1

1

FM- Frequency ModulationPM - Phase Modulation

EELE445-14Lecture 30

DSB-SC, AM, FM and PM

( )

[ ]

∫ ∞−=

=

±=

+=

=

tf

p

dmjDc

tmjDc

c

c

c

eAtg

eAtg

tmjtmAtg

tmAtg

tmAtg

σσ )(

)(

)(

)(

)(ˆ)()(

)(1)(

)()(

:EnvelopeComplex FM

:EnvelopeComplex PM

:EnvelopeComplex SC-SSB

:EnvelopeComplex AM

:EnvelopeComplex SC-DSB

2

FM and PM

[ ] [ ])(cos)(Re)(

)()(

)()( )()(

ttAetgts

AtgtR

eAetRtg

cctj

c

tjc

tj

c θωω

θθ

+==

→

==

==

:signal modulated-angle dTransmitte

constantis power constant ais envelope real the

EnvelopeComplex

FM and PM

[ ] [ ]

voltHzD

voltradD

dmDt

voltradD

tmDtttAetgts

f

f

t

f

p

p

cctj c

π

σσθ

θθωω

2

sec

)()(

)()()(cos)(Re)(

=

−≡

=

≡

=+==

∫ ∞−

constant modulation or deviation frequency

:FM for

constant modulation or ysensitivit phase

:PM for

:signal modulated-angle dTransmitte

3

FM and PM

Relationship between mf(t) and mp(t):

σσ dmDD

tm

dttdm

DD

tm

t

fp

fp

p

f

pf

∫ ∞−=

⎥⎦

⎤⎢⎣

⎡=

)()(

)()(

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

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

4

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

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

Instantaneous Frequency

πθ

πω

π

θωψψ

2)()(

21)(

21)(

))(cos()(cos)(cos)()(

tfdt

tdψttf

ttAPMorFMtA

ttRts

cii

cc

c

•

+=⎥⎦⎤

⎢⎣⎡==

+===

:is Hz in frequencyous instantane The

5

FM and PM differences

radianstmDt p )()( =θ

PM: instantaneous phase deviation of the carrier phase is proportional to the amplitude of m(t)

voltradiansinDp

•Modulation Constant• Modulation sensitivity• Phase sensitivity

ππθ

πψ

ωθωψ

2)(

2)(

2)()(

)()()(

tmDftf

dttdtf

tmDtttt

pcci

pcc

••

+=+==

+=+=

: Hz in frequencyous Instantane

:radians in phaseous Instantane

FM and PM differencesFM: instantaneous frequency deviation from the carrier frequency is proportional to m(t)

radiansdmDtt

f ∫ ∞−= ααθ )()(

ππθ

πψ

ααωθωψ

2)(

2)(

2)()(

)()()(

tmDftf

dttdtf

dmDtttt

fcci

t

fcc

+=+==

+=+=

•

∞−∫

: Hz in frequencyous instantane The

:radians in phaseous instantane The

sec−voltradiansinD f

6

FM and PM differencesFM: instantaneous frequency deviation from the carrier frequency is proportional to m(t)

)(21)(

21)()( tmDtftftf fcid πθ

π==−≡

•

voltHz

voltradKD

voltradiansKD

ff

pp

π2sec

=−

⇒=

⇒=Modulation Constants

FM

[ ]

m(t) of bandwidth the is indexmodulationfrequency

deviationfrequencypeak

deviationfrequency

BBF

tmV

VDdt

tdF

dttdftftf

f

p

pf

cid

Δ=≡

=

=⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡=Δ≡

⎥⎦⎤

⎢⎣⎡=−=≡

β

πθ

π

θπ

)(max21)(

21max

)(21)()(

7

PM and digital modulation

[ ][ ]

s

p

T duration, symbol one in change phase pk-pk the is 2 where

:index modulation the signals Digital For

then deviation, frequency peak same the have signals FM and PM the that such set signal sinusoidal a is m(t) when

:note

index modulation phase

deviationphasepeak deviation phase

θπθ

ββ

θβ

θθθ

Δ

Δ≡

=

Δ=≡

=

==Δ≡≡

2

)(max

)(max)(

h

tmV

VDtt

fp

p

pp

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

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

8

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

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

FM from PM and PM from FM

9

FM/PM s(t) waveforms

FM and PM with m(t)=cos(2πfmLet

For PM

For FM

Define the modulation indices:

10

FM and PM Signals

Define the modulation indices:

