Question

3. One period of a signal is given by the following equation: +1 1 0<t <3 x(t) = 3 NI+ 3 st 35 5 st 57 N Hint: Use the heaviside function in MATLAB to define x(t) for each time interval. Compute and plot for two periods the approximations of x(t) using 1. Complex Exponential Fourier Series computing 7 and 15 terms 2. Trigonometric Fourier Series computing 11 and 17 terms Note: You should get two figures at the end where each figure has three subfigures as shown below 0.5 Original function X(t) Approximated (t) using x terms 0.5 09 0 Approximated x(t) using x terms

Answer 1

1)Script:

T=7;

w0=2*pi/T;

t=linspace(0,T,1000);

xt=(((t/2)+1).*(heaviside(t)-heaviside(t-3)))+((3-(t/2)).*(heaviside(t-3)-heaviside(t-5)))+(2.*(heaviside(t-5)-heaviside(t-7)));

tt=linspace(0,2*T,2*length(t));

x=[xt xt];

figure;subplot(311)

plot(tt,x);grid;xlabel('t');ylabel('x(t)')

title('Original function x(t)')

%complex exponential Fouier series

i=1;

for K=[7 15]

i=i+1;

m=0;

for k=-K:1:K

m=m+1;

C(m)=(1/T)*trapz(t,xt.*exp(-j*k*w0*t));

end

k=-K:1:K;

%plot x(t) using fs

m=0;

for tt=linspace(0,2*T,2*length(t));

m=m+1;

xNt(m)=sum(C.*(exp(j*k*w0*tt)));

end

tt=linspace(0,2*T,2*length(t));

subplot(3,1,i)

plot(tt,xNt,'r');grid;

xlabel('t');

ylabel('x(t)')

title(sprintf('Approximated x(t) using %d tersm',K))

end

Original function x(t) 2 1.5 (7) 0.5 2 12 14 8 10 t Approximated x(t) using 7 tersm 2 1.5 1 0 2 4 8 12 14 6 10 t Approximated x(t) using 15 tersm 3 N x(t) 2 8 LO 12 14 t

2)Script:

clc;close all;clear all;

T=7;w0=2*pi/T;

t=linspace(0,T,1000);

xt=(((t/2)+1).*(heaviside(t)-heaviside(t-3)))+((3-(t/2)).*(heaviside(t-3)-heaviside(t-5)))+(2.*(heaviside(t-5)-heaviside(t-7)));

t1=linspace(0,2*T,2*length(t));

x=[xt xt];

figure;

subplot(311);plot(t1,x,'b')

xlabel('t');grid;

ylabel('x(t)')

title('Original signal x(t)')

%Trigonometric Fourier series coefficients

i=1;

for K=[11 17];

i=i+1;

a0=(1/T).*trapz(t,xt);

k=0;

for n=1:1:K

k=k+1;

an(k)=(trapz(t,xt.*cos(n*w0*t)));

end

an=2*an/T;

k=0;

for n=1:1:K

k=k+1;

bn(k)=(trapz(t,xt.*sin(n*w0*t)));

end

bn=2*bn/T;

n=1:1:K;

%Reconstruction of original signal

k=0;

for t1=linspace(0,2*T,2*length(t))

k=k+1;

Xr(k)=a0+sum((an.*(cos(n.*w0*t1)))+(bn.*(sin(n*w0*t1))));

end

subplot(3,1,i)

t1=linspace(0,2*T,2*length(t));

plot(t1,Xr,'r')

xlabel('t');grid;

ylabel('x(t)')

title(sprintf('Approximated x(t) using %d terms',K))

end

Original signal x(t) 2 x(t) 1.5 0.5 2 8 10 12 14 Approximated x(t) using 11 terms 2 21.5 1 0 2 4 8 12 14 6 10 t Approximated x(t) using 17 terms 3 N x(t) 2 8 LO 12 14 t