| Line 32: | Line 32: | ||
===== - Properties of the Continuous-time Fourier Transform ===== | ===== - Properties of the Continuous-time Fourier Transform ===== | ||
| + | |||
| + | |||
{| border="1" class="wikitable" | {| border="1" class="wikitable" | ||
|- | |- | ||
| − | ! | + | ! name |
| − | ! | + | ! Property |
| − | + | |- | |
| − | |- | + | |
| + | | Linearity | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Time Shifting | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Frequency Shifting | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Conjugation | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Scaling | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Multiplication | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Convolution | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Differentiation | ||
| + | | | ||
| + | |- | ||
| + | |||
| + | | Parseval's Relation | ||
| + | | | ||
| + | |- | ||
Revision as of 16:23, 14 November 2018
CTFT of periodic signals and some properties with proofs
- Fourier series of periodic signals
| 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 | Property |
|---|---|
| Linearity | |
| Time Shifting | |
| Frequency Shifting | |
| Conjugation | |
| Scaling | |
| Multiplication | |
| Convolution | |
| Differentiation | |
| Parseval's Relation |
