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Convolution theorem

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the convolution theorem states that under suitable conditions the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of a <a href="Convolution" title="Convolution">convolution</a> of two functions (or <a href="Signal" title="Signal">signals</a>) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., <a href="Time_domain" title="Time domain">time domain</a>) equals point-wise multiplication in the other domain (e.g., <a href="Frequency_domain" title="Frequency domain">frequency domain</a>). Other versions of the convolution theorem are applicable to various <a href="List_of_Fourier-related_transforms" title="List of Fourier-related transforms">Fourier-related transforms</a>.

<div class="mw-heading mw-heading2"><h2 id="Functions_of_a_continuous_variable">Functions of a continuous variable</h2></div>

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

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

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</math><img src="./b1e5ff65a28eed29d36ddae9c6ae3b596fd14370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle u(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x)}">

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

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</math><img src="./b371a381e15c71d8fc4ec43cf14b156f02a0d35a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.267ex; height:2.843ex;" alt="{\displaystyle v(x)}" loading="lazy"> with <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{-\infty }^{\infty }u(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{-\infty }^{\infty }v(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \end{aligned}}}</annotation>

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</math><img src="./703940743483974572c895a7d66ec110b7defe64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:48.651ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{-\infty }^{\infty }u(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{-\infty }^{\infty }v(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \end{aligned}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> denotes the Fourier transform <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a>. The transform may be normalized in other ways, in which case constant scaling factors (typically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi }}}</annotation>

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</math><img src="./7a9b009153bbbb3273a7e7279cb6b084fd650a80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.43ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2\pi }}}" loading="lazy">) will appear in the convolution theorem below. The convolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is defined by:

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<annotation encoding="application/x-tex">{\displaystyle r(x)=\{u*v\}(x)\triangleq \int _{-\infty }^{\infty }u(\tau )v(x-\tau )\,d\tau =\int _{-\infty }^{\infty }u(x-\tau )v(\tau )\,d\tau .}</annotation>

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</math><img src="./8314d97f6d495649c11749a1926cde4f3ede3ec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.822ex; height:6.009ex;" alt="{\displaystyle r(x)=\{u*v\}(x)\triangleq \int _{-\infty }^{\infty }u(\tau )v(x-\tau )\,d\tau =\int _{-\infty }^{\infty }u(x-\tau )v(\tau )\,d\tau .}" loading="lazy"></dd></dl>

In this context the <a href="Asterisk" title="Asterisk">asterisk</a> denotes convolution, instead of standard multiplication. The <a href="Tensor_product" title="Tensor product">tensor product</a> symbol <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"> is sometimes used instead.

The convolution theorem states that:<a href="#cite_note-McGillem-1">[1]</a><a href="#cite_note-Weisstein-2">[2]</a>: eq.8

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<annotation encoding="application/x-tex">{\displaystyle R(f)\triangleq {\mathcal {F}}\{r\}(f)=U(f)V(f).\quad f\in \mathbb {R} }</annotation>

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</math><img src="./b5767b4e2ee4760af8c24be188a4261d7806d761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.336ex; height:3.009ex;" alt="{\displaystyle R(f)\triangleq {\mathcal {F}}\{r\}(f)=U(f)V(f).\quad f\in \mathbb {R} }" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.1a</td></tr></tbody></table>

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Applying the inverse Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{-1},}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{-1},}</annotation>

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</math><img src="./b601ddd4a698c1e9a5649743221abf4d0e6d90eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.983ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}^{-1},}" loading="lazy"> produces the corollary:<a href="#cite_note-Weisstein-2">[2]</a>: eqs.7, 10

<div class="equation-box" style="margin: ;padding: 6px; border-width:px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table">Convolution theorem

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<annotation encoding="application/x-tex">{\displaystyle r(x)=\{u*v\}(x)={\mathcal {F}}^{-1}\{U\cdot V\}.}</annotation>

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</math><img src="./32160df3e9fe2f17118efce881a5301603d7753b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.057ex; height:3.176ex;" alt="{\displaystyle r(x)=\{u*v\}(x)={\mathcal {F}}^{-1}\{U\cdot V\}.}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.1b</td></tr></tbody></table>

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The theorem also generally applies to multi-dimensional functions.

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<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Multi-dimensional derivation of Eq.1</div>

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Consider functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v}">

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</math><img src="./7e66f4b32a0181923cc1337a5634f38241e5c697.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}" loading="lazy"> in <a href="Lp_space" title="Lp space">Lp</a>-space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} ^{n}),}">

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} ^{n}),}</annotation>

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</math><img src="./e9f117d852749a7454257e54a1c7c8cb2e861fa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.99ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} ^{n}),}" loading="lazy"> with Fourier transforms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V}">

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

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</math><img src="./7681409ec5fffdb272f536757c1211fe0151a9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.604ex; height:2.509ex;" alt="{\displaystyle U,V}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{\mathbb {R} ^{n}}u(x)e^{-i2\pi f\cdot x}\,dx,\quad f\in \mathbb {R} ^{n}\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{\mathbb {R} ^{n}}v(x)e^{-i2\pi f\cdot x}\,dx,\end{aligned}}}</annotation>

