Phase and Frequency Modulation

Transcription

1 Angle Modulaion

2 Phase and Frequency Modulaion Consider a signal of he form x c = A c cos 2π f c + φ θ i ( ) where A c and f c are consans. The envelope is a consan so he message canno be in he envelope. I mus insead lie in he variaion of he cosine argumen wih ime. Le θ i! 2π f c + φ be he insananeous phase. Then x c = A c cos( θ i ) = A c Re e jθ i. conains he message and his ype of modulaion is called angle or exponenial modulaion. If φ he modulaion = k p m so ha x c = A c cos 2π f c + k p m where k p is he deviaion consan or phase is called phase modulaion PM modulaion index.

3 Phase and Frequency Modulaion Think abou wha i means o modulae he phase of a cosine. The oal argumen of he cosine is 2π f c + φ, an angle wih unis of radians (or degrees). When φ = 0, we simply have a cosine and he angle 2π f c is a linear funcion of ime. Think of his angle as he angle of a phasor roaing a a consan angular velociy. Now add he effec of he phase modulaion φ. The modulaion adds a "wiggle" o he roaing phasor wih respec o is posiion when i is unmodulaed. The message is in he variaion of he phasor's angle wih respec o he consan angular velociy of he unmodulaed cosine. Unmodulaed Cosine Modulaed Cosine 2πf c φ() 2πf c

4 Phase and Frequency Modulaion = 2π f c in which f c is a cyclic The oal argumen of an unmodulaed cosine is θ c frequency. The ime derivaive of 2π f c is 2π f c. We could also express he argumen in radian frequency form as θ c = ω c. Is ime derivaive is ω c. Therefore one way d of defining he cyclic frequency of an unmodulaed cosine is as ( 2π d θ c ). Now le's apply his same idea o a modulaed cosine whose argumen is θ c = 2π f c + φ. Is ime derivaive is 2π f c + d ( d φ ). Now we define insananeous frequency as f! d ( 2π d θ c ) = 2π 2π f + d c ( d φ ) = f + d ( c 2π d φ ). I is imporan o draw a disincion beween insananeous frequency f and specral frequency f. They are definiely no he same. Le x c = cos( 2π f c + φ ). I has a Fourier ransform X c ( f ). Specral frequency f is he independen variable in X c ( f ) bu f = f c + d ( 2π d φ ). Some Fourier ransforms of phase and frequency modulaed signals laer will make his disincion clearer.

5 Phase and Frequency Modulaion If we make he variaion of he insananeous frequency of a sinusoid be direcly proporional o he message we are doing frequency modulaion ( FM). If dφ d = k f m hen k f is he frequency deviaion consan in radians/second per uni of m. In frequency modulaion f = f c + f d m, where f d = k f 2π is he frequency deviaion consan in Hz per uni of m. In frequency modulaion herefore φ = k f m λ dλ + φ 0 0 x c = 2π f d m λ dλ + φ 0 = A c cos 2π f c + 2π f d m( λ)dλ + φ 0 0 0, 0 So PM and FM are very similar. The difference is beween inegraing he message signal before phase modulaing or no inegraing i..

6 Phase and Frequency Modulaion Phase Modulaing Signal x m () Time, (μs) Phase-Modulaed Signal - k p = 5 x c () Time, (μs) 200 Phase in Radians θ c () Time, (μs) Insananeous Frequency in MHz f() Time, (μs)

7 Phase and Frequency Modulaion Phase Modulaing Signal x m () Time, (μs) Phase-Modulaed Signal - k p = 5 x c () Time, (μs) 200 Phase in Radians θ c () Time, (μs).5 Insananeous Frequency in MHz f() Time, (μs)

8 Phase and Frequency Modulaion Frequency Modulaing Signal x m () Time, (μs) Frequency-Modulaed Signal - f d = 500,000 x c () Time, (μs) 200 Phase in Radians θ c () Time, (μs).5 Insananeous Frequency in MHz f() Time, (μs)

9 Phase and Frequency Modulaion 2 Frequency Modulaing Signal x m () Time, (μs) Frequency-Modulaed Signal - f d = 500,000 x c () Time, (μs) 200 Phase in Radians θ c () Time, (μs) 2 Insananeous Frequency in MHz f() Time, (μs)

10 Phase and Frequency Modulaion Frequency-Modulaed Signal - f d = 500,000 x c () Time, (μs).5 Insananeous Frequency in MHz f() Time, (μs) X c ( f ) f (MHz)

11 Phase and Frequency Modulaion x() Message x() Message x c () Phase-Modulaed Carrier x c () Frequency-Modulaed Carrier

12 Phase and Frequency Modulaion x() Message x() Message x c () Phase-Modulaed Carrier x c () Frequency-Modulaed Carrier

13 Phase and Frequency Modulaion ( ) For phase modulaion x c = A c cos 2π f c + k p m For frequency modulaion x c = A c cos 2π f c + 2π f d m( λ)dλ There is no simple expression for he Fourier ransforms of hese signals in he general case. Using cos( x +y) = cos( x)cos( y) sin( x)sin( y) we can wrie for PM x c = A c cos( 2π f c )cos( k p m ) sin( 2π f c )sin k p m 0 ( ) and for FM x c = A c cos( 2π f c )cos 2π f d m( λ)dλ sin 2π f ( c )sin 2π f d x( λ)dλ 0 0 ( under he assumpion ha φ( 0 ) = 0).

14 Phase and Frequency Modulaion If k p and f d are small enough, cos k p m and cos 2π f d Then for PM x c 0 m( λ)dλ and sin 2π f d and sin( k p m ) k p m A c cos( 2π f c ) k p m 0 m( λ)dλ 2π f d m( λ)dλ. sin 2π f c and for FM x c A c cos( 2π f c ) 2π f d sin( 2π f c ) m( λ)dλ 0 These approximaions are called narrowband PM and narrowband FM. 0

15 = A m cos( 2π f m ) hen M( f ) = ( A m / 2) δ ( f f m ) + δ ( f + f m ) If he informaion signal is a sinusoid m For PM, x c A c cos 2π f c X c k p A m cos 2π f m ( f ) ( A c / 2) δ ( f f c ) + δ f + f c For FM, x c X c Phase and Frequency Modulaion A c cos 2π f c 2π f d A m 2π f m sin 2π f c ( f ) ( A c / 2) δ ( f f c ) + δ ( f + f c ) and, in he narrowband approximaion, δ ( f + f c f m ) + δ ( f + f c + f m ) δ ( f f c f m ) δ ( f f c + f m ) ja m k p 2 sin( 2π f c )sin 2π f m δ ( f + f c + f m ) + δ ( f f c + f m ) A m f d δ f + f c f m 2 f m δ f f c f m

16 Phase and Frequency Modulaion Unmodulaed Carrier, A c =, f c = A c cos(ω c ) A c k A m cos(ω m )sin(ω c ) p Time, (s) x 0-6 Quadraure Componen, A m =, f m = , k p = Time, (s) x 0-6 Narrowband PM Time, (s) x x Insananeous Frequency f PM () Time, (s) x 0-6

17 Phase and Frequency Modulaion Unmodulaed Carrier, A c =, f c = A c cos(ω c ) (A c fa m /f m )sin(ω m )sin(ω c ) d Time, (s) x 0-6 Quadraure Componen, A =, f = , f = m m d Time, (s) x 0-6 Narrowband FM Time, (s) x x Insananeous Frequency f FM () Time, (s) x 0-6

18 Phase and Frequency Modulaion Narrowband PM and FM Specra for Tone Modulaion Tone-Modulaed PM Tone-Modulaed FM ( f ) X c ( f ) X c π f -f -f c f c c -f m -f c +f m f c -f m f c +f -f c f c m -f c -f m -f c +f m f c -f m f c +f m X c ( f ) X c ( f ) π f f f -π -π

19 Phase and Frequency Modulaion If he informaion signal is a sinc, x = sinc( 2W ) hen X( f ) = ( / 2W )rec( f / 2W ) and, in he narrowband approximaion, For PM, X c ( f ) ( A c / 2) δ ( f f c ) + δ ( f + f c ) For FM, X c ( f ) ( A c / 2) δ ( f f c ) + δ f + f c j k p 2W rec f + f c f m k f 2W ( / 2W ) rec ( f f c ) / 2W rec( ( f + f c ) / 2W ) rec f f c f + f c f f c ( / 2W )

20 Phase and Frequency Modulaion Narrowband PM and FM Specra for a Sinc Message Sinc-Modulaed PM Sinc-Modulaed FM ( f ) X c ( f ) X c f c W f -f c f f c -f c + W f c W f c + W f c W c f c + W f c W f c f f c + W X c π ( f ) X c ( f ) π f f -π -π

21 Phase and Frequency Modulaion

22 If he narrowband approximaion is no adequae we mus deal wih he more complicaed wideband case. In he case of one modulaion we can handle PM and FM wih basically he same analysis echnique if we use he following convenions: For FM, φ Phase and Frequency Modulaion = A m sin 2π f m x = 2π f d x λ φ = 2π A m ω m, PM, FM A m cos 2π f m dλ = 2π f d A m cos( 2π f m λ)dλ 0 0 f d sin( 2π f m ) = A m f m Then, for PM and FM, φ = β sin 2π f m Then x c = A c cos( β sin 2π f m )cos 2π f c f d sin( 2π f m ), where β! k pa m, PM ( A m / f m ) f d, FM sin β sin( 2π f m ) sin 2π f c

23 Phase and Frequency Modulaion In x c = A c cos( β sin( 2π f m ) )cos 2π f c cos β sin 2π f m sin 2π f c sin β sin( 2π f m ) ( ) and sin β sin( 2π f m ) are periodic wih fundamenal period / f m. We can now use wo resuls from applied mahemaics Abramowiz and Segun, page 36 cos( zsin( θ )) = J 0 z sin zsin( θ ) cos( 2kθ ) = J 0 z + 2 J 2k z k= = 2 J 2k+ z + 2 J k z k= k even sin ( 2k +)θ = 2 J k ( z)sin kθ k=0 Adaping hem o our case cos( β sin( 2π f m ) ) = J 0 β = 2 J k β sin zsin( 2π f m ) k= k odd + 2 J k β k= k even sin 2kπ f m k= k odd cos 2kπ f m cos kθ

24 Phase and Frequency Modulaion x c x c x c = A c J 0 β = A c = A c + 2 J k β k= k even cos 2kπ f m cos 2π f c 2 J k β k= k odd J 0 ( β )cos( 2π f c ) + 2 J k ( β )cos( 2π f c )cos 2kπ f m k= k even 2 J k ( β )sin( 2π f c )sin( 2kπ f m ) k= k odd J 0 J k β k= k odd ( β )cos 2π f c + J k ( β ) cos 2π f c kf m k= k even cos 2π f c kf m ( ) cos 2π ( f c + kf m ) sin 2kπ f m ( ) + cos 2π ( f c + kf m ) This can also be wrien in he more compac form, x c = A c J k ( β )cos 2π ( f c + kf m ) k= sin 2π f c

25 Phase and Frequency Modulaion

26 Phase and Frequency Modulaion cos( π) cos( π) 0.289cos( π) cos( π) 0.007cos( π) J k ( β ) A c =, f c = 0 MHz A m =, f m = 0 5, f d = β=2 J 4 ( β) J 2 ( β) -0.4 J 5 ( β) J 3 ( β) J ( β) β J 0 J ( β) β=2 ( β) β J 2 J 3 ( β) J 4 ( β) β J cos( π) cos( π) cos( π) 0.289cos( π) cos( π) cos( π)

27 Phase and Frequency Modulaion A c =, f c = 0 MHz A m =, f m = 0 5, f d = β=2 J 4 ( β) J 2 ( β) J 0 J ( β) ( β) β J 2 J 3 ( β) J 4 ( β) β J 5 J k ( β ) J 5 ( β) J 3 ( β) J ( β) β= 2 β β=2 f (MHz)

28 Phase and Frequency Modulaion Now, o find he specrum of x c ake he Fourier ransform of x c. X c ( f ) = ( A c / 2) J k ( β) δ f ( f c + kf m ) k= + δ ( f + ( f c + kf m )) The impulses in he specrum exend in frequency all he way o infiniy. Bu beyond β f m he impulse srenghs die rapidly. For pracical purposes he bandwidh is approximaely 2β f m.

29 Phase and Frequency Modulaion Wideband FM Specrum for Cosine-Wave Modulaion X c ( f ) β = 8 β f m X c π ( f ) f c f f

30 Phase and Frequency Modulaion Tone PM, β = 5, φ = 5, A m =, f m = X c,pm ( f ) Δ Tone FM, β = 5, f = 5, A m =, f m = X c,fm ( f ) Δ Tone PM, β = 0, φ Δ = 5, A = 2, f = m m X ( f ) c,pm Tone FM, β = 0, f Δ = 5, A = 2, f = m m X ( f ) c,fm f f f f 80 Tone PM, β = 5, φ Δ = 5, A =, f = 2 m m X c,pm ( f ) f Tone FM, β = 2.5, f Δ = 5, A =, f = 2 m m X c,fm ( f ) f 50 Tone PM, β = 0, φ = 5, A m = 2, f m = 2 X c,pm ( f ) f Δ Tone FM, β = 5, f = 5, A m = 2, f m = 2 X c,fm ( f ) Δ f

31 Transmission Bandwidh The bandwidh required for ransmiing an FM signal is heoreically infinie. Tha is, an infinie bandwidh would be required o ransmi an FM signal perfecly, even if he modulaing signal is bandlimied. Forunaely, in pracical sysems, perfecion is no required and we can ge by wih a finie bandwidh. Wih one modulaion, he bandwidh required depends on he modulaion index β. The specral line magniudes fall off rapidly a posiive frequencies for which f f c > β f m. So for one modulaion he bandwidh required for ransmission would be approximaely 2β f m. In he narrowband case when β is very small we canno exacly follow his rule because we would have no modulaion a all. So here is a "floor" of a leas 2f m.

32 Transmission Bandwidh For he general case, Carson's rule is a handy approximaion ha says B 2( D +)W, where D = peak frequency deviaion bandwidh of m = f d W m max If D <<, hen B 2W. This is he narrowband case. If D >>, hen B 2DW = f d m max. This is he wideband case.

33 Generaion and Deecion of FM and PM The mos direc and sraighforward way of generaing FM is o use a device known as a volage-o-frequency converer (VCO). One way his can be done is by varying wih ime he capaciance in an LC parallel resonan oscillaor. Le he capaciance be he capaciance of a varacor diode in parallel wih anoher capacior forming C = C 0 C x. The ime-varying LC resonan frequency is f = 2π d d θ ( ) d d θ = L C = LC 0 C x C 0 We can use he formula (Abramowiz and Segun, page 5), ( + x) α = + αx + α α 2! o wrie d d θ ( ) = LC 0 C x C 0 /2 = ( α 2) x 2 + α α 3! + LC 0 2 x 3 +! C C 0 x C C 0 x 2 +!

34 Generaion and Deecion of FM and PM If C x is "small enough", hen d d θ θ = 2π f c + 2π C Since x + C LC x 0 2 C and 0 f c x( λ)dλ 2C 0. This is in he form of FM wih f d = C f c. 2C 0, he approximaion is good o wihin one percen if C / C 0 < So, aking ha as an upper limi, f d = C 2C 0 ha usually causes no design problems. f c f c. This is a pracical resul + N : RFC DC Block Tuned Circui + x() V B C v () Varacor C L x C () Oscillaor

35 Generaion and Deecion of FM and PM Anoher mehod for generaing FM is o use a phase modulaor, which produces PM, bu inegrae he message before applying i o he phase modulaor. A narrowband phase modulaor can be made by simulaing he narrowband approximaion x c = A c cos( 2π f c ) A c k p xsin( 2π f c ). k p x() x c () A c sin( 2πf c ) +90 A c cos( 2πf c )

36 Generaion and Deecion of FM and PM A hird mehod for generaing FM is called indirec FM. Firs, inegrae he message x. Then use he inegral of he message x( λ)dλ T as he inpu signal o a narrowband phase modulaor wih a carrier frequency f c. This produces a signal wih insananeous frequency f = f c + k p 2πT x. Narrowband Frequency Modulaor x() T Phase Modulaor f () () Frequency f 2 () f Muliplier n RF Power Amp x c () f c f LO

37 Generaion and Deecion of FM and PM Nex frequency-muliply he narrowband FM signal by a facor of n. This moves he carrier frequency o nf c, creaing a signal wih insananeous frequency f 2 = nf c + n k p 2πT x ( ). The effecive value of he frequency deviaion is now f d = n k p. This changes he range of frequency variaion bu no he rae of 2πT frequency variaion. Then, if needed, shif he enire FM specrum o whaever carrier frequency is required and amplify for ransmission. Narrowband Frequency Modulaor x() T Phase Modulaor f () () Frequency f 2 () f Muliplier n RF Power Amp x c () f c f LO

38 Generaion and Deecion of FM and PM There are four common mehods of deecing FM:. FM-o-AM Conversion Followed by Envelope Deecion 2. Phase-Shif Discriminaion 3. Zero-Crossing Deecion 4. Frequency Feedback FM-o-AM conversion can be done by ime-differeniaing he modulaed signal. Le x c = A c cos( θ c ) wih θ! = 2π f c + f d x. Then!x c = A! c θ sin( θ c ) = 2π A c f c + f d x sin( θ c ±80 ). The message can hen be recovered by an envelope deecor. x c () Limier LPF d/d Envelope Deecor DC Block y D ()

