clc; clear all; close all;

% Define the function f(x)
k = 1;
f = @(x) -k*(-pi < x & x < 0) + k*(0 < x & x < pi);

% Define the period and the time range to plot
T = 2*pi;
x = -7*pi:0.001:7*pi;
n = 5;  % Number of terms
L = pi; % Half period
fs = zeros(size(x)); % Initialize Fourier series approximation

% Compute the periodic signal using the modulo function
Fp = f(mod(x - pi, T) - pi);

% Fourier series approximation
a0 = 1/(2*L) * integral(f, -L, L); % Compute a0
an = zeros(1, n); % Initialize an coefficients
bn = zeros(1, n); % Initialize bn coefficients

% Compute an and bn coefficients
for i = 1:n
    an(i) = 1/L * integral(@(x) f(x) .* cos(i * pi * x / L), -L, L);  % Compute an
    bn(i) = 1/L * integral(@(x) f(x) .* sin(i * pi * x / L), -L, L);  % Compute bn
    fs = fs + an(i) * cos((i * pi * x) / L) + bn(i) * sin((i * pi * x) / L); % Add terms to Fourier series
end

fs = a0 + fs; % Final Fourier series with a0 included

% Plot the original function and the Fourier series approximation
figure;
plot(x, Fp, 'b-', 'LineWidth', 2);
hold on;
plot(x, fs, 'r-', 'LineWidth', 2);
title('Fourier Series Approximation for f(x)');
xlabel('x');
ylabel('f(x) and Fourier Series Approximation');
legend('Original Function', 'Fourier Series Approximation');
grid on;

%Question2
clc;clear all; close all;

% Define the function f(x)
k = 1;
f = @(x) 0*(-2<x & x<-1)+k*(-1<x & x<1)+0*(1<x & x<2);

% Define the period and the time range to plot
T = 4; % Period of the function f(x)
x = -10:0.001:10; % Time range

% Compute the periodic signal using the modulo function
Fp = f(mod(x-2, T) -2);

% Fourier series approximation
n = 5; % Number of terms
L = 2; % Period
a0 = 1/(2*L)*integral(f, -L, L); % Compute a0
an = zeros(1, n); % Initialize an coefficients
bn = zeros(1, n); % Initialize bn coefficients
fs = zeros(size(x));

% Compute an and bn coefficients
for i=1:n
    an(i) = 1/L * integral(@(x) f(x).*cos(i*pi*x/L), -L, L);
    bn(i) = 1/L * integral(@(x) f(x).*sin(i*pi*x/L), -L, L);
    fs = fs + an(i)*cos((i*pi*x)/L) + bn(i)*sin((i*pi*x)/L);

end
fs = a0 + fs; % Final Fourier series


% Plot the original function and the Fourier series approximation
figure;
plot(x, Fp, 'b-', 'LineWidth', 2);
hold on;
plot(x, fs, 'r-', 'LineWidth', 2);
title('Fourier Series Approximation for f(x)');
xlabel('x');
ylabel('f(x) and Fourier Series Approximation');
legend('Original Function', 'Fourier Series Approximation');
grid on;