Question

What is the Z-transform of a unit impulse function $\delta[n]$?

• A. $z$
• B. $\dfrac{1}{z}$
• C. $1$
• D. $\dfrac{1}{1-z^{-1}}$

Hint:

$\delta[n]$ → Z-transform = 1 → Only value at $n=0$ contributes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Z-transform of a discrete-time signal $x[n]$ is defined as the sum of $x[n] z^{-n}$ over all $n$.
Step 2: Key Formula or Approach:
\[ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \]
Step 3: Detailed Explanation:
The unit impulse function $\delta[n]$ is defined as $1$ only at $n = 0$ and $0$ elsewhere.
Applying the transform:
\[ X(z) = \delta[0] z^{-0} + \delta[1] z^{-1} + \dots \]
\[ X(z) = 1 \cdot 1 + 0 + 0 \dots = 1 \]
Step 4: Final Answer:
The Z-transform of $\delta[n]$ is $1$.