Question

The Z-transform of \(\delta(n)\) is

• A. \(0\)
• B. \(1\)
• C. \(z^{-1}\)
• D. \(\dfrac{1}{1-z^{-1}}\)

Hint:

Important Z-transforms: \[ \boxed{ \delta(n)\;\longleftrightarrow\;1 } \] \[ \boxed{ u(n)\;\longleftrightarrow\;\frac{1}{1-z^{-1}} } \]

Answer 1

The Correct Option is B

Step 1: Recall the definition of the Z-transform. The Z-transform is \[ X(z)=\sum_{n=-\infty}^{\infty}x(n)z^{-n}. \] For \[ x(n)=\delta(n), \] only the term at \[ n=0 \] is non-zero.

Step 2: Evaluate the summation. \[ X(z) = \delta(0)z^{0} = 1. \] Hence, \[ \boxed{1} \] Therefore, \[ \boxed{(B)} \] is the correct answer.