Frequency Modulation. Dr. HweePink Tan


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1 Frequency Modulaion Dr. HweePink 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 Phaselocked Loop (PLL) Nonlinear 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) radedoff wih increased sysem complexiy Angle modulaion offers pracical means of radingoff 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 Nonlinear 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 sidefrequency Lower sidefrequency Resulan
10 Comparison wih AM AM Signal carrier Upper sidefrequency Lower sidefrequency Resulan Narrow band FM Signal Resulan carrier Lower sidefrequency Upper sidefrequency
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 ff 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 Nonlinear PLL model Sinusoidal nonlineariy 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 Nonlinear 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 [Openloop 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 Phaselocked 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 holdin frequency range, f H f H (K o =1) f H (K o =3)
31 Secondorder 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 Secondorder 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 Nonlinear 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 Nonlinear 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 Nonlinear effecs in FM sysems To separae ou desired FM signal f c +f+w < 2f c W2f =>f c > 3f + 2W Apply bandpass filer [f c fw, 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. nonlineariies f c W f c f c +W 2f c W 2f c 2f c +W 2f 4f
36 Summary Unlike AM, FM is nonlinear modulaion process Specral analysis is more difficul Developed insigh by sudying singleone FM Carson s rule for ransmission bandwidh B T = 2f(1+1/) PhaseLocked Loop for frequency demodulaion
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