Question

Consider a discrete time signal given by $x\left[n\right]=(-0.25{)}^n+(0.5{)}^{n}u[-n-1]$. The region of convergence of its Z-transform would be

• A. the region inside the circle of radius 0.5 and centered at origin.
• B.
  the entire Z-plane.
• C. the region outside the circle of radius 0.25 and centered at origin.
• D. the annular region between the two circles both centered at origin and having radii 0.25 and 0.5.

Answer 1

The correct option is D the annular region between the two circles both centered at origin and having radii 0.25 and 0.5.

$x\left[n\right]=(-0.25{)}^{n}u(n)+(0.5{)}^{n}u(-n-1)$ signal x[n] is sum of two signals, one is right sided $\left[\right(0.25{)}^{n}u\left(n\right)]$ and other is left sided $\left[\right(0.5{)}^{n}u(-n-1)]$. The right sided signal will have pole at location with magnitude 0.25. So, ROC is $|z|>\mathrm{0.25.}$ The left sided signal will have pole at location with magnitude 0.5.
$\text{So, ROC is |z| < 0.5.}$

$\text{So, ROC of X(z) (Z transform of x(n) will be)}$
$0.25<|z|<\mathrm{0.5.}$