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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

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