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. . . . A possibly interesting aside:
We do a lab with some FFT analysis of sound or a
function generator. Even with a good signal
generator the FFT usually produces a Gaussian-like
distribution of frequencies about the "correct"
one. This is because none of the frequencies
included in the FFT analysis match the actual
frequency. That is if the signal is a 103 Hz and
the FFT's frequencies include 100 HZ and 105 Hz,
but nothing in between, the analysis gives a
non-zero amplitude to both 100 and 105 Hz and
usually a bit in 90, 95, 110, and 115.
Thanks
Roger Haar U of AZ
*****************************************
"Edmiston, Mike" wrote:
I have not carefully studied all the posts in this thread, so I
apologize if I say something redundant.
Ludwik says he has a square-wave pulse that is f(x) = 7 between x=2
and
x=5. He then says he solves for the fourier cosine coefficients by
integrating from x=2 to x=5.
John Mallinckrodt suggested Ludwik cannot use cosines alone because
the
function Ludwik is working on is not an even function. John suggested
moving the square wave pulse so it is centered about zero.
It seems to me John's suggestion would yield an entirely different
situation, that is, the two situations are not even close to
equivalent.
Fourier theory assumes the function you are trying to express (as a
fourier series) is a periodic function. If your "function" is
constant
at f(x) = 7 from x=2 to x=5 (and zero from x=0 to x=2), then when you
try to integrate over only that domain (from 2 to 5) you are getting
the
same result as if you had integrated from 0 to 5. That means the
periodic function you are trying to express actually runs from 0 to 5,
then repeats. That is, the periodic function is f(x)=0 from x = 0 to
2,
5 to 7, 10 to 12, etc. and f(x)=7 from x = 2 to 5, 7 to 10, 12 to 15,
etc.
We would indeed need both sine and cosine terms to represent this
(because it is neither even nor odd), except it is also possible to
express it as only sines or only cosines if a phase constant is
specified for each wavenumber. But to find the phase constants you
first find both the sine and cosine coefficients, so even if you want
to
express it as a cosine series you first find both the sine and cosine
coefficients.
If we move the f(x) = 7 "pulse" to center it on the origin, such that
f(x) = 7 for x between -1.5 and +1.5, and we integrate this from -1.5
to
1.5, then we don't really have a pulse. This is just a constant
function. a(0) = 7 and all other a(j) and b(j) are zero.
I also don't understand what Ludwik is doing when he talks about k =
6.1
and k = 5.9. Doesn't fourier analysis involve integer wave numbers?