BIBO stability

A system is bounded-in bounded-out (BIBO) stable if a bounded input will result in a bounded output.

A system with unit sample response $h[n]$ will be BIBO stable if and only if $h[n]$ is absolutely summable. That is:

$$ \sum_{n=-\infty}^{\infty} |h[n]| < \infty $$

A BIBO stable system is guaranteed to have a Fourier transform, since:

$$ |H(j\omega)| = |\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt| \leq \int_{-\infty}^{\infty} |h(t)| dt < \infty $$

We can use this property to conclude that the region of convergence of any BIBO stable system must include the Fourier transform. In the z-domain, the Fourier transform corresponds to a unit circle in the complex z-plane, and in the Laplace domain, the Fourier transform corresponds to the imaginary axis.