# Different Definitions for Inverse Fourier Transform

Started by September 22, 2006
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 Yates                  % "With time with what you've learned,
%% Fuquay-Varina, NC            %  they'll kiss the ground you walk
%%% 919-577-9882                %  upon."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO

Randy Yates skrev:
> 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?

Without having read all the books (only browsed Papulis', some
10 years ago), I would guess it has to do with convenience.

As is usual when implementing the DFT/IDFT pair, squeeze
all the cumbersome scaling factors into the least used transform,
the inverse. The gain is a lot less scribbling to do during calculus;
the expense is that Parseval's identity misses by a factor
1/sqrt(2 pi) or so.

My \$1/2pi...

Rune


"Rune Allnor" <allnor@tele.ntnu.no> writes:
> [...]
> Without having read all the books (only browsed Papulis', some
> 10 years ago), I would guess it has to do with convenience.

Hi Rune,

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" ...
--
%  Randy Yates                  % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC            %  you still wander the fields of your
%%% 919-577-9882                %  sorrow."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO

Randy,

What does the '\' in '1/(2*\pi)' mean?

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.

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? : )

Dirk

Dirk Bell
DSP Consultant

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 Yates                  % "With time with what you've learned,
> %% Fuquay-Varina, NC            %  they'll kiss the ground you walk
> %%% 919-577-9882                %  upon."
> %%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO


"dbell" <bellda2005@cox.net> writes:

> Randy,
>
> What does the '\' in '1/(2*\pi)' mean?

Hi Dirk,

It is a LaTeX-ism - anything preceded by a backslash ("\") is a
command in LaTeX, and all the greek letters are formed by "\"
followed by the letter name ("\pi" for lower case, "\Pi" for
upper case).

> 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.

> 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
>
> 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 Yates                  % "With time with what you've learned,
>> %% Fuquay-Varina, NC            %  they'll kiss the ground you walk
>> %%% 919-577-9882                %  upon."
>> %%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
>

--
%  Randy Yates                  % "Remember the good old 1980's, when
%% Fuquay-Varina, NC            %  things were so uncomplicated?"
%%% 919-577-9882                % 'Ticket To The Moon'
%%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra

Randy Yates skrev:
> "Rune Allnor" <allnor@tele.ntnu.no> writes:
> > [...]
> > Without having read all the books (only browsed Papulis', some
> > 10 years ago), I would guess it has to do with convenience.
>
> Hi Rune,
>
> 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" ...

You do. Every time you choose to use w (lowercase omega)
instead of 2*pi*f. The only effect is to save writing.

Ah, yes, the factor 2*pi may have to do whether you integrate
w from -1 to 1 rather than f between -pi and pi.

Rune


"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?)  :-)


"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

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>


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