In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the convolution theorem states that under suitable conditions the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of two functions (or
signals) is the
pointwise product
In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be m ...
of their Fourier transforms. More generally, convolution in one domain (e.g.,
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
) equals point-wise multiplication in the other domain (e.g.,
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
). Other versions of the convolution theorem are applicable to various
Fourier-related transforms.
Functions of a continuous variable
Consider two functions
and
with
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s
and
:
where
denotes the Fourier transform
operator. The transform may be normalized in other ways, in which case constant scaling factors (typically
or
) will appear in the convolution theorem below. The convolution of
and
is defined by:
In this context the
asterisk
The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star.
Computer scientists and mathematicians often voc ...
denotes convolution, instead of standard multiplication. The
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
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.
This theorem also holds for the ]Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
, the two-sided Laplace transform and, when suitably modified, for the Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used ...
and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
defined over locally compact abelian groups.
Periodic convolution (Fourier series coefficients)
Consider -periodic functions and which can be expressed as periodic summations:
and
In practice the non-zero portion of components and are often limited to duration but nothing in the theorem requires that. The Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
coefficients are:
where denotes the Fourier series integral.
* The pointwise 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 Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
. 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
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
(DTFT) operator. Consider two sequences