Question

$\text{A cascade system having the impulse responses}$
$h_1\left(n\right)\text{={1, - 1} and}{h}_2\left(n\right)\text{= {1, 1}}$
$\uparrow$ $\uparrow$
$\text{is shown in the figure below, where symbol}\uparrow \text{denotes the time origin.}$

$\text{The input sequence x(n) for which the}$
$\text{cascade system produces an output sequence}$
$\text{y(n) = {1, 2, 1, -1, -2, -1} is}$
$\uparrow$

Answer 1

Now, h(n) = overall system impulse response
$=h_1\left(n\right)\ast h_2\left(n\right)$
$\text{= {1, -1}*{1, 1}}$
$\uparrow$ $\uparrow$
$\text{={1, 0, -1}}\rightleftharpoons H\left(z\right)=1-z^{-2}$
$\uparrow$
As we know,
$H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}\Rightarrow X\left(z\right)=\frac{Y\left(z\right)}{H\left(z\right)}$

$=\frac{1+2z^{-1}+z^{-2}-z^{-3}-2z^{-4}-z^{-5}}{1-z^{-2}}$

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

$\Rightarrow$$\text{x(n)={ 1, 2, 2, 1}}$
$\uparrow$