GATE 2015 EE – SET 1 Question 35

Solution :

The region of Convergence (R_{0}C) in the z plane is determined  by  $$\left | z \right |$$= $$\left | e^{s} \right |\Rightarrow e^{\sigma} $$ ( a circle)

General relations

For $$x\left [ x \right ]$$ = $$a^{n}u\left [ n \right ] ROC \rightarrow \left | z \right |> \left | a \right |$$

$$x\left [ x \right ] $$ = $$ -a^{n}u\left [- n-1 \right ] ROC \rightarrow \left | z \right |< \left | a \right | $$

$$ x\left [ x \right ]$$=$$(-0.25)^u{n}$$ u\left [ n \right ] + $$(0.5)^{n} $$u [-n-1]

$$\left | Z \right |> \left | -0.25 \right | \left | z \right |< \left | 0.5 \right |$$

ROC = $$ROC_{1}\bigcap ROC_{2}$$

0.25 < $$\left | z \right |$$< 0.5