1Lecture 2829FM Frequency ModulationPM Phase ModulationEE445102FM and PMDp is the phase sensitivity or phase modulation constant3FM and PMfor FM:Relationship between mf(t) and mp(t):4Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–8 Angle modulator circuits. RFC = radiofrequency choke.5FM and PM6Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–9 FM with a sinusoidal baseband modulating signal.7FM and PM8FM and PM differencesθ (t) = Dpm(t) ⇒ phase is proportional to m(t)instantaneous frequencydeviation from thecarrier is proportional to m(t)θ (t) = D f ∫−t∞ m(α)dαPM:FM:fi(t) − fc = D f m(t) ⇒voltHzD KvoltradiansD Kf fp p= ⇒= ⇒ModulationConstants9FM from PMPM from FM10FM from PMPM from FM11FM and PM SignalsMaximum phase deviation in PM:Maximum frequency deviation in FM:12ExampleLetFor PMFor FMDefine the modulation indices:13ExampleDefine the modulation indices:14Sine Wave ExampleThen15Spectrum Characteristics of FM• FMPM is exponential modulationLetRe( ) (2 sin(2 ))( ) cos(2 sin(2 ))j f t f tcc c mA e c mu t A f t f tπ β ππ β π+== +φ(t) = β sin(2πfmt)u(t) is periodic in fmwe may therefore use the Fourier series16Spectrum Characteristics of FM• FMPM is exponential modulationRe( ) (2 sin(2 ))( ) cos(2 sin(2 ))j f t f tcc c mA e c mc t A f t f tπ β ππ β π+== +c(t) is periodic in fmwe may therefore use the Fourier series17Spectrum Characteristics withSinusoidal Modulationu(t) is periodic in fmwe may therefore use the Fourier series18Jn Bessel Function19Jn Bessel Function20Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.stop 32921Lecture 29FM Frequency ModulationPM Phase Modulation(continued)EE4451022Narrowband FM•Only the Jo and J1 terms are significant•Same Bandwidth as AM•Using Eulers identity, and φ(t) fm, increasing fm does not increase Bc much•Bc is linear with fm for PM26Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.27Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.28Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.29When m(t) is a sum of sine waves30When m(t) is a sum of sine waves31Sideband PowerSignal Amplitude: Ac := 1VModulating frequency: fm := 1KHzCarrier peak deveation: Δf 2.4KHz :=Modulation index: β Δffm:=β = 2.4Reference equation: x t ( )− ∞∞nA∑ ⎡ ⎣ c⋅Jn n ( ) ,β ⋅cos⎡ ⎣( ) ωc + n⋅ωm ⋅t⎤ ⎦⎤ ⎦=Power in the signal: PcAc22 1 ⋅ Ω:= Pc = 0.5 WCarsons rule bandwidth: BW 2 := ⋅( ) β + 1 ⋅fm BW 6.8 10 × 3 1s=Order of significant sidbands predicted by Carsons rule: n round := ( ) β + 1n 3 =Power as a function of number of sidebands: Psum( ) k− kknA( ) c⋅Jn n ( ) ,β 2∑ 2 1 ⋅ Ω=:=Percent of power predicted by Carsons rule:Psum( ) nPc⋅100 = 99.11832Sideband Power0 0.5 1 1.5 2 2.5 3050100PERCENT OF TOTAL POWERPsum( ) kPc⋅100k33Sideband Powerk 0 10 := ..Jk := β Jn k ( ) ,β = 2.4Pk := ( ) Jk 2n 3 =P0 21nj∑ Pj=+ ⋅ = 0.991J00 1 2 3 4 5 6 7 8 9102.508·1030.520.4310.1980.0640.0163.367·1035.927·1049.076·1051.23·1051.496·106= P00 