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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 representationof a narrowband signal, which is discussed in detail in Appendi! $. In linear modulation, bothslit) and s!t) are low"pass signals that are linearly related to the message signal mit).Indeed, depending on how these two components of sit) are defined, we may identifythree types of linear modulation involving a single message signal1. $ouble sideband-suppressed carrier!$S%-S&) modulation, where only the upper andlower sidebands are transmitted.2. Sin'le sideband !SS%) modulation,where only one sideband #the lower sideband orthe upper sideband& is transmitted.3. (esti'ial sideband !(S%) modulation,where only a vestige #i.e., trace& of one of thesidebands and a correspondingly modified version of the other sideband aretransmitted.Table $.) presents a summary of the definitions of these three special forms of linearmodulation.There are two importantpoints to note from Table $.)). The in"phase component slit) is solely dependent on the message signal mit).$. The quadrature components!t) is a filtered version of mit). The spectral modification of the modulated wave sit) is solely due to s!t).To be more specific, the role of the quadrature component #if present& is merely to interferewith the in"phase component, so as to reduce or eliminate power in one of the sidebandsof the modulated signal sit), depending on how the quadrature component is defined.94 CK.u>TER2 I'l CONTlNUOtJS-WAVEMODULATIONITABLE 2.1 Differentforms of linear modulationIn-Phase QuadratureComponent ComponentType of Modulation slt) sQ{t) CommentsDSB-SC m(t) 0 m(t) =messae s!"alSSB#$%a) U&&e' s!(e)a"( !m(t) 1m(t) m(t) =*!l)e't t'a"s+,'m,+ m(t)t'a"sm!tte(%)) L,-e' s!(e)a"( trn(t) -!m(t)t'a"sm!tte(VSB#.%a) Vest!e ,+ l,-e' s!(e)a"( }m(t) ~n'(t)r " ) ~ -~ ,+ t.e +!l'e' ,+t'a"sm!tte( +'e/ue"01 'es&,"se HQ(f)%)) Vest!e ,+ u&&e' !m(t) -~ '(t)(ue t, m(t).s!(e)a"( t'a"sm!tte(2,' t.e (e+!"!t!,",+ HQ(f),see E/. %3.45)$2,' t.e mat.emat!0al(es0'!&t!,",+ s!"les!(e)a"( m,(ulat!,"6 see7',)lem3.45.I! i II DOUBLE SIDEBAND-SU77RESSED CARRIER %DSB-SC) MODUlATIONT.!s +,'m ,+ l!"ea' m,(ulat!," !s e"e'ate( )1 us!" a produt modulator t.at s!m&l1mult!&l!es t.e messae s!"al mIt) )1 t.e 0a''!e' -a8e ! C"#($%Tft),as !llust'ate( !" 2!u'e$a& S&e0!+!0all16 -e -'!tesIt) =Acm(t) C"#($%Tft) (2.8)m(t)Basebandsi gnal m(r)DSBS!m"d#la$ed %a&es(t) 9 Acm(/) 0,s(27T/cl)!arri erti 0,s (2ft/el)%:) (b)6%6)'(ase re&ersals(c)2I;URE3.< (a) Bl,0= (!a'am ,+ &',(u0t m,(ulat,'. (b) Base)a"( s!"al. (e) DSB-SC m,(u>late( -a8e.2.3 Linear Modulation Schemes 95,11(0)S(f)~iAcM(o)- - - - L_ D ~- - - _ - - - - - _ - - - - - J _ _ _ _ J _ _ _ 7_ _.! ! _ _ fL-:~_j 0 L:~_jM(f)(a) (b)FIGURE2.6 (a) Spectrum of baseband sina!. (b) Spectrum of "S#-S$ modu!ated %a&e.Fiure 2.Sc s'o%s t'e modu!ated sina! s(t) for t'e arbitrar( messae %a&eform of Fiure2.S". )'e modu!ated sina! s(t) underoes a #hase re$ersal %'ene&er t'e messae sina!m(t) crosses *ero. $onse+uent!(, t'e en&e!ope of a "S#-S$modu!ated sina! is differentfrom t'e messae sina!- t'is is un!i.e t'e case of an /0 %a&e t'at 'as a percentaemodu!ation !ess t'an 122percent.From E+uation 32.45, t'e Fourier transform of s(t) is obtained asS(f)=~%c&M(f - !c) + M(f + fell (2.9)For t'e case %'en t'e baseband sina! m(t) is !imited to t'e inter&a! - 6:s- f :76, as inFiure 2.'a! %e t'us find t'at t'e spectrum S(f) of t'e "S#-S$%a&e s(t) is as i!!ustratedin Fiure 2.'". E8cept for a c'ane in sca!e factor, t'e modu!ation process simp!( translatest'e spectrum of t'e baseband sina! b( (fc. 9f course, t'e transmission band%idt' re:+uired b( "S#-S$modu!ation is t'e same as t'at for amp!itude modu!ation,name!(, 2W.III COHERENTDETECTION)'e baseband sina! m(t) can be uni+ue!( reco&ered from a "S#-S$ %a&e s(t) b( firstmu!tip!(in s(t) %it' a !oca!!( enerated sinusoida! %a&e and t'en !o%-pass fi!terin t'eproduct,as in Fiure 2.;. It is assumed t'at t'e !oca! osci!!ator sina! is e8act!( co'erentor s(nc'roni*ed, in bot' fre+uenc( and p'ase, %it' t'e carrier %a&e c(t) used in t'e prod:uct modu!ator to enerate s(t). )'is met'od of demodu!ation is .no%n as coherent detec)tion or s*nchronous demodulation.It is instructi&e to deri&e co'erentdetection as a specia! case of t'e more enera!demodu!ation process usin a !oca! osci!!ator sina! of t'e same fre+uenc( but arbitrar(p'ase difference < p , measured %it' respect to t'e carrier %a&e c(t). )'us, denotin t'e !oca!s(t)FIGURE2.; $o'erent detector for demodu!atin "S#-S$ modu!ated %a&e.96 CHAPTER 2B ' ! CONTINUOUS-WAVE MODULATIONV(flFIGURE 2.8 Illustrating the spectrum of a product modulator output with a DSB-SC modulatedwave as input.oscillator signal by A ; COS(27Tfct + r P ) ,and using Equation (.!" for the DSB-SCwave s(t),we find that the product modulator output in #igure .$ isv(t)=A ; COS(27Tlct + ,p )s(t)=A .A ; COS(27Tlct) COS(27Tlct + ,p)m(t)% %=&A .A ; COS(47Tfct ' ,p)m(t) + &A .A ; cos,p m(t)(.%(")he first term in Equation (.%(" represents a DSB-SC modulated signal with a carrierfrequency *n whereas the second term is proportionalto the baseband signal m(t). )hisis further illustrated by the spectrum V(f)shown in #igure. .!+ where it is assumed thatthe baseband signal m(t) is limited to the interval - ,& s-- I& s--,. It is therefore apparentthat the first term in Equation (.%(" is removed by the low-pass filter in #igure .$+provided that the cut-off frequency of this filter is greater than , but less than 21e - ,.)his requirementis satisfied by choosing fc >,. .t the filter output we then obtain asignal given by2.!!")he demodulated signal vo(t) is therefore proportional to m(t) when the phase error /0is a constant.)he amplitude of this demodulated signal is ma1imum when ,p =(+ andit is minimum (2ero" when ,p =7T/2. )he 2ero demodulated signal+ which occurs for,p =#7T/2, represents the quadrature ulleffect of the coherent detector. )hus the phaseerror $ Pin the local oscillator causes the detector output to be attenuated by a factor equalto cos ,p. A% long as the phase error ,p is constant+the detector provides an undisrorredversion of the original baseband signal m(t). In practice+ however+ we usually find that thephase error ,p varies randomly with time+ due to random variations in the communicationchannel. )he result is that at the detector output+the multiplying factor cos ,p also variesrandomly with time+ which is obviously undesirable. )herefore+ provision must be madein the system to maintain the local oscillator in the receiver in perfect synchronism+ in bothfrequency and phase+ with the carrier wave used to generate the DSB-SCmodulated signalin the transmitter. )he resulting system comple1ity is the price that must be paid forsuppressing the carrier wave to save transmitter power.&! 'I C3S).S RECEIVER3ne method of obtaining a practical synchronous receiver system+ suitable for demodu4lating DSB-SC waves+ is to use the Costas rece!ver" shown in #igure .5. )his receiver2.3 Li_r Modulati....Schemes 97[-channelDS8-SC signalA, cos (2"'/,t) m(t)Q-channelFIGURE 2.9 Costas receiver.consists of two coherent detectors supplied with the same input signal namel! the incom"ing #$%-$C wave A, cos(2'1Tlct)m(t), &ut with individual local oscillator signals that arein phase 'uadrature with respect to each other. (he fre'uenc! of the local oscillator isad)usted to &e the same as the carrier fre'uenc!e! which is assumed *nown a"riori. (hedetector in the upper path is referred to as the i#$"hase cohere#t detector or l$cha##el,and that in the lower path is referred to as the %uadrature$"hase cohere#t detector or&$cha##el. (hese two detectors are coupled together to form a #e'ati(e )eed*ac+ s!stemdesigned in such a wa! as to maintain the local oscillator s!nchronous with the carrierwave.