Question

Execute the function, using 11 as Non, and 8 as Noff.

function fourier_example(Non,Noff)

%this function demonstrates fourier analysis and synthesis for a simple

%periodic rectangular wave of varying duty cycle.

%Inputs: Non= integer number of seconds per period that the signal is on

% Noff = integer number of seconds per period the signal is off

t=-30:.001:30; %time scale

Nharm = 20;

%number of harmonics to calculate

%construct a periodic square-wave with desired duty cycle

x=zeros(size(t));

for k1=-Nharm:Nharm

x = x + rectpuls( (t-Non/2 - k1*(Non+Noff))/Non );

end

D0 = Non/(Non+Noff); %dc value

T0=Non+Noff; %fundamental period

f0=1/T0;

%plot the original signal

figure(1); clf

subplot(2,1,1)

plot(t,x,'b','LineWidth',2)

grid on

axis([-30 30 -1 2])

title( ['Original periodic signal T0 = ' num2str(T0) ] ) xlabel('magnitude')

ylabel('time (sec.)')

%Now compute the Fourier coefficients.

%Since Matlab doesn't allow negative array indexing, we define

% D0, Dpos and Dneg for convenience.

%We will also assume we know the solution to the integral and program it

%directly into our code for convenience.

for k=1:Nharm

Dpos(k) = (exp(-j*k*2*pi*f0*Non) - exp(-j*k*2*pi*f0*0)) / (-j*k*2*pi); Dneg(k) = (exp(j*k*2*pi*f0*Non) - exp(j*k*2*pi*f0*0)) / (j*k*2*pi);

end

figure(2);clf

subplot(2,1,1)

hold on

stem(-(1:Nharm),abs(Dneg),'b','LineWidth',2)

stem(0,abs(D0),'b','LineWidth',2)

stem((1:Nharm),abs(Dpos),'b','LineWidth',2)

grid on

hold off

xlabel('Harmonic Index')

title('Magnitude Spectrum')

subplot(2,1,2)

hold on

stem(-(1:Nharm),angle(Dneg),'b','LineWidth',2)

stem(0,angle(D0),'b','LineWidth',2)

stem((1:Nharm),angle(Dpos),'b','LineWidth',2)

grid on

hold off

xlabel('Harmonic Index')

title('Phase Spectrum (in radians)')

%Now reconstruct the original signal from sinusoids using Fourier synthesis

x_recover=D0*ones(size(x)); %start with dc component

for k=1:Nharm

e1 = exp(j*k*2*pi*f0*t);

e2 = exp(-j*k*2*pi*f0*t);

x_recover=x_recover + Dpos(k)*e1 + Dneg(k)*e2;

figure(1)

subplot(2,1,2)

plot(t,x,'b','LineWidth',2)

grid on

hold on

plot(t,x_recover,'r','LineWidth',2)

hold off

axis([-30 30 -1 2])

title( ['Reconstructed Signal with ' num2str(k) ' harmonics.' ] )

xlabel('magnitude')

ylabel('time (sec.)')

pause(1)

end

Answer 1

%Main Script

%Call for function fourier_example using 11 as Non, 8 as Noff
Non = 11;
Noff = 8;
fourier_example(Non, Noff);

%Function definition

function fourier_example(Non,Noff)
%this function demonstrates fourier analysis and synthesis for a simple
%periodic rectangular wave of varying duty cycle.
%Inputs: Non= integer number of seconds per period that the signal is on
% Noff = integer number of seconds per period the signal is off

t=-30:.001:30; %time scale
Nharm = 20;

%number of harmonics to calculate
%construct a periodic square-wave with desired duty cycle
x=zeros(size(t));
for k1=-Nharm:Nharm
x = x + rectpuls( (t-Non/2 - k1*(Non+Noff))/Non );
end

D0 = Non/(Non+Noff); %dc value
T0=Non+Noff; %fundamental period
f0=1/T0;

%plot the original signal
figure(1);
clf
subplot(2,1,1)
plot(t,x,'b','LineWidth',2)
grid on
axis([-30 30 -1 2])
title( ['Original periodic signal T0 = ' num2str(T0) ] )
xlabel('magnitude')
ylabel('time (sec.)')

%Now compute the Fourier coefficients.
%Since Matlab doesn't allow negative array indexing, we define
% D0, Dpos and Dneg for convenience.
%We will also assume we know the solution to the integral and program it
%directly into our code for convenience.

for k=1:Nharm
Dpos(k) = (exp(-j*k*2*pi*f0*Non) - exp(-j*k*2*pi*f0*0)) / (-j*k*2*pi); Dneg(k) = (exp(j*k*2*pi*f0*Non) - exp(j*k*2*pi*f0*0)) / (j*k*2*pi);
end

figure(2);
clf
subplot(2,1,1)
hold on
stem(-(1:Nharm),abs(Dneg),'b','LineWidth',2)
stem(0,abs(D0),'b','LineWidth',2)
stem((1:Nharm),abs(Dpos),'b','LineWidth',2)
grid on
hold off
xlabel('Harmonic Index')
title('Magnitude Spectrum')
subplot(2,1,2)
hold on
stem(-(1:Nharm),angle(Dneg),'b','LineWidth',2)
stem(0,angle(D0),'b','LineWidth',2)
stem((1:Nharm),angle(Dpos),'b','LineWidth',2)
grid on
hold off
xlabel('Harmonic Index')
title('Phase Spectrum (in radians)')
  
%Now reconstruct the original signal from sinusoids using Fourier synthesis
x_recover=D0*ones(size(x)); %start with dc component
for k=1:Nharm
e1 = exp(j*k*2*pi*f0*t);
e2 = exp(-j*k*2*pi*f0*t);
x_recover=x_recover + Dpos(k)*e1 + Dneg(k)*e2;
figure(1)
subplot(2,1,2)
plot(t,x,'b','LineWidth',2)
grid on
hold on
plot(t,x_recover,'r','LineWidth',2)
hold off
axis([-30 30 -1 2])
title( ['Reconstructed Signal with ' num2str(k) ' harmonics.' ] )
xlabel('magnitude')
ylabel('time (sec.)')
pause(1)
end
end

Screenshot of Plot: