Worked Examples

WORKED EXAMPLES

(1) Find the Z-transform and region of convergence of

Hence Z [u (n)] is convergent if z lies between the annulus as shown shaded in figure. Hence, ROC is 2 <z < 4.

(2) Find the Z-transform and region of convergence of

u (n) = ⁿC_k, n ≥k

This series is convergence for | 1/z | < 1 i.e., for z> 1.

Hence ROC is ❘z❘> 1.

(3) Find Z-1 {(z − 5)⁻³] when ❘z❘ > 5. Determine the region of  convergence.

Solution :

The region of convergence is the exterior of the circle |z❘ = 5 i.e., with centre at origin and of radius 5.

(4) Find the Z-transform and the radius of convergence of u (n) = 2ⁿ, n < 0

Solution :

Let u (n) = 0 for n ≥ 0 we have

(5) Find the Z-transform and the radius of convergence of

Solution :

The above series is convergent for all values of z.

Hence, ROC is the entire z-plane.