Editing Convolution theorem

{{Short description|Theorem in mathematics}}
In [[mathematics]], the '''convolution theorem''' states that under suitable conditions the [[Fourier transform]] of a [[convolution]] of two functions (or [[signal]]s) 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 [[List of Fourier-related transforms|Fourier-related transforms]].

== Functions of a continuous variable ==
Consider two functions <math>u(x)</math> and <math>v(x)</math> with [[Fourier transform]]s <math>U</math> and <math>V</math>:

:<math>\begin{align}
U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{-\infty}^{\infty}u(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R}\\
V(f) &\triangleq \mathcal{F}\{v\}(f) = \int_{-\infty}^{\infty}v(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R}
\end{align}</math>

where <math>\mathcal{F}</math> denotes the '''Fourier transform [[Operator (mathematics)|operator]]'''.  The transform may be normalized in other ways, in which case constant scaling factors (typically <math>2\pi</math> or <math>\sqrt{2\pi}</math>) will appear in the convolution theorem below.  The convolution of <math>u</math> and <math>v</math> is defined by:

:<math>r(x) = \{u*v\}(x) \triangleq \int_{-\infty}^{\infty} u(\tau) v(x-\tau)\, d\tau = \int_{-\infty}^{\infty} u(x-\tau) v(\tau)\, d\tau.</math>

In this context the [[asterisk]] denotes convolution, instead of standard multiplication. The [[tensor product]] symbol <math>\otimes</math> is sometimes used instead.

The '''convolution theorem''' states that''':'''<ref name=McGillem/><ref name=Weisstein/>{{rp|eq.8}}

{{Equation box 1
|indent=:|cellpadding=0|border=0|background colour=white
|equation={{NumBlk||
<math>R(f) \triangleq \mathcal{F}\{r\}(f) = U(f) V(f). \quad f \in \mathbb{R}</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1a}} }} }}

Applying the inverse Fourier transform <math>\mathcal{F}^{-1},</math> produces the corollary''':'''<ref name=Weisstein/>{{rp|eqs.7,10}}

{{Equation box 1|title='''Convolution theorem'''
|indent=|cellpadding=6|border=|border colour=#0073CF|background colour=#F5FFFA
|equation={{NumBlk|:|
<math>r(x) = \{u*v\}(x) = \mathcal{F}^{-1}\{U\cdot V\}.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1b}} }} }}

The theorem also generally applies to multi-dimensional functions.

{{Collapse top|title=Multi-dimensional derivation of Eq.1}}
Consider functions <math>u,v</math> in [[Lp space|L<sup>''p''</sup>]]-space <math>L^1(\mathbb{R}^n),</math> with Fourier transforms <math>U,V</math>''':'''

:<math>
\begin{align}
U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{\mathbb{R}^n} u(x) e^{-i 2 \pi f \cdot x} \, dx, \quad f \in \mathbb{R}^n\\
V(f) &\triangleq \mathcal{F}\{v\}(f) = \int_{\mathbb{R}^n} v(x) e^{-i 2 \pi f \cdot x} \, dx,
\end{align}
</math>

where <math>f\cdot x</math> indicates the [[dot product|inner product]] of '''<math>\mathbb{R}^n</math>:''' &nbsp; <math>f\cdot x = \sum_{j=1}^{n} {f}_j x_j,</math> &nbsp; and &nbsp; <math>dx = \prod_{j=1}^{n} d x_j.</math>

The [[convolution]] of <math>u</math> and <math>v</math> is defined by''':'''

:<math>r(x) \triangleq \int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau.</math>

Also''':'''

:<math>\iint |u(\tau)v(x-\tau)|\,dx\,d\tau=\int \left( |u(\tau)| \int |v(x-\tau)|\,dx \right) \,d\tau = \int |u(\tau)|\,\|v\|_1\,d\tau=\|u\|_1 \|v\|_1.</math>

Hence by [[Fubini's theorem]] we have that <math>r\in L^1(\mathbb{R}^n)</math> so its Fourier transform <math>R</math> is defined by the integral formula''':'''

:<math>
\begin{align}
R(f) \triangleq \mathcal{F}\{r\}(f) &= \int_{\mathbb{R}^n} r(x) e^{-i 2 \pi f \cdot x}\, dx\\
&= \int_{\mathbb{R}^n} \left(\int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau\right)\, e^{-i 2 \pi f \cdot x}\, dx.
\end{align}
</math>

Note that <math>|u(\tau)v(x-\tau)e^{-i 2\pi f \cdot x}|=|u(\tau)v(x-\tau)|,</math>&nbsp; Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration)''':'''

:<math>
\begin{align}
R(f) &= \int_{\mathbb{R}^n} u(\tau)
\underbrace{\left(\int_{\mathbb{R}^n} v(x-\tau)\ e^{-i 2 \pi f \cdot x}\,dx\right)}_{V(f)\ e^{-i 2 \pi f \cdot \tau}}\,d\tau\\
&=\underbrace{\left(\int_{\mathbb{R}^n} u(\tau)\ e^{-i 2\pi f \cdot \tau}\,d\tau\right)}_{U(f)}\ V(f).
\end{align}
</math>
{{Collapse bottom}}

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 group]]s.

=== Periodic convolution (Fourier series coefficients) ===

Consider <math>P</math>-periodic functions <math>u_{_P}</math> &nbsp; and &nbsp; <math>v_{_P},</math> which can be expressed as [[periodic summation]]s:

:<math>u_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} u(x-mP)</math> &nbsp; and &nbsp; <math>v_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} v(x-mP).</math>

In practice the non-zero portion of components <math>u</math> and <math>v</math> are often limited to duration <math>P,</math> but nothing in the theorem requires that.

The [[Fourier series]] coefficients are:

:<math>\begin{align}
   U[k] &\triangleq \mathcal{F}\{u_{_P}\}[k] = \frac{1}{P} \int_P u_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}; \quad \quad \scriptstyle \text{integration over any interval of length } P\\
   V[k] &\triangleq \mathcal{F}\{v_{_P}\}[k] = \frac{1}{P} \int_P v_{_P}(x) e^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}
 \end{align}</math>

where <math>\mathcal{F}</math> denotes the '''Fourier series integral'''.

* The product:  <math>u_{_P}(x)\cdot v_{_P}(x)</math> is also <math>P</math>-periodic, and its Fourier series coefficients are given by the [[Convolution#Discrete convolution|discrete convolution]] of the <math>U</math> and <math>V</math> sequences:

:<math>\mathcal{F}\{u_{_P}\cdot v_{_P}\}[k] = \{U*V\}[k].</math>

*The convolution:

:<math>\begin{align}
\{u_{_P} * v\}(x)\ &\triangleq \int_{-\infty}^{\infty} u_{_P}(x-\tau)\cdot v(\tau)\ d\tau\\
&\equiv \int_P u_{_P}(x-\tau)\cdot v_{_P}(\tau)\ d\tau; \quad \quad \scriptstyle \text{integration over any interval of length } P
\end{align}</math>

is also <math>P</math>-periodic, and is called a '''[[periodic convolution]]'''.

{{Collapse top|title=Derivation of periodic convolution}}
:<math>\begin{align}
\int_{-\infty}^\infty u_{_P}(x - \tau) \cdot v(\tau)\,d\tau
  &= \sum_{k=-\infty}^\infty \left[\int_{x_o+kP}^{x_o+(k+1)P} u_{_P}(x - \tau) \cdot v(\tau)\ d\tau\right] \quad x_0 \text{ is an arbitrary parameter}\\
  &=\sum_{k=-\infty}^\infty \left[\int_{x_o}^{x_o+P} \underbrace{u_{_P}(x - \tau-kP)}_{u_{_P}(x - \tau), \text{ by periodicity}} \cdot v(\tau + kP)\ d\tau\right] \quad \text{substituting } \tau \rightarrow \tau+kP\\
  &=\int_{x_o}^{x_o+P} u_{_P}(x - \tau) \cdot \underbrace{\left[\sum_{k=-\infty}^\infty v(\tau + kP)\right]}_{\triangleq \ v_{_P}(\tau)}\ d\tau
\end{align}</math>
{{Collapse bottom}}

