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== {{destruir|Irrelevancia. LA VENGANZA DE ALVARO}}[[[[Image:Bill.jpg]]|right|150px|Taza]]
== Tabla de Propiedades de la transformada de Fourier ==
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<math>\begin{align}
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& \mathbb{F}[f(t)]=F(\omega )=\int\limits_{-\infty }^{\infty }{f(t).e^{-j\omega t}\partial t} \\
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& \mathbb{F}^{-1}[F(\omega )]=f(t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega ).e^{+j\omega t}\partial \omega } \\
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\end{align}</math>
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[[Category:Wikilibros|Café]]
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{| class="wikitable"
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|Linearidad||<math>\mathbb{F}\left[ \alpha f(t)+\beta g(t) \right]=\alpha F(\omega )+\beta G(\omega )</math>
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|Dualidad||<math>\mathbb{F}[f(t)]=F(\omega )\to \mathbb{F}[F(t)]=2\pi f(-\omega )</math>
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|Cambio de escala||<math>\mathbb{F}[f(at)]=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right)</math>
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|Transformada de la conjugada||<math>\mathbb{F}[f^{*}(t)]=F^{*}(-\omega )</math>
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|Translacion en el tiempo||<math>\mathbb{F}[f(t-t_{0})]=e^{-j\omega t_{0}}F(\omega )</math>
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|Translacion en frecuencia||<math>\mathbb{F}[e^{+j\omega _{0}t}f(t)]=F(\omega -\omega _{0})</math>
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|Derivacion en el tiempo||<math>\mathbb{F}\left[ \frac{\partial ^{n}f(t)}{\partial t^{n}} \right]=\left( j\omega \right)^{n}F(\omega )</math>
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|Derivacion en la frecuencia||<math>\mathbb{F}\left[ \left( -jt \right)^{n}f(t) \right]=\frac{\partial ^{n}F(\omega )}{\partial \omega ^{n}}</math>
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|Transformada de la integral||<math>\mathbb{F}\left[ \int\limits_{-\infty }^{t}{f(\tau )\partial \tau } \right]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega )</math>
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|Transformada de la Convolucion||
<math>\begin{align}
& \mathbb{F}\left[ f(t)*g(t) \right]= \\
& \mathbb{F}\left[ \int\limits_{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau } \right]=F(\omega )G(\omega ) \\
\end{align}</math>
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|Teorema de Parseval||<math>\int\limits_{-\infty }^{\infty }{\left| f(t) \right|^{2}\partial t=\frac{1}{2\pi }}\int\limits_{-\infty }^{\infty }{\left| F(\omega ) \right|^{2}\partial \omega }</math>
|}
Demostraciones:
{{Copyvio}}
== Dualidad ==
<math>\begin{align}
& f(t)=\mathbb{F}^{-1}\left[ F(\omega ) \right]=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )e^{j\omega t}\partial \omega } \\
& f(-t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )e^{-j\omega t}\partial \omega }\xrightarrow[t\leftrightarrow \omega ]{}f(-\omega )=\frac{1}{2\pi }\underbrace{\int\limits_{-\infty }^{\infty }{F(t)e^{-j\omega t}\partial t}}_{\mathbb{F}[F(t)]}\to 2\pi f(-\omega )=\mathbb{F}[F(t)] \\
\end{align}</math>
== Cambio de escala ==
<math>\begin{align}
& \mathbb{F}[f(at)]=\int\limits_{-\infty }^{\infty }{f(at)e^{-j\omega t}\partial t} \\
& \left\{ \begin{align}
& u=at\leftrightarrow t=\frac{u}{a} \\
& \partial u=a\partial t\to \partial t=\frac{\partial u}{a} \\
\end{align} \right\}\to \int\limits_{u=-\infty }^{u=\infty }{f(u)e^{-j\omega \frac{u}{a}}\partial {u}/{a}\;\to \left\{ \begin{align}
