Question

Problem 2: A periodic signalxit) is shown below A =10, T-4 sec. -T Write a MATLAB script to plot the signal, using enough points to get a smooth curve. Compute the Fourier series coefficients for the signal (if you can find them in the text, that is ok). Plot the single-sided or double-sided spectra for each signal. Include enough frequencies in the plots to adequately represent the frequency content of the signals. Plot partial sums of the Fourier series for the signal x(t) (terms 1 through N in the infinite series). Decide what N should be to get a get a good reproduction of the corresponding x(t).

Answer 1

Problem:2

a)For N=6

1)Function script:(save it as ctfss.m and negelct the error during execution)

function [y,t]=ctfss(an,T,N)
k=0;
w0=2*pi/T;
n=-N:1:N
for t=0:0.01:T
k=k+1;
xN(k)=sum(an.*(exp(j*n*w0*t)));
end
t=0:0.01:T
y=xN
endfunction

2)MAIN SCRIPT: save it in any name as you wish and execute after the execution of ctfss,m

clc;
clear all;
close all;
T=4;t=0:0.01:T;A=10;
x=(A*(1-(t/4)))
t1=linspace(-20,20,10*length(t))
x1=[x x x x x x x x x x]
subplot(3,2,[1,2])
plot(t1,x1,'r')
xlabel('t')
ylabel('x(t)')
title('Periodic signal x(t)')
subplot(323)
plot(t,x,'r')
xlabel('t')
ylabel('x(t)')
title('one period of x(t)')

%To caclculate fourier series coefficients of x(t)
w0=2*pi/T;N=6
t=0:0.01:T
k=0;
for n=-N:1:N
k=k+1;
a(k)=(trapz(t,x.*exp(-j*n*w0*t)));
end
a=a/T
n=-N:1:N
subplot(324)
stem(n,abs(a))
title('Amplitude spectra')
xlabel('k..>')
ylabel('|ak|')

subplot(325)
stem(n,angle(a))
title('Phase spectra')
xlabel('k..>')
ylabel('<ak>')

%output xN(t) for N=6
[xN,t]=ctfss(a,T,N)
subplot(326)
plot(t,xN,'r',t,x,'b')
title('xN(t) for N=6')
xlabel('t')
ylabel('xN(t)')

Periodic signal x(t) 10 20 10 -10 -20 Amplitude spectra one period of x(t) 4 4 6 -42 0 4 3 xN(t) for N-6 Phase spectra 12 10 2 4 3 6 4

b)N=64

clc;
clear all;
close all;
T=4;t=0:0.01:T;A=10;
x=(A*(1-(t/4)))
t1=linspace(-20,20,10*length(t))
x1=[x x x x x x x x x x]
subplot(3,2,[1,2])
plot(t1,x1,'r')
xlabel('t')
ylabel('x(t)')
title('Periodic signal x(t)')
subplot(323)
plot(t,x,'r')
xlabel('t')
ylabel('x(t)')
title('one period of x(t)')

%To caclculate fourier series coefficients of x(t)
w0=2*pi/T;N=64
t=0:0.01:T
k=0;
for n=-N:1:N
k=k+1;
a(k)=(trapz(t,x.*exp(-j*n*w0*t)));
end
a=a/T
n=-N:1:N
subplot(324)
stem(n,abs(a))
title('Amplitude spectra')
xlabel('k..>')
ylabel('|ak|')

subplot(325)
stem(n,angle(a))
title('Phase spectra')
xlabel('k..>')
ylabel('<ak>')

%output xN(t) for N=64
[xN,t]=ctfss(a,T,N)
subplot(326)
plot(t,xN,'r',t,x,'b')
title('xN(t) for N=64')
xlabel('t')
ylabel('xN(t)')

Periodic signal x(t) 10 -20 -10 10 20 one period of x(t) Amplitude spectra 4 ー3 -50 100 50 3 4 100 Phase spectra XN(t) for N-64 2 12 10 4 50 50 3 100 -100