Transcript
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PLH22_GE2_2010_v2.pdf
PLH22_Ergasia2_ORIETAS.pdf
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.. 22 /
.. 22 (2010-11)2
( 4,6,7)
: 2 , , .
2
(7) 3,4,5 ( ) 3,5
: -- ( ).: Fourier Fourier .
:x( x) x( x A )/ /
1 DSB, FM
. : 1/2/0809 sinc , 0x t a at a .
:
(i) DSB cA
c
f .
(ii) FM cA
c
f . f
k .
:
()
(i) (ii).() DSB .
a ;
Nyquist;()
(i) (ii).
:
.
.
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.. 22 /
2 Fourier. : 2/2007, 4/5/0708, 5/1/0910
x(t) 1 1
( ) cos 24 4
o
o o
X f ft u f u ft t
, (to=1/16)
()
x(t).
() To x(t)
100KHz DSB
SSB .
o .
: x(t)
.
3
. :2/0809/2
tftx 11 2cos2)( 2 2( ) 4cos 2x t f t , (f1>f2),
2)2()(
21
txtxty . y t
(f
) 20 Nyquist.
()
y n n Y f .
()
PCM 1
40PCM
B f .
.
:
. ,
PCM.
4 ,
. : 4/2/0910.
:
41( ) sin (1000 )x t c t , 1500| |2( ) 1000
tx t e , 3 ( ) sin (500 )sin(500 )x t c t t
,
(FDMA) U(pper)SSB
fci. (i=1,2,3) .
10kHz 20KHz. xi(t) ,
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fcut, Xi(fcut) 10% Xi(0).
() .
()
( ).
() fci 1Hz
Xi-1(f) ( fc1)
.
: () Fourier
.
.: g(t), h(t) g(t)
( G f a ) h(t) b (
H f b )
,
*G f H f a b
5 - Fourier, .
x(t) Fourier:
2000 2000( )
1000 2000 2000
f f fX f
1000 3000 Hz 0 2000 Hz 0 1000 Hz
2000 3000 Hz y(t).
() x(t) y(t) .() y(t),
, (
xi(t)+xj(t)) (
cos 2i i iC t f t ).
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.. 22 /
w1 (t)
z1(t)z(t)
w(t)
y2(t)
y1(t)
x(t)
H1(f)
H2(f)
C1(t)
C2(t)
H3(f)
H4(f)
y(t)
x
x
+
C1(t) C2(t).
:. ()
( )
y(t).
6 (PM,
FM), . : 2/ - , 1/2/0910
1 21 2
50 50100 cos 1000 sin 2 sin 2
2 2ax t t f t f t
f f
, 1 25f f .
()
( 1f ) . ,
( 1f ).
() ( 1 2,f f )
(PM) 25prad
KVolt
.
() ( 1 2,f f )
(FM) sec50f
rad
kVolt
.
() (),
2 100f Hz . : Carson
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.. 22 /
. , PM FM .
: :
1 2max cos 2 cos 2 1 1 2f t f t
7 ,
,
PCM. : . 7/2/0910
1MHz.
(FDMA) ( ):
1x t 1 100W a KHz
(DSB) .
2x t 2 150W KHz
(FM)
D=.
( ) 3x t 3 50W a KHz
(PCM). PCM
30dB.
4x t 4 50W a KHz
(PCM).
4x t
PCM 0,5% peak-to-peak 4x t .
()
.()
. : 3x t
PCM (
L ) (
decibel) 21010 logSNR L . 4x t o .
peak-to-peak x t
( )
max minp px x x
maxx
minx . ,
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.. 22 /
0cos 2x t A B f t , , peak-to-peak max min 2p px x x A B A B B
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.. 22 /
: 1 14
()4
()6
()
4
2 11
()5
()6
3 9
() 4
()5
4 15
()
5
()5
()5
5 14
()6
()8
6 24
()7
()6
()6
()5
7 13
()8
()5
100
10,
.
1. - 09 2011, 23:59.
2. (e-mail) (.. MS-Word).
3. 14 2011 site .. ,
http://p-comp.di.uoa.gr/eap/index.html. !!!
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.:56795 2 11/01/2011 1
1.
()
i) DSB ,
( ) cos(2 )c c cx t A f t :
( ) ( ) cos(2 )DSB c cx t x t A f t .
( ) sinc( ) cos(2 )DSB c cx t A a at f t
ii) FM ,
( ) cos(2 )c c cx t A f t :
( ) cos 2 ( )t
FM c c fx t A f t k x d
( ) cos 2 sinc( )
t
FM c c fx t A f t k a a d
()
sinc( ) rect( )F
t f
1sinc( ) rect( )
fFat
a a
( )x t
:
1( ) sinc( ) rect( ) rect( ) ( )
f fFx t a at a X f
a a a , (>0)
( )x t 1,
0 0f .
cf a .
