Convolution theorem

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. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let and be two functions with convolution . (Note that the asterisk denotes convolution in this context, not standard multiplication. The tensor product symbol is sometimes used instead.)

If denotes the Fourier transform operator, then and are the Fourier transforms of and , respectively. Then

where denotes point-wise multiplication. It also works the other way around:

By applying the inverse Fourier transform , we can write:

and:

The relationships above are only valid for the form of the Fourier transform shown in the Proof section below. The transform may be normalized in other ways, in which case constant scaling factors (typically or ) will appear in the relationships above.

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.

This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity. With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced from to , using big O notation. This can be exploited to construct fast multiplication algorithms, as in Multiplication algorithm § Fourier transform methods.

ProofEdit

The proof here is shown for a particular normalization of the Fourier transform. As mentioned above, if the transform is normalized differently, then constant scaling factors will appear in the derivation.

Let   belong to the Lp-space  . Let   be the Fourier transform of   and   be the Fourier transform of  :

 

where the dot between   and   indicates the inner product of  . Let   be the convolution of   and  

 

Also

 

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):

 

Substituting   yields  . Therefore

 
 
 

These two integrals are the definitions of   and  , so:

 

QED.

Convolution theorem for inverse Fourier transformEdit

A similar argument, as the above proof, can be applied to the convolution theorem for the inverse Fourier transform;

 
 
 

and:

 

Functions of discrete variable sequencesEdit

By similar arguments, it can be shown that the discrete convolution of sequences   and   is given by:

 

where DTFT represents the discrete-time Fourier transform.

An important special case is the circular convolution of   and   defined by   where   is a periodic summation:

 

It can then be shown that:

 

where DFT represents the discrete Fourier transform.

The proof follows from § Periodic data, which indicates that   can be written as:

 

The product with   is thereby reduced to a discrete-frequency function:

  (also using § Sampling the DTFT).

The inverse DTFT is:

 

QED.

Convolution theorem for Fourier series coefficientsEdit

Two convolution theorems exist for the Fourier series coefficients of a periodic function:

  • The first convolution theorem states that if   and   are in  , the Fourier series coefficients of the 2π-periodic convolution of   and   are given by:
 [A]
where:
 
  • The second convolution theorem states that the Fourier series coefficients of the product of   and   are given by the discrete convolution of the   and   sequences:
 

See alsoEdit

NotesEdit

  1. ^ The scale factor is always equal to the period, 2π in this case.

ReferencesEdit

  • Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4
  • Weisstein, Eric W. "Convolution Theorem". MathWorld.
  • Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved November 19, 2010

Additional resourcesEdit

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