Question

Consider the z-transform G(z)=(z²+0.2z+0.1)(z²−z+0.5)​/(z²+0.3z−0.18)(z²−2z+4) There are four possible nonoverlapping regions of convergence (ROCs) of this z-transform. Discuss the type of inverse z-transform (left-sided, right-sided, or two-sided sequences) associated with each of the four ROCs. It is not necessary to compute the exact inverse transform.

Answer 1

The z-transform G(z) has four possible ROCs which determine the type of the inverse z-transform: right-sided, left-sided, two-sided, or unstable (entire z-plane excluding poles).

Step-by-step explanation:

The z-transform G(z) given by the expression (z²+0.2z+0.1)(z²−z+0.5)/(z²+0.3z−0.18)(z²−2z+4) has associated regions of convergence (ROCs) which define the type of the inverse z-transform. The ROCs are nonoverlapping and based on the location of the poles of G(z), four possible types of inverse z-transform can be identified:

• Right-sided sequence: The ROC is outside the outermost pole on the z-plane.
• Left-sided sequence: The ROC is inside the innermost pole on the z-plane.
• Two-sided sequence: The ROC is between two poles on the z-plane.
• A fourth potential ROC, which does not result in a stable system, could be the entire z-plane excluding the poles.

For stable and causal systems, the ROC is typically to the right of the rightmost pole. However, we do not compute the exact inverse in this discussion.