The corresponding convolution theorem is''':'''

{{Equation box 1
|indent=|cellpadding=0|border=0|background colour=white
|equation={{NumBlk|:|
<math>\mathcal{F}\{u_{_P} * v\}[k] =\ P\cdot U[k]\ V[k].</math> &nbsp; &nbsp;
|{{EquationRef|Eq.2}} }} }}

<!--{{math proof|title=Derivation of Eq.2| proof = -->
{{Collapse top|title=Derivation of Eq.2}}
:<math>\begin{align}
\mathcal{F}\{u_{_P} * v\}[k] &\triangleq \frac{1}{P} \int_P \left(\int_P u_{_P}(\tau)\cdot v_{_P}(x-\tau)\ d\tau\right) e^{-i 2\pi k x/P} \, dx\\
&= \int_P u_{_P}(\tau)\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k x/P} dx\right) \, d\tau\\
&= \int_P u_{_P}(\tau)\ e^{-i 2\pi k \tau/P}
\underbrace{\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k (x-\tau)/P} dx\right)}_{V[k], \quad \text{due to periodicity}} \, d\tau\\
&=\underbrace{\left(\int_P\ u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} d\tau\right)}_{P\cdot U[k]}\ V[k].
\end{align}</math>
{{Collapse bottom
}}

== 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 <math>\mathcal{F}</math> denotes the '''[[discrete-time Fourier transform]]''' (DTFT) operator.  Consider two sequences <math>u[n]</math> and <math>v[n]</math> with transforms <math>U</math> and <math>V</math>:

:<math>\begin{align}
U(f) &\triangleq \mathcal{F}\{u\}(f) = \sum_{n=-\infty}^{\infty} u[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}, \\
V(f) &\triangleq \mathcal{F}\{v\}(f) = \sum_{n=-\infty}^{\infty} v[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}.
\end{align}</math>

The {{slink|Convolution#Discrete convolution|nopage=y}} of <math>u</math> and <math>v</math> is defined by''':'''

:<math>r[n] \triangleq (u * v)[n] = \sum_{m=-\infty}^\infty u[m]\cdot v[n - m] = \sum_{m=-\infty}^\infty u[n-m]\cdot v[m].</math>

The [[Convolution#Discrete convolution|'''convolution theorem''']] for discrete sequences is''':'''<ref name=Proakis/><ref name=Oppenheim/>{{rp|p.60 (2.169)}}

{{Equation box 1
|indent=|cellpadding=0|border=0|background colour=white
|equation={{NumBlk|:|
<math>R(f) = \mathcal{F}\{u * v\}(f) =\ U(f) V(f).</math> &nbsp; &nbsp;
|{{EquationRef|Eq.3}} }} }}

=== Periodic convolution ===
<math>U(f)</math> and <math>V(f),</math> as defined above, are periodic, with a period of 1.  Consider <math>N</math>-periodic sequences <math>u_{_N}</math> and <math>v_{_N}</math>''':'''

:<math>u_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} u[n-mN]</math> &nbsp; and &nbsp; <math>v_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} v[n-mN], \quad n \in \mathbb{Z}.</math>

These functions occur as the result of sampling <math>U</math> and <math>V</math> at intervals of <math>1/N</math> and performing an inverse '''[[discrete Fourier transform]]''' (DFT) on <math>N</math> samples  (see {{slink|Discrete-time_Fourier_transform#Sampling_the_DTFT|nopage=y}}).  The discrete convolution''':'''

:<math>\{u_{_N} * v\}[n]\ \triangleq \sum_{m=-\infty}^{\infty} u_{_N}[m]\cdot v[n-m] \equiv \sum_{m=0}^{N-1} u_{_N}[m]\cdot v_{_N}[n-m]</math>

is also <math>N</math>-periodic, and is called a '''[[periodic convolution]]'''. Redefining the <math>\mathcal{F}</math> operator as the <math>N</math>-length DFT, the corresponding theorem is:<ref name="Rabiner" /><ref name="Oppenheim" />{{rp|p. 548}}

