Fourier Transform

$F(\omega)$ is a continuous function of $\omega$ called the spectrum of $f(t)$.

Used for aperiodic signals. Applied in telecommunications and transient circuit analysis where circuit parameters are frequency-dependent.

Inverse Transform

Properties

• Linearity
  $\mathcal{F}\{af(t) + bg(t)\} = aF(\omega) + bG(\omega)$

• Time shift
  $\mathcal{F}\{f(t - t_0)\} = e^{-j\omega t_0} F(\omega)$

• Differentiation
  $\mathcal{F}\{f'(t)\} = j\omega\, F(\omega)$

• Time scaling
  $\mathcal{F}\{f(at)\} = \dfrac{1}{|a|} F\!\left(\dfrac{\omega}{a}\right)$

• Convolution
  $\mathcal{F}\{(f * g)(t)\} = 2\pi\, F(\omega)\, G(\omega)$