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FM and PM Lecture 28-29 for FM: FM- Frequency Modulation PM - Phase Modulation Relationship between mf(t) and mp(t): EE445-10 1 3 FM and PM Figure 5–8 Angle modulator circuits. RFC = radio-frequency choke. Dp is the phase sensitivity or phase modulation constant 2 4 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 5 7 Figure 5–9 FM with a sinusoidal baseband modulating signal. FM and PM differences PM: phase is proportional to m(t) θ (t) = Dpm(t) ⇒ FM: t θ (t) = D f m(α)dα ∫−∞ instantaneous frequency deviation from the fi (t) − fc = D f m(t) ⇒ carrier is proportional to m(t) radians Dp = K p ⇒ Modulation volt Constants Hz D f = K f ⇒ 6 volt 8 Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 FM from PM FM and PM Signals PM from FM Maximum phase deviation in PM: Maximum frequency deviation in FM: 9 11 FM from PM Example PM from FM 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(2πfmt) u(t) = Ac cos(2πfct + β sin(2πfmt)) j(2πfct+β sin(2πfmt)) = Re()Ace u(t) is periodic in fm we may therefore use the Fourier series 13 15 Sine Wave Example Spectrum Characteristics of FM Then • FM/PM is exponential modulation c(t) = Ac cos(2πfct + β sin(2πfmt)) j(2πfct+β sin(2πfmt)) = Re()Ace c(t) is periodic in fm we may therefore use the Fourier series 14 16 Spectrum Characteristics with J Bessel Function Sinusoidal Modulation n 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 20 Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Narrowband FM as a Phaser Lecture 29 FM- Frequency Modulation AM PM - Phase Modulation (continued) EE445-10 NBFM 21 23 Wideband FM from Narrowband Narrowband FM FM •Only the Jo and J1 terms are significant •Same Bandwidth as AM s(t) s(t) n •Using Eulers identity, and φ(t)<<1: ωc (s(t)) n x ωc βFM n x βFM •The Output Carrier frequency = n x fc •The output modulation index = n x β Notice the sidebands are “sin”, not “cos” as in AM c •The output bandwidth increases according to Carson’s Rule 22 24 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for Effective Bandwidth- Carson’s Rule various modulation indexes. for Sine Wave Modulation Where β is the modulation index fm is the sinusoidal modulation frequency •Notice for FM, if kfa>> fm, increasing fm does not increase Bc much •Bc is linear with fm for PM 25 27 Couch, Digital and Analog Communication Systems, Seventh Edition ©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. Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes. 26 28 Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Couch, Digital and Analog Communication Systems, Seventh Edition ©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 Carrier peak deveation: Δf:= 2.4KHz Δf Modulation index: β := β = 2.4 fm ∞ Reference equation: () xt() ∑ ⎣⎡Ac⋅Jn n,β ⋅cos⎣⎡()ωc + n⋅ωm ⋅t⎦⎤⎦⎤ n = − ∞ 2 Ac Power in the signal: Pc := Pc = 0.5 W 21⋅Ω Carsons rule bandwidth: BW:= 2⋅()β + 1 ⋅f 3 1 m BW= 6.8× 10 s Order of significant sidbands predicted by Carsons rule: n:= round()β + 1 n3= k 2 ()Ac⋅Jn() n ,β Power as a function of number of sidebands: P ()k := sum ∑ 21⋅Ω n = − k 29 Psum()n 31 Percent of power predicted by Carsons rule: ⋅100 = 99.118 Pc When m(t) is a sum of sine waves Sideband Power PERCENT OF TOTAL POWER 100 Psum()k ⋅100 50 Pc 0 0 0.5 1 1.5 2 2.5 3 k 30 32 Sideband Power Sideband Power k010:= .. β := 0.6 n1:= Jk :=Jn() k ,β 2 β = 2.4 Pk := ()Jk n3= Wj := Jn() j,β 2 Xj := ()Wj 0 0 0 2.508·10-3 0 6.288·10-6 n 0.912 ⎛ 0.832 ⎞ ⎛ ⎞ ⎜ ⎟ n 1 0.52 1 0.271 ⎜ ⎟ P0 + 2⋅ Pj = 0.991 ⎜ 0.082 ⎟ 2 0.431 2 0.186 ∑ ⎜ 0.287 ⎟ X0 + 2⋅ Xj = 0.996 3 0.198 3 0.039 j = 1 ⎜ − 3 ⎟ ∑ ⎜ 0.044 ⎟ j 1 4 0.064 4 4.135·10-3 ⎜ 1.907× 10 ⎟ = J = P = ⎜ ⎟ 5 0.016 5 2.638·10-4 W = − 3 X = ⎜ ⎟ ⎜ 4.4× 10 ⎟ − 5 6 3.367·10-3 6 1.134·10-5 ⎜ 1.936× 10 ⎟ 7 5.927·10-4 7 3.513·10-7 ⎜ − 4 ⎟ 3.315× 10 ⎜ − 7 ⎟ 8 9.076·10-5 8 8.237·10-9 ⎜ ⎟ ⎜ 1.099× 10 ⎟ 9 1.23·10-5 9 1.513·10-10 ⎜ ⎟ − 5 ⎜ − 10 ⎟ 10 1.496·10-6 10 2.238·10-12 ⎝ 1.995× 10 ⎠ ⎝ 3.979× 10 ⎠ 33 35 filename: fmsidebands.mcd Sideband Power avo 09/21/04 FM/PM modulation index: set toπ /2 for peak last edit date:2/27/07 phase dev ofπ /2 set to Δf/fm for frequency modulation. spectru Ac := 1 is the same for sinewave 4 79 modulation. fc := 010⋅ j05:=.. β := 0.1 n1:= 0 Fm:= 10 Modulating frequency- single sinewave Vj := Jn() j,β 2 x M := Uj := ()Vj 10 n:= roundM()+ 1 2 * n is the number of significant sidebands per Carsons rule Bandwidth:= 2⋅ n⋅Fm 0.998 0.995 n9= Modulation_index:= M ⎜⎛ ⎟⎞ ⎜⎛ ⎟⎞ 0.05 − 3 ⎜ ⎟ ⎜ 2.494× 10 ⎟ n ⎡ n ⎤ ⎢ ⎡ k ⎤⎥ ⎜ − 3 ⎟ ⎜ ⎟ U + 2⋅ U = 1 Si() f := Ac⋅ ()J0() M ⋅δ ()ff, c + ⎣Jn() k, M ⋅δ ⎣⎡ff,()c + kFm⋅ ⎦⎤ + ()−1 ⋅δJn() k, M ⋅ ⎣⎡ff,()c − kFm⋅ ⎦⎤⎦ 1.249× 10 − 6 0 j ⎢ ∑ ⎥ ⎜ ⎟ ⎜ 1.56× 10 ⎟ ∑ ⎣ k = 1 ⎦ V = ⎜ − 5 ⎟ U = ⎜ ⎟ j = 1 2.082× 10 − 10 Bf():= δ ⎡ff, + ()Fmn0+ ⋅ ⎤ + δ ff, − nFm⋅ ff:= − ()Fmn1+ ⋅ , f − nFm⋅ .. f + ()Fmn1+ ⋅ ⎜ ⎟ ⎜ 4.335× 10 ⎟ ⎣ c ⎦ ()c c ()c ⎣⎡ c ⎦⎤ ⎜ − 7 ⎟ ⎜ ⎟ − 14 ⎜ 2.603× 10 ⎟ ⎜ 6.775× 10 ⎟ ⎜ − 9 ⎟ ⎜ ⎟ ⎝ 2.603× 10 ⎠ ⎝ 0 ⎠ Bandwidth= 18 Modulation_index= 7.9 34 36 M=.4, Sideband Level =M/2 for Narrowband FM M=2.4, Carrier Null Single Sided Spectrum Single Sided Spectrum Bessel Functions Bessel Functions 0.6 1 1 Modulation_index0.8 0.8 Modulation_index 0.8 0.4 0.6 0.6 0.4 0.4 0.6 0.2 0.2 0.2 0 0.4 0 0 0.2 Peak Volts 0.2 Peak Volts 0.4 0.2 0.4 0.2 0.6 0.6 0.8 0 0.8 0.4 1 1 0246810 0246810 0.2 0.6 Carrier J0 2 1012 Carrier J0 4 2024 1st Sidebands J1 1st Sidebands J1 Spectrum Spectrum 2nd Sidebands J2 2nd Sidebands J2 37 39 M=.9, Sideband Level =M/2 for Narrowband FM M=3.8, first sideband null Single Sided Spectrum Single Sided Spectrum Bessel Functions 1 Bessel Functions 0.6 1 1 0.8Modulation_index 0.8 Modulation_index 0.4 0.6 0.6 0.4 0.4 0.5 0.2 0.2 0.2 0 0 0 0.2 Peak Volts 0.2 Peak Volts 0.4 0 0.4 0.2 0.6 0.6 0.8 0.8 0.4 1 1 0246810 0246810 0.5 0.6 Carrier J0 3 2 10123 Carrier J0 6 4 20246 1st Sidebands J1 1st Sidebands J1 Spectrum Spectrum 2nd Sidebands J2 2nd Sidebands J2 38 40 M=5.1, second sideband null Power vs BW, M=0.9 ⎝ ⎠ M= 0.9 % power vs bandwidth 100 Fm= 1 Hz Single Sided Spectrum Bessel Functions 0.4 n 1 Bandwidth= 4 Hz 0.8 Modulation_index 0.6 99.5 0.2 0.4 PMn(), ⋅100 = 99.958 2 0.2 PMk(), Ac ⋅100 2 0 0 A 2 c 99 0.2 Peak Volts 2 0.4 0.6 0.2 0.8 98.5 1 0246810 0.4 Carrier J0 8 6 4 202468 1st Sidebands J1 Spectrum 2nd Sidebands J2 98 1 1.5 2 2.5 3 3.5 4 k Number of Sideband pairs 41 43 Power vs BW, M=0.1 Power vs BW, M=2.4 ⎝ ⎠ ⎛ 2 n ⎞ J0() M 2 second term includes power in +Jn M= 2.4 PMn(), := ⎜ + Jn() k, M ⎟ ⎜ 2 ∑ ⎟ and power in -Jn, i.e the upper and % power vs bandwidth ⎝ k = 1 ⎠ lower sideband pairs 100 Fm= 1 Hz M= 0.1 n % power vs bandwidth Bandwidth= 8 Hz Fm= 1 Hz n 90 Bandwidth= 2 Hz 99.99995 PMn(), ⋅100 = 99.945 2 99.9999 PMk(), 80 Ac PMn(), ⋅100 ⋅100 = 100 2 2 A 2 PMk(), Ac c ⋅10099.99985 2 2 2 Ac 2 70 99.9998 99.99975 60 99.9997 50 99.99965 12345678 1 1.2 1.4 1.6 1.8 k k Number of Sideband pairs Number of Sideband pairs 42 44 Figure 5–16 Angle-modulated system with preemphasis and deemphasis.
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