Question

The ROC of the given signal $x\left[n\right]=2^{-nsgn\left[n\right]}$ is

• A. $|z|>2$
• B. $|z|>\frac{1}{2}$
• C. $0.5<|z|<2$
• D. No common ROC

Answer 1

The correct option is C $0.5<|z|<2$

$x\left[n\right]=\{\begin{array}{ccc}{2}^n& n<0& [\because \text{signum function is (-1) for}n<0]\\ 2^{-n}& n>0& [\because \text{signum function is (1) for}n>0]\\ 1& n=0& \end{array}$

$x\left[n\right]=\underset{\text{Entire z plane}}{\underset{}{\delta \left(n\right)}}+\underset{|z|<2}{\underset{}{2^{n}u(-n-1)}}+\underset{|z|>0.5}{\underset{}{\left(2^{-1}{)}^{n}u\right(-n-1)}}$

$\therefore ROC:0.5<|z|<2$