convolution theorem

Consider two functions $u(x)$ and $v(x)$ with Fourier transforms $U$ and $V$:

:$\backslash begin\{align\}$

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}

where $\backslash mathcal\{F\}$ denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically $2\backslash pi$ or $\backslash sqrt\{2\backslash pi\}$) will appear in the convolution theorem below. The convolution of $u$ and $v$ is defined by:

:$r(x)\; =\; \backslash \{u*v\backslash \}(x)\; \backslash triangleq\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; u(\backslash tau)\; v(x-\backslash tau)\backslash ,\; d\backslash tau\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; u(x-\backslash tau)\; v(\backslash tau)\backslash ,\; d\backslash tau.$

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol $\backslash otimes$ is sometimes used instead.

The convolution theorem states that:{{rp|eq.8}}

{{Equation box 1

|indent=:|cellpadding=0|border=0|background colour=white

|equation={{NumBlk||

$R(f)\; \backslash triangleq\; \backslash mathcal\{F\}\backslash \{r\backslash \}(f)\; =\; U(f)\; V(f).\; \backslash quad\; f\; \backslash in\; \backslash mathbb\{R\}$    

|{{EquationRef|Eq.1a}} }} }}

Applying the inverse Fourier transform $\backslash mathcal\{F\}^\{-1\},$ produces the corollary:{{rp|eqs.7,10}}

{{Equation box 1|title=Convolution theorem

|indent=|cellpadding=6|border=|border colour=#0073CF|background colour=#F5FFFA

|equation={{NumBlk|:|

$r(x)\; =\; \backslash \{u*v\backslash \}(x)\; =\; \backslash mathcal\{F\}^\{-1\}\backslash \{U\backslash cdot\; V\backslash \}.$    

|{{EquationRef|Eq.1b}} }} }}

The theorem also generally applies to multi-dimensional functions.

{{Collapse top|title=Multi-dimensional derivation of Eq.1}}

Consider functions $u,v$ in L^p-space $L^1(\backslash mathbb\{R\}^n),$ with Fourier transforms $U,V$:

:$$

\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}

where $f\backslash cdot\; x$ indicates the inner product of $\backslash mathbb\{R\}^n$:   $f\backslash cdot\; x\; =\; \backslash sum\_\{j=1\}^\{n\}\; \{f\}\_j\; x\_j,$   and   $dx\; =\; \backslash prod\_\{j=1\}^\{n\}\; d\; x\_j.$

The convolution of $u$ and $v$ is defined by:

:$r(x)\; \backslash triangleq\; \backslash int\_\{\backslash mathbb\{R\}^n\}\; u(\backslash tau)\; v(x-\backslash tau)\backslash ,\; d\backslash tau.$

Also:

:$\backslash iint\; |u(\backslash tau)v(x-\backslash tau)|\backslash ,dx\backslash ,d\backslash tau=\backslash int\; \backslash left(\; |u(\backslash tau)|\; \backslash int\; |v(x-\backslash tau)|\backslash ,dx\; \backslash right)\; \backslash ,d\backslash tau\; =\; \backslash int\; |u(\backslash tau)|\backslash ,\backslash |v\backslash |\_1\backslash ,d\backslash tau=\backslash |u\backslash |\_1\; \backslash |v\backslash |\_1.$

Hence by Fubini's theorem we have that $r\backslash in\; L^1(\backslash mathbb\{R\}^n)$ so its Fourier transform $R$ is defined by the integral formula:

:$$

\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}

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

:$$

\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}

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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.

Consider $P$-periodic functions $u\_\{\_P\}$   and   $v\_\{\_P\},$ which can be expressed as periodic summations:

:$u\_\{\_P\}(x)\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; u(x-mP)$   and   $v\_\{\_P\}(x)\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; v(x-mP).$

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

The Fourier series coefficients are:

:$\backslash begin\{align\}$

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}

where $\backslash mathcal\{F\}$ denotes the Fourier series integral.

• The product: $u\_\{\_P\}(x)\backslash cdot\; v\_\{\_P\}(x)$ is also $P$-periodic, and its Fourier series coefficients are given by the discrete convolution of the $U$ and $V$ sequences:

:$\backslash mathcal\{F\}\backslash \{u\_\{\_P\}\backslash cdot\; v\_\{\_P\}\backslash \}[k]\; =\; \backslash \{U*V\backslash \}[k].$

• The convolution:

:$\backslash begin\{align\}$

\{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}

is also $P$-periodic, and is called a periodic convolution.

{{Collapse top|title=Derivation of periodic convolution}}

:$\backslash begin\{align\}$

\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}

{{Collapse bottom}}

The corresponding convolution theorem is:

{{Equation box 1

|indent=|cellpadding=0|border=0|background colour=white

|equation={{NumBlk|:|

$\backslash mathcal\{F\}\backslash \{u\_\{\_P\}\; *\; v\backslash \}[k]\; =\backslash \; P\backslash cdot\; U[k]\backslash \; V[k].$    

|{{EquationRef|Eq.2}} }} }}

{{Collapse top|title=Derivation of Eq.2}}

:$\backslash begin\{align\}$

\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}

{{Collapse bottom

}}

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now $\backslash mathcal\{F\}$ denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences $u[n]$ and $v[n]$ with transforms $U$ and $V$:

:$\backslash begin\{align\}$

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}

The {{slink|Convolution#Discrete convolution|nopage=y}} of $u$ and $v$ is defined by:

:$r[n]\; \backslash triangleq\; (u\; *\; v)[n]\; =\; \backslash sum\_\{m=-\backslash infty\}^\backslash infty\; u[m]\backslash cdot\; v[n\; -\; m]\; =\; \backslash sum\_\{m=-\backslash infty\}^\backslash infty\; u[n-m]\backslash cdot\; v[m].$

The convolution theorem for discrete sequences is:{{rp|p.60 (2.169)}}

{{Equation box 1

|indent=|cellpadding=0|border=0|background colour=white

|equation={{NumBlk|:|

$R(f)\; =\; \backslash mathcal\{F\}\backslash \{u\; *\; v\backslash \}(f)\; =\backslash \; U(f)\; V(f).$    

|{{EquationRef|Eq.3}} }} }}

$U(f)$ and $V(f),$ as defined above, are periodic, with a period of 1. Consider $N$-periodic sequences $u\_\{\_N\}$ and $v\_\{\_N\}$:

:$u\_\{\_N\}[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; u[n-mN]$   and   $v\_\{\_N\}[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; v[n-mN],\; \backslash quad\; n\; \backslash in\; \backslash mathbb\{Z\}.$

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

:$\backslash \{u\_\{\_N\}\; *\; v\backslash \}[n]\backslash \; \backslash triangleq\; \backslash sum\_\{m=-\backslash infty\}^\{\backslash infty\}\; u\_\{\_N\}[m]\backslash cdot\; v[n-m]\; \backslash equiv\; \backslash sum\_\{m=0\}^\{N-1\}\; u\_\{\_N\}[m]\backslash cdot\; v\_\{\_N\}[n-m]$

is also $N$-periodic, and is called a periodic convolution. Redefining the $\backslash mathcal\{F\}$ operator as the $N$-length DFT, the corresponding theorem is:{{rp|p. 548}}

{{Equation box 1

|indent=|cellpadding=0|border=0|background colour=white

|equation={{NumBlk|:|

$\backslash mathcal\{F\}\backslash \{u\_\{\_N\}\; *\; v\backslash \}[k]\; =\backslash \; \backslash underbrace\{\backslash mathcal\{F\}\backslash \{u\_\{\_N\}\backslash \}[k]\}\_\{U(k/N)\}\; \backslash cdot\; \backslash underbrace\{\backslash mathcal\{F\}\backslash \{v\_\{\_N\}\backslash \}[k]\}\_\{V(k/N)\},\; \backslash quad\; k\; \backslash in\; \backslash mathbb\{Z\}.$    

|{{EquationRef|Eq.4a}} }} }}

And therefore:

{{Equation box 1

|indent=|cellpadding=0|border=0|background colour=white

|equation={{NumBlk|:|

$\backslash \{u\_\{\_N\}\; *\; v\backslash \}[n]\; =\backslash \; \backslash mathcal\{F\}^\{-1\}\backslash \{\backslash mathcal\{F\}\backslash \{u\_\{\_N\}\backslash \}\; \backslash cdot\; \backslash mathcal\{F\}\backslash \{v\_\{\_N\}\backslash \}\backslash \}.$    

|{{EquationRef|Eq.4b}} }} }}

Under the right conditions, it is possible for this $N$-length sequence to contain a distortion-free segment of a $u*v$ convolution. But when the non-zero portion of the $u(n)$ or $v(n)$ sequence is equal or longer than $N,$ some distortion is inevitable.  Such is the case when the $V(k/N)$ 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 $u$ and $v$ sequences whose non-zero duration is less than or equal to $N,$ a final simplification is:

{{Equation box 1

|title=Circular convolution

|indent=|cellpadding=6 |border= |border colour=#0073CF |background colour=#F5FFFA

|equation={{NumBlk|:|

$\backslash \{u\_\{\_N\}\; *\; v\backslash \}[n]\; =\backslash \; \backslash mathcal\{F\}^\{-1\}\backslash \{\backslash mathcal\{F\}\backslash \{u\backslash \}\; \backslash cdot\; \backslash mathcal\{F\}\backslash \{v\backslash \}\backslash \}.$    

|{{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 {{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}}

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:

:$$

\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}

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

:$$

\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)}

:$$

\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).

The product with $V(f)$ is thereby reduced to a discrete-frequency function:

:$$

\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}

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

:$\backslash scriptstyle\{\backslash rm\; DFT\}$

\displaystyle {\{u_{_N} * v\}[k]}

= \left(\scriptstyle{\rm DFT}

\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right).

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

:$$

\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}

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

Here, "$\backslash cdot$" represents the Hadamard product, and "$*$" represents a convolution between the two matrices.

:$\backslash begin\{align\}$

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\

&\mathcal{F}\{u \cdot v\}= \mathcal{F}\{u\}*\mathcal{F}\{v\}

\end{align}

so that

:$\backslash begin\{align\}$

&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}

The convolution theorem extends to tempered distributions.

Here, $v$ is an arbitrary tempered distribution:

:$\backslash begin\{align\}$

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\

&\mathcal{F}\{\alpha \cdot v\}= \mathcal{F}\{\alpha\}*\mathcal{F}\{v\}.

\end{align}

But $u\; =\; F\backslash \{\backslash alpha\backslash \}$ must be "rapidly decreasing" towards $-\backslash infty$ and $+\backslash infty$ in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if $\backslash alpha\; =\; F^\{-1\}\backslash \{u\backslash \}$ is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.{{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}}{{cite book | last=Barros-Neto | first=José | title=An Introduction to the Theory of Distributions | publisher=Dekker | location=New York, NY | year=1973}}{{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}}

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 $1$ are smooth "slowly growing" ordinary functions. If, for example, $v\backslash equiv\backslash operatorname\{\backslash text\{Ш\}\}$ is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, $u\backslash equiv\backslash delta$ is the Dirac delta then $\backslash alpha\; \backslash equiv\; 1$ is constantly one and these equations yield the Dirac comb identity.

• Moment-generating function of a random variable

{{notelist-ua}}

{{reflist|1|refs=

{{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}}

{{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}}

{{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}}

{{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}}

{{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

}}

}}

{{refbegin}}

• {{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}}

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

• Johns Hopkins University's Java-aided simulation: http://www.jhu.edu/signals/convolve/index.html

Category:Theorems in Fourier analysis

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