$[$Adapted from M&I Textbook, Ch$.2,$ Problem $24($Modified$)].$

An LTI system whose output signal $y[n]$ depends only on current and past input samples $x[n]$

and does not depend on any past output samples $y[n]$ has difference equation of the form

$y[n]=b_{0}x[n]+b_{1}x[n-1]+b_{2}x[n-2]+$dots$+b_{M}x[n-M]={\displaystyle \underset{k=0}{\overset{M}{}}}{b}_{k}x[n-k]$

where $b_k$ are constant coefficients. An example of such system is the one that generates at any

time $n$ an output $y[n]$ that is the average of the $M+1$ successive input samples $($called moving

average filter$),$ that is$,y[n]=\frac{1}{M+1}{\displaystyle \underset{k=0}{\overset{M}{}}}x[n-k].$ By definition, the system's impulse response

$h[n]=y[n]$ when $x[n]=[n],$ that is$,h[n]=\frac{1}{M+1}{\displaystyle \underset{k=0}{\overset{M}{}}}[n-k]($see Example $2\mathrm{.}7$ on pp$.$

$67-68$ of the Textbook$).$

i$)$ For the case $M=4,$ write down the system difference equation and sketch the corresponding

impulse response $h[n].$ Is this system causal? Is it stable? State reasons for your answers.

ii$)$ For the case $M=50,$ complete parts $($a$)$ and $($b$)$ of Problem $24,$ Ch$.2$ of the textbook for $y_1[n]$

only $($skip $y_2[n]).$ The referenced data file djw$6576.$txt is posted on Canvas under the $9/3/2024$

date. You will need to download that file in your working directory. Use the Matlab command

'load' in your program to load the file in memory.

Include your program, properly labeled output plots, and type any comments on the answer in

your solutions file. Do not use screen shots. Copy and paste the program text and the figures

$($select the Copy Figure option under Edit at the top of a Matlab figure window$)$ in an MS Word

doc or Google doc or any other formatted document generating program you are familiar with to

generate clean and legible solution document..