1 2 3 4 5 6 7 8 9106.288·1060.2710.1860.0394.135·1032.638·1041.134·1053.513·1078.237·1091.513·10102.238·1012=34Sideband Powerj 0 5 := β .. := 0.1n 1 :=Vj := Jn j ( ) ,βUj := ( ) Vj 2U0 21nj∑ Uj=+ ⋅ = 1V0.9980.051.249 10 × − 32.082 10 × − 52.603 10 × − 72.603 10 × − 9⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠= U0.9952.494 10 × − 31.56 10 × − 64.335 10 × − 106.775 10 × − 140⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=35Sideband Powerβ := 0.6 n 1 :=Wj := Jn j ( ) ,β Xj := ( ) Wj 2X0 21nj∑ Xj=+ ⋅ = 0.996W0.9120.2870.0444.4 10 × − 33.315 10 × − 41.995 10 × − 5⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠= X0.8320.0821.907 10 × − 31.936 10 × − 51.099 10 × − 73.979 10 × − 10⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=36Bandwidth 18 = Modulation_index 7.9 =f fB f ( ) := δ⎡ ⎣f f , c + ( ) Fm n 0 + ⋅ ⎤ ⎦ + δ( ) f f , c − n Fm ⋅ := c − ( ) Fm n 1 + ⋅ ,( ) fc − n Fm ⋅ ..⎡ ⎣fc + ( ) Fm n 1 + ⋅ ⎤ ⎦Si f ( ) Ac ( ) J0 M ( ) ⋅δ( ) f f , c1nk∑ ⎡ ⎣Jn k M ( ) , ⋅δ⎡ ⎣f f ,( ) c + k Fm ⋅ ⎤ ⎦ + ( ) −1 k⋅ δ Jn k M ( ) , ⋅ ⎡ ⎣f f ,( ) c − k Fm ⋅ ⎤ ⎦⎤ ⎦=+⎡⎢⎢⎣⎤⎥⎥⎦:= ⋅n 9 = Bandwidth 2 n := ⋅ ⋅Fm Modulation_index M :=n roundM 1 := ( ) + 2 n is the number of significant sidebands per Carsons ruleMx10:=Fm 10 := 0 Modulating frequency single sinewavefc:= 0 10 ⋅ 4Ac:= 179FMPM modulation index : set toπ2 for peakphase dev of π2set toΔffm for frequency modulation. spectruis the same for sinewavemodulation.filename: fmsidebands.mcdavo 092104last edit date:2270737M=.4, Sideband Level =M2 for Narrowband FM0.2 2 1 0 1 200.20.40.60.8SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index38M=.9, Sideband Level =M2 for Narrowband FM0.5 3 2 1 0 1 2 300.51SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index39M=2.4, Carrier Null0.6 4 2 0 2 40.40.200.20.40.6SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index40M=3.8, first sideband null0.6 6 4 2 0 2 4 60.40.200.20.40.6SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index41M=5.1, second sideband null0.4 8 6 4 2 0 2 4 6 80.200.20.4SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index42Power vs BW, M=0.1second term includes power in +Jnand power in Jn, i.e the upper andlower sideband pairsP M n ( ) , J0 M ( )221nk∑ Jn k M ( ) , 2=+⎛⎜⎜⎝⎞⎟⎟⎠:=99.999651 1.2 1.4 1.6 1.899.999799.9997599.999899.9998599.999999.99995% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 0.1 =Fm 1 = HzBandwidth 2 = HzP M n ( ) ,Ac22⋅100 = 10043Power vs BW, M=0.9⎝ ⎠98 1 1.5 2 2.5 3 3.5 498.59999.5100% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 0.9 =Fm 1 = HzBandwidth 4 = HzP M n ( ) ,Ac22⋅100 = 99.95844Power vs BW, M=2.4⎝ ⎠50 1 2 3 4 5 6 7 860708090100% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 2.4 =Fm 1 = HzBandwidth 8 = HzP M n ( ) ,Ac22⋅100 = 99.94545Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–16 Anglemodulated system with preemphasis and deemphasis.46Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–16 Anglemodulated system with preemphasis and deemphasis.47AM vs FM• FM capture effect: A phenomenon, associated with FMreception, in which only the stronger of two signals at ornear the same frequency will be demodulated.– The complete suppression of the weaker signal occurs at thereceiver limiter, where it is treated as noise and rejected.– When both signals are nearly equal in strength, or are fadingindependently, the receiver may switch from one to the other.• Bandwith: BAM=2 x fm BFM >= 2 x fm use Carson’s Rule• The Receiver IF amplifier is change to a LimitingAmplifier for FM– FM rejects amplitude noise such as lightening and man madenoise• The FM demodulator may be a PLL, Ratio Detector,Foster Sealy Discriminator, or slope detector.
FM and PM for FM: Lecture 28-29 FM- Frequency Modulation PM - Phase Modulation Relationship between mf(t) and mp(t): EE445-10 Figure 5–8 Angle modulator circuits RFC = radio-frequency choke FM and PM Dp is the phase sensitivity or phase modulation constant Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 FM and PM FM and PM Figure 5–9 FM with a sinusoidal baseband modulating signal FM and PM differences PM: θ (t ) = D p m(t ) ⇒ phase is proportional to m(t) FM: t θ (t ) = D f ∫ m(α )dα −∞ f i (t ) − f c = D f m(t ) ⇒ Modulation Constants Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 instantaneous frequency deviation from the carrier is proportional to m(t) radians volt Hz Df = K f ⇒ volt Dp = K p ⇒ FM from PM PM from FM FM and PM Signals Maximum phase deviation in PM: Maximum frequency deviation in FM: 11 FM from PM PM from FM Example Let For PM For FM Define the modulation indices: 10 12 Example Spectrum Characteristics of FM Define the modulation indices: • FM/PM is exponential modulation Let φ (t ) = β sin(2πf mt ) u (t ) = Ac cos(2πf c t + β sin(2πf mt )) ( = Re Ac e j ( 2πf ct + β sin( 2πf mt )) ) u(t) is periodic in fm we may therefore use the Fourier series 13 Sine Wave Example 15 Spectrum Characteristics of FM Then • FM/PM is exponential modulation c(t ) = Ac cos( 2πf c t + β sin(2πf mt )) ( = Re Ac e j ( 2πf ct + β sin( 2πf mt )) ) c(t) is periodic in fm we may therefore use the Fourier series 14 16 Spectrum Characteristics with Sinusoidal Modulation Jn Bessel Function u(t) is periodic in fm we may therefore use the Fourier series 17 19 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes Jn Bessel Function stop 3/29 18 Couch, Digital and Analog Communication Systems, Seventh Edition 20 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 Narrowband FM as a Phaser Lecture 29 FM- Frequency Modulation PM - Phase Modulation (continued) AM EE445-10 NBFM 21 23 Wideband FM from Narrowband FM Narrowband FM •Only the Jo and J1 terms are significant •Same Bandwidth as AM •Using