I can't wrap my mind around the concept of filtering and the unit circle ? any help?
BIBO stability of LTI systems implies that their impulse response is absolutely summable, that is, 
\begin{equation}
\sum_{n=-\infty}^{+\infty}|h(n)| &lt; +\infty
\end{equation}

That exact same relationship is a sufficient condition for the Fourier Transform of the impulse response - the so-called Frequency Response - to converge.

Convergence of the frequency response means that it takes finite values, that is, 
$$|H(e^{j\omega})|&lt;+\infty \Longrightarrow \sum_{n=-\infty}^{+\infty}|h(n)|&lt;+\infty$$

The frequency response is the evaluation of the Z-Transform on the unit circle: $$H(e^{j\omega})= H(z)\Big|_{z=e^{j\omega}}$$

Moreover, a pole is a point on the complex plane where the Z-Transform's magnitude $|H(z)|$ takes an infinite value: if $z=z_p$ is a pole, then $|H(z_p)| = +\infty$.
Given that the Fourier Transform is a "version" of the Z-Transform on the unit circle, if there is a pole on the unit circle, the frequency response does not converge. If it does not converge, the system is not BIBO-stable. 

The last two statements hold because 
\begin{equation}
\sum_{n=-\infty}^{+\infty}|h(n)| &lt; +\infty
\end{equation}
does not hold.