How can i get the fourier series of any given function

Since there is no direct way of finding/computing the fourier seris of a function, e.g. cos(x), how would I do this?

Answer 1

[Friedel Hartmann]

clear
syms x f(x) k fh(x) p(x)  ph(x)  % a0 = a0 Formel /2

iTerme = 8;

sco = zeros(1,iTerme);
cco = zeros(1,iTerme);

% T = 2*pi, omega = 1

f(x) = piecewise(x <= pi/4, -1/(2*pi) * x, pi/4 < x,-1/(2*pi) * x + 1); % Function 

for k=1:iTerme
  sco(k) = 1/pi * int(sin(k * x) * (f(x)),[0,2*pi]); % coefficients
  cco(k) = 1/pi * int(cos(k * x) * (f(x)),[0,2*pi]);
end

cnum = 1:iTerme;  % Table with coefficients
Tabelle.num = cnum';
Tabelle.scof = sco';
Tabelle.ccof = cco';
T = struct2table(Tabelle)

T = 8×3 table
    num       scof         ccof   
    ___    __________    _________

     1        0.22508     -0.22508
     2     9.8393e-18     -0.15915
     3      -0.075026    -0.075026
     4      -0.079577            0
     5      -0.045016     0.045016
     6     3.2798e-18     0.053052
     7       0.032154     0.032154
     8       0.039789            0

fplot(f(x),[0,2*pi]), title('Original'), yline(0);

figure

a0G = vpa(1/2 * 2/(2*pi) * int(f(x),[0,2*pi]),3)

a0G = 

0.375

fh(x) = 1/(2*pi) * int(f(x),[0,2*pi]); % 'Mittelwert' 1. Term ist a0
for k = 1:iTerme
    fh(x) = fh(x) + sco(k) * sin(k * x) + cco(k) * cos(k * x);
end

fplot(fh(x),[0,2*pi]), title('Fourier series'), yline(0);

vpa(fh(0),3)

ans = 

0.046

p(x) = x/(2*pi); % 2nd Function

for k=1:iTerme
  sco(k) = 1/pi * int(sin(k * x) * (p(x)),[0,2*pi]);
  cco(k) = 1/pi * int(cos(k * x) * (p(x)),[0,2*pi]);
end

cnum = 1:iTerme;
Tabelle.num = cnum';
Tabelle.scof = sco';
Tabelle.ccof = cco';
T = struct2table(Tabelle)

T = 8×3 table
    num      scof       ccof
    ___    _________    ____

     1      -0.31831     0  
     2      -0.15915     0  
     3       -0.1061     0  
     4     -0.079577     0  
     5     -0.063662     0  
     6     -0.053052     0  
     7     -0.045473     0  
     8     -0.039789     0  

fplot(p(x),[0,2*pi]), title('Original'), yline(0);

figure

a0L = vpa(1/2 * 2/(2*pi) * int(p(x),[0,2*pi]),3)

a0L = 

0.5

ph(x) = 1/pi * int(p(x),[0,2*pi]) * 1/(2); % 'Mittelwert' 1. Term ist a0/2
for k = 1:iTerme
  ph(x) = ph(x) + sco(k) * sin(k * x) + cco(k) * cos(k * x);
end

fplot(ph(x),[0,2*pi]), title('Fourier series'), yline(0);

[Friedel Hartmann]

The series starts with a constant term, the average value, called 'Mittelwert' The routine first calculates the average value, assigns this number to ph(x) and then begins the loop for k = 1:iTerme etc. and adds the following sin and cos functions.