Different Definitions for Inverse Fourier Transform

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.