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</math><img src="./4d907799d3682481d3c4b15170ebd87e6b6d6e72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:49.557ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{\mathbb {R} ^{n}}u(x)e^{-i2\pi f\cdot x}\,dx,\quad f\in \mathbb {R} ^{n}\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{\mathbb {R} ^{n}}v(x)e^{-i2\pi f\cdot x}\,dx,\end{aligned}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./abd5bfd5169b615983ab89830ca285739be9112f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.287ex; height:2.509ex;" alt="{\displaystyle f\cdot x}" loading="lazy"> indicates the <a href="Dot_product" title="Dot product">inner product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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<annotation encoding="application/x-tex">{\displaystyle f\cdot x=\sum _{j=1}^{n}{f}_{j}x_{j},}</annotation>

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</math><img src="./13b92442d5842e6b5e6f2ce52f6262fe52df4c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.063ex; height:7.176ex;" alt="{\displaystyle f\cdot x=\sum _{j=1}^{n}{f}_{j}x_{j},}" loading="lazy">   and   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx=\prod _{j=1}^{n}dx_{j}.}">

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<annotation encoding="application/x-tex">{\displaystyle dx=\prod _{j=1}^{n}dx_{j}.}</annotation>

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

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is defined by:

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<annotation encoding="application/x-tex">{\displaystyle r(x)\triangleq \int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau .}</annotation>

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</math><img src="./9d2104bdd12346d8556f666b969e1e6a2edcb9b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.451ex; height:5.676ex;" alt="{\displaystyle r(x)\triangleq \int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau .}" loading="lazy"></dd></dl>

Also:

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<annotation encoding="application/x-tex">{\displaystyle \iint |u(\tau )v(x-\tau )|\,dx\,d\tau =\int \left(|u(\tau )|\int |v(x-\tau )|\,dx\right)\,d\tau =\int |u(\tau )|\,\|v\|_{1}\,d\tau =\|u\|_{1}\|v\|_{1}.}</annotation>

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</math><img src="./2e00d7e0eef523b5e1c44bdad869baf506adaf2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.892ex; height:6.176ex;" alt="{\displaystyle \iint |u(\tau )v(x-\tau )|\,dx\,d\tau =\int \left(|u(\tau )|\int |v(x-\tau )|\,dx\right)\,d\tau =\int |u(\tau )|\,\|v\|_{1}\,d\tau =\|u\|_{1}\|v\|_{1}.}" loading="lazy"></dd></dl>

Hence by <a href="Fubini's_theorem" title="Fubini's theorem">Fubini's theorem</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in L^{1}(\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle r\in L^{1}(\mathbb {R} ^{n})}</annotation>

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</math><img src="./6bddab9cabd837c443e110234c18fc6562af878d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.232ex; height:3.176ex;" alt="{\displaystyle r\in L^{1}(\mathbb {R} ^{n})}" loading="lazy"> so its Fourier transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is defined by the integral formula:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R(f)\triangleq {\mathcal {F}}\{r\}(f)&=\int _{\mathbb {R} ^{n}}r(x)e^{-i2\pi f\cdot x}\,dx\\&=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau \right)\,e^{-i2\pi f\cdot x}\,dx.\end{aligned}}}</annotation>

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</math><img src="./003615289e661249b5ef2e87901e5e6f49e5d3b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:58.772ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}R(f)\triangleq {\mathcal {F}}\{r\}(f)&=\int _{\mathbb {R} ^{n}}r(x)e^{-i2\pi f\cdot x}\,dx\\&=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau \right)\,e^{-i2\pi f\cdot x}\,dx.\end{aligned}}}" loading="lazy"></dd></dl>

Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u(\tau )v(x-\tau )e^{-i2\pi f\cdot x}|=|u(\tau )v(x-\tau )|,}">

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<annotation encoding="application/x-tex">{\displaystyle |u(\tau )v(x-\tau )e^{-i2\pi f\cdot x}|=|u(\tau )v(x-\tau )|,}</annotation>

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</math><img src="./9e64ef32970c79c6bbff50806f7f880aa5d4269a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.86ex; height:3.176ex;" alt="{\displaystyle |u(\tau )v(x-\tau )e^{-i2\pi f\cdot x}|=|u(\tau )v(x-\tau )|,}" loading="lazy">  Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R(f)&=\int _{\mathbb {R} ^{n}}u(\tau )\underbrace {\left(\int _{\mathbb {R} ^{n}}v(x-\tau )\ e^{-i2\pi f\cdot x}\,dx\right)} _{V(f)\ e^{-i2\pi f\cdot \tau }}\,d\tau \\&=\underbrace {\left(\int _{\mathbb {R} ^{n}}u(\tau )\ e^{-i2\pi f\cdot \tau }\,d\tau \right)} _{U(f)}\ V(f).\end{aligned}}}</annotation>

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</math><img src="./35e9c8b29e77f6461fcaa36ec3db49a846b09641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:46.445ex; height:20.509ex;" alt="{\displaystyle {\begin{aligned}R(f)&=\int _{\mathbb {R} ^{n}}u(\tau )\underbrace {\left(\int _{\mathbb {R} ^{n}}v(x-\tau )\ e^{-i2\pi f\cdot x}\,dx\right)} _{V(f)\ e^{-i2\pi f\cdot \tau }}\,d\tau \\&=\underbrace {\left(\int _{\mathbb {R} ^{n}}u(\tau )\ e^{-i2\pi f\cdot \tau }\,d\tau \right)} _{U(f)}\ V(f).\end{aligned}}}" loading="lazy"></dd></dl>

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

This theorem also holds for the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a>, the <a href="Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a> and, when suitably modified, for the <a href="Mellin_transform" title="Mellin transform">Mellin transform</a> and <a href="Hartley_transform" title="Hartley transform">Hartley transform</a> (see <a href="Mellin_inversion_theorem" title="Mellin inversion theorem">Mellin inversion theorem</a>). It can be extended to the Fourier transform of <a href="Abstract_harmonic_analysis" class="mw-redirect" title="Abstract harmonic analysis">abstract harmonic analysis</a> defined over <a href="Locally_compact_abelian_group" title="Locally compact abelian group">locally compact abelian groups</a>.

