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Different Definitions for Inverse Fourier Transform

Started by Randy Yates September 22, 2006
Randy Yates wrote:

> It's not completely standard, but I've usually seen a factor of > 1/(2*\pi) in front of the integral for the inverse Fourier transform. > I believe the theore says that this factor must be somewhere in the > round-trip journey from forward transform to revers. It could be > 1/sqrt(2*\pi) in front of both, or whatever.
The one I learned for physics problems is that the quantity that has the 2pi in it is the one that gets divided by 2pi. For example, using t and omega (angular frequency in radians/sec), the omega transform gets the 1/(2pi).
> [papoulis] has the present in his definition of the autocorrelation > function as the inverse transform of the power spectral density. > However, both [proakiscomm] and [garcia] omit this factor.
As far as I know, other than what I wrote above, there is no standard, and no reason to prefer one over the other. -- glen
Randy Yates <yates@ieee.org> writes:

> It's not completely standard, but I've usually seen a factor of > 1/(2*\pi) in front of the integral for the inverse Fourier transform. > I believe the theore says that this factor must be somewhere in the > round-trip journey from forward transform to revers. It could be > 1/sqrt(2*\pi) in front of both, or whatever. > > [papoulis] has the present in his definition of the autocorrelation > function as the inverse transform of the power spectral density. > However, both [proakiscomm] and [garcia] omit this factor. > > Why do these people use this [incorrect] form of the inverse transform? > > --Randy
Folks, I realized that there is a difference of "f" versus "omega" in these two forms, so perhaps I am wrong and not Leon-Garcia or Proakis (ya' think?). I still have some analyzing to do, but I wanted to retract my "accusations." -- % Randy Yates % "So now it's getting late, %% Fuquay-Varina, NC % and those who hesitate %%% 919-577-9882 % got no one..." %%%% <yates@ieee.org> % 'Waterfall', *Face The Music*, ELO http://home.earthlink.net/~yatescr
Randy Yates wrote:

(snip regarding FFT and IFFT)

> If that's true, it's appalling! Why don't we just leave out the bothersome > 2\pi in the exponent argument as well? It'd be more "convenient" ...
The advantages of using radians per second instead of cycles per second. The usual Gaussian units (also called CGS) for electromagnetism have 4pi in most of the formulae. There is another system called Lorentz-Heaviside units which is pretty much the same without all the 4pi terms. I have known exams where one was allowed to use any consistent set of units and the grader had to follow the result. -- glen
Dilip V. Sarwate wrote:

(snip regarding the signs in FFT and IFFT)

> Perhaps Randy believes that there are two square roots of -1, and > that some people use one root while others use the other root (which > accounts for the swapping of the signs?) :-)
I have seen some discussion about the j used for complex numbers for EE problems being equal to -i as used in physics. That is supposed to be related to EE's looking first at signals as a function of time, and second as a function of distance, where physics usually does it the other way around. With the solution to the wave equation as exp(ikx-iwt). That is, exp(ikx)exp(-iwt), where EE would use exp(jwt) for the time dependent phasor. -- glen
glen herrmannsfeldt skrev:
> Dilip V. Sarwate wrote: > > (snip regarding the signs in FFT and IFFT) > > > Perhaps Randy believes that there are two square roots of -1, and > > that some people use one root while others use the other root (which > > accounts for the swapping of the signs?) :-) > > I have seen some discussion about the j used for complex numbers for EE > problems being equal to -i as used in physics. That is supposed to be > related to EE's looking first at signals as a function of time, and > second as a function of distance, where physics usually does it the > other way around. With the solution to the wave equation as > exp(ikx-iwt). That is, exp(ikx)exp(-iwt), where EE would use > exp(jwt) for the time dependent phasor.
Yep indeed. Those sorts of details caused me some 1 year extra in my PhD work. I could never use the maths/physics formulae with my EE signal processing, I had to work through every single formula to make sure I got it right. As I had to work through the petroleum seismic formulas, where depth below the sea surface was expressed as a z axis pointing downwards, and the earthquake material where the earth is modeled as a sphere with depth expressed as a radius ponting outwards... Real "fun". Rune
Rune Allnor wrote:
(snip)

> Those sorts of details caused me some 1 year extra in my > PhD work. I could never use the maths/physics formulae > with my EE signal processing, I had to work through > every single formula to make sure I got it right. As I had to > work through the petroleum seismic formulas, where depth > below the sea surface was expressed as a z axis pointing > downwards, and the earthquake material where the earth is > modeled as a sphere with depth expressed as a radius > ponting outwards...
The two letter state abbreviations used for boat registration are different than the ones used for mail. Boats had them first, and the post office didn't follow the existing standard. -- glen
Randy Yates wrote:

> It's not completely standard, but I've usually seen a factor of > 1/(2*\pi) in front of the integral for the inverse Fourier transform. > I believe the theore says that this factor must be somewhere in the > round-trip journey from forward transform to revers. It could be > 1/sqrt(2*\pi) in front of both, or whatever. > >
Probably because it is just a constant scale factor. That 1/sqrt(2pi) also depends on the angular system used.