Consider the four dynamical systems described by the following difference equations:

$y(n)=-y(n-1)+(n)$

$y(n+1)=-y(n)-0\mathrm{.}5y(n-1)+(n)+(n-1)$

$y(n+1)=2y(n)-y(n-1)$

$y(n)=2(n)+(n-1)+0\mathrm{.}5(n-2)$

In the above, $(k)=1$ if $k=0$;$0$ otherwise.

$($a$)$ For the four systems find the response $y(k)$ for $k=0,1,2,$dots,$10.$ You may need initial conditions; if so specify for which values of $k$ initial conditions are needed and set all required initial conditions to $(k+1).$ Plot the responses $y(k)$ you obtain.

$($b$)$ Using the responses and plots from $($a$)$ determine which systems are asymptotically stable, critically stable and unstable.

$($c$)$ Which systems are of the MA $($Moving Average$)$ type and which ones of the AR $($Autoregressive$)$ type? Notice that in general a system can be of both types, i$.$e$.$ ARMA.