Thanks Allen, Esteban and Sebastian. My main thought error was thinking that negating the exponent was the complex equivalent of flipping the sign of a non-complex sinusoid (sin and -sin). Of course it isn’t. e^-a isn’t the same as -e^a. The real part of a complex sinusoid and its complex conjugate are the same, they only rotate in different directions.
And so the minus is to negate that rotation in the complex plane. Correct me if I’m wrong, of course. Stijn > On 5 Oct 2015, at 15:51, Allen Downey <dow...@allendowney.com> wrote: > > In Chapter 7 of Think DSP, I develop the DFT in a way that might help with > this: > > http://greenteapress.com/thinkdsp/html/thinkdsp008.html > <http://greenteapress.com/thinkdsp/html/thinkdsp008.html> > > If you think of the inverse DFT as matrix multiplication where the matrix, M, > contains complex exponentials as basis vectors, the (forward) DFT is the > multiplication by the inverse of M. Since M is unitary, its inverse is its > conjugate transpose. The conjugation is the source of the negative sign, > when you write the DFT in summation form. > > Allen > > > > On Mon, Oct 5, 2015 at 9:28 AM, Stijn Frishert <stijnfrish...@gmail.com > <mailto:stijnfrish...@gmail.com>> wrote: > Hey all, > > 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. > > Stijn Frishert > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> > https://lists.columbia.edu/mailman/listinfo/music-dsp > <https://lists.columbia.edu/mailman/listinfo/music-dsp> > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp
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