FM- Frequency Modulation PM - Phase Modulation EELE445-14 Lecture 30 1 DSB-SC, AM, FM and PM DSB - SC Complex Envelope : g(t) = Ac m(t) AM Complex Envelope : g(t) = Ac ()1+ m(t) SSB - SC Complex Envelope : g(t) = Ac []m(t) ± jmˆ (t) jDpm(t) PM Complex Envelope : g(t) = Ac e t jD f ∫−∞ m(σ )dσ FM Complex Envelope : g(t) = Ac e 1 FM and PM jθ (t) jθ (t) g(t) = R(t)e = Ac e Complex Envelope R(t) = g(t) = Ac the real envelope is a constant → power is constant Transmitted angle - modulated signal : jωct s(t) = Re[]g(t)e = Ac cos[]ωct +θ (t) FM and PM Transmitted angle - modulated signal : jωct s(t) = Re[]g(t)e = Ac cos[]ωct +θ (t) for PM : θ (t) = Dpm(t) rad D ≡ phase sensitivity or modulation constant p volt t for FM : θ (t) = D f m(σ )dσ ∫−∞ rad D ≡ frequency deviation or modulation constant f volt − sec Hz D = 2π f volt 2 FM and PM Relationship between mf(t) and mp(t): Dp ⎡dmp (t)⎤ m f (t) = ⎢ ⎥ D f ⎣ dt ⎦ D f t mp (t) = m f (σ )dσ ∫−∞ Dp Figure 5–8 Angle modulator circuits. RFC = radio-frequency choke. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 3 Figure 5–8 Angle modulator circuits. RFC = radio-frequency choke. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Instantaneous Frequency s(t) = R(t)cosψ (t) = Ac cosψ (t) FM or PM = Ac cos(ωct +θ (t)) The instantaneous frequency in Hz is : • 1 1 ⎡dψ(t)⎤ θ (t) fi (t) = ωi (t) = = fc + 2π 2π ⎣⎢ dt ⎦⎥ 2π 4 FM and PM differences PM: instantaneous phase deviation of the carrier phase is proportional to the amplitude of m(t) θ (t) = D pm(t) radians radians •Modulation Constant D in • Modulation sensitivity p volt • Phase sensitivity Instantaneous phase in radians : ψ (t) = ωct +θ (t) = ωct + Dpm(t) • • dψ (t) θ (t) D m(t) Instantaneous frequency in Hz : f (t) = = f + = f + p i 2πdt c 2π c 2π FM and PM differences FM: instantaneous frequency deviation from the carrier frequency is proportional to m(t) t θ (t) = D f m(α )dα radians ∫−∞ radians D in f volt − sec t The instantaneous phase in radians : ψ (t) = ω t +θ(t) = ω t + D m(α)dα c c f ∫ −∞ • dψ (t) θ (t) D m(t) The instantaneous frequency in Hz : f (t) = = f + = f + f i 2πdt c 2π c 2π 5 FM and PM differences FM: instantaneous frequency deviation from the carrier frequency is proportional to m(t) 1 • 1 f (t) ≡ f (t) − f = θ (t) = D m(t) d i c 2π 2π f radians D = K ⇒ Modulation p p volt Constants rad Hz D = K ⇒ = 2π f f volt − sec volt FM 1 ⎡dθ (t)⎤ frequency deviation ≡ fd (t) = fi (t) − fc = 2π ⎣⎢ dt ⎦⎥ ⎧ 1 ⎡dθ (t)⎤⎫ 1 peak frequency deviation ≡ ΔF = max = D V ⎨ ⎢ ⎥⎬ f p ⎩2π ⎣ dt ⎦⎭ 2π Vp = max[]m(t) ΔF frequency modulationindex ≡ β = B is the bandwidth of m(t) f B 6 PM and digital modulation phase deviation ≡ θ (t) peak phase deviation ≡ Δθ = max[]θ (t) = DpVp Vp = max[]m(t) phase modulation index ≡ βp = Δθ note : when m(t) is a sinusoidal signal set such that the PM and FM signals have the same peak frequency deviation, then β p = β f For Digital signals the modulation index : 2Δθ h ≡ π where 2Δθ is the pk - pk phase change in one symbol duration, Ts Figure 5–9 FM with a sinusoidal baseband modulating signal. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 7 Figure 5–9 FM with a sinusoidal baseband modulating signal. