Why is my input and output waveform not same? One is from matlab using Fourier series approximation for input, one is from multisim using resistor and capacitor as low pass. Is my matlab coding wrong? Or is my multisim wrong? Below shows my matlab coding. Please teach me$.$ Thank you. Below is my matlab coding:

f$0=1000$e$3$; $\%$ Fundamental frequency $($Hz$)$

T$0=1/$ f$0$; $\%$ Fundamental period

omega$0=2*$ pi $*$ f$0$; $\%$ Angular frequency

fc $=480$e$3$; $\%$ Cutoff frequency of the low$-$pass filter $($Hz$)$

$\%$Number of periods I want, so that i can easily manage the x$-$axis here,

$\%$make it show more periods or less periods

N$\_$T$0=2$;

$\%$Frequency domain display limit

Freq$\_$x$\_$axis $=10$e$6$;

$\%$ Time vector, x$-$axis, for time domain

t $=-$N$\_$T$0*$T$0$ : $1$e$-9$ : N$\_$T$0*$T$0$;

DC$\_$Component $=1/2$;

AC$\_$Component $=$ zeros$($size$($t$))$;

for n $=1$:$2$:$100$

bn $=2/($n $*$ pi$)$;

AC$\_$Component $=$ AC$\_$Component $+$ bn $*$ sin$($n $*$ omega$0*$ t$)$;

end

x$\_$t $=$ DC$\_$Component $+$ AC$\_$Component;

$\%$ Compute the Frequency Domain, X$($w$)$

$\%$Building a new axis, for frequency

N $=$ length$($t$)$; $\%$ Number of samples

$\%$t$(2)$ means $2$nd element of the array

Fs $=1/($t$(2)-$ t$(1))$; $\%$ Sampling frequency

$\%$The new x$-$axis array$/$vector is

f $=(-$N$/2$: $($N$/2)-1)*($Fs $/$ N$)$;

$\%$Get the input into frequency domain

X$\_$f $=$ fft$($x$\_$t$)$;

$\%$Shift the DC component to the center

X$\_$f $=$ fftshift$($X$\_$f$)$;

$\%$Normalize the X$\_$f

X$\_$f $=$ X$\_$f $/$ N;

$\%$The magnitude of Low Pass Filter Transfer function, in frequency domain

H$\_$f$\_$mag $=1./$ sqrt$(1+($f $/$ fc$).^2)$;

$\%$Calculate the output, filtered, in frequency domain or apply the filter to the frequency domain representation of x$($t$)$

Y$\_$f $=$ H$\_$f$\_$mag $.*($X$\_$f$)$; $\%$not abs$($x$\_$f$),$ cuz $20$khz should show about same graph

$\%$Convert the output to the time domain

$\%$By doing inverse fourier transform

y$\_$t $=$ ifft$($ifftshift$($Y$\_$f$))*$ N;

$\%---------------------------------------------------------------------------$

$\%--$All plots put together here to easily manage and modify all the plots$---$

$\%---------------------------------------------------------------------------$

figure;

$\%$Input, in the Time Domain

$\%$input of filter

plot$($t$,$ x$\_$t$,\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

hold on

$\%$output of filter

plot$($t$,$ abs$($y$\_$t$),\text{'}$b$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$(\text{'}$Fourier Series Approximation of input $($Time Domain$)\text{'})$;

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

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

$\%$ Set y$-$axis limits

clear ylim $\%$need this clear ylim or it bugged one, can't work as expected

ylim$([-0\mathrm{.}51\mathrm{.}25])$;

figure;

$\%$Input, in the Time Domain

subplot$(3,2,1)$;

$\%$input of filter

plot$($t$,$ x$\_$t$,\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

hold on

$\%$output of filter

plot$($t$,$ abs$($y$\_$t$),\text{'}$b$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$(\text{'}$Fourier Series Approximation of input $($Time Domain$)\text{'})$;

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

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

$\%$ Set y$-$axis limits

clear ylim $\%$need this clear ylim or it bugged one, can't work as expected

ylim$([-0\mathrm{.}51\mathrm{.}25])$;

$\%$Frequency domain, use the f as x axis

$\%$Input, in the Frequency Domain

subplot$(3,2,2)$;

plot$($f$,$ abs$($X$\_$f$),\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

$\%$Since making the x$-$axis has label kHz$,$ then the x$-$axis need to be divided

$\%$by $1$k$,$ but don't do this, because matlab there auto shows $10$e$5$ something,

$\%$then gets very messy!!!

title$(\text{'}$Fourier Series Approximation of magnitude of input $($Frequency Domain$)\text{'})$;

xlabel$(\text{'}$Frequency $($Hz$)\text{'})$;

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

xlim$([-$Freq$\_$x$\_$axis Freq$\_$x$\_$axis$])$;

$\%$ylim$([-1010])$;

grid on;

$\%$H$($f$),$ Low Pass Transfer function, in Frequency Domain

$\%$Compared to that sketched by desmos, the points, this is confirmed correct

subplot$(3,2,4)$;

plot$($f$,$ H$\_$f$\_$mag, $\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$(\text{'}|$H$($f$)|($Frequency Domain$)\text{'})$;

xlabel$(\text{'}$Frequency $($Hz$)\text{'})$;

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

xlim$([-$Freq$\_$x$\_$axis Freq$\_$x$\_$axis$])$;

grid on;

$\%$Filtered Signal in$,$ Time Domain

subplot$(3,2,5)$;

plot$($t$,$ abs$($y$\_$t$),\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$;

title$(\text{'}$Filtered Signal $($Time Domain$)\text{'})$;

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

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

clear ylim

ylim$([-0\mathrm{.}51\mathrm{.}25])$;

$\%$Filtered Signal in$,$ Frequency Domain

subplot$(3,2,6)$;

plot$($f$,$ abs$($Y$\_$f$),\text{'}$k$\text{'},$ 'LineWidth', $1\mathrm{.}5)$; $\%$ Time$-$domain plot of filtered signal

title$(\text{'}$Filtered Signal $($Frequency Domain$)\text{'})$;

xlabel$(\text{'}$Frequency $($Hz$)\text{'})$;

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

xlim$([-$Freq$\_$x$\_$axis Freq$\_$x$\_$axis$])$;

$\%$clear ylim

ylim$=([011\mathrm{.}5])$;