approximating Fourier integrals with discrete Fourier transforms
Suppose we have a continuous function
for which we want to numerically
compute the continuous Fourier transform
:
Assume that either the function is supported inside the compact rectangle , or that the Fourier integral is well approximated by truncating the domain to .
We develop an approximation to
based on the discrete Fourier transform
(which can be computed efficiently using the fast Fourier transform
algorithm
), and analyze the error in making this approximation.
Derivation
Performing the linear substitution
we find that
where
and denotes the Fourier transform of on the unit cube .
Let denote the vector with components
, for .
The discrete Fourier transform
of is
and approximates since it is a Riemann sum (step size ) for the Fourier integral.
Therefore, for , we have the approximation:
Analysing the error
Assume that can be represented by a pointwise-convergent11
Since the discrete Fourier transform requires sampling of points,
convergence of the Fourier series is not sufficient. Also, the Fourier series
may only be conditionally convergent. If there are difficulties in assuring
its convergence, Fejér summation can be
used instead in this analysis.
Fourier series:
where
We have the exact formula:
and we wish to know what error is introduced
if we were to replace
by the discrete version of the inner product .
Expand as:
Since is if for all , and is otherwise, the discrete inner product reduces to:
In other words, the approximation entails replacing
the true Fourier coefficient
with the same coefficient
plus other Fourier coefficients
corresponding to higher
frequencies, in multiples of .
The magnitude of the approximation error
for
thus depends on how fast the partial sums
of the Fourier coefficients decay to zero.
Title | approximating Fourier integrals with discrete Fourier transforms |
---|---|
Canonical name | ApproximatingFourierIntegralsWithDiscreteFourierTransforms |
Date of creation | 2013-03-22 16:28:23 |
Last modified on | 2013-03-22 16:28:23 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 9 |
Author | stevecheng (10074) |
Entry type | Derivation |
Classification | msc 42B10 |
Classification | msc 65T50 |
Classification | msc 42A38 |
Related topic | DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom |