| |
$f(t)$ |
$F(\omega)$ |
| Linearity |
$a f_{1}(t) + b f_{2}(t)$ |
$a F_{1}(\omega) + b F_{2}(\omega)$ |
| Time Scaling |
$f(at)$ |
$\dfrac{1}{|\,a\,|}F\left(\dfrac{\omega}{a}\right)$ |
| Time Shifting |
$f(t-t_0)$ |
$e^{-j\omega t_0} F(\omega)$ |
| Frequency Shifting |
$e^{j\omega_0 t} . f (t)$ |
$F(\omega - \omega_0)$ |
| Time Reversal |
$f (-t)$ |
$F(-\omega)$ |
| Time Multiplication |
${t^n}f(t)$ |
${j^n}\dfrac{d}{d{\omega ^n}}X(\omega )$ |
| Differential |
$\dfrac{df (t)}{dt}$ |
$j\omega . F(\omega)$ |
| Nth Differential |
$\dfrac{d^n f (t)} {dt^n }$ |
$(j \omega)^n . F(\omega)$ |
| Integration |
$\displaystyle \int_{ - \infty }^t {f(t )dt }$ |
$\dfrac{F(\omega )} {j\omega } + \pi F(0)\delta (\omega )$ |
| Multiplication |
$f_{1}(t). f_{2}(t)$ |
$\dfrac{1}{2 \pi} F_{1}(\omega)\star F_{2}(\omega)$ |
| Convolution |
$f_{1}(t) * f_{2}(t)$ |
$F_{1}(\omega) \cdot F_{2}(\omega)$ |