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Convolution

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Convolution_(disambiguation)" class="mw-disambig" title="Convolution (disambiguation)">Convolution (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a> (in particular, <a href="Functional_analysis" title="Functional analysis">functional analysis</a>), convolution is a <a href="Operation_(mathematics)" title="Operation (mathematics)">mathematical operation</a> on two <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> that produces a third function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}">

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<annotation encoding="application/x-tex">{\displaystyle f*g}</annotation>

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</math><img src="./de088e4a3777d3b5d2787fdec81acd91e78a719e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}" loading="lazy">, as the <a href="Integral" title="Integral">integral</a> of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see <a href="#Properties">commutativity</a>). Graphically, it expresses how the 'shape' of one function is modified by the other.

Some features of convolution are similar to <a href="Cross-correlation" title="Cross-correlation">cross-correlation</a>: for real-valued functions, of a continuous or discrete variable, convolution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}">

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<annotation encoding="application/x-tex">{\displaystyle f*g}</annotation>

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</math><img src="./de088e4a3777d3b5d2787fdec81acd91e78a719e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}" loading="lazy"> differs from cross-correlation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\star g}">

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<annotation encoding="application/x-tex">{\displaystyle f\star g}</annotation>

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</math><img src="./371d3161cd7e094182a184d7601b49880228385c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\star g}" loading="lazy"> only in that either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

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</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"> is reflected about the y-axis in convolution; thus it is a cross-correlation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(-x)}</annotation>

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</math><img src="./0940c1b2e8d4161580c3bff2362ff7bf397c75b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.063ex; height:2.843ex;" alt="{\displaystyle g(-x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(-x)}</annotation>

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</math><img src="./9b64f0e1491306a6f567b373ae27f024cff95cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.226ex; height:2.843ex;" alt="{\displaystyle f(-x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

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</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy">.<a href="#cite_note-1">[A]</a> For complex-valued functions, the cross-correlation operator is the <a href="Hermitian_adjoint" title="Hermitian adjoint">adjoint</a> of the convolution operator.

Convolution has applications that include <a href="Probability" title="Probability">probability</a>, <a href="Statistics" title="Statistics">statistics</a>, <a href="Acoustics" title="Acoustics">acoustics</a>, <a href="Spectroscopy" title="Spectroscopy">spectroscopy</a>, <a href="Signal_processing" title="Signal processing">signal processing</a> and <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a>, <a href="Geophysics" title="Geophysics">geophysics</a>, <a href="Engineering" title="Engineering">engineering</a>, <a href="Physics" title="Physics">physics</a>, <a href="Computer_vision" title="Computer vision">computer vision</a> and <a href="Differential_equation" title="Differential equation">differential equations</a>.<a href="#cite_note-2">[1]</a>

The convolution can be defined for functions on <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> and other <a href="Group_(mathematics)" title="Group (mathematics)">groups</a> (as <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a>). For example, <a href="Periodic_function" title="Periodic function">periodic functions</a>, such as the <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a>, can be defined on a <a href="Circle" title="Circle">circle</a> and convolved by <a href="Periodic_convolution" class="mw-redirect" title="Periodic convolution">periodic convolution</a>. (See row 18 at <a href="DTFT" class="mw-redirect" title="DTFT">DTFT § Properties</a>.) A discrete convolution can be defined for functions on the set of <a href="Integer" title="Integer">integers</a>.

Generalizations of convolution have applications in the field of <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>, and in the design and implementation of <a href="Finite_impulse_response" title="Finite impulse response">finite impulse response</a> filters in signal processing.

Computing the <a href="Inverse_function" title="Inverse function">inverse</a> of the convolution operation is known as <a href="Deconvolution" title="Deconvolution">deconvolution</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}">

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<annotation encoding="application/x-tex">{\displaystyle f*g}</annotation>

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</math><img src="./de088e4a3777d3b5d2787fdec81acd91e78a719e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}" loading="lazy">, denoting the operator with the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

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<annotation encoding="application/x-tex">{\displaystyle *}</annotation>

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</math><img src="./8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">.<a href="#cite_note-3">[B]</a> It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of <a href="Integral_transform" title="Integral transform">integral transform</a>:

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .}</annotation>

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</math><img src="./7206fe9d1b6e0ad341016c78c3f939e0d3c1d14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.175ex; height:6.009ex;" alt="{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .}" loading="lazy"></dd></dl>

An equivalent definition is (see <a href="#Properties">commutativity</a>):

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .}</annotation>

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</math><img src="./a0fe2ad24b9d9b5f10a06a2580c4a80f1c1e87ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.175ex; height:6.009ex;" alt="{\displaystyle (f*g)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .}" loading="lazy"></dd></dl>

While the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is used above, it need not represent the time domain. At each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, the convolution formula can be described as the area under the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation>

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</math><img src="./bcba00f11285b589b0ff57beeaf118defab2cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}" loading="lazy"> weighted by the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

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</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> shifted by the amount <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> changes, the weighting function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

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</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> emphasizes different parts of the input function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation>

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</math><img src="./bcba00f11285b589b0ff57beeaf118defab2cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}" loading="lazy">; If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is a positive value, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

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</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

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</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> that slides or is shifted along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">-axis toward the right (toward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

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<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

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</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">) by the amount of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, while if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is a negative value, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

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</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

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</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> that slides or is shifted toward the left (toward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">) by the amount of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|}">

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<annotation encoding="application/x-tex">{\displaystyle |t|}</annotation>

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</math><img src="./fa9b1439497e4de838a6b1bcf724ef7a8fe48147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.133ex; height:2.843ex;" alt="{\displaystyle |t|}" loading="lazy">.

For functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> <a href="Support_(mathematics)" title="Support (mathematics)">supported</a> on only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation>

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</math><img src="./8dc2d914c2df66bc0f7893bfb8da36766650fe47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}" loading="lazy"> (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:

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</mrow>

</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .}</annotation>

</semantics>

</math><img src="./d852585370b174cf685990d9b33a8ae615a3adaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.371ex; height:6.176ex;" alt="{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .}" loading="lazy"></dd></dl>

For the multi-dimensional formulation of convolution, see <a href="#Domain_of_definition">domain of definition</a> (below).

<div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3></div>

A common engineering notational convention is:<a href="#cite_note-4">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(t)*g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)},}</annotation>

</semantics>

</math><img src="./9a4c134c1adaa4ed3965fb4a3adacc56ba7c06c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:34.015ex; height:10.009ex;" alt="{\displaystyle f(t)*g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)},}" loading="lazy"></dd></dl>

which has to be interpreted carefully to avoid confusion. For instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)*g(t-t_{0})}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle f(t)*g(t-t_{0})}</annotation>

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</math><img src="./da85a8202af7a2978ff30530efb5477cc2852334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.621ex; height:2.843ex;" alt="{\displaystyle f(t)*g(t-t_{0})}" loading="lazy"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t-t_{0})}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(t-t_{0})}</annotation>

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</math><img src="./234cb6bf010ce9f18c6fdbfef1a75d87661c52ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.782ex; height:2.843ex;" alt="{\displaystyle (f*g)(t-t_{0})}" loading="lazy">, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t-t_{0})*g(t-t_{0})}">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(t-t_{0})*g(t-t_{0})}</annotation>

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</math><img src="./c225f7585abf969b8bd4319b078d10f44437abde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.356ex; height:2.843ex;" alt="{\displaystyle f(t-t_{0})*g(t-t_{0})}" loading="lazy"> is in fact equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t-2t_{0})}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(t-2t_{0})}</annotation>

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</math><img src="./792889872f217f9e95757a97da74f00cf27f2e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.944ex; height:2.843ex;" alt="{\displaystyle (f*g)(t-2t_{0})}" loading="lazy">.<a href="#cite_note-5">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Relations_with_other_transforms">Relations with other transforms</h3></div>

Given two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

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</math><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation>

</semantics>

</math><img src="./b84f700860ee7af27797d11ddfad3d185eb7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}" loading="lazy"> with <a href="Two-sided_Laplace_transform" title="Two-sided Laplace transform">bilateral Laplace transforms</a> (two-sided Laplace transform)

</mrow>

</mrow>

</msubsup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u}</annotation>

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</math><img src="./f1dbb8cf7929e70105bbdb54fb36ea32e72d4656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.079ex; height:6.009ex;" alt="{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u}" loading="lazy"></dd></dl>

and

</mrow>

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</msubsup>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v}</annotation>

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</math><img src="./77ce3fd4721a6a260e4fdbf8d82187735c9e3d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.456ex; height:6.009ex;" alt="{\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v}" loading="lazy"></dd></dl>

respectively, the convolution operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(t)}</annotation>

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</math><img src="./aa3f22d92993aba0b3be3ff9238f5e5373c7ca7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle (f*g)(t)}" loading="lazy"> can be defined as the <a href="Inverse_Laplace_transform" title="Inverse Laplace transform">inverse Laplace transform</a> of the product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(s)}</annotation>

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</math><img src="./d633e75b0c0edb9b5cf174df6f79f4b90634718b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.641ex; height:2.843ex;" alt="{\displaystyle F(s)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation>

</semantics>

</math><img src="./cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}" loading="lazy">.<a href="#cite_note-6">[4]</a><a href="#cite_note-7">[5]</a> More precisely,

<mtr>

<mtd>

</mtd>

<mtd>

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</mrow>

</msubsup>

<msup>

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</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

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</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

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</msup>

</mrow>

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</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}}</annotation>

</semantics>

</math><img src="./23204082e6504f44f9fa88d8449b0d6da90575e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:50.546ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}}" loading="lazy"></dd></dl>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=u+v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=u+v}</annotation>

</semantics>

</math><img src="./d6cab55051537427d67893e108a4f322a84b540e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.236ex; height:2.176ex;" alt="{\displaystyle t=u+v}" loading="lazy">, then

<mtr>

<mtd>

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<mtd>

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</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

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</msup>

</mrow>

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</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munder>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(f*g)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(f*g)(t)\ {\text{d}}t.\end{aligned}}}</annotation>

</semantics>

</math><img src="./861a917e43e207e2fd7349149fc4cbe9b789c433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:47.952ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(f*g)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(f*g)(t)\ {\text{d}}t.\end{aligned}}}" loading="lazy"></dd></dl>

Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)\cdot G(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(s)\cdot G(s)}</annotation>

</semantics>

</math><img src="./97cdabf7cb0072ab5628fd96f82627819fe7743a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:2.843ex;" alt="{\displaystyle F(s)\cdot G(s)}" loading="lazy"> is the bilateral Laplace transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(t)}</annotation>

</semantics>

</math><img src="./aa3f22d92993aba0b3be3ff9238f5e5373c7ca7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle (f*g)(t)}" loading="lazy">. A similar derivation can be done using the <a href="Laplace_transform" title="Laplace transform">unilateral Laplace transform</a> (one-sided Laplace transform).

The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system theory</a> for a derivation of convolution as the result of LTI constraints. In terms of the <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a <a href="Transfer_function" title="Transfer function">transfer function</a>). See <a href="Convolution_theorem" title="Convolution theorem">Convolution theorem</a> for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.

<div class="mw-heading mw-heading2"><h2 id="Visual_explanation">Visual explanation</h2></div>

<td>

<div><ol style="margin-left:1.6em;"><li>Express each function in terms of a <a href="Free_variables_and_bound_variables" title="Free variables and bound variables">dummy variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau .}</annotation>

</semantics>

</math><img src="./871bb01391136d3551c8ea59059e106be2a403cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.849ex; height:1.676ex;" alt="{\displaystyle \tau .}" loading="lazy"></li><li>Reflect one of the functions: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(\tau )}</annotation>

</semantics>

</math><img src="./f3ca3806d8f1456510d15896379772656cd465da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.127ex; height:2.843ex;" alt="{\displaystyle g(\tau )}" loading="lazy"> → <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau ).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(-\tau ).}</annotation>

</semantics>

</math><img src="./1654ed026191fb24a5f4e237101c1cccabd82110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.582ex; height:2.843ex;" alt="{\displaystyle g(-\tau ).}" loading="lazy"></li><li>Add an offset of the independent variable, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, which allows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

</semantics>

</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> to slide along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">-axis. If t is a positive value, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

</semantics>

</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

</semantics>

</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> that slides or is shifted along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">-axis toward the right (toward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

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</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">) by the amount of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is a negative value, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

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<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

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</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation>

</semantics>

</math><img src="./7270aba4a907afb2c1e074efac5f90ccc6ede41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}" loading="lazy"> that slides or is shifted toward the left (toward <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

</semantics>

</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">) by the amount of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |t|}</annotation>

</semantics>

</math><img src="./fa9b1439497e4de838a6b1bcf724ef7a8fe48147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.133ex; height:2.843ex;" alt="{\displaystyle |t|}" loading="lazy">.</li><li>Start <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

</semantics>

</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"> and slide it all the way to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

</semantics>

</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">. Wherever the two functions intersect, find the integral of their product. In other words, at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, compute the area under the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation>

</semantics>

</math><img src="./bcba00f11285b589b0ff57beeaf118defab2cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}" loading="lazy"> weighted by the weighting function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau ).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t-\tau ).}</annotation>

</semantics>

</math><img src="./21f5e8326628cba63e490e29f105c1c54ab3393f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.454ex; height:2.843ex;" alt="{\displaystyle g(t-\tau ).}" loading="lazy"></li></ol></div>

The resulting <a href="Waveform" title="Waveform">waveform</a> (not shown here) is the convolution of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

</semantics>

</math><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> is a <a href="Unit_impulse" class="mw-redirect" title="Unit impulse">unit impulse</a>, the result of this process is simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation>

</semantics>

</math><img src="./b84f700860ee7af27797d11ddfad3d185eb7af0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}" loading="lazy">. Formally:

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\delta (\tau )g(t-\tau )\,d\tau =g(t)}</annotation>

</semantics>

</math><img src="./f9484cb49fecb504704370b2ad56a83766e18903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.369ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }\delta (\tau )g(t-\tau )\,d\tau =g(t)}" loading="lazy"></dd></dl>

</td>

<td>

</td></tr>

<tr>

<td>In this example, the red-colored "pulse", <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ g(\tau ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ g(\tau ),}</annotation>

</semantics>

</math><img src="./8682ec88f51b5ebb0dda735983ec3165d825382c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.355ex; height:2.843ex;" alt="{\displaystyle \ g(\tau ),}" loading="lazy"> is an <a href="Even_function" class="mw-redirect" title="Even function">even function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ g(-\tau )=g(\tau )\ ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\ g(-\tau )=g(\tau )\ ),}</annotation>

</semantics>

</math><img src="./652e3c0430a4931e2265d6947e582e557882021e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.779ex; height:2.843ex;" alt="{\displaystyle (\ g(-\tau )=g(\tau )\ ),}" loading="lazy"> so convolution is equivalent to correlation. A snapshot of this "movie" shows functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation>

</semantics>

</math><img src="./3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation>

</semantics>

</math><img src="./bcba00f11285b589b0ff57beeaf118defab2cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}" loading="lazy"> (in blue) for some value of parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t,}</annotation>

</semantics>

</math><img src="./4ea3ad87830a1055c7b85c04cf940cfd3b847ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" loading="lazy"> which is arbitrarily defined as the distance along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> axis from the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =0}</annotation>

</semantics>

</math><img src="./4422051052da869dc5b1f0e1cfb06a045ee0c36a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau =0}" loading="lazy"> to the center of the red pulse. The amount of yellow is the area of the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )\cdot g(t-\tau ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\tau )\cdot g(t-\tau ),}</annotation>

</semantics>

</math><img src="./b778e710d0b6bed2b0b5beed59be539e9dfdde41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.423ex; height:2.843ex;" alt="{\displaystyle f(\tau )\cdot g(t-\tau ),}" loading="lazy"> computed by the convolution/correlation integral. The movie is created by continuously changing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and recomputing the integral. The result (shown in black) is a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t,}</annotation>

</semantics>

</math><img src="./4ea3ad87830a1055c7b85c04cf940cfd3b847ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}" loading="lazy"> but is plotted on the same axis as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ,}</annotation>

</semantics>

</math><img src="./26d6cc28c28ff4ff88402f47f2a99e583e9e045f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.849ex; height:2.009ex;" alt="{\displaystyle \tau ,}" loading="lazy"> for convenience and comparison.

