8.3: Transformadas comunes de Fourier
- Page ID
- 86328
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Propiedades comunes de CTFT
| 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\]