AW: Harmonix content formula

[Ethan Duni]

>Well, yes pain, but if you can get an analytical result for a problem
>this is of much more value then a numerical solution.
>E.g. the spectra for PWM pulses can be given with just one formula,
>on the numeric side you would have to compute "solutions" for
>a couple of duty cycles and still you would have not all of
>the information.

-yes, i absolutely agree, it's just hard to keep that in mind when you have
ten minutes left of the final exam and you are staring at half of a page of
integrals of complex exponentials and fifth-order polynomials. :]

>Sometimes numerical solvers are completely wrong (not in the FFT case).
>So using numerical methods requires that you allready know how
>the result will look like in order to reject wrong "solutions".

-hmm, couldn't we even argue that numerical solutions are always, one some
level, "wrong", because they necessarily involve approximations and, being
done in computers, finite-resolution numbers? i wonder how much progress has
been made in getting computers to perform symbolic (and hence truly
"correct") operations on the level of fourier transforms.. be interesting.. 

>I only know that the basic concept is very old and Fourier seems to have
>given some "shape" to it.

-yeah, and that reminds me.. i have from Amara's Wavelet page
(http://www.amara.com/current/wavelet.html#Fourier):

>From "The Hartley Transform" 

     by Ronald N. Bracewell. (Oxford Press, 1st ed, 1986, page 6)

     "When the FFT was brought into the limelight by Cooley and Tukey in
1965 it had an enthusiastic reception in the populous world of electrical
signal analysis as the news spread via tutorial articles and special issues
of journals.  This ferment occasioned mild surprise in the world of
numerical analysis, where related techniques were already known. Admirable
sleuthing by M.T. Heideman, C.S. Burrus, and D.H. Johnson (to appear in
'Archive for History of the Exact Sciences') has now traced the origins of
the method back to a paper of C.F. Gauss (1777-1855) written in 1805, where
he says, 'Experience will teach the user that this method will greatly
lessen the tedium of mechanical calculation.' " 

which i thought was pretty fascinating..

but i digress

Ethan