Consider a discrete time signal given by x[n] = (

Consider a discrete time signal given by
x[n] = (– 0.25)ⁿ u[n] + (0.5)ⁿ u[ – n – 1]
The region of convergence of its Z-transform would be

1. the region inside the circle of radius 0.5 and centered at origin.
2. the region outside the circle of radius 0.25 and centered at origin.
3. the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.
4. the entire Z plane.

x[n] = (– 0.25)ⁿ u[n] + (0.5)ⁿ u[– n – 1] = x₁ [n] + x₂[n]
x₁ [n] = (– 0.25)ⁿ u[n] (+ve sided sequence)

x₁[z] =	z	=	z
	z - ( -0.25)		z + 0.25

ROC₁ : |z| > |– 0.25|
|z| > 0.25
x₂ [n] = (0.5)ⁿ u[– n – 1] (– ve sided sequence)

x₂(z) =	z
	z - 0.5

ROC₂ :
| z| < | 0.5|
| z| < 0.5
Therefore required ROC for given signal
0.5 > | | > 0.25