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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