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
Thanks in advance.