The mapping between Z and complex frequency

We are familiar with the fact that real values of $\omega$correspond to complex values of $Z=e^{i\omega}$.Now let us look at complex values of $\omega$:

$$Z \eq \Re Z + i \Im Z \eq e^{i(\Re \omega + i\Im \omega)} \e... ...mega}\ e^{i\Re \omega} \eq {\rm amplitude}\ e ^ {i {\rm phase}}$$ (44)

Thus, when $\Im\omega\gt$, |Z|<1. In words, we transform the upper half of the $\omega$-plane to the interior of the unit circle in the Z-plane. Likewise, the stable region for poles is the lower half of the $\omega$-plane, which is the exterior of the unit circle. Figure 12 shows the transformation. Some engineering books choose a different sign convention ($Z=e^{-i\omega}$), but I selected the sign convention of physics.

 

Z
Figure 12 Left is the complex $\omega$-plane with axes $(x,y)=(\Re\omega_0,\Im\omega_0)$. Right is the Z-plane with axes $(x,y)=(\Re Z_0,\Im Z_0)$. The words ``Convergent'' and ``Divergent'' are transformed by $Z=e^{i\omega}$.