(o understand the operation of this receiver suppose that the local oscillator signalis of the same phase as the carrier wave Ac cos(2'1T ,) used to generate the incoming#$%-$C wave. Under these conditionswe find thatthe I-channeloutputcontains thedesired demodulated signal m(t),whereas the Q-channel output is +ero due to the 'uad"rature null effect of the Q-channel. $uppose ne,t that the local oscillator phase drifts fromits proper value &! a small angle ," radians. (he I-channel output will remain essentiall!unchanged &ut there will now &e some signal appearing at the Q-channel output whichis proportionalto sin ," =," for small ,". (his Q-channel output will have the same polarit!as the I-channel output for one direction of local oscillator phase drift and opposite po"larit! for the opposite direction of local oscillator phase drift. (hus &! com&ining the $ and Q-channel outputs in a "hase discrimi#ator -which consists of a multiplier followed&! a low-pass filter. as shown in Figure 2.9 a #C control signal is o&tained that auto"maticall! corrects for local phase errors in the (olta'e$co#trolled oscillator.It is apparentthat phase control in the Costas receiver ceases with modulation andthat phase-loc* has to &e reesta&lished with the reappearance of modulation. (his is nota serious pro&lem when receiving voice transmission &ecause the loc*-up process normall!occurs so rapidl! that no distortion is percepti&le.II!IQU/#R/(URE-C/RRIER0U1(I21E3I4G(he 'uadrature null effect of the coherentdetector ma! also &e put to good use in theconstruction of the so-called %uadrature$carrier multi"le-i#' or %uadrature$am"litude98 CHAPTER2 " CONTINUOUS-WAVEMODULATIONMessagesignal ml(r}MultipJe.x~dsignal I)Multiplexedsignal set)Messagesignal m2(t)(a) (b)Quadrature-carrier multile!i"# $%$tem& (a) Tra"$mitter& (b) Recei'er& (I)URE*+,-modulation.QAM)& T/i$ $c/eme e"a0le$ t12 DS3-SCm2dulated 1a'e$ .re$ulti"# 4r2mt/e alicati2" 24 t12 /%$icall% independent me$$a#e $i#"al$) t2 2ccu% t/e $ame c/a""el0a"d1idt/5a"d %et it all21$ 42r t/e $earati2" 24 t/e t12 me$$a#e $i#"al$ at t/e recei'er2utut& It i$ t/ere42re a bandwidth-conservation scheme.A 0l2c6 dia#ram 24 t/e 7uadrature-carrier multile!i"# $%$tem i$ $/21" i" (i#ure*&,-& T/e tra"$mitter art 24 t/e $%$tem5 $/21" i" (i#ure 2.10a, i"'2l'e$t/e u$e 24 t12$earate r2duct m2dulat2r$t/at are $ulied 1it/ t12 carrier 1a'e$ 24 t/e $ame 4re87ue"c% 0ut di44eri"# i" /a$e 0% -9- de#ree$& T/e tra"$mitted $i#"al sit) c2"$i$t$ 24 t/e$um 24 t/e$e t12 r2duct m2dulat2r2utut$5 a$ $/21" 0%sit) =A.m,(t) cos(2'T/f) + Acm2(t) sin(21Tf!t) .*&,*)1/ere m,(t)a"d m2(t) de"2te t/e t12 di44ere"t me$$a#e $i#"al$ alied t2 t/e r2ductm2dulat2r$& T/u$ sit) 2ccuie$ a c/a""el 0a"d1idt/24 * "# ce"tered at t/e carrier 4re87ue"c% f:, 1/ere "# i$ t/e me$$a#e 0a"d1idt/ 24 m,(t) 2r m2(t). Acc2rdi"# t2 E7uati2".*&,*)5 1e ma% 'ie1 Acm,(t)a$ t/e i"-/a$e c2m2"e"t24 t/e multile!ed 0a"d-a$$$i#"al sit) a"d -Acm2(t) a$ it$ 7uadrature c2m2"e"t&T/e recei'er art 24 t/e $%$tem i$ $/21" i" (i#ure 2.10b. T/e multile!ed $i#"al sit)i$ alied $imulta"e2u$l% t2 t12 $earate c2/ere"t detect2r$ t/at are $ulied 1it/ t12l2cal carrier$ 24 t/e $ame 4re7ue"c% 0ut di44eri"# i" /a$e 0% - 9-de#ree$& T/e 2utut 24t/e t2 detect2r i$ Acm,(t), 1/erea$ t/e 2utut 24 t/e 02tt2m detect2r i$ Acm2(t). (2r t/e$%$temt2 2erate $ati$4act2ril%5 it i$ im2rta"t t2 mai"tai" t/e c2rrect /a$e a"d 4re7ue"c%relati2"$/i$ 0et1ee" t/e l2cal 2$cillat2r$ u$ed i" t/e tra"$mittera"d recei'er art$ 24 t/e$%$tem&T2 mai"tai" t/i$ $%"c/r2"i:ati2"5 1e ma% $e"d a pilot si$nal 2ut$ide t/e a$$0a"d24 t/e m2dulated $i#"al& I" t/i$ met/2d5 t/e il2t $i#"al t%icall% c2"$i$t$ 24 a l21-21er$i"u$2idal t2"e 1/2$e 4re7ue"c% a"d /a$e are related t2 t/e carrier 1a'e cit)% at t/erecei'er5 t/e il2t $i#"al i$ e!tracted 0% mea"$ 24 a $uita0l% tu"ed circuit a"d t/e" tra"$8lated t2 t/e c2rrect 4re7ue"c% 42r u$e i" t/e c2/ere"t detect2r& SINGL!SI"#$N"M%"&l$'I%NIn$i"#le-$ide0a"d m2dulati2"52"l% t/e uer 2r l21er $ide0a"d i$ tra"$mitted&We ma%#e"erate $uc/ a m2dulated 1a'e 0% u$i"# t/e fre&uenc'-discrimination method t/at c2"8$i$t$ 24 t12 $ta#e$;2.3 LiHear Modulatron Schemes 99Ii> The first stage is a product modulator,which generates a DSB-SCmodulated wave.'" The second stage is a band-pass filter, which is designed to pass one of the sidebandsof this modulated wave and suppress the other.From a practical viewpoint the most severe reuirement of SSBgeneration using the fre!uenc" discrimination method arises from the unwanted sideband. The nearest freuenc"component of the unwanted sideband is separated from the desired sideband b" twice thelowest freuenc" componentof the message #modulating$ signal. The implication here isthat for the generation of an SSBmodulated signal to be possible, the message spectrummust have an energ" gap centered at the origin, as illustrated in Figure l.lla. This reuire!ment is naturall" satisfied b" voice signals, whose energ" gap is about %&& '( wide #i.e.,it e)tends from - *&& to +*&& Hz). Thus, assuming that the upper sideband is retained,the spectrum of the SSBmodulated signal is as shown in Figure l.llb.+n designing the band-pass filter used in the freuenc"-discriminator for generatinga SSB-modulated wave, we must meet the three basic reuirements,Il> The desired sideband lies inside the passband of the filter.'" The unwanted sideband lies inside the stopband of the filter.IS >The filter's transition band, which separates the passband from the stopband, is twicethe lowest freuenc" component of the message signal.This -ind of freuenc" discrimination usuall" reuires the use of highl" selective filters,which can onl" be reali(ed in practice b" means of cr"stal resonators.To demodulate a SSBmodulated signal s(t),we ma" use a coherent detector, whichmultiplies s(t)b" a locall" generated carrier and then low-pass filters the product.Thismethod of demodulationassumes perfect s"nchronism between the oscillator in the co!herent detector and the oscillator used to suppl" the carrier wave in the transmitter. Thisreuirement is usuall" met in one of two wa"s,I> . low-power pilot carrieris transmitted in addition to the selected sideband.Il>. highl" stable oscillator, tuned to the same freuenc" as the carrier freuenc", isused in the receiver.Inthe latter method, it is inevitable that there would be some phase error cp in the localoscillator output with respect to the carrier wave used to generate the incoming SSBmod!ulated wave. The effect of this phase error is to introduce a phase distortion in the de!modulated signal, where each freuenc" component of the original message signal under!goes a constant phase shift cpo This phase distortion is tolerable in voice communications,lM(f)1-_~.'b~----~~O~L-----~~f" -fa fa fb/ / / .., +0 '-- Energy gap(a)IS (!)IoF+12345.66 (a) Spectrum of a message signal m(t) with an energ" gap of width 2f. centeredon the origin. (b) Spectrum of corresponding SSB signal containing the upper sideband.(b)100 CHAPTER2 El CONTINUOUS-WAVEMODUlATIONbecause the human ea !s elat!"el# !nsens!t!"e t$ %hase &!st$t!$n' In%at!cula( the %es)ence $* %hase &!st$t!$n +!"es !se t$ a Donald Duck voice effect. In the tansm!ss!$n $*mus!c an& "!&e$ s!+nals( $n the $the han&( the %esence $* th!s *$m $* ,a"e*$m &!st$)t!$n !s uttel# unacce%table'!II VESTI-IA. SIDE/AND MODUlATIONIn vestigial sideband (VSB) modulation, $ne $* the s!&eban&s !s %at!all# su%%esse& an&a "est!+e $* the $the s!&eban& !s tansm!tte& t$ c$m%ensate *$ that su%%ess!$n' A%$%ulameth$& *$ +eneat!n+ a VS/-m$&ulate& ,a"e !s t$ 0use the frequency discriminationmethod. 1!st( ,e +eneate a DS/-SCm$&ulate& ,a"e an& then %ass !t th$u+h a ban&)%ass *!lte( as sh$,n !n 1!+ue 2'122 !t !s the s%ec!al &es!+n $* the ban&-%ass *!lte that&!st!n+u!shes VS/m$&ulat!$n *$m SS/m$&ulat!$n' Assum!n+ that a "est!+e $* the l$,es!&eban& !s tansm!tte&( the *e3uenc# es%$nse H(f) $* the ban&-%ass *!lte ta4es the *$msh$,n !n 1!+ue 2'15' T$ s!m%l!*# mattes( $nl# the es%$nse *$ %$s!t!"e *e3uenc!es !ssh$,n hee' Th!s *e3uenc# es%$nse !s n$mal!6e&( s$ that at the ca!e *e3uenc# h,eha"e IH(fc) I =1/2. The !m%$tant *eatue t$ n$te *$m 1!+ue 2'15 !sthat the cut$**%$t!$n $* the *e3uenc# es%$nse a$un& the ca!e *e3uenc# Ie e7h!b!ts odd symmetry.That !s( !ns!&e the tans!t!$n !nte"al Ie - Iv es I II sI,+ fv the *$ll$,!n+ t,$ c$n&!t!$nsae sat!s*!e&81. The sum $* the "alues 9* the ma+n!tu&e es%$nse IH(f) I at an# t,$ *e3uenc!ese3uall# &!s%lace& ab$"e an& bel$, fc !s un!t#'2' The %hase es%$nse a+:H:*;; !s l!nea' That !s( H(f) sat!s*!es the c$n&!t!$nH(f - fe) + H(f+ lel =1 *$ -Ws Is W :2'15;N$te als$ that $uts!&e the *e3uenc# ban& $* !nteest :!'e'( I fl>! c + W), the *e3uenc#es%$nse H(f) ma# ha"e an ab!ta# s%ec!*!cat!$n' Acc$&!n+l#( the tansm!ss!$n ban&-,!&th $* VS/m$&ulat!$n !s 'BT =W< fv (2.14),hee W!s the messa+e ban&,!&th( an& fv !s the ,!&th $* the "est!+!al s!&eban&'Acc$&!n+ t$ Table 2'1( the VS/m$&ulate& ,a"e !s &esc!be& !n the t!me &$ma!n as1 1's(t) =82 ,m(t) cos(!"rI#):!: 82 d$$%(t) s!n:2=I>; :2'1?;,hee the %lus s!+n c$es%$n&s t$ the tansm!ss!$n $* a "est!+e $* the u%%e s!&eban&(an& the m!nus s!+n c$es%$n&s t$ the tansm!ss!$n $* a "est!+e $* the l$,e s!&eban&' Thes!+nal m%(t) !n the 3ua&atue c$m%$nent $* s(t) !s $bta!ne& b# %ass!n+ the messa+e s!+nalMessagesignal met)VSBmodla!ed"a#ec c$s (!%nfct)$a%%ie% "a#e1I-URE2'12 1!lte!n+ scheme *$ the +eneat!$n $* VS/m$&ulate& ,a"e'2.3 Linear ModulationSchemes 101lH(jJIFIGURE2.13 Magnitude response of VSBfiter! on" the positi#e$fre%uenc" portion is sho&n.m(t) through a fiter &hose fre%uenc" response HQ(f) satisfies the foo&ing re%uirement'see(ro)em 2.20*+HQ(f) =;[H(f - fc) - H(f + fc)] for $ , :s f :s , '2.1-*Figure 2.1. dispa"s a pot of the fre%uenc" response HQ(f), scaed )" 1/!.0he roe of the%uadrature component determined )" HQ(/) is to interfere &ith the in$phase componentin E%uation (2.15) so as to partia" reduce po&er in one of the side)ands of the moduated&a#e 5(t)and retain simp" a #estige of the other side)and1 as desired.It is of interest to note that SSBmoduationma" )e #ie&ed as a specia case ofVSBmoduation.Specifica"1 &hen the #estigia side)and is reduced to 2ero 'i.e.1 &e setI; =0*1 the moduated &a#e 5(t)of E%uation (2.15) ta3es the imiting form of a singe4side)and moduated &a#e.iii 0E5EVISI67 SIG785S8discussion of #estigia side)and moduation &oud )e incompete &ithout a mention ofits roe in commercia tee#ision '0V* )roadcasting. 0he e9act detais of the moduationformat used to transmit the #ideo signa characteri2ing a 0V s"stemare infuenced )" t&ofactors+1. 0he #ideo signa e9hi)its a arge )and&idth and significant o&$fre%uenc" content1&hich suggest the use of #estigia side)and moduation.2. 0he circuitr" used for demoduation in the recei#er shoud )e simpe and thereforeine9pensi#e! this suggests the use of en#eope detection1 &hich re%uires the additionof a carrier to the VSB$moduated &a#e.FIGURE2.. Fre%uenc" response of a fiter for producing the %uadrature component of theVSBmoduated &a#e.102 CHAPI'ER2 .. CONTINUOUS-WAVEMODUlATIONWith regard to point 1, however, it should be stressed that although there is indeeda basic desire to conserve bandwidth, in commercial TV broadcastingthe transmittedsignal is not quite VSBmodulated.The reason is that at the transmitter the power levelsare high, with the result that it would be expensive to rigidly control the filtering of sidebands. !nstead, a VSB filter is inserted in each receiver, where the power levels are low.The overall performance is the same as conventional vestigial"sideband modulation, exceptfor some wasted power and bandwidth.These remar#s are illustrated in $igure %.1&. !nparticular, $igure 2.15a shows the ideali'ed spectrum of a transmitted TV signal. Theupper sideband, %& percent of the lower sideband, and the picture carrier are transmitted.The frequency response of the VSBfilter used to do the required spectrum shaping in thereceiver is shown in $igure 2.15b.The channel bandwidth used for TV broadcasting in (orth )merica is * +,', asindicated in $igure 2.1Sb. This channel bandwidth not only accommodates the bandwidthrequirementof the VSBmodulated video signal but also provides for the accompanyingsound signal that modulates a carrier of its own. The values presented on the frequencyaxis in $igures 2.15a and 2.15b pertain to a specific TV channel. )ccording to this figure,the picture carrier frequency is at &&.%& +,', and the sound carrier frequency is at &-..&+,'. (ote, however, that the informationcontent of the TV signal lies in a basebandspectrum extending from 1.%& +,' below the picture carrier to /.& +,' above it.With regard to point %, the use of envelope detection 0applied to a VSBmodulated""! 111r"""" /.& +,' """2oi !3" 4.%& +,'414..&+,'1 1.4~~"e.~1 4.&'0alPicturecarrierSouncarriero 56"7,8r""9&/8""""""8"56:8&*;;"""&:8S;;""" . ncreasing the width of the vestigial sideband to reduce m#(t).Both methods are in fact used in practice. . n commercial TV broadcasting'the width ofthe vestigial sideband #which is about 1.(& 234' or one!sixth of a full sideband) is determined to 5eep the distortion due to m#(t)within tolerable limits when the percentagemodulation is nearly %11.I2.4 Frequency TranslationThe basic operation involved in single!sideband modulation is in facta form of freq(encytranslation, which is why single!sideband modulation is sometimes referred to as freq(encyc)an*in*, mi+in*,or )eterodynin*. This operation is dearly illustrated in the spectrum ofthe signal shown in 6igure 2.! ! , compared to that of the original message signal in 6igure2.! ! a. Specifically' we see that a message spectrum occupying the band from fa to f , forpositive frequencies in 6igure 2.! ! ais shifted upward by an amount equal to the carrierfrequency fe in 6igure 2.! ! ,, and the message spectrum for negative frequencies is translated downward in a symmetric fashion.The idea of frequency translationdescribed herein may be generali4ed as follows.Suppose that we have a modulatedwave SI(t) whose spectrum is centered on a carrierfrequency fl,and the requirement is to translate it upward in frequency such that its carrierfrequency is changed from fl to a new value kThis requirement may be accomplishedusing the mi+er shown in 6igure $.%7. Specifically' the mi+er is a device that consists of aproduct modulator followed by a band!pass filter.104 CHAPTER2 I i I 1 CONTI NUOUS-WAVEMODUlATI ONModulated wave '1(t)with carrier frequency 11,'(t)Band-a!!"i lte#Modulated wave ,,(I)with carrier frequency fzAI COS (2~f,t)FIGURE2.16 Block diagram of mixer.To explain the action of the mixer, conider the it!ation depicted in Fig!re 2.1",#here, for the p!rpoe of ill!tration,it i a!med that the mixer inp!t s,(t) i an $%ignal #ith carrier fre&!enc' (1and )and#idth 2 W. *art (a) of Fig!re 2.1" dipla' the$% pectr!m +tilla!ming that (1$ ,. *art(b) of the fig!re dipla' the pectr!mS'(!) of the re!lting ignal s'(t) at the prod!ct mod!lator o!tp!t.The ignal s'{t)ma' )e -ie#ed a the !m of t#o mod!lated component. one com/ponent repreented )' the haded pectr!m in Fig!re 2.17b, and the other componentrepreented )' the !nhaded pectr!m in thi fig!re. 0epending on #hether the incomingcarrier fre&!enc' (,i tranlated !p#ard or do#n#ard, #e ma' identif' t#o differentir!ation, a decri)ed here.Up conversion. In thi cae the tranlatedcarrier fre&!enc' (2 i greater than theincoming carrier fre&!enc' (1and the re&!ired local ocillator fre&!enc' fl i there/fore defined )'or/2=/,+flfl=/2-/'%''2 (3 .%f&'%()&'%(456(b)FIGURE2.1" (a) +pectr!m of mod!lated ignal s,(t)at the mixer inp!t. (h) +pectr!m of thecorreponding ignal '4e6 at the o!tp!t of the prod!ct mod!lator in the mixer.2.5 Freqt.ency-DiruionMultiplexing 105The unshaded part of the spectrum in Figure 2.17b defines the wanted modulatedsignal S2(t), and the shaded part of this spectrum defines the image signal associatedwith S2(t). For obvious reasons, the mixer in this case is referred to as a frequencyup converter.Down conversion. In this second case the translated carrier frequency [2 is smallerthan the incoming carrier frequency fh and the required oscillator frequency Ii istherefore defined byorThe picture we have this time is the reverse of that pertaining to up conversion. Inparticular,the shaded part of the spectrum in Figure 2.17b defines the wanted modulated signal S2(t), and the unshaded part of this spectrum defines the associatedimage signal. The mixer is now referred to as a frequency!ownconverter. Note thatin this case the translated carrier frequency f2 has to be larger than "(i.e., one halfof the bandwidth of the modulated signal to avoid sideband overlap.The purpose of the band!pass filter in the mixer of Figure ".