In the z-transform context, what does the final value theorem relate?

In the z-transform context, what does the final value theorem relate?

• A. The final steady-state value of x[n] as n→∞ to the limit of z X(z)/(z-1) as z→1.
• B. The initial value x[0].
• C. The sum of all samples.
• D. The derivative of the sequence at infinity.

Answer: (A)

The final value theorem shows how the long-run (steady-state) value of a discrete-time sequence x[n] can be read directly from its Z-transform X(z) by looking at a limit as z approaches 1. When x[n] converges to a constant L and the ROC is appropriate (the system is stable and no problematic poles block the unit circle), the constant L is obtained from the limit of z X(z)/(z − 1) as z → 1. The idea is that near z = 1, X(z) has a behavior dominated by the tail x[n] → L, and the factor z/(z − 1) isolates that tail value as z approaches 1. This is the standard way to extract the final steady-state value from the transform, provided the usual conditions hold.

The final value theorem shows how the long-run (steady-state) value of a discrete-time sequence x[n] can be read directly from its Z-transform X(z) by looking at a limit as z approaches 1. When x[n] converges to a constant L and the ROC is appropriate (the system is stable and no problematic poles block the unit circle), the constant L is obtained from the limit of z X(z)/(z − 1) as z → 1. The idea is that near z = 1, X(z) has a behavior dominated by the tail x[n] → L, and the factor z/(z − 1) isolates that tail value as z approaches 1. This is the standard way to extract the final steady-state value from the transform, provided the usual conditions hold.