( ) ( ) cos(2 )DSB c cx t x t A f t :
( ) ( ) rec ct rect (os(2 ) )2
c c cDSS BD B c c
Fx t
A f f f f X f
a ax t A f t
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.:56795 2 11/01/2011 2
. 1.1
2caf
2caf
cf 2caf
cf 2caf
2cA
( )DBSX f
. 1.1
0
Nyquist
maxmin 2 2 2c af f f
max
min
1 1
2 cT
f f a
()
( )x t 2x
aW Hz . ,
i)
DSB 2 22DSB x
aW W Hz
DSBW a Hz
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.:56795 2 11/01/2011 3
ii)
FM :
2 1 2 1x xFMW D W D W (1)
o D
2
f x f x
x x
k A k A
f
xA a x xf W
2 22
f f f
x
k a k a k DaW
(1)
22 1
f
FMa
kW
1f
FM a Hz
kW
2.
()
( )x t 1 1
( ) cos 24 4
o
o o
X f ft u f u ft t
-
0t 1/16
( ) cos 2 4 4oX f ft u f u f 4 4 rect 8f
u f u f
( ) cos 2 rect8
o
fX f ft
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.:56795 2 11/01/2011 4
( )x t -
, 0 0f
=8. . 2.1
( ) cos(2 )c c
x t f t
4u f
4u f
rect 8
f
( )X f
. 2.1
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.:56795 2 11/01/2011 5
( )X f 4 Hz2
x
aW
( Nyquist) :
min 2 8 Hzxf W
( )x t . :
)0 02 2
0 01 0 1
( ) ( )( ) ) ( )
2 2cos(2
j ft j ftt t t t e e F
f x tX ft
) rect( ) sinc( )F
f t
2 2(f)=rect 8sinc(8 ) (t)8
f FX t x
) 1 2 1 2( ) ( ) ( ) ( ) ( ) ( )FX f X f X f x t x t x t ( )
, 1 2 1 2( ) ( ) ( )cos 2 rect ( ) ( )8
o
f FX f X f X f ft x t x t
0 0( ) ( ) 8sinc(8 ) ( )2
t t t t t x t
:
0 0 0 0( ) ( )
8sinc(8 ) 4 sinc (8 ) ( ) sinc (8 ) ( )
2
t t t t t x t t t x t t t
0 0( ) ( ) ( )f t t t f t t
0 0 0 04 sinc(8 ) ( ) sinc(8 ) ( ) 4 sinc(8 8 ) sinc(8 8 )t t t t t t t t t t
, 0t 1/16
1 1( ) 4 sinc 8 sinc 8
2 2x t t t
()
DSB 3cos(2 100 10 )t
( )x t , ( )x t
0 100 kHzf . ( )x t ,
4 Hz, [(-100,004)kHz ,(99,996)kHz]
[99,996kHz, 100,004)kHz]. SSB ( )x t .
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.:56795 2 11/01/2011 6
0 100 kHzf .
0 0( )HPFH f u f f u f f ,
0
( ) 1 rect2
HPF
fH f
f
5( ) 1 rect
2 10HPF
fH f
2.2 ( )DSBX f , ( )HPFH f
( )SSBX f .
( )DSBX f
( )HPFH f
( )SSBX f
100 kHz-100 kHz
-100,004 kHz -99,996 kHz 99,996 kHz 100,004 kHz
0
u f f 0
u f f
. 2.2
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.:56795 2 11/01/2011 7
( )HPF
h f
( ) ( )HPF HPFF
fH h f .
:
1 ( )F
t
rect( ) sinc( )F
f t
5 5
5rect 2 10 sinc(2 10 )
2 10
f Ft
5 5( ) 2 10 sinc(2( ) 10 ) ( )HPF HPFt t
FfH h f
3.
1 2
21
( ) (2 )2
( ) 2 cos 2 2 4cos 22
ty t x t x
fy t f t t
1 2f f ( )y t 1max 2f f
min
f
( Nyquist)
1 1maxmin 2 2 2 4f f f f . , y t
1min 8020f f f
()
( )f n :
( ) ( )y n y t , 1
1 1
80t nT n n
f f
.
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.:56795 2 11/01/2011 8
21
1 1
1 1( ) 2 cos 2 2 4cos 2
280 80
fy n f n n
f f
1 2
1 1
1 1( ) 2 cos 2 4cos 240 160
y n f n f nf f
( )Y f ( )y t ,
( )Y f .
