The continuous Fourier transform (CFT) of a function
and its inverse are defined by
A numerical approximation of the CFT requires evaluating a large number of integrals, each with a different integrand, since the values of this integral for a large range of
are needed.
The FFT can be effectively applied to this problem as follows. Let us assume that
is zero outside the interval
. Let
be the sample spacing in
for the
input values of
, which are assumed to be centered at zero, where
is even. The values of
and
are chosen at the beginning in this procedure so the range of the interval
is changing and depends on these parameters. The abscissas for the input data are
,
. Then we can write
We define
,
. This is necessary for the above expression to be in the form of the DFT, denoted here by
:
The sample spacing (i.e., the resolution) of the result is fixed at the value
as soon as one specifies the number
of sample points and their interval
. The above definition of the DFT is equivalent to the
Mathematica command
Fourier[list, FourierParameters->{1,-1}].
Usually, comparable sample spacing intervals in
and
are required. Then, one must put
, or
. It is clear that if one wishes to obtain accurate, high-resolution results using this procedure, then it may be necessary to set
very large.
More details can be found in D. H. Bailey and P. N. Swarztrauber, "A Fast Method for the Numerical Evaluation of Continuous Fourier and Laplace Transform,"
SIAM Journal on Scientific Computing,
15(5), 1994 pp. 1105–1110.
You can choose the number of data points
(an integer power of two) from the radio button menu. The appropriate value of the output sample interval
for a given
is displayed after choosing the input sample spacing
. To show the sample points in the initial function (every fourth sample point is showed) and that the CFT in this algorithm is obtained in the form of points, check "points".
It is very convincing to fix a sample spacing
from the popup menu and decrease the number of sampling points
: you can then observe the deterioration of the accuracy of the Fourier transform.
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