{{Equation box 1
|indent=|cellpadding=0|border=0|background colour=white
|equation={{NumBlk|:|
<math>\mathcal{F}\{u_{_N} * v\}[k] =\ \underbrace{\mathcal{F}\{u_{_N}\}[k]}_{U(k/N)} \cdot \underbrace{\mathcal{F}\{v_{_N}\}[k]}_{V(k/N)}, \quad k \in \mathbb{Z}.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.4a}} }} }}

And therefore''':'''

{{Equation box 1
|indent=|cellpadding=0|border=0|background colour=white
|equation={{NumBlk|:|
<math>\{u_{_N} * v\}[n] =\ \mathcal{F}^{-1}\{\mathcal{F}\{u_{_N}\} \cdot \mathcal{F}\{v_{_N}\}\}.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.4b}} }} }}

Under the right conditions, it is possible for this <math>N</math>-length sequence to contain a distortion-free segment of a <math>u*v</math> convolution.  But when the non-zero portion of the <math>u(n)</math> or <math>v(n)</math> sequence is equal or longer than <math>N,</math> some distortion is inevitable.&nbsp; Such is the case when the <math>V(k/N)</math> sequence is obtained by directly sampling the DTFT of the infinitely long {{slink|Hilbert transform|Discrete Hilbert transform|nopage=y}} [[impulse response]].{{efn-ua
|1=An example is the [[MATLAB]] function, '''[http://www.mathworks.com/help/toolbox/signal/ref/hilbert.html;jsessionid=67ed4e69e9729363548abed31054 hilbert(u,N)]'''.}}

For <math>u</math> and <math>v</math> sequences whose non-zero duration is less than or equal to <math>N,</math> a final simplification is:
{{Equation box 1
|title='''[[Circular convolution]]'''
|indent=|cellpadding=6 |border= |border colour=#0073CF |background colour=#F5FFFA
|equation={{NumBlk|:|
<math>\{u_{_N} * v\}[n] =\ \mathcal{F}^{-1}\{\mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\}.</math> &nbsp; &nbsp;
|{{EquationRef|Eq.4c}} }} }}

This form is often used to efficiently implement numerical convolution by [[computer]].  (see {{slink|Convolution|Fast convolution algorithms|nopage=y}} and {{slink|Circular_convolution|Example|nopage=y}})

As a partial reciprocal, it has been shown <ref>{{cite book |last1=Amiot |first1=Emmanuel |title=Music through Fourier Space |series=Computational Music Science |date=2016 |publisher=Springer |location=Zürich |isbn=978-3-319-45581-5 |page=8 |doi=10.1007/978-3-319-45581-5 |s2cid=6224021 |url=https://link.springer.com/book/10.1007/978-3-319-45581-5 |ref=Theorem 1.11}}</ref>
that any linear transform that turns convolution into a product is the DFT (up to a permutation of coefficients).

{{Collapse top|title=Derivations of Eq.4}}
A time-domain derivation proceeds as follows''':'''

:<math>
\begin{align}
\scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k] &\triangleq \sum_{n=0}^{N-1} \left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot v_{_N}[n-m]\right) e^{-i 2\pi kn/N}\\
&= \sum_{m=0}^{N-1} u_{_N}[m] \left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi kn/N}\right)\\
&= \sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N}
\underbrace{
\left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi k(n-m)/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\quad \scriptstyle \text{due to periodicity}}\\
&= \underbrace{
\left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]}
\left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right).
\end{align}
</math>

A frequency-domain derivation follows from {{slink|DTFT|Periodic data|nopage=y}}, which indicates that the DTFTs can be written as''':'''

:<math>
\mathcal{F}\{u_{_N} * v\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k]\right)\cdot \delta\left(f-k/N\right). \quad \scriptstyle \mathsf{(Eq.5a)}
</math>

:<math>
\mathcal{F}\{u_{_N}\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \delta\left(f-k/N\right).
</math>

The product with <math>V(f)</math> is thereby reduced to a discrete-frequency function''':'''

:<math>
\begin{align}
\mathcal{F}\{u_{_N} * v\}(f) &= G_{_N}(f) V(f) \\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(f)\cdot \delta\left(f-k/N\right)\\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(k/N)\cdot \delta\left(f-k/N\right)\\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right) \cdot \delta\left(f-k/N\right), \quad \scriptstyle \mathsf{(Eq.5b)}
\end{align}
</math>

where the equivalence of <math>V(k/N)</math> and <math>\left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right)</math> follows from {{slink|DTFT|Sampling the DTFT|nopage=y}}.  Therefore, the equivalence of (5a) and (5b) requires:

:<math>\scriptstyle{\rm DFT}
\displaystyle {\{u_{_N} * v\}[k]}
 = \left(\scriptstyle{\rm DFT}
\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right).</math>

<br>We can also verify the inverse DTFT of (5b)''':'''

:<math>
\begin{align}
(u_{_N} * v)[n] & = \int_{0}^{1} \left(\frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \delta\left(f-k/N\right)\right)\cdot e^{i 2 \pi f n} df \\
& = \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \underbrace{\left(\int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\right)}_{\text{0, for} \ k\ \notin\ [0,\ N)} \\
& = \frac{1}{N} \sum_{k=0}^{N-1} \bigg(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\bigg)\cdot e^{i 2 \pi \frac{n}{N} k}\\
&=\ \scriptstyle{\rm DFT}^{-1} \displaystyle \bigg( \scriptstyle{\rm DFT}\displaystyle \{u_{_N}\}\cdot \scriptstyle{\rm DFT}\displaystyle \{v_{_N}\} \bigg).
\end{align}
</math>
{{Collapse bottom}}

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

Here, "<math>\cdot</math>" represents the [[Hadamard product (matrices)|Hadamard product]], and "<math>*</math>" represents a convolution between the two matrices.

:<math>\begin{align}
&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\
&\mathcal{F}\{u \cdot v\}= \mathcal{F}\{u\}*\mathcal{F}\{v\}
\end{align}</math>

so that

:<math>\begin{align}
&u*v= \mathcal{F}^{-1}\left\{\mathcal{F}\{u\}\cdot\mathcal{F}\{v\}\right\}\\
&u \cdot v= \mathcal{F}^{-1}\left\{\mathcal{F}\{u\}*\mathcal{F}\{v\}\right\}
\end{align}</math>

==Convolution theorem for tempered distributions==

The convolution theorem extends to [[Distribution (mathematics)#Convolution versus multiplication|tempered distributions]]. 
Here, <math>v</math> is an arbitrary tempered distribution:

:<math>\begin{align}
&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\
&\mathcal{F}\{\alpha \cdot v\}= \mathcal{F}\{\alpha\}*\mathcal{F}\{v\}.
\end{align}</math>

But <math>u = F\{\alpha\}</math> must be "rapidly decreasing" towards <math>-\infty</math> and <math>+\infty</math> in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if <math>\alpha = F^{-1}\{u\}</math> is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.<ref>{{cite book | last=Horváth | first=John | author-link=John Horvath (mathematician) | title=Topological Vector Spaces and Distributions | publisher=Addison-Wesley Publishing Company | location=Reading, MA | year=1966}}</ref><ref>{{cite book | last=Barros-Neto | first=José | title=An Introduction to the Theory of Distributions | publisher=Dekker | location=New York, NY | year=1973}}</ref><ref>{{cite book | last=Petersen | first=Bent E. | title=Introduction to the Fourier Transform and Pseudo-Differential Operators | publisher=Pitman Publishing | location=Boston, MA | year=1983}}</ref>

In particular, every compactly supported tempered distribution, such as the [[Dirac delta function|Dirac delta]], is "rapidly decreasing". Equivalently, [[Bandlimiting|bandlimited functions]], such as the function that is constantly <math>1</math> are smooth "slowly growing" ordinary functions. If, for example, <math>v\equiv\operatorname{\text{Ш}}</math> is the [[Dirac comb]] both equations yield the [[Poisson summation formula]] and if, furthermore, <math>u\equiv\delta</math> is the Dirac delta then <math>\alpha \equiv 1</math> is constantly one and these equations yield the [[Dirac comb#Dirac-comb identity|Dirac comb identity]].

== See also ==
* [[Moment-generating function]] of a [[random variable]]

==Notes==
{{notelist-ua}}

== References ==
{{reflist|1|refs=
<ref name=McGillem>
{{cite book |last1=McGillem |first1=Clare D. |last2=Cooper |first2=George R. |title=Continuous and Discrete Signal and System Analysis
 |page=118 (3–102) |publisher=Holt, Rinehart and Winston
 |edition=2 |date=1984 |isbn=0-03-061703-0}}
</ref>
<ref name=Proakis>
{{Citation |last1=Proakis |first1=John G. |last2=Manolakis |first2=Dimitri G. |title=Digital Signal Processing: Principles, Algorithms and Applications |page=297 |place=New Jersey |publisher=Prentice-Hall International |year=1996 |edition =3 |language=en |id=sAcfAQAAIAAJ |isbn=9780133942897 |bibcode=1996dspp.book.....P |url-access=registration |url=https://archive.org/details/digitalsignalpro00proa}}
</ref>
<ref name=Rabiner>
{{cite book |last1=Rabiner |first1=Lawrence R. |author1-link=Lawrence Rabiner |last2=Gold |first2=Bernard |date=1975 |title=Theory and application of digital signal processing |page=59 (2.163) |location=Englewood Cliffs, NJ |publisher=Prentice-Hall, Inc. |isbn=978-0139141010 |url-access=registration |url=https://archive.org/details/theoryapplicatio00rabi}}
</ref>
<ref name=Weisstein>
{{cite web |last1=Weisstein |first1=Eric W. |title=Convolution Theorem |url=https://mathworld.wolfram.com/ConvolutionTheorem.html |website=From MathWorld--A Wolfram Web Resource |access-date=8 February 2021}}
</ref>
<ref name=Oppenheim>
{{cite book
 |last1=Oppenheim
 |first1=Alan V.
 |author-link=Alan V. Oppenheim
 |last2=Schafer
 |first2=Ronald W.
 |author2-link=Ronald W. Schafer
 |last3=Buck
 |first3=John R.
 |title=Discrete-time signal processing
 |year=1999
 |publisher=Prentice Hall
 |location=Upper Saddle River, N.J.
 |isbn=0-13-754920-2
 |edition=2nd
 |url-access=registration
 |url=https://archive.org/details/discretetimesign00alan
}}
</ref>
}}
{{refbegin}}

== Further reading ==
*{{citation |first=Yitzhak |last=Katznelson |title=An introduction to Harmonic Analysis|year=1976|publisher=Dover |isbn=0-486-63331-4}}
*{{citation |first1=Bing |last1=Li |first2=G. Jogesh |last2=Babu |chapter=Convolution Theorem and Asymptotic Efficiency |title=A Graduate Course on Statistical Inference |location=New York |publisher=Springer |year=2019 |isbn=978-1-4939-9759-6 |pages=295–327 }}
*{{citation |last=Crutchfield |first=Steve |url=http://www.jhu.edu/signals/convolve/index.html |title=The Joy of Convolution |work=Johns Hopkins University |date=October 9, 2010 |access-date=November 19, 2010}}
{{refend}}

== Additional resources ==
For a visual representation of the use of the convolution theorem in [[signal processing]], see:

*[[Johns Hopkins University]]'s [[Java (software platform)|Java]]-aided simulation: http://www.jhu.edu/signals/convolve/index.html

[[Category:Theorems in Fourier analysis]]
[[Category:Articles containing proofs]]

[[de:Faltung (Mathematik)#Faltungstheorem 2]]
[[fr:Produit de convolution]]