1 1( ) 2 cos 2 2y t f t 22 ( ) 4cos 2 2f
y t t
1 1( ) ( )F
y t Y f , 2 2( ) ( )F
y t Y f 1 2( ) ( ) ( )F
Y f Y f Y f
Fourrier
1 1 1 1 12
( ) 2 cos 2 2 2 2 ( )2
Fy t f t f f f f Y f
2 22
2 2
4( ) 4cos 2 ( )
2 2 2 2
f fFfy t t f f Y f
2 21 12
( ) 2 2 22 2 2
f fY f f f f f f f
( ) ( )m
Y f f Y f mf
2 21 1 12
2 2 22 2 2
( ) 80m
f fY f f mf f f mf f mf f mf f f
()
2
1log
2PCMB Lf (1)
L . (1)
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.:56795 2 11/01/2011 9
12 2 2
1
2 2 40log log log 1
80PCM
B fL L L
f f
2L
.
4.
()
4 2 21( ) sinc (1000 ) sinc (1000 ) sinc (1000 )x t t t t
sinc( ) triF
t f
1sinc(1000 ) tri
1000 1000
fFt ,
2 2 1 1sinc (1000 ) sinc (1000 ) tri tri1000 1000 1000 1000
f fFt t
1tri
1000 1000
f 1000triW (
) ,
1( )x t
1 tri trixW W W
12000xW
Fourrier 2( )x t
61500| |
2 2 2 2 2 2
2 1500 3 101000 1000
1500 4 1500 4
t Fef f
2( )x t
2xW
, 3 ( ) sinc(500 )sin(500 )x t t t
1sinc(500 ) rect
500 500
fFt
1
sin(500 ) 250 2502
Ft f f
j 0 02 500 250f t t f
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.:56795 2 11/01/2011 10
1 1
sinc(500 ) sin(500 ) rect 250 250500 500 2
fFt t f f
j
rect 250W ,
250W ( 250
Hz -250 Hz) ,
3( )x t
rect3xW W W
3500xW
4.1 .
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.:56795 2 11/01/2011 11
1,33( f=0)
0,133
( f=fcut)
Fcut=717 Hz
-2000 2000
Wx1
Wx2
500500
Wx3
. 4.1
rect 1434f
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.:56795 2 11/01/2011 12
()
1
( )x t
6
2 2 21
3 10
1500 4xW
f
0f 1x
W 6
21
3 101,33
1500xW
, 10% 1,33 0,133
62
6
2 2 2
3 101500
3 10 0,133
0,133 1500 4 2cut
cutff
717 Hzcutf
rect1434
f
. . 4.1 (
)
()
U(pper)SSB,
. 2000 Hz,
717 Hz 500 Hz . 10 kHz,
12 kHz. 1 kHz
13 kHz 13,717
kHz. 14,717 kHz
15,217 kHz.
4,217 kHz. . 4.2
.
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.:56795 2 11/01/2011 13
10 kHz 12 kHz 13 kHz 13,717
kHz
14,717
kHz
15,217
kHz
Wmx1 Wmx2Wmx3
. 4.2
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.:56795 2 11/01/2011 14
5.
()
2000 2000
( )1000 2000 2000
f f fX f
.
. 2000 Hz
2000 Hz , 0 Hz.
2000 Hz. ( )X f , ( )Y f
( )x t , ( )y t . 5.1
( )X f
( )Y f
. 5.1
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.:56795 2 11/01/2011 15
( )x t
1( )H f - 3( )H f 2 ( )H f 4 ( )H f .
.
i) 1( )H f - 3( )H f
To ( )x t 1( )H f ,
1000 Hz 1( ) 1000 1000H f u f u f
1( ) 1 rect 1 rect2 1000 2000
f fH f
1( )H f
1( )y t .
1( )y t DSB 1 2 cos 2 1000c t t .
1( )y t
0 1000 kHzf ( 0 1000 kHzf ). 1c t 2,
( )W f ,
2 . . 5.2 ( )X f ,
1( )H f , 1( )Y f , 1C f ( )W f .
( )w t
3( )H f 0 2000 Hz
3
2000 2000
2 2( ) rect rect2000 2000
f fH f
3( )H f 2
1( )w t . . 5.4 ( )W f , 3( )H f 1( )W f
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.:56795 2 11/01/2011 16
. 5.2
( )X f
( )1
H f
( )1
Y f
( )
1
fC
( )fW
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.:56795 2 11/01/2011 17
. 5.3
( )X f
( )2
H f
( )2
fY
( )2
C f
( )Z f
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.:56795 2 11/01/2011 18
ii) 2 ( )H f - 4 ( )H f
To ( )x t 2 ( )H f ,
1000 Hz
2 ( ) rect2 1000
fH f
2 ( )H f
2 ( )y t .
2 ( )y t DSB 2 2 cos 2 2000c t t .
