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 | ||
| Frequency Shifting | ||
| Conjugation | ||
| Scaling | ||
| Multiplication | ||
| Convolution | ||
| Differentiation | ||
| Parseval's Relation |
