3. Carrier Modulation Analog

Transcription

1 3. Carrier Modulaion Analog Modulaion is he process of using an informaion signal (such as voice or music) o aler some propery of a higher frequency waveform which can hen be efficienly radiaed by reasonably small anenna. Waveforms a differen radio frequencies can simulaneously carry a pluraliy of informaion signals. The informaion signal ampliude is used o vary eiher he carrier ampliude (A c ) or he carrier phase angle (ω c + θ). The unmodulaed carrier is normally a sinusoid described by: c() = A c cos(ω c + θ ) where: A c = peak ampliude of he carrier ω c = πf c = carrier frequency θ = carrier phase a = 3.1 AMPLITUDE MODULATION Consider a carrier sinusoid c () = Accosω cha is muliplied by a facor which varies beween and 1; a process ha can be simply realized wih a poeniomeer. For rapidly flucuaing informaion signals, he poeniomeer is replaced wih an elecronic muliplier ha implemens a gain coefficien (again consrained beween and 1). For his inroducion, we assume he gain coefficien (or modulaion) o be always posiive and of he form k () = A + Am cosω m where k () 1. In broadcas AM radio, he modulaion signal (or gain) is always posiive as i is in Figure 3-1. We will see ha he broadcas signal has ransmied carrier wih double sideband (DSB-TC). Ampliude modulaion is, in fac, a non-linear process and his is evidenced by he generaion of new frequencies. However, i is ofen called linear modulaion since he oupu carrier ampliude varies in one-o-one correspondence wih he message signal. Figure 3-1 Muliplicaion of a uni ampliude carrier by he facor k() Sinusoidal modulaion of a carrier We now pu aside he requiremen for a posiive gain coefficien and focus on he produc of wo sinusoids. In his secion, we consider modulaing signals ha have zero average value and, as a resul, here is no carrier componen in he produc.

2 Inroducion o Communicaion Sysems A Mulimedia workbook 35 The oupu produc has new componen frequencies (ω c - ω m ) and (ω c + ω m ) ha are differen from eiher of he inpu frequencies. Signals in he modulaion process are illusraed wih double sided frequency specra, a represenaion consisen wih complex exponenial jθ θ represenaion of he signals. Using he Euler ideniy cosθ = e + e j, he produc s () = Amcosωm Accosω cis illusraed below in erms of exponenial componens. We can hen resae he produc s() in cosine form as 5. AA c m cos( ωc + ωm) + cos( ωc ωm) consisen wih he rigonomeric ideniy cos x cos y = 5. cos( x+ y) + cos( x y). [ ], a resul [ ] A m ω m ω m A m ω Modulaion A e ωm e + ( ) m j jωm m() = A m cos ω m A c ωc ωc A c ω Carrier A e ωc + e ( ) c j jωc c() = A c cos ω c s() AA c m 4 (ωc+ω m ) (ωc ω m ) AA c m 4 ωc+ω m ωc ω m [ ] AA e ( ωc+ ωm) e ( ωc ωm) ω e ( c ωm) ω e c+ ωm ( ) 4 c m j j j j Figure 3- Muliplicaion of A m cosω m and A c cosω c Example 3.1 Produc of Sinusoids A 6 khz cosine waveform wih peak ampliude 1 vols is muliplied by a 4 vol peak cosine waveform wih frequency 3 khz. Deermine he frequency and ampliude of all produc componens and deermine heir normalized powers. Illusrae he ampliude specrum and normalized power specrum on a wo-sided frequency axis. 3 ( 1Vcosπ6 1 ) ( 4Vcosπ3 ) 3 3 = Vcosπ Vcosπ V 1 W 1 V 1 W 1 V 1 V 1 W 1 W f (khz) 597 khz, Vp = V, PN = W 63 khz, Vp = V, PN = W Drill Problem Ampliude Modulaion - For a carrier signal c()=1vcosϖ and he following modulaion signals, deermine he sinusoidal componen ampliudes (in vols) and componen frequencies (in khz). Modulaion Signal A1 F1 A F A3 F3 A4 F4 cos π4 4cos π11 + 4cosπ cos π 4 + cos π 8 Checksum Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

3 Chaper 3: Carrier Modulaion - Analog On/Off modulaion of a carrier Double-sided specral represenaions faciliae rapid undersanding of he specra resuling from signal muliplicaion. The example below shows muliplicaion of a carrier sinusoid by an on/off gaing signal. (Noe ha on/off modulaion is used in opical ransmission sysems). The gaing (or modulaion) signal is represened by a low frequency square waveform wih levels and 1 and is specrum has a cluser of componens near zero frequency. Through he muliplicaion process, he posiive frequency componens of he squarewave ranslae o an upper sideband (USB) wih frequencies greaer han ha of he carrier while negaive frequency componens ranslae o he lower sideband (LSB) wih frequencies less han he carrier. Example 3. On-Off Swiched Sinusoid - A 16 khz sinusoid Acosω c is muliplied by a 1 khz repeiive on/off square waveform wih 5% duy cycle. Illusrae he ime waveform of his gaed signal. Also illusrae he Fourier series for he wo inpu signals, W(f) and S(f), and he oupu signal, W S (f). vols w() = Acosω c A uniless nt s() = T n = (ms) (ms) Aδ(f + f c ) W( f ) Aδ(f f c ) - f c f c f (Hz) -16 khz 16 khz S(f) vols f (khz) A vols w s () = s()w() (ms).5 A W s (f) AS(f + f c ).319 A AS(f f c ).64 A vols - f c -.16 A.5 A.319 A.64 A f c f (Hz) -.16 A Drill Problem 3. On/Off Modulaion - a) In he 1 khz baseband on/off gaing waveform of Example 3., wha posiive frequency has exponenial componen ampliude -.16, b) In he oupu signal, w s (), wha are he highes posiive componen frequencies wih ampliudes.64 A and.35 A, and c) if A = 4 V, wha is he magniude of he oupu componen a 13 khz? a) khz b1) khz b) khz c) V Checksum = Specral Translaion & Convoluion Ampliude modulaion of a sinusoidal carrier can be viewed as ranslaing he baseband modulaion signal o a higher frequency. In he oupu produc specrum, here is a copy of he wo-sided baseband specrum cenered abou he carrier frequency hus we have frequency ranslaed he baseband signal. Each modulaion frequency exponen has been increased by he carrier frequency. Ch3-AM-6d3.doc /3/6

