In mathematics, the convolution theorem states that under suitable conditions
the Fourier transform of a convolution is the pointwise product of Fourier
transforms. In other words, convolution in one domain equals point-wise
multiplication in the other domain. Versions of the convolution theorem are
true for various Fourier-related transforms. Let and be two functions
with convolution . Let denote the Fourier transform operator, so and are
the Fourier transforms of and , respectively. Then
where denotes point-wise multiplication. It also works the other
way around: By applying the inverse Fourier
transform , we can write: and:
Note that the relationships above are only valid for the form of the Fourier
transform shown in the Proof section below. The transform may be normalized
in other ways, in which case constant scaling factors will appear in the
relationships above. This theorem also holds for the Laplace
transform, the two-sided Laplace transform and, when suitably modified,
for the Mellin transform and Hartley transform. It can be extended to the
Fourier transform of abstract harmonic analysis defined over locally compact
abelian groups. This formulation is especially useful
for implementing a numerical convolution on a computer: The standard convolution
algorithm has quadratic computational complexity. With the help of the
convolution theorem and the fast Fourier transform, the complexity of the
convolution can be reduced to O(n log n). This can be exploited to construct
fast multiplication algorithms. Proof
The proof here is shown for a particular normalization of the Fourier transform.
As mentioned above, if the transform is normalized differently, then constant
scaling factors will appear in the derivation.
Let f, g belong to L1(Rn). Let be the Fourier transform of and be the
Fourier transform of : where the dot between x and ν indicates
the inner product of Rn. Let be the convolution of and
Now notice that Hence by Fubini's theorem we have that
so its Fourier transform is defined by the integral formula
Observe that and hence by the argument above we may apply Fubini's theorem
again: Substitute ; then , so:
These two integrals are the definitions of and , so:
QED. Convolution theorem for inverse Fourier
transform With similar argument as the above
proof, we have the convolution theorem for the inverse Fourier transform.
and: Functions of discrete variable sequences
By similar arguments, it can be shown that the discrete convolution of
sequences and is given by: where DTFT represents the discrete-time
Fourier transform. An important special case is the
circular convolution of and defined by where is a periodic summation:
It can then be shown that: where DFT represents the discrete
Fourier transform. The proof follows from DTFT#Periodic
data, which indicates that can be written as:
The product with is thereby reduced to a discrete-frequency function:
The inverse DTFT is: QED.
References Katznelson, Yitzhak, An introduction to
Harmonic Analysis, Dover, ISBN 0-486-63331-4
Weisstein, Eric W., "Convolution Theorem", MathWorld.
Crutchfield, Steve, "The Joy of Convolution", Johns Hopkins University,
retrieved November 19, 2010 Additional resources
For visual representation of the use of the convolution theorem in signal
processing, see: Johns Hopkins University's Java-aided
simulation: http:www.jhu.educonvolve/index.html