<div class="mw-heading mw-heading3"><h3 id="Periodic_convolution_(Fourier_series_coefficients)">Periodic convolution (Fourier series coefficients)</h3></div>

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">-periodic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{_{P}}}">

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

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</math><img src="./a4ed2fd4bdcb8f1768cb93d1ecdd09a257cafca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.728ex; height:2.343ex;" alt="{\displaystyle u_{_{P}}}" loading="lazy">   and   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{_{P}},}">

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

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</math><img src="./7f0a4bc18c30310f0065d2736aa9b740e31cd3b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.173ex; height:2.343ex;" alt="{\displaystyle v_{_{P}},}" loading="lazy"> which can be expressed as <a href="Periodic_summation" title="Periodic summation">periodic summations</a>:

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<annotation encoding="application/x-tex">{\displaystyle u_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }u(x-mP)}</annotation>

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</math><img src="./f3997193040504fb7d0cd973cbe1965e26395dfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.671ex; height:6.843ex;" alt="{\displaystyle u_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }u(x-mP)}" loading="lazy">   and   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }v(x-mP).}">

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<annotation encoding="application/x-tex">{\displaystyle v_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }v(x-mP).}</annotation>

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</math><img src="./ddc35a9959b087ea6ea897e6ef15b831efe7966f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.914ex; height:6.843ex;" alt="{\displaystyle v_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }v(x-mP).}" loading="lazy"></dd></dl>

In practice the non-zero portion of components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> are often limited to duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P,}">

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

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</math><img src="./bd35af9d5901e795c83d9f519ac73264e74fa595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.392ex; height:2.509ex;" alt="{\displaystyle P,}" loading="lazy"> but nothing in the theorem requires that.

The <a href="Fourier_series" title="Fourier series">Fourier series</a> coefficients are:

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<mtext>integration over any interval of length </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U[k]&\triangleq {\mathcal {F}}\{u_{_{P}}\}[k]={\frac {1}{P}}\int _{P}u_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\\V[k]&\triangleq {\mathcal {F}}\{v_{_{P}}\}[k]={\frac {1}{P}}\int _{P}v_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}</annotation>

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</math><img src="./51b732f1fe5d5074f423f1b96c4eea07631284ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:88.253ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}U[k]&\triangleq {\mathcal {F}}\{u_{_{P}}\}[k]={\frac {1}{P}}\int _{P}u_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\\V[k]&\triangleq {\mathcal {F}}\{v_{_{P}}\}[k]={\frac {1}{P}}\int _{P}v_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> denotes the Fourier series integral.

<ul><li>The product: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{_{P}}(x)\cdot v_{_{P}}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle u_{_{P}}(x)\cdot v_{_{P}}(x)}</annotation>

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</math><img src="./8e0219f88c3a89c48e07c432e59dd45bf63e5331.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.211ex; height:3.009ex;" alt="{\displaystyle u_{_{P}}(x)\cdot v_{_{P}}(x)}" loading="lazy"> is also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">-periodic, and its Fourier series coefficients are given by the <a href="Convolution#Discrete_convolution" title="Convolution">discrete convolution</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{u_{_{P}}\cdot v_{_{P}}\}[k]=\{U*V\}[k].}</annotation>

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<ul><li>The convolution:</li></ul>

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<mtext>integration over any interval of length </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{u_{_{P}}*v\}(x)\ &\triangleq \int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \\&\equiv \int _{P}u_{_{P}}(x-\tau )\cdot v_{_{P}}(\tau )\ d\tau ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\end{aligned}}}</annotation>

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</math><img src="./128f2029c16f9a5db5689ae5a6c29adb6f2d90f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:74.092ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\{u_{_{P}}*v\}(x)\ &\triangleq \int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \\&\equiv \int _{P}u_{_{P}}(x-\tau )\cdot v_{_{P}}(\tau )\ d\tau ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\end{aligned}}}" loading="lazy"></dd></dl>

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">-periodic, and is called a <a href="Periodic_convolution" class="mw-redirect" title="Periodic convolution">periodic convolution</a>.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Derivation of periodic convolution</div>

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<mtext> is an arbitrary parameter</mtext>

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<mtext> by periodicity</mtext>

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<mtext>substituting </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\,d\tau &=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}+kP}^{x_{o}+(k+1)P}u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \right]\quad x_{0}{\text{ is an arbitrary parameter}}\\&=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}}^{x_{o}+P}\underbrace {u_{_{P}}(x-\tau -kP)} _{u_{_{P}}(x-\tau ),{\text{ by periodicity}}}\cdot v(\tau +kP)\ d\tau \right]\quad {\text{substituting }}\tau \rightarrow \tau +kP\\&=\int _{x_{o}}^{x_{o}+P}u_{_{P}}(x-\tau )\cdot \underbrace {\left[\sum _{k=-\infty }^{\infty }v(\tau +kP)\right]} _{\triangleq \ v_{_{P}}(\tau )}\ d\tau \end{aligned}}}</annotation>