4 Inroducion o Communicaion Sysems A Mulimedia workbook 37 When he carrier signal iself has several frequency componens, ranslaed copies of he baseband specrum may overlap. Several combinaions of inpu frequencies can resul in he same oupu frequency. Upon muliplicaion, he frequency exponens add and hus he oupu specrum may be calculaed by convoluion of inpu signal specra. Convoluion has been previously demonsraed in he calculaion of PDF for he arihmeic sum of signal volages. 3. RADIO BROADCAST AMPLITUDE MODULATION (AM-DSB-TC) In AM broadcas radio, he carrier ampliude increases and decreases in proporion o he informaion signal volage. The modulaion volage is always posiive and hus here are no polariy reversals in he modulaed carrier. This forma is known as ampliude modulaion wih double sideband and ransmied carrier (AM-DSB-TC) or simply AM. A broadcas radio sysem needs simple, low cos, receivers (since here are many of hem). By ransmiing a large carrier componen, low-cos envelope or diode deecors can be used in he receivers. So he exra cos in ransmiing carrier power is balanced by reduced receiver cos. In Norh America, AM broadcas carrier frequencies are spaced a 1 khz inervals ranging from 54 khz o 17 khz. Originally, modulaion signals were resriced o less han 5 khz resuling in 1 khz ransmission bandwidh in each channel. Recenly adoped sandards now allow signal frequencies up o 1 khz wih khz ransmission bandwidh. The obvious inerference beween channels occupying he same or an adjacen frequency band is conrolled by geographic separaion beween ransmiers. The maximum allowed unmodulaed ransmier power is 5 kw and he modulaing waveform is consrained o be posiive by limiing he (negaive) modulaion index o µ =.9 (9%). The majoriy of AM radio saions are licensed for 1 kw or less and, in many cases, ransmier power is reduced a nigh. Ampliude modulaion is modeled by he muliplicaion of a consan carrier wih a posiively biased baseband informaion signal (i never becomes negaive). In he illusraion below, we assume an informaion signal of u cosω m where he modulaion index u =.5. x() = cosω m s() = (1 + µcosω m )A c cosω c x() = 1 + m() khz khz c() = A c cosω c 1 V -1 V 15V 1V 5V Figure 3-3 Ampliude modulaion waveforms Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

5 Chaper 3: Carrier Modulaion - Analog AM signal specra We assume an informaion signal m() wih frequency, f m (smaller han carrier frequency, f c ). The ransmied produc waveform has ampliude variaion proporional o modulaing signal ampliude. For a sinusoidal informaion signal, we have 1. + µ cos ω m and he modulaion index, µ, mus be less han uniy. For oher informaion signals, he index, µ, is defined in erms of he mos negaive excursion of he baseband signal i.e. µ = m () min. Specral analysis reveals a large carrier componen plus an upper sideband (USB) one and a lower sideband (LSB) one. As µ is increased, sideband componens increase while he carrier componen remains consan. This leads o he descripion: ampliude modulaion wih double sideband and ransmied carrier (AM-DSB-TC). Sideband and carrier componens are eviden by expanding he produc s() as s () = ( 1 + µ cos ω A ) cos ω m c c µ A = A cos + c µ A c ωc cos( ωc ωm) + c cos( ωc + ωm) 3.. AM power efficiency Only he sidebands carry informaion so he power efficiency of AM ransmission is calculaed as he raio of sideband power o oal ransmied power. This is expressed below for sinusoidal modulaion and, a he upper limi, we have µ = 1 resuling in 33% efficiency. A Toal Transmier Power = P + P + P = c + ( uac / ) + ( uac / ) C USB LSB. P + + Power Efficiency = USB PLSB u 8 u 8 u = = P + P + P C USB LSB + u + u u. Review Virual Laboraory on ampliude modulaion and observe 4 vol added dc componen. Modulaion volage range is hen 4 V ±.5 V. a) Deermine he larges posiive peak of he oupu. b) Calculae he modulaion index. c) The specrum analyzer shows oupu carrier a.8 Vrms. Calculae rms sideband ampliude (compare wih observaion) Checksum = 8. Drill Problem 3.3 AM Power Efficiency - For a carrier signal c()= Vp cosϖ6 and sinusoidal modulaion, complee he following able. Refer o Example 3.3. Carrier Vp Pc (kw) µ Vp s ( ) Poal (kw) Vp/rms η (%) PEP (kw) Checksum 1 V V V V Checksum Ch3-AM-6d3.doc /3/6

6 Inroducion o Communicaion Sysems A Mulimedia workbook 39 Example 3.3 Average Power, Peak Power, and Peak Envelope Power (PEP) A 1 V, 5 khz carrier wih normalized power 5 kw is ampliude modulaed by a 5 khz sinusoid using modulaion index µ =.6. a) Illusrae he ime waveform and, during he modulaion cycle, illusrae he maximum carrier ampliude and calculae he insananeous peak power. b) Deermine he sideband ampliudes, he power in he sidebands and he oal average power. c) Calculae average normalized power when he envelope is a is maximum (his is known as peakenvelope-power or PEP). a) 16 V 1 V 4 V B µ = A B µ =.6 A Insananeous peak power 5,6 was Peak Envelope Power (PEP) in his region (vols) Ac = 1 V µac/ single-sided specrum µac/ = 3 V LSB USB f (khz) b) LSB V p = (.6)1/ = 3 V P N = V p / = 45 was Carrier V p = 1 V P N = V p / = 5, was USB V p = (.6)1/ = 3 V P N = V p / = 45 was Toal power = 5,9 was c) PEP = (16 V) / = 1,8 was (normalized) 3..3 AM efficiency for realisic signals Typical informaion signals conain many frequency componens and he efficiency is much less for hese compound signals. For he compound modulaion signal, m(), he ransmied AM signal is expressed as: s () = Accos ωc+ ma ( ) ccosω c. The ime averaged oupu power and sideband power efficiency are compued as follows ave c c c c c c P = s () = A cos ω + m() A cos ω + m ( ) A cos ω and P ave ( ) Ac = + m 1 ( ) if ω c >> ω m and m () = Power Efficiency sideband power = = average oal power m () Ac / m () = + A 1 + m / () ( 1 m () ) c. Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

7 Chaper 3: Carrier Modulaion - Analog 4 For he specific case of sinusoidal modulaion, m() = µ cos ω m and m () = µ. For a compound signal wih peak/rms =.5, efficiency is less han 15% a he loudes volume (µ = 1). Wih normal AM broadcas modulaion µ < 1 and efficiency may be considerably less han 1%. Example 3.4: calculae he power efficiency when ransmiing a compound signal using AM-DSB-TC wih m() min = -.9 and he modulaing signal peak/rms =.8. m ( 9. ) 81. () = 8 = = (. ) (. ) 8 m () (. ) Efficiency = = =. 919 = 9. % 1 + m () (. 15) 3..4 AM Demodulaors (Deecors) Coheren Deecor - Recovering he modulaion signal m() direcly from he A M modulaed signal, s(), using a synchronized oscillaor is called synchronous deecion, coheren deecion, or homodyne (single frequency) deecion. The receiver is known as a zero-if or direc conversion receiver. Coheren deecion requires a receiver local oscillaor ha is synchronized in frequency o he carrier used a he ransmier. The ampliude modulaed signal, s() = [1 + m()]a c cos(ω c ), is ranslaed o baseband by muliplicaion wih he local oscillaor, cos(ω c + θ), where θ is a small phase offse, ideally equal o zero. The muliplier oupu is [ ] r 1 () = [1 + m()] A c cos(ω c ) cos(ω c + θ) = [ 1+ m ( )] A cosθ+ cos( ω + θ ). The erm wih wice he carrier frequency is removed by he receiver oupu low-pass filer, and he filered oupu becomes r d () = [1 + m()] A c cosθ. c c [1+m()] s() = [1+m()]A c cos(ω c ) S(f) A c cos(ω c ) s() e -jω c e +jω c e +jω c r() r 1 () = [1+m()]A c cos(ω c )cos(ω c +θ) = [1+m()]A c cos(θ) +[1+m()]A c cos(ω c +θ) r 1 () -f c f c Received baseband signal f cos(ω c +θ) r d () =[1+m()]A c cosθ Figure 3-4 Coheren AM Deecor Ch3-AM-6d3.doc /3/6

8 Inroducion o Communicaion Sysems A Mulimedia workbook 41 Diode Deecor (Recifier) - Demodulaion (or deecion) of AM signals is mos economically accomplished by diode deecors or envelope deecors, however, hese deecors require a large carrier componen in he ransmied AM signal. Boh mehods are an alernaive o synchronous (i.e. coheren) demodulaion of he previous secion. Diode and envelope deecors are someimes referred o as self-homodyne or self-coheren deecors since he received carrier componen is direcly used in demodulaion. If AM is applied o a diode and a resisor circui he negaive par of he AM wave will be eliminaed and he oupu across he resisor, r (), is a recified version of he AM signal. Recificaion can be modeled as muliplicaion by a 1 and squarewave signal, k(), a he same frequency and phase as he carrier. The recified oupu is: r () = {[ 1+ m()] Accos ωc} k( ) r () = [ 1+ m( )] Accosωc + cosωc cos3ωc+ cos 5ωc... π r () = [ 1+ m( )] Ac cos c cos c cos4 c cos 6 c ω ω ω ω π 7 When he recified oupu r () is applied o a lowpass filer wih cuoff frequency f bw < ω c /π, he filer oupu is [1 + m()]a c /π, and all he oher componens wih frequencies higher han f bw are suppressed. The dc erm is blocked by a capacior o give he desired oupu m()a c /π. Noe ha recifier Figure 3-5 Recifier Deecor (adaped from Sremler) deecion is, in effec, synchronous deecion performed wihou using a local carrier. The large carrier conen in he received signal makes his possible. If a full-wave recifier is used, he equivalen muliplicaion is by +1 and 1 and he oupu ampliude is doubled as compared o a half wave recifier. -3f c P( f ) 3f c -f c f c r() S( f ) r 1 () e + jω c e +jω c cos(ω c +θ) -fc f c f Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

9 Chaper 3: Carrier Modulaion - Analog 4 Envelope Deecor The key poin of ineres in an AM signal is ha he posiive ouer envelope of s AM () has he exac shape of m(). If we could consruc a receiver ha follows he envelope of he posiive signal peaks, hen m() can be recovered. An envelope deecor as shown in Figure 3-6 has higher oupu ampliude han he previously described diode deecor. During a posiive cycle of he AM wave, he diode is forward biased and capacior charges up o he peak value. Ripple in he oupu occurs Figure 3-6 Envelope deecor (adaped from Sremler) when he capacior discharges hrough he resisor during he ime period beween peaks of he AM wave. The ripple can be decreased by increasing he carrier frequency or by increasing he RC ime consan. The oupu lowpass filer can remove any remaining ripple. If he RC ime consan is oo large, he capacior volage canno follow he rae of decrease in he envelope and here will be disorion in he shape of he demodulaed waveform. If he modulaing frequency is much less han he carrier frequency, he ime consan should be seleced as follows: 1 f bw << << f πrc c where f bw = f m (max). Observe he diode deecor in he Virual Experimens CD. a) Wha is he primary baseband oupu frequency when he modulaion has no dc componen (suppressed carrier)? b) Wha is he primary baseband oupu frequency when he maximum dc componen is applied a he ransmier? c) Why are here only odd harmonics of he oupu baseband frequency if here is no dc componen (suppressed carrier)? 3..5 Superheerodyne receiver To separae radio saions, receivers were iniially equipped wih a unable highly selecive band-pass filer. In hese uned radio frequency (TRF) receivers, several resonan circuis were used o realize a band-pass filer wih fla gain over he signal bandwidh bu unforunaely i was difficul o proporionaely une all resonan circuis over he enire A M frequency band. This problem was overcome in he super-heero-dyne (Lain & Greek for abovedifferen-force/frequency ) receiver, where a higher frequency local oscillaor (LO) is muliplied wih (mixed wih or modulaed by) he incoming desired signal. This produces an inermediae Ch3-AM-6d3.doc /3/6

10 Inroducion o Communicaion Sysems A Mulimedia workbook 43 frequency (IF) signal ha sill conains he informaion signal and can be seleced by a fixed frequency fla filer. The following diagram illusraes he specral ampliudes a several poins in a superheerodyne receiver (he verical axis has unis of V Hz ). News Counry 65 khz 6 khz Oldies 6 khz Vols / Hz RF filer khz Counry News 6 khz 65 khz Oldies 6 khz a) b) -155 khz +155 khz -155 khz +155 khz Local Oscillaor 155 khz f (khz) f (khz) Oldies Counry khz c) Counry -455 khz Oldies -435 khz Oldies 435 khz Counry 455 khz Counry 1655 khz Oldies 1675 khz IF Filer khz f (khz) d) Counry -455 khz Counry 455 khz e) & f) -455 khz +455 khz Baseband Filer - 5 khz f (khz) f (khz) Figure 3-7 Signal specra in a superheerodyne receiver. The desired frequency relaionship in he mixer is f LO f RF = f IF where f LO is he frequency of he local oscillaor, f IF is he inermediae frequency (IF), and f RF is he carrier frequency of he desired incoming RF signal. By using a variable frequency LO, any desired RF signal can be ranslaed o he IF frequency. All frequency componens are ranslaed by he same amoun so he sidebands are no changed (excep for reversal) by he ranslaion. The advanage of his mehod is ha he highly selecive accurae band-pass filering can be done a he inermediae frequency (IF). The accurae filer is fixed in frequency and here is no requiremen o une i. Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

11 Chaper 3: Carrier Modulaion - Analog 44 a) RF Amp. & Filer coarse bandpass filer (Pre-selecor) Modulaor (Mixer) b) c) IF Amp. & Filer accurae bandpass filer khz Local Oscillaor Diode Deecor d) e) Audio Amp & Filer Speaker f) Figure 3-8 Superheerodyne receiver. Example 3.5 An envelope deecor is used o demodulae a 455 khz inermediae frequency DSB-TC signal o a baseband signal ha has bandwidh 5 khz. The envelope deecor resisance is 1, ohms. Wha is a reasonable value for he capacior? Answer - To preven excessive ripple in he oupu, he cu-off frequency (or bandwidh fb ) of he RC nework should be much less han he IF frequency. To preven excessive disorion, he bandwidh should be much greaer han he highes modulaing frequency. fm(max) << fb << fif Selec geomeric fb = fm,max fif = 47. 7kHz mean: 1 = 47. 7kHz 1 πrc C = = 334 pf 4 3 π The problem of image frequency in superheerodyne receivers In a superheerodyne receiver, he LO frequency is greaer han he desired RF frequency and he IF frequency is he difference beween hem. (i.e. f IF = f LO f RF ) Unforunaely i is possible for an image signal (wih frequency greaer han he LO) o also resul in a mixer oupu a he IF frequency. (i.e. f IF = f IMAGE f LO ) The unwaned signal a he image frequency, afer down conversion o IF, will overlay he desired signal and resul in inerference. I is herefore imporan o preven image frequencies from reaching he mixer and his is he reason for he RF pre-selecor filer. I is used o remove signals a f IMAGE and allow signals a f RF. The raio of pre-selecor filer gain a he wo frequencies is known as he image rejecion raio. Ch3-AM-6d3.doc /3/6

12 Inroducion o Communicaion Sysems A Mulimedia workbook 45 Local Oscillaor Radio Frequency khz News RF filer 91 khz khz khz f (khz) 156 Jazz f (khz) Inermediae Frequency Eliminaed by RF filer Eliminaed by RF filer f (khz) Figure 3-9 Removal of image frequency signals by he RF filer. Example 3.6 Image Frequency - a commercial AM radio saion wih carrier frequency 54 khz is being received by a superheerodyne receiver wih 455 khz inermediae frequency. Wha is he local oscillaor (LO) frequency and wha is he image frequency? Wha is he significance of he image frequency? Soluion - f f = f f = = 995kHz RF LO IF LO f f = f f = = 145 khz IM LO IF IM Any signal a he image frequency would be ranslaed o he IF frequency and would inerfere wih recepion of he desired signal. Image frequency signals mus be removed by he RF filer before reaching he mixer. Here, he image frequency is 91 khz ( x 455) higher han he desired RF signal. 3.3 DOUBLE SIDEBAND SUPPRESSED CARRIER (DSB-SC) A double sideband suppressed carrier (DSB-SC) signal is essenially an AM signal ha has no discree carrier componen. When he baseband informaion signal wih specrum M(f) is muliplied by a uni ampliude sinusoid a carrier frequency, f c, he following specrum resuls. Ac SDSB( f) = [ M ( f + f c) + M ( f f c) ] DSB-SC wases no power on ransmiing a carrier componen hus he modulaion efficiency is 1%. For a given peak ransmier volage, he sideband ampliudes can be wice as large (and sideband power 4 imes larger) as for AM-DSB-TC. Despie his advanage, DSB-SC is no generally suiable for broadcas ransmission since he exac carrier frequency and phase is required for demodulaion and his presens a difficul problem Demodulaion of DSB-SC Direc demodulaion of DSB-SC signals o recover m() requires coheren deecion (synchronous deecion); he receiver local oscillaor mus be exacly he same frequency (and phase) as he modulaed carrier ha would be received from he ransmier. This is no always possible (especially if he baseband signal has periods of silence). A receiver using direc coheren deecion is also known as a zero-if receiver. Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

13 Chaper 3: Carrier Modulaion - Analog 46 M( f) V Hz a) Baseband -f b f b -f c +f c f (khz) b) DSB-SC USB LSB LSB USB -f c f c f c + f b -f c +f -f c c +f c f (khz) c) USB LSB LSB USB -f c -f b f b Figure 3-1 Specra in modulaion and coheren ( zero-if ) demodulaion of DSB-SC. f c f (khz) A homodyne, direc conversion, receiver has some advanages over he superheerodyne receiver. I eliminaes he IF filer, he image rejecion filer and he need for addiional oscillaors. I does, however, require he receiver o synhesize an accurae local oscillaor wih small phase error θ (ideally zero). m() Accos(ωc) s() s() =m()accos(ωc) assume uniy gain channel r() r 1 () =m()accos(ωc)cos(ωc+θ) r 1 () =m()ac{cosθ + cos(ωc+θ)} r 1 () LO = cos(ωc+θ) Figure 3-11 Zero-IF (homodyne) receiver for DSB-SC signals. r d () =m()accosθ As he phase error θ increases, he oupu ampliude diminishes and, if θ is 9 degrees, he produc demodulaor oupu is zero! In order o ensure minimum phase error, he local oscillaor (LO) signal can be developed using a phase locked loop incorporaing he square of he received signal. This fails, however, during periods of silence when he sidebands become zero. To permi he use of a low-cos diode deecor, we can re-inser he locally generaed carrier signal o form DSB-TC. This mehod avoids he relaively cosly muliplier/mixer illusraed in Figure Applicaion of DSB-SC wihin a FM sereo broadcas signal An ineresing applicaion of DSB-SC is in commercial FM sereo broadcasing. Original frequency modulaion sysems were monaural and hey carried one Hz - 15 khz audio signal. Ch3-AM-6d3.doc /3/6

