In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency 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 (typically or ) will appear in the convolution theorem below. The convolution of and is defined by:
Applying the inverse Fourier transform , produces the corollary:[b]
The theorem also generally applies to multi-dimensional functions.
Multi-dimensional derivation of Eq.1
Consider functions in Lp-space , with Fourier transforms :
where indicates the inner product of : and
The convolution of and is defined by:
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):
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.
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 convolution:
is also -periodic, and is called a periodic convolution. The corresponding convolution theorem is:
Derivation of Eq.2
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 (DTFT) operator. Consider two sequences and with transforms and :
The § Discrete convolution of and is defined by:
Consider -periodic sequences and :
In practice the non-zero portion of components and are often limited to duration but nothing in the theorem requires that. The discrete convolution:
is also -periodic, and is called a periodic convolution. In this case, the operator can be redefined as the much simpler -length Discrete Fourier transform (DFT). And the corresponding theorem is:[d]
For and sequences whose non-zero duration is less than or equal to N, a final simplification is:
This form is especially useful for implementing a numerical convolution on a computer. (see § Fast convolution algorithms) Under certain conditions, a sub-sequence of is equivalent to linear (aperiodic) convolution of and which is usually the desired result. (see § Example)
Derivations of Eq.4
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 and follows from § Sampling the DTFT. Therefore, the equivalence of (5a) and (5b) requires:
Convolution theorem for inverse Fourier transform
There is also a convolution theorem for the inverse Fourier transform:
Convolution theorem for tempered distributions
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.
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For a visual representation of the use of the convolution theorem in signal processing, see: