# 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**

### 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 products⊗product**

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**

### Mellin inversion theorem

**inverse Mellin transformInverse two-sided Laplace transform**

### 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.