Stability cont.

• The transfer function
  $$H(z) = \frac{1}{1 - Re^{j\omega T}z^{-1}}$$
  has a single pole at $Re^{j\omega T}$.

• If $R > 1$ the pole of $H(z)$ moves outside the unit circle, and the impulse response $h(n)$ has an exponentially increasing amplitude ( $\vert h(n)\vert = R^n$). That is, it is unstable.

• A pole lying on the unit circle is considered marginally stable, that is, it neither decays nor grows in amplitude. In physically modelling synthesis, marginally stable poles often occur in lossless systems such as ideal vibrating strings.

• Therefore, to verify stability, we may find the roots of the denominator polynomial and ensure their roots are less than 1.