39 Generaion and Deecion of FM and PM The "differeniaor" in FM-o-AM deecion need no be a rue differeniaor. All ha is really needed is a frequency response magniude ha has a linear (or almos linear) slope over he bandwidh of he FM signal. Jus below and jus above resonance a uned circui resonaor has an almos linear magniude dependence on frequency. This ype of deecion is commonly called slope deecion. H( f ) Almos Linear Slope f f c f 0

40 Generaion and Deecion of FM and PM The lineariy of slope deecion can be improved by using wo resonan circuis insead of only one. This ype of circui is called a balanced discriminaor. f 0 > f c + x C + () C C L L K x() f c f f 0 < f c

41 Consider a cosine of he form x Phase and Frequency = Acos 2π f 0 + φ. The phase of his cosine is θ = 2π f 0 + φ and φ is is phase shif. Firs consider he case φ Then x and θ = Acos 2π f 0 = 2π f 0. = 0.!"# Acos! 0 " x A T 0 The cyclic frequency of his cosine is f 0. Also, he firs ime derivaive of θ T 0 # / f 0 # 2 / 0 is 2π f 0. So one way of defining cyclic frequency is as he firs derivaive of phase, divided by 2π. I hen follows ha phase is he inegral of frequency.

42 Phase and Frequency If x = Acos( 2π f 0 ) and θ = 2π f 0. Then a graph of phase versus ime would be a sraigh line hrough he origin wih slope 2π f 0.!"# Acos! 0 " x A θ () T 0 T 0 # / f 0 # 2 / 0 2πf 0

43 Le x = Acos θ cyclic frequency is frequency f. Phase and Frequency and le θ 2π x d d θ = 2π f 0 u = 2π f 0 ramp = f 0 u " # "#! Acos 0 u"# A. Then he. Call is insananeous cyclic T 0 f 0 f() T 0! / f 0! 2 / 0 2πf 0 () θ

44 Now le x Phase and Frequency = Acos 2π ( u + u( ) ) = u + u( ) and he phase is = 2π ramp + ramp( ) cyclic frequency is f θ. Then he insananeous. " " " ### x"#! Acos 2 u"#$ u A 2 f ()!! 2 θ () 2π

45 Le x = A sin 2π f 0 +θ The produc is x x 2 Phase Discriminaion = A 2 cos 2π f 0 + φ = A A 2 sin 2π f 0 +θ and le x 2 Using a rigonomeric ideniy, and x x 2 = A A 2 2 x x 2 = A A 2 sin φ. cos( 2π f 0 + φ ). θ +θ + sin( 4π f 0 + φ ) 2 sin φ θ A A 2 2 A sin( 2π f +θ() ) 0 sin( φ() θ () )+ sin 4π f 0 +φ()+θ ( ()) LPF A A 2 2 sin φ() θ ( ()) A A 2 / () () A 2 cos( 2π f 0 +φ() )

46 Volage-Conrolled Oscillaors A volage - conrolled oscillaor (VCO) is a device ha acceps an analog volage as is inpu and produces a periodic waveform whose fundamenal frequency depends on ha volage. Anoher common name for a VCO is "volage-o-frequency converer". The waveform is ypically eiher a sinusoid or a recangular wave. A VCO has a free-running frequency f v. When he inpu analog volage is zero, he fundamenal frequency of he VCO oupu signal is f v. The oupu frequency of he VCO is f VCO = f v + K v v in where K v is a gain consan wih unis of Hz/V.

47 Phase-Locked Loops A phase - locked loop (PLL) is a device used o generae a signal wih a fixed phase relaionship o he carrier in a bandpass signal. An essenial ingredien in he locking process is an analog phase comparaor. A phase comparaor produces a signal ha depends on he phase difference beween wo bandpass signals. One sysem ha accomplishes his goal is an analog muliplier followed by a lowpass filer. Le he wo bandpass signals be x r = A c cos( 2π f c + φ ) and e 0 = A v sin( 2π f c +θ ) and le he oupu signal from he phase comparaor be e d. x r () ()= A c cos 2π f c + φ LPF -K ed d e 0 () ()= A v sin 2π f c +θ

48 x r x r e d e 0 Phase-Locked Loops = K d A c A v cos 2π f c + φ e 0 = K da c A v 2 = K A A d c v 2 sin φ sin 2π f c +θ sin θ ( ) ( φ ) + sin 4π f c + φ +θ ( θ ) = K A A d c v e K d A c A d v 2 () 2 sin( ψ ) x r ( ()) ()= A c cos 2π f c + φ LPF -K ed d e 0 ( ()) ()= A v sin 2π f c +θ

