Get Convolution Theorem essential facts below. View Videos or join the Convolution Theorem discussion. Add Convolution Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
Convolution Theorem
Theorem that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms
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.
Periodic convolution (Fourier series coefficients)
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:
is also -periodic, and is called a periodic convolution. The corresponding convolution theorem is:
Derivation of Eq.2
Functions of a discrete variable (sequences)
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. Consider two sequences and with transforms and :
This form is especially useful for implementing a numerical convolution on a computer. (see § Fast convolution algorithms) Under certain conditions, a sub-sequence of is equivalent to linear (aperiodic) convolution of and which is usually the desired result. (see § Example)
Derivations of Eq.4
A time-domain derivation proceeds as follows:
A frequency-domain derivation follows from § Periodic data, which indicates that the DTFTs can be written as:
The product with is thereby reduced to a discrete-frequency function:
where the equivalence of and follows from § Sampling the DTFT. Therefore, the equivalence of (5a) and (5b) requires:
We can also compute the inverse DTFT of (5b):
Convolution theorem for inverse Fourier transform
There is also a convolution theorem for the inverse Fourier transform:
but must be "rapidly decreasing" towards and
in order to guarantee the existence of both, convolution and multiplication product.
Equivalently, if is a smooth "slowly growing"
ordinary function, it guarantees the existence of both, multiplication and convolution product.
.[4][5][6]
In particular, every compactly supported tempered distribution,
such as the Dirac Delta, is "rapidly decreasing".
Equivalently, bandlimited functions, such as the function that is constantly
are smooth "slowly growing" ordinary functions.
If, for example, is the Dirac comb both equations yield the Poisson Summation Formula and if, furthermore, is the Dirac delta then is constantly one and these equations yield the Dirac comb identity.
^
McGillem, Clare D.; Cooper, George R. (1984). Continuous and Discrete Signal and System Analysis (2 ed.). Holt, Rinehart and Winston. p. 118 (3-102). ISBN0-03-061703-0.
Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN0-486-63331-4
Li, Bing; Babu, G. Jogesh (2019), "Convolution Theorem and Asymptotic Efficiency", A Graduate Course on Statistical Inference, New York: Springer, pp. 295-327, ISBN978-1-4939-9759-6
Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved 2010
Additional resources
For a visual representation of the use of the convolution theorem in signal processing, see: