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# 8.3: Transformadas comunes de Fourier

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Mesa$$\PageIndex{1}$$
Señal de dominio de tiempo Señal de dominio de frecuencia Condición
$$e^{-(a t)} u(t)$$ $$\frac{1}{a+j \omega}$$ $$a>0$$
$$e^{at}u(−t)$$ $$\frac{1}{a-j \omega}$$ $$a>0$$
$$e^{−(a|t|)}$$ $$\frac{2a}{a^2+\omega^2}$$ $$a>0$$
$$te^{−(at)}u(t)$$ $$\frac{1}{(a+j \omega)^2}$$ $$a>0$$
$$t^ne^{−(at)}u(t)$$ $$\frac{n !}{(a+j \omega)^{n+1}}$$ $$a>0$$
$$\delta(t)$$ $$1$$
$$1$$ $$2 \pi \delta(\omega)$$
$$e^{j \omega_0 t}$$ $$2 \pi \delta\left(\omega-\omega_{0}\right)$$
$$\cos (\omega_0 t)$$ $$\pi\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)$$
$$\sin (\omega_0 t)$$ $$j \pi\left(\delta\left(\omega+\omega_{0}\right)-\delta\left(\omega-\omega_{0}\right)\right)$$
$$u(t)$$ $$\pi \delta(\omega)+\frac{1}{j \omega}$$
sgn ($$t)$$ $$\frac{2}{j \omega}$$
$$\cos \left(\omega_{0} t\right) u(t)$$ $$\frac{\pi}{2}\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)+\frac{j \omega}{\omega_{0}^{2}-\omega^{2}}$$
$$\sin \left(\omega_{0} t\right) u(t)$$ $$\frac{\pi}{2 j}\left(\delta\left(\omega-\omega_{0}\right)-\delta\left(\omega+\omega_{0}\right)\right)+\frac{\omega_{0}}{\omega_{0}^{2}-\omega^{2}}$$
$$e^{-(a t)} \sin \left(\omega_{0} t\right) u(t)$$ $$\frac{\omega_{0}}{(a+j \omega)^{2}+\omega_{0}^{2}}$$ $$a>0$$
$$e^{-(a t)} \cos \left(\omega_{0} t\right) u(t)$$ $$\frac{a+j \omega}{(a+j \omega)^{2}+\omega_{0}^{2}}$$ $$a>0$$
$$u(t+\tau)-u(t-\tau)$$ $$2 \tau \frac{\sin (\omega \tau)}{\omega \tau}=2 \tau \operatorname{sinc}(\omega t)$$
$$\frac{\omega_{0}}{\pi} \frac{\sin \left(\omega_{0} t\right)}{\omega_{0} t}=\frac{\omega_{0}}{\pi} \operatorname{sinc}\left(\omega_{0}\right)$$ $$u\left(\omega+\omega_{0}\right)-u\left(\omega-\omega_{0}\right)$$
\ (\ begin {array} {l}
\ izquierda (\ frac {t} {\ tau} +1\ derecha)\ izquierda (u\ izquierda (\ frac {t} {\ tau} +1\ derecha) -u\ izquierda (\ frac {t} {\ tau}\ derecha)\ derecha)\
\ izquierda (-\ frac {t} {\ tau}\ +1 derecha)\ izquierda (u\ izquierda (\ frac {t} {\ tau}\ derecha) -u\ izquierda (\ frac {t} {\ tau} -1\ derecha)\ derecha) =\\
\ operatorname {triag}\ left (\ frac {t} {2\ tau}\ derecha)
\ end {array}\)
$$\tau \operatorname{sinc}^{2}\left(\frac{\omega \tau}{2}\right)$$
$$\frac{\omega_{0}}{2 \pi} \operatorname{sinc}^{2}\left(\frac{\omega_{0} t}{2}\right)$$ \ (\ begin {array} {l}
\ izquierda (\ frac {\ omega} {\ omega_ {0}} +1\ derecha)\ izquierda (u\ izquierda (\ frac {\ omega} {\ omega_ {0}} +1\ derecha) -u\ izquierda (\ frac {\ omega} {\ omega_ {0}}\ derecha)\ derecha) +\\ izquierda (-
\ frac {\ omega} {\ omega_ {0}} +1\ derecha)\ izquierda (u\ izquierda (\ frac {\ omega} {\ omega_ {0}}\ derecha) -u\ izquierda (\ frac {\ omega} {\ omega} {\ omega_ {0}} -1\ derecha)\ derecha) =\\
\ nombreoperador {triag}\ izquierda (\ frac {\ omega} {2\ omega_ {0}}\ derecha)
\ end {array}\)
$$\sum_{n=-\infty}^{\infty} \delta(t-n T)$$ $$\omega_{0} \sum_{n=-\infty}^{\infty} \delta\left(\omega-n \omega_{0}\right)$$ $$\omega_0 = \frac{2 \pi}{T}$$
$$e^{-\frac{t^{2}}{2 \sigma^{2}}}$$ $$\sigma \sqrt{2 \pi} e^{-\frac{\sigma^{2} \omega^{2}}{2}}$$

triag [n] es la función triangular para valores reales arbitrarios$$n$$.

\ [\ nombreoperador {triag} [\ mathrm {n}] =\ left\ {\ begin {array} {ll}
1+n &\ text {if} -1\ leq n\ leq 0\\
1-n &\ text {if} 0<n\ leq 1\\
0 &\ text {de lo contrario}
\ end {array}\ derecho. \ nonumber\]

This page titled 8.3: Transformadas comunes de Fourier is shared under a CC BY license and was authored, remixed, and/or curated by Richard Baraniuk et al..