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</math><img src="./f1575bd74f86366816ffc18d0fabd05dd5cd1269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.859ex; margin-bottom: -0.313ex; width:102.623ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\,d\tau &=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}+kP}^{x_{o}+(k+1)P}u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \right]\quad x_{0}{\text{ is an arbitrary parameter}}\\&=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}}^{x_{o}+P}\underbrace {u_{_{P}}(x-\tau -kP)} _{u_{_{P}}(x-\tau ),{\text{ by periodicity}}}\cdot v(\tau +kP)\ d\tau \right]\quad {\text{substituting }}\tau \rightarrow \tau +kP\\&=\int _{x_{o}}^{x_{o}+P}u_{_{P}}(x-\tau )\cdot \underbrace {\left[\sum _{k=-\infty }^{\infty }v(\tau +kP)\right]} _{\triangleq \ v_{_{P}}(\tau )}\ d\tau \end{aligned}}}" loading="lazy"></dd></dl>

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

The corresponding convolution theorem is:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{u_{_{P}}*v\}[k]=\ P\cdot U[k]\ V[k].}</annotation>

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</math><img src="./9d4ee3375f365aa800cc34e56a6bf89e0a90ed3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.718ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\{u_{_{P}}*v\}[k]=\ P\cdot U[k]\ V[k].}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.2</td></tr></tbody></table>

</div>

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Derivation of Eq.2</div>

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<mtext>due to periodicity</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\{u_{_{P}}*v\}[k]&\triangleq {\frac {1}{P}}\int _{P}\left(\int _{P}u_{_{P}}(\tau )\cdot v_{_{P}}(x-\tau )\ d\tau \right)e^{-i2\pi kx/P}\,dx\\&=\int _{P}u_{_{P}}(\tau )\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi kx/P}dx\right)\,d\tau \\&=\int _{P}u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}\underbrace {\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi k(x-\tau )/P}dx\right)} _{V[k],\quad {\text{due to periodicity}}}\,d\tau \\&=\underbrace {\left(\int _{P}\ u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}d\tau \right)} _{P\cdot U[k]}\ V[k].\end{aligned}}}</annotation>

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</math><img src="./252a143162a38ecc1ee2c1780ed231edf64ed597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.838ex; width:72.172ex; height:32.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}\{u_{_{P}}*v\}[k]&\triangleq {\frac {1}{P}}\int _{P}\left(\int _{P}u_{_{P}}(\tau )\cdot v_{_{P}}(x-\tau )\ d\tau \right)e^{-i2\pi kx/P}\,dx\\&=\int _{P}u_{_{P}}(\tau )\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi kx/P}dx\right)\,d\tau \\&=\int _{P}u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}\underbrace {\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi k(x-\tau )/P}dx\right)} _{V[k],\quad {\text{due to periodicity}}}\,d\tau \\&=\underbrace {\left(\int _{P}\ u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}d\tau \right)} _{P\cdot U[k]}\ V[k].\end{aligned}}}" loading="lazy"></dd></dl>

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<div class="mw-heading mw-heading2"><h2 id="Functions_of_a_discrete_variable_(sequences)">Functions of a discrete variable (sequences)</h2></div>

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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

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</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> denotes the <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT) operator. Consider two sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u[n]}">

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<annotation encoding="application/x-tex">{\displaystyle u[n]}</annotation>

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</math><img src="./e6f1362207606428a09d907db25527859eab6ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle u[n]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v[n]}">

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<annotation encoding="application/x-tex">{\displaystyle v[n]}</annotation>

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</math><img src="./7cbc20699310c20ae48d376fc4f6365c67966405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.816ex; height:2.843ex;" alt="{\displaystyle v[n]}" loading="lazy"> with transforms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\sum _{n=-\infty }^{\infty }u[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} ,\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\sum _{n=-\infty }^{\infty }v[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} .\end{aligned}}}</annotation>

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</math><img src="./244dd8c93040378c9c8849ab6483c6582445d1b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:49.025ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\sum _{n=-\infty }^{\infty }u[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} ,\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\sum _{n=-\infty }^{\infty }v[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} .\end{aligned}}}" loading="lazy"></dd></dl>

The <a href="Convolution#Discrete_convolution" title="Convolution">§ Discrete convolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is defined by:

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<annotation encoding="application/x-tex">{\displaystyle r[n]\triangleq (u*v)[n]=\sum _{m=-\infty }^{\infty }u[m]\cdot v[n-m]=\sum _{m=-\infty }^{\infty }u[n-m]\cdot v[m].}</annotation>

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</math><img src="./22aac1f7102502f6b39153be6aacd9bce5fc7cdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.969ex; height:6.843ex;" alt="{\displaystyle r[n]\triangleq (u*v)[n]=\sum _{m=-\infty }^{\infty }u[m]\cdot v[n-m]=\sum _{m=-\infty }^{\infty }u[n-m]\cdot v[m].}" loading="lazy"></dd></dl>

The <a href="Convolution#Discrete_convolution" title="Convolution">convolution theorem</a> for discrete sequences is:<a href="#cite_note-Proakis-3">[3]</a><a href="#cite_note-Oppenheim-4">[4]</a>: p.60 (2.169)

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<annotation encoding="application/x-tex">{\displaystyle R(f)={\mathcal {F}}\{u*v\}(f)=\ U(f)V(f).}</annotation>