49 Phase-Locked Loops e d depends on boh he phase difference and A c and A v. We can make he dependence on hese ampliudes go away if we firs hard limi he signals, urning hem ino fixed-ampliude square waves. Anoher benefi of hardlimiing is ha he muliplicaion becomes a swiching operaion and he error signal e d is now a linear funcion of ψ over a wider range. x r () Hard Limier Swiching Circui LPF ed e d () () e 0 Hard Limier 90 90

50 Phase-Locked Loops From he block diagram of he phase-locked loop below i is clear ha E v ( s) = F( s)e d ( s), where F( s) is he ransfer funcion of he loop-filer-loop-amplifier combinaion. The VCO convers volage o frequency and phase is he inegral of frequency. Tha is why he VCO is represened as an inegraor wih volage in and phase ou. Phase Comparaor () + () sin( ) K v K d A c A v 2 ev e d Loop Amplifier Loop Filer VCO

51 Phase-Locked Loops Phase-locked loops operae in wo modes, acquisiion and racking. When a PLL is urned on i mus firs acquire a phase lock and hereafer i mus rack he phase changes in he incoming signal. The acquisiion of a phase lock mus be described by he non-linear model of he PLL in which he phase discriminaor has a sine ransfer funcion. In he racking mode, he phase error is ypically small, he sine funcion can be approximaed by is argumen and he model of he PLL becomes linear.

52 Phase-Locked Loops Non-linear PLL Model for Acquisiion () + () Phase Comparaor sin( ) K v K d A c A v 2 ev e d Loop Amplifier Loop Filer VCO Phase Comparaor Linear PLL Model for Tracking () + () K v K d A c A v 2 ev e d Loop Amplifier Loop Filer VCO

53 Phase-Locked Loops In he racking mode Θ( s) = K A A d c v Φ( s) Θ s 2 I follows ha H( s) = Θ( s) Φ( s) = K F( s) s + K F s Ψ( s) = Φ( s) Θ( s), herefore G s F( s) K v s. where K = K A A K d c v v 2 = Ψ( s) Φ( s) = Φ( s) Θ( s) Φ( s). = H( s). Phase Comparaor () + () K v K d A c A v 2 ev e d Loop Amplifier Loop Filer VCO

54 Phase-Locked Loops Le he phase deviaion of he incoming signal φ be of he general form φ = π R 2 + 2π f Δ +θ 0 u. Then dφ 2π d = ( R + f Δ )u, a frequency ramp plus a frequency sep. Then Φ( s) = 2π R + 2π f Δ + θ 0 s 3 s 2 s. Using he final value heorem of he Laplace ransform, he seady sae phase error beween he incoming signal and he VCO oupu signal is limψ = lim Then H( s) = s 2π R + 2π f Δ + θ 0 s 0 s 3 s 2 s G ( s ). Now le F s K ( s 2 + as + b) s 3 + K ( s 2 + as + b) and G s s 3 + K = s 3 = s2 + as + b s 2. ( s 2 + as + b).

55 Phase-Locked Loops Then he seady-sae phase error is limψ = lim. s θ 0s 2 + 2π f Δ s + 2π R s 0 s 3 + K s 2 + as + b Is value depends on he form of he inpu signal's phase deviaion and he order of he loop filer. Seady Sae Error, limψ PLL Order θ 0 0 f Δ = 0 R = 0 θ 0 0 f Δ 0 R = 0 θ 0 0 f Δ 0 R 0 ( a = 0,b = 0) 0 2π f Δ K 2( a 0,b = 0) a 0,b 0 2π R 0 0 0

56 Phase-Locked Loops So a firs-order PLL can rack a phase sep wih zero error and a frequency sep wih a finie error. A second-order PLL can rack a frequency sep wih zero error and a frequency ramp wih a finie error. A hird-order PLL can rack a frequency sep and a frequency ramp wih zero error. When he error is finie, is size can be made arbirarily small by making K large. However, his also increases he loop bandwidh, making he signal-o-noise raio worse. (More in Chaper 8.)

