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 in a discrete-time sequence is tied to how its Z-transform behaves near z = 1. If the sequence settles to a finite constant, the long-term value can be obtained by taking a limit of a suitable function of X(z) as z approaches 1. In the form used here, that relationship is expressed as the limit of z X(z) divided by (z − 1) as z → 1. This limit isolates the steady-state contribution from the transform, because it effectively cancels the part of X(z) that corresponds to transient behavior and leaves the constant final value.

In other words, the final steady-state value is read off directly from how X(z) behaves around z = 1, provided the system is stable and the unit circle lies in the region of convergence. The other options describe different notions (initial value, total sum, or a derivative-like quantity) that do not capture the long-term steady-state behavior.