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</math><img src="./aa963b44b0fe9992cca85e5c37c52f5618bb9881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.013ex; height:2.843ex;" alt="{\displaystyle R(f)={\mathcal {F}}\{u*v\}(f)=\ U(f)V(f).}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.3</td></tr></tbody></table>

</div>

<div class="mw-heading mw-heading3"><h3 id="Periodic_convolution">Periodic convolution</h3></div>

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

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</math><img src="./30d83bfe71c38f5a9d5be65332943723fe08c0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle U(f)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(f),}">

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

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</math><img src="./a212575864d2f5156a3823983cdad5f0aa5ea202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.522ex; height:2.843ex;" alt="{\displaystyle V(f),}" loading="lazy"> as defined above, are periodic, with a period of 1. Consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">-periodic sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{_{N}}}">

<msub>

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

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</math><img src="./aa1eaf648a52fd0a6233e13ac35a40c477ceb77f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.911ex; height:2.343ex;" alt="{\displaystyle u_{_{N}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{_{N}}}">

<msub>

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

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</math><img src="./bdd93868a8d30d480b1ba57ace2f01673cd9a13f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.709ex; height:2.343ex;" alt="{\displaystyle v_{_{N}}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle u_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }u[n-mN]}</annotation>

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</math><img src="./ed44f4cf6f7e56baa871eef261cd01f53855dd07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.271ex; height:6.843ex;" alt="{\displaystyle u_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }u[n-mN]}" loading="lazy">   and   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }v[n-mN],\quad n\in \mathbb {Z} .}">

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<annotation encoding="application/x-tex">{\displaystyle v_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }v[n-mN],\quad n\in \mathbb {Z} .}</annotation>

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</math><img src="./62b3cb372f16be04589aa1f446455da9d8bcd8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.656ex; height:6.843ex;" alt="{\displaystyle v_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }v[n-mN],\quad n\in \mathbb {Z} .}" loading="lazy"></dd></dl>

These functions occur as the result of sampling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> at intervals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/N}">

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

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</math><img src="./aa5c2544725c51dfe75eea07ee1f487feb8664c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.389ex; height:2.843ex;" alt="{\displaystyle 1/N}" loading="lazy"> and performing an inverse <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT) on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> samples (see <a href="Discrete-time_Fourier_transform#Sampling_the_DTFT" title="Discrete-time Fourier transform">§ Sampling the DTFT</a>). The discrete convolution:

<msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \{u_{_{N}}*v\}[n]\ \triangleq \sum _{m=-\infty }^{\infty }u_{_{N}}[m]\cdot v[n-m]\equiv \sum _{m=0}^{N-1}u_{_{N}}[m]\cdot v_{_{N}}[n-m]}</annotation>

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</math><img src="./907f89efecfca8b8c1519230c0efe93f721d0ad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.824ex; height:7.343ex;" alt="{\displaystyle \{u_{_{N}}*v\}[n]\ \triangleq \sum _{m=-\infty }^{\infty }u_{_{N}}[m]\cdot v[n-m]\equiv \sum _{m=0}^{N-1}u_{_{N}}[m]\cdot v_{_{N}}[n-m]}" loading="lazy"></dd></dl>

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

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">-periodic, and is called a <a href="Periodic_convolution" class="mw-redirect" title="Periodic convolution">periodic convolution</a>. Redefining the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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

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</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> operator as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">-length DFT, the corresponding theorem is:<a href="#cite_note-Rabiner-5">[5]</a><a href="#cite_note-Oppenheim-4">[4]</a>: p. 548

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<mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{u_{_{N}}*v\}[k]=\ \underbrace {{\mathcal {F}}\{u_{_{N}}\}[k]} _{U(k/N)}\cdot \underbrace {{\mathcal {F}}\{v_{_{N}}\}[k]} _{V(k/N)},\quad k\in \mathbb {Z} .}</annotation>

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</math><img src="./cbf1eab2ab63c1b6db4fc8f33c8a4a30b1391f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:47.086ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}\{u_{_{N}}*v\}[k]=\ \underbrace {{\mathcal {F}}\{u_{_{N}}\}[k]} _{U(k/N)}\cdot \underbrace {{\mathcal {F}}\{v_{_{N}}\}[k]} _{V(k/N)},\quad k\in \mathbb {Z} .}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.4a</td></tr></tbody></table>

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And therefore:

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<annotation encoding="application/x-tex">{\displaystyle \{u_{_{N}}*v\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{u_{_{N}}\}\cdot {\mathcal {F}}\{v_{_{N}}\}\}.}</annotation>

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</math><img src="./cd918e288950dc7554d3cb3e478d441022b16afc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.035ex; height:3.343ex;" alt="{\displaystyle \{u_{_{N}}*v\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{u_{_{N}}\}\cdot {\mathcal {F}}\{v_{_{N}}\}\}.}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.4b</td></tr></tbody></table>

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Under the right conditions, it is possible for this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">-length sequence to contain a distortion-free segment of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u*v}">

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

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</math><img src="./181ac3b003e9033e2012d7cde681b75d62f150df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.652ex; height:1.676ex;" alt="{\displaystyle u*v}" loading="lazy"> convolution. But when the non-zero portion of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(n)}">

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

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</math><img src="./9c4ffafb6eafd4c62d8eb2f2477e169d24adb1c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle u(n)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(n)}">

