Publicado en por Edison Del Rosario

Transformadas de Fourier – Tabla

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) 1a+jω \frac{1}{a + j \omega} a>0
2 eat μ (-t) 1a−jω \frac{1}{a - j \omega} a>0
3 e-a|t| 2aa2+ω2 \frac{2a}{a^2+ \omega ^2} a>0
4 t e-at μ (t) 1(a+jω)2 \frac{1}{(a+j \omega)^2} a>0
5 tn  e-at μ (t) n!(a+jω)n+1 \frac{n!}{(a+j \omega)^{n+1}} a>0
6a δ(t) 1
6b δ(t-t0) e-jωt0
7 1 2πδ(ω) 2\pi \delta (\omega)
8 ejω0t 2πδ(ω−ω0) 2\pi \delta (\omega-\omega_0)
9 cos (ω0 t) π[δ(ω−ω0)+δ(ω+ω0)] \pi [\delta (\omega - \omega_0) +\delta (\omega + \omega_0)]
10 sin (ω0 t) π[δ(ω+ω0)−δ(ω−ω0)] \pi [\delta (\omega + \omega_0) -\delta (\omega - \omega_0)]
11a μ(t) πδ(ω)+1jω \pi \delta (\omega ) +\frac{1}{j \omega }
11b μ(-t) πδ(ω)−1jω \pi \delta (\omega ) - \frac{1}{j \omega }
12 sgn (t) 2jω\frac{2}{j \omega}
13 cos (ω0 t) μ(t) π2[δ(ω−ω0)+δ(ω+ω0)]+jωω02−ω2\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) π2j[δ(ω−ω0)−δ(ω+ω0)]+ω0ω02−ω2\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) ω0(a+jω)2+ω02 \frac{\omega_0}{(a+j\omega)^2 + \omega_0^2} a>0
16 e-at cos (ω0 t) μ(t) a+jω(a+jω)2+ω02 \frac{a + j\omega}{(a+j\omega)^2 + \omega_0^2} a>0
17 rect(1τ) rect \Big(\frac{1}{\tau}\Big) τsinc(ωτ2) \tau sinc \Big( \frac{\omega \tau}{2} \Big)
18 Wπsinc(Wt) \frac{W}{\pi} sinc (Wt) rect(ω2W) rect \Big(\frac{\omega}{2W} \Big)
19 Δ(tτ) \Delta \Big( \frac{t}{\tau} \Big) τ2sinc2(ωτ4) \frac{\tau}{2}sinc ^2 \Big( \frac{\omega \tau}{4} \Big)
20 W2πsinc2(Wt2) \frac{W}{2\pi} sinc ^2 \Big(\frac{Wt}{2} \Big) Δ(ω2W) \Delta \Big(\frac{\omega}{2W} \Big)
21 ∑n=−∞∞δ(t−nT) \sum_{n=- \infty}^{\infty} \delta(t-nT) ω0∑n=−∞∞δ(ω−nω0) \omega_0 \sum_{n=-\infty}^{\infty} \delta(\omega-n \omega_0) ω0=2πT \omega_0 = \frac{2 \pi}{T}
22 e−t22σ2 \frac{e^{-t^2}}{2 \sigma ^2} σ2πe−σ2ω2/2 \sigma \sqrt{2 \pi} e^{-\sigma^2 \omega^2 /2}
23 1a2+t2 \frac{1}{a^2 + t^2} e−a∣ω∣ e ^{-a|\omega|}
24 e−at2 e ^{-at^2} πae−ω2/4a \sqrt{\frac{\pi}{a}}e ^{-\omega^2 /4a} a>0
25 pa(t)={1∣t∣<a0∣t∣>a p_a(t) = \begin{cases} 1 & |t|<a \\ 0 & |t|>a \end{cases} 2asin⁡(ωa)(ωa) 2a\frac{\sin (\omega a)}{(\omega a)}

Transformada Fourier – Propiedades

Series de Fourier de señales periódicas con Python