FM- Frequency Modulation PM - Phase Modulation

Home , Baseband, Frequency modulation, Modulation, Phase modulation, Radio, Radio frequency

EELE445-14 Lecture 30

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.

3 Figure 5–8 Angle modulator circuits. RFC = radio-frequency choke.

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

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.

7 Figure 5–9 FM with a sinusoidal baseband modulating signal.

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)

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

NBFM- Narrowband Frequency Modulation WBFM - Wideband Frequency Modulation Carson’s Bandwidth Rule

EELE445-14 Lecture 31

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

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.

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

18 Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for Various modulation indexes.

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. spectrum Ac := 1 is the same for sinewave 4 79 modulation. fc := 010⋅

0 Fm:= 10 Modulating frequency- single sinewave x M := 10 n:= round() M+ 1 2 * n is the number of significant sidebands per Carsons rule Bandwidth:= 2⋅ n⋅Fm n9= Modulation_index:= M

⎡ n ⎤ k Si() f := A ⋅⎢ ()J0() M ⋅δ ff, + ⎡ Jn() k, M ⋅δ ⎡ff, + kFm⋅ ⎤ + ()−1 ⋅δJn() k, M ⋅ ⎡ff, − kFm⋅ ⎤ ⎤ ⎥ c ⎢ ()c ∑ ⎣ ⎣ ()c ⎦ ⎣ ()c ⎦ ⎦ ⎥ ⎣ k = 1 ⎦

Bf():= δ ff, + ()Fmn0+ ⋅ + δ ff, − nFm⋅ ⎣⎡ c ⎦⎤ ()c ff:= c − ()Fmn1+ ⋅ ,()fc − nFm⋅ .. ⎣⎡ fc + ()Fmn1+ ⋅ ⎦⎤

Bandwidth= 18 Modulation_index= 7.9

β=.4, Sideband Level =β/2 for Narrowband FM

Single Sided Spectrum Bessel Functions 1

Modulation_index0.8 0.8 0.6

0.4 0.6 0.2

0 0.4

0.2 Volts Peak

0.4 0.2

0.6

0.8 0

1 0246810 0.2 Carrier J0 2 1012 1st Sidebands J1 Spectrum 2nd Sidebands J2

23 β=.9, Sideband Level =β/2 for Narrowband FM

Single Sided Spectrum Bessel Functions 1 1

0.8Modulation_index

0.6

0.4 0.5 0.2

0.2 Volts Peak

0.4 0

0.6

0.8

1 0246810 0.5 Carrier J0 3 2 10123 1st Sidebands J1 Spectrum 2nd Sidebands J2

β=2.4, Carrier Null

Single Sided Spectrum Bessel Functions 0.6 1

0.8 Modulation_index 0.4 0.6

0.4 0.2 0.2

0 0

0.2 Peak Volts

0.4 0.2

0.6

0.8 0.4

1 0246810 0.6 Carrier J0 4 20 24 1st Sidebands J1 Spectrum 2nd Sidebands J2

24 β=3.8, first sideband null

Single Sided Spectrum Bessel Functions 0.6 1

0.8 Modulation_index 0.4 0.6

0.4 0.2 0.2

0 0

0.2 Peak Volts

0.4 0.2

0.6

0.8 0.4

1 0246810 0.6 Carrier J0 6 4 20246 1st Sidebands J1 Spectrum 2nd Sidebands J2

β=5.1, second sideband null

Single Sided Spectrum Bessel Functions 0.4 1

0.8 Modulation_index

0.6 0.2 0.4

0.2

0 0

0.2 Volts Peak

0.4

0.6 0.2

0.8

1 0246810 0.4 Carrier J0 8 6 4 202468 1st Sidebands J1 Spectrum 2nd Sidebands J2

25 Power vs BW, β=0.1

⎛ 2 n ⎞ J0() M 2 second term includes power in +Jn PMn(), := ⎜ + Jn() k, M ⎟ ⎜ 2 ∑ ⎟ and power in -Jn, i.e the upper and ⎝ k = 1 ⎠ lower sideband pairs M= 0.1 % power vs bandwidth Fm= 1 Hz n Bandwidth= 2 Hz 99.99995

99.9999 PMn(), ⋅100 = 100 2 PMk(), Ac ⋅10099.99985 2 2 Ac 2 99.9998

99.99975

99.9997

99.99965 1 1.2 1.4 1.6 1.8 k Number of Sideband pairs

Power vs BW, β=0.9

⎝ ⎠ M= 0.9 % power vs bandwidth 100 Fm= 1 Hz n Bandwidth= 4 Hz

99.5 PMn(), ⋅100 = 99.958 2 PMk(), Ac ⋅100 2 A 2 c 99 2

98.5

98 1 1.5 2 2.5 3 3.5 4 k Number of Sideband pairs

26 Power vs BW, β=2.4

⎝ ⎠ M= 2.4 % power vs bandwidth 100 Fm= 1 Hz n Bandwidth= 8 Hz

PMn(), ⋅100 = 99.945 2 PMk(), 80 Ac ⋅100 2 2 Ac 2 70

50 12345678 k Number of Sideband pairs