</td>

<td>

</td></tr>

<tr>

<td>In this depiction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation>

</semantics>

</math><img src="./bcba00f11285b589b0ff57beeaf118defab2cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}" loading="lazy"> could represent the response of a <a href="Resistor-capacitor_circuit" class="mw-redirect" title="Resistor-capacitor circuit">resistor-capacitor circuit</a> to a narrow pulse that occurs at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =0.}</annotation>

</semantics>

</math><img src="./536a344c98711768c9d2055a50e50c62e69b5fac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.11ex; height:2.176ex;" alt="{\displaystyle \tau =0.}" loading="lazy"> In other words, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\tau )=\delta (\tau ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(\tau )=\delta (\tau ),}</annotation>

</semantics>

</math><img src="./ecb1ed1bb9b4152e2dda12a3ab518e94a5d5a35e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.932ex; height:2.843ex;" alt="{\displaystyle g(\tau )=\delta (\tau ),}" loading="lazy"> the result of convolution is just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t).}</annotation>

</semantics>

</math><img src="./db88c28d6c644c905a4e12de7971c3d1eed37540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.574ex; height:2.843ex;" alt="{\displaystyle f(t).}" loading="lazy"> But when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(\tau )}</annotation>

</semantics>

</math><img src="./f3ca3806d8f1456510d15896379772656cd465da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.127ex; height:2.843ex;" alt="{\displaystyle g(\tau )}" loading="lazy"> is the wider pulse (in red), the response is a "smeared" version of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t).}</annotation>

</semantics>

</math><img src="./db88c28d6c644c905a4e12de7971c3d1eed37540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.574ex; height:2.843ex;" alt="{\displaystyle f(t).}" loading="lazy"> It begins at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=-0.5,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=-0.5,}</annotation>

</semantics>

</math><img src="./122a1d7a1cc954c319923b4e430df85a9a1edd36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.365ex; height:2.509ex;" alt="{\displaystyle t=-0.5,}" loading="lazy"> because we defined <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> as the distance from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =0}</annotation>

</semantics>

</math><img src="./4422051052da869dc5b1f0e1cfb06a045ee0c36a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau =0}" loading="lazy"> axis to the center of the wide pulse (instead of the leading edge).

</td>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Historical_developments">Historical developments</h2></div>

One of the earliest uses of the convolution integral appeared in <a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">D'Alembert</a>'s derivation of <a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a> in Recherches sur différents points importants du système du monde, published in 1754.<a href="#cite_note-8">[6]</a>

Also, an expression of the type:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(u)\cdot g(x-u)\,du}</annotation>

</semantics>

</math><img src="./1f1f152a830ce18ae1fa1716f0d88244597d5dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.035ex; height:5.676ex;" alt="{\displaystyle \int f(u)\cdot g(x-u)\,du}" loading="lazy"></dd></dl>

is used by <a href="Sylvestre_Fran%C3%A7ois_Lacroix" title="Sylvestre François Lacroix">Sylvestre François Lacroix</a> on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.<a href="#cite_note-9">[7]</a> Soon thereafter, convolution operations appear in the works of <a href="Pierre_Simon_Laplace" class="mw-redirect" title="Pierre Simon Laplace">Pierre Simon Laplace</a>, <a href="Jean-Baptiste_Joseph_Fourier" class="mw-redirect" title="Jean-Baptiste Joseph Fourier">Jean-Baptiste Joseph Fourier</a>, <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in <a href="German_language" title="German language">German</a>), composition product, superposition integral, and <a href="John_Renshaw_Carson" title="John Renshaw Carson">Carson</a>'s integral.<a href="#cite_note-10">[8]</a> Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses.<a href="#cite_note-11">[9]</a><a href="#cite_note-12">[10]</a>

The operation:

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,}</annotation>

</semantics>

</math><img src="./ebda18d1386dc4edffcc831ad0bf4393156fb452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.41ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,}" loading="lazy"></dd></dl>

is a particular case of composition products considered by the Italian mathematician <a href="Vito_Volterra" title="Vito Volterra">Vito Volterra</a> in 1913.<a href="#cite_note-13">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Circular_convolution">Circular convolution</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Circular_convolution" title="Circular convolution">Circular convolution</a></div>

When a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{T}}</annotation>

</semantics>

</math><img src="./56d8eb15ca707256c0af06ec582d34853c7e4ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.498ex; height:2.009ex;" alt="{\displaystyle g_{T}}" loading="lazy"> is periodic, with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, then for functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*g_{T}}</annotation>

</semantics>

</math><img src="./b02c2270d194a1ab42521472b5c969e641d3931b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.972ex; height:2.509ex;" alt="{\displaystyle f*g_{T}}" loading="lazy"> exists, the convolution is also periodic and identical to:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g_{T})(t)\equiv \int _{t_{0}}^{t_{0}+T}\left[\sum _{k=-\infty }^{\infty }f(\tau +kT)\right]g_{T}(t-\tau )\,d\tau ,}</annotation>

</semantics>

</math><img src="./46ca67ae76bc1e6841511aa12fab10aed9cb970d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.633ex; height:7.509ex;" alt="{\displaystyle (f*g_{T})(t)\equiv \int _{t_{0}}^{t_{0}+T}\left[\sum _{k=-\infty }^{\infty }f(\tau +kT)\right]g_{T}(t-\tau )\,d\tau ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> is an arbitrary choice. The summation is called a <a href="Periodic_summation" title="Periodic summation">periodic summation</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{T}}</annotation>

</semantics>

</math><img src="./56d8eb15ca707256c0af06ec582d34853c7e4ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.498ex; height:2.009ex;" alt="{\displaystyle g_{T}}" loading="lazy"> is a periodic summation of another function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*g_{T}}</annotation>

</semantics>

</math><img src="./b02c2270d194a1ab42521472b5c969e641d3931b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.972ex; height:2.509ex;" alt="{\displaystyle f*g_{T}}" loading="lazy"> is known as a circular or cyclic convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">.

And if the periodic summation above is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{T}}</annotation>

</semantics>

</math><img src="./30c170d388d33f69f7bbfb345b3bb50599d15f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.529ex; height:2.509ex;" alt="{\displaystyle f_{T}}" loading="lazy">, the operation is called a periodic convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{T}}</annotation>

</semantics>

</math><img src="./30c170d388d33f69f7bbfb345b3bb50599d15f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.529ex; height:2.509ex;" alt="{\displaystyle f_{T}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{T}}</annotation>

</semantics>

</math><img src="./56d8eb15ca707256c0af06ec582d34853c7e4ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.498ex; height:2.009ex;" alt="{\displaystyle g_{T}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Discrete_convolution">Discrete convolution</h2></div>

For complex-valued functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> defined on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> of integers, the discrete convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is given by:<a href="#cite_note-14">[12]</a>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[m]g[n-m],}</annotation>

</semantics>

</math><img src="./ea98dff26dac2459282e10b7c7e4f5e5b6c91dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.16ex; height:6.843ex;" alt="{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[m]g[n-m],}" loading="lazy"></dd></dl>

or equivalently (see <a href="#Properties">commutativity</a>) by:

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[n-m]g[m].}</annotation>

</semantics>

</math><img src="./c98a8db58f6ced1a80968fe0f2c99a7a81e782f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.16ex; height:6.843ex;" alt="{\displaystyle (f*g)[n]=\sum _{m=-\infty }^{\infty }f[n-m]g[m].}" loading="lazy"></dd></dl>

The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two <a href="Polynomial" title="Polynomial">polynomials</a>, then the coefficients of the <a href="Polynomial_multiplication" class="mw-redirect" title="Polynomial multiplication">ordinary product of the two polynomials</a> are the convolution of the original two sequences. This is known as the <a href="Cauchy_product" title="Cauchy product">Cauchy product</a> of the coefficients of the sequences.

Thus, when g is non-zero over a finite interval [-M,+M] (representing, for instance, a <a href="Finite_impulse_response" title="Finite impulse response">finite impulse response</a>), a finite summation may be used:<a href="#cite_note-15">[13]</a>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m].}</annotation>

</semantics>

</math><img src="./fddacde29cbcb3c6fca263493335c31a4d2ebce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.244ex; height:7.509ex;" alt="{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m].}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Circular_discrete_convolution">Circular discrete convolution</h3></div>

When a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{_{N}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{_{N}}}</annotation>

</semantics>

</math><img src="./004ae13a50553e9818e9c72da3fa4c63e540c0cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.69ex; height:2.343ex;" alt="{\displaystyle g_{_{N}}}" loading="lazy"> is periodic, with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N,}</annotation>

</semantics>

</math><img src="./b2285a1804b7fdcac187d155af09aff63152dd56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle N,}" loading="lazy"> then for functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,}</annotation>

</semantics>

</math><img src="./9e9687ea22c0f310582e97ee5f6c6a5fca28203d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g_{_{N}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*g_{_{N}}}</annotation>

</semantics>

</math><img src="./b0e55b54474ccd9dfcf4245d5a225215cda943b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.163ex; height:2.843ex;" alt="{\displaystyle f*g_{_{N}}}" loading="lazy"> exists, the convolution is also periodic and identical to:

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g_{_{N}})[n]\equiv \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{f}[m+kN]\right)g_{_{N}}[n-m].}</annotation>

</semantics>

</math><img src="./5e79e433507d93d1d4beb971829368cbcce26f96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.853ex; height:7.509ex;" alt="{\displaystyle (f*g_{_{N}})[n]\equiv \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{f}[m+kN]\right)g_{_{N}}[n-m].}" loading="lazy"></dd></dl>

The summation on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is called a <a href="Periodic_summation" title="Periodic summation">periodic summation</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>

</semantics>

</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{_{N}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{_{N}}}</annotation>

</semantics>

</math><img src="./004ae13a50553e9818e9c72da3fa4c63e540c0cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.69ex; height:2.343ex;" alt="{\displaystyle g_{_{N}}}" loading="lazy"> is a periodic summation of another function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,}</annotation>

</semantics>

</math><img src="./81f2986cd965e404a1ee33ec84baee5c43da47fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g_{_{N}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*g_{_{N}}}</annotation>

</semantics>

</math><img src="./b0e55b54474ccd9dfcf4245d5a225215cda943b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.163ex; height:2.843ex;" alt="{\displaystyle f*g_{_{N}}}" loading="lazy"> is known as a <a href="Circular_convolution" title="Circular convolution">circular convolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g.}</annotation>

</semantics>

</math><img src="./23a3f421f58ef3bc6f9ec70e883e1496ff871e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}" loading="lazy">

When the non-zero durations of both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> are limited to the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,N-1],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,N-1],}</annotation>

</semantics>

</math><img src="./2af8a24e932d2fbcfabac7f435d28167f1787aa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.203ex; height:2.843ex;" alt="{\displaystyle [0,N-1],}" loading="lazy">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g_{_{N}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*g_{_{N}}}</annotation>

</semantics>

</math><img src="./b0e55b54474ccd9dfcf4245d5a225215cda943b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.163ex; height:2.843ex;" alt="{\displaystyle f*g_{_{N}}}" loading="lazy"> reduces to these common forms:

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</style><table role="presentation" class="numblk" id="math_Eq.1" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(f*g_{N}\right)[n]&=\sum _{m=0}^{N-1}f[m]g_{N}[n-m]\\&=\sum _{m=0}^{n}f[m]g[n-m]+\sum _{m=n+1}^{N-1}f[m]g[N+n-m]\\[2pt]&=\sum _{m=0}^{N-1}f[m]g[(n-m)_{\bmod {N}}]\\[2pt]&\triangleq \left(f*_{N}g\right)[n]\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(f*g_{N}\right)[n]&=\sum _{m=0}^{N-1}f[m]g_{N}[n-m]\\&=\sum _{m=0}^{n}f[m]g[n-m]+\sum _{m=n+1}^{N-1}f[m]g[N+n-m]\\[2pt]&=\sum _{m=0}^{N-1}f[m]g[(n-m)_{\bmod {N}}]\\[2pt]&\triangleq \left(f*_{N}g\right)[n]\end{aligned}}}</annotation>

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</math><img src="./2ea93687815cb3c2fe0ef1acee64c01b50f9e421.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.838ex; width:59.474ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}\left(f*g_{N}\right)[n]&=\sum _{m=0}^{N-1}f[m]g_{N}[n-m]\\&=\sum _{m=0}^{n}f[m]g[n-m]+\sum _{m=n+1}^{N-1}f[m]g[N+n-m]\\[2pt]&=\sum _{m=0}^{N-1}f[m]g[(n-m)_{\bmod {N}}]\\[2pt]&\triangleq \left(f*_{N}g\right)[n]\end{aligned}}}" loading="lazy">        </td> <td></td> <td class="nowrap"><a href="#math_Eq.1">Eq.1</a></td></tr></tbody></table>

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The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*_{N}g}">

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<annotation encoding="application/x-tex">{\displaystyle f*_{N}g}</annotation>

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</math><img src="./6f3248f70edc54a4c0294f977a21df44f64f30cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.281ex; height:2.509ex;" alt="{\displaystyle f*_{N}g}" loading="lazy"> for cyclic convolution denotes convolution over the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of <a href="Modular_arithmetic" title="Modular arithmetic">integers modulo N</a>.

