Question

Consider a signal $x\left[n\right]={\left(\frac{1}{2}\right)}^{n}1\left[n\right]$, where $1\left[n\right]=0$ if $n<0$, and $1\left[n\right]=1$ if $n\ge 0$. The z-transform of x[n - k], k > 0 is $\frac{z^{-k}}{1-\frac{1}{2}{z}^{-1}}$ with region of convergence being

• A. $|z|<2$
• B. $|z|>2$
• C. $|z|<\frac{1}{2}$
• D. $|z|>\frac{1}{2}$

Answer 1

The correct option is D $|z|>\frac{1}{2}$

$x\left(n\right)={\left(\frac{1}{2}\right)}^{n}u\left(n\right)$
$ROCofx\left(n\right):\left|z\right|>\frac{1}{2}$

$x(n-k)\rightleftharpoons X\left(z\right)=\frac{z^{-k}}{1-\frac{1}{2}{z}^{-1}},$

$ROCofx(n-k):\left|z\right|>\frac{1}{2}$

$Forx(n-k)ROCwillbe\left|z\right|>\frac{1}{2}$