#$ is to pass the wantedmodulated signal S2(t)and eliminate the associated image signal. This ob%ective is achievedby aligning the midband frequency of the filter with the translated carrier frequency f2 andassigning it a bandwidth equal to that of the incoming modulated signal &I (t).It is importantto note that mixing is a linear operation.'ccordingly, the relation ofthe sidebands of the incoming modulated wave to the carrier is completely preserved atthe mixer output.I2.5Frequency-Division Multiplexing'nother importantsignal processing operation is multiple#ing,whereby a number of independent signals can be combined into a composite signal suitable for transmission overa common channel. (oice frequencies transmitted over telephone systems, for example,range from )** to )#** +,. To transmit a number of these signals over the same channel,the signals must be -ept apart so that they do not interfere with each other, and thus theycan be separated at the receiving end. This is accomplished by separating the signals eitherin frequency or in time. The technique of separating the signals in frequency is referred toas frequency!ivision multiple#ing (F./, whereas the technique of separating the signalsin time is called time!ivision multiple#ing (T./. Inthis section, we discuss F./ systems, and T./ systems are discussed in 0hapter ).' bloc- diagram of an F./ system is shown in Figure ".#1. The incoming messagesignals are assumed to be of the low!pass type, but their spectra do not necessarily havenon,ero values all the way down to ,ero frequency. Following each signal input, we haveshown a low!pass filter, which is designed to remove high!frequency components that donot contribute significantly to signal representation but are capable of disturbing othermessage signals that share the common channel. These low!pass filters may be omittedonly if the input signals are sufficiently band limited initially. The filtered signals are applied106 CHAPTER 2 iii CONTINLJOLJS-WAVEMODLJLATIONMessage Low-passinputs filters ModulatorsBand-passfiltersBand-passfilters DemodulatorsLow-pass Messagefilters outputsN NTransmitterReceiverFIGLJRE 2.18 Block di!"# o$ FDM %&%'(#.'o #od)l'o"% '*' %*i$' '*( $"(+)(,c& ",!(% o$ '*( %i!,l% %o % 'o occ)-& #)')ll&(.cl)%i/( $"(+)(,c& i,'("/l%. T*( ,(c(%%"& c""i(" $"(+)(,ci(% ,((d(d 'o -("$o"# '*(%($"(+)(,c& '",%l'io,% "( o0'i,(d $"o#c""i(" %)--l&. Fo" '*( #od)l'io,1 2( #&)%( ,&o,( o$ '*( #('*od% d(%c"i0(d i, -"(/io)% %(c'io,% o$ '*i% c*-'(". Ho2(/("1 '*(#o%' 2id(l& )%(d #('*od o$ #od)l'io, i, $"(+)(,c&-di/i%io, #)l'i-l(.i,! i% %i,!l( %id(30,d #od)l'io,1 2*ic*1 i, '*( c%( o$ /oic( %i!,l%1 "(+)i"(%0,d2id'* '*' i% -3-"o.i#'(l& (+)l 'o '*' o$ '*( o"i!i,l /oic( %i!,l.l, -"c'ic(1 (c* /oic( i,-)' i% )%)ll&%%i!,(d0,d2id'* o$ 4 kH5. T*( 0,d--%% $il'("% $ollo2i,! '*( #od)l'o"% "( )%(d'o "(%'"ic' '*( 0,d o$ (c* #od)l'(d 2/( 'o i'% -"(%c"i0(d ",!(. T*( "(%)l'i,! 0,d3-%% $il'(" o)'-)'% "( ,(.' co#0i,(d i, -"ll(l 'o $o"# '*( i,-)' 'o '*( co##o, c*,,(l.A' '*( "(c(i/i,! '("#i,l10,k o$ 0,d--%% $il'("%1 2i'* '*(i" i,-)'% co,,(c'(d i, -"3ll(l1 i% )%(d 'o %(-"'( '*( #(%%!( %i!,l% o,$"(+)(,c&-occ)-,c& 0%i%. Fi,ll&1 '*(o"i!i,l #(%%!( %i!,l% "( "(co/("(d 0& i,di/id)l d(#od)l'o"%. No'( '*' '*( FDM%&%'(# %*o2, i, Fi!)"( 2.18 o-("'(% i, o,l& o,( di"(c'io,. To -"o/id( $o" '2o-2&'",%#i%%io,1 % i, '(l(-*o,&1 $o" (.#-l(1 2( */( 'o co#-l('(l& d)-lic'( '*( #)l'i3-l(.i,! $cili'i(%1 2i'* '*( co#-o,(,'% co,,(c'(d i, "(/("%( o"d(" ,d 2i'* '*( %i!,l2/(% -"oc((di,! $"o# "i!*' 'o l($'..,. ExAMPLE 2.1T*( -"c'icl i#-l(#(,''io, o$ , FDM %&%'(# )%)ll& i,/ol/(% #,& %'(-% o$ #od)l'io,,d d(#od)l'io,1 % ill)%'"'(d i, Fi!)"( 2.16. T*( $i"%' #)l'i-l(.i,! %'(- co#0i,(% 12 /oic(i,-)'% i,'obasic group, 2*ic* i% $o"#(d 0& */i,! '*( nth i,-)' #od)l'(c""i(" at$"(+)(,c& t;=78 + 4n kH51 2*("( n =11 21 ... 1 12. T*( lo2(" %id(0,d% "( '*(, %(l(c'(d0& 0,d--%% $il'("i,! ,d co#0i,(d 'o $o"#!"o)- o$ 12 lo2(" %id(0,d% 9o,( $o" (c*/oic( i,-)':. T*)% '*( 0%ic !"o)- occ)-i(% '*( $"(+)(,c& 0,d 78 'o 188 kH5. T*( ,(.' %'(-i, '*( FDM *i(""c*& i,/ol/(% '*( co#0i,'io, o$ $i/( 0%ic !"o)-% i,'osupergroup. T*i%i% cco#-li%*(d 0& )%i,! '*( nth !"o)- 'o #od)l'(c""i(" o$ $"(+)(,c& t: =;n 109Modulatingwave(a) (b)FIGURE2.20 Illustrating the relationship between frequency modulation and phase modulation.(a) Scheme for generating an F wa!e by using a phase modulator. (b) Scheme for generating a" wa!e by using a frequency modulator.on the assumption that m(t) is a !oltage wa!eform. Integrating Equation #2.2$% withrespect to time and multiplying the result by 21T, we get9i(t)=21TfJ + 21Tkf & ' m(T) dr (2.25)where( for con!enience( we ha!e assumed that the angle of the unmodulated carrierwa!e is )ero at t =*.+he frequency,modulated signal is therefore described in thetime domain by#2.2-%. consequence of allowing the angle 9i(t) to become dependent on the message signalm(t) as in Equation #2.22% or on its integral as in Equation #2.2/% is that the zero crossingsof a " signal or F signal no longer ha!e a perfect regularity in their spacing0 )erocrossings refer to the instants of time at which a wa!eform changes from a negati!e to apositi!e !alue or !ice !ersa. +his is one important feature that distinguishes both " andF signals from an . signal. .nother important difference is that the en!elope of a "or F signal is constant #equal to the carrier amplitude%( whereas the en!elope of an .signal is dependent on the message signal.1omparing Equation #2.22% with #2.2-% re!eals that an F signal may be regardedas a " signal in which the modulating wa!e is J~m(T) dr in place of m(t). +his meansthat an F signal can be generated by first integrating m(t) and then using the result asthe input to a phase modulator( as in Figure 2.20a. 1on!ersely( a " signal can be gen3erated by first differentiating m( t) and then using the result as the input to a frequencymodulator(as in Figure 2.20. 4e may thus deduce all the properties of " signals fromthose of F signals and !ice !ersa. 5enceforth( we concentrate our attentionon Fsignals.L 2.7Frequency Modulation+he F signal s!t) defined by Equation #2.2-% is a nonlinear function of the modulatingsignal m(t), which ma6es frequency modulation a non"inear modu"ation #rocess. 1onse3quently( unli6e amplitude modulation( the spectrum of an F signal is not related in asimple manner to that of the modulating signal0 rather( its analysis is much more difficultthan that of an . signal.110 CHAPTIlR2 aCONTINUOUS-WAVEMODULATIONHow then can we tackle the spectral analysis o an !" si#nal$ We propose to pro%i&ean e'pirical answer to this i'portant()estion *y procee&in# in the ollowin# 'anner+, We consi&er the si'plest case possi*le- na'ely- that o a sin#le-tone 'o&)lation thatpro&)ces a narrow*an& !" si#nal., We ne/t consi&er the 'ore #eneral case also in%ol%in# a sin#le-tone 'o&)lation-*)tthis ti'e the !" si#nal is wi&e*an&.We co)l&- o co)rse- #o on an& consi&er the 'ore ela*orate case o a ')lti tone !" si#nal.Howe%er- we propose not to &o so- *eca)se o)r i''e&iate o*0ecti%e is to esta*lish ane'pirical relationship *etween the trans'ission *an&wi&th o an !" si#nal an& the 'es1sa#e *an&wi&th. As we shall s)*se()ently see- the two-sta#e spectral analysis &escri*e&here pro%i&es )s with eno)#h insi#ht to propose a sol)tion to the pro*le'.Consi&er then a sin)soi&al 'o&)latin# si#nal &eine& *ym(t) =Am COS(271'fmt) 22.234The instantaneo)s re()ency o the res)ltin# !" si#nal e()alsf;(t)=fe+ kfAm cos(271'fmt)=t; + !:1f cos(271'fmt)22.254where22.264The ()antity af is calle& the frequency deviation, representin# the 'a/i')' &epart)reo the instantaneo)s re()ency o the !" si#nal ro' the carrier re()ency feA )n&a1'ental characteristic o an !" si#nal is that the re()ency &e%iation af is proportionalto the a'plit)&e o the 'o&)latin# si#nal an& is in&epen&ent o the 'o&)lation re()ency.Usin# E()ation 22.254- the an#le !i(t)o the !" si#nal is o*taine& asOi(t) =237 8 " : /;(1 ') dr=27#fct+ ~: sin(271'$;nt)22.9:4The ratio o the re()ency &e%iation af to the 'o&)lation re()ency i; is co''only calle&the modu%ation inde& o the !" si#nal. We &enote it *y ; 9- an& so write22.97 4an&22.924!ro' E()ation 22.924 we see that- in a physical sense- the para'eter 29represents the phase&e%iation o the !" si#nal- that is- the 'a/i')' &epart)re o the an#le Oi(t) ro' thean#le 271'fet o the )n'o&)late& carrier< hence- ; 9 is 'eas)re& in ra&ians.The !" si#naI itsel is #i%en *y22.9942.7 Frequency Modulatw.. 