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 FM from PM and PM from FM 8 FM/PM s(t) waveforms FM and PM with m(t)=cos(2πfm Let For PM For FM Define the modulation indices: 9 FM and PM Signals Define the modulation indices: FM and PM Signals Then 10 Spectrum Characteristics of FM • FM/PM is exponential modulation Let φ(t) = β sin( 2πfmt) u(t) = Ac cos( 2πf ct + β sin(2πfmt)) j(2πfct+β sin( 2πfmt )) = Re()Ace u(t) is periodic in fm we may therefore use the Fourier series Spectrum Characteristics of FM • FM/PM is exponential modulation 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 11 Spectrum with Sinusoidal Modulation g(t) = e jβ sin(2πfmt) u(t) is periodic in fm we may therefore use the Fourier series Jn Bessel Function 12 Jn Bessel Function TABLE 5–2 FOUR-PLACE VALUES OF THE BESSEL FUNCTIONS Jn (β) 13 TABLE 5–3 ZEROS OF BESSEL FUNCTIONS: VALUES FOR β WHEN Jn(β) = 0 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes. Ac J1(β = 1) cos( 2π ( f c + 1 f1 )t) Ac J 0 (β = 1) cos( 2πfct) Ac J −1 (β = 1) cos( 2π ( f c − 1 f1 )t) Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 14 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 NBFM- Narrowband Frequency Modulation WBFM - Wideband Frequency Modulation Carson’s Bandwidth Rule EELE445-14 Lecture 31 30 15 Narrowband FM •Only the Jo and J1 terms are significant •Same Bandwidth as AM •Using Eulers identity, and φ(t)<<1: Notice the sidebands are “sin”, not “cos” as in AM Narrowband FM as a Phaser AM NBFM 16 Frequency Multiplication: Wideband FM from Narrowband FM si(t) so(t) n ωc (s(t)) n x ωc βFM n x βFM jφ (t ) j2πfct n jnφ (t ) j2πnf ct so (t) = Re(e e ) = Re(e e ) t nφ(t) = nD m(λ )dλ f ∫ −∞ β fmout = nβ f min •The Output Carrier frequency = n x fc •The output modulation index = n x βfm •The output bandwidth increases according to Carson’s Rule Effective Bandwidth- Carson’s Rule 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 17 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes. 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. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 18 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for Various modulation indexes. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 When m(t) is a sum of sine waves 19 When m(t) is a sum of sine waves Sideband Power 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: Psum()k := ∑ 21⋅Ω n = − k Psum()n Percent of power predicted by Carsons rule: ⋅100 = 99.118 Pc 20 Power vs Bandwidth PERCENT OF TOTAL POWER 100 Psum()k ⋅100 50 Pc 0 0 0.5 1 1.5 2 2.5 3 k Sideband Power β=2.4 k010:= .. Jk :=Jn() k , β 2 β = 2.4 Pk := ()Jk n3= 0 0 0 2.508·10-3 0 6.288·10-6 n 1 0.52 1 0.271 P0 + 2⋅ P j = 0.991 2 0.431 2 0.186 ∑ 3 0.198 3 0.039 j = 1 J = 4 0.064 P = 4 4.135·10-3 5 0.016 5 2.638·10-4 6 3.367·10-3 6 1.134·10-5 7 5.927·10-4 7 3.513·10-7 8 9.076·10-5 8 8.237·10-9 9 1.23·10-5 9 1.513·10-10 10 1.496·10-6 10 2.238·10-12 21 Sideband Power β=0.1 j05:=.. β := 0.1 n1:= V := Jn() j , β j 2 Uj := ()Vj 0.998 0.995 ⎜⎛ ⎟⎞ ⎜⎛ ⎟⎞ 0.05 − 3 ⎜ ⎟ ⎜ 2.494× 10 ⎟ n ⎜ − 3 ⎟ ⎜ ⎟ U + 2⋅ U = 1 1.249× 10 − 6 0 j ⎜ ⎟ ⎜ 1.56× 10 ⎟ ∑ V = ⎜ − 5 ⎟ U = ⎜ ⎟ j = 1 2.082× 10 − 10 ⎜ ⎟ ⎜ 4.335× 10 ⎟ ⎜ − 7 ⎟ ⎜ ⎟ 2.603× 10 − 14 ⎜ ⎟ ⎜ 6.775× 10 ⎟ ⎜ − 9 ⎟ ⎜ ⎟ ⎝ 2.603× 10 ⎠ ⎝ 0 ⎠ Sideband Power β=0.6 β := 0.6 n1:= W j := Jn() j , β 2 Xj := ()W j 0.912 ⎛ 0.832 ⎞ ⎛ ⎞ ⎜ ⎟ n ⎜ ⎟ ⎜ 0.082 ⎟ ⎜ 0.287 ⎟ X0 + 2⋅ Xj = 0.996 ⎜ − 3 ⎟ ∑ ⎜ 0.044 ⎟ ⎜ 1.907× 10 ⎟ j = 1 ⎜ ⎟ W = − 3 X = ⎜ − 5 ⎟ ⎜ 4.4× 10 ⎟ ⎜ 1.936× 10 ⎟ ⎜ − 4 ⎟ 3.315× 10 ⎜ − 7 ⎟ ⎜ ⎟ ⎜ 1.099× 10 ⎟ ⎜ ⎟ − 5 ⎜ − 10 ⎟ ⎝ 1.995× 10 ⎠ ⎝ 3.979× 10 ⎠ 22 filename: fmsidebands.mcd avo 09/21/04 FM/PM modulation index: set to π/2 for peak last edit date:2/27/07 phase dev of π/2 set to Δf/fm for frequency modulation.