Convolution theorem


In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions is the product of their Fourier transforms. More generally, convolution in one domain equals point-wise multiplication in the other domain. Other versions of the convolution theorem are applicable to various Fourier-related transforms.

Functions of a continuous variable

Consider two functions and with Fourier transforms and :
where denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors will appear in the convolution theorem below. The convolution of and is defined by:
In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol is sometimes used instead.
The convolution theorem states that:
Applying the inverse Fourier transform produces the corollary:
The theorem also generally applies to multi-dimensional functions.
Consider functions in Lp-space with Fourier transforms :
where indicates the inner product of : and
The convolution of and is defined by:
Also:
Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula:
Note that Hence by the argument above we may apply Fubini's theorem again :
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.

Periodic convolution (Fourier series coefficients)

Consider -periodic functions and which can be expressed as periodic summations:
In practice the non-zero portion of components and are often limited to duration but nothing in the theorem requires that.
The Fourier series coefficients are:
where denotes the Fourier series integral.
  • The product: is also -periodic, and its Fourier series coefficients are given by the discrete convolution of the and sequences:
  • The convolution:
is also -periodic, and is called a periodic convolution.
The corresponding convolution theorem is''':'''

Functions of a discrete variable (sequences)

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now denotes the discrete-time Fourier transform operator. Consider two sequences and with transforms and :
The of and is defined by:
The convolution theorem for discrete sequences is''':'''

Periodic convolution

and as defined above, are periodic, with a period of 1. Consider -periodic sequences and :
These functions occur as the result of sampling and at intervals of and performing an inverse discrete Fourier transform on samples . The discrete convolution:
is also -periodic, and is called a periodic convolution. Redefining the operator as the -length DFT, the corresponding theorem is:
And therefore:
Under the right conditions, it is possible for this -length sequence to contain a distortion-free segment of a convolution. But when the non-zero portion of the or sequence is equal or longer than some distortion is inevitable. Such is the case when the sequence is obtained by directly sampling the DTFT of the infinitely long impulse response.
For and sequences whose non-zero duration is less than or equal to a final simplification is:
This form is often used to efficiently implement numerical convolution by computer.
As a partial reciprocal, it has been shown
that any linear transform that turns convolution into a product is the DFT.
A time-domain derivation proceeds as follows:
A frequency-domain derivation follows from, which indicates that the DTFTs can be written as:
The product with is thereby reduced to a discrete-frequency function:
where the equivalence of and follows from. Therefore, the equivalence of and requires:
We can also verify the inverse DTFT of ''':'''

Convolution theorem for inverse Fourier transform

There is also a convolution theorem for the inverse Fourier transform:
Here, "" represents the Hadamard product, and "" represents a convolution between the two matrices.
so that

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.

Additional resources

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