Communication System Analysis

Transcription

1 Communicaion Sysem Analysis

2 Communicaion Sysems A naïve, absurd communicaion sysem 12/29/10 M. J. Robers - All Righs Reserved 2

3 Communicaion Sysems A beer communicaion sysem using elecromagneic waves o carry informaion 12/29/10 M. J. Robers - All Righs Reserved 3

4 Communicaion Sysems Problems Anenna inefficiency a audio frequencies All ransmissions from all ransmiers are in he same bandwidh, hereby inerfering wih each oher Soluion Frequency muliplexing using modulaion 12/29/10 M. J. Robers - All Righs Reserved 4

5 Communicaion Sysems Double-Sideband Suppressed-Carrier (DSBSC) Modulaion Modulaor y( ) = x( )cos( 2πf c ) 12/29/10 M. J. Robers - All Righs Reserved 5

6 Communicaion Sysems Double-Sideband Suppressed-Carrier (DSBSC) Modulaion Y( f ) = X f ( ) 1 2 δ f f c ( ) + δ ( f + f c ) [ ] Frequency muliplexing is using a differen carrier frequency f c for each ransmier. 12/29/10 M. J. Robers - All Righs Reserved 6

7 Communicaion Sysems Double-Sideband Suppressed-Carrier (DSBSC) Modulaion Typical signal received by an anenna Synchronous Demodulaion 12/29/10 M. J. Robers - All Righs Reserved 7

8 Communicaion Sysems Double-Sideband Transmied-Carrier (DSBTC) Modulaion () () ( y = K + m x Ac cos 2π f c ) Modulaor m = 1 12/29/10 M. J. Robers - All Righs Reserved 8

9 Communicaion Sysems Double-Sideband Transmied-Carrier (DSBTC) Modulaion Carrier Carrier 12/29/10 M. J. Robers - All Righs Reserved 9

10 Communicaion Sysems Double-Sideband Transmied-Carrier (DSBTC) Modulaion Envelope Deecor 12/29/10 M. J. Robers - All Righs Reserved 10

11 Communicaion Sysems Double-Sideband Transmied-Carrier (DSBTC) Modulaion 12/29/10 M. J. Robers - All Righs Reserved 11

12 Communicaion Sysems Single-Sideband Suppressed-Carrier (SSBSC) Modulaion 12/29/10 M. J. Robers - All Righs Reserved 12

13 Communicaion Sysems Single-Sideband Suppressed-Carrier (SSBSC) Modulaion 12/29/10 M. J. Robers - All Righs Reserved 13

14 Angle Modulaion Ampliude modulaion varies he carrier ampliude in proporion o he informaion signal. Angle modulaion varies he carrier phase angle in proporion o he informaion signal. Le he carrier be of he form A c cos ω c form y θ c ( ) and le he modulaed carrier be of he ( ) = A c cos( θ c ( ) ) or y( ) = A c cos( ω c + Δθ ( ) ) where ( ) = ω c + Δθ ( ) and ω c = 2π f c. If Δθ ( ) = k p x( ) where x( ) is he informaion signal his kind of angle modulaion is called phase modulaion (PM). 12/29/10 M. J. Robers - All Righs Reserved 14

15 Angle Modulaion In an unmodulaed carrier he radian frequency is ω c. If we differeniae he sinusoidal argumen ω c of an unmodulaed carrier wih respec o ime we ge he consan ω c. So one way of defining he radian frequency of a sinusoid is as he derivaive of he argumen of he sinusoid. We could similarly define cyclic frequency as he derivaive of he argumen divided by 2π. If we apply ha definiion o he modulaed angle ( ) = ω c + Δθ θ c frequency or ω f ( ) = 1 2π ( ) we ge a funcion of ime ha is defined as insananeous ( ) = d d θ c ( ) d d θ c ( ) = ω c + d d ( ( ) ) = f c + 1 2π ( Δθ ( ) ) radian frequency d d ( Δθ ( ) ) cyclic frequency 12/29/10 M. J. Robers - All Righs Reserved 15