Circular convolution arises most often in the context of fast convolution with a <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) algorithm.

<div class="mw-heading mw-heading3"><h3 id="Fast_convolution_algorithms">Fast convolution algorithms</h3></div>

In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in <a href="Multiplication" title="Multiplication">multiplication</a> of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (<a href="#CITEREFKnuth1997">Knuth 1997</a>, §4.3.3.C; <a href="#CITEREFvon_zur_GathenGerhard2003">von zur Gathen & Gerhard 2003</a>, §8.2).

<a href="#math_Eq.1">Eq.1</a> requires N arithmetic operations per output value and N2 operations for N outputs. That can be significantly reduced with any of several fast algorithms. <a href="Digital_signal_processing" title="Digital signal processing">Digital signal processing</a> and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.

The most common fast convolution algorithms use <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) algorithms via the <a href="Discrete_Fourier_transform#Circular_convolution_theorem_and_cross-correlation_theorem" title="Discrete Fourier transform">circular convolution theorem</a>. Specifically, the <a href="Circular_convolution" title="Circular convolution">circular convolution</a> of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output. Other fast convolution algorithms, such as the <a href="Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen algorithm</a> or the Mersenne transform,<a href="#cite_note-Rader1972-16">[14]</a> use fast Fourier transforms in other <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>. The Winograd method is used as an alternative to the FFT.<a href="#cite_note-17">[15]</a> It significantly speeds up 1D,<a href="#cite_note-18">[16]</a> 2D,<a href="#cite_note-19">[17]</a> and 3D<a href="#cite_note-20">[18]</a> convolution.

If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.<a href="#cite_note-Madisetti1999-21">[19]</a> Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the <a href="Overlap%E2%80%93save_method" title="Overlap–save method">overlap–save method</a> and <a href="Overlap%E2%80%93add_method" title="Overlap–add method">overlap–add method</a>.<a href="#cite_note-Juang2004-22">[20]</a> A hybrid convolution method that combines block and <a href="Finite_impulse_response" title="Finite impulse response">FIR</a> algorithms allows for a zero input-output latency that is useful for real-time convolution computations.<a href="#cite_note-Gardner1994-23">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Domain_of_definition">Domain of definition</h2></div>

The convolution of two complex-valued functions on Rd is itself a complex-valued function on Rd, defined by:

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(x)=\int _{\mathbf {R} ^{d}}f(y)g(x-y)\,dy=\int _{\mathbf {R} ^{d}}f(x-y)g(y)\,dy,}</annotation>

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</math><img src="./75e7e753e7ec3d165472b6362e1152d898c008b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.268ex; height:5.676ex;" alt="{\displaystyle (f*g)(x)=\int _{\mathbf {R} ^{d}}f(y)g(x-y)\,dy=\int _{\mathbf {R} ^{d}}f(x-y)g(y)\,dy,}" loading="lazy"></dd></dl>

and is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g:

<div class="mw-heading mw-heading3"><h3 id="Compactly_supported_functions">Compactly supported functions</h3></div>

If f and g are <a href="Compact_support" class="mw-redirect" title="Compact support">compactly supported</a> <a href="Continuous_function" title="Continuous function">continuous functions</a>, then their convolution exists, and is also compactly supported and continuous (<a href="#CITEREFHörmander1983">Hörmander 1983</a>, Chapter 1). More generally, if either function (say f) is compactly supported and the other is <a href="Locally_integrable_function" title="Locally integrable function">locally integrable</a>, then the convolution f∗g is well-defined and continuous.

Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [a, +∞) (or both supported on [−∞, a]).

<div class="mw-heading mw-heading3"><h3 id="Integrable_functions">Integrable functions</h3></div>

The convolution of f and g exists if f and g are both <a href="Lebesgue_integral" title="Lebesgue integral">Lebesgue integrable functions</a> in <a href="Lp_space" title="Lp space">L1(Rd)</a>, and in this case f∗g is also integrable (<a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, Theorem 1.3). This is a consequence of <a href="Fubini's_theorem#Tonelli's_theorem" title="Fubini's theorem">Tonelli's theorem</a>. This is also true for functions in L1, under the discrete convolution, or more generally for the <a href="#Convolutions_on_groups">convolution on any group</a>.

Likewise, if f ∈ L1(Rd)  and  g ∈ Lp(Rd)  where 1 ≤ p ≤ ∞,  then  f*g ∈ Lp(Rd),  and

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<annotation encoding="application/x-tex">{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}.}</annotation>

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</math><img src="./b76f8c931c10d6caf9d2a2ba919e39bc5ce90717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.877ex; height:3.009ex;" alt="{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}.}" loading="lazy"></dd></dl>

In the particular case p = 1, this shows that L1 is a <a href="Banach_algebra" title="Banach algebra">Banach algebra</a> under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).

More generally, <a href="Young's_convolution_inequality" title="Young's convolution inequality">Young's inequality</a> implies that the convolution is a continuous bilinear map between suitable Lp spaces. Specifically, if 1 ≤ p, q, r ≤ ∞ satisfy:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1,}</annotation>

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</math><img src="./adbb8ff3d28a9eca3f097f605d952fec6a1680f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.591ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1,}" loading="lazy"></dd></dl>

then

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<annotation encoding="application/x-tex">{\displaystyle \left\Vert f*g\right\Vert _{r}\leq \left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q},}</annotation>

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</math><img src="./066d66a77d5ed9f7dd9aa16dc9b5e9bec9c80d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.986ex; height:3.176ex;" alt="{\displaystyle \left\Vert f*g\right\Vert _{r}\leq \left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q},}" loading="lazy"></dd></dl>

so that the convolution is a continuous bilinear mapping from Lp×Lq to Lr.

The Young inequality for convolution is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 < p, q, r < ∞, there exists a constant Bp,q < 1 such that:

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<annotation encoding="application/x-tex">{\displaystyle \left\Vert f*g\right\Vert _{r}\leq B_{p,q}\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q}.}</annotation>

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</math><img src="./bc4e7ef522adae20e9b92cca1126f713aa2290d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.023ex; height:3.176ex;" alt="{\displaystyle \left\Vert f*g\right\Vert _{r}\leq B_{p,q}\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in L^{p},\ g\in L^{q}.}" loading="lazy"></dd></dl>

The optimal value of Bp,q was discovered in 1975<a href="#cite_note-24">[22]</a> and independently in 1976,<a href="#cite_note-25">[23]</a> see <a href="Brascamp%E2%80%93Lieb_inequality" title="Brascamp–Lieb inequality">Brascamp–Lieb inequality</a>.

A stronger estimate is true provided 1 < p, q, r < ∞:

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<annotation encoding="application/x-tex">{\displaystyle \|f*g\|_{r}\leq C_{p,q}\|f\|_{p}\|g\|_{q,w}}</annotation>

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</math><img src="./9c8b015b1ca7cc3c1965aad267f9980fac45bc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.647ex; height:3.009ex;" alt="{\displaystyle \|f*g\|_{r}\leq C_{p,q}\|f\|_{p}\|g\|_{q,w}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|g\|_{q,w}}">

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</math><img src="./0dda2a107a3d00d180f3fe7d44f96d60979bfe1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.064ex; height:3.009ex;" alt="{\displaystyle \|g\|_{q,w}}" loading="lazy"> is the <a href="Lp_space#Weak_Lp" title="Lp space">weak Lq</a> norm. Convolution also defines a bilinear continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}}</annotation>

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</math><img src="./88e02aabe272c293a1a38234f777c7e5229ab3f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.127ex; height:2.343ex;" alt="{\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<p,q,r<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle 1<p,q,r<\infty }</annotation>

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</math><img src="./6e3a9274fba0fef933778d434e42fdde9fa8dda9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.039ex; height:2.509ex;" alt="{\displaystyle 1<p,q,r<\infty }" loading="lazy">, owing to the weak Young inequality:<a href="#cite_note-26">[24]</a>

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<annotation encoding="application/x-tex">{\displaystyle \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.}</annotation>

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</math><img src="./8915308c4fb7b0bdca3217867e64a6c367370366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.548ex; height:3.009ex;" alt="{\displaystyle \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Functions_of_rapid_decay">Functions of rapid decay</h3></div>

In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are <a href="Rapidly_decreasing_function" class="mw-redirect" title="Rapidly decreasing function">rapidly decreasing functions</a>, then so is the convolution f∗g. Combined with the fact that convolution commutes with differentiation (see <a href="#Properties">#Properties</a>), it follows that the class of <a href="Schwartz_function" class="mw-redirect" title="Schwartz function">Schwartz functions</a> is closed under convolution (<a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>, Theorem 3.3).

<div class="mw-heading mw-heading3"><h3 id="Distributions">Distributions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Distribution_(mathematics)" title="Distribution (mathematics)">Distribution (mathematics)</a></div>

If f is a smooth function that is <a href="Support_(mathematics)#Compact_support" title="Support (mathematics)">compactly supported</a> and g is a distribution, then f∗g is a smooth function defined by

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<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{d}}{f}(y)g(x-y)\,dy=(f*g)(x)\in C^{\infty }(\mathbb {R} ^{d}).}</annotation>

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</math><img src="./32aa909b04d9dbc3a7b9f8f62522b56772868bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.59ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{d}}{f}(y)g(x-y)\,dy=(f*g)(x)\in C^{\infty }(\mathbb {R} ^{d}).}" loading="lazy"></dd></dl>

More generally, it is possible to extend the definition of the convolution in a unique way with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle f*(g*\varphi )=(f*g)*\varphi }</annotation>

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</math><img src="./983c2bc301f63665918893b996d200248d893d51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.325ex; height:2.843ex;" alt="{\displaystyle f*(g*\varphi )=(f*g)*\varphi }" loading="lazy"></dd></dl>

remains valid in the case where f is a distribution, and g a compactly supported distribution (<a href="#CITEREFHörmander1983">Hörmander 1983</a>, §4.2).

<div class="mw-heading mw-heading3"><h3 id="Measures">Measures</h3></div>

The convolution of any two <a href="Borel_measure" title="Borel measure">Borel measures</a> μ and ν of <a href="Bounded_variation" title="Bounded variation">bounded variation</a> is the measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu *\nu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu *\nu }</annotation>

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</math><img src="./cca828cbddba41df82298158087e06c76b44c233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.176ex;" alt="{\displaystyle \mu *\nu }" loading="lazy"> defined by (<a href="#CITEREFRudin1962">Rudin 1962</a>)

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</mrow>

</msup>

</mrow>

</msub>

<msub>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbf {R} ^{d}}f(x)\,d(\mu *\nu )(x)=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x+y)\,d\mu (x)\,d\nu (y).}</annotation>

</semantics>

</math><img src="./5570e51b3cfeb01fbbf46025e36359dccada10bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.471ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbf {R} ^{d}}f(x)\,d(\mu *\nu )(x)=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x+y)\,d\mu (x)\,d\nu (y).}" loading="lazy"></dd></dl>

In particular,

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<annotation encoding="application/x-tex">{\displaystyle (\mu *\nu )(A)=\int _{\mathbf {R} ^{d}\times \mathbf {R} ^{d}}1_{A}(x+y)\,d(\mu \times \nu )(x,y),}</annotation>

</semantics>

</math><img src="./cd6b6ece916077fdced9fb2821929f42b22ae780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.66ex; height:5.676ex;" alt="{\displaystyle (\mu *\nu )(A)=\int _{\mathbf {R} ^{d}\times \mathbf {R} ^{d}}1_{A}(x+y)\,d(\mu \times \nu )(x,y),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset \mathbf {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subset \mathbf {R} ^{d}}</annotation>

</semantics>

</math><img src="./73f7a3a2453062ac192e8f099919359a5c0d7637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.937ex; height:2.676ex;" alt="{\displaystyle A\subset \mathbf {R} ^{d}}" loading="lazy"> is a measurable set and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{A}}</annotation>

</semantics>

</math><img src="./d1a15eaa9285cd4654e86a76f3318c6ab2aad95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle 1_{A}}" loading="lazy"> is the <a href="Indicator_function" title="Indicator function">indicator function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.

This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.

The convolution of measures also satisfies the following version of Young's inequality

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\mu *\nu \|\leq \|\mu \|\|\nu \|}</annotation>

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</math><img src="./a5a031232d3f3687f628da6c2a0a1a87d2aec84c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.536ex; height:2.843ex;" alt="{\displaystyle \|\mu *\nu \|\leq \|\mu \|\|\nu \|}" loading="lazy"></dd></dl>

where the norm is the <a href="Total_variation" title="Total variation">total variation</a> of a measure. Because the space of measures of bounded variation is a <a href="Banach_space" title="Banach space">Banach space</a>, convolution of measures can be treated with standard methods of <a href="Functional_analysis" title="Functional analysis">functional analysis</a> that may not apply for the convolution of distributions.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_properties">Algebraic properties</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Convolution_algebra" class="mw-redirect" title="Convolution algebra">Convolution algebra</a></div>

The convolution defines a product on the <a href="Linear_space" class="mw-redirect" title="Linear space">linear space</a> of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative <a href="Associative_algebra" title="Associative algebra">associative algebra</a> without <a href="Identity_element" title="Identity element">identity</a> (<a href="#CITEREFStrichartz1994">Strichartz 1994</a>, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are <a href="Closure_(mathematics)" title="Closure (mathematics)">closed</a> under the convolution, and so also form commutative associative algebras.

