Definition of Fourier transform

[jollage]

Hi All,

Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition.

My question is can I define Fourier transform as $\int$f(x)e$^{2\pi ix \varsigma}$dx instead, i.e., with the minus sign removed, as the forward Fourier transform? The backward one is the one with the minus sign. So the definition is the opposite to the definition on the wiki page.

Can I define this? Will the so-transformed frequency domain still bear the physical meanings as we usually talk about?

Thanks in advance. Any comment will help.

Jo

 

[I like Serena]

Welcome to PF, jollage!

Yep. You can do that.
Fourier transforms are defined haphazardly as you may already have noticed.
Changing the sign or the constants does not change the way it operates, nor the physical meaning.

 

[marcusl]

Just be aware that the result you get might differ from one found the other way.

 

[jollage]

OK, thank you for confirming this. This is great. I guess I could move on with this definition.

 

[DrGreg]

It just means that what you call a positive frequency, everyone else calls a negative frequency, and vice-versa. If you are dealing with real-valued functions only (i.e. not complex), it won't make much difference, because in that case the negative-frequency spectrum is just a mirror image of the positive-frequency spectrum.

 

[Dale]

Hi All,

Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition.

My question is can I define Fourier transform as $\int$f(x)e$^{2\pi ix \varsigma}$dx instead, i.e., with the minus sign removed, as the forward Fourier transform? The backward one is the one with the minus sign. So the definition is the opposite to the definition on the wiki page.

See equations 15 and 16 here:
http://mathworld.wolfram.com/FourierTransform.html

To get a "general" Fourier transform there are two free parameters that you can set. Different groups use different choices of those free parameters as their "standard", but it is all just a matter of convention.