2 ( )y t
0 2000 kHzf (0 2000 kHzf ). 2c t 2,
( )Z f . . 5.3
( )X f , 2 ( )H f , 2 ( )Y f , 2C f ( )Z f .
( )z t 4 ( )H f
a 2000 Hz
4( ) 2000 2000H f u f u f
4 ( ) 1 rect 1 rect2 2000 4000
f fH f
4 ( )H f ( )z t , (U)SSB
, 1( )z t . . 5.4 ( )Z f , 4 ( )H f
1( )Z f .
, 1( )w t , 1( )z t -
( )y t , ( )Y f . 5.1
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.:56795 2 11/01/2011 19
. 5.4
( )W f
3 ( )H f
( )Z f
4 ( )H f
1( )Z f
1( )W f
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.:56795 2 11/01/2011 20
6.
()
1 21 2
50 50100 cos 1000 sin 2 sin 22 2
ax t t f t f tf f
(1),
( ) cos(2 ( ))a a cx t A f t t .
ax t
100 VaA
10002 cf
500
Hzcf
2 ( )cf t t
1 21 2
50 50( ) sin 2 sin 2
2 2t f t f t
f f .
aif
1 2
1 2
1 1 2 2
1 2
1 2
1 2
1 ( )
2
1 50 501000 sin 2 sin 2
2 2 2
1 50 501000 2 cos 2 2 cos 2
2 2 2
50025cos 2 25cos 2
50025 cos 2 cos 2
( )
( )
ai
ai
d tf
dt
f t f t f tf f
f f t f f tf f
f t f t
f t f t
t
t
D :
max
max
a
aix
fD
f
(1), 1maxf f
1 2max max 25 cos 2 cos 2aif f t f t
1 2max cos 2 cos 2 1 1 2f t f t
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max 50aif
, 1
50D
f -
max2 ( 1)W D f
max 1
1
502 ( 1) 2 ( 1)
ax xaW D f f
f
1100 2xaW f
()
()
( ) cos cos(( ))a a aa p xxx t A t k tA ,
( ) cos( )x a xxx t tA
(1)
1 2
1 2
1 2
1 2
1 2
1 2
50 50cos( sin 2 sin 2
2 2
50 50sin 2 sin 2
2 2cos(
50 50sin 2 sin 2
2 2cos(
25
)
)
)
p a
ap
a
x
x
x
x
x
x
k f t f t f f
f t f tf f
k
f t f tf f
t
t
t
A
A
A
1 21 2
1 1( ) sin 2 sin 2xx t f t f t
f f
()
(F)
( ) cos ( ) ( )t
a a a f xx t A dt k x
,
( ) ( ) ( )
t
x xx t dx
(1)
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.:56795 2 11/01/2011 22
1 2
1 2
1 2
1 2
50 50( ) ( ) sin 2 sin 2
2 2
50 50sin 2 sin 2
2 2( ) ( )
t
t
f
f
x
x
d f t f t f f
f t f tf f
d
k
k
x
x
1 21 2
50 50sin 2 sin 2
2 2( )
50x
f t f tf fd
x tdt
1 1 2 21 2
50 502 cos 2 2 cos 2
2 2( )
50x
f f t f f tf f
x t
1 2( ) cos 2 cos 2xx t f t f t
()
2 100 Hzf 1 2( ) cos 2 cos 2xx t f t f t
( ) cos 2 500 cos 2 100xx t t f t
1
2
11500
1 5
100
T
T ,
1 25T T T
1sec
100T
500 Hz.
minf ( Nyquist)
min 2 max 1000 Hzff
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.:56795 2 11/01/2011 23
7.
()
DSB - ,
12 2 100 kHzDSBW W a
200 kHzDSBW a
(FM)
D,
22 ( 1)FMW D W
2( 1)150 kHzFMW a
PCM 21010 logSNR L ,
30SNR dB
2 210 1030 10 log 3 log 1000 31,622dB L L L (
2, 32=25)
32L
23
1
2logPCMB f L , max2 2 50 kHz=100 kHzf f f a a ,
23
1100 32
2logPCMB a
3 250 kHzPCMB a
0,5%2
1%pp ppV V
1000,01
pp pp
pp
V VL L
V
L
2, L=128 (128=27)
24
1
2logPCMB f L , max2 2 50 kHz=100 kHzf f f a a ,
24
1100 128
2logPCMB a
4350 kHzPCMB a
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.:56795 2 11/01/2011 24
()
6
3 4
3 6
3 5
5
5
10 Hz
200 300 300 250 350 10 Hz 10 Hz
1100 10 Hz 7 10 Hz
7 10 Hz
11 10 Hz
DSB FM PCM PCMW W B B
a a a a
a
a
0,636a
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