Inverse Fourier Transform | Sahithyan's Notes

Used to recover original function from its Fourier transform.

f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} \, \text{d}\omega

Provided that the integral converges.

Convergence Conditions

The inverse Fourier transform exists when $F(\omega)$ is absolutely integrable:

\int_{-\infty}^{\infty} |F(\omega)| \, \text{d}\omega < \infty

Properties

Linearity
$\mathcal{F}^{-1}\{aF(\omega) + bG(\omega)\} = af(t) + bg(t)$
Time shift
$\mathcal{F}^{-1}\{F(\omega) e^{-i\omega t_0}\} = f(t - t_0)$
Frequency shift
$\mathcal{F}^{-1}\{F(\omega - \omega_0)\} = e^{i\omega_0 t} f(t)$
Duality
$\mathcal{F}^{-1}\{F(t)\} = \frac{1}{2\pi} f(-\omega)$

Comparison with Inverse Laplace Transform

| Property | Inverse Fourier | Inverse Laplace | | ----------- | ------------------------------- | --------------------------------------------- | --- | ----------- | | Integration | Along the real $\omega$ -axis | Along a vertical line $\text{Re}(s) = c$ | | Domain | Bilateral (all $t$ ) | Unilateral ( $t \geq 0$ ) | | Convergence | Requires absolute integrability | Requires $F(s)$ to decay as $| s | \to \infty$ | | Typical use | Signal analysis, filters | System response, ODEs with initial conditions |