Propiedades de la transformada de Fourier

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Tabla de Propiedades de la transformada de Fourier[editar]

\begin{align}
  & \mathbb{F}[f(t)]=F(\omega )=\int\limits_{-\infty }^{\infty }{f(t)\cdot e^{-j\omega t}\partial t} \\ 
 & \mathbb{F}^{-1}[F(\omega )]=f(t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )\cdot e^{+j\omega t}\partial \omega } \\ 
\end{align}

Linealidad \mathbb{F}\left[ \alpha f(t)+\beta g(t) \right]=\alpha F(\omega )+\beta G(\omega )
Dualidad \mathbb{F}[f(t)]=F(\omega )\to \mathbb{F}[F(t)]=2\pi f(-\omega )
Cambio de escala \mathbb{F}[f(at)]=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right)
Transformada de la conjugada \mathbb{F}[f^{*}(t)]=F^{*}(-\omega )
Translacion en el tiempo \mathbb{F}[f(t-t_{0})]=e^{-j\omega t_{0}}F(\omega )
Translacion en frecuencia \mathbb{F}[e^{+j\omega _{0}t}f(t)]=F(\omega -\omega _{0})
Derivacion en el tiempo \mathbb{F}\left[ \frac{\partial ^{n}f(t)}{\partial t^{n}} \right]=\left( j\omega  \right)^{n}F(\omega )
Derivacion en la frecuencia \mathbb{F}\left[ \left( -jt \right)^{n}f(t) \right]=\frac{\partial ^{n}F(\omega )}{\partial \omega ^{n}}
Transformada de la integral \mathbb{F}\left[ \int\limits_{-\infty }^{t}{f(\tau )\partial \tau } \right]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega )
Transformada de la Convolucion

\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}

Teorema de Parseval \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 }

Demostraciones:

Dualidad[editar]

\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}

Cambio de escala[editar]

\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}

Transformada de la conjugada[editar]

\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 )

Translación en el tiempo[editar]

\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}

Translacion en frecuencia[editar]

Analogamente:

\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}

Derivacion en el tiempo[editar]

\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}

Derivacion en la frecuencia[editar]

Analogamente:

\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}

Convolucion[editar]

Debido a que va a ser necesario utilizarlo, definamos primeramente la convolucion de dos señales:

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 }

Demostracion de conmutativilidad:

\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} \\ 
 & \partial \tau _{0}=-\partial \tau \text{ } \\ 
\end{align} \right\} \\ 
 & \int\limits_{\tau _{0}=\infty }^{\tau _{0}=-\infty }{f(t-\tau _{0})g(\tau _{0})\left( -\partial \tau _{0} \right)=}\int\limits_{\tau _{0}=-\infty }^{\tau _{0}=\infty }{f(t-\tau _{0})g(\tau _{0})\partial \tau _{0}} \\ 
\end{align}

Integracion en el tiempo[editar]

\begin{align}
  & \mathbb{F}\left[ \int\limits_{-\infty }^{t}{f(\tau )\partial \tau } \right]\to f(t)*u(t)=\int_{\tau =-\infty }^{\tau =\infty }{f(\tau )u(t-\tau )\partial \tau }\to u(t-\tau )=\left\{ \begin{align}
  & 1,t-\tau \ge 0 \\ 
 & 0,t-\tau <0 \\ 
\end{align} \right.\to  \\ 
 & t-\tau \ge 0\to t\ge \tau \Rightarrow \int_{\tau =-\infty }^{\tau =\infty }{f(\tau )\underbrace{u(t-\tau )}_{\begin{smallmatrix} 
 u(t-\tau )=1\to  \\ 
 t-\tau >0\to \tau <t 
\end{smallmatrix}}\partial \tau }=\int_{-\infty }^{t}{f(\tau )\partial \tau } \\ 
 & \mathbb{F}\left[ \int\limits_{-\infty }^{t}{f(\tau )\partial \tau } \right]=\mathbb{F}\left[ f(t)*u(t) \right]=F(\omega ).\left[ \frac{1}{j\omega }+\pi \delta (\omega ) \right]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega ) \\ 
\end{align}

Transformada de la convolucion[editar]

\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]=\int\limits_{-\infty }^{\infty }{\left[ \int\limits_{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau } \right]e^{-j\omega t}\partial t=} \\ 
 & \int\limits_{-\infty }^{\infty }{\int\limits_{-\infty }^{\infty }{f(\tau )g(t-\tau )e^{-j\omega t}\partial t\partial \tau }}=\int\limits_{-\infty }^{\infty }{f(\tau )\underbrace{\int\limits_{-\infty }^{\infty }{g(t-\tau )e^{-j\omega t}\partial t}}_{G(\omega )e^{-j\omega \tau }}}\partial \tau =\int\limits_{-\infty }^{\infty }{f(\tau )G(\omega )e^{-j\omega \tau }\partial \tau }= \\ 
 & G(\omega )\int\limits_{-\infty }^{\infty }{f(\tau )e^{-j\omega \tau }\partial \tau }=G(\omega )\cdot F(\omega ) \\ 
\end{align}

