Aim
To find the fourier series coefficients using MATLAB and plotting them as well.
Code
% Defining the periodic signals
T = 2*pi; % Period of the signal
t = linspace(-T, T, 1000);
% Number of harmonics to compute
N = 10;
% Initialize coefficient arrays
a_0 = 0; % DC Component
% Initialize coefficient arrays for each signal
a_n_square = zeros(1, N);
b_n_square = zeros(1, N);
a_n_sawtooth = zeros(1, N);
b_n_sawtooth = zeros(1, N);
a_n_sine = zeros(1, N);
b_n_sine = zeros(1, N);
a_n_cosine = zeros(1, N);
b_n_cosine = zeros(1, N);
% Define and analyze square wave
signal = square(t);
for n = 1:N
a_n_square(n) = (2/T) * trapz(t, signal .* cos(n * (2*pi/T) * t));
b_n_square(n) = (2/T) * trapz(t, signal .* sin(n * (2*pi/T) * t));
end
% Define and analyze sawtooth wave
signal = sawtooth(t, 0.5);
for n = 1:N
a_n_sawtooth(n) = (2/T) * trapz(t, signal .* cos(n * (2*pi/T) * t));
b_n_sawtooth(n) = (2/T) * trapz(t, signal .* sin(n * (2*pi/T) * t));
end
% Define and analyze sine wave
signal = sin(t);
for n = 1:N
a_n_sine(n) = (2/T) * trapz(t, signal .* cos(n * (2*pi/T) * t));
b_n_sine(n) = (2/T) * trapz(t, signal .* sin(n * (2*pi/T) * t));
end
% Define and analyze cosine wave
signal = cos(t);
for n = 1:N
a_n_cosine(n) = (2/T) * trapz(t, signal .* cos(n * (2*pi/T) * t));
b_n_cosine(n) = (2/T) * trapz(t, signal .* sin(n * (2*pi/T) * t));
end
% Display results for each signal
disp('Fourier Series Coefficients for Square Wave:')
disp(['a_0: ', num2str(a_0)])
for n = 1:N
disp([num2str(n), ' ', num2str(a_n_square(n)), ' ', num2str(b_n_square(n))]);
end
disp('Fourier Series Coefficients for Sawtooth Wave:')
disp(['a_0: ', num2str(a_0)])
for n = 1:N
disp([num2str(n), ' ', num2str(a_n_sawtooth(n)), ' ', num2str(b_n_sawtooth(n))]);
end
disp('Fourier Series Coefficients for Sine Wave:')
disp(['a_0: ', num2str(a_0)])
for n = 1:N
disp([num2str(n), ' ', num2str(a_n_sine(n)), ' ', num2str(b_n_sine(n))]);
end
disp('Fourier Series Coefficients for Cosine Wave:')
disp(['a_0: ', num2str(a_0)])
for n = 1:N
disp([num2str(n), ' ', num2str(a_n_cosine(n)), ' ', num2str(b_n_cosine(n))]);
end
% Plot the original signals
figure;
subplot(2,2,1);
plot(t, square(t));
title('Square Wave');
xlabel('Time (t)');
ylabel('Signal');
grid on;
subplot(2,2,2);
plot(t, sawtooth(t, 0.5));
title('Sawtooth Wave');
xlabel('Time (t)');
ylabel('Signal');
grid on;
subplot(2,2,3);
plot(t, sin(t));
title('Sine Wave');
xlabel('Time (t)');
ylabel('Signal');
grid on;
subplot(2,2,4);
plot(t, cos(t));
title('Cosine Wave');
xlabel('Time (t)');
ylabel('Signal');
grid on;
% Plot Fourier coefficients
figure;
subplot(2,2,1);
stem(1:N, a_n_square, 'r', 'DisplayName', 'a_n (cosine)');
hold on;
stem(1:N, b_n_square, 'b', 'DisplayName', 'b_n (sine)');
title('Fourier Coefficients for Square Wave');
xlabel('Harmonic Number (n)');
ylabel('Coefficient Value');
grid on;
subplot(2,2,2);
stem(1:N, a_n_sawtooth, 'r', 'DisplayName', 'a_n (cosine)');
hold on;
stem(1:N, b_n_sawtooth, 'b', 'DisplayName', 'b_n (sine)');