57 Phase-Locked Loops A firs-order PLL can be used for demodulaion of angle-modulaed signals bu a second-order PLL has some advanages and is more common in pracice. Therefore, in F( s) = s2 + as + b, make b = 0. s 2 Then F( s) = + a. This can be implemened as he signal plus a imes s he inegral of he signal. Wih his F( s), H( s) = Θ( s) Φ( s) = K ( s + a) ( s + a) and = Ψ( s) Φ( s) = s 2 s 2 + K G s, a second-order ransfer funcion. Expressing s 2 + K ( s + a) his ransfer funcion in a sandard second-order sysem form, G( s) = ω 0 = s 2 s 2 + 2ζω 0 s + ω 0 2, where ζ = 2 K a K a is he radian naural frequency. is he damping facor and

58 Phase-Locked Loop Saes Inpu and Feedback Signals a Same Frequency Locked in Quadraure No Locked - Inpu and Feedback Signals in Phase LPF K a ()= K a sin( ε () ) y LPF K a ()= K a sin( ε () ) y VCO VCO K v K v No Locked - Inpu and Feedback Signals 80 Ou of Phase LPF K a ()= K a sin( ε () ) y VCO K v

59 Phase-Locked Loop Saes Inpu and Feedback Signals a Differen Frequencies Inpu Frequency > Feedback Frequency Inpu Frequency < Feedback Frequency LPF K a LPF K a VCO VCO K v K v

60 Phase-Locked Loops For DSB signals, which do no have ransmied carriers, Cosas invened a sysem o synchronize a local oscillaor and also do synchronous deecion. The incoming signal is x c = xcos ω c wih bandwidh 2W. I is applied o wo phase discriminaors, main and quad, each consising of a muliplier followed by a LPF and an amplifier. The local oscillaors ha drive hem are 90 ou of phase so ha he oupu signal from he main phase discriminaor is xsin( ε ss ) and he oupu signal from he quad phase discriminaor is xcos( ε ss ). x()sin ( ε ss ) Main PD x()cos ( ω c ) cos( ω c ε ss + 90 ) 90 VCO y ss = T 2 S sin 2ε x ( ss ) T Error Signals Quad PD x()cos ( ε ss ) Oupu

61 Phase-Locked Loops The VCO conrol volage y ss is he ime average of he produc of xsin( ε ss ) and xcos ε ss y ss = T ( 2 x2 ) sin 0 or y ss = x 2 λ cos( ε ss )sin( ε ss )dλ which is T + sin( 2ε ss ) = T 2 S sin 2ε x ( ss ). When he angular error ε ss is zero, y ss does no change wih ime, he loop is locked and he oupu signal from he quad phase discriminaor is xcos( ε ss ) = x because ε ss = 0. Main PD x()sin ( ε ss ) x()cos ( ω c ) cos( ω c ε ss + 90 ) 90 VCO y ss = T 2 S sin 2ε x ( ss ) T Error Signals Quad PD x()cos ( ε ss ) Oupu

62 Inerference Le he oal received signal a a receiver be v = A c cos( ω c ) + A i cos ( ω c + ω i ) + φ i where he firs erm represens he desired signal and he second erm represens inerference. Also define ρ! A i / A c and θ i! ω i + φ i. Then v = A c cos( ω c ) + ρ cos( ω c +θ i ) = A c cos( ω c ) + ρ cos ( ω c)cos( θ i ) sin( ω c )sin( θ i ) v = A { c + ρ cos( θ i ) cos( ω c ) ρsin( θ i )sin( ω c ) } ( ) The in-phase componen is A c + ρ cos θ i A c ρsin( θ i )sin( ω c ). The envelope is A v 2 = A c + ρ cos θ i + ρ 2 sin θ i ρsin θ o he desired signal is φ v = an i + ρ cos θ i cos ω c and he quadraure componen is ( ) = A c + ρ 2 + 2ρ cos θ i ( ).. The phase relaive

63 Inerference The envelope and phase of he oal received signal A v = A c + ρ 2 + 2ρ cos θ i ρsin θ ( ) and φ v = an i + ρ cos θ i show ha he effec of he inerference on he received signal is o creae boh ampliude and phase modulaion. If ρ <<, hen A v or A c + 2ρ cos θ i A v ( ) A c + ρ cos θ i A c + ρ cos ω i + φ i and φ v and φ v ρsin( ω i + φ i ) ( ) ρsin( θ i ) an ρsin θ i This resul has he form of AM one modulaion wih µ = ρ and simulaneous PM or FM one modulaion wih β = ρ. If ρ >>, hen A v = ρa c + 2ρ cos( ω i + φ i ) ρa c + ρ cos( ω i + φ i ) and φ v = ω i + φ i

