Plotting Fourier Series Expansion

I was given a problem of the Fourier series expansion of with a periodicity interval . I found that the Fourier series expansion of such function with given periodicity interval is . However, I am new to MATLAB and my knowledge is limited to simple plotting. Suppose that my answer is correct, how to subplot and its Fourier series expansion using loop that generates the n-order of the expansion? I searched in this community and actually find a really good plotting, but I don't understand how it works. Thanks in advance!

[Hanif]

Here, I am attaching the complete works on how do I find the Fourier series expansion just in case it would help anyone who want to help me to make the subplot (see in problem 1).

[Hanif]

I also want to share my attempt on this:

x = linspace(-10,10,100);
y = 1-(4*cos(x))/pi - (4*cos(3*x))/(9*pi) - (4*cos(5*x))/(25*pi);
plot(x,y,'r-');

Of course, it is very simple because I am new to MATLAB.

Answer 1

[Paul]

Hi Hanif,

Maybe this will help.

First, the problem statement says that one period spans -1 < x <= 1 (I don't think it matter if using < or <=)

x = linspace(-1,1,100);

Instead of using your problem, let's do something simpler, like 1 - 2*sum(n=1:5,n*x)

fsum = 0*x;  % initialize
for n = 1:5
    fsum = fsum + n*x;
end
plot(x,1-2*fsum)

I think you can adapt that to your problem. Of course we can't compute an infinite sum, so you'll have to pick an upper bound for the sum over n.

[Hanif]

Hi Paul,

Thank you for your help.

This actually a very good example to understand the basic for a MATLAB newcomer like me. I have tried my problem on your program.

x = linspace(-10,10,100);
fsum = 4*cos((pi*x)/(pi^2));  % initialize
for n = 2:10
    fsum = fsum + 4*cos((2*n-1)*pi*x)/(pi^2*(2*n-1)^2);
end
y = 1-fsum;
plot(x,y,'r-')

The result was good. However, is there a way to improve this program such that it plots the Fourier series expansion of for each n, i.e., for , for , and so on? Let's say I want to stop at , so I have five graphs in one plot.

[Paul]

Use plot inside the loop, with hold set to on. Also, the initial value for fsum was incorrect. And still need to define x over the correct interval

% x = linspace(-10,10,100);  should be from -1 to 1
% fsum = 4*cos((pi*x)/(pi^2));  % initialize, should be 4*cos(pi*x)/pi^2;
% for n = 2:10
%     fsum = fsum + 4*cos((2*n-1)*pi*x)/(pi^2*(2*n-1)^2);
% end
% y = 1-fsum;
% plot(x,y,'r-')

x = linspace(-1,1,100);  % changed this line
fsum = 0*x; % initialize
figure;
hold on;
for n = 1:10  % start loop at 1
    fsum = fsum + 4*cos((2*n-1)*pi*x)/(pi^2*(2*n-1)^2);
    plot(x,1-fsum)
end

The sum seems like it's biased up by 0.5. Also, as your code showed, the reconstruction is not periodic with period = 2. Recheck derivation?

[Hanif]

I see. I actually tried to put the plot function into the loop, but it did not work because I forgot to put the hold-on command. As you also mentioned, I haven't check my derivation again, but I will do! Thank you for your helps, Paul.