All answers are given in MATLAB code, for each part provide screenshots of your MATLAB code, Figrues$/$Command window output, to show the code works. You are provided a demo Z$-$transform MATLAB file, you MUST use it$,$ make chnages accordingly to it based on the requirement of the question. The demo code is provided below:

clear

clc

sympref$(\text{'}$HeavisideAtOrigin$\text{'},1)$;

syms z n;

$\%$ function to use

xn $=$ heaviside$($n$)+2^$n$*$heaviside$($n$)$

xn $=3*(1/2)^$n$*$heaviside$($n$)-5*(1/3)^$n$*$heaviside$($n$)$

$\%\%$ z transform

XZ $=$ ztrans$($xn$)$

$\%$ Extract numerator and denominator

$[$num$,$ den$]=$ numden$($XZ$)$;

rational$\_$form $=$ num$/$den $\%$ rational form

$\%\%$ partial fraction expansion

$\%$ numerator and denominator to polynomial form

b $=$ sym$2$poly$($num$)$;

a $=$ sym$2$poly$($den$)$;

$\%$ partial$-$fraction expansion coefficient

$[$r$,$p$,$k$]=$ residue$($b$,$a$)$

$\%$ build partial fraction expansion form

$\%$ f$($n$)=$ sum$($ r$($i$)/($z$-$p$($i$)))+$ k

$\%$ k is polynomial of z

f $=0$;

for i $=1$:size$($r$,1)\%$ column vector

f $=$ f $+$ r$($i$)/($z $-$ p$($i$))$;

end

t $=$ size$($k$,2)$; $\%$ row vector

for i $=1$:t

f $=$ f $+$ k$($i$)*$n$^($t$-$i$)$;

end

PFE $=$ vpa$($f$,4)\%$ display partial fraction expansion form

$\%\%$ inverse z transform

$\%$ with partial fraction method

fn $=0$;

for i $=1$:size$($r$,1)$

fi $=$ iztrans$($r$($i$)/($z $-$ p$($i$)))$; $\%$ inverse z transform for each term

fn $=$ fn $+$ fi;

end

$\%$ build k term

fi $=0$;

t $=$ size$($k$,2)$;

for i $=1$:t

fi $=$ fi $+$ k$($i$)*$n$^($t$-$i$)$;

end

ifi $=$ iztrans$($fi$)$; $\%$ inverse z transform for k term

fn $=$ fn $+$ ifi;

$\%$ compare with build$-$in method

xn$\_$PFE $=$ vpa$($fn$,4)\%$ izt by PFE

xn$\_$izt $=$ iztrans$($XZ$)$

sympref$(\text{'}$HeavisideAtOrigin$\text{'},$'default'$)$;