111Depending on the value of the modulation indexf 3 ,we may distinguish two cases offrequency modulation:l> Narrowband FM, for which f 3 is small compared to one radian."" Wideband FM, for which f 3 is large compared to one radian.These two cases are considered next, in that order.!illN!!"#$ND F!%&'%N()M"D'*T+"N(onsider %quation ,-.../, which defines an FM signal resulting from the use of a sinusoidalmodulating signal. %xpanding this relation, we getssuming that the modulation index f 3 is small compared to one radian,we may use thefollowing approximations:cos0f.sin(21T/mt)]=1and1ence, %quation ,-..2/ simplifies to,-..3/%quation ,-..3/ defines the approximate form of a narrow4and FM signal produced 4y asinusoidal modulating signal !n "#$(21T/mt). From this representation we deduce the mod5ulator shown in 4loc6 diagram form in Figure -.-1. This modulator involves splitting thecarrier wave c "#$(21T/ct) into two paths. "ne path is direct7 the other path contains a89:degree phase8shifting networ6 and a product modulator, the com4ination of whichgenerates a D;:3:13 T-e +al)e of n(a2 +a#!es *!$- $-e(o%)la$!on !n%e2 5 6 an% can &e %e$e#(!ne% #ea%!l' f#o( $a&)la$e% +al)es of $-e ;esself)nc$!on I n (/').Ta&le 232 s-o*s $-e $o$al n)(&e# of s!"n!f!can$ s!%e f#e,)enc!es . *-!c- !s %#a*n as a &es$ f!$ $-#o)"- $-e se$ of /o!n$s o&$a!ne% &' )s!n" Ta&le 2323In !")#e 232> *e no$e $-a$ as $-e (o%)la$!on !n%e2 5 6 !s !nc#ease%. $-e &an%*!%$- occ)/!e%A;?E 232 )umber of si&n ifI Can t sidee#uen $ies of a ideban d FMsi&n al for!ar %in &adulation in de*+odulation I n de*f6)umber of ,i&n ifi$an t ,ide -r e#uen $ies2n ma*:31:36:3@13:23:5. .1:3:2:3:6 :3:2AA6816285.B:2.7 FrequencyModulation 119402 -------------------------------------f 3FIGURE2.26 Universal curve for evaluating the 1 percentbandwidth of an FM wave.by the significant side frequencies drops toward thatover which the carrier frequencyactually deviates. This means that small values of the modulation index { 3 are relativelymore extravagantin transmission bandwidth than are the larger values of (3 .onsider next the more general case of an arbitrary modulating signal mit) with itshighest frequency component denoted by !. The bandwidth required to transmit an"# signal generated by this modulating signal is estimated by using a worst$case tone%modulation analysis. &pecifically' we first determine the so$called deviation ratio(' definedas the ratio of the frequency deviation )*' which corresponds to the maximum possibleamplitude of the modulation signal mit),to the highest modulation frequency !+ theseconditions represent the extreme cases possible. The deviation ratio( plays the same rolefor nonsinusoidal modulation that the modulation index { 3 plays for the caseof sinusoidalmodulation. Then' replacing { 3 by ( and replacing 1m with !' we may use arson,s rulegiven by -quation (..//0 or the universal curve of "igure ...1 to obtain a value for thetransmission bandwidth of the "# signal. "rom a practical viewpoint' arson,s rule some%what underestimates the bandwidth requirement of an "# system' whereas using the universal curve of "igure ...1 yields a somewhat conservative result. Thus' the choice of atransmission bandwidth that lies between the bounds provided by these two rules of thumbis acceptable for most practical purposes.2+3 EXAMPLE ..3*n 4orth )merica'the maximumvalue of frequency deviation 111 is fixed at 5/ 678 forcommercial"# broadcasting by radio. If we ta6e the modulation frequency ! =9/ 678'which is typically the :maximum: audio frequency of interest in "# transmission' we findthat the correspondingvalue of the deviation ratio is( =5/ =/9/Using arson,s rule of -quation (..//0' replacing{ 3 by (' and replacing t;by !' the ap%proximate value of the transmission bandwidth of the "# signal is obtained asBT ; .(5/ + 9/0 ; 9 The modulated signal sit) transmitted by each system has the same aerage power.I> The channel noise (it) has the same aerage power measured in the message bandwidth W.SN(f)rN"2 r B T- > - jr.3.r..... .....4--~~~fL,~------OL------L_f,Lc~---fF%&5*' 2.34 %deali6edcharacteristic of band.pass filtered noise.132 CHAPTER2 III CONTINUOUS-WAVE MODUlATIONOutputNoisew(t)FIGURE 2.35 The baseband transmission model, assuming a message signal of bandwidth \V,used for calculating the channel signal-to-noise ratio.Accordingly, as a frame of reference we define the channel signal-to-noise ratio, (SN!c,as the ratio of the average power of the modulated signal to the average power of channelnoise in the message bandwidth, both measured at the receiver input. This definition isillustrated in "igure 2.35."or the #ur#ose of com#aring different continuous-wa$e (%&! modulation systems,we normalize the recei$er #erformance by di$iding the out#ut signal-to-noise ratio by thechannel signal-to-noise ratio. &e thus define a figure of merit for the recei$er as follows'. . (SN!o"igure of merit =(SN!c%learly, the higher the $alue of the figure of merit, the better will the noise #erformanceof the recei$er be. The figure.of merit may e(ual one, be less than one, or be greater thanone, de#ending on the ty#e of modulationused, which will become a##arent from thediscussion that follows.(2.81)2.11 Noise in Linear ReceiversUsing Coherent Detection"rom Sections 2.2 and 2.3 we recall that the demodulationof an am#litude-modulatedwa$e de#ends on whether the carrier is su##ressed or not. &hen the carrier is su##ressedwe usually re(uire the use of coherent detection, in which case the recei$er is linear. )nthe other band, when the am#litude modulation includes transmission of the carrier, de*modulation is accom#lished sim#ly by using an en$elo#e detector, in which case the re*cei$er is nonlinear. Inthis section we study the effect of noise on the #erformance of alinear recei$er. The more difficult case of a nonlinear recei$er is deferred to Section 2.12.%onsider the case of +S,-S%modulation "igure 2.3- shows the model of a +S,-S%recei$er using a coherent detector. The use of coherent detection re(uires multi#licationof the filtered signal x(t)by a locally generated sinusoidal wa$e cos(27rfct)and then low*#ass filtering the #roduct. To sim#lify the analysis, we assume that the am#litude of thelocally generated sinusoidal wa$e is unity. "or this demodulation scheme to o#erate sat*isfactorily, howe$er, it is necessary that the local oscillator be synchroni.ed both in #haseand in fre(uency with the oscillator generating the carrier wa$e in the transmitter.&eassume that this synchroni.ation has been achie$ed.The +S,-S%com#onent of the filtered signal x(t) is e/#ressed as(2.82)where Ac cos(27rfct)is the sinusoidal carrier wa$e and m(t) is the message signal. Inehee/#ression for s(t) in 0(uation (2.12! we ha$e included a system-dependentscaling factor%, the #ur#ose of which is to ensure that the signal com#onent s(t) is measured in the sameunits as the additi$e noise com#onent n(t). &e assume that m(t) is the sam#le function of2.11 Noisein Linear Receivers UsingCoherent Detection 133DSB-SCsignal sfr)y(t)FIGliRE2.36 Model of DSB-SC receiver using coherent detection. sttionr! "rocess of #ero $en% &hose "o&er s"ectrl densit! SM(f) is li$ited to $'i$u$ fre(uenc! W; tht is% W is the messagebandwidth. )he verge "o&er * of the$essge signl is the totl re under the curve of "o&er s"ectrl densit!