<dl><dt><a href="Commutativity" class="mw-redirect" title="Commutativity">Commutativity</a></dt>

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<annotation encoding="application/x-tex">{\displaystyle f*g=g*f}</annotation>

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</math> Proof: By definition: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau }">

</mrow>

</mrow>

</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau }</annotation>

</semantics>

</math> Changing the variable of integration to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=t-\tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=t-\tau }</annotation>

</semantics>

</math><img src="./e9f136ba801f6bd00ee985e98cec37235490289b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.31ex; height:2.176ex;" alt="{\displaystyle u=t-\tau }" loading="lazy"> the result follows.</dd></dl>

<dl><dt><a href="Associativity" class="mw-redirect" title="Associativity">Associativity</a></dt>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f*(g*h)=(f*g)*h}</annotation>

</semantics>

</math> Proof: This follows from using <a href="Fubini's_theorem" title="Fubini's theorem">Fubini's theorem</a> (i.e., double integrals can be evaluated as iterated integrals in either order).</dd></dl>

<dl><dt><a href="Distributivity" class="mw-redirect" title="Distributivity">Distributivity</a></dt>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f*(g+h)=(f*g)+(f*h)}</annotation>

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</math> Proof: This follows from linearity of the integral.</dd></dl>

<dl><dt>Associativity with scalar multiplication</dt>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a(f*g)=(af)*g}</annotation>

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</math> for any real (or complex) number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">.</dd></dl>

<dl><dt><a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">Multiplicative identity</a></dt>

<dd>No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a <a href="Dirac_delta" class="mw-redirect" title="Dirac delta">delta distribution</a> (a unitary impulse, centered at zero) or, at the very least (as is the case of L1) admit <a href="Nascent_delta_function" class="mw-redirect" title="Nascent delta function">approximations to the identity</a>. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*\delta =f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f*\delta =f}</annotation>

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</math> where δ is the delta distribution.</dd></dl>

<dl><dt>Inverse element</dt>

<dd>Some distributions S have an <a href="Inverse_element" title="Inverse element">inverse element</a> S−1 for the convolution which then must satisfy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}*S=\delta }">

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}*S=\delta }</annotation>

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</math> from which an explicit formula for S−1 may be obtained.<div class="paragraphbreak" style="margin-top:0.5em"></div>The set of invertible distributions forms an <a href="Abelian_group" title="Abelian group">abelian group</a> under the convolution.</dd></dl>

<dl><dt>Complex conjugation</dt>

<mover>

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</mover>

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<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}</annotation>

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</math></dd></dl>

<dl><dt>Time reversal</dt>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(t)=r(t)*s(t),}</annotation>

</semantics>

</math><img src="./8fbd129f70374adf631f1dc644be0f3424bae759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.095ex; height:2.843ex;" alt="{\displaystyle q(t)=r(t)*s(t),}" loading="lazy">  then  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(-t)=r(-t)*s(-t).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(-t)=r(-t)*s(-t).}</annotation>

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</math><img src="./8eea4c6a9c15a99ebfe72c6318f9f15e9ce8f9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.52ex; height:2.843ex;" alt="{\displaystyle q(-t)=r(-t)*s(-t).}" loading="lazy"></dd></dl>

Proof (using <a href="Convolution_theorem" title="Convolution theorem">convolution theorem</a>):

<mover>

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</mrow>

</mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle q(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(f)=R(f)S(f)}</annotation>

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</math><img src="./722e66a5fc38d9f3e4172fcabea15f512efdb45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.241ex; height:4.176ex;" alt="{\displaystyle q(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(f)=R(f)S(f)}" loading="lazy">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle q(-t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(-f)=R(-f)S(-f)}</annotation>

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</math><img src="./bd992c7ec988946c331d8fa87e0e40722d7776db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.473ex; height:4.176ex;" alt="{\displaystyle q(-t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \ Q(-f)=R(-f)S(-f)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}q(-t)&={\mathcal {F}}^{-1}{\bigg \{}R(-f)S(-f){\bigg \}}\\&={\mathcal {F}}^{-1}{\bigg \{}R(-f){\bigg \}}*{\mathcal {F}}^{-1}{\bigg \{}S(-f){\bigg \}}\\&=r(-t)*s(-t)\end{aligned}}}</annotation>

</semantics>

</math><img src="./ea9a28d15b953bfb8b8f9a7617eb6c80c26eb870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:40.271ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}q(-t)&={\mathcal {F}}^{-1}{\bigg \{}R(-f)S(-f){\bigg \}}\\&={\mathcal {F}}^{-1}{\bigg \{}R(-f){\bigg \}}*{\mathcal {F}}^{-1}{\bigg \{}S(-f){\bigg \}}\\&=r(-t)*s(-t)\end{aligned}}}" loading="lazy">

</blockquote>

<dl><dt>Relationship with differentiation</dt>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)'=f'*g=f*g'}</annotation>

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</math> Proof:</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f*g)'&={\frac {d}{dt}}\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau ){\frac {\partial }{\partial t}}g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )g'(t-\tau )\,d\tau =f*g'.\end{aligned}}}</annotation>

</semantics>

</math><img src="./66b93baa12332ebd7d6695d406989c4d4e0f29e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.336ex; margin-bottom: -0.336ex; width:40.377ex; height:18.343ex;" alt="{\displaystyle {\begin{aligned}(f*g)'&={\frac {d}{dt}}\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau ){\frac {\partial }{\partial t}}g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )g'(t-\tau )\,d\tau =f*g'.\end{aligned}}}" loading="lazy"></dd></dl>

<dl><dt>Relationship with integration</dt>

</mrow>

</mrow>

</msubsup>

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<annotation encoding="application/x-tex">{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau ,}</annotation>

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</math><img src="./2ac0792eba9d5398066050439c3b9d396a1e5729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.481ex; height:3.676ex;" alt="{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau ,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau ,}">

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau ,}</annotation>

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</math><img src="./076433e19b9727e7a36e6449fee47d2ae0789f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.792ex; height:3.676ex;" alt="{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau ,}" loading="lazy"> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau .}">

</mrow>

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</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau .}</annotation>

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</math></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Integration">Integration</h3></div>

If f and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:<a href="#cite_note-27">[25]</a>

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<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbf {R} ^{d}}(f*g)(x)\,dx=\left(\int _{\mathbf {R} ^{d}}f(x)\,dx\right)\left(\int _{\mathbf {R} ^{d}}g(x)\,dx\right).}</annotation>

</semantics>

</math><img src="./8a07e9f9da38f7b4fa906822b10e0e1a1201e2fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.555ex; height:6.176ex;" alt="{\displaystyle \int _{\mathbf {R} ^{d}}(f*g)(x)\,dx=\left(\int _{\mathbf {R} ^{d}}f(x)\,dx\right)\left(\int _{\mathbf {R} ^{d}}g(x)\,dx\right).}" loading="lazy"></dd></dl>

This follows from <a href="Fubini's_theorem" title="Fubini's theorem">Fubini's theorem</a>. The same result holds if f and g are only assumed to be nonnegative measurable functions, by <a href="Fubini's_theorem#Tonelli's_theorem" title="Fubini's theorem">Tonelli's theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Differentiation">Differentiation</h3></div>

In the one-variable case,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g=f*{\frac {dg}{dx}}}</annotation>

</semantics>

</math><img src="./24718e5d4c48f46fb073d3fd014121d452cc3822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.525ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g=f*{\frac {dg}{dx}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}}</annotation>

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</math><img src="./6e80537df4f7e8d3d157ed7d50514cfe0c04b91f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}}" loading="lazy"> is the <a href="Derivative" title="Derivative">derivative</a>. More generally, in the case of functions of several variables, an analogous formula holds with the <a href="Partial_derivative" title="Partial derivative">partial derivative</a>:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x_{i}}}(f*g)={\frac {\partial f}{\partial x_{i}}}*g=f*{\frac {\partial g}{\partial x_{i}}}.}</annotation>

</semantics>

</math><img src="./98f666b52bb37620cc7e18d342d61f8591e41c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.877ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial }{\partial x_{i}}}(f*g)={\frac {\partial f}{\partial x_{i}}}*g=f*{\frac {\partial g}{\partial x_{i}}}.}" loading="lazy"></dd></dl>

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of f and g is differentiable as many times as f and g are in total.

These identities hold for example under the condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence of <a href="Young's_convolution_inequality" title="Young's convolution inequality">Young's convolution inequality</a>. For instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g.}</annotation>

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</math><img src="./14a9256a10d33eb0dbdcc95836446973b226b042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.218ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g.}" loading="lazy"></dd></dl>

These identities also hold much more broadly in the sense of tempered distributions if one of f or g is a

<a href="Distribution_(mathematics)#Convolution_versus_multiplication" title="Distribution (mathematics)">rapidly decreasing tempered distribution</a>, a

compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.

In the discrete case, the <a href="Difference_operator" class="mw-redirect" title="Difference operator">difference operator</a> D f(n) = f(n + 1) − f(n) satisfies an analogous relationship:

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<annotation encoding="application/x-tex">{\displaystyle D(f*g)=(Df)*g=f*(Dg).}</annotation>

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</math><img src="./5bcad070b728ebe92558d5b3b39c6a82bebc8bdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.812ex; height:2.843ex;" alt="{\displaystyle D(f*g)=(Df)*g=f*(Dg).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Convolution_theorem">Convolution theorem</h3></div>

The <a href="Convolution_theorem" title="Convolution theorem">convolution theorem</a> states that<a href="#cite_note-28">[26]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}</annotation>

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</math><img src="./2774439962f8fbdd02622e126adaeb4cd09e5833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.516ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{f\}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{f\}}</annotation>

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</math><img src="./399889a7c37400ec52d43a4d0ca72efadf05b0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{f\}}" loading="lazy"> denotes the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Convolution_in_other_types_of_transformations">Convolution in other types of transformations</h4></div>

Versions of this theorem also hold for the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a>, <a href="Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a>, <a href="Z-transform" title="Z-transform">Z-transform</a> and <a href="Mellin_transform" title="Mellin transform">Mellin transform</a>.

<div class="mw-heading mw-heading4"><h4 id="Convolution_on_matrices">Convolution on matrices</h4></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation>

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</math><img src="./6a1cc103563219127f59aec7ed9327a3595566dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}" loading="lazy"> is the <a href="DFT_matrix" title="DFT matrix">Fourier transform matrix</a>, then

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}\left(C^{(1)}x\ast C^{(2)}y\right)=\left({\mathcal {W}}C^{(1)}\bullet {\mathcal {W}}C^{(2)}\right)(x\otimes y)={\mathcal {W}}C^{(1)}x\circ {\mathcal {W}}C^{(2)}y}</annotation>

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</math><img src="./ef62613d6f6f800199fc323867abcebfaf9eeb8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.025ex; height:4.843ex;" alt="{\displaystyle {\mathcal {W}}\left(C^{(1)}x\ast C^{(2)}y\right)=\left({\mathcal {W}}C^{(1)}\bullet {\mathcal {W}}C^{(2)}\right)(x\otimes y)={\mathcal {W}}C^{(1)}x\circ {\mathcal {W}}C^{(2)}y}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bullet }">

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<annotation encoding="application/x-tex">{\displaystyle \bullet }</annotation>

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</math><img src="./3576c2406959ee194a6fc55c34b5ee9f6ffbb715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \bullet }" loading="lazy"> is <a href="Khatri%E2%80%93Rao_product#Face-splitting_product" title="Khatri–Rao product">face-splitting product</a>,<a href="#cite_note-slyusar-29">[27]</a><a href="#cite_note-slyusar1-30">[28]</a><a href="#cite_note-DIPED-31">[29]</a><a href="#cite_note-slyusar2-32">[30]</a><a href="#cite_note-general-33">[31]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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<annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation>

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</math><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"> denotes <a href="Kronecker_product" title="Kronecker product">Kronecker product</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

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<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>

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</math><img src="./99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> denotes <a href="Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a> (this result is an evolving of <a href="Count_sketch" title="Count sketch">count sketch</a> properties<a href="#cite_note-ninh-34">[32]</a>).

This can be generalized for appropriate matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ,\mathbf {B} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ,\mathbf {B} }</annotation>

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</math><img src="./88643597666fbf6379537d6b2c2fa05a006fb77e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.954ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} ,\mathbf {B} }" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}\left((\mathbf {A} x)\ast (\mathbf {B} y)\right)=\left(({\mathcal {W}}\mathbf {A} )\bullet ({\mathcal {W}}\mathbf {B} )\right)(x\otimes y)=({\mathcal {W}}\mathbf {A} x)\circ ({\mathcal {W}}\mathbf {B} y)}</annotation>

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</math><img src="./13fd00f82adb390f15a2e5e7774132f823a5867e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.922ex; height:2.843ex;" alt="{\displaystyle {\mathcal {W}}\left((\mathbf {A} x)\ast (\mathbf {B} y)\right)=\left(({\mathcal {W}}\mathbf {A} )\bullet ({\mathcal {W}}\mathbf {B} )\right)(x\otimes y)=({\mathcal {W}}\mathbf {A} x)\circ ({\mathcal {W}}\mathbf {B} y)}" loading="lazy"></dd></dl>

from the properties of the <a href="Face-splitting_product" class="mw-redirect" title="Face-splitting product">face-splitting product</a>.

<div class="mw-heading mw-heading3"><h3 id="Translational_equivariance">Translational equivariance</h3></div>

The convolution commutes with translations, meaning that

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<annotation encoding="application/x-tex">{\displaystyle \tau _{x}(f*g)=(\tau _{x}f)*g=f*(\tau _{x}g)}</annotation>

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</math><img src="./975d3da8a0b4b18f91154f1a166919ccb8835382.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.959ex; height:2.843ex;" alt="{\displaystyle \tau _{x}(f*g)=(\tau _{x}f)*g=f*(\tau _{x}g)}" loading="lazy"></dd></dl>

where τxf is the translation of the function f by x defined by

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<annotation encoding="application/x-tex">{\displaystyle (\tau _{x}f)(y)=f(y-x).}</annotation>

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</math><img src="./93083b3b6e337e1fc5555be17cec4e810674b2ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.4ex; height:2.843ex;" alt="{\displaystyle (\tau _{x}f)(y)=f(y-x).}" loading="lazy"></dd></dl>

If f is a <a href="Schwartz_function" class="mw-redirect" title="Schwartz function">Schwartz function</a>, then τxf is the convolution with a translated Dirac delta function τxf = f ∗ τx δ. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.

Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds

<dl><dd>Suppose that S is a bounded <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> acting on functions which commutes with translations: S(τxf) = τx(Sf) for all x. Then S is given as convolution with a function (or distribution) gS; that is Sf = gS ∗ f.</dd></dl>

Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of <a href="Time-invariant_system" title="Time-invariant system">time-invariant systems</a>, and especially <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system theory</a>. The representing function gS is the <a href="Impulse_response" title="Impulse response">impulse response</a> of the transformation S.

