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Concept
Convolution theorem
Summary
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.
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:
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.
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.
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 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. The corresponding convolution theorem is:
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.
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