# Approximation of Periodic Parabolic Function by Fourier Series!

I've just tried to approximate the periodic-parabolic signal by Fourier Series. I know, this sounds a bit strange. I am just trying to figure out relationship between Fourier Series and Taylor Expansion.

First of all, I generated parabolic pulse train and its period $$2\pi$$. On the other hand, time duration of parabolic pulse train is $$10\pi$$. After that, I've just tried to approximate this signal by Fourier Series. Here is my simple code on Matlab.

  TimeRes=1/(2*pi);  % Time Resolution
t1=0:TimeRes:2*pi-TimeRes; % One Period
SinglePeriodVector = zeros(1,length(t1));
SinglePeriodVector = t1.*(2*pi-t1);
Signal = repmat(SinglePeriodVector, [1, 5]);
t2=0:TimeRes:5*(2*pi-TimeRes)+TimeRes;
t2=[t2 t2(end)];

%% Fourier Series Approximation for Even Signals
a0=(2/3)*pi^2;
FS=zeros(size(t2));
N=1;

for n=1:2:N
FS=a0+FS-4*cos(n*t2/(n^2))
end


After first run, I successfully composed to signal. Here is the result. • Please edit your question to show your Fourier series for your repetitive signal in math. If I'm reading your code right, then, first, you have a typo (1/pi * n in one spot, 1/n * pi in another), and second, you don't have the Fourier series for your given curve. Sep 18, 2021 at 18:30