A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a <a href="Continuous_linear_operator" title="Continuous linear operator">continuous linear operator</a> with respect to the appropriate <a href="Topology" title="Topology">topology</a>. It is known, for instance, that every continuous translation invariant continuous linear operator on L1 is the convolution with a finite <a href="Borel_measure" title="Borel measure">Borel measure</a>. More generally, every continuous translation invariant continuous linear operator on Lp for 1 ≤ p < ∞ is the convolution with a <a href="Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform" title="Distribution (mathematics)">tempered distribution</a> whose <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is bounded. To wit, they are all given by bounded <a href="Fourier_multiplier" class="mw-redirect" title="Fourier multiplier">Fourier multipliers</a>.

<div class="mw-heading mw-heading2"><h2 id="Convolutions_on_groups">Convolutions on groups</h2></div>

If G is a suitable <a href="Group_(mathematics)" title="Group (mathematics)">group</a> endowed with a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> λ, and if f and g are real or complex valued <a href="Lebesgue_integral" title="Lebesgue integral">integrable</a> functions on G, then we can define their convolution by

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(x)=\int _{G}f(y)g\left(y^{-1}x\right)\,d\lambda (y).}</annotation>

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</math><img src="./5fead2d71959ecfcb746e93ac826b5f70ffbef70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.496ex; height:5.676ex;" alt="{\displaystyle (f*g)(x)=\int _{G}f(y)g\left(y^{-1}x\right)\,d\lambda (y).}" loading="lazy"></dd></dl>

It is not commutative in general. In typical cases of interest G is a <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="Topological_group" title="Topological group">topological group</a> and λ is a (left-) <a href="Haar_measure" title="Haar measure">Haar measure</a>. In that case, unless G is <a href="Unimodular_group" class="mw-redirect" title="Unimodular group">unimodular</a>, the convolution defined in this way is not the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}">

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<annotation encoding="application/x-tex">{\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}</annotation>

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</math><img src="./835b2fb75357463bff2287a03a36dc801dcdd971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.814ex; height:3.176ex;" alt="{\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}" loading="lazy">. The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:

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<annotation encoding="application/x-tex">{\displaystyle L_{h}(f*g)=(L_{h}f)*g.}</annotation>

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</math><img src="./ea963980f21681d5885efb77d57d8362baeba976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.066ex; height:2.843ex;" alt="{\displaystyle L_{h}(f*g)=(L_{h}f)*g.}" loading="lazy"></dd></dl>

Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former.

On <a href="Locally_compact_abelian_group" title="Locally compact abelian group">locally compact abelian groups</a>, a version of the <a href="Convolution_theorem" title="Convolution theorem">convolution theorem</a> holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The <a href="Circle_group" title="Circle group">circle group</a> T with the Lebesgue measure is an immediate example. For a fixed g in L1(T), we have the following familiar operator acting on the <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> L2(T):

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<annotation encoding="application/x-tex">{\displaystyle T{f}(x)={\frac {1}{2\pi }}\int _{\mathbf {T} }{f}(y)g(x-y)\,dy.}</annotation>

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</math><img src="./244a83c58e39d00f2fbc81e8db82b474def6a6fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.996ex; height:5.676ex;" alt="{\displaystyle T{f}(x)={\frac {1}{2\pi }}\int _{\mathbf {T} }{f}(y)g(x-y)\,dy.}" loading="lazy"></dd></dl>

The operator T is <a href="Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">compact</a>. A direct calculation shows that its adjoint T* is convolution with

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<annotation encoding="application/x-tex">{\displaystyle {\bar {g}}(-y).}</annotation>

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</math><img src="./f88c7673d5fef9debb0f92916e2d2b34a80fbb66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:2.843ex;" alt="{\displaystyle {\bar {g}}(-y).}" loading="lazy"></dd></dl>

By the commutativity property cited above, T is <a href="Normal_operator" title="Normal operator">normal</a>: T* T = TT* . Also, T commutes with the translation operators. Consider the family S of operators consisting of all such convolutions and the translation operators. Then S is a commuting family of normal operators. According to <a href="Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">spectral theory</a>, there exists an orthonormal basis {hk} that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have

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<annotation encoding="application/x-tex">{\displaystyle h_{k}(x)=e^{ikx},\quad k\in \mathbb {Z} ,\;}</annotation>

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</math><img src="./86c5ac0374f88eaaa5c7c073b192c501facf301a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.596ex; height:3.176ex;" alt="{\displaystyle h_{k}(x)=e^{ikx},\quad k\in \mathbb {Z} ,\;}" loading="lazy"></dd></dl>

which are precisely the <a href="Character_(mathematics)" title="Character (mathematics)">characters</a> of T. Each convolution is a compact <a href="Multiplication_operator" title="Multiplication operator">multiplication operator</a> in this basis. This can be viewed as a version of the convolution theorem discussed above.

A discrete example is a finite <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of order n. Convolution operators are here represented by <a href="Circulant_matrices" class="mw-redirect" title="Circulant matrices">circulant matrices</a>, and can be diagonalized by the <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>.

A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional <a href="Unitary_representation" title="Unitary representation">unitary representations</a> form an orthonormal basis in L2 by the <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a>, and an analog of the convolution theorem continues to hold, along with many other aspects of <a href="Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> that depend on the Fourier transform.

<div class="mw-heading mw-heading2"><h2 id="Convolution_of_measures">Convolution of measures</h2></div>

Let G be a (multiplicatively written) topological group.

If μ and ν are <a href="Radon_measure" title="Radon measure">Radon measures</a> on G, then their convolution μ∗ν is defined as the <a href="Pushforward_measure" title="Pushforward measure">pushforward measure</a> of the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> and can be written as<a href="#cite_note-35">[33]</a>

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<annotation encoding="application/x-tex">{\displaystyle (\mu *\nu )(E)=\iint 1_{E}(xy)\,d\mu (x)\,d\nu (y)}</annotation>

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</math><img src="./fe62b48789da39bb701ab7c9c2b143d5392522d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.017ex; height:5.676ex;" alt="{\displaystyle (\mu *\nu )(E)=\iint 1_{E}(xy)\,d\mu (x)\,d\nu (y)}" loading="lazy"></dd></dl>

for each measurable subset E of G. The convolution is also a Radon measure, whose <a href="Total_variation" title="Total variation">total variation</a> satisfies

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<annotation encoding="application/x-tex">{\displaystyle \|\mu *\nu \|\leq \left\|\mu \right\|\left\|\nu \right\|.}</annotation>

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</math><img src="./a41adf360e1121af62a5f22bd15b61f336e18627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.957ex; height:2.843ex;" alt="{\displaystyle \|\mu *\nu \|\leq \left\|\mu \right\|\left\|\nu \right\|.}" loading="lazy"></dd></dl>

In the case when G is <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> with (left-)<a href="Haar_measure" title="Haar measure">Haar measure</a> λ, and μ and ν are <a href="Absolute_continuity" title="Absolute continuity">absolutely continuous</a> with respect to a λ, <a href="Radon%E2%80%93Nikodym_theorem" title="Radon–Nikodym theorem">so that each has a density function</a>, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions. In fact, if either measure is absolutely continuous with respect to the Haar measure, then so is their convolution.<a href="#cite_note-36">[34]</a>

If μ and ν are <a href="Probability_measure" title="Probability measure">probability measures</a> on the topological group (R,+), then the convolution μ∗ν is the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of the sum X + Y of two <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> <a href="Random_variable" title="Random variable">random variables</a> X and Y whose respective distributions are μ and ν.

<div class="mw-heading mw-heading2"><h2 id="Infimal_convolution">Infimal convolution</h2></div>

In <a href="Convex_analysis" title="Convex analysis">convex analysis</a>, the infimal convolution of proper (not identically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

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<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

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</math><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy">) <a href="Convex_function" title="Convex function">convex functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\dots ,f_{m}}">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{1},\dots ,f_{m}}</annotation>

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</math><img src="./d6f067fd333142a9525e2bb6d75e68879724992b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f_{1},\dots ,f_{m}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> is defined by:<a href="#cite_note-37">[35]</a>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

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<msub>

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</msub>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (f_{1}*\cdots *f_{m})(x)=\inf _{x}\{f_{1}(x_{1})+\cdots +f_{m}(x_{m})|x_{1}+\cdots +x_{m}=x\}.}</annotation>

</semantics>

</math>

It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the <a href="Legendre_transform" class="mw-redirect" title="Legendre transform">Legendre transform</a>:

<msup>

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</msup>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \varphi ^{*}(x)=\sup _{y}(x\cdot y-\varphi (y)).}</annotation>

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</math>

We have:

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<msup>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle (f_{1}*\cdots *f_{m})^{*}(x)=f_{1}^{*}(x)+\cdots +f_{m}^{*}(x).}</annotation>

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</math>

<div class="mw-heading mw-heading2"><h2 id="Bialgebras">Bialgebras</h2></div>

Let (X, Δ, ∇, ε, η) be a <a href="Bialgebra" title="Bialgebra">bialgebra</a> with comultiplication Δ, multiplication ∇, unit η, and counit ε. The convolution is a product defined on the <a href="Endomorphism_algebra" class="mw-redirect" title="Endomorphism algebra">endomorphism algebra</a> End(X) as follows. Let φ, ψ ∈ End(X), that is, φ, ψ: X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as the composition

<mover>

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<annotation encoding="application/x-tex">{\displaystyle X\mathrel {\xrightarrow {\Delta } } X\otimes X\mathrel {\xrightarrow {\phi \otimes \psi } } X\otimes X\mathrel {\xrightarrow {\nabla } } X.}</annotation>

</semantics>

</math><img src="./56ed4b13e3cbafbf266a4148d3e6c163916e45e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-top: -0.345ex; width:32.115ex; height:4.009ex;" alt="{\displaystyle X\mathrel {\xrightarrow {\Delta } } X\otimes X\mathrel {\xrightarrow {\phi \otimes \psi } } X\otimes X\mathrel {\xrightarrow {\nabla } } X.}" loading="lazy"></dd></dl>

The convolution appears notably in the definition of <a href="Hopf_algebra" title="Hopf algebra">Hopf algebras</a> (<a href="#CITEREFKassel1995">Kassel 1995</a>, §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that

<msub>

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<annotation encoding="application/x-tex">{\displaystyle S*\operatorname {id} _{X}=\operatorname {id} _{X}*S=\eta \circ \varepsilon .}</annotation>

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</math><img src="./c25a3eb0496e23e10635b403c8aec7b721f093c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.178ex; height:2.676ex;" alt="{\displaystyle S*\operatorname {id} _{X}=\operatorname {id} _{X}*S=\eta \circ \varepsilon .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Convolution and related operations are found in many applications in science, engineering and mathematics.

<ul><li><a href="Convolutional_neural_network" title="Convolutional neural network">Convolutional neural networks</a> apply multiple cascaded convolution kernels with applications in <a href="Machine_vision" title="Machine vision">machine vision</a> and <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>.<a href="#cite_note-38">[36]</a><a href="#cite_note-39">[37]</a> Though these are actually cross-correlations rather than convolutions in most cases.<a href="#cite_note-40">[38]</a></li>

<li>In non-<a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural-network</a>-based <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a>

<ul><li>In <a href="Digital_image_processing" title="Digital image processing">digital image processing</a> convolutional filtering plays an important role in many important <a href="Algorithm" title="Algorithm">algorithms</a> in <a href="Edge_detection" title="Edge detection">edge detection</a> and related processes (see <a href="Kernel_(image_processing)" title="Kernel (image processing)">Kernel (image processing)</a>)</li>

<li>In <a href="Optics" title="Optics">optics</a>, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is <a href="Bokeh" title="Bokeh">bokeh</a>.</li>

<li>In image processing applications such as adding blurring.</li></ul></li>

<li>In digital data processing

<ul><li>In <a href="Analytical_chemistry" title="Analytical chemistry">analytical chemistry</a>, <a href="Savitzky%E2%80%93Golay_smoothing_filter" class="mw-redirect" title="Savitzky–Golay smoothing filter">Savitzky–Golay smoothing filters</a> are used for the analysis of spectroscopic data. They can improve <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> with minimal distortion of the spectra</li>

<li>In <a href="Statistics" title="Statistics">statistics</a>, a weighted <a href="Moving_average" title="Moving average">moving average</a> is a convolution.</li></ul></li>

<li>In <a href="Acoustics" title="Acoustics">acoustics</a>, <a href="Reverberation" title="Reverberation">reverberation</a> is the convolution of the original sound with <a href="Echo_(phenomenon)" class="mw-redirect" title="Echo (phenomenon)">echoes</a> from objects surrounding the sound source.

<ul><li>In digital signal processing, convolution is used to map the <a href="Impulse_response" title="Impulse response">impulse response</a> of a real room on a digital audio signal.</li>

<li>In <a href="Electronic_music" title="Electronic music">electronic music</a> convolution is the imposition of a <a href="Spectrum" title="Spectrum">spectral</a> or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other.<a href="#cite_note-41">[39]</a></li></ul></li>

<li>In <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a>, the convolution of one function (the <a href="Signal_(electrical_engineering)" class="mw-redirect" title="Signal (electrical engineering)">input signal</a>) with a second function (the impulse response) gives the output of a <a href="Linear_time-invariant_system" title="Linear time-invariant system">linear time-invariant system</a> (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred.</li>

<li>In <a href="Physics" title="Physics">physics</a>, wherever there is a <a href="Linear_system" title="Linear system">linear system</a> with a "<a href="Superposition_principle" title="Superposition principle">superposition principle</a>", a convolution operation makes an appearance. For instance, in <a href="Spectroscopy" title="Spectroscopy">spectroscopy</a> line broadening due to the Doppler effect on its own gives a <a href="Normal_distribution" title="Normal distribution">Gaussian</a> <a href="Spectral_line_shape" title="Spectral line shape">spectral line shape</a> and collision broadening alone gives a <a href="Cauchy_distribution" title="Cauchy distribution">Lorentzian</a> line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a <a href="Voigt_function" class="mw-redirect" title="Voigt function">Voigt function</a>.

<ul><li>In <a href="Time-resolved_spectroscopy#Time-resolved_fluorescence_spectroscopy" title="Time-resolved spectroscopy">time-resolved fluorescence spectroscopy</a>, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.</li>

<li>In <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a>, the <a href="Large_eddy_simulation" title="Large eddy simulation">large eddy simulation</a> (LES) <a href="Turbulence_model" class="mw-redirect" title="Turbulence model">turbulence model</a> uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost.</li></ul></li>

<li>In <a href="Probability_theory" title="Probability theory">probability theory</a>, the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of the sum of two <a href="Independent_(probability)" class="mw-redirect" title="Independent (probability)">independent</a> <a href="Random_variable" title="Random variable">random variables</a> is the convolution of their individual distributions.

