Question

A sequence x(n) with the z-transform $X\left(z\right)=z^4+z^2-2z+2-3z^{-4}$ is applied as an input to a linear, time invariant system with the impulse response $h\left(n\right)=2\delta (n-3)$ where
$\delta \left(n\right)=$ $\{\begin{array}{cc}1,& n=0\\ 0,& otherwise\end{array}$

The output at n = 4 is

• A.
  2
• B.
  zero
• C.
  - 6
• D.
  - 4

Answer 1

The correct option is B

zero
$Y\left(z\right)=H\left(z\right)X\left(z\right)$

$H\left(z\right)=2z^{-3}$

$\therefore Y\left(z\right)=2z^{-3}(z^4+z^2-2z+2-3z^{-4})$

$=2(z+z^{-1}-2z^{-2}+2z^{-3}-3z^{-7})$

Taking inverse of z-transform

$y\left(n\right)=2\left[\delta \right(n+1)+\delta (n-1)-2\delta (n-2)+2\delta (n-3)-3\delta (n-7\left)\right]$

$\text{At n = 4, y(4) = 0}$