In communication, x$[$k$]$ may represent the message sequence in a binary form being transmitted to another location. In Matlab, you can simulate the coin tossing experiment with a random number generator command rand. If you type rand in Matlab command window, it will return a random number within $0$ and $1.$

$1.$ Find a way to generate a random sequence x$[$k$]$ taking values either $1$ or $1$ and count the ratio between the number of $1$s and the number of $(1)$s in a sequence of length $105.$ You may find sign$($rand$\mathrm{.}5)$ helpful.

Now consider a system with input x$[$k$]$ and output y$[$k$]$ being related by

y$[$k$]=$ay$[$k$1]+$x$[$k$],$y$[1]=0$y$[$k$]=$ay$[$k$1]+$x$[$k$],$y$[1]=0(1)$

$2.$ Compute the unit impulse response and the unit step response for a$=0\mathrm{.}5$a$=0\mathrm{.}5$ and plot them.

$3.$ Find the system transfer function and decide the range of aa that the system is stable.

When a random sequence x$[$k$]$ is the input, the output y$[$k$]$ is also random.

$4.$ If the input sequence x$[$k$]$ can only be $1$ or $1,$ determine the range of y$[$k$]$ as a function of aa for the stable system given by $(1).$