& a>0\to \text{ Se queda como esta} \\
& a<0\to \int\limits_{\infty }^{-\infty }{f(u)e^{-j\omega \frac{u}{a}}{\partial u}/{-a}\;} \\
\end{align} \right.} \\
& \mathbb{F}[f(at)]=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right) \\
\end{align}</math>
== Transformada de la conjugada ==
<math>\mathbb{F}\left[ \overline{f(t)} \right]=\mathbb{F}[f^{*}(t)]=\int\limits_{-\infty }^{\infty }{f^{*}}(t)\cdot e^{-j\omega t}\partial t=\left[ \underbrace{\int\limits_{-\infty }^{\infty }{f}(t)\cdot e^{+j\omega t}\partial t}_{F(-\omega )} \right]^{*}=F^{*}(-\omega )</math>
== Translación en el tiempo ==
<math>\begin{align}
& \mathbb{F}[f(t-t_{0})]=e^{-j\omega t_{0}}F(\omega ) \\
& \mathbb{F}[f(t-t_{0})]=\int\limits_{-\infty }^{\infty }{f(t-t_{0})e^{-j\omega t}\partial t}\to \left\{ \begin{align}
& u=t-t_{0}\leftrightarrow t=u+t_{0} \\
& \partial u=\partial t \\
\end{align} \right\} \\
& \mathbb{F}[f(t-t_{0})]=\int\limits_{-\infty }^{\infty }{f(u)e^{-j\omega \left( u+t_{0} \right)}\partial u=}e^{-j\omega t_{0}}\overbrace{\int\limits_{-\infty }^{\infty }{f(u)e^{-j\omega u}\partial u}}^{F(\omega )} \\
\end{align}</math>
== Translacion en frecuencia ==
Analogamente:
<math>\begin{align}
& \mathbb{F}[\underbrace{e^{+j\omega _{0}t}f(t)}_{g(t)}]=F(\omega -\omega _{0}) \\
& g(t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega -\omega _{0})e^{j\omega t}\partial \omega \to \left\{ \begin{align}
& u=\omega -\omega _{0}\leftrightarrow \omega =u+\omega _{0} \\
& \partial u=\partial \omega \\
\end{align} \right\}}\to \frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(u)e^{j\left( u+\omega _{0} \right)t}\partial u} \\
& g(t)=e^{j\omega _{0}t}\overbrace{\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(u)e^{jut}\partial u}}^{f(t)} \\
\end{align}</math>
== Derivacion en el tiempo ==
<math>\begin{align}
& \mathbb{F}\left[ \frac{\partial ^{n}f(t)}{\partial t^{n}} \right]=\left( j\omega \right)^{n}F(\omega ) \\
& \mathbb{F}[f^{'}(t)]=\int\limits_{-\infty }^{\infty }{f^{'}(t)e^{-j\omega t}\partial t\to \int{u\partial v=u\cdot v-\int{v\partial u\to }}\left\{ \begin{align}
& u=e^{-j\omega t}\text{ }\partial u=-j\omega \cdot e^{-j\omega t}\partial t \\
& \partial v=f^{'}(t)\partial t\text{ }v=f(t) \\
\end{align} \right\}} \\
& \mathbb{F}[f^{'}(t)]=\underbrace{e^{-j\omega t}\left. f(t) \right|_{-\infty }^{\infty }}_{0}+\int{j\omega \cdot f(t)e^{-j\omega t}}\partial t \\
& \underset{t\to \pm \infty }{\mathop{\lim }}\,\text{ }f(t)=0\text{ si }f(t)\text{ continua y abs}\text{. integrable} \\
\end{align}</math>
== Derivacion en la frecuencia ==
Analogamente:
<math>\begin{align}
& \mathbb{F}\left[ \underbrace{\left( -jt \right)^{n}f(t)}_{g(t)} \right]=\frac{\partial ^{n}F(\omega )}{\partial \omega ^{n}} \\
& g(t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F^{'}(\omega )e^{+j\omega t}\partial \omega }\to \int{u\partial v=u\cdot v-\int{v\partial u\to }}\left\{ \begin{align}
& u=e^{+j\omega t}\text{ }\partial u=jt\cdot e^{+j\omega t}\partial \omega \\
& \partial v=F^{'}(\omega )\partial \omega \text{ }v=F(\omega ) \\
\end{align} \right\}\to \\
& g(t)=\frac{1}{2\pi }\underbrace{e^{+j\omega t}\left. F(\omega ) \right|_{-\infty }^{\infty }}_{0}-\frac{1}{2\pi }\int{jt\cdot F(\omega )e^{+j\omega t}}\partial \omega \\
\end{align}</math>
== Convolucion ==
Debido a que va a ser necesario utilizarlo, definamos primeramente la convolucion de dos señales:
<math>f(t)*g(t)=\int\limits_{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau }=\int\limits_{-\infty }^{\infty }{g(\tau )f(t-\tau )\partial \tau }</math>
Demostracion de conmutativilidad:
<math>\begin{align}
& \int\limits_{\tau =-\infty }^{\tau =\infty }{f(\tau )g(t-\tau )\partial \tau }\to \left\{ \begin{align}
& \tau _{0}=t-\tau \text{ }\leftrightarrow \tau =t-\tau _{0} \\
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