>the reason why it's merely convention is that if the minus sign was swapped >between the forward and inverse Fourier transform in all of the literature and >practice, all of the theorems would work the same as they do now.
Note that in some other areas they do actually use other conventions. It's been a while since I've looked at it but IIRC in areas like geophysics they have the signs swapped around. Also there are different conventions about where to put the normalization constants (on the analysis side, or on the synthesis side, or take the square root and include it on both). Those make a bit more difference for some of the theorems like Parseval, but again it all works the same you just gotta be careful to be consistent. E On Mon, Oct 5, 2015 at 2:52 PM, robert bristow-johnson < r...@audioimagination.com> wrote: > On 10/5/15 5:40 PM, robert bristow-johnson wrote: > >> >> about an hour ago i posted to this list and it hasn't shown up on my end. >> >> > okay, something got lost in the aether. i am reposting this: > > > On 10/5/15 9:28 AM, Stijn Frishert wrote: > >> In trying to get to grips with the discrete Fourier transform, I have a >> question about the minus sign in the exponent of the complex sinusoids you >> correlate with doing the transform. >> >> The inverse transform doesn’t contain this negation and a quick search on >> the internet tells me Fourier analysis and synthesis work as long as one of >> the formulas contains that minus and the other one doesn’t. >> >> So: why? If the bins in the resulting spectrum represent how much of a >> sinusoid was present in the original signal (cross-correlation), I would >> expect synthesis to use these exact same sinusoids to get back to the >> original signal. Instead it uses their inverse! How can the resulting >> signal not be 180 phase shifted? >> >> This may be text-book dsp theory, but I’ve looked and searched and >> everywhere seems to skip over it as if it’s self-evident. >> > > > hi Stijn, > > so just to confuse things further, i'll add my 2 cents that i had always > thought made it less confusing. (but people have disabused me of that > notion.) > > first of all, it's a question oft asked in DSP circles, like the USENET > comp.dsp or, more recently at Stack Exchange (not a bad thing to sign up > and participate in): > > > http://dsp.stackexchange.com/questions/19004/why-is-a-negative-exponent-present-in-fourier-and-laplace-transform > > > > in my opinion, the answer to your question is one word: "convention". > > the reason why it's merely convention is that if the minus sign was > swapped between the forward and inverse Fourier transform in all of the > literature and practice, all of the theorems would work the same as they do > now. > > the reason for that is that the two imaginary numbers +j and -j are, > qualitatively, *exactly* the same even though they are negatives of each > other and are not zero. (the same cannot be said for +1 and -1, which are > qualitatively different.) both +j and -j are purely imaginary and have > equal claim to squaring to become -1. > > so, by convention, they chose +j in the inverse Fourier Transform and -j > had to come out in the forward Fourier transform. they could have chosen -j > for the inverse F.T., but then they would need +j in the forward F.T. > > so why did they do that? in signal processing, where we are as comfortable > with negative frequency as we are with positive frequency it's because if > you want to represent a single (complex) sinusoid at an angular frequency > of omega_0 with an amplitude of 1 and phase offset of zero, it is: > > > e^(j*omega_0*t) > > so, when we represent a periodic signal with fundamental frequency of > omega_0>0 (that is, the period is 2*pi/omega_0), it is: > > +inf > x(t) = SUM X[k] * e^(j*k*omega_0*t) > k=-inf > > > each frequency component is at frequency k*omega_0. for positive > frequencies, k>0, for negative, k<0. > > > to extract the coefficient X[m], we must multiply x(t) by > e^(-j*m*omega_0*t) to cancel the factor e^(j*m*omega_0*t) in that term > (when k=m) in that summation, and then we average. the m-th term is now DC > and averaging will get X[m]. all of the other terms are AC and averaging > will eventually make those terms go to zero. so only X[m] is left. > > that is conceptually the basic way in which Fourier series or Fourier > transform works. (discrete or continuous.) > > > but, we could do the same thing all over again, this time replace every > occurrence of +j with -j and every -j with +j, and the same results will > come out. the choice of +j in the above two expressions is one of > convention. > > > > > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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