Different Definitions for Inverse Fourier Transform
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From: Randy Yates on 22 Sep 2006 11:40 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}", publisher = "Addison-Wesley", 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(a)ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
From: Rune Allnor on 22 Sep 2006 11:54 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
From: Randy Yates on 22 Sep 2006 12:00 "Rune Allnor" <allnor(a)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(a)ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
From: dbell on 22 Sep 2006 12:02 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}", > publisher = "Addison-Wesley", > 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(a)ieee.org> % '21st Century Man', *Time*, ELO > http://home.earthlink.net/~yatescr
From: Randy Yates on 22 Sep 2006 12:15
"dbell" <bellda2005(a)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}", >> publisher = "Addison-Wesley", >> 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(a)ieee.org> % '21st Century Man', *Time*, ELO >> http://home.earthlink.net/~yatescr > -- % Randy Yates % "Remember the good old 1980's, when %% Fuquay-Varina, NC % things were so uncomplicated?" %%% 919-577-9882 % 'Ticket To The Moon' %%%% <yates(a)ieee.org> % *Time*, Electric Light Orchestra http://home.earthlink.net/~yatescr |