Question

If $x\left(n\right)$ is right sided discrete time signal and $X\left(z\right)$ is z-transform of $x\left(n\right)$. If $X\left(z\right)=\frac{z}{z-0.1}$, then value of $\sum _{n=-\infty }^{\infty }nx\left(n\right)$ is ______.

• A. 0.12

Answer 1

The correct option is A 0.12
Given that $x\left(n\right)⟷X\left(Z\right)$

and $X\left(z\right)=\frac{z}{z-0.1}$

and $nx\left(n\right)\stackrel{Z.T}{⟷}\frac{-ZdX\left(z\right)}{dz}$

$nx\left(n\right)\stackrel{Z.T}{⟷}\frac{0.1z}{(z-0.1{)}^2}$

So $\frac{0.1z}{(z-0.1{)}^2}=\sum _{n=-\infty }^{\infty }nx\left(n\right)Z^{-n}$

Putting z=1 in above equation we get

$\sum _{n=-\infty }^{\infty }nx\left(n\right)=\frac{0.1}{(1-0.1{)}^2}=\frac{0.1}{0.81}=0.1234$