<ul><li>In <a href="Kernel_density_estimation" title="Kernel density estimation">kernel density estimation</a>, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian.<a href="#cite_note-FOOTNOTEDiggle1985-42">[40]</a></li></ul></li>

<li>In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm.</li>

<li>In structural reliability, the reliability index can be defined based on the convolution theorem.

<ul><li>The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the <a href="Joint_distribution_function" class="mw-redirect" title="Joint distribution function">joint distribution function</a>. In fact, the joint distribution function can be obtained using the convolution theory.<a href="#cite_note-FOOTNOTEGhasemiNowak2017-43">[41]</a></li></ul></li>

<li>In <a href="Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a>, simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, some physical quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>

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</math><img src="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> is calculated as a convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation>

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</math><img src="./6019bb70c912e59e9d5f442e9217517743ed4831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}" loading="lazy"> with a weighting function, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> denotes the neighbors of particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.<a href="#cite_note-1992ARA&A..30..543M-44">[42]</a></li>

<li>In <a href="Fractional_calculus" title="Fractional calculus">Fractional calculus</a> convolution is instrumental in various definitions of fractional integral and fractional derivative.</li></ul>

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<ul><li><a href="Analog_signal_processing" title="Analog signal processing">Analog signal processing</a></li>

<li><a href="Circulant_matrix" title="Circulant matrix">Circulant matrix</a></li>

<li><a href="Convolution_for_optical_broad-beam_responses_in_scattering_media" title="Convolution for optical broad-beam responses in scattering media">Convolution for optical broad-beam responses in scattering media</a></li>

<li><a href="Convolution_power" title="Convolution power">Convolution power</a></li>

<li><a href="Convolution_quotient" title="Convolution quotient">Convolution quotient</a></li>

<li><a href="Deconvolution" title="Deconvolution">Deconvolution</a></li>

<li><a href="Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a></li>

<li><a href="Generalized_signal_averaging" title="Generalized signal averaging">Generalized signal averaging</a></li>

<li><a href="List_of_convolutions_of_probability_distributions" title="List of convolutions of probability distributions">List of convolutions of probability distributions</a></li>

<li><a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system theory#Impulse response and convolution</a></li>

<li><a href="Multidimensional_discrete_convolution" title="Multidimensional discrete convolution">Multidimensional discrete convolution</a></li>

<li><a href="Scaled_correlation" title="Scaled correlation">Scaled correlation</a></li>

<li><a href="Titchmarsh_convolution_theorem" title="Titchmarsh convolution theorem">Titchmarsh convolution theorem</a></li>

<li><a href="Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a> (convolutions can be considered a Toeplitz matrix operation where each row is a shifted copy of the convolution kernel)</li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet transform</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Reasons for the reflection include:

<ul><li>It is necessary to implement the equivalent of the pointwise product of the Fourier transforms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">.</li>

<li>When the convolution is viewed as a <a href="Moving-average_model" title="Moving-average model">moving weighted average</a>, the weighting function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(-x)}</annotation>

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</math><img src="./0940c1b2e8d4161580c3bff2362ff7bf397c75b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.063ex; height:2.843ex;" alt="{\displaystyle g(-x)}" loading="lazy">, is often specified in terms of another function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

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</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy">, called the <a href="Impulse_response" title="Impulse response">impulse response</a> of a <a href="Linear_time-invariant_system#Impulse_response_and_convolution" title="Linear time-invariant system">linear time-invariant system</a>.</li></ul>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> The symbol <style data-mw-deduplicate="TemplateStyles:r886049734">

/* start https://en.wikipedia.org/ */

.mw-parser-output .monospaced{font-family:monospace,monospace}

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</style>U+2217 ∗ ASTERISK OPERATOR is different than U+002A * ASTERISK, which is often used to denote complex conjugation. See <a href="Asterisk#Mathematical_typography" title="Asterisk">Asterisk § Mathematical typography</a>.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBahriAshinoVaillancourt2013" class="citation journal cs1">Bahri, Mawardi; Ashino, Ryuichi; Vaillancourt, Rémi (2013). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/25493611.pdf">"Convolution Theorems for Quaternion Fourier Transform: Properties and Applications"</a> (PDF). Abstract and Applied Analysis. 2013: 1–10. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2013%2F162769">10.1155/2013/162769</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201021001150/https://core.ac.uk/download/pdf/25493611.pdf">Archived</a> (PDF) from the original on 2020-10-21. Retrieved 2022-11-11.</cite>

</li>

<cite id="CITEREFSmith1997" class="citation book cs1">Smith, Stephen W (1997). <a rel="nofollow" class="external text" href="https://dspguide.com/ch13/2.htm">"13.Convolution"</a>. The Scientist and Engineer's Guide to Digital Signal Processing (1 ed.). California Technical Publishing. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-9660176-3-3</bdi>. Retrieved 22 April 2016.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFIrwin1997" class="citation book cs1"><a href="J._David_Irwin" title="J. David Irwin">Irwin, J. David</a> (1997). "4.3". The Industrial Electronics Handbook (1 ed.). Boca Raton, FL: CRC Press. p. 75. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-8343-9</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFDifferential_Equations_(Spring_2010)" class="citation web cs1">Differential Equations (Spring 2010), MIT 18.03. <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-21-convolution-formula/">"Lecture 21: Convolution Formula"</a>. MIT Open Courseware. MIT. Retrieved 22 December 2021.</cite><code class="cs1-code">{{cite web}}</code>: CS1 maint: numeric names: authors list (link)

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/transfer-system-and-weight-functions-greens-formula/MIT18_03SCF11_s30_5text.pdf">"18.03SC Differential Equations Fall 2011"</a> (PDF). Green's Formula, Laplace Transform of Convolution. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150906102242/https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/transfer-system-and-weight-functions-greens-formula/MIT18_03SCF11_s30_5text.pdf">Archived</a> (PDF) from the original on 2015-09-06.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Dominguez-Torres, p 2

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Dominguez-Torres, p 4

</li>

<cite id="CITEREFR._N._Bracewell2005" class="citation cs2">R. N. Bracewell (2005), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v2SqL0zCrwcC&pg=PA172">"Early work on imaging theory in radio astronomy"</a>, in W. T. Sullivan (ed.), The Early Years of Radio Astronomy: Reflections Fifty Years After Jansky's Discovery, Cambridge University Press, p. 172, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-61602-7</bdi></cite>

</li>

<cite id="CITEREFJohn_Hilton_Grace_and_Alfred_Young1903" class="citation cs2">John Hilton Grace and Alfred Young (1903), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NIe4AAAAIAAJ&pg=PA40">The algebra of invariants</a>, Cambridge University Press, p. 40</cite>

</li>

<cite id="CITEREFLeonard_Eugene_Dickson1914" class="citation cs2">Leonard Eugene Dickson (1914), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LRGoAAAAIAAJ&pg=PA85">Algebraic invariants</a>, J. Wiley, p. 85, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4297-0042-9</bdi></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)

</li>

According to

[Lothar von Wolfersdorf (2000), "Einige Klassen quadratischer Integralgleichungen",

Sitzungsberichte der Sächsischen Akademie der Wissenschaften zu Leipzig,

Mathematisch-naturwissenschaftliche Klasse, volume 128, number 2, 6–7], the source is Volterra, Vito (1913),

"Leçons sur les fonctions de linges". Gauthier-Villars, Paris 1913.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFDamelinMiller2011">Damelin & Miller 2011</a>, p. 219

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFPressFlanneryTeukolskyVetterling1989" class="citation book cs1">Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/numericalrecipes0000unse/page/450">Numerical Recipes in Pascal</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/numericalrecipes0000unse/page/450">450</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-37516-9</bdi>.</cite>

</li>

<li id="cite_note-Rader1972-16"><a href="#cite_ref-Rader1972_16-0">^</a> <cite id="CITEREFRader1972" class="citation journal cs1">Rader, C.M. (December 1972). "Discrete Convolutions via Mersenne Transforms". IEEE Transactions on Computers. 21 (12): 1269–1273. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FT-C.1972.223497">10.1109/T-C.1972.223497</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1939809">1939809</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFWinograd1980" class="citation book cs1">Winograd, Shmuel (January 1980). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/book/10.1137/1.9781611970364">Arithmetic Complexity of Computations</a>. Society for Industrial and Applied Mathematics. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611970364">10.1137/1.9781611970364</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-163-9</bdi>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFLyakhovNagornovSemyonovaAbdulsalyamova2023" class="citation journal cs1">Lyakhov, P. A.; Nagornov, N. N.; Semyonova, N. F.; Abdulsalyamova, A. S. (June 2023). <a rel="nofollow" class="external text" href="https://link.springer.com/10.1134/S1054661823020074">"Reducing the Computational Complexity of Image Processing Using Wavelet Transform Based on the Winograd Method"</a>. Pattern Recognition and Image Analysis. 33 (2): 184–191. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS1054661823020074">10.1134/S1054661823020074</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1054-6618">1054-6618</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:259310351">259310351</a>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFWuFanCaoWang2021" class="citation journal cs1">Wu, Di; Fan, Xitian; Cao, Wei; Wang, Lingli (May 2021). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/9373543">"SWM: A High-Performance Sparse-Winograd Matrix Multiplication CNN Accelerator"</a>. IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 29 (5): 936–949. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTVLSI.2021.3060041">10.1109/TVLSI.2021.3060041</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1063-8210">1063-8210</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:233433757">233433757</a>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFMittalVibhu2021" class="citation journal cs1">Mittal, Sparsh; Vibhu (May 2021). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S1383762121000400">"A survey of accelerator architectures for 3D convolution neural networks"</a>. Journal of Systems Architecture. 115: 102041. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.sysarc.2021.102041">10.1016/j.sysarc.2021.102041</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:233917781">233917781</a>.</cite>

</li>

<li id="cite_note-Madisetti1999-21"><a href="#cite_ref-Madisetti1999_21-0">^</a> <cite id="CITEREFSelesnickBurrus1999" class="citation book cs1">Selesnick, Ivan W.; Burrus, C. Sidney (1999). "Fast Convolution and Filtering". In Madisetti, Vijay K. (ed.). Digital Signal Processing Handbook. CRC Press. p. Section 8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4200-4563-5</bdi>.</cite>

</li>

<li id="cite_note-Juang2004-22"><a href="#cite_ref-Juang2004_22-0">^</a> <cite id="CITEREFJuang" class="citation web cs1">Juang, B.H. <a rel="nofollow" class="external text" href="https://users.ece.gatech.edu/~juang/4270/BHJ4270-21.pdf">"Lecture 21: Block Convolution"</a> (PDF). EECS at the Georgia Institute of Technology. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040729204137/https://users.ece.gatech.edu/~juang/4270/BHJ4270-21.pdf">Archived</a> (PDF) from the original on 2004-07-29. Retrieved 17 May 2013.</cite>

</li>

<li id="cite_note-Gardner1994-23"><a href="#cite_ref-Gardner1994_23-0">^</a> <cite id="CITEREFGardner1994" class="citation journal cs1">Gardner, William G. (November 1994). <a rel="nofollow" class="external text" href="https://cs.ust.hk/mjg_lib/bibs/DPSu/DPSu.Files/Ga95.PDF">"Efficient Convolution without Input/Output Delay"</a> (PDF). Audio Engineering Society Convention 97. Paper 3897. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150408211312/https://cs.ust.hk/mjg_lib/bibs/DPSu/DPSu.Files/Ga95.PDF">Archived</a> (PDF) from the original on 2015-04-08. Retrieved 17 May 2013.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFBeckner1975" class="citation journal cs1"><a href="William_Beckner_(mathematician)" title="William Beckner (mathematician)">Beckner, William</a> (1975). "Inequalities in Fourier analysis". Annals of Mathematics. Second Series. 102 (1): 159–182. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970980">10.2307/1970980</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970980">1970980</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFBrascampLieb1976" class="citation journal cs1">Brascamp, Herm Jan; <a href="Elliott_H._Lieb" title="Elliott H. Lieb">Lieb, Elliott H.</a> (1976). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2876%2990184-5">"Best constants in Young's inequality, its converse, and its generalization to more than three functions"</a>. <a href="Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>. 20 (2): 151–173. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0001-8708%2876%2990184-5">10.1016/0001-8708(76)90184-5</a>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFReedSimon1975">Reed & Simon 1975</a>, IX.4

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Convolution.html">"Convolution"</a>. mathworld.wolfram.com. Retrieved 2021-09-22.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConvolutionTheorem.html">"From MathWorld--A Wolfram Web Resource"</a>.</cite>

</li>

<li id="cite_note-slyusar-29"><a href="#cite_ref-slyusar_29-0">^</a> <cite id="CITEREFSlyusar1996" class="citation journal cs1">Slyusar, V. I. (December 27, 1996). <a rel="nofollow" class="external text" href="https://slyusar.kiev.ua/en/IZV_1998_3.pdf">"End products in matrices in radar applications"</a> (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130811122444/https://slyusar.kiev.ua/en/IZV_1998_3.pdf">Archived</a> (PDF) from the original on 2013-08-11.</cite>

</li>

<li id="cite_note-slyusar1-30"><a href="#cite_ref-slyusar1_30-0">^</a> <cite id="CITEREFSlyusar1997" class="citation journal cs1">Slyusar, V. I. (1997-05-20). <a rel="nofollow" class="external text" href="https://slyusar.kiev.ua/ICATT97.pdf">"Analytical model of the digital antenna array on a basis of face-splitting matrix products"</a> (PDF). Proc. ICATT-97, Kyiv: 108–109. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130811112059/https://slyusar.kiev.ua/ICATT97.pdf">Archived</a> (PDF) from the original on 2013-08-11.</cite>

</li>

<li id="cite_note-DIPED-31"><a href="#cite_ref-DIPED_31-0">^</a> <cite id="CITEREFSlyusar1997" class="citation journal cs1">Slyusar, V. I. (1997-09-15). <a rel="nofollow" class="external text" href="https://slyusar.kiev.ua/DIPED_1997.pdf">"New operations of matrices product for applications of radars"</a> (PDF). Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130811113217/https://slyusar.kiev.ua/DIPED_1997.pdf">Archived</a> (PDF) from the original on 2013-08-11.</cite>

