[SS] Fourier Transform Table

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}}$

 

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1. 같이보면 좋은 자료들

2023.10.05 - [분류 전체보기] - [SS] The Spectrum of Continuous Time Signals

[SS] The Spectrum of Continuous Time Signals

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