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

Concept: The Z-transform of a discrete-time signal $x[n]$ is defined as: \[ X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \]
Step 1: Unit impulse definition.
\[ \delta[n] = \begin{cases} 1, & n = 0
0, & n \neq 0 \end{cases} \]
Step 2: Apply Z-transform.
\[ X(z) = \sum_{n=-\infty}^{\infty} \delta[n] z^{-n} \] Only $n=0$ contributes: \[ X(z) = z^{0} = 1 \]
Step 3: Conclusion.
Thus, the Z-transform of $\delta[n]$ is 1.