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On 05-Oct-2015 11:18 PM, music-dsp-requ...@music.columbia.edu wrote: > > Send music-dsp mailing list submissions to > music-dsp@music.columbia.edu > > To subscribe or unsubscribe via the World Wide Web, visit > https://lists.columbia.edu/mailman/listinfo/music-dsp > or, via email, send a message with subject or body 'help' to > music-dsp-requ...@music.columbia.edu > > You can reach the person managing the list at > music-dsp-ow...@music.columbia.edu > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of music-dsp digest..." > > > Today's Topics: > > 1. Re: Fourier and its negative exponent (Sebastian Roos) > 2. Re: Fourier and its negative exponent (Stijn Frishert) > 3. Re: Fourier and its negative exponent (Esteban Maestre) > 4. Re: Fourier and its negative exponent (Esteban Maestre) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Mon, 05 Oct 2015 23:38:43 +0700 > From: Sebastian Roos <s...@realtimeonly.com> > To: music-dsp@music.columbia.edu > Subject: Re: [music-dsp] Fourier and its negative exponent > Message-ID: <5612a793.3090...@realtimeonly.com> > Content-Type: text/plain; charset=UTF-8; format=flowed > > Since e^(-jw) equals 1/(e^(jw)), the same sinusoids are used, just > reverting what the other transformation did. No phase shift involved. > > Sebastian > > > Stijn Frishert 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 > > https://lists.columbia.edu/mailman/listinfo/music-dsp > > > ------------------------------ > > Message: 2 > Date: Mon, 5 Oct 2015 19:06:47 +0200 > From: Stijn Frishert <stijnfrish...@gmail.com> > To: music-dsp@music.columbia.edu > Subject: Re: [music-dsp] Fourier and its negative exponent > Message-ID: <25a2920e-a652-483e-aca1-61191ab14...@gmail.com> > Content-Type: text/plain; charset="utf-8" > > 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 > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: > <https://lists.columbia.edu/pipermail/music-dsp/attachments/20151005/6dd52adb/attachment-0001.html> > > > ------------------------------ > > Message: 3 > Date: Mon, 5 Oct 2015 20:11:54 +0300 > From: Esteban Maestre <este...@ccrma.stanford.edu> > To: music-dsp@music.columbia.edu > Subject: Re: [music-dsp] Fourier and its negative exponent > Message-ID: <5612af5a.40...@ccrma.stanford.edu> > Content-Type: text/plain; charset="windows-1252"; Format="flowed" > > Hi again, > > You can see the Fourier Transform as a projection. Finding projections > can be seen as computing inner products. Inner products of complex > numbers (or functions) involve complex-conjugating one of the numbers > (functions). > > Here's an alternative read: > > https://sites.google.com/site/butwhymath/fourier-analysis/the-fourier-transform > > > Cheers, > Esteban > > On 10/5/2015 8:06 PM, Stijn Frishert wrote: > > 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 > >> <mailto: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 > >> > >> 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 > >> > >> > >> _______________________________________________ > >> dupswapdrop: music-dsp mailing list > >> music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> > >> 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 > > -- > > Esteban Maestre > CIRMMT/CAML - McGill Univ > MTG - Univ Pompeu Fabra > http://ccrma.stanford.edu/~esteban > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: > <https://lists.columbia.edu/pipermail/music-dsp/attachments/20151005/a8cb1cca/attachment-0001.html> > > > ------------------------------ > > Message: 4 > Date: Mon, 5 Oct 2015 20:47:57 +0300 > From: Esteban Maestre <este...@ccrma.stanford.edu> > To: music-dsp@music.columbia.edu > Subject: Re: [music-dsp] Fourier and its negative exponent > Message-ID: <5612b7cd.4070...@ccrma.stanford.edu> > Content-Type: text/plain; charset="windows-1252"; Format="flowed" > > By the way: complex-conjugate does not mean it rotates in opposite > direction; check out this picture: > > http://www.eetasia.com/STATIC/ARTICLE_IMAGES/200902/EEOL_2009FEB04_DSP_RFD_NT_01c.gif > > > Rotation in opposite direction happens with negative frequencies. > > Cheers, > Esteban > > On 10/5/2015 8:06 PM, Stijn Frishert wrote: > > 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 > >> <mailto: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 > >> > >> 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 > >> > >> > >> _______________________________________________ > >> dupswapdrop: music-dsp mailing list > >> music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> > >> 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 > > -- > > Esteban Maestre > CIRMMT/CAML - McGill Univ > MTG - Univ Pompeu Fabra > http://ccrma.stanford.edu/~esteban > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: > <https://lists.columbia.edu/pipermail/music-dsp/attachments/20151005/b20911b6/attachment.html> > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp > > ------------------------------ > > End of music-dsp Digest, Vol 4, Issue 6 > *************************************** _______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
