4.5 Ampliude modulaion: AM 4.55. DSB-SC ampliude modulaion (which is summarized in Figure 21) is easy o undersand and analyze in boh ime and frequency domains. However, analyical simpliciy is no always accompanied by an equivalen simpliciy in pracical implemenaion. Message (modulaing signal) 2cos 2 f Modulaor c 1 Figure 21: DSB-SC modulaion. Problem: The (coheren) demodulaion of DSB-SC signal requires he receiver o possess a carrier signal ha is synchronized wih he incoming carrier. This requiremen is no easy o achieve in pracice because he modulaed signal may have raveled hundreds of miles and could even suffer from some unknown frequency shif. 4.56. f a carrier componen is ransmied along wih he DSB signal, demodulaion can be simplified. (a) The received carrier componen can be exraced using a narrowband bandpass filer and can be used as he demodulaion carrier. (There is no need o generae a carrier a he receiver.) (b) f he carrier ampliude is sufficienly large, he need for generaing a demodulaion carrier can be compleely avoided. This will be he focus of his secion. 63
Definiion 4.57. For AM, he ransmied signal is ypically defined as x AM () = (A + m ()) cos (2πf c ) = A cos (2πf c ) carrier 4.58. Specrum of x AM (): + m () cos (2πf c ) sidebands Basically he same as ha of DSB-SC signal excep for he wo addiional impulses (discree specral componen) a he carrier frequency ±f c. This is why we say he DSB-SC sysem is a suppressed carrier sysem. Definiion 4.59. Consider a signal A() cos(2πf c ). f A() varies slowly in comparison wih he sinusoidal carrier cos(2πf c ), hen he envelope E() of A() cos(2πf c ) is A(). 4.60. Envelope of AM signal: For AM signal, A() = A + m() and See Figure 22. E() = A + m(). (a) f, A() > 0, hen E() = A() = A + m() The envelope has he same shape as m(). We can deec he desired signal m() by deecing he envelope (envelope deecion). (b) f, A() < 0, hen E() A(). The envelope shape differs from he shape of m() because he negaive par of A + m() is recified. This is referred o as phase reversal and envelope disorion. 64
192 AMPLTUDE MODULATONS AND DEMODULATONS Figure 4.8 AM signal and is envelope. m(1) ~np A+ m(l) > 0 far all 1 A (b) A + m(1) ':{> 0 far all 1 }L\------- 1~ (~ 1-- En velope ~A+ m(l) Envelope la + m()l / (d) (e) Figure 22: AM signal and is envelope [5, Fig 4.8] and m() canno be recovered from he envelope. Consequenly, demodulaion of rpam () in Fig. 4.8d amouns o sim ple envelope deecion. Thus, he condiion for envelope deecion of an AM signal is Definiion 4.61. The posiive consan µ max (envelope of he sidebands) A + m() ::: 0 for all (4.9a) max m () f m() ::: 0 for all/, hen A = 0 already sai sfies condiion ( 4.9a). n hi s case here is no need o add any carrier because he envelope of he DSB-SC = signal m() cos We is rn() = m p, and such a max DSB-SC (envelope signal can be deeced of he by envelope carrier) deecion. n he max following A discussion we Aassume ha 111() _ 0 for all ; ha is, m() can be negaive over some range of. is called he modulaion index. m p max ivessage Signals m () wih Ze1 o Offse: Le ±mp be he maximum and he minimum va lues of m(), res peci vely (Fig. 4.8). This means ha m() ::: - mp. Hence, he condiion of envelope deecion (4.9a) is equi valen o m () (4.9b) By he way m p is defined, he message m() is beween ±m p. Thus, he minimum carrier ampliude required for he viabiliy of envelope deecion is mp This is quie clear from Fig. 4.8. We define he modulaion index fj- as The quaniy µ 100% is ofen referred o as he percen modulaion. mp fj- = A (4.10a) 65
Example 4.62. Consider a sinusoidal (pure-one) message m() = A m cos(2πf m ). Suppose A = 1. Then, µ = A m. Figure 23 shows he effec of changing he value modulaion index on he modulaed signal. 1.5 50% Modulaion 0.5 0 0.5 1.5 2 Time 100% Modulaion Envelope Modulaed Signal 0 2 Time 2.5 150% Modulaion 0.5 0 0.5 1 2.5 Time Figure 23: Modulaed signal in sandard AM wih sinusoidal message 4.63. should be noed ha he raio ha defines he modulaion index compares he maximum of he wo envelopes. n oher references, he noaion for he AM signal may be differen bu he idea (and he corresponding moivaion) ha defines he modulaion index remains he same. n [3, p 163], i is assumed ha m() is already scaled or normalized o have a magniude no exceeding uniy ( m() 1) [3, p 163]. There, x AM () = A c (1 + µm ()) cos (2πf c ) = A c cos (2πf c ) + A c µm () cos (2πf c ). carrier sidebands 66
m p = 1 The modulaion index is hen n [14, p 116], max (envelope of he sidebands) max (envelope of he carrier) ( x AM () = A c 1 + µ m () ) m p The modulaion index is hen max (envelope of he sidebands) max (envelope of he carrier) 4.64. Power of he ransmied signals. = max A c µm () max A c = A cµ A c = µ. cos (2πf c ) = A c cos (2πf c ) + A c µ m () cos (2πf c ) m p. carrier = max A c µ m() m p max A c (a) n DSB-SC sysem, recall, from 4.36, ha, when wih f c sufficienly large, we have x() = m() cos(2πf c ) P x = 1 2 P m. } {{ } sidebands = A c µ mp m p A c Therefore, all ransmied power are in he sidebands which conain message informaion. (b) n AM sysem, x AM () = A cos (2πf c ) + m () cos (2πf c ). carrier sidebands f we assume ha he average of m() is 0 (no DC componen), hen he specrum of he sidebands m() cos(2πf c +θ) and he carrier A cos(2πf c + θ) are non-overlapping in he frequency domain. Hence, when f c is sufficienly large P x = 1 2 A2 + 1 2 P m. = µ. 67
Efficiency: For high power efficiency, we wan small m 2 p µ 2 P m. By definiion, m() m p. Therefore, m2 p P m 1. Wan µ o be large. However, when µ > 1, we have phase reversal. So, he larges value of µ is 1. The bes power efficiency we can achieved is hen 50%. Conclusion: a leas 50% (and ofen close o 2/3[3, p. 176]) of he oal ransmied power resides in he carrier par which is independen of m() and hus conveys no message informaion. 4.65. An AM signal can be demodulaed using he same coheren demodulaion echnique ha was used for DSB. However, he use of coheren demodulaion negaes he advanage of AM. Noe ha, concepually, he received AM signal is he same as DSB- SC signal excep ha he m() in he DSB-SC signal is replaced by A() = A + m(). We also assume ha A is large enough so ha A() 0. Recall he key equaion of swiching demodulaor (52): LPF{A() cos(2πf c ) 1[cos(2πf c ) 0]} = 1 A() (53) π We noed before ha his echnique requires he swiching o be in sync wih he incoming cosine. 68
4.66. Demodulaion of AM Signals via recifier deecor: The receiver will firs recover A + m() and hen remove A. When, A() 0, we can replace he swiching demodulaor by he recifier demodulaor/deecor. n which case, we suppress he negaive par of A() cos(2πf c ) using a diode (half-wave recifier: HWR). Here, we define a HWR o be a memoryless device whose inpuoupu relaionship is described by a funcion f HWR ( ): { x, x 0, f HWR (x) = 0, x < 0. This is mahemaically equivalen o a swiching demodulaor in (52) and (53). is in effec synchronous deecion performed wihou using a local carrier [4, p 167]. This mehod needs A() 0 so ha he sign of A() cos(2πf c ) will be he same as he sign of cos(2πf c ). The dc erm A π m()/π. may be blocked by a capacior o give he desired oupu 69
AMPLTUDE MODULATONS AND DEMODULATONS ure 4.10 ifier deecor AM. [a+ m()] cos we ' VR() -_f /[A + m()] " rr [A + 111(1)] - -;-[A + m(1)] ~ ' / [A + m(l)] cos we Low-pass filer Figure 24: Recifier deecor for AM [5, Fig. 4.10]. signal is muliplied by w(). Hence, he half-wave recified oupu vr() is Figure 4.11 Envelope deecor for AM. 4.4 Bandwidh-Efficien Ampliude Modulaions 197 VR() ={[A+ m()] COS We) w() (4.12) =[A+ m()] cos We [ ~ + ~ (cos (Ve- ~co s 3we + ~co s Swe- )] (4.13) l AM signal c = -[A+ m()] +oher erms of higher frequencies (4.14) ][ When vr() is applied o a low-pass filer of cuoff (a) B Hz, he oupu is [A+ m()]jn, and all he oher erms of frequencies higher han B Hz are suppressed. The de Envelope erm deecor Ajn oupu may be blocked by a capac ior (Fig. 4.10) o give he desired oupu m() j n. The oupu can be doubled by RC oo large \ using a full-wave K 'f<k~ recifi er. -- ~.. En ve lop~.--... '( K "' "" is ineresing o noe ha because of he muliplicaion wih ll'(l), recifier deecion is in effec synchronous deecion ~ performed -~. wihou using a local,.. carrier. < i""!'-- The high carrier conen in AM ensures ha is zero crossings are periodic and he informaion abou frequency and phase of he carrier a he ransmier is buil in o he AM signal iself.,., ~ ~" W'~ Envelope Deecor: fn an enve lope deecor, he oupu of he deecor follows he envelope of he modulaed signal. The simple circui shown in Fig. 4. lla funcions as an envelope deecor. On he posiive cycle... of he inpu signal, he inpu grows and may exceed he charged vo lage on he capaciy... - ' vc(), urning on he diode and allowing he capacior C o charge up o he peak volage.. of he inpu signal cycle. As he inpu... signal falls below his... peak value, i falls quickly below he capacior (b) volage (which is very nearly he peak volage), hus causing he diode o open. The capacior now di scharges hrough he resisor R a a slow rae (wih a ime consan RC). During he nex posiive cycle, he same drama repeas. As he a declining envelope. Figure 25: Capacior Envelope discharge deecor beween for posii AM ve [5, peaks Fig. causes 4.11]. a ripple signal of inpu signal rises above he capacior volage, he diode conducs again. The capacior again frequency We in he oupu. This ripple can be reduced by choosing a larger ime consan charges o he peak value of his (new) cycle. The capacior discharges slowly during he cuoff RC so ha he capacior discharges very lile beween he posiive peaks (RC» /eve). f period. RC were made oo large, however, i would be impossible for he capac ior volage o follow During a fas each declining posiive e nvelope cycle, (Fig. he 4.11 capacior b). Because 70charges he maximum up o rae he of peak AM volage envelope decline of he inpu signal and is hen dominaed decays by slowly he bandwidh unil he B of nex he posiive message cycle, signal m as (r ) shown, he design in Fig. crierion 4.ll b. of Thus, RC he should be oupu volage vc(), closely follows he (rising) envelope of he inpu AM signal. Equally imporan, he slow capaciy discharge via he resisor R a ll ows he capaciy vo lage o follow
4.67. Demodulaion of AM signal via envelope deecor: Design crierion of RC: 2πB 1 RC 2πf c. The envelope deecor oupu is A + m() wih a ripple of frequency f c. The dc erm can be blocked ou by a capacior or a simple RC high-pass filer. The ripple may be reduced furher by anoher (low-pass) RC filer. 4.68. AM Trade-offs: (a) Disadvanages: Higher power and hence higher cos required a he ransmier The carrier componen is wased power as far as informaion ransfer is concerned. Bad for power-limied applicaions. (b) Advanages: Coheren reference is no needed for demodulaion. Demodulaor (receiver) becomes simple and inexpensive. For broadcas sysem such as commercial radio (wih a huge number of receivers for each ransmier), any cos saving a he receiver is muliplied by he number of receiver unis. i is more economical o have one expensive high-power ransmier and simpler, less expensive receivers. (c) Conclusion: Broadcasing sysems end o favor he rade-off by migraing cos from he (many) receivers o he (fewer) ransmiers. 4.69. References: [3, p 198 199], [5, Secion 4.3] and [13, Secion 3.1.2]. 71