</li>

<li id="cite_note-slyusar2-32"><a href="#cite_ref-slyusar2_32-0">^</a> <cite id="CITEREFSlyusar1998" class="citation journal cs1">Slyusar, V. I. (March 13, 1998). <a rel="nofollow" class="external text" href="https://slyusar.kiev.ua/FACE.pdf">"A Family of Face Products of Matrices and its Properties"</a> (PDF). Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz.- 1999. 35 (3): 379–384. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02733426">10.1007/BF02733426</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119661450">119661450</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130811113935/https://slyusar.kiev.ua/FACE.pdf">Archived</a> (PDF) from the original on 2013-08-11.</cite>

</li>

<li id="cite_note-general-33"><a href="#cite_ref-general_33-0">^</a> <cite id="CITEREFSlyusar2003" class="citation journal cs1">Slyusar, V. I. (2003). <a rel="nofollow" class="external text" href="https://slyusar.kiev.ua/en/IZV_2003_10.pdf">"Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels"</a> (PDF). Radioelectronics and Communications Systems. 46 (10): 9–17. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130811125643/https://slyusar.kiev.ua/en/IZV_2003_10.pdf">Archived</a> (PDF) from the original on 2013-08-11.</cite>

</li>

<li id="cite_note-ninh-34"><a href="#cite_ref-ninh_34-0">^</a> <cite id="CITEREFNinhPagh2013" class="citation conference cs1">Ninh, Pham; <a href="Rasmus_Pagh" title="Rasmus Pagh">Pagh, Rasmus</a> (2013). Fast and scalable polynomial kernels via explicit feature maps. SIGKDD international conference on Knowledge discovery and data mining. Association for Computing Machinery. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2487575.2487591">10.1145/2487575.2487591</a>.</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> Hewitt and Ross (1979) Abstract harmonic analysis, volume 1, second edition, Springer-Verlag, p 266.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> Hewitt and Ross (1979), Theorem 19.18, p 272.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFR._Tyrrell_Rockafellar1970" class="citation cs2"><a href="R._Tyrrell_Rockafellar" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a> (1970), Convex analysis, Princeton University Press</cite>

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFZhangSoonYeFuh2020" class="citation journal cs1">Zhang, Yingjie; Soon, Hong Geok; Ye, Dongsen; Fuh, Jerry Ying Hsi; Zhu, Kunpeng (September 2020). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/8913613">"Powder-Bed Fusion Process Monitoring by Machine Vision With Hybrid Convolutional Neural Networks"</a>. IEEE Transactions on Industrial Informatics. 16 (9): 5769–5779. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTII.2019.2956078">10.1109/TII.2019.2956078</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1941-0050">1941-0050</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:213010088">213010088</a>.</cite>

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> <cite id="CITEREFChervyakovLyakhovDeryabinNagornov2020" class="citation journal cs1">Chervyakov, N.I.; Lyakhov, P.A.; Deryabin, M.A.; Nagornov, N.N.; Valueva, M.V.; Valuev, G.V. (September 2020). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S092523122030583X">"Residue Number System-Based Solution for Reducing the Hardware Cost of a Convolutional Neural Network"</a>. Neurocomputing. 407: 439–453. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neucom.2020.04.018">10.1016/j.neucom.2020.04.018</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:219470398">219470398</a>. <q>Convolutional neural networks represent deep learning architectures that are currently used in a wide range of applications, including computer vision, speech recognition, time series analysis in finance, and many others.</q></cite>

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFAtlas,_Homma,_and_Marks" class="citation journal cs1">Atlas, Homma, and Marks. <a rel="nofollow" class="external text" href="https://papers.nips.cc/paper/1987/file/98f13708210194c475687be6106a3b84-Paper.pdf">"An Artificial Neural Network for Spatio-Temporal Bipolar Patterns: Application to Phoneme Classification"</a> (PDF). Neural Information Processing Systems (NIPS 1987). 1. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210414091306/https://papers.nips.cc/paper/1987/file/98f13708210194c475687be6106a3b84-Paper.pdf">Archived</a> (PDF) from the original on 2021-04-14.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> Zölzer, Udo, ed. (2002). DAFX:Digital Audio Effects, p.48–49. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0471490784</bdi>.

</li>

<li id="cite_note-FOOTNOTEDiggle1985-42"><a href="#cite_ref-FOOTNOTEDiggle1985_42-0">^</a> <a href="#CITEREFDiggle1985">Diggle 1985</a>.

</li>

<li id="cite_note-FOOTNOTEGhasemiNowak2017-43"><a href="#cite_ref-FOOTNOTEGhasemiNowak2017_43-0">^</a> <a href="#CITEREFGhasemiNowak2017">Ghasemi & Nowak 2017</a>.

</li>

<li id="cite_note-1992ARA&A..30..543M-44"><a href="#cite_ref-1992ARA&A..30..543M_44-0">^</a> <cite id="1992ARA&A..30..543M" class="citation journal cs1">Monaghan, J. J. (1992). <a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ARA&A..30..543M">"Smoothed particle hydrodynamics"</a>. Annual Review of Astronomy and Astrophysics. 30: 543–547. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ARA&A..30..543M">1992ARA&A..30..543M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.aa.30.090192.002551">10.1146/annurev.aa.30.090192.002551</a>. Retrieved 16 February 2021.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBracewell1986" class="citation cs2">Bracewell, R. (1986), The Fourier Transform and Its Applications (2nd ed.), McGraw–Hill, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986ftia.book.....B">1986ftia.book.....B</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-116043-4</bdi></cite>.</li>

<li><cite id="CITEREFDamelinMiller2011" class="citation cs2">Damelin, S.; Miller, W. (2011), The Mathematics of Signal Processing, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1107601048</bdi></cite></li>

<li><cite id="CITEREFDiggle1985" class="citation cs2">Diggle, P. J. (1985), "A kernel method for smoothing point process data", Journal of the Royal Statistical Society, Series C, 34 (2): 138–147, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2347366">10.2307/2347366</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347366">2347366</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116746157">116746157</a></cite></li>

<li>Dominguez-Torres, Alejandro (Nov 2, 2010). "Origin and history of convolution". 41 pgs. <a rel="nofollow" class="external free" href="https://slideshare.net/Alexdfar/origin-adn-history-of-convolution">https://slideshare.net/Alexdfar/origin-adn-history-of-convolution</a>. Cranfield, Bedford MK43 OAL, UK. Retrieved Mar 13, 2013.</li>

<li><cite id="CITEREFGhasemiNowak2017" class="citation cs2">Ghasemi, S. Hooman; Nowak, Andrzej S. (2017), "Reliability Index for Non-normal Distributions of Limit State Functions", Structural Engineering and Mechanics, 62 (3): 365–372, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12989%2Fsem.2017.62.3.365">10.12989/sem.2017.62.3.365</a></cite></li>

<li><cite id="CITEREFGrinshpan2017" class="citation cs2">Grinshpan, A. Z. (2017), "An inequality for multiple convolutions with respect to Dirichlet probability measure", Advances in Applied Mathematics, 82 (1): 102–119, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aam.2016.08.001">10.1016/j.aam.2016.08.001</a></cite></li>

<li><cite id="CITEREFHewittRoss1979" class="citation cs2">Hewitt, Edwin; Ross, Kenneth A. (1979), Abstract harmonic analysis. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115 (2nd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-09434-0</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0551496">0551496</a></cite>.</li>

<li><cite id="CITEREFHewittRoss1970" class="citation cs2">Hewitt, Edwin; Ross, Kenneth A. (1970), Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0262773">0262773</a></cite>.</li>

<li><cite id="CITEREFHörmander1983" class="citation cs2"><a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Hörmander, L.</a> (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-96750-4">10.1007/978-3-642-96750-4</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-12104-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0717035">0717035</a></cite>.</li>

<li><cite id="CITEREFKassel1995" class="citation cs2">Kassel, Christian (1995), <a rel="nofollow" class="external text" href="https://archive.org/details/quantumgroups0000kass">Quantum groups</a>, Graduate Texts in Mathematics, vol. 155, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0783-2">10.1007/978-1-4612-0783-2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94370-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1321145">1321145</a></cite>.</li>

<li><cite id="CITEREFKnuth1997" class="citation cs2"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997), Seminumerical Algorithms (3rd. ed.), Reading, Massachusetts: Addison–Wesley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-89684-2</bdi></cite>.</li>

<li><cite id="CITEREFNariciBeckenstein2011" class="citation book cs1">Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584888666</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a>.</cite></li>

<li><cite id="CITEREFReedSimon1975" class="citation cs2">Reed, Michael; <a href="Barry_Simon" title="Barry Simon">Simon, Barry</a> (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers, pp. xv+361, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-585002-6</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0493420">0493420</a></cite></li>

<li><cite id="CITEREFRudin1962" class="citation cs2"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1962), Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, vol. 12, New York–London: Interscience Publishers, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-52364-X</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0152834">0152834</a></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help).</li>

<li><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7155-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite></li>

<li><cite id="CITEREFSteinWeiss1971" class="citation cs2"><a href="Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a>; Weiss, Guido (1971), <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontofo0000stei">Introduction to Fourier Analysis on Euclidean Spaces</a>, Princeton University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-08078-X</bdi></cite>.</li>

<li><cite id="CITEREFSobolev2001" class="citation cs2">Sobolev, V.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Convolution_of_functions">"Convolution of functions"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>

<li><cite id="CITEREFStrichartz1994" class="citation cs2">Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-8273-4</bdi></cite>.</li>

<li><cite id="CITEREFTitchmarsh1948" class="citation cs2"><a href="Edward_Charles_Titchmarsh" title="Edward Charles Titchmarsh">Titchmarsh, E</a> (1948), Introduction to the theory of Fourier integrals (2nd ed.), New York, N.Y.: Chelsea Pub. Co. (published 1986), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8284-0324-5</bdi></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help).</li>

<li><cite id="CITEREFTrèves2006" class="citation book cs1"><a href="Fran%C3%A7ois_Tr%C3%A8ves" title="François Trèves">Trèves, François</a> (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-45352-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853623322">853623322</a>.</cite></li>

<li><cite id="CITEREFUludag1998" class="citation cs2">Uludag, A. M. (1998), "On possible deterioration of smoothness under the operation of convolution", J. Math. Anal. Appl., 227 (2): 335–358, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1998.6091">10.1006/jmaa.1998.6091</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11693%2F25385">11693/25385</a></cite></li>

<li><cite id="CITEREFvon_zur_GathenGerhard2003" class="citation cs2">von zur Gathen, J.; Gerhard, J . (2003), Modern Computer Algebra, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-82646-2</bdi></cite>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/convolution" class="extiw external" title="wiktionary:convolution">convolution</a> in Wiktionary, the free dictionary.</div></div>

</div>

<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Convolution" class="extiw external" title="commons:Category:Convolution">Convolution</a>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="https://jeff560.tripod.com/c.html">Earliest Uses: The entry on Convolution has some historical information.</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060221234856/https://rkb.home.cern.ch/rkb/AN16pp/node38.html#SECTION000380000000000000000">Convolution</a>, on <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060512020859/https://rkb.home.cern.ch/rkb/titleA.html">The Data Analysis BriefBook</a></li>

<li><a rel="nofollow" class="external free" href="https://jhu.edu/~signals/convolve/index.html">https://jhu.edu/~signals/convolve/index.html</a> Visual convolution Java Applet</li>

<li><a rel="nofollow" class="external free" href="https://jhu.edu/~signals/discreteconv2/index.html">https://jhu.edu/~signals/discreteconv2/index.html</a> Visual convolution Java Applet for discrete-time functions</li>

<li><a rel="nofollow" class="external free" href="https://get-the-solution.net/projects/discret-convolution">https://get-the-solution.net/projects/discret-convolution</a> discret-convolution online calculator</li>

<li><a rel="nofollow" class="external free" href="https://lpsa.swarthmore.edu/Convolution/CI.html">https://lpsa.swarthmore.edu/Convolution/CI.html</a> Convolution demo and visualization in JavaScript</li>

<li><a rel="nofollow" class="external free" href="https://phiresky.github.io/convolution-demo/">https://phiresky.github.io/convolution-demo/</a> Another convolution demo in JavaScript</li>

<li><a href="https://archive.org/details/Lectures_on_Image_Processing" class="extiw external" title="iarchive:Lectures on Image Processing">Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution.</a>, by Alan Peters</li>

<li><a rel="nofollow" class="external free" href="https://archive.org/details/Lectures_on_Image_Processing">https://archive.org/details/Lectures_on_Image_Processing</a></li>

<li><a rel="nofollow" class="external text" href="https://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/kernelmaskoperation/">Convolution Kernel Mask Operation Interactive tutorial</a></li>

<li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Convolution.html">Convolution</a> at <a href="MathWorld" title="MathWorld">MathWorld</a></li>

<li><a rel="nofollow" class="external text" href="https://nongnu.org/freeverb3/">Freeverb3 Impulse Response Processor</a>: Opensource zero latency impulse response processor with VST plugins</li>

<li>Stanford University CS 178 <a rel="nofollow" class="external text" href="https://graphics.stanford.edu/courses/cs178/applets/convolution.html">interactive Flash demo</a> showing how spatial convolution works.</li>

<li><a rel="nofollow" class="external text" href="https://youtube.com/watch?v=IW4Reburjpc">A video lecture on the subject of convolution</a> given by <a href="Salman_Khan_(educator)" class="mw-redirect" title="Salman Khan (educator)">Salman Khan</a></li>

<li><a rel="nofollow" class="external text" href="https://dspguide.com/ch24/6.htm">Example of FFT convolution for pattern-recognition (image processing)</a></li>

<li><a rel="nofollow" class="external text" href="https://betterexplained.com/articles/intuitive-convolution/">Intuitive Guide to Convolution</a> A blogpost about an intuitive interpretation of convolution.</li></ul>

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</style><div id="Artificial_intelligence_(AI)426" style="font-size:114%;margin:0 4em"><a href="Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a> (AI)</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_artificial_intelligence" title="History of artificial intelligence">History</a>

<ul><li><a href="Timeline_of_artificial_intelligence" title="Timeline of artificial intelligence">timeline</a></li></ul></li>

<li><a href="List_of_artificial_intelligence_companies" title="List of artificial intelligence companies">Companies</a></li>

<li><a href="List_of_artificial_intelligence_projects" title="List of artificial intelligence projects">Projects</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Parameter" title="Parameter">Parameter</a>

