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

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

• A. The final steady-state value as n→∞.
• B. The initial value x[0] to the limit of X(z) as z→∞.
• C. The average value of the sequence.
• D. The radius of convergence of X(z).

Answer: (B)

The initial value in the Z-transform sense is read from the high-frequency behavior of the transform. If X(z) = x[0] + x[1] z^{-1} + x[2] z^{-2} + ..., then as z grows large in magnitude, the terms with z^{-n} for n ≥ 1 become negligible, and X(z) approaches x[0]. Therefore x[0] = lim_{z→∞} X(z). This shows how the first sample of the sequence is encoded directly in the limit of the transform as z goes to infinity, assuming the limit exists and the series converges.

This is not about the final steady state, which involves n → ∞ and a different relation, nor about the average value or the radius of convergence, which describe other aspects of the transform.