title('Fourier Coefficients for Sawtooth Wave');
xlabel('Harmonic Number (n)');
ylabel('Coefficient Value');
grid on;
subplot(2,2,3);
stem(1:N, a_n_sine, 'r', 'DisplayName', 'a_n (cosine)');
hold on;
stem(1:N, b_n_sine, 'b', 'DisplayName', 'b_n (sine)');
title('Fourier Coefficients for Sine Wave');
xlabel('Harmonic Number (n)');
ylabel('Coefficient Value');
grid on;
subplot(2,2,4);
stem(1:N, a_n_cosine, 'r', 'DisplayName', 'a_n (cosine)');
hold on;
stem(1:N, b_n_cosine, 'b', 'DisplayName', 'b_n (sine)');
title('Fourier Coefficients for Cosine Wave');
xlabel('Harmonic Number (n)');
ylabel('Coefficient Value');
grid on;
Output
Output Text
Fourier Series Coefficients for Square Wave:
a_0: 0
1 0.004004 2.5465
2 0.004004 -1.2592e-05
3 0.004004 0.84882
4 0.004004 -2.5183e-05
5 0.004004 0.50929
6 0.004004 -3.7776e-05
7 0.004004 0.36377
8 0.004004 -5.0369e-05
9 0.004004 0.28292
10 0.004004 -6.2963e-05
Fourier Series Coefficients for Sawtooth Wave:
a_0: 0
1 -1.6211 -1.2338e-18
2 -4.0081e-06 8.369e-18
3 -0.18013 1.0457e-17
4 -4.0082e-06 -8.8349e-18
5 -0.064847 -1.7256e-18
6 -4.0084e-06 -2.0017e-18
7 -0.033086 -5.3147e-18
8 -4.0086e-06 5.5908e-18
9 -0.020015 4.9696e-18
10 -4.009e-06 3.7272e-18
Fourier Series Coefficients for Sine Wave:
a_0: 0
1 1.0345e-17 2
2 1.182e-18 -9.1174e-18
3 -6.005e-18 -4.3154e-17
4 -4.892e-18 1.0095e-18
5 6.8419e-18 6.4407e-18
6 1.4667e-19 -5.1978e-17
7 -9.7494e-18 8.2068e-17
8 -4.521e-18 -4.0378e-18
9 3.3476e-18 3.3063e-16
10 -1.7946e-18 -4.3398e-18
Fourier Series Coefficients for Cosine Wave:
a_0: 0
1 2 1.0345e-17
2 -1.6345e-16 -2.8058e-17
3 -1.9382e-16 9.2145e-18
4 -1.5848e-16 -1.553e-17
5 -1.4136e-16 8.6278e-19
6 -1.524e-16 -6.4191e-18
7 -3.048e-16 -1.056e-17
8 -1.4301e-16 -2.0707e-19
9 -2.5621e-16 -2.1397e-18
10 -1.3473e-16 -1.8636e-18
>> k057
Fourier Series Coefficients for Square Wave:
a_0: 0
1 0.004004 2.5465
2 0.004004 -1.2592e-05
3 0.004004 0.84882
4 0.004004 -2.5183e-05
5 0.004004 0.50929
6 0.004004 -3.7776e-05
7 0.004004 0.36377
8 0.004004 -5.0369e-05
9 0.004004 0.28292
10 0.004004 -6.2963e-05
Fourier Series Coefficients for Sawtooth Wave:
a_0: 0
1 -1.6211 -1.2338e-18
2 -4.0081e-06 8.369e-18
3 -0.18013 1.0457e-17
4 -4.0082e-06 -8.8349e-18
5 -0.064847 -1.7256e-18
6 -4.0084e-06 -2.0017e-18
7 -0.033086 -5.3147e-18
8 -4.0086e-06 5.5908e-18
9 -0.020015 4.9696e-18
10 -4.009e-06 3.7272e-18
Fourier Series Coefficients for Sine Wave:
a_0: 0
1 1.0345e-17 2
2 1.182e-18 -9.1174e-18
3 -6.005e-18 -4.3154e-17
4 -4.892e-18 1.0095e-18
5 6.8419e-18 6.4407e-18
6 1.4667e-19 -5.1978e-17
7 -9.7494e-18 8.2068e-17
8 -4.521e-18 -4.0378e-18
9 3.3476e-18 3.3063e-16
10 -1.7946e-18 -4.3398e-18
Fourier Series Coefficients for Cosine Wave:
a_0: 0
1 2 1.0345e-17
2 -1.6345e-16 -2.8058e-17
3 -1.9382e-16 9.2145e-18
4 -1.5848e-16 -1.553e-17
5 -1.4136e-16 8.6278e-19
6 -1.524e-16 -6.4191e-18
7 -3.048e-16 -1.056e-17
8 -1.4301e-16 -2.0707e-19
9 -2.5621e-16 -2.1397e-18
10 -1.3473e-16 -1.8636e-18
>> References
Information
- date: 2024.08.30
- time: 12:19