Question

3. For the periodic signal shown below, find the period T and compute the main harmonics. For our purposes, “main” means having at least 2% of the amplitude of the fundamental harmonic. Use MATLAB to plot the signal for two periods. Also plot approximations to the signal using finitely many harmonics..

For the periodic signal shown below, find the period T and compute the main harmonics. For our purposes, "main" means having at least 2% of the amplitude of the fundamental harmonic. Use MATLAB to plot the signal for two periods. Also plot approximations to the signal using finitely many harmonics r(t) 1 23 4 5 6 7 8

Answer 1

Consider the signal x(r) The signal repeats for every 3 seconds Thus, the time period Tis 3 sec Define the signal -0 2<t<3 Determine the Fourier series of the signal n-1 (t)cos nco,tadt Determine the Fourier coefficients tdtdt 312 +(2-1) 2 t)cosnatdt 2π 3t COSt 2πη 2πη 3 ) 4㎡n2 2πη 32nin 3 ) πη

2π 2Tn 2πη Sih ATn sin+COS The Fourier series is, 2770s(2m)-1 2mn 2722 Sin-t , cos The fundamental harmonics are obtained by substituting 1 for n 3) 2π 2π sin-COS _ ㄧ-0.076-0.276-0.1521 cos 21T --+ 0.1316+0.3175sin (t)--0.504 cos-t +0.45 sin-t

Determine the main harmonics 4π 8T 47 8r 3 ) 2π 4π 87 sin_t 87323 = 1-0019 + 0.1378 +0.038 cos 0.0329 +0.157 sint -1+ 2π 2T f,(r) = 0.1 57cos--t + 0.124sin cos( 2π ) + sin ( 4π) sin ( 2π ) + cos ( 47 ) | sin 2πί J, (r)-0.1 59sin1m Write the MATLAB code to plot the signal and its Fourier series f=sig(t) function f-t; f-l; elseif t<=2 f=0 ; f=t-3; elseif t<=5 else f=0; end fplot (@sig, [0 6]) xlabel t ylabel x (t)

1.2 0.8 0.6 0.4 0.2 4 a0-0.5; ft-a0 t 0:0.001:6,; for n-1: 1000 an (3./ (2.*pi*n.*2)) . *cos (2*pi*n/3) (1/ (pi*n)) . *sin (4. *pi.*n/3) 3/ (2.*pi.~2*n.2); ft-fttan. *cos (n.*2.*pi.*t./3) +bn.*sin (n end plot (t,ft) 1.6 1.2 0.8 0.6 0.4 0.2 0.2

MATLAB code:

function f=sig(t)

if t<=1

    f=t;

elseif t<=2

    f=1;

elseif t<=3

   f=0;

elseif t<=4

    f=t-3;

elseif t<=5

    f=1;

else

   f=0;

end

fplot(@sig,[0 6])

xlabel t

ylabel x(t)

---------------

a0=0.5;

ft=a0;

t=0:0.001:6;

for n=1:1000

an=(3./(2.*pi*n.^2)).*cos(2*pi*n/3)+(1/(pi*n)).*sin(4.*pi.*n/3)-3/(2.*pi.^2*n.^2);

bn=(3./(2.*pi.^2.*n.^2)).*sin(2.*pi.*n./3)+(1./(pi.*n)).*cos(4.*pi.*n./3);

ft=ft+an.*cos(n.*2.*pi.*t./3)+bn.*sin(n.*2.*pi.*t./3);

end

plot(t,ft)