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:
- Define the Piecewise function and its range and period.
- Calculate the Fourier series coefficients (an and bn).
- Create a Fourier series approximation using the calculated coefficients.
- 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:
