Discrete$-$time system models:

$($a$)$ Let the output, y$[$n$],$ be the average of the current input, x$[$n$],$ and the most recent past four values of the

input. Write a linear, constant coefficient difference equation that models this input$/$output behavior.

$($b$)$ Express your answer by specifying the parameters $\{$M$,$ N$,$ b$0,...,$ bM $,$ a$0,...,$ aN $\}$ as defined on slide $1\mathrm{.}24.$

y$[$n$]=$

M$$

k$=0$

bkx$[$n $$ k$]$

N$$

k$=1$

aky$[$n $$ k$]$

$($c$)$ Draw a block diagram for the input$-$to$-$output relationship using only memory registers, multipliers and

adders $($see slide $1\mathrm{.}23$ for associated symbols$).$

$($d$)$ Write a difference equation for the AR$(2)$ model, in which the current output value is predicted as a

weighted combination of the previous two output values, plus a white noise input term. Express your

answer by specifying the parameters $\{$M$,$