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.

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


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


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


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 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

-- 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

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 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.
>
> [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
>
> @BOOK{proakiscomm,
>   title = "{Digital Communications}",
>   author = "John~G.~Proakis",
>   publisher = "McGraw-Hill",
>   edition = "fourth",
>   year = "2001"}
>
> @book{garcia,
>   title = "Probability and Random Processes for Electrical Engineering",
>   author = "{Alberto~Leon-Garcia}",
>   year = "1989"}
>
> @book{papoulis,
>   title = "Probability, Random Variables, and Stochastic Processes",
>   author = "{Athanasios~Papoulis}",
>   publisher = "WCB/McGraw-Hill",
>   edition = "Third",
>   year = "1991"}
>
Randy,

Where the factor of two pi shows up depends on which frequeny domain
variable is used.  If omega (radians/sec) is used, the factor shows up
outside the integral and whether is shows up in the forward, inverse, or
both transforms depends on the author's discipline.  If "f" (cycles/sec)
is used, the factor shows up in the exponential but nowhere else.

Brigham has a nice explanation of these issues in section 2-4 of his
book "The Fast Fourier Transform".

Mike

Randy Yates wrote:

<snipped>

>
> > I don't know about the books listed, but there seems to be some
> > correlation with whether the writer is an engineer, physicist, or
> > mathematician.
>
> That's a bit like saying the inverse square law changes depending on
> who you are.

Randy,

I don't think the "square law" comparison holds. By playing with the
scaling you don't change any basic truth.

I think consistency, like you said regarding the frequency sign, takes
care of the problem.  If you move the scaling around, other related
equations/units must also change.

If you think of the inverse FT as representing a signal by a weighted
sum/integral of complex exponentials, then incorporating any common
weighting into the weighting of the exponentials kind of makes
aesthetic sense, with no scale factor out front. A mathematician I know
thinks that is how it should be, he will argue for about for hours if
you are willing.

Some math people seem to like forward and inverse transforms to be as
similar possible and often distribute any special scaling between them.

Check your CRC Tables and see what they use, I don't recall it being
consistent with my engineering texts. Years ago IIRC MathCAD actually
had both scalings in their functions. That was kind of annoying because
it was function dependent, not parameter dependent, and not obvious
from the function name; you had to check Help.

I personally like when I plot 2 different size FFT's of the same sine
wave if I get the same amplitudes. That effectively requires moving the
1/NFFT scaling to the FFT, when it is commonly applied in the IFFT.  If
I am playing with FFT size for analysis I will often scale out the size
dependence so I can better compare the plots.

You just need to keep track of what you are doing.

So what's the problem? : )

Dirk

Dirk Bell
DSP Consultant

>
> > How come you didn't object to the swapping of the negative sign in the
> > imaginary exponents between forward and inverse transforms that you
> > also find in the various DFT/IDFT definitions? : )
>
> Because by and large I have found that folks are consistent, and even
> if they weren't, I'm not sure that would be a violation of the theory
> (as long as one is the opposite sign of the other).
>
> --Randy
>
> >
> > Dirk
> >
> > Dirk Bell
> > DSP Consultant
<snipped>


"Dilip V. Sarwate" <sarwate@YouEyeYouSee.edu> writes:

> "dbell" <bellda2005@cox.net> asked in message
>> Randy,
>> How come you didn't object to the swapping of the negative sign in the
>> imaginary exponents between forward and inverse transforms that you
>> also find in the various DFT/IDFT definitions? : )
>
> 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?)  :-)

Didn't I say that? :)
--
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface,
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO