Propiedades de la transformada de Fourier

Definición de la transformada de Fourier

${\begin{aligned}&\mathbb {F} [f(t)]=F(\omega )=\int \limits _{-\infty }^{\infty }{f(t)\cdot e^{-i\omega t}\partial t}\\&\mathbb {F} ^{-1}[F(\omega )]=f(t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }{F(\omega )\cdot e^{+i\omega t}\partial \omega }\\\end{aligned}}$

Tabla resumen de propiedades

Propiedad	Definición
Linealidad	$\mathbb {F} \left[\alpha f(t)+\beta g(t)\right]=\alpha F(\omega )+\beta G(\omega )$
Dualidad	$\mathbb {F} \left[F(t)\right]=2\pi f(-\omega )$
Cambio de escala	$\mathbb {F} \left[f(at)\right]={\frac {1}{\left|a\right|}}F\left({\frac {\omega }{a}}\right)$
Inversión el tiempo	$\mathbb {F} \left[f^{}(-t)\right]=F^{}(-\omega )$
Traslación en el tiempo	$\mathbb {F} \left[f(t-t_{0})\right]=F(\omega )e^{-j\omega t_{0}}$
Traslación en frecuencia	$\mathbb {F} \left[f(t)e^{j\omega _{0}t}\right]=F(\omega -\omega _{0})$
Derivación en el tiempo	$\mathbb {F} \left[{\frac {\partial ^{n}f(t)}{\partial t^{n}}}\right]=\left(j\omega \right)^{n}F(\omega )$
Derivación 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 convolución	$\mathbb {F} \left[f(t)*g(t)\right]=\mathbb {F} \left[\int \limits _{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau }\right]=F(\omega )G(\omega )$
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

Linealidad

$\mathbb {F} \left[\alpha f(t)+\beta g(t)\right]$	$=$	$\alpha F(\omega )+\beta G(\omega )$
$\mathbb {F} \left[\alpha f(t)+\beta g(t)\right]$	$=$	$\color {Black}\int \limits _{-\infty }^{\infty }{\left(\alpha f(t)+\beta g(t)\right)\cdot e^{-j\omega t}\partial t}=\int \limits _{-\infty }^{\infty }{\left(\alpha f(t)\cdot e^{-j\omega t}\right)+\left(\beta g(t)\cdot e^{-j\omega t}\right)\partial t}=$
	$\color {Black}=$	$\color {Black}\int \limits _{-\infty }^{\infty }{\alpha f(t)\cdot e^{-j\omega t}\partial t}\;+\int \limits _{-\infty }^{\infty }{\beta g(t)\cdot e^{-j\omega t}\partial t}=\alpha \int \limits _{-\infty }^{\infty }{f(t)\cdot e^{-j\omega t}\partial t}\;+\;\beta \int \limits _{-\infty }^{\infty }{g(t)\cdot e^{-j\omega t}\partial t}=\color {Black}\alpha F(\omega )\;+\;\beta G(\omega )$

Dualidades

$\mathbb {F} \left[F(t)\right]$	$=$	$2\pi f(-\omega )$
$\mathbb {F} \left[F(t)\right]$	$=$	$\color {Black}\int \limits _{-\infty }^{\infty }{F(t)\cdot e^{-j\omega t}\partial t}\quad \Rightarrow \quad \mathbb {F} \left[F(-t)\right]=\int \limits _{-\infty }^{\infty }{F(-t)\cdot e^{j\omega t}\partial t}=$
	$\color {Black}=$	$\color {Black}2\pi {\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }{F(-t)\cdot e^{j\omega t}\partial t}=\color {Black}2\pi f(-\omega )$

Cambio de escala

$\mathbb {F} \left[f(at)\right]$	$=$	${\frac {1}{\left|a\right|}}F\left({\frac {\omega }{a}}\right)$
$\mathbb {F} \left[f(at)\right]$	$=$	$\color {Black}\int \limits _{-\infty }^{\infty }{f(at)\cdot e^{-j\omega t}\partial t}\quad \Rightarrow \quad \left\{{\begin{aligned}&u=at\quad \to \quad t={\frac {u}{a}}\\&\partial u=a\partial t\;\to \;\partial t={\frac {\partial u}{a}}\end{aligned}}\right\}\quad \Rightarrow$
	$\color {Black}\Rightarrow$	$\color {Black}\left\{{\begin{aligned}&si\quad a>0\;\to \int \limits _{-\infty }^{\infty }{f(u)\cdot e^{-j\omega {\frac {u}{a}}}{\frac {\partial u}{a}}}\;=\;{\frac {1}{a}}\int \limits _{-\infty }^{\infty }{f(u)\cdot e^{-j{\frac {\omega }{a}}u}\partial u}={\frac {1}{a}}F\left({\frac {\omega }{a}}\right)\\&si\quad a<0\;\to \int \limits _{\infty }^{-\infty }{f(u)\cdot e^{-j\omega {\frac {u}{a}}}{\frac {\partial u}{a}}}\;=\;{\frac {-1}{a}}\int \limits _{-\infty }^{\infty }{f(u)\cdot e^{-j{\frac {\omega }{a}}u}\partial u}={\frac {-1}{a}}F\left({\frac {\omega }{a}}\right)\end{aligned}}\right\}=\color {Black}{\frac {1}{\left|a\right|}}F\left({\frac {\omega }{a}}\right)$

