Trigonometric Fourier Series of Odd Square Wave

M-file:

%% Filename: TrigFSOddSquareWave.m

% Description: m-file to plot Trigonometric Fourier Series

% representation of odd square wave and its frequency

% (amplitude and phase) spectrum.

clear; clc; close all; % clear memory and command window, close all figures

%% Fourier Series representation of square wave for different numbers

% of terms in summation

t = -3:0.001:3; % times as vector over which to plot FS

Nval = [1, 3, 7, 251]; % number of terms for n in four summations

figure(); % open figure in which to plot truncated Fourier Series

for in = 1:4, % % Build four square waves using increasing number of terms

v = 0; % initialize sum to a_0 = 0

subplot(2,2,in);

% "brute force" plot of true square wave

plot([-3, -3, -2, -2, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3], ...

[ 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1], ...

'k-.','LineWidth',2);

hold on;

for n = 1:2:Nval(in), % sum over odd values of n

harm = (4/(n*pi))*sin(n*pi*t);

v = v + harm;

plot(t,harm,'b--','LineWidth',1); % plot each harmonic

end

plot(t,v,'r-','LineWidth',2); % plot sum of all terms/harmonics

grid; hold off;

xlabel('time, sec','FontSize',14); ylabel('FS Approx to v(t)','FontSize',14);

title(['Fourier Series Approx w/',num2str(Nval(in)),' Terms'],'FontSize',14);

end

%% Frequency spectrum of Fourier Series

figure(); % open a figure to plot frequency spectrum

n = 1:2:11; % vector of values for n odd noting omega = n x omega_0

a0 = 0; % DC offset

an = 0; % coefficients of cosines

bn = 4./(n*pi); % coefficients of sines for n odd

A = sqrt(an.^2 + bn.^2); % amplitudes

phi = -atan2(bn,an)*180/pi; % phases

% plot amplitude spectrum

subplot(2,1,1); stem([0, n*pi], [a0, A], 'filled'); grid;

xlabel('\omega (rad/sec)'); ylabel('A_n'); title('Amplitude Spectrum');

% plot phase spectrum

subplot(2,1,2); stem([0, n*pi], [0, phi], 'filled'); grid;

xlabel('\omega (rad/sec)'); ylabel('\phi_n'); title('Phase Spectrum');

Plots generated: