Frequency Modulation. Dr. Hwee-Pink Tan

Transcription

1 Frequency Modulaion Dr. Hwee-Pink Tan hp:// Lecure maerial was absraced from "Communicaion Sysems" by Simon Haykin.

2 Ouline Day 1 Day 2 Day 3 Angle Modulaion Frequency Modulaion (FM) Narrowband and Wideband FM Transmission bandwidh FM Sereo Mix Phase-locked Loop (PLL) Non-linear effecs in FM receivers Summary Tuorial

3 Recall Wha is modulaion? Message m() Highes freq W ransmier Sinusoidal carrier c()= A c cos [2f c ] modulaion Direc ransmission Unsuiable/ inefficien Transmission channel Modulaed signal s()=a()cos () receiver m() demodulaion s() obained by varying characerisic of c() according o m() Ampliude A()<-> Ampliude Modulaion Angle () <-> Angle Modulaion

4 Recall Ampliude Modulaion s() = A c [1+k a m()]cos 2f c Envelope of s() has same shape as m() provided: k a m() <1 f c >>W Easy and cheap o generae s() and reverse

5 Recall Ampliude Modulaion Drawbacks of AM waseful of power ransmission of carrier waseful of bandwidh ransmission bandwidh, B T = 2W Improved resource uilizaion (power or bandwidh) raded-off wih increased sysem complexiy Angle modulaion offers pracical means of rading-off beween power and bandwidh

6 Angle modulaion s() = A c cos[ i ()], f i () = 1 d i () 2 d Phase Modulaion (PM) i () = 2 f c + k p m() s() = A c cos[2 f c + k p m()] Frequency Modulaion (FM) f i () = f c + k f m() s() = A c cos[2 f c + 2k f m()d] 0

7 PM vs FM Frequency modulaor Inegraor Phase modulaor FM wave Modulaing wave, m() A c cos(2f c ) s() = A c cos[2 f c + 2k f m()d] 0 Differeniaor Phase modulaor Frequency modulaor PM wave s() = A c cos[2 f c + k p m()]

8 Frequency modulaion Non-linear funcion of m() FM signal: s() = A c cos[2 f c + 2k f m()d] 0 Consider single one signal: m()=a m cos2f m f i () = f c + k f m() = f c + k f A m cos[2 f m ] = f c + f cos[2 f m ] i () = 2 0 f i ()d = 2 f c + f sin[2 f m ] f m = 2 f c + sin[2 f m ] Frequency deviaion Modulaion index s() = A c cos[2 f c + sin(2 f m )]

9 Narrowband FM (<<1) )] sin(2 )sin[ sin(2 )] sin(2 )cos[ cos(2 ) ( we have :, Expanding f f A f f A s s() m c c m c c = 1 sin(2f m ) ]} ) ( cos[2 ] ) ( {cos[2 2 1 ) cos(2 ) )sin(2 sin(2 ) cos(2 ) s( 1, If f f f f A f A f f A f A m c m c c c c m c c c c + + << carrier Upper side-frequency Lower side-frequency Resulan

10 Comparison wih AM AM Signal carrier Upper side-frequency Lower side-frequency Resulan Narrow band FM Signal Resulan carrier Lower side-frequency Upper side-frequency

11 Wideband FM s() = A c cos[2 f c + sin(2 f m )] f c >> f m = Re[A c exp( j2 f c + j sin(2 f m ))] = Re[ s()exp( j2 f c )] (*) s() = A c exp[ j sin(2 f m )] = A c J n ()exp( j2n f m ) n= nh order Bessel funcion of firs kind Subs. ino (*), and applying FT: S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m

12 Observaions J 0 () = 1 Ampliude of carrier varies wih J 0 () Wih AM, ampliude of carrier = A c J 1 () = 2 J n2 () 0 n= J n 2 () = 1 Special case: << 1 Only J 0 (), J 1 () f c ± f m significan (narrowband FM) nh order Bessel funcion of firs kind S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] = A c J 0 ()( f f c ) + A c 2 J ±1 ()( f f c f m ) + A c 2 J ±2 ()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m

13 Example - fixed f m, variable A m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + Carrier componen Side freq f c ±f m Side freq f c ±2f m f m f m A m

14 Example - fixed A m, variable f m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + f m f m f m

15 Example - fixed A m, variable f m m() = A m cos[2 f m ], f = k f A m, = f f m S(f) = A c J 0 ()( f f c ) + A c 2 J ±1()( f f c f m ) + A c 2 J ±2()( f f c 2 f m ) + f m f m As, number of specral lines wihin f c f < f < f c + f As, he bandwidh of s() approaches he limiing value of 2f!! [Noe: For 1,bandwidh of s() 2 f m (As in AM)]

16 Transmission bandwidh S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] ransmission bandwidh =!!! Bu, effecively, finie number of side frequencies are significan Large : Falls rapidly owards 0 for f-f c >2f Small : Significan sidebands wihin f c ± f m Carson's rule: B T,Carson 2f (1 + 1 )

17 Transmission bandwidh S(f) = A c 2 n= J n ()[( f f c nf m ) + ( f + f c + nf m )] ransmission bandwidh =!!! Bu, effecively, finie number of side frequencies are significan reain up o n max side frequencies s.. J nmax () J 0 () B T = 2n max f m B T,1% = 1 % bandwidh wih = 0.01

