Question

Let $x\left(n\right)=4^{n}u\left(n\right)-b^{2n}u(-n-1)$
If the 'z' transform of x(n) exists, then the condition on b is

• A. $b<2$
• B. $-2<b<2$
• C. No value of 'b' is possible
• D. $b>2$

Answer 1

The correct option is A $b<2$

$x\left(n\right)=4^{n}u\left(n\right)-b^{2n}u(-n-1)$

$x_1\left(n\right)=4^{n}u\left(n\right)$

$X_1\left(z\right)=\frac{1}{1-4z^{-1};}\left|z\right|>4$

$X_2\left(n\right)=-b^{2n}u(-n-1)$

$X_2\left(Z\right)=\frac{1}{1-b^2{z}^{-1}};\left|z\right|<b^2$

Thus for 'z' transform to exist

$4<\left|z\right|<b^2$

$\Rightarrow b^2>4$

$\left|b\right|>2$