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</math><img src="./9c1a80423b464b7437cb4432bd525cf4336e7ec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.332ex; height:2.843ex;" alt="{\displaystyle v(n)}" loading="lazy"> sequence is equal or longer than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,}">

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

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</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"> some distortion is inevitable.  Such is the case when the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(k/N)}">

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

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</math><img src="./2ee0c1eded85fd23d08902e9722ac398569a2a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.034ex; height:2.843ex;" alt="{\displaystyle V(k/N)}" loading="lazy"> sequence is obtained by directly sampling the DTFT of the infinitely long <a href="Hilbert_transform#Discrete_Hilbert_transform" title="Hilbert transform">§ Discrete Hilbert transform</a> <a href="Impulse_response" title="Impulse response">impulse response</a>.<a href="#cite_note-6">[A]</a>

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

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

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> sequences whose non-zero duration is less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,}">

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

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</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"> a final simplification is:

<div class="equation-box" style="margin: ;padding: 6px; border-width:px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"><a href="Circular_convolution" title="Circular convolution">Circular convolution</a>

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<annotation encoding="application/x-tex">{\displaystyle \{u_{_{N}}*v\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\}.}</annotation>

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</math><img src="./6ac7a801e261a01f897517049a2095ee7d745cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.873ex; height:3.343ex;" alt="{\displaystyle \{u_{_{N}}*v\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\}.}" loading="lazy">

</td> <td></td> <td class="nowrap">Eq.4c</td></tr></tbody></table>

</div>

This form is often used to efficiently implement numerical convolution by <a href="Computer" title="Computer">computer</a>. (see <a href="Convolution#Fast_convolution_algorithms" title="Convolution">§ Fast convolution algorithms</a> and <a href="Circular_convolution#Example" title="Circular convolution">§ Example</a>)

As a partial reciprocal, it has been shown <a href="#cite_note-7">[6]</a>

that any linear transform that turns convolution into a product is the DFT (up to a permutation of coefficients).

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Derivations of Eq.4</div>

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A time-domain derivation proceeds as follows:

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<mtext>due to periodicity</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}*v\}[k]&\triangleq \sum _{n=0}^{N-1}\left(\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot v_{_{N}}[n-m]\right)e^{-i2\pi kn/N}\\&=\sum _{m=0}^{N-1}u_{_{N}}[m]\left(\sum _{n=0}^{N-1}v_{_{N}}[n-m]\cdot e^{-i2\pi kn/N}\right)\\&=\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot e^{-i2\pi km/N}\underbrace {\left(\sum _{n=0}^{N-1}v_{_{N}}[n-m]\cdot e^{-i2\pi k(n-m)/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\quad \scriptstyle {\text{due to periodicity}}}\\&=\underbrace {\left(\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot e^{-i2\pi km/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]}\left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right).\end{aligned}}}</annotation>

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</math><img src="./61dd82c63ec339357bb750973d87a819d0490c8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:72.44ex; height:39.843ex;" alt="{\displaystyle {\begin{aligned}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}*v\}[k]&\triangleq \sum _{n=0}^{N-1}\left(\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot v_{_{N}}[n-m]\right)e^{-i2\pi kn/N}\\&=\sum _{m=0}^{N-1}u_{_{N}}[m]\left(\sum _{n=0}^{N-1}v_{_{N}}[n-m]\cdot e^{-i2\pi kn/N}\right)\\&=\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot e^{-i2\pi km/N}\underbrace {\left(\sum _{n=0}^{N-1}v_{_{N}}[n-m]\cdot e^{-i2\pi k(n-m)/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\quad \scriptstyle {\text{due to periodicity}}}\\&=\underbrace {\left(\sum _{m=0}^{N-1}u_{_{N}}[m]\cdot e^{-i2\pi km/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]}\left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right).\end{aligned}}}" loading="lazy"></dd></dl>

A frequency-domain derivation follows from <a href="DTFT" class="mw-redirect" title="DTFT">§ Periodic data</a>, which indicates that the DTFTs can be written as:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{u_{_{N}}*v\}(f)={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}*v\}[k]\right)\cdot \delta \left(f-k/N\right).\quad \scriptstyle {\mathsf {(Eq.5a)}}}</annotation>

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</math><img src="./9dc6053c6711e16180fcaf82615d9ba6d205646c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.807ex; height:7.009ex;" alt="{\displaystyle {\mathcal {F}}\{u_{_{N}}*v\}(f)={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}*v\}[k]\right)\cdot \delta \left(f-k/N\right).\quad \scriptstyle {\mathsf {(Eq.5a)}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{u_{_{N}}\}(f)={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \delta \left(f-k/N\right).}</annotation>

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</math><img src="./42c03475328868e8c4027366eb1a6740a38895e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.659ex; height:7.009ex;" alt="{\displaystyle {\mathcal {F}}\{u_{_{N}}\}(f)={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \delta \left(f-k/N\right).}" loading="lazy"></dd></dl>

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

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

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</math><img src="./a716ecee96f695ac9d7451ab772abd188102cccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.875ex; height:2.843ex;" alt="{\displaystyle V(f)}" loading="lazy"> is thereby reduced to a discrete-frequency function:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\{u_{_{N}}*v\}(f)&=G_{_{N}}(f)V(f)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot V(f)\cdot \delta \left(f-k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot V(k/N)\cdot \delta \left(f-k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right)\cdot \delta \left(f-k/N\right),\quad \scriptstyle {\mathsf {(Eq.5b)}}\end{aligned}}}</annotation>

