Amplitude Modulation

Amplitude ModulationAng Man ShunOctober 30, 2012ReferenceHwei P. Hsu Analog and Digital tone Am cos ωm tLarge Carrier Ac cos ωc tCarrierUnitySimple AMDSB-LCGeneral(Ac m(t)) cos ωc t Ac cos ωc t m(t) cos ωc tMonotone Ac cos ωc t mAcmAccos (ωc ωm ) t cos (ωc ωm ) t22GeneralDSB-SCMonotoneSSB / VSBSimple AMDSB-LCDSB-SCSSB/VSBcos ωc tm(t) cos ωc tmAcmAccos (ωc ωm ) t cos (ωc ωm ) t22mAccos (ωc ωm ) tMonotone2BandwidthAdvantageDisadvantage2ωm Low η2ωmHighη ωmHigh ηSmall BW 1

1IntroductionMessage signal and Carrier Signal in time domain Monotone General Unity Carrier1 jωc t(e e jωc t )2 Larger Carrier Ac cos ωc t Ac (ejωc t e jωc t )2Am jωm t(e e jωm t )Am cos ωm t 2m(t)cos ωc t 2 frequency component in same amplitude Negative frequency is not real, just mathematical ωc ωm , if not ,Aliasing1.1Continuous Wave ModulationCW modulation is just a multiplication, for montone message and larger carrier :(Am cos ωm t) (Ac cos ωc t)By identity cos A cos B cos (A B) cos (A B)2}Am AcAm Ac { j(ωc ωm )t{cos (ωc ωm ) t cos (ωc ωm ) t} e e j(ωc ωm )t ej(ωc ωm )t e j(ωc ωm )t24 One cosine term has 2 frequency spectral line, so 2 cosine terms have 4 spectral line Frequency spectra line translated Negative frequency is not real, it is only mathematical, so although it has 4 spectra, it actuallyhas only 2 spectra1.2The signal’s Fourier TransfromsFor general messageFm(t) M (jωm )CW Modulated signal of general message with large carrierm(t)Ac cos ωc t () F AcAcm(t) ejωc t e jωc t [M (jωm jωc ) M (jωm jωc )]222

2Amplitude ModulationThere are 3 to 4 basic types of AM modulation : Simple AM (DSB-LC) , DSB-SC, SSB, and VSB2.1Simple AM signal in Time DomainAM : Message signal “embed” into the amplitude of carrierSimple AM is also called : Double Sideband Large Carrier (DSB-LC) ModultionFor general message signal m(t) :xAM (t) (Ac m(t)) cos ωc tFor simple monotone message signal m(t) Am cos ωm t :xAM (t) (Ac Am cos ωm t) cos ωc t Ac cos ωc t Am cos ωm t cos ωc t Ac cos ωc t Ac m cos ωm t cos ωc tBy cos A cos B where m AmAccos (A B) cos (A B)2mAc Ac cos ωc t [cos (ωc ωm ) t cos (ωc ωm ) t]2mAcmAccos (ωc ωm ) t cos (ωc ωm ) t22Simple AM / DSB-LC modulated signal is thus Ac cos ωc t xAM (t) (Ac m(t)) cos ωc t Ac cos ωc t m(t) cos ωc t mAcmAc cos (ωc ωm ) t cos (ωc ωm ) t xAM (t) (Ac m(t)) cos ωc t Ac cos ωc t 22 Envelope of modulated signal will follow the message signal m Am100% m 100% Am AcAc m 100% overmodulation Am Ac Requirements for AM / DSB-LC : Ac Am , ωc ωm3GeneralMonotoneMessage

2.2Simple AM signal in Frequency DomainFor general message signalFm(t) M (jωm )Bandwidth [0 , ωm ] ωmThe simple AM signalxAM (t) (Ac m(t)) cos ωc t Ac cos ωc t m(t) cos ωc tThe AM signal’s Fourier TransformF {xAM (t)} F {Ac cos ωc t} F {m(t) cos ωc t}()}} 1 {}1 {1 { Ac F {cos ωc t} F m(t) ejωc t e jωc t Ac F {cos ωc t} F m(t)ejωc t F m(t)e jωc t222ˆ Ac jωtcos ωc teFˆ ˆ m(t)e j(ω ωc )t 1dt 2ˆ m(t)e j(ω ωc )t dt Fthus m(t)e jat M (jω ja)As m(t) M (jωm )Ac 21dt 2()11e j(ω ωc )t e j(ω ωc )t dt M (jω jωc ) M (jω jωc )22XAM (jω) 11Acδ (ω ωc ) M (jω jωc ) M (jω jωc )222 Bandwidth is twice of original bandwidth : BWAM 2W 2ωm AM wave contains 2 sideband with bandwidth of each band as WLSB WU SB W ωm4

