Question

During lab 4, we have seen numerical implementation of Fourier Series for periodic signals. As first part of this assignment, you need to write a Matlab function that would take an array representing a single period of a signal (x), corresponding time array (t), and return the Fourier Series coefficients (Ck) in exponential form. The function should also be able to take two (2) optional input arguments: number of Fourier coefficients (Nk) and plot option (p). Use the template ‘fourier_series_exp.m’ for this problem.

(Template "fourier_series_exp.m")

function [Ck]=fourier_series_exp(x,t,Nk,p)

% Ck = exponential fourier series cofficient
% x = single period of a signal
% t = time corrosponding to 'x'
% Nk = (optional input) number of exponential terms
% p = plotting option ; p=0, no plots, p = 1 plot Ck vs k and reconstructed signal
% dT = t(2)-t(1) = temporal resolution of signal (x)
% T = peiod of signal 'x'
% w0= angular frequency of signal 'x'

dT=t(2)-t(1);
T= dT*length(t);
w0=2*pi/T;   

% Check the number of inputs, 'nargin' returns number of input arguments
if nargin <2
error('Not enough input argument!')
elseif nargin == 2
Nk=101; % you can set any default value you like
p=0; % not plots
elseif nargin ==3
p=0; % not plots
end
k=-floor(Nk/2):floor(Nk/2); % if Nk=11, k=-5:5; if Nk=12, k=-6:6

%% evaluate Ck
%
% % % write this code segment
%
%% plot spectrum and reconstructed signal   
if p==1
% plot abs(Ck) vs k and angle(Ck) vs k
%
% % % write this code segment
%
  
% plot 3 cycles of the signal 'x' and the reconstructed signal
%
% % % write this code segment
%
end

end

Answer 1

function [Ck]=fourier_series_exp(x,t,Nk,p)

% Ck = exponential fourier series cofficient
% x = single period of a signal
% t = time corrosponding to 'x'
% Nk = (optional input) number of exponential terms
% p = plotting option ; p=0, no plots, p = 1 plot Ck vs k and reconstructed signal
% dT = t(2)-t(1) = temporal resolution of signal (x)
% T = peiod of signal 'x'
% w0= angular frequency of signal 'x'

dT=t(2)-t(1);
T= dT*length(t);
w0=2*pi/T;   

% Check the number of inputs, 'nargin' returns number of input arguments
if nargin <2
error('Not enough input argument!')
elseif nargin == 2
Nk=101; % you can set any default value you like
p=0; % not plots
elseif nargin ==3
p=0; % not plots
end
k=-floor(Nk/2):floor(Nk/2); % if Nk=11, k=-5:5; if Nk=12, k=-6:6

%% evaluate Ck
%
Ck = [];
for k=-floor(Nk/2):floor(Nk/2)
Ck1=(1/T)*sum(x.*exp(-j*k*w0*t));
Ck = [Ck, Ck1];
end
k=-floor(Nk/2):floor(Nk/2);
%
%% plot spectrum and reconstructed signal   
if p==1
% plot abs(Ck) vs k and angle(Ck) vs k
%
subplot(2,1,1)
stem(k,abs(Ck),'linewidth',2);
grid on;
xlabel('k');
ylabel('|Ck|');
title('Magnitude Spectrum');

subplot(2,1,2)
stem(k,angle(Ck),'linewidth',2);
grid on;
xlabel('k');
ylabel('\angle Ck');
title('Phase Spectrum');

%
  
% plot 3 cycles of the signal 'x' and the reconstructed signal
%

t = 0:dT:3*T-dT;
x = [x x x];

figure
plot(t,x,'linewidth',2)
grid on;
xlabel('Time, t (s)');
ylabel('Amplitude');
title('Original Waveform');

%
end

end