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2.2 Fourier transforms
Let
be a real or complex function defined on the interval
. Let
be the Fourier transform of
. Let
a power of
, and
The discrete Fourier transform of
is
It is the true Fourier transform of the function
which is an approximation of
. Here we assume that
is linear between
and
, hence we have the exact formula
A FFT would give the value of
for all multiples of
.
Suppose we have
with
an integer and
. Then we have
Developping the last exponential, we get
This means that we would get the exact formula for
if we had all
the FFT's of the functions
In fact only the
between 0 and 20 are needed to get double precision. We
can even reduce this number by taking approximations of the exponential
using Tschebychev polynomials.
We can of course also define Fourier transforms of a function
defined
on an arbitrary interval
, by first computing the Fourier transform of
(defined on
) and multiplying the result by
.
Next: 2.3 Bessel transforms
Up: 2. Theoretical background
Previous: 2.1 Functions
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jmdr
2003-10-01