2.3Double-Sideband Suppressed-Carrier Modulation DSBSCRecall, simple AM / DSB-LC for m(t) Am cos ωm t xAM (t) (Ac m(t)) cos ωc t Ac cos ωc t m(t) cos ωc t General mAcmAc xAM (t) Ac cos ωc t 2 cos (ωc ωm ) t 2 cos (ωc ωm ) tMonotoneMessage Carrier contains no info, so suppresse it to enhance power efficiencyAfter dropping the carrier term, xDSB SC (t) m(t) cos ωc t m(t) cos ωc t General mAcmAc xDSB SC (t) 2 cos (ωc ωm ) t 2 cos (ωc ωm ) tMonotoneMessageFFor general message signal m(t) M (t) , the Fourier Transform is{}F x(t)DSB-SC F {m(t) cos ωc t} F2.4{ejωc t e jωc tm(t)2} Generation & Demodulation of DSB signal51[M (jω jωc ) M (jω jωc )]2

For Simple AM / DSB-LC After mixing : xAM (t) cos ωc t [Ac m(t)] cos2 ωc t [Ac m(t)] After passing LPF :Ac m(t)1 cos 2ωc tA m(t) ccos ωc t222Ac m(t) 22 After passing capacitor to block the DC :m(t)2 Thus, with suitable amplifier, the original signal can be recovered. * Then this demodulator works AcSince carrier is suppressed, so Am 0 , and thus m for DSB-SC signal.2.5Am so modulation index is meaninglessAcSingle-Sideband Suppressed CarrierTo improve power efficiency, further dropping one sidebandFor monotone signal, the DSB-SC signal isxDSB SC (t) mAcmAccos (ωc ωm ) t cos (ωc ωm ) t22After dropping one sideband After LPF LSB After HPF HSBxSSB LSB (t) mAccos (ωc ωm ) t2xSSB U SB (t) mAccos (ωc ωm ) t2Vestigial Sideband SignalSince SSB require a very sharp cut-off filter to remove one sideband, such filter is not easy toimplementThus, the requirement is relaxed by allowing a vestige part : Vestigial Sideband signal6

3Power of the AM Signal3.1Review of Root-Mean-Square valueFor a function A cos ωt , the RMS value is : ˆ Tˆ Tˆ111 T 1 cos 2ωt22(A cos ωt) dt Acos ωtdt AdtRM S (A cos ωt) T 0T 0T 02Since cos θ is orthogonal to 1, so the second integral is zerov ˆ ˆ TuT1Au1 T 1cos 2ωtdt Adt AutT 0 22T 02T2{z} 0ASo the RMS value of a function of A cos ωt is 2Then , recall that simple AM / DSB-LC signal of montone message has the formxAM (t) (Ac m(t)) cos ωc t Ac cos ωc t Ac mAc mcos (ωc ωm ) t cos (ωc ωm ) t22Then the RMS value isAcAc m Ac mxAM,RM S 2 2 22 2Thus the total power , by P A2, isRA2cA2 m2 A2c m2 c 2R8R8RThe power used to transmitte information for simple AM is thus :PT,AM ηAM PInf oPTA2c m2 A2c m2m2 m2 m22m244 2 8R 2 2 8R 2 2 AcAmAmm2 m24 2m22 m2 c c1 2R8R8R44When m 121 66.6%Power Lost33Therefore, simple AM signal is not power-efficienct.ηAM ηAM 33.3% sup ηAMIn summary, for simple AM / DSB-LC signal, The efficiency is limited to 33% The carrier signal is present even if nothing is being transmitted The circuitary is relatively simple (only envelop detector is required ! ) Bandwidth is 2ωm7

For Dobule Sideband Suppressed Carrier of montone message , the wave form isxDSB (t) xAM (Without Carrier) Ac mAc mcos (ωc ωm ) t cos (ωc ωm ) t22Thus, the RMS value isAc m Ac mxDSB,RM S 2 22 2Thus, the Total power isPT,DSB A2c m2 A2c m2 8R8RAnd hence, the power efficiency isPTPInf o 100% (Ideal)PTPTThe power efficiency of DSB singal is very good, but the tradeoff is it require relatively expensivecircuitry in the receiverηDSB In summary, for DSB signal It have must higher power efficiency ( 100% ) But it has same bandwidth as simple AM , 2ωm It require relatively expensive circuitry in the receiverFor SSB singal, a sideband filter, either high pass or low pass, is concatenated to the receiver circuit.For Single Sideband Suppressed Carrier of montone message, the wave form isxSSB (t) Ac mcos (ωc ωm ) t2Thus, the RMS value isAc mxDSB,RM S 2 2Thus, the Total power isPT,SSB A2c m28RAnd hence, the power efficiency isηSSB PTPInf o 100% (Ideal)PTPTIn summary, for SSB signal, It has high power efficiency ( 100% ) It has relatively most expensive circuitry ( An extra sideband filter ) It cut bandwidth in half, BWSSB ωm EN D 8

2 Amplitude Modulation There are 3 to 4 basic types of AM modulation : Simple AM (DSB-LC) , DSB-SC, SSB, and VSB 2.1 Simple AM signal in Time Domain AM : Message signal “embed” into the amplitude of carrier Simple AM is also called : Double Sideband Large Carrier (DSB-LC) Modultion Fo