In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms.
Other versions of the convolution theorem are applicable to various Fourier-related transforms.
The transform may be normalized in other ways, in which case constant scaling factors (typically
is defined by: In this context the asterisk denotes convolution, instead of standard multiplication.
The theorem also generally applies to multi-dimensional functions.
is defined by the integral formula: Note that
Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration): 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.
which can be expressed as periodic summations: In practice the non-zero portion of components
denotes the Fourier series integral.
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 (DTFT) operator.
is defined by: The convolution theorem for discrete sequences is:[3][4]: p.60 (2.169)
: These functions occur as the result of sampling
and performing an inverse discrete Fourier transform (DFT) on
sequence is obtained by directly sampling the DTFT of the infinitely long § Discrete Hilbert transform impulse response.
sequences whose non-zero duration is less than or equal to
This form is often used to efficiently implement numerical convolution by computer.
(see § Fast convolution algorithms and § Example) As a partial reciprocal, it has been shown [6] that any linear transform that turns convolution into a product is the DFT (up to a permutation of coefficients).
A time-domain derivation proceeds as follows: A frequency-domain derivation follows from § Periodic data, which indicates that the DTFTs can be written as: The product with
is thereby reduced to a discrete-frequency function: where the equivalence of
Therefore, the equivalence of (5a) and (5b) requires: We can also verify the inverse DTFT of (5b): There is also a convolution theorem for the inverse Fourier transform: Here, "
so that The convolution theorem extends to tempered distributions.
in order to guarantee the existence of both, convolution and multiplication product.
is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.
[7][8][9] In particular, every compactly supported tempered distribution, such as the Dirac delta, is "rapidly decreasing".
are smooth "slowly growing" ordinary functions.
is the Dirac comb both equations yield the Poisson summation formula and if, furthermore,
is constantly one and these equations yield the Dirac comb identity.
For a visual representation of the use of the convolution theorem in signal processing, see: