Complex Fourier Series of Odd Square Wave

%% Filename: ComplexFSOddSquareWave.m

% Description: m-file to plot complex (exponential) Fourier Series

% representation of an odd square wave.

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

t = linspace(-2,3,1000); % times over which to plot FS

N = [2, 4, 6, 100]; % upper limit (N) (and lower limit (-N)) for n in summation

c0 = 0; % DC component (average value)

%% Build complex Fourier Series representation of signal

figure(1); % open figure in which to plot

% Build square wave using increasing number of terms

for in = 1:4, % loop over four different limits (-N,...,N) of Fourier Series

  f = c0*zeros(1,length(t)); % initialize summation to c0

  subplot(2,2,in); % open one of four subplots

  % brute force original square wave

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

       [ 1,  1, -1, -1, 1, 1, -1, -1, 1, 1],'k-.','LineWidth',2);

  hold on;

  for n = 1:1:N(in), % loop over different lengths of FS

    fpos = ((1 - cos( n*pi))/( j*n*pi))*exp( j*n*pi*t); %compute +n term

    fneg = ((1 - cos(-n*pi))/(-j*n*pi))*exp(-j*n*pi*t); %compute -n term

    f    = f + fpos + fneg; % add terms for pos and neg n to series

    % plot real terms (harmonics) by adding complex terms for pos and neg n

    plot(t, fpos+fneg, 'b--','LineWidth',1);

  end

  plot(t,f,'r-','LineWidth',2); % plot complex FS

  hold off;  

  xlabel('time (sec)', 'FontSize', 14);

  ylabel('Complex FS Approx to f(t)', 'FontSize', 14);

  title(['Fourier Series Approx w/n = ',num2str(-N(in)),', \ldots, ', ...

    num2str(N(in)),' Terms'],'FontSize',14);

end

%% Plot complex frequency spectrum (amplitude and phase) of signal

clear all; % clear memory 

c0 = 0; % DC component (average value)

figure(2); % open figure in which to plot

n  = [-6:-1,1:6]; % values of n for complex spectrum leaving out 0 due to c0 = 0/0

cn = (1 - cos( n*pi))./( j*n*pi); % Fourier coefficients noting 0/0 for n = 0

subplot(2,1,1); % open subplot of amplitude spectrum

stem(0, abs(c0), 'filled', 'LineWidth', 2); % w = 0 component

hold on;

stem(n*pi, abs(cn), 'filled', 'LineWidth', 2); % w = n x pi component

hold off;

xlabel('\omega (rad/sec)', 'FontSize', 14);

ylabel('|c_n|', 'FontSize', 14);

title('Amplitude Spectrum of Complex Fourier Series', 'FontSize', 14);

subplot(2,1,2); % open subplot of phase spectrum

stem(0, angle(c0)*180/pi, 'filled', 'LineWidth', 2); % w = 0 component

hold on;

stem(n*pi, angle(cn)*180/pi, 'filled', 'LineWidth', 2); % w = n x pi component

hold off;

xlabel('\omega (rad/sec)', 'FontSize', 14);

ylabel('\angle c_n (\circ)', 'FontSize', 14);

title('Phase Spectrum of Complex Fourier Series', 'FontSize', 14);