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| align="left" style="padding-left: 0em;" | CTFT of a complex exponential | | align="left" style="padding-left: 0em;" | CTFT of a complex exponential | ||
|- | |- | ||
| − | |<math>x(t)=e^{i\omega_0 t}</math> | + | | <math>a.\text{ } x(t)=e^{i\omega_0 t}</math> |
|- | |- | ||
| − | |<math>X(f)= \mathcal{X}(2\pi f)=2\pi \delta (2\pi f-\omega_0)</math> | + | | <math>X(f)= \mathcal{X}(2\pi f)=2\pi \delta (2\pi f-\omega_0)</math> |
|- | |- | ||
| − | |<math>Since\text{ } k\delta (kt)=\delta (t),\forall k\ne 0</math> | + | | <math>Since\text{ } k\delta (kt)=\delta (t),\forall k\ne 0</math> |
| + | |- | ||
| + | | <math>X(f)=\delta (f-\frac{\omega_0}{2\pi})</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>c.\text{ } x(t)=te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math> | ||
| + | |- | ||
| + | | <math>X(f)= \mathcal{X}(2\pi f)=\left( \frac{1}{a+i2\pi f}\right)^2</math> | ||
|- | |- | ||
| − | |||
|} | |} | ||
Revision as of 15:08, 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 $ |
