Referencia: Lathi Tabla 7.1 p699. Shaum Hsu Tabla 5-2 p223. Oppenheim Tabla 4.2 p329 pdf357
Transformada Fourier – Propiedades
Series de Fourier de señales periódicas con Python
| No | x(t) | X(ω) | |
|---|---|---|---|
| 1 | e-at μ (t) | \frac{1}{a + j \omega} | a>0 |
| 2 | eat μ (-t) | \frac{1}{a - j \omega} | a>0 |
| 3 | e-a|t| | \frac{2a}{a^2+ \omega ^2} | a>0 |
| 4 | t e-at μ (t) | \frac{1}{(a+j \omega)^2} | a>0 |
| 5 | tn e-at μ (t) | \frac{n!}{(a+j \omega)^{n+1}} | a>0 |
| 6a | δ(t) | 1 | |
| 6b | δ(t-t0) | e-jωt0 | |
| 7 | 1 | 2\pi \delta (\omega) | |
| 8 | ejω0t | 2\pi \delta (\omega-\omega_0) | |
| 9 | cos (ω0 t) | \pi [\delta (\omega - \omega_0) +\delta (\omega + \omega_0)] | |
| 10 | sin (ω0 t) | \pi [\delta (\omega + \omega_0) -\delta (\omega - \omega_0)] | |
| 11a | μ(t) | \pi \delta (\omega ) +\frac{1}{j \omega } | |
| 11b | μ(-t) | \pi \delta (\omega ) - \frac{1}{j \omega } | |
| 12 | sgn (t) | \frac{2}{j \omega} | |
| 13 | cos (ω0 t) μ(t) | \frac{\pi}{2} [\delta (\omega - \omega_0) +\delta (\omega + \omega_0)] + \frac{j \omega}{\omega_0^2 - \omega ^2} | |
| 14 | sin (ω0 t) μ(t) | \frac{\pi}{2j} [\delta (\omega - \omega_0) - \delta (\omega + \omega_0)] + \frac{\omega_0}{\omega_0^2 - \omega ^2} | |
| 15 | e-at sin (ω0 t) μ(t) | \frac{\omega_0}{(a+j\omega)^2 + \omega_0^2} | a>0 |
| 16 | e-at cos (ω0 t) μ(t) | \frac{a + j\omega}{(a+j\omega)^2 + \omega_0^2} | a>0 |
| 17 | rect \Big(\frac{1}{\tau}\Big) | \tau sinc \Big( \frac{\omega \tau}{2} \Big) | |
| 18 | \frac{W}{\pi} sinc (Wt) | rect \Big(\frac{\omega}{2W} \Big) | |
| 19 | \Delta \Big( \frac{t}{\tau} \Big) | \frac{\tau}{2}sinc ^2 \Big( \frac{\omega \tau}{4} \Big) | |
| 20 | \frac{W}{2\pi} sinc ^2 \Big(\frac{Wt}{2} \Big) | \Delta \Big(\frac{\omega}{2W} \Big) | |
| 21 | \sum_{n=- \infty}^{\infty} \delta(t-nT) | \omega_0 \sum_{n=-\infty}^{\infty} \delta(\omega-n \omega_0) | \omega_0 = \frac{2 \pi}{T} |
| 22 | \frac{e^{-t^2}}{2 \sigma ^2} | \sigma \sqrt{2 \pi} e^{-\sigma^2 \omega^2 /2} | |
| 23 | \frac{1}{a^2 + t^2} | e ^{-a|\omega|} | |
| 24 | e ^{-at^2} | \sqrt{\frac{\pi}{a}}e ^{-\omega^2 /4a} | a>0 |
| 25 | p_a(t) = \begin{cases} 1 & |t|<a \\ 0 & |t|>a \end{cases} | 2a\frac{\sin (\omega a)}{(\omega a)} | |