Inversión del tiempo

$\mathbb {F} \left[f^{}(-t)\right]=F^{}(-\omega )$

$\mathbb {F} \left[f^{}(-t)\right]=\int _{-\infty }^{\infty }x(-t)e^{-jwt}\partial t\rightarrow \left\{{\begin{matrix}\tau =-t\leftrightarrow t=-\tau \\\partial \tau =-\partial t\leftrightarrow \partial t=-\partial \tau \end{matrix}}\right.$

$\mathbb {F} \left[f^{}(-t)\right]=-\int _{\infty }^{-\infty }x(\tau )e^{-jw(-\tau )}\partial \tau =\overbrace {\int _{-\infty }^{\infty }x(\tau )e^{-j(-w)\tau }\partial \tau } ^{F(-w)}$

$\mathbb {F} \left[f^{}(-t)\right]=F(-w)$

Traslación en el tiempo

${\begin{aligned}&\mathbb {F} [f(t-t_{0})]=e^{-j\omega t_{0}}F(\omega )\\&\mathbb {F} [f(t-t_{0})]=\int \limits _{-\infty }^{\infty }{f(t-t_{0})e^{-j\omega t}\partial t}\to \left\{{\begin{aligned}&u=t-t_{0}\leftrightarrow t=u+t_{0}\\&\partial u=\partial t\\\end{aligned}}\right\}\\&\mathbb {F} [f(t-t_{0})]=\int \limits _{-\infty }^{\infty }{f(u)e^{-j\omega \left(u+t_{0}\right)}\partial u=}e^{-j\omega t_{0}}\overbrace {\int \limits _{-\infty }^{\infty }{f(u)e^{-j\omega u}\partial u}} ^{F(\omega )}\\\end{aligned}}$

Traslacion en frecuencia

Analogamente:

${\begin{aligned}&\mathbb {F} [\underbrace {e^{+j\omega _{0}t}f(t)} _{g(t)}]=F(\omega -\omega _{0})\\&g(t)={\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }{F(\omega -\omega _{0})e^{j\omega t}\partial \omega \to \left\{{\begin{aligned}&u=\omega -\omega _{0}\leftrightarrow \omega =u+\omega _{0}\\&\partial u=\partial \omega \\\end{aligned}}\right\}}\to {\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }{F(u)e^{j\left(u+\omega _{0}\right)t}\partial u}\\&g(t)=e^{j\omega _{0}t}\overbrace {{\frac {1}{2\pi }}\int \limits _{-\infty }^{\infty }{F(u)e^{jut}\partial u}} ^{f(t)}\\\end{aligned}}$

Derivación en el tiempo

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

Derivación en la frecuencia

Analogamente:

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

Convolución

La convolución de dos señales se define como:

$f(t)*g(t)=\int \limits _{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau }=\int \limits _{-\infty }^{\infty }{g(\tau )f(t-\tau )\partial \tau }$

de modo que:

${\begin{aligned}&\int \limits _{\tau =-\infty }^{\tau =\infty }{f(\tau )g(t-\tau )\partial \tau }\to \left\{{\begin{aligned}&\tau _{0}=t-\tau {\text{ }}\leftrightarrow \tau =t-\tau _{0}\\&\partial \tau _{0}=-\partial \tau {\text{ }}\\\end{aligned}}\right\}\\&\int \limits _{\tau _{0}=\infty }^{\tau _{0}=-\infty }{f(t-\tau _{0})g(\tau _{0})\left(-\partial \tau _{0}\right)=}\int \limits _{\tau _{0}=-\infty }^{\tau _{0}=\infty }{f(t-\tau _{0})g(\tau _{0})\partial \tau _{0}}\\\end{aligned}}$

Integración en el tiempo

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

Transformada de la convolución

${\begin{aligned}&\mathbb {F} \left[f(t)*g(t)\right]=\mathbb {F} \left[\int \limits _{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau }\right]=\int \limits _{-\infty }^{\infty }{\left[\int \limits _{-\infty }^{\infty }{f(\tau )g(t-\tau )\partial \tau }\right]e^{-j\omega t}\partial t=}\\&\int \limits _{-\infty }^{\infty }{\int \limits _{-\infty }^{\infty }{f(\tau )g(t-\tau )e^{-j\omega t}\partial t\partial \tau }}=\int \limits _{-\infty }^{\infty }{f(\tau )\underbrace {\int \limits _{-\infty }^{\infty }{g(t-\tau )e^{-j\omega t}\partial t}} _{G(\omega )e^{-j\omega \tau }}}\partial \tau =\int \limits _{-\infty }^{\infty }{f(\tau )G(\omega )e^{-j\omega \tau }\partial \tau }=\\&G(\omega )\int \limits _{-\infty }^{\infty }{f(\tau )e^{-j\omega \tau }\partial \tau }=G(\omega )\cdot F(\omega )\\\end{aligned}}$

Teorema de Parseval

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

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

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

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.

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

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

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