explains it then. Wolfram is a Physicist, so may be he is the one who wrote the above help page. Now we know who to blame :) – Nasser May 20 '20 at 00:06