Propiedades de la transformada de Fourier

Definición de la transformada de Fourier

$\begin{array}{cc}& 𝔽[f(t)]=F(\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)\cdot e^{-i\omega t}\partial t\\ & {𝔽}^{-1}[F(\omega )]=f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\omega )\cdot e^{+i\omega t}\partial \omega \\ \end{array}$

Tabla resumen de propiedades

Propiedad	Definición
Linealidad	$𝔽[\alpha f(t)+\beta g(t)]=\alpha F(\omega )+\beta G(\omega )$
Dualidad	$𝔽[F(t)]=2\pi f(-\omega )$
Cambio de escala	$𝔽[f(at)]=\frac{1}{\left|a\right|}F\left(\frac{\omega }{a}\right)$
Inversión el tiempo	$𝔽[f^{}(-t)]=F^{}(-\omega )$
Traslación en el tiempo	$𝔽[f(t-t_0)]=F(\omega )e^{-j\omega t_0}$
Traslación en frecuencia	$𝔽[f(t)e^{j{\omega }_{0}t}]=F(\omega -{\omega }_0)$
Derivación en el tiempo	$𝔽\left[\frac{{\partial }^{n}f(t)}{\partial t^n}\right]={\left(j\omega \right)}^{n}F(\omega )$
Derivación en la frecuencia	$𝔽[{(-jt)}^{n}f(t)]=\frac{{\partial }^{n}F(\omega )}{\partial {\omega }^n}$
Transformada de la integral	$𝔽\left[\underset{-\mathrm{\infty }}{\overset{t}{\int }}f(\tau )\partial \tau \right]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega )$
Transformada de la convolución	$𝔽[f(t)*g(t)]=𝔽\left[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )\partial \tau \right]=F(\omega )G(\omega )$
Teorema de Parseval	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|f(t)|}^2\partial t=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|F(\omega )|}^2\partial \omega$

Demostraciones

Linealidad

$𝔽[\alpha f(t)+\beta g(t)]$	$=$	$\alpha F(\omega )+\beta G(\omega )$
$𝔽[\alpha f(t)+\beta g(t)]$	$=$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\alpha f(t)+\beta g(t))\cdot e^{-j\omega t}\partial t=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\alpha f(t)\cdot e^{-j\omega t})+(\beta g(t)\cdot e^{-j\omega t})\partial t=$
	$=$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\alpha f(t)\cdot e^{-j\omega t}\partial t+\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\beta g(t)\cdot e^{-j\omega t}\partial t=\alpha \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)\cdot e^{-j\omega t}\partial t+\beta \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)\cdot e^{-j\omega t}\partial t=\alpha F(\omega )+\beta G(\omega )$

Dualidades

$𝔽[F(t)]$	$=$	$2\pi f(-\omega )$
$𝔽[F(t)]$	$=$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(t)\cdot e^{-j\omega t}\partial t\Rightarrow 𝔽[F(-t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(-t)\cdot e^{j\omega t}\partial t=$
	$=$	$2\pi \frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(-t)\cdot e^{j\omega t}\partial t=2\pi f(-\omega )$

Cambio de escala

$𝔽[f(at)]$	$=$	$\frac{1}{\left|a\right|}F\left(\frac{\omega }{a}\right)$
$𝔽[f(at)]$	$=$	$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(at)\cdot e^{-j\omega t}\partial t\Rightarrow \left\{\begin{array}{cc}& u=at\to t=\frac{u}{a}\\ & \partial u=a\partial t\to \partial t=\frac{\partial u}{a}\end{array}\right\}\Rightarrow$
	$\Rightarrow$	$\left\{\begin{array}{cc}& sia>0\to \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)\cdot e^{-j\omega \frac{u}{a}}\frac{\partial u}{a}=\frac{1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)\cdot e^{-j\frac{\omega }{a}u}\partial u=\frac{1}{a}F\left(\frac{\omega }{a}\right)\\ & sia<0\to \underset{\mathrm{\infty }}{\overset{-\mathrm{\infty }}{\int }}f(u)\cdot e^{-j\omega \frac{u}{a}}\frac{\partial u}{a}=\frac{-1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)\cdot e^{-j\frac{\omega }{a}u}\partial u=\frac{-1}{a}F\left(\frac{\omega }{a}\right)\end{array}\right\}=\frac{1}{\left|a\right|}F\left(\frac{\omega }{a}\right)$

Inversión del tiempo

$𝔽[f^{}(-t)]=F^{}(-\omega )$

$𝔽[f^{}(-t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(-t)e^{-jwt}\partial t\to \{\begin{array}{c}\tau =-t\leftrightarrow t=-\tau \\ \partial \tau =-\partial t\leftrightarrow \partial t=-\partial \tau \end{array}$

$𝔽[f^{}(-t)]=-\underset{\mathrm{\infty }}{\overset{-\mathrm{\infty }}{\int }}x(\tau )e^{-jw(-\tau )}\partial \tau =\stackrel{F(-w)}{\overbrace{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(\tau )e^{-j(-w)\tau }\partial \tau }}$

$𝔽[f^{}(-t)]=F(-w)$

Traslación en el tiempo

$\begin{array}{cc}& 𝔽[f(t-t_0)]=e^{-j\omega t_0}F(\omega )\\ & 𝔽[f(t-t_0)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t-t_0)e^{-j\omega t}\partial t\to \left\{\begin{array}{cc}& u=t-t_0\leftrightarrow t=u+t_0\\ & \partial u=\partial t\\ \end{array}\right\}\\ & 𝔽[f(t-t_0)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)e^{-j\omega (u+t_0)}\partial u=e^{-j\omega t_0}\stackrel{F(\omega )}{\overbrace{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)e^{-j\omega u}\partial u}}\\ \end{array}$

Traslacion en frecuencia

Analogamente:

$\begin{array}{cc}& 𝔽[\underset{g(t)}{\underbrace{e^{+j{\omega }_{0}t}f(t)}}]=F(\omega -{\omega }_0)\\ & g(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\omega -{\omega }_0)e^{j\omega t}\partial \omega \to \left\{\begin{array}{cc}& u=\omega -{\omega }_0\leftrightarrow \omega =u+{\omega }_0\\ & \partial u=\partial \omega \\ \end{array}\right\}\to \frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(u)e^{j(u+{\omega }_0)t}\partial u\\ & g(t)=e^{j{\omega }_{0}t}\stackrel{f(t)}{\overbrace{\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(u)e^{jut}\partial u}}\\ \end{array}$

Derivación en el tiempo

$\begin{array}{cc}& 𝔽\left[\frac{{\partial }^{n}f(t)}{\partial t^n}\right]={\left(j\omega \right)}^{n}F(\omega )\\ & 𝔽[f^{\text{'}}(t)]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}^{\text{'}}(t)e^{-j\omega t}\partial t\to \int u\partial v=u\cdot v-\int v\partial u\to \left\{\begin{array}{cc}& u=e^{-j\omega t}\partial u=-j\omega \cdot e^{-j\omega t}\partial t\\ & \partial v=f^{\text{'}}(t)\partial tv=f(t)\\ \end{array}\right\}\\ & 𝔽[f^{\text{'}}(t)]=\underset{0}{\underbrace{e^{-j\omega t}{f(t)|}_{-\mathrm{\infty }}^{\mathrm{\infty }}}}+\int j\omega \cdot f(t)e^{-j\omega t}\partial t\\ & \underset{t\to \pm \mathrm{\infty }}{\lim }f(t)=0\text{ si }f(t)\text{ continua y abs}\text{. integrable}\\ \end{array}$

Derivación en la frecuencia

Analogamente:

$\begin{array}{cc}& 𝔽\left[\underset{g(t)}{\underbrace{{(-jt)}^{n}f(t)}}\right]=\frac{{\partial }^{n}F(\omega )}{\partial {\omega }^n}\\ & g(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{F}^{\text{'}}(\omega )e^{+j\omega t}\partial \omega \to \int u\partial v=u\cdot v-\int v\partial u\to \left\{\begin{array}{cc}& u=e^{+j\omega t}\partial u=jt\cdot e^{+j\omega t}\partial \omega \\ & \partial v=F^{\text{'}}(\omega )\partial \omega v=F(\omega )\\ \end{array}\right\}\to \\ & g(t)=\frac{1}{2\pi }\underset{0}{\underbrace{e^{+j\omega t}{F(\omega )|}_{-\mathrm{\infty }}^{\mathrm{\infty }}}}-\frac{1}{2\pi }\int jt\cdot F(\omega )e^{+j\omega t}\partial \omega \\ \end{array}$

Convolución

La convolución de dos señales se define como:

$f(t)*g(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )\partial \tau =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(\tau )f(t-\tau )\partial \tau$

de modo que:

$\begin{array}{cc}& \underset{\tau =-\mathrm{\infty }}{\overset{\tau =\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )\partial \tau \to \left\{\begin{array}{cc}& {\tau }_0=t-\tau \leftrightarrow \tau =t-{\tau }_0\\ & \partial {\tau }_0=-\partial \tau \\ \end{array}\right\}\\ & \underset{{\tau }_0=\mathrm{\infty }}{\overset{{\tau }_0=-\mathrm{\infty }}{\int }}f(t-{\tau }_0)g({\tau }_0)(-\partial {\tau }_0)=\underset{{\tau }_0=-\mathrm{\infty }}{\overset{{\tau }_0=\mathrm{\infty }}{\int }}f(t-{\tau }_0)g({\tau }_0)\partial {\tau }_0\\ \end{array}$

