Fourier Series Signal Codes

I'm trying to make this graph in its Fourier Series Signal in Trigonometric Form. Can you please correct my codes.

t= [-3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3]; % values of t
xt= [0 0 0 1 0 0 0 1 0 0 0 1 0 0]; % values of x
subplot(2,1,1)% to plot 2 graphs in a single page
plot(t,xt,'LineWidth',1.5)% plotting the original signal
title('Graph of the Original Signal')
xlabel('Time, (t)') % label the x-axis
ylabel('Amplitude, x(t)') % label the y-axis
grid on % putting a grid

subplot (2,1,2) % label the x-axis
fs=100; % trigonometric form
T=2;
w0=2*pi/T;
k=-3:0.001:3;
y=square(w0*k,50);
syms t
N=5;
n=1:N;

a0=(2/T)*(int(1,t,0,1)+int(-1,t,1,2));
an=(2/T)*(int(1*cos(n*w0*t),t,0,1)+int(-1*cos(n*w0*t),t,1,2));
bn=(2/T)*(int(1*sin(n*w0*t),t,0,1)+int(-1*sin(n*w0*t),t,1,2));

F=a0/2;
for i=1:N
   F=F+an(i)*cos(i*w0*k)+bn(i)*sin(i*w0*k);
end
plot(k,F,'LineWidth',1.5) % plotting the fourier signal 
title('Graph of the Fourier Seires Signal in Trigonometric Form')
xlabel('Time, (t)') % label the x-axis
ylabel('Amplitude, x(t)') % label the y-axis
grid on

Answer 1

[Aashray]

Hello Mr. Santa

I went through the shared code and suggest the following changes:

1. Define a sawtooth-like waveform, where x(t) = 0 from t = 0 to 1, then rises to 1 and linearly drops from 1 to 0 in the interval t = 1 to 2. This pattern repeats with period T = 2 using the mod function for periodicity.
2. Use a finely sampled time vector (-3:0.01:3) instead of discrete time points, in order to make the plot more smoother.
3. Remove the square function.
4. Calculate correct Fourier coefficients by integrating the expression (2 - t) over [1, 2], where the signal is non-zero.
5. Clean up the loop to compute the Fourier sum using:
6. F = a0/2 + ∑ [aₙ cos(nω₀t) + bₙ sin(nω₀t)]

Please refer to the following code for better understanding:

% Define the time vector and signal for the sawtooth-like wave
t = -3:0.01:3; % Fine resolution for smooth plotting
T = 2; % Period of the signal
xt = zeros(size(t)); % Initialize the signal
for i = 1:length(t)
    t_mod = mod(t(i), T); % Map time to one period [0, T]
    if t_mod >= 1 && t_mod <= 2
        xt(i) = 2 - t_mod; % Linear decrease from 1 to 0 over t=1 to t=2
    end
end

% Plot the original signal
subplot(2,1,1)
plot(t, xt, 'LineWidth', 1.5)
title('Graph of the Original Signal')
xlabel('Time, t')
ylabel('Amplitude, x(t)')
grid on
axis([-3 3 -0.2 1.2])

% Fourier Series Calculation
w0 = 2 * pi / T; % Fundamental frequency
N = 5; % Number of harmonics
k = -3:0.01:3; % Time vector for Fourier series
F = zeros(size(k)); % Initialize Fourier series output

% Compute a0 (DC component)
syms t
a0 = (2/T) * int(2-t, t, 1, 2); % Integrate over [1,2] where signal is non-zero
a0 = double(a0); % Convert to numerical value
F = F + a0 / 2; % Add DC component

% Compute an and bn coefficients
for n = 1:N
    an = (2/T) * int((2-t) * cos(n*w0*t), t, 1, 2); % Integrate over [1,2]
    bn = (2/T) * int((2-t) * sin(n*w0*t), t, 1, 2); % Integrate over [1,2]
    an = double(an); % Convert to numerical
    bn = double(bn); % Convert to numerical
    F = F + an * cos(n*w0*k) + bn * sin(n*w0*k); % Add nth harmonic
end

% Plot the Fourier series
subplot(2,1,2)
plot(k, F, 'LineWidth', 1.5)
title('Graph of the Fourier Series Signal in Trigonometric Form')
xlabel('Time, t')
ylabel('Amplitude, x(t)')
grid on
axis([-3 3 -0.2 1.2])