<ul><li><a href="Hyperparameter_(machine_learning)" title="Hyperparameter (machine learning)">Hyperparameter</a></li></ul></li>

<li><a href="Loss_functions_for_classification" title="Loss functions for classification">Loss functions</a></li>

<li><a href="Regression_analysis" title="Regression analysis">Regression</a>

<ul><li><a href="Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li>

<li><a href="Double_descent" title="Double descent">Double descent</a></li>

<li><a href="Overfitting" title="Overfitting">Overfitting</a></li></ul></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Clustering</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient descent</a>

<li><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton method</a></li>

<li><a href="Conjugate_gradient_method" title="Conjugate gradient method">Conjugate gradient method</a></li></ul></li>

<li><a href="Backpropagation" title="Backpropagation">Backpropagation</a></li>

<li><a href="Attention_(machine_learning)" title="Attention (machine learning)">Attention</a></li>

<li><a href="Normalization_(machine_learning)" title="Normalization (machine learning)">Normalization</a>

<ul><li><a href="Batch_normalization" title="Batch normalization">Batchnorm</a></li></ul></li>

<li><a href="Activation_function" title="Activation function">Activation</a>

<ul><li><a href="Softmax_function" title="Softmax function">Softmax</a></li>

<li><a href="Sigmoid_function" title="Sigmoid function">Sigmoid</a></li>

<li><a href="Rectifier_(neural_networks)" title="Rectifier (neural networks)">Rectifier</a></li></ul></li>

<li><a href="Gating_mechanism" title="Gating mechanism">Gating</a></li>

<li><a href="Weight_initialization" title="Weight initialization">Weight initialization</a></li>

<li><a href="Regularization_(mathematics)" title="Regularization (mathematics)">Regularization</a></li>

<li><a href="Training%2C_validation%2C_and_test_data_sets" title="Training, validation, and test data sets">Datasets</a>

<ul><li><a href="Data_augmentation" title="Data augmentation">Augmentation</a></li></ul></li>

<li><a href="Prompt_engineering" title="Prompt engineering">Prompt engineering</a></li>

<li><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a>

<ul><li><a href="Q-learning" title="Q-learning">Q-learning</a></li>

<li><a href="State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li>

<li><a href="Imitation_learning" title="Imitation learning">Imitation</a></li>

<li><a href="Policy_gradient_method" title="Policy gradient method">Policy gradient</a></li></ul></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion</a></li>

<li><a href="Latent_diffusion_model" title="Latent diffusion model">Latent diffusion model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregression</a></li>

<li><a href="Adversarial_machine_learning" title="Adversarial machine learning">Adversary</a></li>

<li><a href="Uncanny_valley" title="Uncanny valley">Uncanny valley</a></li>

<li><a href="Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li>

<li><a href="Reflection_(artificial_intelligence)" class="mw-redirect" title="Reflection (artificial intelligence)">Reflection</a></li>

<li><a href="Recursive_self-improvement" title="Recursive self-improvement">Recursive self-improvement</a></li>

<li><a href="Hallucination_(artificial_intelligence)" title="Hallucination (artificial intelligence)">Hallucination</a></li>

<li><a href="Word_embedding" title="Word embedding">Word embedding</a></li>

<li><a href="Vibe_coding" title="Vibe coding">Vibe coding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Machine_learning" title="Machine learning">Machine learning</a>

<ul><li><a href="Prompt_engineering#In-context_learning" title="Prompt engineering">In-context learning</a></li></ul></li>

<li><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Artificial neural network</a>

<ul><li><a href="Deep_learning" title="Deep learning">Deep learning</a></li></ul></li>

<li><a href="Language_model" title="Language model">Language model</a>

<ul><li><a href="Large_language_model" title="Large language model">Large language model</a></li>

<li><a href="Reasoning_language_model" title="Reasoning language model">Reasoning language model</a></li>

<li><a href="Model_Context_Protocol" title="Model Context Protocol">Model Context Protocol</a></li>

<li><a href="Intelligent_agent" title="Intelligent agent">Intelligent agent</a></li>

<li><a href="Artificial_human_companion" title="Artificial human companion">Artificial human companion</a></li>

<li><a href="Humanity's_Last_Exam" title="Humanity's Last Exam">Humanity's Last Exam</a></li>

<li><a href="Artificial_general_intelligence" title="Artificial general intelligence">Artificial general intelligence (AGI)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Audio–visual</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AlexNet" title="AlexNet">AlexNet</a></li>

<li><a href="WaveNet" title="WaveNet">WaveNet</a></li>

<li><a href="Human_image_synthesis" title="Human image synthesis">Human image synthesis</a></li>

<li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Deep_learning_speech_synthesis" title="Deep learning speech synthesis">Speech synthesis</a>

<li><a href="ElevenLabs" title="ElevenLabs">ElevenLabs</a></li></ul></li>

<li><a href="Speech_recognition" title="Speech recognition">Speech recognition</a>

<ul><li><a href="Whisper_(speech_recognition_system)" title="Whisper (speech recognition system)">Whisper</a></li></ul></li>

<li><a href="Facial_recognition_system" title="Facial recognition system">Facial recognition</a></li>

<li><a href="AlphaFold" title="AlphaFold">AlphaFold</a></li>

<li><a href="Text-to-image_model" title="Text-to-image model">Text-to-image models</a>

<ul><li><a href="Aurora_(text-to-image_model)" class="mw-redirect" title="Aurora (text-to-image model)">Aurora</a></li>

<li><a href="Adobe_Firefly" title="Adobe Firefly">Firefly</a></li>

<li><a href="Ideogram_(text-to-image_model)" title="Ideogram (text-to-image model)">Ideogram</a></li>

<li><a href="Imagen_(text-to-image_model)" title="Imagen (text-to-image model)">Imagen</a></li>

<li><a href="Midjourney" title="Midjourney">Midjourney</a></li>

<li><a href="Recraft" title="Recraft">Recraft</a></li>

<li><a href="Stable_Diffusion" title="Stable Diffusion">Stable Diffusion</a></li></ul></li>

<li><a href="Text-to-video_model" title="Text-to-video model">Text-to-video models</a>

<ul><li><a href="Dream_Machine_(text-to-video_model)" title="Dream Machine (text-to-video model)">Dream Machine</a></li>

<li><a href="Runway_(company)#Services_and_technologies" title="Runway (company)">Runway Gen</a></li>

<li><a href="MiniMax_(company)#Hailuo_AI" title="MiniMax (company)">Hailuo AI</a></li>

<li><a href="Kling_(text-to-video_model)" class="mw-redirect" title="Kling (text-to-video model)">Kling</a></li>

<li><a href="Music_and_artificial_intelligence" title="Music and artificial intelligence">Music generation</a>

<ul><li><a href="Riffusion" title="Riffusion">Riffusion</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Text</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="GloVe" title="GloVe">GloVe</a></li>

<li><a href="Llama_(language_model)" title="Llama (language model)">Llama</a></li>

<li><a href="Chinchilla_(language_model)" title="Chinchilla (language model)">Chinchilla AI</a></li>

<li><a href="ChatGPT" title="ChatGPT">ChatGPT</a></li>

<li><a href="Claude_(language_model)" title="Claude (language model)">Claude</a></li>

<li><a href="Gemini_(language_model)" title="Gemini (language model)">Gemini</a>

<ul><li><a href="Gemini_(chatbot)" title="Gemini (chatbot)">chatbot</a></li></ul></li>

<li><a href="LaMDA" title="LaMDA">LaMDA</a></li>

<li><a href="BLOOM_(language_model)" title="BLOOM (language model)">BLOOM</a></li>

<li><a href="Project_Debater" title="Project Debater">Project Debater</a></li>

<li><a href="IBM_Watson" title="IBM Watson">IBM Watson</a></li>

<li><a href="IBM_Watsonx" title="IBM Watsonx">IBM Watsonx</a></li>

<li><a href="IBM_Granite" title="IBM Granite">Granite</a></li>

<li><a href="Huawei_PanGu" title="Huawei PanGu">PanGu-Σ</a></li>

<li><a href="DeepSeek_(chatbot)" title="DeepSeek (chatbot)">DeepSeek</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decisional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AlphaGo" title="AlphaGo">AlphaGo</a></li>

<li><a href="AlphaZero" title="AlphaZero">AlphaZero</a></li>

<li><a href="OpenAI_Five" title="OpenAI Five">OpenAI Five</a></li>

<li><a href="Self-driving_car" title="Self-driving car">Self-driving car</a></li>

<li><a href="MuZero" title="MuZero">MuZero</a></li>

<li><a href="Action_selection" title="Action selection">Action selection</a>

<ul><li><a href="AutoGPT" title="AutoGPT">AutoGPT</a></li></ul></li>

<li><a href="Robot_control" title="Robot control">Robot control</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alan_Turing" title="Alan Turing">Alan Turing</a></li>

<li><a href="Warren_Sturgis_McCulloch" title="Warren Sturgis McCulloch">Warren Sturgis McCulloch</a></li>

<li><a href="Walter_Pitts" title="Walter Pitts">Walter Pitts</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li>

<li><a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li>

<li><a href="Shun'ichi_Amari" title="Shun'ichi Amari">Shun'ichi Amari</a></li>

<li><a href="Kunihiko_Fukushima" title="Kunihiko Fukushima">Kunihiko Fukushima</a></li>

<li><a href="Takeo_Kanade" title="Takeo Kanade">Takeo Kanade</a></li>

<li><a href="Marvin_Minsky" title="Marvin Minsky">Marvin Minsky</a></li>

<li><a href="John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">John McCarthy</a></li>

<li><a href="Nathaniel_Rochester_(computer_scientist)" title="Nathaniel Rochester (computer scientist)">Nathaniel Rochester</a></li>

<li><a href="Allen_Newell" title="Allen Newell">Allen Newell</a></li>

<li><a href="Cliff_Shaw" title="Cliff Shaw">Cliff Shaw</a></li>

<li><a href="Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a></li>

<li><a href="Oliver_Selfridge" title="Oliver Selfridge">Oliver Selfridge</a></li>

<li><a href="Frank_Rosenblatt" title="Frank Rosenblatt">Frank Rosenblatt</a></li>

<li><a href="Bernard_Widrow" title="Bernard Widrow">Bernard Widrow</a></li>

<li><a href="Joseph_Weizenbaum" title="Joseph Weizenbaum">Joseph Weizenbaum</a></li>

<li><a href="Seymour_Papert" title="Seymour Papert">Seymour Papert</a></li>

<li><a href="Seppo_Linnainmaa" title="Seppo Linnainmaa">Seppo Linnainmaa</a></li>

<li><a href="Paul_Werbos" title="Paul Werbos">Paul Werbos</a></li>

<li><a href="Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a></li>

<li><a href="John_Hopfield" title="John Hopfield">John Hopfield</a></li>

<li><a href="J%C3%BCrgen_Schmidhuber" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a></li>

<li><a href="Yann_LeCun" title="Yann LeCun">Yann LeCun</a></li>

<li><a href="Yoshua_Bengio" title="Yoshua Bengio">Yoshua Bengio</a></li>

<li><a href="Lotfi_A._Zadeh" title="Lotfi A. Zadeh">Lotfi A. Zadeh</a></li>

<li><a href="Stephen_Grossberg" title="Stephen Grossberg">Stephen Grossberg</a></li>

<li><a href="Alex_Graves_(computer_scientist)" title="Alex Graves (computer scientist)">Alex Graves</a></li>

<li><a href="James_Goodnight" title="James Goodnight">James Goodnight</a></li>

<li><a href="Andrew_Ng" title="Andrew Ng">Andrew Ng</a></li>

<li><a href="Ilya_Sutskever" title="Ilya Sutskever">Ilya Sutskever</a></li>

<li><a href="Alex_Krizhevsky" title="Alex Krizhevsky">Alex Krizhevsky</a></li>

<li><a href="Ian_Goodfellow" title="Ian Goodfellow">Ian Goodfellow</a></li>

<li><a href="Demis_Hassabis" title="Demis Hassabis">Demis Hassabis</a></li>

<li><a href="David_Silver_(computer_scientist)" title="David Silver (computer scientist)">David Silver</a></li>

<li><a href="Andrej_Karpathy" title="Andrej Karpathy">Andrej Karpathy</a></li>

<li><a href="Ashish_Vaswani" title="Ashish Vaswani">Ashish Vaswani</a></li>

<li><a href="Noam_Shazeer" title="Noam Shazeer">Noam Shazeer</a></li>

<li><a href="Aidan_Gomez" title="Aidan Gomez">Aidan Gomez</a></li>

<li><a href="Mustafa_Suleyman" title="Mustafa Suleyman">Mustafa Suleyman</a></li>

<li><a href="Fran%C3%A7ois_Chollet" title="François Chollet">François Chollet</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Architectures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Neural_Turing_machine" title="Neural Turing machine">Neural Turing machine</a></li>

<li><a href="Differentiable_neural_computer" title="Differentiable neural computer">Differentiable neural computer</a></li>

<li><a href="Transformer_(deep_learning_architecture)" title="Transformer (deep learning architecture)">Transformer</a>

<ul><li><a href="Vision_transformer" title="Vision transformer">Vision transformer (ViT)</a></li></ul></li>

<li><a href="Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network (RNN)</a></li>

<li><a href="Long_short-term_memory" title="Long short-term memory">Long short-term memory (LSTM)</a></li>

<li><a href="Gated_recurrent_unit" title="Gated recurrent unit">Gated recurrent unit (GRU)</a></li>

<li><a href="Echo_state_network" title="Echo state network">Echo state network</a></li>

<li><a href="Multilayer_perceptron" title="Multilayer perceptron">Multilayer perceptron (MLP)</a></li>

<li><a href="Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network (CNN)</a></li>

<li><a href="Residual_neural_network" title="Residual neural network">Residual neural network (RNN)</a></li>

<li><a href="Highway_network" title="Highway network">Highway network</a></li>

<li><a href="Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li>

<li><a href="Autoencoder" title="Autoencoder">Autoencoder</a></li>

<li><a href="Variational_autoencoder" title="Variational autoencoder">Variational autoencoder (VAE)</a></li>

<li><a href="Generative_adversarial_network" title="Generative adversarial network">Generative adversarial network (GAN)</a></li>

<li><a href="Graph_neural_network" title="Graph neural network">Graph neural network (GNN)</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-01" href="https://en.wikipedia.org/wiki/?title=Convolution&oldid=1303690904">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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