Consider a second$-$order discrete$-$time system described by the difference equation

$y(k+2)-\frac{3}{2}y(k+1)+\frac{1}{2}y(k)=r(k+1)+\frac{1}{2}r(k)$

The system is initially relaxed for $k<mn\; id="EditorElement\_6712212">0</mn>$ and is excited by the input

$r(k)=\{\begin{array}{cc}0\text{;},& k=0\\ 1\text{;},& k>0\end{array}$