16 Angle Modulaion In phase modulaion he insananeous radian frequency as a funcion d of ime is ω ( ) = ω c + k p ( x( ) ). If we conrol he derivaive of he d phase wih he informaion signal insead of conrolling he phase direcly wih he informaion signal and d d ( Δθ ( ) ) = k f x( ) ω ( ) = ω c + k f x( ) and f ( ) = f c + k f 2π x This ype of angle modulaion is called frequency modulaion (FM). ( ) 12/29/10 M. J. Robers - All Righs Reserved 16

17 Angle Modulaion 12/29/10 M. J. Robers - All Righs Reserved 17

18 Angle Modulaion 12/29/10 M. J. Robers - All Righs Reserved 18

19 Angle Modulaion For phase modulaion y PM ( ) = A c cos ω c + k p x ( ) ( ) For frequency modulaion y FM ( ) = A c cos ω c + k f x( τ )dτ There is no simple expression for he CTFT's of hese signals in he general case. Using cos( x +y) = cos( x)cos( y) sin( x)sin( y) we can wrie y PM ( ) = A c cos( ω c )cos( k p x( ) ) sin( ω c )sin k p x 0 ( ) ( ) and y FM ( ) = A c cos( ω c )cos k f x( τ )dτ sin( ω c )sin k f x( τ )dτ /29/10 M. J. Robers - All Righs Reserved 19

20 Angle Modulaion If k p and k f are small enough cos k p x( ) and cos k f 0 Then y PM x( τ )dτ 1 and sin k f ( ) A c cos ω c ( ) ( ) 1 and sin( k p x ) k p x 0 ( ) k p x x( τ )dτ k f ( ) ( )sin ω c and y FM ( ) A c cos( ω c ) sin( ω c )k f x( τ )dτ 0 These approximaions are called narrowband PM and narrowband FM and we can find heir CTFT's. x( τ )dτ. 0 ( ) 12/29/10 M. J. Robers - All Righs Reserved 20

21 Angle Modulaion Y PM Y FM or ( ω ) ( A c / 2) { 2π δ ( ω ω c ) + δ ( ω + ω c ) jk p X( ω + ω c ) X( ω ω c ) } X( ω + ω ( ω ) ( A c / 2) 2π δ ( ω ω c ) + δ ( ω + ω c ) k c ) f X ( ω ω c ) ω + ω c ω ω c Y PM ( f ) ( A c / 2) { δ ( f f c ) + δ ( f + f c ) jk p X( f + f c ) X( f f c ) } Y FM ( f ) ( A c / 2) δ ( f f c ) + δ ( f + f c ) k f X( f + f c ) X ( f f c ) 2π f + f c f f c ( on he assumpion ha he average value of x( ) is zero) 12/29/10 M. J. Robers - All Righs Reserved 21

22 Angle Modulaion If he informaion signal is a sinusoid x( ) = A m cos( ω m ) = A m cos( 2π f m ) hen X( f ) = ( A m / 2) δ ( f f m ) + δ ( f + f m ) Y PM ( f ) ( A c / 2) δ ( f f c ) + δ ( f + f c ) ja k δ f + f m p ( c f m ) + δ ( f + f c + f m ) 2 δ ( f f c f m ) δ ( f f c + f m ) Y FM ( f ) ( A c / 2) δ ( f f c ) + δ f + f c ( ) and, in he narrowband approximaion, A m k f 4π f m ( ) δ ( f + f c + f m ) ( ) + δ ( f f c + f m ) δ f + f c f m δ f f c f m 12/29/10 M. J. Robers - All Righs Reserved 22

23 Angle Modulaion Narrowband PM and FM Specra for a Sinusoidal Informaion Signal 12/29/10 M. J. Robers - All Righs Reserved 23

