MATLAB Toolkit for Exponential Fourier Series

%% Author Roberto Rosas-Romero

%% Parameters of Periodic Signal

% Period

T = pi;

% Fundamental Frequency

w = 2 * pi / T;

% Pulse width for the case of a pulse train

% u = 1.5;

%% Integration to compute Fourier Coefficients

% Integration Limits

a = 0;

b = pi;

% Integration to estimate Exponential Fourier Coefficients

syms t;

syms n;

integral = (1/T) * int(exp(-t/2) * exp(-i* n * w* t), t, a, b);

% Defining Dn functions of n

Dn = inline(integral, 'n');

%% Approximation of the original signal by using the Fourier Series

% Time Interval over which the periodic signal is approximated

t = - 2*pi : 0.01 : 2*pi;

% Number of Harmonics used in the approximation

Harmonics = 50;

% Initializing approximation

x = 0;

% Iterative addition of harmonics

for n = -Harmonics : Harmonics

x = x + Dn(n) * exp(i * n * w* t);

end;

%% Plot of the Approximation

subplot(3, 1, 1), plot(t, x, 'r');

%% Computation and Plot of Magnitude Spectrum

% Initialization of vector of frequencies

VectorOfFrequencies = [];

% Initialization of vector of coefficients

VectorOfCoefficients = [];

% Initialization of Vector of Phase

VectorOfPhase = [];

% Initialization of the Power of the Approximation

P = 0;

% Number of Harmonics used in the approximation

Harmonics = 50;

for n = -Harmonics : Harmonics

VectorOfFrequencies = [VectorOfFrequencies; n*w];

VectorOfCoefficients = [VectorOfCoefficients; abs(Dn(n))];

VectorOfPhase = [VectorOfPhase; phase(Dn(n))];

P = P + abs(Dn(n))^2;

end;

% Power in the original signal

Poriginal = (1 - exp(-pi)) / pi;

% Percentage of Power contained in the approximation

Percentage = double(100 * P / Poriginal);

% Plot of the Magnitude Spectrum

hold on; subplot(3, 1, 2), plot(VectorOfFrequencies, VectorOfCoefficients, 'o');

% Plot of the Phase Spectrum

subplot(3, 1, 3), plot(VectorOfFrequencies, VectorOfPhase, 'o');