% s sho&n +!P =i: w SM(f)d],2.-3.)he crrier &ve is sttisticll! inde"endent of the $essge signl. )o e$"hsi#e thisinde"endence% the crrier should includerndo$ "hse tht is unifor$l! distri+uted over2/)rdins. In the defining e(ution for sIt) this rndo$ "hse ngle hs +een o$itted forconvenience of "resenttion. 0sing the result of E'$"le 1./of Ch"ter 1on$odultedrndo$ "rocess% &e $! e'"ress the verge "o&er of the DSB-SC $odulted signlco$"onent sIt) s C2A~PI2. 1ithnoise s"ectrl densit! of 2o 12, the verge noise "o&erin the $essge +nd&idthis e(ul to No. )he chnnel signl-to-noise rtio of theDSB3SC$odultion s!ste$ is thereforeC2!2",S2R.c%DsB=24o&here the constnt C2 in the nu$ertor ensures tht this rtio is di$ensionless.2e't% &e &ish to deter$ine the out"ut signl-to-noise rtio of the s!ste$. 0sing thenrro&+nd re"resenttion of the filtered noise n(t), the totl signl t the coherent detec5tor in"ut $! +e e'"ressed s,2.-6.x(t)=sIt) + nit)=CAe COs(27Tft)m(t) + nI(t) CO!(27T",t) ndt) sin,2/)fct.,2.-7.&here n,(t) nd n#(t) re the in-"hse nd (udrture co$"onents of nIt) &ith res"ect tothe crrier. )he out"ut of the "roduct-$odultor co$"onent of the coherent detector istherefore$(t) =x(t)CO!(27Tft)=%CA.m(t) + %nI(t)& 'CA(111(t) + n,(t)) CO!(*7T"t) + ~n#(t) sin,6/)ht.)he lo&-"ss filter in the coherent detector in Figure 2.36 re$oves the high-fre(uenc!co$"onents of $(t), !ielding the receiver out"ut,(t) =%CA,m(t) + %nI(t) (2.-.)E(ution ,2.-6. indictes the follo&ing81. )he $essge signl mit) nd in-"hse noise co$"onent n/(t) of the filtered noise n(t)""er dditivel! t the receiver out"ut.134 CHAPTER2 " CONTINUOUS-WAVEMODUlATION2. The quadrature co!o"e"tnQ(t) o# the "o$%e nit)$% co!letel& re'ected (& the co)here"t detector.The%e t*o re%ult% are $"de!e"de"t o# the $"!ut %$+"al-to-"o$%e rat$o. Thu%, cohere"t detec)t$o" d$%t$"+u$%he% $t%el# #ro other deodulat$o" tech"$que% $" a" $!orta"t !ro!ert&- Theout!ut e%%a+e co!o"e"t$% u"ut$lated a"d the "o$%e co!o"e"t al*a&% a!!ear% ad)d$t$.el& *$th the e%%a+e, $rre%!ect$.e o# the $"!ut %$+"al-to-"o$%e rat$o.The e%%a+e %$+"al co!o"e"tat the rece$.er out!ut $% CAcm(t)/2. There#ore, thea.era+e !o*er o# th$% co!o"e"ta& (e e/!re%%ed a% C2A~PI4,*here P $% the a.era+e!o*er o# the or$+$"al e%%a+e %$+"al mit) a"d C $% the %&%te-de!e"de"t%cal$"+ #actorre#erred to earl$er.I" the ca%e o# DS0-SCodulat$o", the (a"d-!a%% #$lter $" 1$+ure 2.23 ha% a (a"d)*$dth By equal to 2W $" order to accoodate the u!!er a"d lo*er %$de(a"d% o# theodulated %$+"al sit). It#ollo*% there#ore that the a.era+e !o*er o# the #$ltered "o$%e nit)$% 2WNo. 1ro the d$%cu%%$o" o# "arro*(a"d "o$%e !re%e"ted $" Sect$o" 4.44, *e 5"o*that the a.era+e !o*er o# the 6lo*-!a%%7 $"-!ha%e "o$%e co!o"e"tn,(t) $% the %ae a%that o# the 6(a"d-!a%%7 #$ltered "o$%e nit). S$"ce #ro Equat$o" 62.837 the "o$%e co!o"e"tat the rece$.er out!ut $% nj(t)/2, $t #ollo*% that the a.era+e !o*er o# the "o$%e at the rece$.erout!ut $%69722WNo =tWNoThe out!ut %$+"al-to-"o$%e #or a DS0-SCrece$.er u%$"+ cohere"t detect$o" $% there#oreC2A~PI46SNR7o, DSMC = WNo/262.8:7C2A~P2WNoU%$"+ Equat$o"% 62.8;7 a"d 62.8:7, *e o(ta$" the #$+ure o# er$t6SNRlo$ - 46SNR7c US0-SCNote that the #actor C2 $% coo" to (oth the out!ut a"d cha""el %$+"al-to-"o$%e rat$o%,a"d there#ore ca"cel% out $" e.aluat$"+ the #$+ure o# er$t.1ollo*$"+ throu+h the "o$%e a"al&%$% o# a cohere"t detector #or SS0, *e #$"d that,de%!$te the #u"dae"tal d$##ere"ce% (et*ee" $t a"d the cohere"t detector #or DS0-SCod)ulat$o", the #$+ure o# er$t $% e/actl& the %ae #or (oth o# the< %ee Pro(le 2.;=.The $!orta"t co"clu%$o"% to.(e dra*" #ro the d$%cu%%$o"% !re%e"ted $" th$% %ect$o"a"d Pro(le 2.;= are t*o-#old-62.8874. 1or the %ae a.era+e tra"%$ttedor odulated %$+"al !o*er a"d the %ae a.era+e"o$%e !o*er $" the e%%a+e (a"d*$dth,a cohere"t SS0rece$.er *$ll ha.e e/actl& the%ae out!ut %$+"al-co-"o$%e rat$o a% a cohere"t DS0-SCrece$.er.2. I" (oth ca%e%, the "o$%e !er#ora"ce o# the rece$.er $% e/actl& the %ae a% that o()ta$"ed (& %$!l& tra"%$tt$"+ the e%%a+e %$+"al $" the !re%e"ce o# the %ae cha""el"o$%e. The o"l& e##ect o# the odulat$o" !roce%% $% to tra"%late the e%%a+e %$+"al toa d$##ere"t #reque"c& (a"d to #ac$l$tate $t% tra"%$%%$o" o.er a (a"d-!a%% cha""el.S$!l& !ut, "e$ther DS0-SCodulat$o" "or SS0odulat$o" o##er% the ea"% #or a trade)o## (et*ee" $!ro.ed "o$%e !er#ora"ce a"d $"crea%ed cha""el (a"d*$dth.Th$% $% a %e-r$ou% !ro(le *he" h$+h qual$t& o# rece!t$o" $% a requ$ree"t. .2.12 Noise in AMReceivers UsingEnveu.pe DetectUm 135I~:~2NoiseinAMRecei~ers~g En~elope DetectionThe next noise analysis we perform is for an amplitude modulation (AM) system using anenvelope detetor in the reeiver! as shown in the model of "igure 2#3$# %n a full AM signal!&oth side&ands and the arrier wave are transmitted!as shown &y(2#'()where A,COS(27!,t"is the arrier wave! mit"is the message signal! and #$ is a onstantthat determines the perentage modulation#%n the expression for the amplitude)modulatedsignal omponentsit" given in *+uation (2#'()! we see no need for the use of a salingfator! &eause it is reasona&le to assume that the arrier amplitude Ac has the same unitsas the additive noise omponent#The average power of the arrier omponentin the AM signal sit"is A~%2.Theaverage power of the information)&earing omponent Ac#&m(t" COS(27!ct"is A~#$'(2,where , is the average power of the message signal mit". The average power of the fullAM signal sit" is therefore e+ual to A~(1 + #~'"%2. As for the -./).0system! the averagepower of noise in the message &andwidth is WNo) The hannel signal)to)noise ratio forAM is therefore(.12) =A~(1 + #$'"0!AM 2*NoTo evaluate the output signal)to)noise ratio! we first represent the filtered noise nit"in terms of its in)phase and +uadrature omponents# 3e may therefore define the filteredsignal +(t" applied to the envelope detetor in the reeiver model of "igure 2#3$ as follows:(2#(4)+(t"=sit" + nit"=,A, + A".m(t" + nl(t"- COS(27!,t". n/(t" sin(27!,t"%t is informative to represent the omponents that omprise the signal +(t" &y means ofphasors! as in "igure 2.012. "rom this phasor diagram! the reeiver output is readily o&5tained as(2#(1)3(t" =envelope of +(t"=(,A, + A,#2m(t" + nl(t*+ n~(t"4ll2The signal 3(t" defines the output of an ideal envelope detetor# The phase of +(t" is of nointerest to us! &eause an ideal envelope detetor is totally insensitive to variations in thephase of +(t".The expression defining 3(t" is somewhat omplex and needs to &e simplified in somemanner to permit the derivation of insightful results# .peifially! we would li6e to ap5proximate the output 3(t" as the sum of a message term plus a term due to noise# Ingeneral!this is +uite diffiult to ahieve# 7owever! when the average arrier power is large om)(2#(2)AM signals(f)Output~signalY(f)"%892* 2#3$ Model of AM reeiver#136 CHAPTER2.. CONTINUOUS-WAVE MODUlATIONResultant yet) --- InQ(t) :I(a)(b)FIGURE 2.38 (a) Phasor diagram for AMwave plus narrowband noise for the case of high carrier-to-noise ratio. (b) Phasor diagram for AMwave plus narrowband noise for the case of lowcarrier-to-noise ratio.pared with the average noise power, so that the receiver is operating satisfactorily, thenthe signal term Ac[l +kam(t)] will be large compared with the noise terms nJ(t) and nQ(t),at least most of the time. Then we may approximate the output y(t) as (seeProblem .!"#$(.%The presence of the '( or constant term A, in the envelope detector output y(t) of)*uation (.% is due to demodulation of the transmitted carrier wave. +e may ignorethis term, however, because it bears no relation whatsoever to the message signal m(t). Inany case, it may be removed simply by means of a bloc,ing capacitor. Thus ! we neglectthe '( term A, in )*uation (.%, we find that the remainder has, except for scalingfactors, a form similar to the output of a '-.--(receiver using coherent detection. Accordingly, the output signal-to-noise ratio of an AM receiver using an envelope detectoris approximatelyA2k2p(-/0# 12 _,_a_3,AM 2WNo)*uation (.%4# is, however, valid only if the following two conditions are satisfied$(.%4#". The average noise power is small compared to the average carrier power at theenvelope detector input.. The amplitude sensitivity ka is ad5usted for a percentage modulationless than ore*ual to "66 percent.7sing )*uations (.%6# and (.%4#, we obtain the following figure of merit for amplitudemodulation$(-/0#38 k;P(-/0#c AM"# " + k;PThus, whereas the figure of merit of a '-.--( receiver or that of an --.receiver usingcoherent detection is always unity, the corresponding figure of merit of an AM receiverusing envelope detection is always less than unity. In other words, the noise performanceof a full AM receier is al!ays inferior to that of a "#$-#% receier. This is due to the(.%!#2.12 Noise in A,M Receivers UsingEnvelope Detection 137wastage of transmitter power, which results from transmitting the carrier as a componentof the AM wave.I> EXAMPLE2.4 Single-Tone ModulationConsider the special case of a sinusoidal wave of frequenc fm and amplitude Amas themodulating wave, as shown !m(t) =Am cos(2'Trfmt)The corresponding AM wave iss(t) =A,[l +J . Lcos(2'Trfmt)] cos(2'Trfct)whereJ . L =k.,Am is the modulation factor. The average power of the modulatingwave m(t) is"assuming a load resistor of 1 ohm#Therefore,using $quation"%.&'#, we get"S()#ol"S()#cAM"%.&*#J . L 2=2 +J . L 2+hen J . L =1, which corresponds to 1,, percem modulation, we get a figure of merit equalto 113. This means that, other factors !eing equal, an AM sstem "using envelope detection#must transmit three times as much average power as a suppressed-carrier sstem "using co-herent detection# to achieve the same qualit of noise performance. 4! l! Il lT.)$S./01$22$CT+hen the carrier-to-noise ratio is small compared with unit, the noise term dominatesand the performance of the envelope detector changes completel from that 3ust descri!ed.Inthis case it is more convenient to represent the narrow!andnoise nit)in terms of itsenvelope r(t) and phase I/I(t), as shown !nit)=r(t) cos[2 'lT. fct+ I/I(t)] (2.97)The corresponding phasor diagram for the detector input x(t) =sit) + nit)is shown in2igure 2 . 38b,where we have used the noise envelope as reference, !ecause it is now thedominant term. To the noise phasor r(t)we have added a phasor representing the signalterm Ac[l+ kam(t)],with the angle !etween them !eing equal to the phase I/I(t) of thenoise nit). 4n 2igure 2 . 38b it is assumed that the carrier-to-noise ratio is so low that thecarrier amplitude Ac is small compared with the noise envelope r(t), at least most of thetime. Then we ma neglect the quadrature componentof the signal with respect to thenoise, and thus find from 2igure 2 . 38b that the envelope detector output is(t) =r(t)+ !c cos[l/I(t)]+ !ckam(t) cos[l/I(t)] (2.98)This relation reveals that when the carrier-to-noise ratio is low, the detector output hasno componentstrictl proportional to the message signal mit). The last term of the e5-pression defining (t) contains the message signal mit)multiplied ! noise in the form of138 CHAPTER 2 ., CONTINtJOVS-WAVE MODVLATJONcos[l/I(t)]. From Secto! 1.11 "e rec#$$ t%#t t%e &%#'e I/I(t)o( t%e !#rro")#!* !o'e n(t) '+!(orm$, *'tr)+te* o-er 2.r r#*#!'. It (o$$o"' t%ere(ore t%#t "e %#-e # com&$ete $o''o( !(orm#to! ! t%#t t%e *etector o+t&+t *oe' !ot co!t#! t%e me''#/e '/!#$ m(t) #t #$$.T%e $o'' o( # me''#/e ! #! e!-e$o&e *etector t%#t o&er#te' #t # $o" c#rrer-to-!o'e r#to' re(erre* to #' t%e threshold effect. 0, threshold "e me#! a value of the carrier-to-noiseratio below which the noise performance of a detector deteriorates much more rapidlythan proportionately to the carrier-to-noise ratio. It' m&ort#!t to reco/!1e t%#t e-er,!o!$!e#r *etector 2e./., e!-e$o&e *etector3 e4%)t' # t%re'%o$* e((ect. O! t%e ot%er %#!*,'+c% #! e((ect *oe' not #r'e ! # co%ere!t *etector.Ar/oro+' m#t%em#tc#$ #!#$,'' o( t%e t%re'%o$* e((ect (or t%e /e!er#$ c#'e o( #!AM"#-e ' )e,o!* t%e 'co&e o( t%' )oo5. Int%e !e4t '+)'ecto! "e 'm&$(, m#tter' ),co!'*er!/ t%e c#'e o( #! +!mo*+$#te* c#rrer. De'&te t%' 'm&$(c#to!, "e c#! 't$$*e-e$o& # /re#t *e#$ o( !'/%t !to t%e t%re'%o$* e((ect e4&ere!ce* ! #! e!-e$o&e *etector.General Formulafor 2SNR3o in Envelope etection!Co!'*er #! e!-e$o&e *etector "%o'e !&+t '/!#$ ' *e(!e* ),x(t)="c co'22.r(ct3 + n(t) (2.99)"%ere "c cos(#$rf%t) ' t%e +!mo*+$#te* c#rrer #!* n(t) ' t%e '#m&$e (+!cto! o( )#!*6$mte*, 1ero-me#!, "%te 7#+''#! !o'e N(t). T%e &o"er '&ectr#$ *e!'t, o( N(t) '8NoSN(t) 9 02(or I f - t;I ' &(2.100)ot%er"'eRe&re'e!t!/ t%e !#rro")#!* !o'e n(t) ! term' o( t' !-&%#'e com&o!e!t n/(t) #!*:+#*r#t+re com&o!e!t n'(t)% "e m#, e4&re'' t%e !o', '/!#$ #t t%e *etector !&+t #'(2.101)T%e !o'e com&o!e!t' nI(t) #!* n'(t) #re 1ero-me#!, ;o!t$, 7#+''#!, m+t+#$$, !*e&e!6*e!t $o"-'' r#!*om &roce''e' "t% *e!tc#$ &o"er '&ectr#$ *e!'te' 2'ee E:+#to!1.1 is the difference between the e&pectation of y(t) in thecombined presence of signal and noise and the e&pectation of y(t)in the presence ofnoise alone, as shown b$#o =E'y(t)( ) E'*o(t)(where y(t) is itself defined b$ '(uation (2.105 and yo(t)is defined b$yo(t)=)nf(t + nt(t)(2.10*(2.10+2. "he %e& o#tp#t oisepowe$ is the difference between the mean-s(uare value of thedetector output y(t) and the s(uare of the mean value of y(t), as shown b$var,$(t- =E'y2(t)( ) (Ely(t)()2.n this basis, we define the o#tp#t sig&l)to)oise $&tio as(2.10/s+(#01o =var,$(t-(2.1023rom #ection 1.12, we recall that the envelope detector output due to noise alone is1a$leigh distributed4 that is,y (y2 )-$e&p---2 ,-y.y) =50 2UN"he e&pectation of yo(t)is therefore(2.110otherwiseE'yo(t)( =r~y-y.y) /yf ~ $2 (y2 )= -e&p --- /yo#0 2#03rom the definition of the g&%%& -#ctio for real positive values of the argument 1, wehave(2.111f(& =r2,,)l e&p(-6 /2!e ma$ therefore rewrite '(uation (2.111 asE'yo(t)(=v 1 U N r( ~)=3 UN(2.112(2.117140 CHAPTER1 II CONTINUOUS-lVAVEMODUlATIONwhere we have used the value [(3/2) = VTr12.To calculate the mean signal So at thedetector output,we also need the expectation of y(t). Due to the combined presence ofsignal and noise, we recall from Section 1.13 that y(t)is Rician distributed,as shown by{y (y2 + A~) (A~Y)~ex ---~ I -fy(y)= ~~p 2(T~ 0 (T~for y ;= 0!"#$$%&otherwisewhere Ia ( ) is the modified 'essel function of the first (ind of )ero order !see *ppendix +,ence,{~y2 (y2 + A "& (A y)E[y(t)]=-o (T~ exp - 2(T~ . fa ~ dy!"#$$/&0utting Acy/~ = u and recogni)ing that p = A~/2(T~,we ma1 recast this expectation inthe formE[y(t)]=!"r$" exp!-p& ru2 exp! -:)!o(") duThe integral in 23uation !"#$$4& can be written in a concise form b1 using co#f$u%#t&yp%r'%o(%tr)c fu#ct)o#* see *ppendix %# 5n particular,using the integral representation!"#$$4&!"#$$6&with ( =+, [((!2) =VTr$2 and +2 =1/,p,we ma1 express the expectation of y(t) interms of the confluent h1pergeornetric function $-$(3/21p) asE[y(t)]=~(T.%/0(-p)(,-!1$0))!"#$$7&8e ma1 further simplif1 matters b1 using the following identit1%2p(-u)($-,(a {3u)) =,-,({3 - a {3-u)and so finall1 express the expectation of y(t)in the concise form!"#$$9&!"#$"0&Thus using 23uations !"#$$+& and !"#$"0& in 23uation !"#$04& 1ields the mean outputsignal as!"#$"$&whose dependence on the standard deviation(T. of the noise nit) is testimon1 to theintermingling of signal and noise at the detector output#:ollowing a similar procedure, we ma1 express the mean-s3uare value of the detectoroutput y(t)asE.[y2(t)] = !=y: %2p(-y22~2A~)!a (A3!) dy. ~ " " J o UN UN UN=2(T~(r-,(-11-p))!"#$""&-i2.12 Naise'n4M ReceiversUsingEnvelopeDetection 141Hence using Equations (2.120) and (2.122) in Equation (2.108) yields the mean outputnoise power asvar[y(t) =2!(l"l(-1#1#-$-%) - i(&"l( - ~ ; l; - P )r )"inally' using Equations (2.121) and (2.12() in Equation (2.10)) yields the outputsignal-to-noise ratio *or the envelope detection pro+lem at hand as(2.12()(l"l(- ~ ; l; - P ) 1r(,-.)o = 2( 4I! ) ( l" l( - 1; 1; - P ) )- ( l" l(- ~ ; l; - P ) )Equation (2.12/) is the general *ormula *or the output signal-to-noise o* an envelopedetector whose input consists o* an unmodulated carrier and +and-limited' white 0aussiannoise. 1wo limiting cases o* this general *ormula are o* particular interest2(2.12/)1. #arge carrier- to- noise ratio. "or large p$ we may use the *ollowing asymptotic *or3mula (see 4ppendi5 /)l" l( - ~ ; l; - P ) =2! *or p ~ 00 (2.126)7oreover' the *ollowing identityIFl(-l;l;-p) =1+ P (2.128)+alds e5actly *or all p. 4ccordingly' the use o* Equations (2.126) and (2.128) inEquation (2.12/) yields the *ollowing appro5imate *ormula *or the output signal-to3noise ratio2(,-.9o =p *or p ~ 00 (2.12:)where we have ignored contri+utions due to p1/2 and pOin the numerator o* Equation(2.12/) as +eing su+dominant compared to p *or large p. Equation (2.12:) showsthat *or large carrier-to-noise pt%e envelope detector +ehaves li;e a coherent detector'in that the outputsignal-to-noise ratio is proportional to the input signal-to-noiseratio.2. &'all carrier- to- noise ratio. "or small p$ we have (see 4ppendi5 /)IF1(a;c;-p) =1-: ! _ pc*or p ~ 0 (2.128)Hence' using this asymptotic *ormula'we may appro5imate the outputsignal-to3noise ratio *or small p as! ! p2(,-.)o =18 - 4! !=(.)1p2(2.12))*or p ~ 0where' in the denominator'we have ignored contri+utionsdue to p and p2 as +eingsu+dominantcompared to pO*or small p. Equation (2.12)) shows that *or a smallcarrier-to-noise ratio$ the output signal-to-noise ratio o* the envelope detector is pro3portional to the squared input signal-to-noise ratio.142 CHAPTER2 '" CONTINUOUS-WAVEMODUlATIONCarrier-to-noise ratio,pFIGURE2.39 Output signl-t!-n!is" #ti! !$ n "n%"l!p" &"t"'t!# $!# %#(ing '##i"#-t!-n!is"ratio.T)" '!n'lusi!ns *n $#!+ t)" t*! li+iting 's"s '!nsi&"#"& )"#"in #" t)t n"n%"l!p" &"t"'t!# $%!#s st#!ng signls n& p"nli,"s *"- signls. T)" p)"n!+"n!n !$*"- signls ."ing p"nli,"& .( t)" &"t"'t!# is #"$"##"& t! s weak signal suppression,*)i') is+ni$"stti!n !$ t)" t)#"s)!l& "$$"'t.Using t)" $!#+ul !$ E/uti!n 02.12234 in Figu#" 2.39 *" )%" pl!tt"& t)" !utputsignl-t!-n!is" #ti! 0SNR3! !$ t)" "n%"l!p" &"t"'t!# %"#sus t)" '##i"#-t!-n!is" #ti! pusing t.ult"& %lu"s !$ '!n$lu"nt )(p"#g"!+"t#i' $un'ti!ns. T)is $igu#" ls! in'lu&"s t)"t*! s(+pt!t"s $!# l#g" p n& s+ll p. F#!+ Figu#" 2.39 *" s"" t)t t)" !utput signl4t!-n!is" #ti! &"%it"s $#!+lin"t .")%i!# #!un&'##i"#-t!-n!is" #ti! !$ 15 &60i.".4p =153.I2.13Noise in FMReceiversFinll(4 *" tu#n !u# tt"nti!n t! t)" n!is" nl(sis !$$#"/u"n'( +!&ulti!n 0FM3 s(st"+4$!# *)i') *" us" t)" #"'"i%"# +!&"l s)!*n in Figu#" 2.25. As ."$!#"4 t)" n!is" wit) is+!&"l"& s *)it" Gussin n!is" !$ ,"#! +"n n& p!*"# sp"'t#l &"nsit( No/2. T)"FMsignlsit)OutputsignalNoisewet)FIGURE2.40 M!&"l !$ n FM#"'"i%"#.2.B Noise inFMReceivers 143received FM signal s(t)has a carrier frequency I e and transmission bandwidthBT,suchthat only a negligible amountof power lies outside the frequency band I e ::t: B,I 2forpositive frequencies, and similarly for negative frequencies.As in the AM case, the band-pass filter has a midband frequency ! c and bandwidthBT and therefore passes the FM signal ,essentially withoutdistortion. Ordinarily, BT issmall compared with the midbandfrequency ! C , so thatwe may use the narrowbandrepresentation for n(t),the filtered version of channel noise w(t),in terms of its in-phaseand quadrature components.Inan FM system, the message signal is transmitted by variations of the instantaneousfrequency of a sinusoidal carrier wave, and its amplitude is maintained constant. Therefore,any variations of the carrier amplitude at the receiver input must result from noise orinterference. The amplitude limiter, following the band-pass filter in the receiver model ofFigure 2.!, is used to remove amplitude variations by clipping the modulated wave atthe filter output almost to the "ero a#is. The resulting rectangular wave is rounded off byanother band-pass filter that is an integral part of the limiter, thereby suppressing har$monics of the carrier frequency, Thus, the filter output is again sinusoidal, with an am$plitude that is practically independent of the carrier amplitude at the receiver input.The discriminator in the model of Figure 2.! consists of two components%&. A slope network or differentiator with a purely imaginary frequency response thatvaries linearly with frequency. 't produces a hybrid-modulatedwave in which bothamplitude and frequency vary in accordance with the message signal.2. An envelope detector that recovers the amplitude variation and thus reproduces themessage signal.The slope networ( and envelope detector are usually implemented as integral parts of asingle physical unit.The postdetection filter, labeled )baseband low-pass filter) inFigure 2.!,has abandwidth that is *ust large enough to accommodate the highest frequency component ofthe message signal. This filter removes the out-of-band components of the noise at thediscriminator output and thereby (eeps the effect of the output noise to a minimum.The filtered noise n(t)at the band-pass filter output in Figure 2.! is defined in termsof its in-phase and quadrature components byn(t)=nI (t) cos+2,T',fct-- nQ(t)sin+2,T',fct-.quivalently, we may e#press n(t)in terms of its envelope and phase asn(t)=r(t) cos[ (2' TI ' fct) + I /(t)! +2.&/!-where the envelope isr(t)=[ny(t) + n"(t)#' #2 +2.&/&-and the phase isI /(t) =tan$1[ nQ(t)!nI (t)The envelope r(t)is 0ayleigh distributed, and the phase I /(t)is uniformly distributed over1"r radians +see 2ection &.&2-.The incoming FM signal s(t) is defined by+2.&/2-+2.&//-144 CHAPIER2 " CONTINUOUS-WAVE MODUL4.TIONwhere Ac is the carrier amplitudevj,is the carrier frequency, kf is the frequency sensitivityand mit) is the message signal. Note that,as with the standard AM, in FM there is n~need to introduce a scaling factor in the definition of the modulated signal sit), since it isreasonable to assume that its amplitude Ac has the same units as the additive noise com.ponent nit).To proceed, we define ~B]'"5 "0 -4'#.22Wi'2676~.* 66*/~2/ 66~/*20~2&"~.~228 955i!2*8 -'in~)/#-o)0)&)2)** & / )* )2 )& )0 )/ 2*Carrier-to-noise ratio 10 laglop,dB%$:;3 2{t)]where Au is the '%*litu&e( With ' !$tr$l #$lt'-e v{t) '**lie& t$ the #e$ i*ut. the '-le4> 2(t) is rel'te& t$ v{t) ," the ite-r'lrP2(t)=21Tku f: V {T) d - r /2(0122where ku is the freque!" sesiti#it" $f the #e$. %e'sure& i Hert3 *er #$lt( The $,4e!t$f the *h'se-l$!+e& l$$* is t$ -eer'te ' #e$ $ut*ut r(t) th't h's the s'%e *h'se '-le/e5!e*t f$r the fi5e& &iffere!e $f 67 &e-rees2 's the i*ut 8Msi-'l s{t). The ti%e-#'r"i-*h'se '-le 4> ,{t) !h'r'!teri3i-s{t) %'" ,e &ue t$ %$&ul'ti$ ," ' %ess'-e si-'l mit),i whi!h !'se we wish t$ re!$#er 9 : ,it) '& there," *r$&u!e ' esti%'te $f m{t). In$ther'**li!'ti$s $f the *h'se-l$!+e& l$$*. the ti%e-#'r"i- *h'se '-le 4> ,(t) $f the i!$%i-si-'l sit) %'" ,e ' uw'te& *h'se shift !'use& ," flu!tu'ti$s i the !$%%ui!'ti$!h'el; i this l'tter !'se. we wish t$ track 9 : .(t) s$ 's t$ *r$&u!e ' si-'l with the s'%e*h'se '-le f$r the *ur*$se $f !$heret &ete!ti$ /s"!hr$$us &e%$&ul'ti$2(II MODEL OF THE PHASE-LOCKED )$$r-"T$ &e#el$* ' u&erst'&i- $f the *h'se-l$!+e& l$$*. it is &esir',le t$ h'#e ' md !" $fthe l$$*( We st'rt ," &e#el$*i- ' $lie'r %$&el. whi!h is su,sequetl" lie'ri3e& t$si%*lif" the ''l"sis( A!!$r&i- t$ 8i-ure 2(roblem 2"184 for more details, see the paper by :ennie %12;8'";" 1or detailed description of the superheterodyne recei!er, see the Radio*n)ineerin)+and,-oo. edited by =enney %1258, pp" 12$3 8$12$81'"8" #he qualitati!e study of threshold in en!elope detection presented here follos /onin&%1268, p" ;1'"2" 1or a?ustification of the critical assumption on hich the simplificationpresented in ,qua)tion %2"182' rests, see 0ice %1263 '"13" 1or a detailed discussion of the threshold effect in 1: recei!ers, see the paper by 0ice%1263 ' and the boo6 by -chart