Integración en el tiempo

$\begin{array}{cc}& 𝔽\left[\underset{-\mathrm{\infty }}{\overset{t}{\int }}f(\tau )\partial \tau \right]\to f(t)*u(t)=\underset{\tau =-\mathrm{\infty }}{\overset{\tau =\mathrm{\infty }}{\int }}f(\tau )u(t-\tau )\partial \tau \to u(t-\tau )=\{\begin{array}{cc}& 1,t-\tau \ge 0\\ & 0,t-\tau <0\\ \end{array}\to \\ & t-\tau \ge 0\to t\ge \tau \Rightarrow \underset{\tau =-\mathrm{\infty }}{\overset{\tau =\mathrm{\infty }}{\int }}f(\tau )\underset{\begin{array}{c}u(t-\tau )=1\to \\ t-\tau >0\to \tau <t\end{array}}{\underbrace{u(t-\tau )}}\partial \tau =\underset{-\mathrm{\infty }}{\overset{t}{\int }}f(\tau )\partial \tau \\ & 𝔽\left[\underset{-\mathrm{\infty }}{\overset{t}{\int }}f(\tau )\partial \tau \right]=𝔽[f(t)*u(t)]=F(\omega ).[\frac{1}{j\omega }+\pi \delta (\omega )]=\frac{F(\omega )}{j\omega }+\pi F(0)\delta (\omega )\\ \end{array}$

Transformada de la convolución

$\begin{array}{cc}& 𝔽[f(t)*g(t)]=𝔽\left[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )\partial \tau \right]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )\partial \tau \right]e^{-j\omega t}\partial t=\\ & \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )g(t-\tau )e^{-j\omega t}\partial t\partial \tau =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )\underset{G(\omega )e^{-j\omega \tau }}{\underbrace{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t-\tau )e^{-j\omega t}\partial t}}\partial \tau =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )G(\omega )e^{-j\omega \tau }\partial \tau =\\ & G(\omega )\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )e^{-j\omega \tau }\partial \tau =G(\omega )\cdot F(\omega )\\ \end{array}$

Teorema de Parseval

El teorema de Parseval es una solución particular de la propiedad:

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

$\begin{array}{cc}& \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|f(t)|}^2\partial t=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)f^{*}(t)\partial t\underset{f(t)=g(t)}{\overset{}{\to }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)g^{*}(t)\partial t\to \{f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\omega )e^{j\omega t}\partial \omega \}\\ & \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left[\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\omega )e^{j\omega t}\partial \omega \right]g^{*}(t)\partial t\underset{\begin{array}{c}\text{Cambiando}\\ \text{orden de}\\ \text{integracion}\end{array}}{\overset{}{\to }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\omega )e^{j\omega t}{g}^{*}(t)\partial t\partial \omega =\\ & \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{2\pi }\left[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{g}^{*}(t)e^{j\omega t}\partial t\right]F(\omega )\partial \omega =\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\left[\underset{G(\omega )}{\underbrace{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-j\omega t}\partial t}}\right]}^{*}F(\omega )\partial \omega =\\ & \frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{G}^{*}(\omega )F(\omega )\partial \omega \underset{f(t)=g(t)}{\overset{}{\to }}\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{F}^{*}(\omega )F(\omega )\partial \omega =\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|F(\omega )|}^2\partial \omega \\ \end{array}$

Consejo general

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

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

También podemos aplicar las propiedades en otro orden que esta a continuación:

$\begin{array}{cc}& 𝔽[f(at-b)]?\\ & g(t)=f(at)\to g(t-b)=f\left(a(t-b)\right)=f(at-ab)\ne f(at-b)\\ & g(t-{}^b╱{}_a)=f\left(a(t-{}^b╱{}_a)\right)=f(at-b)\\ & 𝔽\left[g(t-{}^b╱{}_a)\right]=G(\omega )\cdot e^{-j\omega \frac{b}{a}}\Rightarrow g(t)=f(at)\to G(\omega )=\frac{1}{\left|a\right|}F\left(\frac{\omega }{a}\right)\\ & 𝔽[f(at-b)]=𝔽\left[g(t-{}^b╱{}_a)\right]=G(\omega )\cdot e^{-j\omega \frac{b}{a}}=\frac{1}{\left|a\right|}F\left(\frac{\omega }{a}\right)e^{-j\omega \frac{b}{a}}\\ \end{array}$