18 1 percen bandwidh of FM wave As, n max B T,1%

19 1 percen bandwidh of FM wave Small Large Small values of more exravagan in B T han larger!! Pracically, B T,Carson B T B T,1%

20 FM Sereo Transmi wo separae signals via same carrier 2 differen secions of orchesra, e.g., vocalis and accompanis, o give spaial dimension o is percepion Requiremens Mus operae wihin allocaed FM broadcas channels Mus be compaible wih monophonic radio receivers

21 FM Sereo Mux m l +m r : Monophonic recepion m l -m r 2f c =38kHz f c =19kHz m() = [m l () + m r ()] +[m l () m r ()]cos(2[2 f c ]) +K cos(2 f c )

22 FM Sereo Demux m l () + m r () [m l () m r ()]cos(2[2 f c ]) K cos(2 f c ) 2f c =38kHz m() = [m l () + m r ()] +[m l () m r ()]cos(2[2 f c ]) +K cos(2 f c )

23 Phase Locked Loop (PLL) s() = A c sin[2 f c + 1 ()] X Loop filer v() r() = A v cos[2 f c + 2 ()] 2 () = 2k v 0 v()d Volage Conrolled Oscillaor PLL for freq. demod If s() is FM wave, obain m() from v() Require o <=> Phase lock!

24 Phase Locked Loop (PLL) HF :4f c erm e() = { s() = A c sin[2 f c + 1 ()] LF : K sin[ 1 () 2 ()] e ( ) X H(f) Loop filer v() = e( )h( )d r() = A v cos[2 f c + 2 ()] 2 () = 2k v 0 v()d Volage Conrolled Oscillaor Dynamic behavior of PLL d e () = d () 1 2K o d d K o = k m k v A c A v sin[ e ( )]h( )d,

25 Non-linear PLL model Sinusoidal non-lineariy makes i difficul o analyze PLL Dynamic behavior of PLL d e () = d () 1 2K o d d K o = k m k v A c A v sin[ e ( )]h( )d,

26 Non-linear PLL model Assume e () << 1 sin e () e () Linearized behavior of PLL d e () d = d 1 () d 2K o e ( )h( )d, FT can be applied! K o = k m k v A c A v

27 Linear PLL model 1 e ( f ) = 1 + L( f ) ( f ), 1 H ( f ) L( f ) = K o [Open-loop ransfer funcion] jf V( f ) = K o k v H ( f ) e ( f ) = jf k v L( f ) e ( f ) V( f ) = ( jf / k v )L( f ) 1 + L( f ) 1 ( f )

28 Phase-locked Linear PLL V( f ) = ( jf / k )L( f ) v 1 ( f ) 1 + L( f ) e ( f ) = 1 1+ L( f ) 1 ( f ) L( f ) 1 V( f ) jf k v 1 ( f ) v() = 1 2k v d 1 () d e ( f ) 0 Phase lock!!!

29 Phase locked linear PLL as frequency demodulaor If s() is FM signal, 1 () = 2k f 0 m()d v() = k f k v m() If L(f) >>1 Linearized PLL model Phase lock saisfied [ e 0] v() = 1 d 1 () 2k v d Deermines complexiy of PLL Bandwidh of s() >> bandwidh of H(f) [m()]

30 Design of H(f) Firs order H(f)=1 L(f) =K o /f Drawback K o conrols boh loop bandwidh and L(f) >>1 hold-in frequency range, f H f H (K o =1) f H (K o =3)

31 Second-order PLL H ( f ) = 1 + a jf ( jf / f e ( f ) = n ) ( jf / f n ) + ( jf / f n ) 2 ( f ), 1 f n = ak o Naural frequency = K o 4a Damping facor For m() = A m cos2 f m, 1 () = sin(2 f m ) e () = e0 cos(2 f m + )

32 Second-order PLL Wih appropriae choice of (,f n ), we can mainain e small Linear PLL model Rule of humb: Loop should remain locked if e0 (f m =f n )<90 o

33 Non-linear effecs in FM sysems v i () v i () = A c cos[2 f c + ()], () = 2k f 0 m()d memoryless communicaions channel v o () = a 1 v i () + a 2 v i 2 () + a 3 v i 3 () v o () = A 0 + A 1 cos[2 f c + ()] + A 2 cos[4 f c + 2()] + A 3 cos[6 f c + 3()] Assume v i () is FM signal f = frequency deviaion W = highes freq. comp. of m()

34 Non-linear effecs in FM sysems v i () = A c cos[2 f c + ()], () = 2k f 0 m()d Carson s Rule 2f memoryless communicaions channel v o () = A 0 + A 1 cos[2 f c + ()] + A 2 cos[4 f c + 2()] + A 3 cos[6 f c + 3()] f c -W f c f c +W 2f c -W 2f c 2f c +W 2f 4f

35 Non-linear effecs in FM sysems To separae ou desired FM signal f c +f+w < 2f c -W-2f =>f c > 3f + 2W Apply bandpass filer [f c -f-w, f c +f+w] v o '() = (a a 3 A c2 )v i () Unlike AM, FM no affeced by disorion due o channel wih amp. non-lineariies f c -W f c f c +W 2f c -W 2f c 2f c +W 2f 4f

36 Summary Unlike AM, FM is non-linear modulaion process Specral analysis is more difficul Developed insigh by sudying single-one FM Carson s rule for ransmission bandwidh B T = 2f(1+1/) Phase-Locked Loop for frequency demodulaion