Pi Principles i of Communications

Transcription

1 Pi Priniples i o Communiaions i Chaper 3: Analog Modulaion (oninued Texbook: Ch 3, Ch , Ch 6.-6.

2 3E 3. Ee o Noise on AMS Sysems Baseband sysem (a basis or omparison o various modulaion sysems: No arrier demodulaion The reeiver is an ideal LPF wih bandwidh W Noise power a he oupu o he reeiver N W 0 Pn = 0 d = N W 0W Baseband SNR is given by s m ( n( + LPF S = N b PR N W 0 00/0 Meixia SJTU

3 Example: Find he SNR in a baseband sysem wih a bandwidh o 5 khz and wih N 4 0 /= 0 W/Hz. The ransmier power is kw and he hannel aenuaion is 0 0 Soluion: 3 9 P R = 0 0 = 0 Was P R 9 S R 0 = = 4 = N N W b 0 0 = 0log0 0 = 3dB 00/0 Meixia SJTU 3

4 Ee o Noise on DSBSC s ( n( s ( m 0 ( + BPF demod n i ( n ( 0 modulaed signal Inpu o he demodulaor r ( = s ( + n( i s( = A m(os ω = A m (os w+ n(os w n(sin w s Here n i ( is a Gaussian narrow-band noise N / 0 W Sn i ( = 0 oherwise 00/0 Meixia SJTU 4

5 In he demodulaor, he reeived signal is irs muliplied by a loally generaed sinusoid signal r ( os( w+ φ = Am ( os wos( w+ φ + n(os wos( w+ φ n(sin wos( w+ φ s = Am (os φ+ Am (os( w + φ + [ n(os φ+ ns(sin φ] + n( os( w+ φ n( sin( w+ φ [ ] s Assume oheren deeor, we have φ = 0 00/0 Meixia SJTU 5

6 Then he signal is passed hrough a LPF wih bandwidh W Sn ( = S ( ( y( = [ Am ( + n( n i + Sn + i ] where or W The oupu SNR an hus be deined as AP m S Po 4 A Pm = = = N o Pn WN o P 0 n 4 AP m Sine he reeived power o DSBSC in baseband is P R = The oupu SNR an be rewrien as S P S = = DBSSC does no provide any SNR improvemen over a R N odsb WN 0 N b simple baseband sysems 00/0 Meixia SJTU 6

7 Ee o Noise on SSB Modulaed signal: Inpu o he demodulaor s( = Am(os w± Am(sin w N 0 / W / r ( = s ( + n Sn ( = i ( where 0 oherwise = Am ( + n (os w + ± Am ( n (sin w [ ] [ ] s Oupu o LPF: A y ( = m ( + n ( Thereore, he oupu SNR is AP m o APm S P 4 = = = N o Pn o P WN n /0 Meixia SJTU 7

8 Bu in his ase, Thus, P = A P R m S PR S = = N WN N ossb 0 b SNR in an SSB sysem is equivalen o ha o a DSBSC sysem 00/0 Meixia SJTU 8

9 Ee o Noise on Convenional AM Modulaed signal [ ] s( = A + am(os w Inpu o he demodulaor (oheren deeor r ( = s ( + n( i [ ] = A + Aam ( + n(os w n(sin w s Aer mixing and low pass iler y( = A + Aam ( + n( [ ] Removing DC omponen y ( = Aam ( + n ( [ ] 00/0 Meixia SJTU 9

10 In his ase, he reeived signal power Now, we an derive he oupu SNR as AaP m S 4 AaP m = = N oam n WN0 4 A ap a Pm m + S = = η + ap NW N m 0 Modulaion eiieny b P = A + a P R m SNR in onvenional AM is always smaller han ha in baseband. 00/0 Meixia SJTU 0

11 Perormane o Envelope-Deeor Inpu o he envelope-deeor [ ] r ( = A+ Aam ( + n(os w n(sin w Envelope o r( s [ ] V ( = A + Aam ( + n ( + n ( r s I signal omponen is muh sronger han noise V ( A + A am ( + n ( r Aer removing DC omponen, we obain y ( = Aam ( + n( A high SNR, perormane o oheren deeor and envelop deeor is he same 00/0 Meixia SJTU

12 Perormane o Envelope-Deeor I noise power is muh sronger han he signal power ( ( + V A am n n An am r ( = + ( + ( + s ( + ( + ( Ignore s erm ( A ( ( n n( + ns( + + am( n( + ns( V ( n = n ( + ns ( ε An ( + ε + Vn ( + ( + am( Vn ( The sysem is operaing below An ( = Vn ( + ( + am( he hreshold, no meaningul V ( n SNR an be deined. 00/0 Meixia SJTU

13 Exerise Consider ha he message is a WSS r.p M( wih auoorrelaion union RM ( τ = 6sin (0000 τ. I is given ha m ( = 6. max We wan o ransmi his message o a desinaion via a hannel wih a 50dB aenuaion and addiive whie noise wih PSD. Sn ( = N0 /= 0 W/Hz. We also wan o ahieve an SNR a he modulaor oupu o a leas 50dB. Wha is he required ransmied power and he hannel bandwidh i we employ he ollowing modulaion shemes? DSB-SC SSB AM wih modulaion index = /0 Meixia SJTU 3

14 33A 3.3 Angle Modulaion Angle modulaion is eiher phase or requeny o he arrier is varied aording o he message signal The general orm o an angle modulaed wave is [ π θ ] s( = A os + ( where = arrier req, θ( is he ime-varying phase and varied by he message m( The insananeous requeny o s( is ( i = + d θ ( π d 00/0 Meixia SJTU 4

15 Represenaion i o FM and PM signals For phase modulaion (PM, we have θ ( = k pm ( where k p = phase deviaion onsan For requeny modulaion (FM, we have d where k = requeny i( = km( = ( π d θ deviaion onsan The phase o FM is θ ( = πk m( τ dτ 0 00/0 Meixia SJTU 5

16 Disinguishing Feaures o PM and FM No pere regulariy in spaing o zero rossing Zero rossings reer o he ime insans a whih a waveorm hanges beween negaive and posiive values Consan envelop, i.e. ampliude o s( is onsan Relaionship beween PM and FM m( m( inegraor diereniaor m ( d d d m( Phase modulaor A C os( π Frequeny modulaor A C os( π FM wave A [ + k p m( d ] C os π PM wave A C os [ π + πk m( ] Disuss he properies o FM only 00/0 Meixia SJTU 6

17 Example: Sinusoidal id Modulaion Sinusoid modulaing wave m( FM wave d d m( PM wave 00/0 Meixia SJTU 7

18 Example: Square Modulaion Square modulaing wave m( FM wave PM wave 00/0 Meixia SJTU 8

19 FM by a Sinusoidal Signal Consider a sinusoidal modulaing wave m( = Am os(π m Insananeous requeny o resuling FM wave is i ( = + k A os(π = + Δ os(π m m m where Δ = k A mis alled he requeny deviaion, proporional o he ampliude o m(, and independen o m. Hene, he arrier phase is Δ θ( = π ( ( sin( τ dτ = π 0 i m m = βsin( π m Where β = Δ is alled he modulaion index m 00/0 Meixia SJTU 9

20 Example Problem: a sinusoidal modulaing wave o ampliude 5V and requeny khz is applied o a requeny modulaor. The requeny sensiiviy is 40Hz/V. The arrier requeny is 00kHz. Calulae (a he requeny deviaion, and (b he modulaion index Problem: i id l d l i li d 5V d Soluion: Frequeny deviaion Δ = k Am = 40 5 = 00Hz Modulaion index Δ 00 β = = = m 00/0 Meixia SJTU 0

21 Sperum Analysis o Sinusoidal FM Wave The FM wave or sinusoidal modulaion is s( = A = A os os [ π + β sin(π m ] [ β sin( π ] os( π A sin [ β sin( π ] sin( π m m s In-phase omponen I = A os [ β sin( i(π m ] s ( = A sin [ β sin(π ] ( Quadraure-phase omponen Q m Hene, he omplex envelop o FM wave is ~ jβ sin(π m s ( = s ( ( I + jsq = Ae ~ ( reains omplee inormaion abou s( s j [ + β sin(π ] s { A e } [ s e j ] π π Re ~ ( m ( = Re = 00/0 Meixia SJTU

22 ~ s ( where is periodi, an be expanded in Fourier series as n = = ~ j πnm s ( = ne ~ jβ sin(π m s ( = A e n= /( m m /( A m /( ~ s ( e e d [ β sin( i( π m πn m ] d m m /( m j jπn Le x = π m A = π n exp j ( β sin x nx dx π π n-h order Bessel union o he irs kind J n (β is deined as π Jn( β = exp j βsinx nx dx ( π π Hene, = n A J n(β m 00/0 Meixia SJTU

23 ~ Subsiuing n ino s ( ~ s ( = A n= = J n s ( = A J n ( β ( β exp ( j πn Hene, FM wave in ime domain an be represened by s( = A Re Jn( exp j ( + nm n= β [ π ] m [ π ] = A J ( β os ( + n n m n= In requeny-domain, we have A S = ( J n ( β m δ + m n= [ δ ( n + ( + n ] 00/0 Meixia SJTU 3

24 Propery :Narrowband FM (or small β 0.3 Approximaions J 0( β J( β β J ( β 0, n > n Subsiuing above ino s( βa s( A os(π + os + βa os[ π ( m ]? In wha ways do a onveninal AM wave and a narrow band FM wave dier rom eah oher [ π ( ] m J n ( β 0 as β J n ( β = J n( β, n even J n ( β, n odd 00/0 Meixia SJTU 0/4/004 4

25 Propery : Wideband FM (or large β> In heory, s( onains a arrier and an ininie number o siderequeny omponens, wih no approximaions made Propery 3: Consan average power The envelop o FM wave is onsan, so he average power is also onsan, P = / A The average power is also given by A P = J n ( β = n A [ π ( n ] ( = s A J ( β os π + n ( n m n= n= J n ( β = 00/0 Meixia SJTU 5

26 Example Goal: o invesigae how he ampliude A m, and requeny m, o a sinusoidal modulaing wave ae he sperum o FM wave Fixed m and varying A m requeny deviaion Δ = k A m and modulaion index β =Δ/ m are varied.0.0 β = β = 5 Δ Inreasing A m inreases he number o harmonis in he bandwidh Δ 00/0 Meixia SJTU 6

27 Fixed A m and varying m Δ is ixed, bu β is varied.0.0 β = 5 β = Δ Δ Inreasing m dereases he number o harmonis bu a he same ime inreases he spaing beween he harmonis. 00/0 Meixia SJTU 7

28 Eeive Bandwidh o FW Waves Theoreially, FM bandwidh = ininie In praie, or a single one FM wave, when β is large, Bis only slighly greaer han he oal requeny exursion Δ. when β is small, he sperum is eeively limied o [ - m, + m ] Carson s Rule approximaion or single-one modulaing wave o requeny m B Δ + = ( +β m m 00/0 Meixia SJTU 8

29 99% bandwidh approximaion The separaion beween he wo requenies beyond whih none o he side-requenies is greaer han % o he unmodulaed arrier ampliude i.e B n where n max is he max n ha saisies max m J n ( β > 0.0 β n max /0 Meixia SJTU 9

30 A universal urve or evaluaing he 99% bandwidh As β inreases, he bandwidh oupied by he signiian side- requenies drops oward ha over whih he arrier requeny aually deviaes, i.e. B beome less aeed by β /0 Meixia SJTU 30

31 FM by an Arbirary Message Consider an arbirary m( wih highes req omponen W Deine deviaion raio D = Δ / W, where Δ = k max m( D β and W m Carson s rule applies as ( B Δ + W = W + D Carson s rule somewha underesimae he FM bandwidh requiremen, while universal urve yields a somewha onservaive resul Assess FM bandwidh beween he bounds given by Carson s rule and he universal urve 00/0 Meixia SJTU 3

32 Example In norh Ameria, he maximum value o requeny deviaion Δ is ixed a 75kHz or ommerial FM broadasing by raio. I we ake he modulaion requeny W = 5kHz, whih is ypially he maximum audio requeny o ineres in FM ransmission, he orresponding value o he deviaion raio is D = 75/5 = 5 Using Carson s s rule, he approximae value o he ransmission bandwidh o he FM wave is B = (75+5 = 80kHz Using universal urve, B=3Δ 3. = 3. x 75 = 40kHz 00/0 Meixia SJTU 3

33 Exerise 4 ( Assuming ha m ( = 0sin 0, deermine he ransmission bandwidh o an FM modulaed signal wih k = /0 Meixia SJTU 33

34 Generaion o FM waves Dire approah Design an osillaor whose requeny hanges wih he inpu volage => volage-onrolled osillaor (VCO Indire approah Firs generae a narrowband FM signal and hen hange i o a wideband signal Due o he similariy o onvenional AM signals, he generaion o a narrowband FM signal is sraighorward. 00/0 Meixia SJTU 34

35 Generaion o Narrow-band FM Consider a narrow band FM wave s = A os π + φ ( [ ] ( where = πk m( τ 0 φ ( dτ = arrier requeny k = requeny sensiiviy Given φ ( << wih β , we emay use φ ( [ ] [ φ ( ] os sin φ ( Correspondingly, we may approximae s ( as s ( = A os π A sin π φ ( = A os ( ( ( π πk A sin( π 0 m( τ dτ Narrow-band FW wave 00/0 Meixia SJTU 35

36 Narrow-band requeny modulaor m( inegraor Produ Modulaor A sin( π - + Narrow-band + FM wave s ( phase Carrier wave shier A os( π Nex, pass s ( hrough a requeny muliplier, whih onsiss o a non-linear devie and a bandpass iler. Narrow-band FM wave Wideband FM Wave Memoryless Band-pass nonlinear devie iler The inpu-oupu relaionship o he non-linear devie is modeled as n s( = as ( + as ( + K+ a s ( The BPF is used o Pass he FM wave enred a n and wih deviaion nδ and suppress all oher FM spera n 00/0 Meixia SJTU 36

37 Example: requeny muliplier wih n = a p e eque y u p e Problem: Consider a square law devie based requeny muliplier Problem: Consider a square-law devie based requeny muliplier wih ( ( ( s a s a s + = wih Speiy he midband req. and bandwidh o BPF used in he req. + = d m k A s 0 ( os ( τ τ π π muliplier or he resuling req. deviaion o be wie ha a he inpu o he nonlinear devie Soluion: Soluion: = A a A a d m k A a d m k A a s 0 0 ( os ( os ( τ τ π π τ τ π π = d m k A a A a d m k A a 0 0 ( 4 4 os ( os τ τ π π τ τ π π = 00/0 Meixia SJTU 37 Removed by BPF wih BW > Δ = 4Δ

38 Thus, onneing he narrow-band requeny modulaor and he requeny muliplier, we may build he wideband requeny modulaor s = + ( A os π πk m( τ dτ 0 Message Wideband d signal Narrow-band Frequeny FM signal Inegraor phase muliplier modulaor A os( π Crysal-onrolled osillaor = n s( = A + os π πk m( τ dτ k = nk 0 Δ = nδ 00/0 Meixia SJTU 38

39 Mixer = n may no be he desired arrier requeny. The modulaor perorms an up/down onversion o shi he modulaed signal o he desired ener req. This onsiss o a mixer and a BPF s( v ( Band-pass iler v ( os( π l 00/0 Meixia SJTU 39

40 Exerise: A ypial FM ransmier Problem: Given he simpliied blok diagram o a ypial FM ransmier used o ransmi audio signals onaining requenies in he range 00Hz o 5kHz. Problem: Gi h i lii d bl k di i l FM Desired FM wave: = 00MHz, Δ = 75kHz. Se β = 0. in he narrowband phase modulaion o limi harmoni disorion. Speiy he wo-sage requeny muliplier aors n and n 0.MHz 9.5MHz 00/0 Meixia SJTU 40

41 Demodulaion o FM Balaned Frequeny Disriminaor Given FM wave s ( = A + os π πk m τ dτ ( 0 d s ( = A π π km ( sin π π k m ( τ d τ d Hybrid-modulaed wave wih AM and FM H Diereniaor + envelop deeor = FM demodulaor Frequeny disriminaor: a req o ampliude ransorm devie FM wave ( + B / ( + B / Slope irui H ( Slope irui H ( Envelop deeor Envelop deeor jπa, B / + B / ( = jπa, B / + / 0, elsewhere B + - Baseband signal H ( = H( 00/0 Meixia SJTU 4

42 Cirui diagram and requeny response 00/0 Meixia SJTU 4

43 Think Compared wih ampliude modulaion, angle modulaion requires a higher implemenaion omplexiy and a higher bandwidh oupany. Wha is he useulness o angle modulaion sysems? 00/0 Meixia SJTU 43

44 Appliaion: FM Radio broadasing As wih sandard AM radio, mos FM radio reeivers are o super-heerodyne ype Limier disriminaor Baseband low-pass iler Audio ampliier wih de-emphasis Typial req parameers RF arrier range = 88~08 MHz Midband o IF = 0.7MHz IF bandwidh = 00kHz Peak req. deviaion = 75KHz loudspeaker 00/0 Meixia SJTU 44