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</math><img src="./fbfa658cb06c8f39bd4f6c14c0cf49ebb7f614af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.608ex; margin-bottom: -0.23ex; width:75.838ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}\{u_{_{N}}*v\}(f)&=G_{_{N}}(f)V(f)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot V(f)\cdot \delta \left(f-k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot V(k/N)\cdot \delta \left(f-k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right)\cdot \delta \left(f-k/N\right),\quad \scriptstyle {\mathsf {(Eq.5b)}}\end{aligned}}}" loading="lazy"></dd></dl>

where the equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(k/N)}">

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

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</math><img src="./2ee0c1eded85fd23d08902e9722ac398569a2a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.034ex; height:2.843ex;" alt="{\displaystyle V(k/N)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right)}</annotation>

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</math><img src="./867247cbc04ad960f719050702f1b613652470f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.863ex; height:3.009ex;" alt="{\displaystyle \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right)}" loading="lazy"> follows from <a href="DTFT" class="mw-redirect" title="DTFT">§ Sampling the DTFT</a>. Therefore, the equivalence of (5a) and (5b) requires:

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\rm {DFT}}\displaystyle {\{u_{_{N}}*v\}[k]}=\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right).}</annotation>

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</math><img src="./8f7b413f938f761ac2c932b93215c4e166e02071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.319ex; height:3.009ex;" alt="{\displaystyle \scriptstyle {\rm {DFT}}\displaystyle {\{u_{_{N}}*v\}[k]}=\left(\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\right).}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(u_{_{N}}*v)[n]&=\int _{0}^{1}\left({\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\cdot \delta \left(f-k/N\right)\right)\cdot e^{i2\pi fn}df\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\cdot \underbrace {\left(\int _{0}^{1}\delta \left(f-k/N\right)\cdot e^{i2\pi fn}df\right)} _{{\text{0, for}}\ k\ \notin \ [0,\ N)}\\&={\frac {1}{N}}\sum _{k=0}^{N-1}{\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]{\bigg )}\cdot e^{i2\pi {\frac {n}{N}}k}\\&=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle {\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}{\bigg )}.\end{aligned}}}</annotation>

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</math><img src="./d1c651d3df13f237865ee17f554975dbd3136b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.094ex; margin-bottom: -0.244ex; width:77.563ex; height:31.843ex;" alt="{\displaystyle {\begin{aligned}(u_{_{N}}*v)[n]&=\int _{0}^{1}\left({\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\cdot \delta \left(f-k/N\right)\right)\cdot e^{i2\pi fn}df\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]\cdot \underbrace {\left(\int _{0}^{1}\delta \left(f-k/N\right)\cdot e^{i2\pi fn}df\right)} _{{\text{0, for}}\ k\ \notin \ [0,\ N)}\\&={\frac {1}{N}}\sum _{k=0}^{N-1}{\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}[k]{\bigg )}\cdot e^{i2\pi {\frac {n}{N}}k}\\&=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle {\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{u_{_{N}}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{v_{_{N}}\}{\bigg )}.\end{aligned}}}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading2"><h2 id="Convolution_theorem_for_inverse_Fourier_transform">Convolution theorem for inverse Fourier transform</h2></div>

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

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

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

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</math><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy">" represents the <a href="Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a>, and "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

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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">" represents a convolution between the two matrices.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{u\cdot v\}={\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\end{aligned}}}</annotation>

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</math><img src="./37ddecd7286fc974d3451bcde4d7ff808ec5a289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.393ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{u\cdot v\}={\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\end{aligned}}}" loading="lazy"></dd></dl>

so that

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&u*v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\right\}\\&u\cdot v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\right\}\end{aligned}}}</annotation>

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</math><img src="./7b190c5722a0514e83731d0e42be4df3eed7b499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.189ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}&u*v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\right\}\\&u\cdot v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\right\}\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Convolution_theorem_for_tempered_distributions">Convolution theorem for tempered distributions</h2></div>

The convolution theorem extends to <a href="Distribution_(mathematics)#Convolution_versus_multiplication" title="Distribution (mathematics)">tempered distributions</a>.

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

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is an arbitrary tempered distribution:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{\alpha \cdot v\}={\mathcal {F}}\{\alpha \}*{\mathcal {F}}\{v\}.\end{aligned}}}</annotation>

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</math><img src="./c4daa86d0f58079ef77237bd1a2ca428456deca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.356ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{\alpha \cdot v\}={\mathcal {F}}\{\alpha \}*{\mathcal {F}}\{v\}.\end{aligned}}}" loading="lazy"></dd></dl>

But <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=F\{\alpha \}}">

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</math><img src="./48c2cfe3161623b17d9eb22eee8e844efd8f676a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.981ex; height:2.843ex;" alt="{\displaystyle u=F\{\alpha \}}" loading="lazy"> must be "rapidly decreasing" towards <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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

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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"> in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =F^{-1}\{u\}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =F^{-1}\{u\}}</annotation>

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</math><img src="./4da7c0cca19f9aa4a4eb70f69ffd26826bdde9ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.388ex; height:3.176ex;" alt="{\displaystyle \alpha =F^{-1}\{u\}}" loading="lazy"> is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.<a href="#cite_note-8">[7]</a><a href="#cite_note-9">[8]</a><a href="#cite_note-10">[9]</a>

