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Wikipedia / Convolution theorem

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. 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

with Fourier transforms:

denotes the Fourier transform operator is defined by:

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product is sometimes used instead.

The convolution theorem states that:

produces the corollary:

The theorem also generally applies to multi-dimensional functions.

in Lp:

indicates the inner product: and

The convolution is defined by:

Also:

Hence by Fubini's theorem is defined by the integral formula:

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)

and which can be expressed as periodic summations:

but nothing in the theorem requires that.

The Fourier series coefficients are:

denotes the Fourier series integral.

The corresponding convolution theorem is:

Functions of a discrete variable (sequences)

denotes the discrete-time Fourier transform:

The is defined by:

The convolution theorem for discrete sequences is:

Periodic convolution

:

and performing an inverse discrete Fourier transform samples (see ). The discrete convolution:

-periodic, and is called a periodic convolution-length DFT, the corresponding theorem is:

And therefore:

sequence is obtained by directly sampling the DTFT of the infinitely long impulse response.

a final simplification is:

This form is often used to efficiently implement numerical convolution by computer. (see and )

As a partial reciprocal, it has been shown 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 , which indicates that the DTFTs can be written as:

is thereby reduced to a discrete-frequency function:

follows from . Therefore, the equivalence of (5a) and (5b) requires:

We can also verify the inverse DTFT of (5b):

Convolution theorem for inverse Fourier transform

There is also a convolution theorem for the inverse Fourier transform:

" represents the Hadamard product" represents a convolution between the two matrices.

so that

Convolution theorem for tempered distributions

The convolution theorem extends to tempered distributions is an arbitrary tempered distribution:

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 is the Dirac comb both equations yield the Poisson summation formula is constantly one and these equations yield the Dirac comb identity.

See also

Notes

References

Further reading

Additional resources

For a visual representation of the use of the convolution theorem in signal processing, see:

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