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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. 2

Thanks in advance.

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  • $\begingroup$ Apparently, one period of your signal is not strictly parabolic. It is made of a piecewise polynomial signal, parabolic on the first half, and on degree 0 on the second one $\endgroup$
    – Laurent Duval
    Sep 18, 2021 at 15:57
  • $\begingroup$ 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. $\endgroup$
    – TimWescott
    Sep 18, 2021 at 18:30
  • $\begingroup$ Maybe you find this example helpful: tf.uni-kiel.de/matwis/amat/math_for_ms/kap_1/backbone/… $\endgroup$
    – Juha P
    Sep 18, 2021 at 20:19
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    $\begingroup$ I edited the signal type @Laurent Duval $\endgroup$
    – LunaLOVEGOOD
    Sep 21, 2021 at 11:36
  • $\begingroup$ I followed the link you suggest. And obtained at first run. You can see the result. Thanks @JuhaP $\endgroup$
    – LunaLOVEGOOD
    Sep 21, 2021 at 11:41

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