In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their 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, not standard 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 , using big O notation. This can be exploited to construct fast multiplication algorithms, as in Multiplication algorithm § Fourier transform methods.
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 belong to the Lp-space . Let be the Fourier transform of and be the Fourier transform of :
Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula
Note 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.
It can then be shown that:
where DFT represents the discrete Fourier transform.
The proof follows from § 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:
Convolution theorem for Fourier series coefficients
Two convolution theorems exist for the Fourier series coefficients of a periodic function:
- The first convolution theorem states that if and are in , the Fourier series coefficients of the 2π-periodic convolution of and are given by:
- The second convolution theorem states that the Fourier series coefficients of the product of and are given by the discrete convolution of the and sequences:
- The scale factor is always equal to the period, 2π in this case.
For a visual representation of the use of the convolution theorem in signal processing, see: