Fourier synthesis is simple application of coefficients summation either numerically or symbolically, Fourier decomposition requires certain conditions of the input function, you can start with a simple example which is an approximation of a square wave of 2 Hz, using 11 terms :
Fs=150;
t=0:1/Fs:4-1/Fs;
f=2;
x=square(2*pi*f*t);
figure;
plot(t,x);
axis([0 1 -2 2]);
% Approximation with Fourier decomposition
y=0;
N=11;
for r=1:2:N
y=y+sin(2*pi*f*t*r)/r;
end
hold on;
plot(t,y,'r');
xlabel('t');
ylabel('magnitude');
hold off;
