Exercise: Properties of linear discrete time systems

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Consider the following discrete time system

math

where

math

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.

  1. 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.

  2. 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.

  3. 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

    math