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 (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms.
Let and be two functions with convolution . (Note that the asterisk denotes convolution in this context, and not multiplication. The tensor product symbol is sometimes used instead.)
If denotes the Fourier transform operator, then 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:
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 (typically or ) 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 (see Mellin inversion theorem). 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 from to . This can be exploited to construct fast multiplication algorithms.
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
Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula
It should be noted that and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):
Substituting yields . Therefore
These two integrals are the definitions of and , so:
Convolution theorem for inverse Fourier transform
A similar argument, as the above proof, can be applied to the convolution theorem for the inverse Fourier transform;
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:
- (also using Sampling the DTFT).
The inverse DTFT is:
- Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4
- Weisstein, Eric W. "Convolution Theorem". MathWorld.
- Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved November 19, 2010
For a visual representation of the use of the convolution theorem in signal processing, see: