Question

The output sequence y[n] and the pulse response h[n] of a discrete system are given by,
$y\left[n\right]=\{\underset{\uparrow }{1},1,2,-1,3\}$
$h\left[n\right]=\{\underset{\uparrow }{1},2,3\}$
The input sequence x[n] is

• A. $\{\underset{\uparrow }{-1},1,1\}$
• B. $\{\underset{\uparrow }{1},-1,-1\}$
• C. $\{\underset{\uparrow }{1},-1,1\}$
• D. $\{\underset{\uparrow }{1},1,-1\}$

Answer 1

The correct option is C $\{\underset{\uparrow }{1},-1,1\}$

$H\left(z\right)=1+2z^{-1}+3z^{-2}$

$Y\left(z\right)=1+z^{-1}+2z^{-2}-z^{-3}+3z^{-4}$

$\frac{Y\left(z\right)}{X\left(z\right)}=H\left(z\right)$

$\Rightarrow \frac{Y\left(z\right)}{H\left(z\right)}=X\left(z\right)$

$1+2z^{-1}+3z^{-2})1+z^{-1}+2z^{-2}-z^{-3}+3z^{-4}(1-z^{-1}+z^{-1}$

$1+2z^{-1}+3z^{-2}$

$\overline{-z^{-1}-z^{-2}-z^{-3}+3z^{-4}}$

$\underset{–}{-z^{-1}-2z^{-2}-3z^{-3}}$

$z^{-2}+2z^{-3}+3z^{-4}$

$z^{-2}+2z^{-3}+3z^{-4}$

$\overline{\underset{–}{0}}$

$X\left(z\right)=1-z^{-1}+z^{-2}$

$x\left[n\right]=\{\underset{\uparrow }{1},-1,1\}$