Convolution theorem

convolution propertyDirect convolutionFourier convolution
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.wikipedia
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Fourier transform

continuous Fourier transformFourierFourier transforms
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.
Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem).

Convolution

convolvedconvolvingconvolution kernel
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. Let f and g be two functions with convolution f*g.
See Convolution theorem for a derivation of that property of convolution.

Fast Fourier transform

FFTFast Fourier Transform (FFT)Fast Fourier Transforms
With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced from [[Big O notation|]] to [[Big O notation|]], using big O notation.
Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime N, expresses a DFT of prime size N as a cyclic convolution of (composite) size N − 1, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods).

Laplace transform

Laplaces-domainLaplace domain
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).
The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions of

Hartley transform

cas (mathematics)Hartley kernelcos + sin
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).
There is also an analogue of the convolution theorem for the Hartley transform.

Discrete-time Fourier transform

convolution theoremDFTDTFT § Properties
where DTFT represents the discrete-time Fourier transform.
The convolution theorem for sequences is:

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
where DFT represents the discrete Fourier transform.
When data is convolved with a function with wide support, such as for downsampling by a large sampling ratio, because of the Convolution theorem and the FFT algorithm, it may be faster to transform it, multiply pointwise by the transform of the filter and then reverse transform it.

Mathematics

mathematicalmathmathematician
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.

Signal

signalselectrical signalsignaling
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.

Pointwise product

pointwise multiplicationproduct of functionspointwise
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms.

Time domain

time-domaintimetime coordinates
In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).

Frequency domain

frequency-domainFourier spaceFourier domain
In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).

List of Fourier-related transforms

Fourier-related transformFourier-related transforms
Versions of the convolution theorem are true for various Fourier-related transforms.

Function (mathematics)

functionfunctionsmathematical function
Let f and g be two functions with convolution f*g.

Asterisk

*starasterisks
(Note that the asterisk denotes convolution in this context, not standard multiplication.

Tensor product

tensor productsproduct
The tensor product symbol \otimes is sometimes used instead.)

Operator (mathematics)

operatoroperatorsmathematical operators
If \mathcal{F} denotes the Fourier transform operator, then and are the Fourier transforms of f and g, respectively.

Two-sided Laplace transform

bilateral Laplace transformtwo-sided
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).

Mellin transform

Cahen-Mellin integralCahen–Mellin integralMellin
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).

Mellin inversion theorem

inverse Mellin transformInverse two-sided Laplace transform
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).

Harmonic analysis

abstract harmonic analysisFourier theoryharmonic
It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.

Locally compact abelian group

Fourier transform on locally compact Abelian groups
It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.

Computer

computerscomputer systemdigital computer
This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.

Quadratic function

quadraticquadratic polynomialquadratically
This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.

Analysis of algorithms

computational complexitycomplexity analysiscomputationally expensive
This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.