Exercise: Properties of linear discrete time systems
Consider the following discrete time system
where
In this problem, we will explore some of the properties of this discrete time system as a function of the parameters, the initial conditions, and the inputs.
Assume that the off diagonal element
and that there is
no input, 
. Write a closed form expression for the output of the
system from a nonzero initial condition 
and give
conditions on 
and 
under which the output gets smaller as
gets larger.
Now assume that
and write a closed form expression
for the response of the system from a nonzero initial conditions. Given a
condition on the elements of 
under which the output gets smaller as
gets larger.
Write a MATLAB program to plot the output of the system in response to a unit step input,
, 
. Plot the response of your system
with 
and given by