45 FM Radio Sereo Muliplexing ing Sereo muliplexing is a orm o FDM designed o ransmi wo separae + m + signals via he same arrier. r ( + + Widely used in FM broadasing o - send wo dieren elemens o a program (e.g. voalis and Frequeny doubler aompanis in an orhesra so as o give a spaial dimension o is os( perepion by a lisener a he m( = [ ml ( + mr ( ] reeiving end + [ ml ( mr ( ] os(4π The sum signal is le unproessed in is baseband orm + K os(π The dierene signal and a 38-kHz subarrier produe a DSBSC wave = 9kHz The 9-kHz pilo is inluded as a reerene or oheren deeion i 00/0 Meixia SJTU 0/4/ m l ( + m( K π

46 FM-Sereo Reeiver m( Baseband LPF BPF enered a =38kHz m l (+m r ( Baseband LPF + - m l (-m r ( + + m l ( m r ( To wo loudspeakers Frequeny doubler Narrow-band iler uned o =9kHz 00/0 Meixia SJTU 0/4/004 46

47 3.4 Ee o Noise on Angle Modulaion s ( + n ( Blok diagram o an angle demodulaor BPF BW=B r ( = s ( + n ( y ( Demod LPF BW=W S N + w o Inpu o he demodulaor is r ( = s( + n( [ ] = A os ω + φ ( + n ( os ω n ( sin ω s = R (os( ( w+θ ( n n 00/0 Meixia SJTU 47

48 Assume ha he signal is muh larger han he noise ( θ φ r ( A + Rn(os n( ( os w + φ( + g ( θn φ ( θ φ Rn(sin ( ( A + R (os ( ( n n The phase erm an be urher approximaed as Rn ( θr( = φ( + sin θn( φ( A ( Rn ( ns n ( ( R ( y θ e ( A Rn ( φ ( θ y ( θ n ( A E n ( θ ( φ( n 00/0 Meixia SJTU 48

49 Thereore, he oupu o he demodulaor d is d d Rn ( y ( = θ r( = k m( + sin θ n( φ( π d π d A d = km ( + Y n( π d Desired signal Noise ( φ The noise omponen is inversely proporional o he signal ampliude A. (This is no he ase or AM sysem Rn ( Yn( = sin ( θn( φ( = [ ns(os φ( n(sin φ( ] A A = [ ns (os φ n (sin φ ] (Sine φ ( is slowly varying A 00/0 Meixia SJTU 49

50 d Y n( π d The power speral densiy o is S ( S ( S ( N 0 BC = S ( n = A A 0 oherwise 4 π os φ sin φ Y = n + n π A A 4 n s S ( no Sn ( A he oupu o LPF, he noise is limied o he req. range [-W, W] B T x x BT -W W 00/0 Meixia SJTU 50

51 Now we an deermine he oupu SNR in FM Firs, he oupu signal power is s m The oupu noise power is P n o P o = k P N = = N W W 0 0 d W A 3A Then, he oupu SNR is P s 3 k A P 0 m n 0 S = = N P W N W o o 3 = 3β P m S max m ( N ( b 00/0 Meixia SJTU 5

52 Observaions Inreasing he modulaion index β inreases he oupu SNR, in onras o AM Inreasing he bandwidh inreases he oupu SNR. Thereore, angle modulaion provides a way o rade o bandwidh or ransmied power Inreasing he ransmied power inreases oupu SNR in boh FM and AM sysems, bu he mehanisms are oally dieren (explain! Inreasing β up o a erain value improves he perormane, bu anno oninue indeiniely due o he hreshold ee 00/0 Meixia SJTU 5

53 Threshold h Ee There exiss a speii SNR a he inpu o he demodulaor below whih he signal is no disinguishable rom he noise S 0 N 0 FM DSB 0 a 00/0 Meixia SJTU 53 S i N i

54 Comparison o Analog-Modulaion l Bandwidh eiieny SSB is he mos bandwidh eiien, bu anno eeively ransmi DC VSB is a good ompromise PM/FM are he leas avorable sysems Power eiieny FM provides high noise immuniy Convenional AM is he leas power eiien Ease o implemenaion (ransmier and reeiver The simples reeiver sruure is onvenional AM FM reeivers are also easy o implemen DSB-SC and SSB-SC requires oheren deeor and hene is muh more ompliaed. 00/0 Meixia SJTU 54

55 Appliaions SSB-SC: Voie ransmission over mirowave and saellie links VSB-SCSC Widely used in TV broadasing FM High-ideliy radio broadasing Convenional AM AM radio broadasing DSB-SCSC Hardly used in analog signal ransmission! 00/0 Meixia SJTU 55