Principles of Communications Lecture 4: Analog Modulation Techniques (2) Chih-Wei Liu 劉志尉 National Chiao Tung University
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1 Principles o Communicaions Lecure 4: Analog Modulaion Techniques Chih-Wei Liu 劉志尉 Naional Chiao Tung Universiy cwliu@wins.ee.ncu.edu.w
2 Oulines Linear Modulaion Angle Modulaion Inererence Feedback Demodulaors Analog Pulse Modulaion Dela Modulaion and PCM Muliplexing Commun.-Lec4
3 Hilber Transorm 1 > H jsgn, sgn. -1 < 1 Impulse response: h. π Remark: sgnsignum uncion Magniude and phase H j -j h H 1-9 o H 9 o Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 3
4 4 Remarks Hilber ransorm is jus a sysem ha simply phase-shis all requency componens o is inpu by -½π No abs-inegrable, since he value is ininie a Consider h π 1 < >,, e e G α α α sgn lim G α α + I 1 4 ; ] [ j d e e d e e g G j j π α π α π α π α α j j g G π π α π α α α α α + I 1 4 lim ; lim ] [ lim Noe: hen
5 Hilber Transorm Hilber ransorm is a liner sysem m ˆ -jsgn m m ˆ? m x ˆ x h Ex: 1 x τ dτ π τ x cos ω x ˆ sin ω. x sin ω x ˆ cos ω. x e x ˆ j e?inverse H P H H 1 Thus, x ˆ x jω jω sgn ω. ω >,... Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 5
6 Properies o Hilber Transorm 1 Energy or power o x energy or power o xˆ. P Xˆ... X Check specra densiy: X ˆ I[ xˆ ] H X j sgn X X I x is real, hen x and x ˆ are orhogonal. xxd ˆ or energy signals. 1 T Or, lim xxd ˆ or power signals. T T T Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 6
7 xxd ˆ ˆ * P ˆ Generalized Parseval's Theorem j sgn X d j 1 X d 1 X d X xxd X X d ˆ * * X X d X jsgn X d + is even... Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 7
8 3 I m is a low - pass signal and c high - pass, and heir specra do no overlap, hen Hmc { } mc ˆ, where H{} is Hilber ransorm. 1 1 P mc [ M ] [ C '] AB, I I ' M C ' e j π + d M C e dd j π + ' ' ' d M C ' Ex : c cos ω, ω > BW o M H{ m c} m cˆ m sinω -W W Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 8
9 1 Η{ mc } mc π 1 M C e dd π AB, AB, AB, AB, j π + ' '[ ] ' M C j + e dd j π + ' '[ sgn '] ' M C j e dd j π + ' '[ sgn '] ' M e C j e dd jπ j π ' '[ sgn '] ' M e mc ˆ jπ j π ' d C '[ j sgn '] e d ' A W -W W -W B Regions A&B: M and C have nonzero values Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 9
10 ω ω' H e j ω+ ω e ω ω' j { + j } sgn ' +. sgn + '?sgn ' Le + ' u u + ' no inersecion u> sgn + ' 1 sgn ' in A u< sgn + ' 1 sgn ' in B A B sgn+ 1 + u> + u sgn+ -1 Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 1
11 Analyic Signals I x is real Deine x ˆ P x + jx. analyic signal 1 * x Re{ xp } { xp + xp }. Key: I x is a low-pass signal x p has only one-sided specrum Usage: 1 SSB signal Complex envelopes o band-pass signals Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 11
12 X p X, >, < P x x + jxˆ P FT Xp X + j jsgn X [ ] X 1+ sgn Similarly, X x x jxˆ n X n, > X, < -W W X p W Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 1
13 Band-pass Signal BP signal: cenered around in req domain x a cos[ π + θ ] x cos π xsin π R I in-phase quadraure a x + x 1 xi θ an xr R I a : naural envelope. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 13
14 Commun.-Lec4 14 Equivalen Bandpass Signal Represenaion Bandpass signal: ~ X B/ -B/ A X - cos a x θ π + x x a a a x I R sin cos sin sin cos cos cos π π π θ π θ θ π + I + an 1 x x x x a R I I R θ where inphase quadraure called he complex envelop o is ~ where } Re{ ~ } Re{ x x e x e a x j j π θ π + A carrier zero-requency pars o x, lowpass signals complex envelop ~ jx x e a x I R j + θ
15 BP Signals Remarks: 1 In general, a and θ can be wo independen ime-varying signals. I we can exrac hem separaely wihou inererences, we can send wo independen messages. Two independen messages can also be in he orm o x R and x I. They are carried by wo orhogonal carriers, cos and sin. Q: Can we remove he carrier o a BP signal, and obain an equivalen baseband signal? X I X R X Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 15
16 Complex Envelope De: Complex envelope x % x + jx x e R I P jπ I we sar wih a real baseband signal, he corresponding BP signal has ampliude inormaion only one message. I we sar wih a real BP signal, he corresponding baseband signal becomes complex wo messages. The complex envelope o an arbirary BP signal is an equivalen expression ha conains wo messages. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 16
17 Complex Envelope How o conver a BP signal x o is complex x% envelope mahemaically? x x R I jπ e x% xp x e jπ x+ Remarks: and x p do NOT exis in physical world. They are mahemaical models, convenien in mahemaical operaion. Re jx ˆ real complex complex real x% Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 17
18 To show x x p x % 1 Sar wih x xr cos π xi sin π. x ˆ xrsin π xi cos π xrsin π + xicos π. Assume xr and xi baseband signals and use Hilber ransorm Propery 3. xp x + jx ˆ xr[cos π + jsin π ] + x [ jcosπ sin π ] I jπ jπ I x e + jx e R x + jx e x% e jπ jπ [ ]. R jπ Thereore,. I x % x e P Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 18
19 Summary Analyic signal: signal has only posiive specrum x x + jxˆ P Complex envelop: shi analyic signal o he baseband. Real signal is compleely speciied by analyic signal bu complex signal may no no symmeric in req The complex envelop o a BP DSB signal is simply he message isel. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 19
20 Illusraions o x, x, and x % p Commun.-Lec4 cwliu@wins.ee.ncu.edu.w
21 Bandpass Sysems A physical sysem in passband e.g., a channel may have dieren in-phase and quadraure impulse responses. I he BP impulse response is h, is equivalen baseband complex envelope o his BP sysem is: h % h + jh R I Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 1
22 BP Sysems Physical real-valued BP sysem x real BP signal h real-valued BP sysem y real BP signal Mah model equivalen complex-valued represenaion x % complex envelope h % Complex envelope ~ ~ y h ~ x y % complex envelope Commun.-Lec4 cwliu@wins.ee.ncu.edu.w
23 Proo: h he % he % + he % he % + h% e jπ 1 jπ 1 * jπ x Re{ xe % } xe % + x% e. jπ jπ jπ * jπ * jπ Re{ } [ ]. y x h h λ x λ dλ 1 j π 1 λ * [ h% λ e + h% jπ λ 1 j π λ 1 * j π λ λ e ] [ x% λ e + x% λ e ] dλ 1 1 j π * * jπ h% 4 λ x% λ e dλ + h% λ x λ e dλ 4 % 1 * j4π λ jπ 1 * j4π λ jπ + h% λ x λ e dλ e + h% λ x λ e dλ e 4 % 4 %. * j4π λ j4π * Q h% λ x% λ e dλ h% e x% I 1 { H% X% * }. 1 1 % 4 λ % λ λ % λ λ λ 4 % 1 jπ 1 * * jπ h % x % e + h% x% e Re{ jπ } Re{ jπ h % x % e ye % }, ih % x % y %. jπ * * jπ y h x e d + h x e d Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 3 No overlap in req domain
24 Concluding Remarks On p.9, Z&T, 1 y h % x % e jπ Re{ } 1 Re{ jπ ye % } On p.88 eq.314: wihou ½. On p.89 eq.34: wihou ½. y% I is deined wihou ½, hen j x Re{ xe % π } j h Re{ he % π } ~ ~ y h ~ x y% y % h % x % I we ollow he deiniion o Z&T on, hen Eiher way is ine. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 4
25 Commun.-Lec4 5 h R h I h I h R x R x I + - y R + + y I y y y x x x h h h I R I R I R sin cos sin cos sin cos π π π π π π ~ ~ ~ x h y BP Sysem Implemenaion
26 From Analyic Signal o SSB Signal M p : he posiive-requency porion o M M n : he negaiverequency porion o M Apply he requencyranslaion heorem o boh he M p and M n We obain he uppersideband SSB signal and lower-sideband SSB signal, respecively Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 6
27 SSB Demodulaion Coheren deecion wih possibly phase error X c - c c K cos π LPF - c c x c d LPF y d Kcosω c +θ Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 7
28 Le K 4, we have d x 4cos ω + θ AC [ m cos ω ˆ c + m sin ωc ] 4cos ωc + θ Am cos θ + Am cos[ ω + θ] Remarks: I here exis requency and phase error in he local carrier, hen LSB: y mcos[ Δ π + θ] mˆ sin[ Δ π + θ]. D USB: y mcos[ Δ π + θ] + mˆ sin[ Δ π + θ]. D C C c A m ˆ sin θ + Am ˆ sin[ ω + θ ]. C C c Aer LPF, we have LSB: y m cos θ mˆ sin θ. D message cross-alk USB: y mcos θ + mˆ sin θ. D c c Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 8
29 Carrier inserion wih envelope deecor AC e [ m cos ω ˆ c± m sin ωc] + Kcosωc AC AC [ m + K]cos ω ˆ c± m sinωc a cos ω ± b sin ω. c R a + b Deine. 1 b θ an a e R cos θ cos ω ± R sin θsin ω. c c AC AC y ˆ D R m + K + m. AC I K is large enough, hen yd m + K. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 9 c
