| CTFT of a complex exponential |
| $ a.\text{ } x(t)=e^{i\omega_0 t} $ |
| $ X(f)= \mathcal{X}(2\pi f)=2\pi \delta (2\pi f-\omega_0) $ |
| $ Since\text{ } k\delta (kt)=\delta (t),\forall k\ne 0 $ |
| $ X(f)=\delta (f-\frac{\omega_0}{2\pi}) $ |
| $ b.\text{ } x(t)=e^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ |
| $ X(f)= \mathcal{X}(2\pi f)=\frac{1}{a+i2\pi f} $ |
| $ c.\text{ } x(t)=te^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ |
| $ X(f)= \mathcal{X}(2\pi f)=\left( \frac{1}{a+i2\pi f}\right)^2 $ |
