Question

If $X(z) = \frac{1}{1 - z^{-1}}$ with $|z|>1$, then what is the corresponding $x(n)$?

• A. $e^n$
• B. $e^{n^2}$
• C. $u(n)$
• D. $\delta(n)$

Hint:

Memorize standard Z-transform pairs like $u(n) \leftrightarrow \frac{1}{1 - z^{-1}}$, and match the ROC.

Answer 1

The Correct Option is C

Step 1: Recognize the standard Z-transform pair.
We know: \[ Z\{u(n)\} = \sum_{n=0}^{\infty} z^{-n} = \frac{1}{1 - z^{-1}}, \text{for } |z|>1 \] Step 2: Compare given expression.
Given $X(z) = \frac{1}{1 - z^{-1}}$ matches the Z-transform of unit step function $u(n)$.
Conclusion: $\boxed{x(n) = u(n)}$