Lecture 28-29 FM- Frequency Modulation PM

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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):

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FM and PM Figure 5–8 Angle modulator circuits. RFC = radio-frequency choke.

Dp is the phase sensitivity or phase modulation constant

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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:

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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

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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

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Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for various modulation indexes. Jn Bessel Function

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FM- Frequency Modulation AM PM - Phase Modulation (continued)

EE445-10 NBFM

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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

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 ⎠

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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

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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

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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. 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 >= 2 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.

Figure 5–16 Angle-modulated system with preemphasis and deemphasis.