Trigonometric Fourier Series of Even Triangle Wave

M-file:

%% Filename: TrigFSEvenTriangleWave.m

% Description: m-file to plot Trigonometric Fourier Series

% representation of even triangle wave and its frequency

% (amplitude and phase) spectrum.

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

%% Fourier Series representation of triangle wave for different numbers

% of terms in summation

t = -1.5:0.0001:3.5; % times as vector over which to plot FS

Nval = [1, 3, 5, 21]; % number of terms for n in four summations

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

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

v = -1/2; % initialize sum to a_0 = -1/2

subplot(2,2,in);

% "brute force" plot of true triangle wave

plot([ -1.5, -1, 0, 1, 2, 3, 3.5],...

[ -0.5, -1, 0,-1, 0, -1, -0.5], ...

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

hold on;

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

harm = (4/(n*pi)^2)*cos(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 = -1/2; % DC offset

an = 4./(n*pi).^2; % coefficients of cosines for n odd

bn = 0; % coefficients of sines

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], [abs(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], [angle(a0)*180/pi, phi], 'filled'); grid;

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

Plots generated: