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| <math>b.\text{ } x(t)=e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math> | | <math>b.\text{ } x(t)=e^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math> | ||
| − | |- | + | |- |
| <math>X(f)= \mathcal{X}(2\pi f)=\frac{1}{a+i2\pi f}</math> | | <math>X(f)= \mathcal{X}(2\pi f)=\frac{1}{a+i2\pi f}</math> | ||
|- | |- | ||
Revision as of 15:10, 9 September 2010
| 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 $ |