24 Angle Modulaion Narrowband PM and FM Specra for a Sinc Informaion Signal 12/29/10 M. J. Robers - All Righs Reserved 24

25 Angle Modulaion If he narrowband approximaion is no adequae we mus deal wih he more complicaed wideband case. For FM y FM ( ) = A c cos( ω c )cos k f x( τ )dτ sin( ω c )sin k f x( τ )dτ 0 0 If he modulaion is x( ) = A m cos( ω m ), y FM ( ) = A c cos( ω c )cos k f A m sin( ω m ) ω m sin ( ω c )sin k f A m sin ω m ω m Le m = k f A m /ω m, he modulaion index. Then y FM ( ) = A c cos( ω c )cos msin ω m ( ( )) sin ω c ( )sin msin ω m ( ) ( ( )) 12/29/10 M. J. Robers - All Righs Reserved 25

26 Angle Modulaion ( ( )) sin ω c ( ( )) and sin msin( ω m ) In y FM ( ) = A c cos( ω c )cos msin ω m cos msin ω m ( ( )) ( )sin msin ω m ( ) are periodic wih fundamenal period 2π /ω m. Therefore hey can each be expressed as a Fourier ( ) = c c k series. For example, cos msin( ω m ) c c [ k] = ω m 2π 2π /ω m cos msin( ω m ) ha cos( ω c )cos msin ω m k= [ ]e jkω m.wih ( )e jkω m d. I hen follows ( ( )) = 1 2 c c [ k] e j ( kω m +ω ) c + e j( kω m ω ) c. k= The CTFS harmonic funcion can be wrien in he form c c [ k] = ω m 4π π /ω m π /ω m e j m sin ( ω m) kω m + e j m sin ω m ( ) kω m d 12/29/10 M. J. Robers - All Righs Reserved 26

27 Angle Modulaion The inegral c c [ k] = ω m 4π can be evaluaed using J k π /ω m π /ω m ( z) = 1 2π e j m sin ( ω m) kω m + e j m sin ω m ( ) kω m π e j ( zsin( λ) kλ ) dλ where J k π d ( ) is he Bessel funcion of he firs kind of order k. One useful propery of his Bessel funcion is J k ( z) = J k ( z). 12/29/10 M. J. Robers - All Righs Reserved 27

28 Angle Modulaion 12/29/10 M. J. Robers - All Righs Reserved 28

29 Angle Modulaion I can be shown (and is in he ex) ha, for cosine-wave frequency modulaion, Y FM or ( f ) = A c 2 k= J k ( ( )) + J k m ( m)δ f kf m + f c ( ) + δ ( f + f c ) ( ( )) ( )δ f kf m f c J 0 ( m) δ f f c Y FM ( f ) = A c J k ( m)δ ( f ( kf m + f c )) + J k ( m)δ ( f ( kf m f c )) 2 + k=1 + J k ( m)δ ( f ( kf m + f c )) + J k ( m)δ ( f ( kf m f c )) The impulses in he FM specrum exend in frequency all he way o infiniy. Bu beyond mf m (where m is he modulaion index and f m is he cyclic frequency of he modulaing cosine) he impulse srenghs die rapidly. For pracical purposes he bandwidh is approximaely 2mf m. 12/29/10 M. J. Robers - All Righs Reserved 29

30 Angle Modulaion Wideband FM Specrum for Cosine-Wave Modulaion 12/29/10 M. J. Robers - All Righs Reserved 30

31 Discree-Time Modulaion Discree-ime DSBSC modulaion of a sinusoidal [ ] = cos( 2πF 0 n) carrier c n y[ n] = x[ n]c[ n] = x[ n]cos 2πF 0 n ( ) 12/29/10 M. J. Robers - All Righs Reserved 31

32 Discree-Time Modulaion Y( F) = ( 1/ 2) X F F 0 ( ) + X( F + F 0 ) 12/29/10 M. J. Robers - All Righs Reserved 32