Teorema de Parseval[editar]

El teorema de parseval es una solucion particular de la propiedad:

\int\limits_{-\infty }^{\infty }{f(t)g^{*}(t)}\partial t=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )G^{*}(\omega )}\partial \omega

\begin{align}
  & \int\limits_{-\infty }^{\infty }{\left| f(t) \right|^{2}\partial t=\int\limits_{-\infty }^{\infty }{f(t)f^{*}(t)}\partial t\xrightarrow[f(t)=g(t)]{}}\int\limits_{-\infty }^{\infty }{f(t)g^{*}(t)\partial t}\to \left\{ f(t)=\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )e^{j\omega t}\partial \omega } \right\} \\ 
 & \int\limits_{-\infty }^{\infty }{\left[ \frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )e^{j\omega t}\partial \omega } \right]g^{*}(t)\partial t}\xrightarrow[\begin{smallmatrix} 
 \text{Cambiando} \\ 
 \text{orden  de} 
 \\ 
 \text{integracion} 
\end{smallmatrix}]{}\int\limits_{-\infty }^{\infty }{\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F(\omega )}e^{j\omega t}g^{*}(t)\partial t}\partial \omega = \\ 
 & \int\limits_{-\infty }^{\infty }{\frac{1}{2\pi }\left[ \int\limits_{-\infty }^{\infty }{g^{*}(t)e^{j\omega t}\partial t} \right]}F(\omega )\partial \omega =\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{\left[ \underbrace{\int\limits_{-\infty }^{\infty }{g(t)e^{-j\omega t}\partial t}}_{G(\omega )} \right]^{*}}F(\omega )\partial \omega = \\ 
 & \frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{G^{*}(\omega )}F(\omega )\partial \omega \xrightarrow[f(t)=g(t)]{}\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{F^{*}(\omega )}F(\omega )\partial \omega =\frac{1}{2\pi }\int\limits_{-\infty }^{\infty }{\left| F(\omega ) \right|^{2}}\partial \omega  \\ 
\end{align}

Consejo general[editar]

Finalmente, puede ser muy comun que tengamos que aplicar mas de una propiedad para una misma funcion, en ese caso, lo mejor es usar funciones auxiliares y cambios de variable.

\begin{align}
  & Ej: \\ 
 & \mathbb{F}[f(at-b)]? \\ 
 & g(t)=f(t-b)\to g(at)=f(at-b) \\ 
 & \mathbb{F}[g(at)]=\frac{1}{\left| a \right|}G\left( \frac{\omega }{a} \right) \\ 
 & \mathbb{F}[g(t)]=\mathbb{F}[f(t-b)]\to G(\omega )=F(\omega )\cdot e^{-j\omega b}\to G\left( \frac{\omega }{a} \right)=F\left( \frac{\omega }{a} \right)\cdot e^{-j\left( \frac{\omega }{a} \right)b} \\ 
 & \mathbb{F}[g(at)]=\mathbb{F}[f(at-b)]=\frac{1}{\left| a \right|}G\left( \frac{\omega }{a} \right)=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right)\cdot e^{-j\left( \frac{\omega }{a} \right)b} \\ 
 & \mathbb{F}[f(at-b)]=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right)\cdot e^{-j\left( \frac{\omega }{a} \right)b} \\ 
\end{align}

Tambien podemos aplicar las propiedades en otro orden que esta a continuacion:

\begin{align}
  & \mathbb{F}[f(at-b)]? \\ 
 & g(t)=f(at)\to g(t-b)=f\left( a\left( t-b \right) \right)=f(at-ab)\ne f(at-b) \\ 
 & g\left( t-{}^{b}\!\!\diagup\!\!{}_{a}\; \right)=f\left( a\left( t-{}^{b}\!\!\diagup\!\!{}_{a}\; \right) \right)=f\left( at-b \right) \\ 
 & \mathbb{F}\left[ g\left( t-{}^{b}\!\!\diagup\!\!{}_{a}\; \right) \right]=G(\omega )\cdot e^{-j\omega \frac{b}{a}}\Rightarrow g(t)=f(at)\to G(\omega )=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right) \\ 
 & \mathbb{F}[f(at-b)]=\mathbb{F}\left[ g\left( t-{}^{b}\!\!\diagup\!\!{}_{a}\; \right) \right]=G(\omega )\cdot e^{-j\omega \frac{b}{a}}=\frac{1}{\left| a \right|}F\left( \frac{\omega }{a} \right)e^{-j\omega \frac{b}{a}} \\ 
\end{align}