Question

Signal x(n) whose z-transform is given by $X\left(z\right)=log(1-2z^{-1})$ $|z|>2$, then the value of x[2] is ______

• A. -2

Answer 1

The correct option is A -2

$X\left(z\right)=log(1-2z^{-1})$ $|z|>2$

$\frac{dX\left(z\right)}{dz}=\frac{2z^{-2}}{1-2z^{-1}}$

$\frac{-zdX\left(z\right)}{dz}=\frac{-2z^{-1}}{1-2z^{-1}}$

From differentiation property,

$z\left[nx\right(n\left)\right]=\frac{-zdX\left(z\right)}{dz}$

$nx\left[n\right]=z^{-1}[-z\frac{dX\left(z\right)}{dz}]=z^{-1}\left[\frac{-2z^{-1}}{1-2z^{-1}}\right]$

$\therefore nx\left[n\right]=-2\left[2^{n-1}u\right[n-1\left]\right]$

$\therefore x\left[n\right]=\frac{-2^{n}u[n-1]}{n}$

$\therefore x\left[2\right]=\frac{-2^2}{2}=-2$