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
Contents
- Proof
- Convolution theorem for inverse Fourier transform
- Functions of discrete variable sequences
- References
where
By applying the inverse Fourier transform
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 (typically
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 O(n²) 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
where the dot between x and ν indicates the inner product of Rn. Let
Now notice that
Hence by Fubini's theorem we have that
Observe that
Substitute
These two integrals are the definitions of
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
where DTFT represents the discrete-time Fourier transform.
An important special case is the circular convolution of
It can then be shown that:
where DFT represents the discrete Fourier transform.
The proof follows from DTFT#Periodic data, which indicates that
The product with
The inverse DTFT is:
QED.