Question

Find a Fourier series representation in the form x(t)-xp ol + 〉 2 KI k || cos(kat+ X | k |) of a. に! the impulse train and plot the spectrum of the series through the 5th harmonic. Write out the first five terms of the Fourier series of x(t) b. Now, find a Fourier series representation in the form x(t)=X[0] +Σ2k[k] cos(kay + X[k]) of the following (periodic) square wave に! 0 To To/2 To and plot the spectrum of the series through the 5th harmonic. Write out the first five terms of the x(t) and, using Matlab, plot three periods of the Fourier series approximation of x() Fourier series of for k-O 100 with ao-2π and A-1

Answer 1

Qa) It is calculated using the definition of FS

Qb) To solve this, integration property of FS is made used

a) We have X [k] = 1/T_0 integral^T_0/2_-T_0/2 delta (t) e^-j k 2 pi f_0 t dt = 1/T_0 Therefore X [k] = 1/T_0 Forall k X [k] is plotted for first 5 harmonics as below Q b] Let us define a signal Y (t) = d/dt x (t). Then Y (t) is represented as below Y [k] = 1/T_0 integral^T_0/2_-T_0/2 2A [delta (t + T_0/4) - delta (-T_0/4)] e^-j k 2 pi f_0 t dt \r\n Y [k] = 2A/T_0 [e^+ j k pi f_0 t_0/2 - e^-j k pi f_0 T_0/2] = 2 A/T_0 times 2 j times sin (pi k f_0 T_0/2) = 4 Aj/T_0 sin (k pi/2) Because f_0 = 1/T_0 Now since Y (t) = d/dt x (t): Using Fourier series properties we can write X [k] = Y [k]/j k 2 pi f_0 = 4 Aj sin (k pi/2)/T_0 times j k 2 pi f_0 X [k] =
The required MATLAB code is as below

clc
X1=[2/(5*pi),0,-2/(3*pi),0,2/pi,0,2/pi,0,-2/(3*pi),0,2/(5*pi)]; % this array contains values of FS of x(t)
n1=-5:1:5;
figure(1) % this plots spectrum of x(t) up to first 5 harmonics
stem(n1,X1);
title('plot of Fourier series of x(t)');
xlabel('k');
ylabel('X[k]');

% now X[k] is constructed for -100<k<100
k=1;
j=sqrt(-1);
for n=-100:1:100
if (n==0)
X(k)=0;
else
X(k)=(2*sin(n*pi/2))/(n*pi);
end
k=k+1;
end

%calculation of reconstructed x(t) from its FS coefficients

tp=1;
for tt=0:0.01:5
s=0;
for k=-100:1:100
s=s+X(k+101)*exp(+j*2*pi*k*tt);
end
xp(tp)=s;
tp=tp+1;
end

tt=0:0.01:5;
figure(2) % plot of reconstructed x(t)
plot(tt,real(xp));
title('plot of reconstructed x(t)from X[k]');
xlabel('t');
ylabel('xp(t)');

The plots are shown below