Eulers identity, and φ(t) fm, increasing fm does not increase Bc much •Bc is linear with fm for PM 25 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes Couch, Digital and Analog Communication Systems, Seventh Edition Couch, Digital and Analog Communication Systems, Seventh Edition 27 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes 26 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 Couch, Digital and Analog Communication Systems, Seventh Edition 28 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 Sideband Power When m(t) is a sum of sine waves Signal Amplitude: Ac := 1V Modulating frequency: fm := 1KHz Δf := 2.4KHz Carrier peak deveation: Modulation index: β := Δf β = 2.4 fm ∞ Reference equation: x( t) ⎡⎣Ac ⋅Jn( n , β ) ⋅cos ⎡⎣( ω c + n⋅ω m) ⋅t⎤⎦⎤⎦ ∑ n= −∞ Power in the signal: Carsons rule bandwidth: Pc := Ac Pc = 0.5 W ⋅1Ω BW := ⋅( β + 1) ⋅fm BW = 6.8 × 10 k Power as a function of number of sidebands: s n := round( β + 1) Order of significant sidbands predicted by Carsons rule: Psum( k) := ∑ ( Ac⋅Jn( n , β ) ) 2 ⋅1 Ω n= −k 29 Percent of power predicted by Carsons rule: Psum( n) Pc n=3 31 ⋅100 = 99.118 Sideband Power When m(t) is a sum of sine waves PERCENT OF TOTAL POWER 100 P sum( k) Pc ⋅100 50 30 0.5 1.5 k 2.5 32 Sideband Power k := 10 Jk := Jn( k , β ) Pk := ( Jk) J= Sideband Power β := 0.6 β = 2.4 W j := Jn( j , β ) n=3 0 2.508·10-3 0.52 6.288·10-6 0.271 0.431 0.186 0.198 0.064 0.039 4.135·10-3 P= 0.016 2.638·10-4 3.367·10-3 1.134·10-5 5.927·10-4 3.513·10-7 9.076·10-5 1.23·10-5 8.237·10-9 1.513·10-10 10 1.496·10-6 10 2.238·10-12 n P0 + ⋅ n := ∑ P j = 0.991 j= X j := ( W j) 0.832 ⎛⎜ ⎟⎞ 0.082 ⎜ ⎟ ⎜ −3 ⎟ ⎜ 1.907 × 10 ⎟ X=⎜ −5 ⎟ ⎜ 1.936 × 10 ⎟ ⎜ −7 ⎟ ⎜ 1.099 × 10 ⎟ ⎜ 3.979 × 10− 10 ⎟ ⎝ ⎠ ⎞ ⎛ 0.912 ⎜ ⎟ ⎜ 0.287 ⎟ ⎜ 0.044 ⎟ ⎜ ⎟ −3 W= ⎜ 4.4 × 10 ⎟ ⎜ −4⎟ ⎜ 3.315 × 10 ⎟ ⎜ −5⎟ ⎝ 1.995 × 10 ⎠ n X0 + ⋅ ∑ Xj = 0.996 j= 33 35 filename: fmsidebands.mcd avo 09/21/04 last edit date:2/27/07 Sideband Power Ac := 79 j := fc := 0⋅10 β := 0.1 Vj := Jn( j , β ) n := Fm := 10 Uj := ( V j) M := 10 ⎛⎜ ⎞⎟ 0.05 ⎜ ⎟ ⎜ −3⎟ ⎜ 1.249 × 10 ⎟ V=⎜ −5⎟ ⎜ 2.082 × 10 ⎟ ⎜ −7⎟ ⎜ 2.603 × 10 ⎟ ⎜ 2.603 × 10− ⎟ ⎝ ⎠ ⎛⎜ ⎞⎟ ⎜ 2.494 × 10− ⎟ ⎜ ⎟ ⎜ 1.56 × 10− ⎟ U=⎜ ⎟ ⎜ 4.335 × 10− 10 ⎟ ⎜ ⎟ − 14 ⎜ 6.775 × 10 ⎟ ⎜ ⎟ ⎝ ⎠ n U0 + ⋅ ∑ ⎡ Uj = j= * n is the number of significant sidebands per Carsons rule Bandwidth:= 2⋅n⋅Fm n= 0.995 Modulating frequency- single sinewave x n := round(M + 1) 0.998 FM/PM modulation index : set toπ /2 for peak phase dev ofπ /2 set toΔ f/fm for frequency modulation spectru is the same for sinewave modulation ⎢ ⎣ n ( ) ∑ Si(f) := Ac ⋅⎢ (J0(M)) ⋅δ f, fc + k=1 ( ⎤ ⎡ Jn(k , M) ⋅δ ⎡f, f + k⋅Fm ⎤ + (−1)k⋅Jn(k , M) ⋅δ ⎡f, f − k⋅Fm ⎤ ⎤ ⎥ )⎦ )⎦ ⎦ ⎥ ⎣ ⎣ (c ⎣ (c ⎦ ) B(f) := δ ⎡ f, fc + (n + 0) ⋅Fm⎤ + δ f, fc − n⋅Fm ⎣ Bandwidth= 18 34 ⎦ Modulation_index := M ( ) f := fc − (n + 1) ⋅Fm, fc − n⋅Fm ⎡ fc + (n + 1) ⋅Fm⎤ ⎣ ⎦ Modulation_index = 7.9 36 M=.4, Sideband Level =M/2 for Narrowband FM M=2.4, Carrier Null Single Sided Spectrum Bessel Functions Single Sided Spectrum Bessel Functions 0.6 Modulation_index 0.8 Modulation_index 0.8 0.8 0.4 0.6 0.6 0.4 0.6 Peak Volts 0.2 0.2 0.4 0.2 0.2 0.4 0.4 0.6 0.2 0.6 0.8 0.2 0.2 Peak Volts 0.4 0.2 Carrier J0 1st Sidebands J1 2nd Sidebands J2 0.4 0.8 10 1 10 0.6 Carrier J0 1st Sidebands J1 2nd Sidebands J2 Spectrum 2 Spectrum 37 39 M=.9, Sideband Level =M/2 for Narrowband FM M=3.8, first sideband null Single Sided Spectrum Bessel Functions Single Sided Spectrum Bessel Functions 1 0.6 Modulation_index 0.8 0.8 0.6 0.6 0.4 0.4 Modulation_index 0.4 0.5 0.2 0.4 0.2 0.4 0.6 0.2 0.6 0.8 Peak Volts 0.2 0.2 Peak Volts 0.2 0.4 0.8 Carrier J0 1st Sidebands J1 2nd Sidebands J2 10 0.5 1 Spectrum 38 Carrier J0 1st Sidebands J1 2nd Sidebands J2 10 0.6 2 Spectrum 40 M=5.1, second sideband null Power vs BW, M=0.9 ⎝ ⎠ M = 0.9 % power vs bandwidth Fm = 100 Bessel Functions Single Sided Spectrum 0.4 n Hz Bandwidth= Hz Modulation_index 0.8 0.6 99.5 0.2 P (M , n) 0.4 Peak Volts 0.2 0.2 P ( M , k) Ac Ac ⋅100 ⋅100 = 99.958 99 0.4 0.2 0.6 98.5 0.8 10 0.4 Carrier J0 1st Sidebands J1 2nd Sidebands J2 2 Spectrum 98 1.5 2.5 3.5 k Number of Sideband pairs 41 43 Power vs BW, M=0.1 ⎛ J0(M)2 P(M , n) := ⎜ ⎜ ⎝ n + ∑ ⎞ Jn(k , M) ⎟ ⎟ ⎠ k=1 Power vs BW, M=2.4 ⎝ ⎠ M = 2.4 second term includes power in +Jn and power in -Jn, i.e the upper and lower sideband pairs % power vs bandwidth % power vs bandwidth Fm = n n Hz Bandwidth= Hz Hz 90 Bandwidth= Hz 99.99995 Fm = 100 M = 0.1 P (M , n) 99.9999 P ( M , k) Ac 2 P (M , n) P ( M , k) ⋅100 = 100 Ac Ac ⋅10099.99985 ⋅100 Ac 80 ⋅100 = 99.945 70 99.9998 99.99975 60 99.9997 99.99965 50 1.2 1.4 1.6 1.8 k Number of Sideband pairs k Number of Sideband pairs 42 44 Figure 5–16 Angle-modulated system with preemphasis and deemphasis AM vs FM • FM capture effect: A phenomenon, associated with FM reception, in which only the stronger of two signals at or near the same frequency will be demodulated – The complete suppression of the weaker signal occurs at the receiver limiter, where it is treated as noise and rejected – When both signals are nearly equal in strength, or are fading independently, the receiver may switch from one to the other • Bandwith: BAM=2 x fm BFM >= x fm use Carson’s Rule • The Receiver IF amplifier is change to a Limiting Amplifier for FM – FM rejects amplitude noise such as lightening and man made noise • The FM demodulator may be a PLL, Ratio Detector, Foster Sealy Discriminator, or slope detector Couch, Digital and Analog Communication Systems, Seventh Edition 45 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 Figure 5–16 Angle-modulated system with preemphasis and deemphasis Couch, Digital and Analog Communication Systems, Seventh Edition 46 ©2007 Pearson Education, Inc All rights reserved 0-13-142492-0 47