Fourier series calculating n, an and bn as well as plotting it?

Hi all,

I need to compare the fourier series of given function and then plot and compare it to its orignal function

It is a piecewise function:

f(x) = 2 - (x/2)^2 if 0<= x <2

1 if -2<= x <0

the fourier series has a periodic function of 4

Answer 1

[Divit]

Hi Ben,

To calculate the Fourier series coefficients (an and bn) and plot the Fourier series approximation for the given piecewise function, you'll need to perform the following steps:

1. Define the Piecewise function and its range and period.
2. Calculate the Fourier series coefficients (an and bn).
3. Create a Fourier series approximation using the calculated coefficients.
4. Plot and compare the original function with its Fourier series approximation.

You can refer to the following MATLAB code:

% Define the period and the range for x
T = 4;      % Period of the periodic function
x = -2:0.01:2; % Range of x values

% Define the given piecewise function
f = @(x) (2 - (x/2).^2).*(x >= 0 & x < 2) + (1).*(x >= -2 & x < 0);

% Number of terms in the Fourier series
N = 10;

% Calculate the Fourier series coefficients (an and bn)
a0 = (1/T) * integral(@(x) f(x), 0, T);
an = zeros(1, N);
bn = zeros(1, N);
for n = 1:N    
    an(n) = (1/T) * integral(@(x) f(x).*cos(n*pi*x/2), 0, T);
    bn(n) = (1/T) * integral(@(x) f(x).*sin(n*pi*x/2), 0, T);
end

% Create the Fourier series approximation
F = a0/2;
for n = 1:N
    F = F + an(n) * cos(n*pi*x/2) + bn(n) * sin(n*pi*x/2);
end

% Plot the original function and its Fourier series approximation
figure;
plot(x, f(x), 'b', 'LineWidth', 2, 'DisplayName', 'Original Function');
hold on;
plot(x, F, 'r--', 'LineWidth', 2, 'DisplayName', 'Fourier Series Approximation');
xlabel('x');
ylabel('f(x)');
title('Comparison of Original Function and Fourier Series Approximation');
legend('show');
grid on;

You can adjust the value of N to include more or fewer terms in the series for a better approximation.

To know more about Fourier Series, you can refer to the following documentation link:

• https://www.mathworks.com/help/curvefit/fourier.html