CTFT of periodic signals and some properties with proofs
| Function | CTFT | Proof |
|---|---|---|
| $ sin(\omega_0t) $ | $ \frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0)) $ | |
| $ cos(\omega_0t) $ | $ \pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0)) $ | |
| $ e^{j\omega_0t} $ | $ 2\pi\delta(\omega - \omega_0) $ | |
| $ \sum_{k=-\infty}^{\infty}u(t+5k) - u(t-1+5k) $ |
| Name | $ x(t) \longrightarrow \ $ | $ \mathcal{X}(\omega) $ |
|---|---|---|
| Linearity | $ ax(t) + by(t) \ $ | $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $ |
| Time Shifting | $ x(t-t_0) \ $ | $ e^{-j\omega t_0}X(\omega) $ |
| Frequency Shifting | $ e^{j\omega_0 t}x(t) $ | $ \mathcal{X} (\omega - \omega_0) $ |
| Conjugation | $ x^{*}(t) \ $ | $ \mathcal{X}^{*} (-\omega) $ |
| Scaling | $ x(at) \ $ | $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $ |
| Multiplication | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta $ |
| Convolution | $ x(t)*y(t) \ $ | $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $ |
| Differentiation | $ tx(t) \ $ | $ j\frac{d}{d\omega} \mathcal{X} (\omega) $ |
| Parseval's Relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $ |
