Re: [music-dsp] Fourier and its negative exponent

[Theo Verelst]

Think of the Fast Fourier Transform as computing the inner product of a piece of signal 
(the length of the transform) with all kinds of basis functions: the various frequencies 
that can "fit" in the interval. Without going into engineering basics, you can take a sine 
and a cosine as a basis function got each frequency that "fits" in the chosen interval (so 
a sine the exact length of the interval, two sines that exactly fit the chosen interval, 3 
full sines, etc., until you have a sine that isn't distinguishable anymore because it's 
frequency is too high because it's peaks appear shorter together than the elements of you 
time vector. As it appears, it is a good idea to take sine and cosines, because than you 
can proof/make credible that there is one precise and only one FFT transformed signal that 
can be added to your time "sample" or signal data point vector, which given certain 
accuracies is also a bi-jective mapping. The idea of measuring the sine and cosine 
correlation is as an ancient EE trick (probably in mechanics and physics before that) 
connected with the idea that each signal (given certain conditions) can be seen as an 
addition of sine waves, where there's phase associated with each component. Writing that 
as a complex number is a trick, it has in this case not really to do with the transform, 
it's a short way of writing things. I assure you there's a lot of mathematical hassle with 
complex numbers and matrices possible you might not want to get into, because it is often 
not very insightful.


T.
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Re: [music-dsp] Fourier and its negative exponent

[Esteban Maestre]

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 > 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 
> 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
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--

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CIRMMT/CAML - McGill Univ
MTG - Univ Pompeu Fabra
http://ccrma.stanford.edu/~esteban

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Re: [music-dsp] Fourier and its negative exponent

[Esteban Maestre]

"does not mean" > "does mean"

Esteban

On 10/5/2015 8:47 PM, Esteban Maestre wrote:
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  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 
 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
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--

Esteban Maestre
CIRMMT/CAML - McGill Univ
MTG - Univ Pompeu Fabra
http://ccrma.stanford.edu/~esteban  


--

Esteban Maestre
CIRMMT/CAML - McGill Univ
MTG - Univ Pompeu Fabra
http://ccrma.stanford.edu/~esteban

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Re: [music-dsp] Fourier and its negative exponent

[robert bristow-johnson]

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)  =   SUMX[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."



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Re: [music-dsp] Fast convolution with synthesis window

[Earl Vickers]

rbj wrote:

> why would you *want* to use a synthesis window if you're doing OLA 
> fast-convolution?

Good question.

> [ ] it might be a very nice way to have a changing filter kernel and 
> have it sound nice in the transitions.  

Good answer!

Yes, I’m doing time-varying filtering. Just wondering if there were
any tricks I’m missing.

Thanks for the suggestions,

Earl
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Re: [music-dsp] Fourier and its negative exponent

[robert bristow-johnson]

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."



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Re: [music-dsp] test (sorry)

[Stijn Frishert]

Hi Robert,

Your mail (the first copy) was well received and is still ringing through my 
mind. Especially the part about -j and +j having equal claim to square to -1 is 
an eye opener. I'm still thinking about the consequences and need to write some 
stuff out on paper, but I'll get back to it. That's why it's taking some time 
to react ;)

And thanks to Theo as well. The beauty and curse of the Fourier is that there 
are so many ways to look at it, right!

Stijn

2015/10/05 23:40、robert bristow-johnson  のメッセージ:

> 
> 
> about an hour ago i posted to this list and it hasn't shown up on my end.
> 
> 
> -- 
> 
> r b-j  r...@audioimagination.com
> 
> "Imagination is more important than knowledge."
> 
> 
> 
> ___
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[music-dsp] test (sorry)

[robert bristow-johnson]

about an hour ago i posted to this list and it hasn't shown up on my end.


--

r b-j  r...@audioimagination.com

"Imagination is more important than knowledge."



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Re: [music-dsp] test (sorry)

[robert bristow-johnson]

On 10/5/15 5:58 PM, Stijn Frishert wrote:
> Your mail (the first copy) was well received and is still ringing through my 
> mind. Especially the part about -j and +j having equal claim to square to -1 
> is an eye opener.
check out "Imaginary unit" at Wikipedia.

>  I'm still thinking about the consequences and need to write some stuff out 
> on paper, but I'll get back to it. That's why it's taking some time to react 
> ;)

i wasn't worried about your (or anyone else's) reaction. you get what
you pay for. but it did *not* show up on my email, so i thought the
internet somehow ate it.


-- 

r b-j  r...@audioimagination.com

"Imagination is more important than knowledge."



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Re: [music-dsp] Fourier and its negative exponent

[Ethan Duni]

>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."
>
>
>
> ___
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>
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Re: [music-dsp] Fast convolution with synthesis window

[robert bristow-johnson]

On 10/4/15 12:36 PM, Earl Vickers wrote:

rbj wrote:


why would you *want* to use a synthesis window if you're doing OLA
fast-convolution?

Good question.


[ ] it might be a very nice way to have a changing filter kernel and
have it sound nice in the transitions.

Good answer!

Yes, I’m doing time-varying filtering. Just wondering if there were
any tricks I’m missing.


dunno.  just make sure that N >= B + L - 1

where N = FFT length
  B = length of the non-zero values of the window
  L = FIR length


Thanks for the suggestions,


here is another.  i presume your filter kernels are computed in advance 
(and you select one or another for each fast-convolution frame)?  maybe 
it doesn't matter, but whatever they are (they are the DFT of the 
zero-padded h[n]), you must make sure that they are the DFT of an h[n] 
that is no longer than N-B+1.


earlier i wrote here about (and i have a pdf document that spells it 
out) what the optimal size of N (where N is a power of 2) given what the 
FIR length L is.  it was done for a rectangular window (with no overlap 
of the windows, even though there is always overlap of the frames), but 
i think the results are similar for 50% overlapped windows.  it would be 
half as efficient, but i think the optimal N would be the same.


just FYI.



--

r b-j  r...@audioimagination.com

"Imagination is more important than knowledge."



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[music-dsp] [ANN] Announcing the JUCE Summit, the world's first C++ audio conference

[Jean-Baptiste Thiebaut]

Hi all,

We would like to announce that JUCE will host its first Summit in London,
on November 19th and 20th 2015. Click here for full details
.

*Theme*: This conference will focus on all aspects of C++ audio, from
developing plug-ins to real-time embedded DSP and mobile audio.

*High performance audio on Android: *This has been a long awaited feature:
Android has finally cut down latency! We will host a workshop with a Google
engineer to showcase the latest developments, and in particular how to use
JUCE to get your existing code run on Android devices.

*Guest program: *We have invited guest speakers David Zicarelli, CEO of
Cycling'74 , and Pete Goodliffe, lead developer at
AKAI , to share their experience developing music
software. More speakers to be announced shortly.

*Participate*: Anyone is invited to submit a proposal for a talk. The
deadline for sending proposals is the 9th of October, and a proposal
template can be found here
.
The selected speakers will have their registration fees waived.

*Register:* Full registration costs £345, and includes lunch for both days.
Early bird registration are available for £245 and ends on 23rd October. We
also offer 20% discount to students.

We look forward to welcoming you in London!

Best regards,
Jean-Baptiste Thiebaut

Product Manager, JUCE
http://www.juce.com
JUCE is a subsidiary of ROLI 
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Re: [music-dsp] Fourier and its negative exponent

[Esteban Maestre]

HI Stijn,

That "minus" comes from complex-conjugate (of Euler's formula). To find 
the projection coefficients (Fourier Transform), in each of the terms in 
the summation one computes the inner product of two complex vectors: the 
complex sinusoid you are "testing", and its complex-conjugate. The 
resulting complex number (each bin is a complex number) will not only 
tell you "how much of a sinusoid was present in the original signal", 
but also its relative phase.


This is an excellent read:

https://ccrma.stanford.edu/~jos/st/

Cheers,
Esteban





On 10/5/2015 4:28 PM, 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
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--

Esteban Maestre
CIRMMT/CAML - McGill Univ
MTG - Univ Pompeu Fabra
http://ccrma.stanford.edu/~esteban

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[music-dsp] Fourier and its negative exponent

[Stijn Frishert]

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
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Re: [music-dsp] Fourier and its negative exponent

[Allen Downey]

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 
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
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Re: [music-dsp] Fourier and its negative exponent

[Esteban Maestre]

On 10/5/2015 6:15 PM, Esteban Maestre wrote:

the complex sinusoid you are "testing", and its complex-conjugate

Sorry:

I mean "your signal and the complex sinusoid your are testing".

Esteban

--

Esteban Maestre
CIRMMT/CAML - McGill Univ
MTG - Univ Pompeu Fabra
http://ccrma.stanford.edu/~esteban

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