64 Inerference In he weak inerference case A v A c + ρ cos( ω i + φ i ) and φ v ρsin( ω i + φ i ) if we demodulae wih an envelope, phase or frequency demodulaor we ge ( wih φ i = 0) Envelope Deecor: K D + ρ cos ω i Phase Deecor: K D ρsin ω i Frequency Deecor: K D ρ f i cos ω i For AM or PM demodulaion he demodulaed signal srengh is proporional o ρ. For FM demodulaion he demodulaed signal srengh is porporional o he produc of ρ and f i. Ampliude FM AM or PM W f i

65 Inerference The effecs of inerference on FM signals increases wih frequency. So one way o reduce he effec is o lowpass filer he demodulaed oupu. Of course his also lowpass filers he message, an undesirable oucome. To avoid he lowpass filering effec on he message a echnique called preemphasis is ofen used. The higher frequency pars of he message are preemphasized before ransmission by passing hem hrough a preemphasis filer wih frequency response H pe ( f ) ha amplifies he higher frequencies more han he lower frequencies. Then, afer ransmission and frequency demodulaion, he demodulaed signal is passed hrough a deemphasis filer whose frequency response is H de ( f ) = H pe ( f ).

66 Inerference A ypical deemphasis filer has a frequency response H de ( f ) = + jf / B de in which B de is less han he cuoff frequency of he normal sharp-cuoff lowpass filer ha deermines he bandwidh. Tha makes he corresponding preemphasis filer have a frequency response H pe ( f ) = + jf / B de.

67 Inerference A phenomenon ha mos people have experienced in receiving FM signals is he so-called capure effec. Suppose here are wo FM saions, boh ransmiing in he same bandwidh and of approximaely equal signal srengh a he receiver. Their signal srenghs will flucuae some causing one o be sronger for a ime and hen he oher. The sronger signal will "capure" he receiver for a shor ime and will dominae he demodulaed signal. Bu hen laer he oher signal will dominae and capure he receiver. The wo saions swich back and forh and he lisener hears a ime-muliplexed version of boh signals. To keep he mah simple, assume we have one unmodulaed carrier and one modulaed carrier. This is exacly he "inerfering sinusoid" case we analyzed earlier wih he resuls wih θ i A v = A c + ρ 2 + 2ρ cos θ i ρsin θ ( ) and φ v = an i + ρ cos θ i = φ i, he phase modulaion of he inerfering signal.

68 A v = A c + ρ 2 + 2ρ cos θ i The demodulaed signal is hen y D Using d ( dz an z) y D y D y D = = = φ v = d d = ρsin φ + i + ρ cos φ i Inerference ρsin θ ( ) and φ v = an i + ρ cos θ i ρsin φ an i + ρ cos φ i ( ). and he chain rule of differeniaion, 2 + z ( ) ( ) + ρ cos ( φ i ) ρ cos( φ i ) φ! + ρsin( φ i )ρsin φ i ρ cos( φ i ) ρ cos( φ i ) + ρ 2 2 φ + ρ cos( φ i ) + ρ 2 sin 2 ( φ i )! ( ) φ + ρ 2 + 2ρ cos( φ i )! = α ( ρ,φ i ) φ! = ρ ρ + cos ( φ i ) + ρ 2 + 2ρ cos( φ i ) = ρ ρ + cos φ i where α ρ,φ i ( ) φ!

69 y D = α ρ,φ i Inerference φ! facor suggess ha he inerference may be inelligible if α ( ρ,φ i ) and y D!φ The!φ is relaively consan wih ime. If ρ >>, hen α ρ,φ i. Bu we wish o examine he case in which he wo signals are approximaely equal in srengh, implying ha ρ. = ρ ρ + cos ( φ i ) + ρ 2 + 2ρ cos( φ i ) = α ρ,φ i, φ i = 0 + 2nπ ρ / + ρ ρ 2 / + ρ 2 ρ / ρ, φ i = π / 2 + nπ, φ i = π + 2nπ, n an ineger 0.5 ρ = 0.9 α ( ρ, φ ) ρ = φ

70 α ( ρ, φ ) y D Inerference = α ( ρ,φ i ) φ!, α ( ρ,φ i ) = ρ ρ + cos ( φ i ) 0.5!φ. + ρ 2 + 2ρ cos φ i ( ) As ρ, α 0.5 and y D For ρ <, he srengh of he demodulaed inerference depends mosly on he peak-o-peak value of α 2ρ α p p = α ( ρ,0) α ( ρ,π ) = ( ρ) 2 The inerference effec is small-o-negligible for ρ < 0.7 and he inerference ρ = 0. capures he demodulaed oupu signal when ρ > 0.7. ρ = 0.9 α p-p (ρ ) φ ρ