[Maxima-discuss] Fourier Transform , almost symbolic

[Edward M. <qui...@gm...>]

This relates to sampled data , being quite
common nowadays.

 If we look at the Analytical Fourier Transform of
a rectangular pulse we obtain the now very
familiar Sinc function .
 Using the shift theorem for a train of rectangular pulses , all of the
same width ,
we obtain a series of complex exponential
terms ; all multiplied by the Sinc function for
the single centered rectangular pulse.

 Symbolically simplifying the complex exponential series we arrive at a
complex
Fourier function that completely describes
the train of , variable amplitude rectangular
pulses . This exhibits all of the properties
of the continuous Fourier Transform for
such a series .

 This I have implemented , in wxMaxima ;
the Inverse Fourier Transform might be
derived in a similar way , whilst recognizing
that the pulse width is the reciprocal of that
in the other domain.
 I have yet to complete the Inverse Fourier
Transform , I ran out of motivation.
  The inverse fft of the Symbolic Fourier Transform appears to return a
correct numerical
representation of the original pulses.

  Is this a valid approach , from first principles
it appears to be valid , of course the digital
computation might have a few surprises.