| CT Fourier Transform Pairs and Properties (frequency $ \omega $ in radians per time unit) | |
|---|---|
| Definition CT Fourier Transform and its Inverse | |
| CT Fourier Transform | $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt $ |
| Inverse DT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega\, $ |
| CT Fourier Transform Pairs | ||||
|---|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | ||
| CTFT of a unit impulse | $ \delta (t)\ $ | $ 1 \! \ $ | ||
| CTFT of a shifted unit impulse | $ \delta (t-t_0)\ $ | $ e^{jwt_0} \ \ $ | ||
| CTFT of a complex exponential | $ e^{jw_0t}\ $ | $ 2\pi \delta (\omega - \omega_0) \ $ | ||
| $ e^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \frac{1}{a+j\omega} \ $ | |||
| $ te^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \left( \frac{1}{a+j\omega}\right)^2 \ $ | |||
| CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $ | ||
| CTFT of a sine | $ sin(\omega_0 t) \ $ | $ \frac{\pi}{j} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] \ $ | ||
| CTFT of a rect | $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ | $ \frac{2 \sin \left( T \omega \right)}{\omega} \ $ | ||
| CTFT of a sinc | $ \frac{2 \sin \left( W t \right)}{\pi t } \ $ | $ \left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $ | ||
| CTFT of a periodic function | $ \sum^{\infty}_{k=-\infty} a_{k}e^{jkw_{0}t} \ $ | $ 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ $ | ||
| CTFT of an impulse train | $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ | $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) \ $ | ||
| CT Fourier Transform Properties | |||
|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| multiplication property | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x(t)*y(t) \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x(-t) $ | $ \ X(-\omega) $ | |
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ |
