Question

If the z-transform of a discrete time signal x[n] is denoted as X(z), then the z-transform of x[n - 2] and x[n/2] will be, respectively,
1. z^-2 X(z), 2X (2z)
2. z² X(z), X(2z)
3. X (z - 2), X(z/2)
4. z^-2 X(z), X(z²)

Answer 1

Correct Answer - Option 4 : z^-2 X(z), X(z²)

Concept:

The z-transform of a unit impulse function x[n] = δ [n] is given as:

X(z) = 1

Also, the time-shifting affects the z-transform as:

x[n - n0] ↔ z -n0 X(z)

Scaling:

x[n/a] ↔ X(z^a)

Calculation:

Given that,

Z transform of x[n] is X(z)

⇒Z transform of x[n - 2] = z^-2 X(z)

And, Z transform of x[n/2]

⇒ x[n/2] ↔ X[z²]