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0. Fourier Transform Table
$x(t)$ | $X(\omega)$ | ||
1 | $e^{-a t}u(t), a>0$ | $\frac{1}{a+j\omega}$ | ref. |
2 | $e^{a t} u(-t) , a>0$ | $\frac{1}{a-j\omega}$ | |
3 | $e^{-a \vert t\vert}, a>0$ | $\frac{2a}{a^2+\omega^2}$ | |
4 | $te^{-a t}u(t), a>0$ | $\frac{1}{ (a+j\omega)^2}$ | |
5 | $t^ne^{-a t}u(t), a>0$ | $\frac{n!}{ (a+j\omega)^{n+1}}$ | |
6 | $\delta(t)$ | $1$ | ref. |
7 | $1$ | $2\pi \delta(\omega)$ | ref. |
8 | $e^{j\omega_0t}$ | $2\pi \delta (\omega-\omega_0) $ | ref. |
9 | $\text{sgn}(t)$ | $\frac{2}{j\omega}$ | ref. |
10 | $u(t)$ | $\pi \delta(\omega) +\frac{1}{j\omega}$ | ref. |
11 | $\cos \omega_0 t$ | $\pi \left[ \delta(\omega+\omega_0) + \delta (\omega-\omega_0) \right]$ | ref. |
12 | $\sin \omega_0 t$ | $j\pi \left[ \delta(\omega+\omega_0) - \delta (\omega-\omega_0) \right]$ | ref. |
13 | $\cos \omega_0 t u(t)$ | $\frac{\pi}{2} \left[ \delta(\omega+\omega_0) + \delta (\omega-\omega_0) \right] +\frac{j\omega}{\omega_0^2-\omega^2}$ | |
14 | $\sin \omega_0 t u(t)$ | $\frac{\pi}{2j} \left[ \delta(\omega+\omega_0) - \delta (\omega-\omega_0) \right] +\frac{\omega_0}{\omega_0^2-\omega^2}$ | |
15 | $e^{-at} \sin \omega_0t u(t), a>0$ | $ \frac{\omega_0}{(a+j\omega)^2+\omega_0^2}$ | |
16 | $e^{-at} \cos \omega_0t u(t), a>0$ | $ \frac{a+j\omega}{(a+j\omega)^2+\omega_0^2}$ | |
17 | $ \displaystyle \sum_{n=-\infty}^{\infty} \delta(t-nT_s)$ | $ \displaystyle \omega_0 \sum_{n=-\infty}^{\infty} \delta(\omega-n\omega_0) , \omega_0=\frac{2\pi}{T_s}$ | ref. |
18 | $ \text{rect}_\tau (t)$, 너비가 $\tau$인 rectangle pulse |
$ \tau \text{sinc}\left ( \frac{\omega \tau}{2\pi}\right)$ | ref. |
19 | $ \frac{W}{2\pi} \text{sinc}\left ( \frac{W}{2\pi}t\right)$ | $\text{rect}_W (\omega)$, 너비가 $W$인 rectangle pulse |
ref. |
20 | $ \Delta \left ( \frac{t}{\tau}\right)$ 기울기가 $\frac{1}{\tau}$인 삼각파 |
$ \tau \text{sinc}^2 \left ( \frac{\omega \tau}{2\pi}\right)$ | |
21 | $ e^{-\frac{t^2}{2\sigma ^2}}$ | $ \sigma \sqrt{\pi} e^{-\frac{\sigma^2\omega^2}{2}}$ |
1. 같이보면 좋은 자료들
2023.10.05 - [분류 전체보기] - [SS] The Spectrum of Continuous Time Signals
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