14 Inroducion o Communicaion Sysems A Mulimedia workbook 47 Sereo broadcas in FM sysems now uses a baseband signal wih Hz - 53 khz frequency range o carry wo audio signals. To be compaible he previous monaural sysem, he sum of he lef and righ (L+R) audio signals is broadcas in he Hz 15 khz frequency range. The difference of he lef and righ audio signals (L-R) is placed a a higher (inaudible) frequency range in he baseband signal. The L-R signal ampliude modulaes a 38 khz sub-carrier producing a lower sideband in he frequency range khz and an upper sideband in he frequency range khz. The 38 khz sub-carrier is no broadcas so ha (almos) all of he combined signal volage relaes o he audio signals. In effec, he L-R audio signal is relocaed in he frequency domain by muliplying is ime waveform by a pure sinusoid a f c = 38 khz. This is expressed as S DSB () = m L-R ()A c cosω c. For example, a pure sinusoidal L-R modulaion would resul in frequencies f c ± f m and no componen a he carrier frequency f c. To avoid specral overlap wih he L+R baseband signal, he 38 khz sub-carrier is seleced such ha f c > f m. In addiion o he main sereo program carried in he L+R and L-R signals, some broadcas signals include a background music program in he frequency range khz. This is shown in Figure 3-1 as subsidiary communicaions. L+R M ( f ) L+R L-R LSB M ( f-f c ) L-R USB Opional Subsidiary Communicaions f (khz) Figure 3-1 Specrum of sub-carrier DSB-SC sereo signal. m L () L khz R - 15 khz m R () + - Σ Σ m L () + m R () L+R 38 khz Cos ωc L-R 15 frequency divider f (khz) 19 khz + + Σ + f (khz) 13.5 ±.1 MHz m b () - 75 khz s() FM Transmier 38 Figure 3-13 DSB-SC modulaor for sereo broadcas on FM. The broadcas frequency allocaion, or in oher words, he allowed bandwidh for commercial FM ransmission is ± 1 khz abou he carrier. For example, a licensed ransmier migh have an allowed ransmission bandwidh of 13.1±.1 MHz. The combined audio signal mus no resul in a frequency deviaion ha would cause his bandwidh o be exceeded. Since Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

15 Chaper 3: Carrier Modulaion - Analog 48 he receiver signal o noise raio (SNR) improves wih he degree of modulaion, i is advanageous o remove any non-informaion componen such as a baseband subcarrier. The receiver, however, requires a large 38 khz local oscillaor o perform coheren demodulaion. This is addressed by sending a small ampliude 19 khz pilo one ha he receiver uses o synhesize he required 38 khz. The 19 khz pilo signal represens only a small porion of he baseband volage used o modulae he FM ransmier so he SNR a he receiver is no significanly reduced. A he receiver, he FM broadcas signal is iniially demodulaed o a signal wih 75 khz bandwidh. The 75 khz signal is hen separaed by filers o form a L+R baseband signal and a L-R signal in DSB-SC forma. A 38 khz local oscillaor signal is used o demodulae he L-R signal by synchronous demodulaion as shown in Figure Alernaely he local oscillaor (LO) oupu can be added o he L-R signal o form DSB-TC ha can be subsequenly demodulaed by an envelope deecor. r() 13.5 ±.1 MHz 15 khz LPF L+R + Σ + L - 15 khz FM Receiver db() - 75 khz 19 khz BPF & Doubler 3-53 khz 38 khz 15 khz L-R + - Σ - 15 khz R Figure 3-14 DSB-SC Demodulaor for sub-carrier DSB-SC sereo signal Using he Virual Experimens CD, observe he specrum analyzer display showing he demodulaed specrum of a FM sereo broadcas signal. This specrum shows a baseband signal exending up o 15 khz plus a DSB-SC signal conaining L-R informaion. a) Observe he pilo frequency and deermine he bandwidh of he unused specrum surrounding i. b) Observe he receiver s 38 khz carrier reference and deermine i s ampliude in db relaive o he received pilo a 19 khz. c) Pause he movie and observe he similariy of he baseband specrum o he sidebands abou he 38 khz carrier Ch3-AM-6d3.doc /3/6

16 Inroducion o Communicaion Sysems A Mulimedia workbook SINGLE SIDEBAND AMPLITUDE MODULATION (SSB) The mos power and bandwidh efficien linear modulaion scheme is single sideband ampliude modulaion. Boh sidebands in DSB-SC have equal ampliude and are symmeric in frequency abou he carrier; he informaion conained in one sideband is also in he oher. Transmiing only one sideband halves he ransmission bandwidh and, when compared o AM- DSB-TC, power is saved by no ransmiing he carrier and he redundan sideband Filer mehod of generaing SSB One mehod of generaing a SSB signal is o firs generae a DSB-SC signal and hen remove one of he sidebands wih a filer. Eiher he upper or lower sideband can be ransmied. The following diagram implies an ideal bandpass filer wih perfecly sharp cuoff a f c, however, real filers have a ransiion frequency range beween sopband and passband. Since his ransiion is generally in proporion o he cuoff frequency, i is advanageous o filer a a relaively low inermediae frequency. For he example of voice signals wih 3 34 Hz bandwidh, he DSB-SC signal has 6 Hz separaion beween he sidebands and he SSB filer ransiion mus occur over less han 6 Hz. a) Baseband M ( f ) V Hz Modulae f b -f b -f c +f c f (khz) b) DSB-SC USB LSB LSB USB Filer -f c f c - f b f c Filer f (khz) c) SSB-SC USB USB Demodulae -f c ~ f c f c +f b f (khz) M ( f ) +flo -f LO -f LO +f LO f LO = f c Baseband Filer -f c -f b f b f c f (khz) Figure 3-15 Specra in modulaion and coheren demodulaion of SSB-SC Demodulaion of SSB using a mixer Demodulaion of SSB requires an almos-coheren local oscillaor a he receiver. Phase shif and even a small amoun of frequency error can someimes be oleraed. When he receiver s local oscillaor has a small frequency offse relaive o he ransmier frequency, he demodulaed informaion signal incurs a sligh frequency shif. This would no be olerable for elevision signals, however a modes ranslaion is accepable in voice ransmission sysems such as in amaeur radio, ciizen band radio and in elephony. The 8.8 and 33.6 kb/s voiceband modems Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

17 Chaper 3: Carrier Modulaion - Analog 5 can olerae up o 7 Hz of frequency ranslaion. Early elephone sysems used SSB for mulichannel microwave ransmission and a pilo one enabled he receiver o synhesize a series of almos synchronous demodulaion carriers for each SSB voice channel. SSB demodulaion is accomplished wih a mixer (muliplier) in he same way as for DSB sysems. Alernaely, if replica of he ransmier carrier is insered a he receiver, hen an envelope or diode deecor may be used in he same way as for broadcas AM. V Hz USB USB f (khz) -f c f c -(f LO - f ) f LO - f -(f LO - f ) f LO - f Demodulae ~ M ( f ) SSB-SC R(f) -(f c - f ) - f f f c - f Figure 3-16 SSB demodulaion wih LO frequency error Envelope Deecion of SSB The mos economical mehod of deecing a SSB-SC signal is o re-inser he missing carrier and hen use a diode or envelope deecor. As illusraed in Figure 3-17, a large carrier componen is required o minimize he inheren disorion in g1 (). If he added carrier, represened by A r, has large ampliude relaive o he sideband ampliude, hen g () x() = Re{ g () + A }. 1 r I is also possible o have a single sideband signal wih ransmied carrier. Alhough his form is no power efficien, i allows he direc use of a self-homodyne diode deecor while sill conserving ransmission bandwidh. The reduced bandwidh has advanages for opical fiber ransmission where propagaion delay varies wih opical frequency. r Resulan g 1 () is he sum of carrier and USB one Added carrier Ar Ar+x() g 1 () ω m g Upper r () sideone x() g 1 () Figure 3-17 Phasor diagram showing carrier added o received upper sideone SSB demodulaion wih muliplier r() r 1 () LO = cosωc m() ~ r() r () 1 m() ~ SSB demodulaion wih diode deecor Σ + + Ar cosωc Figure 3-18 SSB demodulaion wih muliplier and wih diode deecion. Ch3-AM-6d3.doc /3/6

18 Inroducion o Communicaion Sysems A Mulimedia workbook 51 Using he Virual Experimens CD, observe he phasor diagrams, specrum analyzer and scope display for he SSB-SC demo as receiver carrier is added for he purpose of diode or envelope deecion. a) Wha envelope change do you observe as one sideband is removed. b) Is his compensaed by doubling he ransmied sideband ampliude? c) For equivalen mod. index, any difference beween DSB & SSB envelope? SSB generaion using he Phase Shif Mehod The phase shif SSB modulaor illusraed below produces a single sideband oupu wihou he use of filers o eliminae one sideband. The sine carrier c ( ) is easily produced from he cosine carrier source c () since, for sinusoidal modulaion, a similar simple phase shif can be used. In he illusraion, rigonomeric ideniies verify he producion of an upper sideone a ωc + ωm. (When boh summaion signs are posiive, he lower sideone is produced.) This modulaion concep is useful in microwave or opical sysems o frequency shif a relaively narrowband informaion signal. m() = cosω m c() = cosωc -9 º Σ -9 º m ˆ ( ) = sinω m Analyic Signals c ˆ ( ) = sinω c cos(ω c -ω m ) + cos(ω c +ω m ) DSB-SC DSB-SC + - cos(ω c -ω m ) - cos(ω c +ω m ) Figure 3-19 Phase shif mehod of generaing SSB. s() = cos(ω c +ω m ) When he informaion signal m () is compound (i.e. has many componen frequencies), a Hilber ransform filer (shaded in Figure 3-19) is employed o phase-shif all frequency componens of he signal by 9 degrees wihou changing he ampliudes. The informaion signal m () and is Hilber ransform m ( ), form he analyic signal g () = m () + jm ( ). On a doublesided frequency scale, his analyic signal has only posiive frequency componens. In digial signal processing, boh in-phase and quadraure signal componens are reained and his analyic form faciliaes digial signal processing for phase roaion or frequency shif. Analyic signals are have boh real and imaginary componens (i.e. wo signal lines are required). The analyic signal x p () is characerized by a specrum ha has only posiive frequency componens and is zero valued for all negaive frequencies. Similarly, he signal x n () is Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

19 Chaper 3: Carrier Modulaion - Analog 5 zero valued for posiive frequencies. An analyic signal is defined in erms of he real signal and is Hilber ransform. x() -9º ^x() Re Im x p () = x() + jx() ^ xp() Verical Deflecion Horizonal Deflecion xp() roaes CCW xn() would roae CW Figure 3- Componens of an analyic signal and he X-Y oscilloscope display. The analyic signal x p () can be viewed on an oscilloscope and, providing ha he frequency is low enough, he couner-clockwise roaion can be observed. Similarly, he signal x n () which has only negaive frequencies and will resul in a clockwise roaion. Example 3.7: Analyic signals Given he real signal x () = Acosωrand is Hilber ransform x ( ) = Asinω (for posiive frequencies ransform shifs by 9º or j ), wrie expressions for x p () and for x n (). r Soluion: x () = cosω + jsin ω = e p r r jω r and x () = cosω jsin ω = e n r r jω r Implemenaion of he Hilber Transform* The single sideband modulaor requires a 9-degree phase shif and his migh be implemened wih an inegraor or differeniaor (wih negaive sign). Unforunaely, hese circuis do no mainain consan gain when he frequency is changed. The required funcion is known as a Hilber ransform filer having he ransfer funcion H( f) = j sgn( f). Thus H(f) = -j for posiive frequency, H(f) = +j for negaive frequency and H(f) = for zero frequency. The impulse response for he Hilber ransform filer is given by 1 1 h () = I [ H( f)] = I [ j sgn( f)] = 1 π. If we assume he ime funcion x() wih Fourier ransform X(f), he corresponding Hilber ransformed ime funcion x ( ) is given by: x ( ) =I 1 1 [ H( f) X( f)] =I [ j sgn( f) X( f)] or x ( ) = x () h () = x () 1 π One approximaion o a Hilber ransform filer uses a apped delay line. I is reasonably accurae over a range of frequencies bu fails a low frequencies when he half period of he inpu frequency exceeds he oal ime delay in he filer and i also fails a high frequencies when he half period is less han he ime delay beween he aps (see sampling heory). A apped delay line Hilber ransform filer wih 6 aps is illusraed below. Ch3-AM-6d3.doc /3/6

20 Inroducion o Communicaion Sysems A Mulimedia workbook 53 The apped delay line can be viewed as a disribued differeniaor. The wo cenral aps conribue he difference x ( ) + x ( T d ) where x( ) is a delayed version of he inpu x(). The pair of aps ouside he cenral pair, increase he gain for signals of long period and decrease he filer gain for signals wih shor period. Thus he gain, which would increase wih frequency for he simple differeniaor, becomes uniform over a wider range of frequencies as more aps are added. h() Td Td Td Td Td 1/π x() Td 1/3π π 1/5π -1/5-1/ /3 +1/5 S 1/π -1/5π y() -1/3π Y(f) H(f)X(f) -1/π when Td 1/(f) 5Td Figure 3-1 Approximaion o Hilber ransform filer. Anoher approximaion o he Hilber ransform is he phase shif circui illusraed below and which is used for SSB ransmission of voice signals. While i does no provide he Hilber ransform of x(), i does provide wo (differenial) oupu signals ha have a 9 degree phase relaion over he voice frequency range 3 33 Hz. The circui conains 8 phase shifer sages, each affecing a differen band of frequencies and in oal covering he voice frequency range. Wihin he operaing frequency range, he four oupus are equally spaced a 9 degree inervals. Alhough m() is somewha differen han x(), he resuling SSB signal is quie saisfacory for ransmission of voice and voiceband daa. A B C D E F G H Td 3Td 5Td x() +1 1 kω 1 nf 1 kω 47 nf 1 kω 33 nf 1 kω nf 1 kω 15 nf 1 kω 1 nf 1 kω 6.8nF 1 kω 3.3 nf m()+ m()+ -1 -x() 1 kω 1 nf 1 kω 1 nf 1 kω 47 nf 1 kω 47 nf 1 kω 33 nf 1 kω 33 nf 1 kω nf 1 kω nf 1 kω 15 nf 1 kω 15 nf 1 kω 1 nf 1 kω 1 nf 1 kω 6.8nF 1 kω 6.8nF 1 kω 3.3 nf 1 kω 3.3 nf m()- m()- 1 kω 1 nf 1 kω 47 nf 1 kω 33 nf 1 kω nf 1 kω 15 nf 1 kω 1 nf 1 kω 6.8nF 1 kω 3.3 nf A B C D E F G H Figure 3- Phase Shifer wih 9 degree oupus (Couresy of Criical Telecom Inc.) Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

21 Chaper 3: Carrier Modulaion - Analog Frequency Translaion using Analyic Signals* j Muliplicaion of he analyic modulaing signal mp()= e ω m by he analyic carrier j signal cp()= e ω c jω yields he oupu sp e m jω j e c ( ω () e c+ ωm) = = where all hree signals have only posiive frequency componens. Thus he posiive frequencies of he modulaing signal are direcly ranslaed o a higher frequency as is required for SSB ransmission. In rigonomeric form, we muliply mp() = cosωm+ jsinω m by he carrier cp() = cosωc+ jsinω c o ge he produc which is also an analyic signal sp() = cosωccosωm sinωcsinωm+ jsinωccosωm+ jcosωcsinω m. The firs wo erms form he real par of he produc (he par ha would be ransmied by an anenna) and his simplifies o s () = cos( ωc + ω m) as illusraed in Figure Wih a mulifrequency modulaing signal, m (), each frequency componen mus be shifed by 9 degrees (his is where we need he Hilber ransform) o form m ( ) and permi he consrucion of he analyic signal mp () = m () + jm (). The oupu sp () = s () + js (), where s () = mc () () mc ( ) ( ) and s ( ) = mc ( ) () + mc () ( ), is calculaed by 4 mulipliers as illusraed in Figure 3-3. The use of analyic signals is especially convenien in digial signal processing (DSP) receivers. Signals are mainained in complex form during he processing and filers are no required o eliminae image frequency, double frequency or sideband componens. x() -9 º Re Im x() ^x() xp() = x() + jx() ^ x() c() ^x() ^c() + - Σ s () c() -9 º Re Im c() ^c() cp() = c() + jc() ^ x() ^c() ^x() c() + + Σ ^s () Figure 3-3 Two analyic signals and a complex muliplier Drill Problem 3.4 Placeholder - For a carrier signal c()= Vp cosϖ6 and sinusoidal modulaion, complee he following able. Refer o Example 3.3. Carrier Vp Pc (kw) µ Vp s ( ) Poal (kw) Vp/rms η (%) PEP (kw) Checksum 1 V V V V Checksum Ch3-AM-6d3.doc /3/6

22 Inroducion o Communicaion Sysems A Mulimedia workbook VESTIGIAL SIDEBAND MODULATION (VSB-TC) Vesigial sideband ransmission is a widely used sandard for broadcas and cable elevision (TV). I is a compromise ha combines he benefis of DSB-TC wih he bandwidh conservaion of SSB. Vesigial sideband (VSB) signals are relaively easy o generae and have only slighly greaer (5-15%) bandwidh han an equivalen SSB signal. The broadcas elevision signal has one full sideband plus a small vesige of he oher sideband. A low frequencies, where componens from boh sidebands are received, ransmission is equivalen o DSB wih excellen reproducion of he low frequency componens ha are essenial for TV receiver synchronizaion. As wih oher forms of DSB ransmission, he receiver requires a coheren local oscillaor and hus broadcas elevision ransmis a large carrier o allow envelope deecion in he receiver. Picure (AM) Filer Picure MHz MHz V Hz 4. f (MHz) Sound (FM) Picure (AM) Filer Sound Picure (VSB) Broadcas Signal Channel - Picure (VSB) Sound Sound Picure Picure Received & Filered Signal Sound Sound +11 MHz Picure Picure Sound MHz Picure MHz MHz Picure Picure Diode Demodulaed Signal Received Picure Signal Figure 3-4 Modulaion and demodulaion in VSB broadcas elevision. Receiver IF Signal Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

23 Chaper 3: Carrier Modulaion - Analog 56 In Figure 3-4, a - 4. MHz baseband elevision picure signal is used o DSB-TC modulae a channel carrier of frequency 55.5 MHz. The lower sideband of he resuling signal is runcaed by removing componens less han 54 MHz. The vesige of he lower sideband is aenuaed saring.75 MHz below he carrier and is compleely aenuaed a frequencies more han 1.5 MHz below he carrier. The associaed elevision sound signal, which frequency modulaes a carrier cenered a MHz, is added o he VSB picure signal prior o broadcas. A he receiver, a filer is applied o specrally shape he signal such ha M( f fc) + M( f + fc) = MDSB( f) where he specral densiy of he equivalen DSB signal is represened by MDSB ( f). Wih his shaping, he demodulaed video signals overlap a baseband o provide uniform ransmission gain hroughou he frequency range. 3.6 QUADRATURE AMPLITUDE MODULATION (QAM) By using quadraure ampliude modulaion (QAM), wo independen modulaion signals can be ransmied in he same bandwidh ha would be used by a single DSB ransmission. The QAM scheme provides specral efficiency equal o SSB bu i does no require accurae filers or he use of he Hilber ransform.. For he majoriy of recenly developed sysems, QAM has been he choice. We assume uni ampliude informaion signals x() and y() and modulaion indicies µ i and µ q such ha mi() = µ ix() and mq() = µ qy(). Two baseband signals are separaed a he receiver hrough synchronous deecion using wo quadraure phase carriers. QAM modulaion and demodulaion are illusraed in Figure mi() c() = cosωc mq() sinωc -9 º [1+ mi()]cosωc DSB-TC DSB-SC + Σ + mq()sinωc s() [1+ mi()](1+cosωc) +mq()sinωc r() C.R. -9 º cosωc sinωc mq()(1- cosωc) + [1+ mi()]sinωc ri() = 1+mi() rq() = mq() Figure 3-5 QAM Transmier and Receiver. A he receiver, a carrier recovery circui develops a demodulaor carrier ha is ideally equal o he ransmier carrier in frequency and in phase (i.e. coheren). Phase error (i.e. θ ) in he demodulaor carrier (LO) will resul in crossalk beween he wo channels. We assume no channel aenuaion and ha he received signal is idenical o he ransmied signal. r () = s () = ( 1+ mi()cos ) ωc+ mq()sin ωc In he upper (in-phase) porion of he receiver, he local oscillaor (wih phase error) is cos( ωc + θ) = ( cosθcosωc sinθsinωc). The produc demodulaor oupu hen becomes [( 1 + mi()cos ) ωc+ mq()sin ωc ] ( cosθcosωc sinθsinω c). Ch3-AM-6d3.doc /3/6

24 Inroducion o Communicaion Sysems A Mulimedia workbook 57 This produc has erms a wice he carrier frequency plus erms a baseband. Afer he low-pass filer, only baseband erms remain in he in-phase oupu r i (). [ ] [ ] ri() = 1+ mi()cos θ mq()sin θ In he lower or quadraure porion of he receiver, he local oscillaor (wih phase shif) is sin( ωc + θ) = ( cosθsinωc sinθcosωc) and he produc demodulaor oupu becomes [( 1 + mi()cos ) ωc+ mq() Acsin ωc] ( cosθsinωc+ sinθcosω c) Afer he low-pass filer, his resuls in [ ] rq() = [ mq()cos ] θ mi()sin θ. When θ =, he wo signals can be ransmied independenly over he same bandwidh. When here is a phase error, he in-phase oupu includes some of he quadraure signal and vice versa. Example 3.8: Assume a QAM sysem wih receiver phase error φ =.1 rad = 5.7 Calculae he crossalk, X(dB), beween channels. Noe ha ri() = 1+ mi()cos θ mq()sin θ [ ] [ ] Soluion: XdB ( ) = log ( sin θ / cosθ)= log (. 993/. 995)=. db 1 1 A widespread example of QAM ransmission is found in sereo AM broadcas. The block diagram of Figure 3-6 incorporaes he same in-phase and quadraure modulaors as in Figure 3-5. A he receiver, he local oscillaor is derived from a phase locked loop (PLL) ha exracs he carrier componen from he received signal. To mainain compaibiliy wih exising AM receivers, broadcas QAM uilizes wo auomaic gain conrol (AGC) elemens. A he ransmier, he gain conrol loop scales back he resulan ampliude so ha is envelope is equal o he in-phase modulaion signal 1+L+R. Exising AM receivers can demodulae he signal (as usual) wih an envelope deecor. A sereo AM receiver envelope deecs he AM signal hen uses a feedback conrol loop o scale up he QAM signal prior o demodulaion. The amoun of downscaling and upscaling is dependen on he magniude of he L-R signal and ideally he produc of he wo operaions (αβ) is uniy. 1 + L + R (1 + L + R) 1 + L + R DSB-TC Envelope Deecor Envelope Deecor c L - R -9 DSB-SC AGC AGC PLL -9 (L - R) Figure 3-6 Sereo AM sysem. Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6

25 Chaper 3: Carrier Modulaion - Analog PHASE MODULATION (PM) Improved immuniy o noise is he principle advanage of phase modulaion (PM) or frequency modulaion (FM). We begin wih a sudy of phase modulaion since his can be direcly compared wih noise immuniy in AM. Phase modulaion is widely used in digial modulaion (such as QPSK), however, i is rarely used in analog sysems because FM is easier o demodulae. A well-known example is broadcas FM which uses frequency modulaion for signal frequencies up o.1 khz, however, phase modulaion is used for remaining signal frequencies up o 15 khz since PM has superior performance. In PM and FM (ogeher known as angle modulaions), we increase ransmission bandwidh and gain he benefi of improved noise performance - somehing ha was no possible wih AM. Wih phase modulaion, carrier phase varies direcly wih he baseband signal volage m(); he carrier ampliude remains consan. We define a modulaion coefficien kp in unis of radians/vol so carrier phase θ() = kpm() and he ransmied signal s() is hen s() = A c cos [ωc + θ()] = A c cos [ωc + kpm()]. We also define a modulaion index β p ha indicaes he peak phase deviaion θ p (in radians). Phasor Diagram β p = Time Waveforms βp = (one radian) 57.3 π θ () V -π (- one radian) Figure 3-7 Two examples of wideband phase modulaion (second is adaped from Couch-6). Drill Problem Phase Modulaion In he waveforms illusraed in he above righ-hand figure, assume ha he carrier frequency is 1 MHz, he carrier ampliude is 5 vols and he peak modulaion volage is 1.5 vols. Answer he following quesions wih a precision of decimal places. a) Deermine kp, he gain coefficien of phase modulaion. b) Deermine he phase advance (in radians) when he modulaion volage is +.75 V. c) Wha is he modulaion frequency (in MHz)? Checksum Ch3-AM-6d3.doc /3/6

26 Inroducion o Communicaion Sysems A Mulimedia workbook Narrowband Phase Modulaion When he modulaion index β is small compared o one radian, we have narrowband phase modulaion. A hese low levels of modulaion, he PM specrum is composed primarily of carrier plus wo firs order sidebands. The specrum resembles ha of ampliude modulaion and, for small phase deviaions, a good approximaion o PM is obained by summing an in-phase carrier wih componens generaed by DSB-SC modulaion of a quadraure carrier. Figure 3-8 illusraes he case when sinusoidal modulaion is applied; he small amoun of incidenal ampliude variaion in he oupu s() is negleced. V cos ω c c() Narrowband Phase Modulaor + + s() Phasor Diagram βp =.1 (no o scale) Sum.1 cos ω m m() θ = -9 x xy y 5.7 max Resulan Lower fm side one Carrier ( vols) fm Upper side one (1 vol) Figure 3-8 Narrowband phase modulaion (β =.1). In Figure 3-8 i is insrucive o consider he alernae case where here is no phase shif and he sideband componens add in-phase wih he carrier resuling in ampliude modulaion. In hese wo cases, he AM index µ is numerically equal o he PM index β p. Early angle modulaion sysems used narrowband angle modulaion as a base and generaed wideband modulaion hrough frequency muliplicaion (i.e. using non-linear 3 rd or 5 h harmonic generaion followed by filers). In his indirec mehod, he modulaion index increased in proporion o he muliplicaion facor Wideband Phase Modulaion The appeal of phase modulaion lies in is abiliy o improve SNR in noisy channels - bu his occurs only for large modulaion indices (β p > 1) and his can grealy increase ransmission bandwidh. We iniially digress from his noise reducion objecive o sudy PM specral properies. Phase modulaion is a non-linear modulaion process and he specrum of a PM signal is no a simple replica of he modulaing signal specrum - paricularly a higher levels of modulaion. To sudy specra of wideband phase modulaion, we reurn o he mahemaical expression s() = A c cos [ωc + kpm()]. This can also be expressed as s()= Re{A c e jθ() e jω c } wih complex envelope g() = A c e jθ() where θ() = k p m(). As in ampliude modulaion, he baseband specrum of he complex envelope ceners abou ωc and we can simply focus on specral analysis Ch3-AM-6d3.doc D.E. Dodds, Univ. of Saskachewan, Canada /3/6