16.362 Signal and System I Amplitude modulation with a complex exponential carrier.
15
16.362 Signal and System I tude modulation with a complex exponential carrier t j c e t c ) ( t j c e t x t y ) ( ) ( ' )) ' ( ( ) ' ( 2 1 ) ( d j C j X j Y ) ( 2 ) ( c j C ) ( ' ) ' ( ) ' ( 2 1 ) ( c c j j X d j X j Y ) ( j X 1 ) ( j Y c 1
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Transcript of 16.362 Signal and System I Amplitude modulation with a complex exponential carrier.
16.362 Signal and System I Amplitude modulation with a complex exponential carrier
tj cetc )(
tj cetxty )()(
'))'(()'(
2
1)(
djCjXjY
)(2)( cjC
)(
')'()'(2
1)(
c
c
jjX
djXjY
)( jX
1
)( jY
c
1
16.362 Signal and System I
)(tx
Implementation
tj cetxty )()(
)sin()()cos()()( ttjxttxty cc
x
)sin( tc
x
)cos( tc
)(Re ty
)(Im ty
16.362 Signal and System I Amplitude modulation with a Sinusoidal carrier
)cos()()( ttytw c
)cos()()( ttxty c )()()( ccjC
)(2
1)(
2
1)( cc jjXjjXjW
)( jX
1
)( jW
c2
2
1
c2
Mc
)2cos(12
1)(
)(cos)()( 2
ttx
ttxtw
c
c
)2(
4
1)(
4
1)(
4
1)2(
4
1
)(2
1)(
2
1)(
cc
cc
jjXjXjXjjX
jjYjjYjW
16.362 Signal and System I Synchronous demodulation
)cos()( ttc c
)cos()()( ttxty c)()()( ccjC
)(2
1)(
2
1)( cc jjXjjXjY
)( jX
1
)( jY
c
2
1
c)( jY
cc
2
1
16.362 Signal and System I
)(tx
Implementation
)cos()()( ttxty c
)cos( tc
x)(ty
x
)cos( tc
co
2
co
)( jH
Low pass filter
)(tw
16.362 Signal and System I Asynchronous demodulation
)cos()()( ttxAty c
)()(2
1)()(
2
1)( cccc jjAjjXjjAjjXjY
)( jX
1
)( jY
c
2
1
c
)( cA )( cA
16.362 Signal and System I
Asynchronous )( jY
c
2
1
c
)( cA )( cA
Synchronous
)( jY
c
2
1
c
16.362 Signal and System I
Asynchronous
Synchronous
16.362 Signal and System I Asynchronous demodulation
)cos()()( ttxAty c
)()(2
1)()(
2
1)( cccc jjAjjXjjAjjXjY
)( jY
c
2
1
c
)( cA )( cA
ss
16.362 Signal and System I Frequency-division multiplexing
)(1 tx x
)cos( 1tc
)(2 tx x
)cos( 2tc
)(txn x
)cos( tcn
+ )(tw
)( jW
1c1c 2c 3c2c3c
16.362 Signal and System I Frequency-division multiplexing -- demodulation
1
cn
)(1 jH
Bandpass filter Low pass filter
1
Mx
)cos( tcn
M
1
cn
)(txn
16.362 Signal and System I Single-sideband sinusoidal amplitude modulation
)( jW
cc
0,
0,)(
j
jjH
)(1 tx
x
)cos( 1tc
x
+
)sin( 1tc
16.362 Signal and System I Single-sideband sinusoidal amplitude modulation
)( jY
cc
0,
0,)(
j
jjH
)(1 tx
x
)cos( 1tc
x
+
)sin( 1tc
16.362 Signal and System I Single-sideband sinusoidal amplitude modulation
)( jY
2/
Single-sideband
double-sideband
)( jX
16.362 Signal and System I clear all;clf;deltat = 0.001;T = 1;t = 0:deltat:20-deltat;alpha1 = -10;x1 = exp(alpha1.*t);wc = 2*pi*10;y1 = x1.*cos(wc.*t);dm = 0.01;om = -16*2*pi:dm:2*16*pi;for m=1:length(om) % Fourier transmform Ym = x1.*exp(-j*om(m).*t); X1m(m) = trapz(t,Ym); if om(m)>=0 X1pm(m) = j*X1m(m); else X1pm(m) = -j*X1m(m); endendfor m=1:length(om) % Fourier transmform Ym = y1.*exp(-j*om(m).*t); Y1m(m) = trapz(t,Ym);endfor k=1:length(t) %inverse fourier transfrom Y1k = X1pm.*exp(j*om*t(k)); x1p(k) = trapz(om,Y1k);endx1p = x1p./(2*pi);y1p = x1p.*sin(wc.*t);y = y1p+y1;for m=1:length(om) % Fourier transmform Ym = y.*exp(-j*om(m).*t); Ypm(m) = trapz(t,Ym);end
plot(t, y1);hold onplot(t,real(y1p),'r');hold off;figure(2)plot(om/(2*pi),abs(X1m));hold on;plot(om/(2*pi),abs(X1pm),'y');plot(om/(2*pi),abs(Y1m),'r'); plot(om/(2*pi),abs(Ypm),'g');hold off;