In particular, every compactly supported tempered distribution, such as the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta</a>, is "rapidly decreasing". Equivalently, <a href="Bandlimiting" title="Bandlimiting">bandlimited functions</a>, such as the function that is constantly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> are smooth "slowly growing" ordinary functions. If, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\equiv \operatorname {\text{Ш}} }">

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</math><img src="./158dd07b75aaa22005378ccf2f4fe1cd58b79e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.643ex; height:3.343ex;" alt="{\displaystyle v\equiv \operatorname {\text{Ш}} }" loading="lazy"> is the <a href="Dirac_comb" title="Dirac comb">Dirac comb</a> both equations yield the <a href="Poisson_summation_formula" title="Poisson summation formula">Poisson summation formula</a> and if, furthermore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\equiv \delta }">

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</math><img src="./42a11ce0c1bbaa5cd4448c194c8c923a1440b5c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle u\equiv \delta }" loading="lazy"> is the Dirac delta then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \equiv 1}">

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</math><img src="./fb47720cbb746dd0bcdadf72628f0417dbcf199a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha \equiv 1}" loading="lazy"> is constantly one and these equations yield the <a href="Dirac_comb#Dirac-comb_identity" title="Dirac comb">Dirac comb identity</a>.

<ul><li><a href="Moment-generating_function" title="Moment-generating function">Moment-generating function</a> of a <a href="Random_variable" title="Random variable">random variable</a></li></ul>

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

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> An example is the <a href="MATLAB" title="MATLAB">MATLAB</a> function, <a rel="nofollow" class="external text" href="http://www.mathworks.com/help/toolbox/signal/ref/hilbert.html;jsessionid=67ed4e69e9729363548abed31054">hilbert(u,N)</a>.

</li>

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

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

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFMcGillemCooper1984" class="citation book cs1">McGillem, Clare D.; Cooper, George R. (1984). Continuous and Discrete Signal and System Analysis (2 ed.). Holt, Rinehart and Winston. p. 118 (3–102). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-03-061703-0</bdi>.</cite>

</li>

<cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConvolutionTheorem.html">"Convolution Theorem"</a>. From MathWorld--A Wolfram Web Resource. Retrieved 8 February 2021.</cite>

</li>

<cite id="CITEREFProakisManolakis1996" class="citation cs2">Proakis, John G.; Manolakis, Dimitri G. (1996), <a rel="nofollow" class="external text" href="https://archive.org/details/digitalsignalpro00proa">Digital Signal Processing: Principles, Algorithms and Applications</a> (3 ed.), New Jersey: Prentice-Hall International, p. 297, <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/1996dspp.book.....P">1996dspp.book.....P</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780133942897</bdi>, sAcfAQAAIAAJ</cite>

</li>

<cite id="CITEREFOppenheimSchaferBuck1999" class="citation book cs1"><a href="Alan_V._Oppenheim" title="Alan V. Oppenheim">Oppenheim, Alan V.</a>; <a href="Ronald_W._Schafer" title="Ronald W. Schafer">Schafer, Ronald W.</a>; Buck, John R. (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan">Discrete-time signal processing</a> (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-754920-2</bdi>.</cite>

</li>

<cite id="CITEREFRabinerGold1975" class="citation book cs1"><a href="Lawrence_Rabiner" title="Lawrence Rabiner">Rabiner, Lawrence R.</a>; Gold, Bernard (1975). <a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00rabi">Theory and application of digital signal processing</a>. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 59 (2.163). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0139141010</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="Theorem_1.11" class="citation book cs1">Amiot, Emmanuel (2016). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-319-45581-5">Music through Fourier Space</a>. Computational Music Science. Zürich: Springer. p. 8. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-45581-5">10.1007/978-3-319-45581-5</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-45581-5</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6224021">6224021</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFHorváth1966" class="citation book cs1"><a href="John_Horvath_(mathematician)" title="John Horvath (mathematician)">Horváth, John</a> (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBarros-Neto1973" class="citation book cs1">Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFPetersen1983" class="citation book cs1">Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.</cite>

</li>

</ol></div>

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

<ul><li><cite id="CITEREFKatznelson1976" class="citation cs2">Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-63331-4</bdi></cite></li>

<li><cite id="CITEREFLiBabu2019" class="citation cs2">Li, Bing; Babu, G. Jogesh (2019), "Convolution Theorem and Asymptotic Efficiency", A Graduate Course on Statistical Inference, New York: Springer, pp. 295–327, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4939-9759-6</bdi></cite></li>

<li><cite id="CITEREFCrutchfield2010" class="citation cs2">Crutchfield, Steve (October 9, 2010), <a rel="nofollow" class="external text" href="http://www.jhu.edu/signals/convolve/index.html">"The Joy of Convolution"</a>, Johns Hopkins University, retrieved November 19, 2010</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="Additional_resources">Additional resources</h2></div>

For a visual representation of the use of the convolution theorem in <a href="Signal_processing" title="Signal processing">signal processing</a>, see:

<ul><li><a href="Johns_Hopkins_University" title="Johns Hopkins University">Johns Hopkins University</a>'s <a href="Java_(software_platform)" title="Java (software platform)">Java</a>-aided simulation: <a rel="nofollow" class="external free" href="http://www.jhu.edu/signals/convolve/index.html">http://www.jhu.edu/signals/convolve/index.html</a></li></ul></div><div><div class="zim-footer">

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