PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

444 Practical MATLAB® Applications for Engineers

The FM spectrums can be obtained by the expansion of the time function in terms of

the Bessel functions, illustrated as follows:

Awtkmt Ajkwtjk wwt ww[cos( 00 
( ))] { ( )cos 0  100 ( )[cos( m) cos( mm

mm

m

t

jk w w t w w t

jk w w t

)]

( )[cos( ) cos( ) ]

( )[cos( )





20 0

30

22

3



ccos( ) ]

( )[cos( ) cos( ) ]

( )[cos

wwt

jk w w t w w t

jk

m

mm

n

0

40 0

3

44







...

(()cos()]}w nwt 00  mmw nwt

where wm is the highest frequency of m(t), and jn(k) denotes the Bessel function of the

fi rst kind and nth order (referred to the command besselj in MATLAB).

In general, the jn(k)s are negligible for n > k, if k > 1.

MATLAB Solution

% Script file : telecom _ signals

syms t w m _ t c _ t AM FM fm

m _ t = 3*cos(5*t);

c _ t = cos(50*t);

disp (‘*********************************************************************’)

disp (‘************************* R E S U L T S

*******************************’)

disp (‘The spectrum of the communication signals are:’)

disp(‘*********************************************************************’)

Cw _ spectrum _ of _ carrier=fourier(c _ t)

disp (‘*********************************************************************’)

Mw_spectrum_of_message = fourier(m_t)

disp(‘**********************************************************************’)

MCw _ spectrum _ carry _ plus _ message = fourier(m _ t+c _ t)

disp(‘**********************************************************************’)

AM _ DSB _ SC _ spectrum _ =fourier(c _ t*m _ t)

disp(‘**********************************************************************’)

AM _ DSB _ WC _ spectrum _ =fourier(c _ t*(m _ t+5))

disp(‘**********************************************************************’)

fm=besselj(3,0)*cos(50*t)+besselj(3,1)*(cos((50+5)*t)-cos((50-5)*t));

fm=fm+besselj(3,2)*(cos((50+5*2)*t)+cos((50-5*2)*t))+besselj(3,3)*(cos

((50+5*3)*t)- c os((50-5*3)*t));

fm= fm+besselj(3,4)*(cos((50+5*4)*t)+cos((50-5*4)*t));

fm=fm+besselj(3,5)*(cos((50+5*5)*t)-cos((50-5*5)*t));

spectrum _ fm=fourier(fm);

FM _ spectrum=vpa(spectrum _ fm,3)

disp(‘**********************************************************************’)

figure(1)

subplot(3,2,1);

fplot(‘3*cos(5*t)’,[0 3 -6 6]);

title(‘TIME DOMAIN’);

ylabel(‘message(t)’);

subplot(3,2,2)

w_mes=[-5 5];

mag _ mes=[3*pi 3*pi];

stem(w_mes,mag_mes)

title(‘FREQUENCY DOMAIN’)