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 | Property | |
---|---|---|
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 |