Convolution theorem


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 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.
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:
and:
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 from Big O notation| to Big O notation|, using big O notation. This can be exploited to construct fast multiplication algorithms, as in.

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 belong to the Lp-space. Let be the Fourier transform of and be the Fourier transform of :
where the dot between and indicates the inner product of . Let be the convolution of and
Also
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 :
Substituting yields. Therefore
These two integrals are the definitions of and , so:
QED.

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;
and:

Convolution theorem for tempered distributions

The convolution theorem extends to
tempered distributions.
Here, is an arbitrary tempered distribution
but must be "rapidly decreasing" towards and
in order to guarantee the existence of both, convolution and multiplication product.
Equivalently, if is a smooth "slowly growing"
ordinary function, it guarantees the existence of both, multiplication and convolution product.
In particular, every compactly supported tempered distribution,
such as the Dirac Delta, is "rapidly decreasing".
Equivalently, bandlimited functions, such as the function that is constantly
are smooth "slowly growing" ordinary functions.
If, for example, is the Dirac comb both equations yield the Poisson Summation Formula and if, furthermore, is the Dirac delta then is constantly one and these equations yield the Dirac comb identity.

Functions of discrete variable sequences

The analogous convolution theorem for discrete sequences and is:
where DTFT represents the discrete-time Fourier transform.
There is also a theorem for circular and periodic convolutions:
where and are periodic summations of sequences and :
The theorem is:
where DFT represents an N-length Discrete Fourier transform.
And therefore:
For and sequences whose non-zero duration is less than or equal to, a final simplification is:
Under certain conditions, a sub-sequence of is equivalent to linear convolution of and, which is usually the desired result. And when the transforms are efficiently implemented with the Fast Fourier transform algorithm, this calculation is much more efficient than linear convolution.

Convolution theorem for Fourier series coefficients

Two convolution theorems exist for the Fourier series coefficients of a periodic function:

Additional resources

For a visual representation of the use of the convolution theorem in signal processing, see: