I$\text{'}$m plotting a fourier series graph. Below are my matlab codings. But they seem so stupid $($hoewever I still achive the things I want, Just the code is so stupid that it takes $3$ for loop and $3$ if$,$ how can i modify it so that it only take $1$ loop and $3$ if$,$ to make it more efficient, while producing the same result $($my graph$)?$

$\%$BEJ$20203$ Pre$-$project Task $2$

T$0=1$;

tau $=0\mathrm{.}5$;

t $=-10$:$0\mathrm{.}01$:$10$;

figure;

$\%$ Define the function for coefficient x$\_$n

$\%$Use @$($n$)$ because later I can sub n with something using x$\_$n$($n$)$

Coefficient$\_$of$\_$individual$\_$term $=$ @$($n$)($tau $/$ T$0)*$ sinc$($n $*$ tau $/$ T$0)$;

$\%$If don't understand, gemini answer me better than the matlab documentation

$\%$Method $2,$ directly open a new matlab to test to see what I get,

$\%$ then I $\%$would understand

Final$\_$x$\_$t $=$ zeros$($size$($t$))$;

$\%$n $=+-3$

for n $=-20$:$20$; $\%$Later ask, how to implement using $1$ for, and $3$ if$?$

$\%$because this loop would sub in n $=0,$

$\%$ this would cause incorrectness!

$\%$ i want it to skip when n $=0$

$\%$So$,$ only if when the n is not $0,$ execute the following!

if abs$($n$)<=5$

Individual$\_$term$\_$of$\_$x$\_$t $=($tau $/$ T$0)*$ sinc$($n $*$ tau $/$ T$0)*$ exp$(1$j $*2*$ pi $*$ n $*$ t $/$ T$0)$;

Final$\_$x$\_$t $=$ Final$\_$x$\_$t $+$ Individual$\_$term$\_$of$\_$x$\_$t;

$\%$The plot function, put inside this $"$if$",$ or it would show nothing

$\%$subplot put first, before the plot function, or it would bug! $($Found

$\%$by trial and error, and deduction, just now the title won't show,

$\%$observation!

subplot$(4,1,1)$

plot$($t$,$ real$($Final$\_$x$\_$t$))$;

title$(\text{'}$x$($t$)$ for n $=0$ to $5\text{'})$;

xlabel$(\text{'}$Time $($s$)\text{'})$;

ylabel$(\text{'}$Amplitude$\text{'})$;

grid on;

end $\%$end the if statement

end $\%$end the for loop

$\%$n $=+-10$

$\%$Reset again the x$\_$t$,$ put here, or the following one, would produce

$\%$incorrect result, because the Final$\_$x$\_$t would have value already

Final$\_$x$\_$t $=$ zeros$($size$($t$))$;

for n $=-20$:$20$;

if abs$($n$)<=10\%$If don't want calculate n $=0,$ then put if n~$=0$

Individual$\_$term$\_$of$\_$x$\_$t $=($tau $/$ T$0)*$ sinc$($n $*$ tau $/$ T$0)*$ exp$(1$j $*2*$ pi $*$ n $*$ t $/$ T$0)$;

Final$\_$x$\_$t $=$ Final$\_$x$\_$t $+$ Individual$\_$term$\_$of$\_$x$\_$t;

subplot$(4,1,2)$

plot$($t$,$ real$($Final$\_$x$\_$t$))$;

title$(\text{'}$x$($t$)$ for n $=0$ to $10\text{'})$;

xlabel$(\text{'}$Time $($s$)\text{'})$;

ylabel$(\text{'}$Amplitude$\text{'})$;

grid on;

end

end

$\%$n $=+-20$

Final$\_$x$\_$t $=$ zeros$($size$($t$))$;

for n $=-20$:$20$;

if abs$($n$)<=20$

Individual$\_$term$\_$of$\_$x$\_$t $=$ Coefficient$\_$of$\_$individual$\_$term$($n$)*$ exp$(1$j $*2*$ pi $*$ n $*$ t $/$ T$0)$;

Final$\_$x$\_$t $=$ Final$\_$x$\_$t $+$ Individual$\_$term$\_$of$\_$x$\_$t;

subplot$(4,1,3)$

plot$($t$,$ real$($Final$\_$x$\_$t$))$;

title$(\text{'}$x$($t$)$ for n $=0$ to $20\text{'})$;

xlabel$(\text{'}$Time $($s$)\text{'})$;

ylabel$(\text{'}$Amplitude$\text{'})$;

grid on;

end

end

$\%$ How to plot the coefficients x$\_$n on the fourth row?

subplot$(4,1,4)$;

$\%$How to declare n $=-20,-19,...+20?$ Use mass declaration, $\%$mass declaration, of n values,

$\%$ and very fast declaration, from n $=-20,$ to $20,$ this would become an array

n$\_$values $=-20$:$20$;

$\%$Apply the function, Coefficient$\_$of$\_$individual$\_$term, on every elements of the n$\_$values, which is a array

$\%$Or in other words, sub in all the n$\_$values$=-20$ : $20,$ into the function of Coefficient$\_$of$\_$individual$\_$term

$\%$It's mass substitution of element array into the function $,$ so use arrayfun, instead of sub

$\%$But why? And the outcome is what? although this is intermediate, still need the outcome!

x$\_$n$\_$values $=$ arrayfun$($Coefficient$\_$of$\_$individual$\_$term, n$\_$values$)$;

$\%$How to do discrete plotting? Use stemplot, only got bubble,

stem$($n$\_$values, abs$($x$\_$n$\_$values$))$; $\%$Do they need all to be absolute value according to theory?

title$(\text{'}$Magnitude of Coefficients X$\_$n$\text{'})$;

xlabel$(\text{'}$n$\text{'})$;

ylabel$(\text{'}|$X$\_$n$|\text{'})$;

grid on;

$\%\%$Actually below don't need!

$\%\%$Apply if$-$else for conditional coefficient plotting

$\%$ for idx $=1$:length$($n$\_$values$)$

$\%$ n $=$ n$\_$values$($idx$)$;

$\%$ if abs$($n$)<=20$

$\%\%$ Only plot coefficients for n within range

$\%$ coeff$\_$value $=$ abs$($Coefficient$\_$of$\_$individual$\_$term$($n$))$;

$\%$ else

$\%$ coeff$\_$value $=0$; $\%$ Default or out$-$of$-$range handling

$\%$ end

$\%$ end