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CONTINUOUS-WAVEMODULATIONIn this chapter we study continuous-wave modulation, which is basic to the operation ofanalog communication systems. The chapter is divided into two related parts. In the firstpart we study the time-domain and frequency-domain descriptions of two basic families ofcontinuous-wave modulation: -~ Amplitude modulation, in which the amplitude of a sinusoidal carrier is varied inaccordance with an incoming message signal.~ Angle modulation, in which the instantaneous frequency or phase of the sinusoidal carrieris varied in accordance with the message signal.The second part of the chapter focuses on the effects of channel noise on the performanceof the receivers pertaining to these m~dulation schemes.Advantages and disadvantages of the different methods of continuous-wavemodulation are highlighted in light of the material presented herein.I2.1 IntroductionThe purpose of a communication system is to transmit information-bearing signals througha communication channelseparating the transmitter from the receiver. Informationbearing signals are also referred to as baseband signals. The term baseband is used todesignate the band of frequencies representing the original signal as delivered by a sourceof information. The proper use of the communication channel requires a shift of the rangeof baseband frequencies into other frequency ranges suitable for transmission,and a corresponding shift bac to the original frequency range after reception. !or e"ample, a radiosystem must operate with frequencies of #$ %& and upward, whereas the baseband signalusually contains frequencies in the audio frequency range, and so some form of frequencyband shifting must be used for the system to operate satisfactorily. A shift of the range offrequencies in a signal is accomplished by using modulation, which is defined as the processby which some characteristic of a carrier is varied in accordance with a modulating wave(signal). A common form of the carrier is a sinusoidal wave, in which case we spea of acontinuous-wave modulation! process. The baseband signal is referred to as the modulating wave, and the result of the modulation process is referred to as the modulated wave.'odulation is performed at the transmitting end of the communication system. At thereceiving end of the system, we usually require the original baseband signal to be restored.This is accomplished by using a process nown as demodulation,which is the reverse ofthe modulation process.In basic signal-processing terms, we thus find that the transmitterof an analog communication system consists of a modulatorand the receiver consists of a demodulator,as882.1 Introductien 89MessagesignalModulatedwave(b)Sinusoidalcarrierwave(a)FIGURE2.1 Components of a continuous-wave modulation system: (a) transmitter and (b)receiver.depicted in Fi!ure 2.1. Inaddition to t"e si!nal received from t"e transmitter t"e receiverinput includes c"annel noise. #"e de!radation in receiver performance due to c"annel noiseis determined $y t"e type of modulation used.In t"is c"apter we study two families of continuous-wave %C&' modulation systemsnamely amplitude modulation and angle modulation. In amplitude modulationt"e am(plitude of t"e sinusoidal carrier wave is varied in accordance wit" t"e $ase$and si!nal. Inan!le modulation t"e an!le of t"e sinusoidal carrier wave is varied in accordance wit" t"e$ase$and si!nal. Fi!ure 2.2 displays t"e waveforms of amplitude-modulated and an!le(modulated si!nals for t"e case of sinusoidal modulation.)arts (a) and (b) of t"e fi!ures"ow t"e sinusoidal carrier and modulatin! waves respectively. )arts (e ) and (d) s"ow t"en n n O O A O A A A n A n n A O A n o nv v v v v v v v v v v v v v v v v v v v(a)~(b)%el(d)#ime r+e-FIGURE2.2 Illustratin! *Mand FM si!nals produced $y a sin!le tone. %a+ Carrier wave. (b),inusoidal modulatin! si!ual. %el *mplitude-modulated si!ual. (d) Fre-uency-modulated si!nal.90 CHAPTER 2 .. CONTINUOUS-WAVEMODUlATIONcorresponding amplitude-modulated and frequency-modulated waves, respectively; frequency modulation is a form of angle modulation.This figure clearly illustrates the basicdifferences between amplitude modulation and angle modulation,which are discussed inwhat follows.I2.2 Amplitude ModulationConsider a sinusoidal carrier wave cit) defined by(2.!where Ae is the carrier amplitude and t;is the carrier frequency. To simplify the e"positionwithout affecting the results obtained and conclusions reached, we have assumed that thephase of the carrier wave is #ero in $quation (2.!. %et mit) denote the baseband signalthat carries the specification of the message. The source of carrier wave cit) is physicallyindependent of the source responsible for generating mit). Amplitude modulation (AM) isdefined as a process in which the amplitude of the carrier wave crt) is varied about a meanvalue, linearly with the baseband signal m(t). An amplitude-modulated (&'! wave maythus be described, in its most general form, as a function of time as follows(sit) =Ac[l+ kam(t) cos(!"#$fct) (2.2!where ka is a constant called the amplitude sensitivity of the modulator responsible for thegeneration of the modulated signal sit). Typically, the carrier amplitude Ac and the messagesignal mit) are measured in volts, in which case ka is measured in volr).*igure !.%a shows a baseband signal mit), and *igures !.%b and !.%c show the corresponding &' wave sit) for two values of amplitude sensitivity k; and a carrier amplitude&Ae= volt. +e observe that the envelope of sit) has essentially the same shape as the..baseband signal mit) provided that two requirements are satisfied(1. The amplitude of kam(t) is always less than unity, that is,I kam(t) I .0 herein lies the imp!rtan"e !) the iea !) "negati(e" )re+uen"ies.2. F!r p!siti(e )re+uen"ies, the p!rti!n !) the spe"trum !) an %& 'a(e l*ing a$!(e the"arrier )re+uen"* fe is re)erre t! as the uppe! sideband 'hereas the s*mmetri"p!rti!n $el!' fe is re)erre t! as the lo"e! sideband. F!r negati(e )re+uen"ies, theupper sie$an is represente$* the p!rti!n !) the spe"trum $el!' - fe an thel!'er sie$an $* the p!rti!n a$!(e - fc. 1he "!niti!n fe >. ensures thatthe sie$ans ! n!t !(erlap.92 CHAYrER 1 .. CONTINUOUS-WAVEMODUlATIONM(f) S(f)M(O)~~(f-f"(a)(b)FIGURE 1.4 (a) Spectrum of baseband signal. (b) Spectrum of AM wave.3. For positive frequencies, the highest frequenc componentof the AM wave equalsI e + !, and the lowest frequenc component equals t; - !. "he difference betweenthese two frequencies defines the transmission bandwidth By for an AM wave, whichis e#actl twice the message bandwidth W,that is,BT= 2W $2.%&i ! ' II '()"*+S A,-.(M("A"(/,S /F AM0.("*-+ M/-*.A"(/,Amplitude modulation is the oldest method of performing modulation. (ts greatest virtueis the simplicit of implementation12 (n the transmitter,amplitude modulation is accomplished using a nonlinear device.For e#ample, in the switching modulator discussed in 0roblem 2.3, the combinedsum of the message signal and carrier wave is applied to a diode, with the carrieramplitude being large enough to swing across the characteristic curve of the diode.Fourier analsis of the voltage developed across a resistive load reveals the generationof an AM component, which ma be e#tracted b means of a band3pass filter.,..Inthe receiver, amplitude demodulation is also accomplished using a nonlinear de4vice. For e#ample, we ma use a simple and et highl effective circuit 5nown as thee nve loede te ctor, which is discussed in 0roblem 2.6. "he circuit consists of a diodeconnected in series with the parallel combinationof a capacitor and load resistor.Some version of this circuit is found in most commercial AM radio receivers. 0ro4vided that the carrier frequenc is high enough and the percentage modulation is lessthan 177 percent, the demodulator output developed across the load resistor is nearlthe same as the envelope of the incoming AM wave, hence the name 8envelopedetector.8)ecall, however, that transmitted power and channel bandwidth are our two primarcommunication resources, and the should be used efficientl. Inthis conte#t, we find thatthe standard form of amplitude modulation defined in +quation $2.2& suffers from twoma9or limitations11. !mlitude modulation is waste fulof owe r""he carrier wave cit) is completelindependent of the information3bearing signal mit)" "he transmission of the carrierwave therefore represents a waste of power, which means that in amplitude modu4lation onl a fraction of the total transmitted power is actuall affected b mit)"2.3 Linear MotLUationScJ.emes 932. Amplitude modulation is wasteful of bandwidth. The upper and lower sidebands ofan AM wave are uniquely related to each other by virtue of their symmetry aboutthe carrier frequency; hence, given the magnitude and phase spectra of either sideband, we can uniquely determine the other. This means that insofar as the transmission of information is concerned, only one sideband is necessary, and the communication channel therefore needs to provide only the same bandwidth as the basebandsignal. In light of this observation,amplitude modulation is wasteful of bandwidthas it requires a transmission bandwidth equal to twice the message bandwidth.To overcome these limitations,we must make certain modificationssuppress thecarrier and modify the sidebands of the AM wave. These modifications naturally result inincreased system comple!ity. In effect, we trade system comple!ity for improved use ofcommunicationresources. The basis of this trade"off is linear modulation, which is discussed in the ne!t section. In a strict sense, full amplitude modulation does not qualify aslinear modulation because of the presence of the carrier wave.I2.3Linear Modulation SchemesInits most general form, linear modulation is defined by#$.%&where slit) is the in-phase component of the modulated wave sit), and s!t) is its "uad#rature component.'quation #$.%& is recogni(ed as the canonical