30 Time-Domain SSB Waveorm The SSB waveorm is complicaed in general m cosω1.4cosω1 +.9cos3ω1 m ˆ sinω1.4sin ω1+.9sin 3ω1 AC x [ cos ˆ c m ωc± msin ωc] R cos + ω θ c AC R m + mˆ where. 1 m ˆ θ ± an m Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 3
31 SSB Time Signals Commun.-Lec4 31
32 Vesigial-SideBand Modulaion VSB SSB has advanages over DSB and AM. Problems relaed o implemenaion o pracical SSB sysems: 1 imperec Hilber iler sharp cuo, poor low-requency componens, and 3 loss o carrier. Soluions? An inermediae scheme beween DSB and SSB, i.e., do no cu ou hal o he specrum compleely. We call he new scheme VSB. hmhang/ee, Commun.-Lec4 NCTU Oc 9 cwliu@wins.ee.ncu.edu.w 3 3
33 DSB vs. VSB Specra Concep: Include a small amoun o he oher sideband symmeric iler. M+ c M- c M - c c M+ c H M- c H - c c Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 33
34 Requiremens on H VSB iler Carlson, Fig4.4-8 Coheren Demod Carlson, Fig4.5-4 Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 34
35 VSB Demodulaion x d Coheren Deecion c LPF y d cosω c Xc [ M + c + M c] H VSB signal D I{ d } 1 1 [ M + c + M ] H + c + [ M + M c] H c 1 M [ H + c + H c] M + c H + c + M c H c. Aer LPF, 1 YD M [ H + c + H c ]. LP Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 35
36 One soluion: H is mean-shied conjugae anisymmeric abou c. Explicily, le H β be an LP anisymmeric iler; i.e., H β -H β - and H β or > β. U H U c H c β + H β + c or > c or < H β H 1/ -β -1/ β c -β 1/ c c +β U- c Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 36
37 Time-domain Signal o VSB Oen, carrier is added o ransmied signal. This is similar o SSB wih carrier inserion. We only need o use an envelope deecor a he receiver. Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 37
38 Transmied TV Specrum Example: Color TV; Video: VSB; Chrominance: I/Q; Audio: FM Commun.-Lec4 38
39 Frequency Translaion Goal: Translae a BP signal o a new carrier req. Mixing: The process o muliplying a BP signal by a periodic signal. hmhang/ee, Commun.-Lec4 NCTU Oc 9 cwliu@wins.ee.ncu.edu.w 39 39
40 Mixing Mixing wih ω 1 +ω [Original a ω 1 ] ω -ω 1 ±ω 1 + ω ω 1 ±ω 1 + ω -ω 1 -ω -ω 1 ω 1 ω -ω ω 1 +ω Problem? Two inpu signals are ranslaed o he same req.! Image requency Inpu 1 a ω 1 ±ω 1 +ω ω and ω 1 +ω Inpu a ω 1 +ω ±ω 1 +ω ω and ω 1 +3ω Same ω! hmhang/ee, Commun.-Lec4 NCTU Oc 9 cwliu@wins.ee.ncu.edu.w 4 4
41 Image Frequency Commun.-Lec4 41
42 Superheerodyne Receiver A ypical A broadcas receiver should perorm he ollowing uncions in addiion o demodulaion. 1. Carrier-requency uning: selec he desired signal channel. Filering: separae he desired signal rom oher modulaed signals. 3. Ampliicaion: compensae or ransmission loss. Carlson, p.88 The super-heerodyne superhe receiver ulils he above requiremens wihou using a high-gain unable bandpass iler. by E.H. Armsrong Remark: I is diicul o consruc a narrow bandpass iler a high requency Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 4
43 Superheerodyne Receiver I has wo ampliicaion and ilering secions prior o demodulaion: 1 RF: radio req; IF: inermediae req A B C D Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 43
44 A. A he anenna: The desired signal wih carrier ω c. undesired image requency signals: ω c +ω IF or ω c -ω IF B. Aer he RF iler: Only he desired signal a ω c can go hrough a wide BPF ilers ou he image requency wih ω IF. Some near-by channels can go hrough bu will be ilered ou a IF C. Aer he mixer: The desired signal a ω IF. There are oher near-by channels D. Aer he IF iler: Only he desired signal a ω IF can go hrough. a narrow BPF. Oher channels are ilered ou Remarks: IF req is almos ixed Carlson, p.89 AM: IF~455 KHz; IFBW~1K; RF: MHz FM: IF~1.7 MHz; IFBW~K; RF: MHz Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 44
45 High-Side Tuning Two possible LO reqs. Choose he one wih smaller LO uning range. AM range: 54kHz ~ 16kHz; IF: 455kHz Low-side uning: ω LO ω c - ω IF Range: 85kHz 54kHz 455kHz ~ 1145kHz 16kHz 455kHz 1 : High-side uning: ω LO ω c + ω IF Range: 995kHz 54kHz + 455kHz ~ 55kHz 16kHz + 455kHz 1 :.7 Commun.-Lec4 cwliu@wins.ee.ncu.edu.w 45
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