FM and PM SignalsThen

11

Spectrum Characteristics of FM

• FM/PM is exponential modulationLet

( )))2sin(2(Re

))2sin(2cos()(tftfj

c

mcc

mceA

tftfAtuπβπ

πβπ+=

+=

)2sin()( tft mπβφ =

u(t) is periodic in fmwe may therefore use the Fourier series

Spectrum Characteristics of FM

• FM/PM is exponential modulation

( )))2sin(2(Re

))2sin(2cos()(tftfj

c

mcc

mceA

tftfAtuπβπ

πβπ+=

+=

u(t) is periodic in fmwe may therefore use the Fourier series

12

Spectrum with Sinusoidal Modulation

u(t) is periodic in fmwe may therefore use the Fourier series

)2sin()( tfj metg πβ=

Jn Bessel Function

13

Jn Bessel Function

TABLE 5–2 FOUR-PLACE VALUES OF THE BESSEL FUNCTIONS Jn (β)

14

TABLE 5–3 ZEROS OF BESSEL FUNCTIONS: VALUES FOR β WHEN Jn(β) = 0

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.

))1(2cos()1()2cos()1(

))1(2cos()1(

11

0

11

tffJAtfJA

tffJA

cc

cc

cc

−==

+=

− πβπβπβ

15

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 forvarious modulation indexes.

30

NBFM- Narrowband Frequency ModulationWBFM - Wideband FrequencyModulation Carson’s Bandwidth Rule

EELE445-14Lecture 31

16

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

17

Frequency Multiplication:Wideband FM from Narrowband FM

(s(t))nsi(t)ωcβFM

so(t)n x ωcn x βFM

•The Output Carrier frequency = n x fc•The output modulation index = n x βfm•The output bandwidth increases according to Carson’s Rule

min

2)(2)(

)()(

)Re()Re()(

ffmout

t

f

tnfjtjnntfjtjo

n

dmnDtn

eeeets cc

ββ

λλφ

πφπφ

=

=

==

∫∞−

Effective Bandwidth- Carson’s Rule for Sine Wave ModulationWhere β is the modulation

index fm is the sinusoidalmodulation frequency

•Notice for FM, if kfa>> fm, increasing fm does not increase Bc much•Bc is linear with fm for PM

18

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 forvarious 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 forvarious modulation indexes.

19

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.

When m(t) is a sum of sine waves

20

When m(t) is a sum of sine waves

Sideband PowerSignal Amplitude: Ac 1V:=

Modulating frequency: fm 1KHz:=

Carrier peak deveation: Δf 2.4KHz:=

Modulation index: βΔffm

:=β 2.4=

Reference equation: x t( )

∞−

∞

n

Ac Jn n β,( )⋅ cos ωc n ωm⋅+( ) t⋅⎡⎣ ⎤⎦⋅⎡⎣ ⎤⎦∑=

Power in the signal: PcAc

2

2 1⋅ Ω:= Pc 0.5 W=

Carsons rule bandwidth: BW 2 β 1+( )⋅ fm⋅:= BW 6.8 103×

1s

=

Order of significant sidbands predicted by Carsons rule: n round β 1+( ):= n 3=

Power as a function of number of sidebands: Psum k( )

k−

k

n

Ac Jn n β,( )⋅( )2

2 1⋅ Ω∑=

:=

Percent of power predicted by Carsons rule: Psum n( )

Pc100⋅ 99.118=

21

Power vs Bandwidth

0 0.5 1 1.5 2 2.5 30

50

100PERCENT OF TOTAL POWER

Psum k( )

Pc100⋅

k

Sideband Power β=2.4

k 0 10..:=Jk Jn k β,( ):= β 2.4=Pk Jk( )2

:=n 3=

P0 2

1

n

j

P j∑=

⋅+ 0.991=

J

0012345678910

-32.508·100.52

0.4310.1980.0640.016

-33.367·10-45.927·10-59.076·10-51.23·10-61.496·10

= P

0012345678910

-66.288·100.2710.1860.039

-34.135·10-42.638·10-51.134·10-73.513·10-98.237·10

-101.513·10-122.238·10

=

22

Sideband Power β=0.1

j 0 5..:= β 0.1:= n 1:=

Vj Jn j β,( ):=Uj Vj( )2

:=

U0 2

1

n

j

Uj∑=

⋅+ 1=

V

0.998

0.05

1.249 10 3−×

2.082 10 5−×

2.603 10 7−×

2.603 10 9−×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= U

0.995

2.494 10 3−×

1.56 10 6−×

4.335 10 10−×

6.775 10 14−×

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

Sideband Power β=0.6

β 0.6:= n 1:=

W j Jn j β,( ):= Xj W j( )2:=

X0 2

1

n

j

Xj∑=

⋅+ 0.996=

W

0.912

0.287

0.044

4.4 10 3−×

3.315 10 4−×

1.995 10 5−×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

= X

0.832

0.082

1.907 10 3−×

1.936 10 5−×

1.099 10 7−×

3.979 10 10−×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

23

Modulation_index 7.9=Bandwidth 18=

f fc n 1+( ) Fm⋅− fc n Fm⋅−( ), fc n 1+( ) Fm⋅+⎡⎣ ⎤⎦..:=B f( ) δ f fc n 0+( ) Fm⋅+,⎡⎣ ⎤⎦ δ f fc n Fm⋅−,( )+:=

Si f( ) Ac J0 M( )( ) δ f fc,( )⋅

1

n

k

Jn k M,( ) δ f fc k Fm⋅+( ),⎡⎣ ⎤⎦⋅ 1−( )k Jn k M,( )⋅ δ f fc k Fm⋅−( ),⎡⎣ ⎤⎦⋅+⎡⎣

⎤⎦∑

=

+⎡⎢⎢⎣

⎤⎥⎥⎦

⋅:=

Modulation_index M:=n 9=Bandwidth 2 n⋅ Fm⋅:=

2 * n is the number of significant sidebands per Carsons rulen round M 1+( ):=

Mx

10:=

Modulating frequency- single sinewaveFm 100:=

fc 0 104⋅:=

Ac 1:=79

FM/PM modulation index: set to π/2 for peak phase dev of π/2set to Δf/fm for frequency modulation. spectrumis the same for sinewavemodulation.

filename: fmsidebands.mcd avo 09/21/04last edit date:2/27/07

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

2 1 0 1 20.2

0

0.2

0.4

0.6

0.8

Spectrum

Single Sided Spectrum

Peak

Vol

ts

0 2 4 6 8 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J01st Sidebands J12nd Sidebands J2

Bessel Functions

Modulation_index

24

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

3 2 1 0 1 2 30.5

0

0.5

1

Spectrum

Single Sided Spectrum

Peak

Vol

ts

0 2 4 6 8 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J01st Sidebands J12nd Sidebands J2

Bessel Functions

Modulation_index

β=2.4, Carrier Null

4 2 0 2 40.6

0.4

0.2

0

0.2

0.4

0.6

Spectrum

Single Sided Spectrum

Peak

Vol

ts

0 2 4 6 8 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J01st Sidebands J12nd Sidebands J2

Bessel Functions

Modulation_index

25

β=3.8, first sideband null

6 4 2 0 2 4 60.6

0.4

0.2

0

0.2

0.4

0.6

Spectrum

Single Sided Spectrum

Peak

Vol

ts

0 2 4 6 8 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J01st Sidebands J12nd Sidebands J2

Bessel Functions

Modulation_index

β=5.1, second sideband null

8 6 4 2 0 2 4 6 80.4

0.2

0

0.2

0.4

Spectrum

Single Sided Spectrum

Peak

Vol

ts

0 2 4 6 8 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J01st Sidebands J12nd Sidebands J2

Bessel Functions

Modulation_index

26

Power vs BW, β=0.1

second term includes power in +Jn and power in -Jn, i.e the upper and lower sideband pairs

P M n,( )J0 M( )2

21

n

k

Jn k M,( )2∑=

+⎛⎜⎜⎝

⎞⎟⎟⎠

:=

1 1.2 1.4 1.6 1.899.99965

99.9997

99.99975

99.9998

99.99985

99.9999

99.99995

% power vs bandwidth

Number of Sideband pairs

P M k,( )

Ac2

2

100⋅

n

k

M 0.1=

Fm 1= Hz

Bandwidth 2= Hz

P M n,( )

Ac2

2

100⋅ 100=

Power vs BW, β=0.9

⎝ ⎠

1 1.5 2 2.5 3 3.5 498

98.5

99

99.5

100% power vs bandwidth

Number of Sideband pairs

P M k,( )

Ac2

2

100⋅

n

k

M 0.9=

Fm 1= Hz

Bandwidth 4= Hz

P M n,( )

Ac2

2

100⋅ 99.958=

27

Power vs BW, β=2.4

⎝ ⎠

1 2 3 4 5 6 7 850

60

70

80

90

100% power vs bandwidth

Number of Sideband pairs

P M k,( )

Ac2

2

100⋅

n

k

M 2.4=

Fm 1= Hz

Bandwidth 8= Hz

P M n,( )

Ac2

2

100⋅ 99.945=