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Re: Fourier transforms



1) If one takes the harmonic Fourier Series expansion of a real periodic
function expressed in terms of sines and cosines, and simply uses Euler's
theorem , e(inkx) = cos(nkx) +i sin(nkx), to eliminate trigonometric
functions in favor of complex exponentials, one obtains a series of terms in
e(inkx) and e(-inkx). Here n is always positive. However, one can then
choose to allow n to be both negative and positive and express the same
information as a sum of terms in e(inkx) with n ranging from - to +
infinity. This is a mathematical convenience, invoking the use of both
negative wavelengths/frequencies and imaginary numbers, and not adding any
new physical implications.

2) Note the resulting complex Fourier series can also be defined and used in
its own right as the expansion of a complex, periodic function.

3) Passing to the limit of an infinite period produces the Fourier Transform
involving a continuum of wavelenghts/frequencies.

4) The above also can be extended to a Digital Fourier Transform which
generates discrete samples of the Fourier Transform (at discrete
wavelengths/frequencies) from a sampled series of the original function ( in
x/t ).

5) For computational purposes there are the Fast Fourier Transform and the
Digital Fast Fourier Transform.


Bob Sciamanda
Physics, Edinbobo U of PA (em)
http://www.velocity.net/~trebor
trebor@velocity.net