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### Transcript of FM- Frequency Modulation PM - Phase PM and digital modulation [] [] s p where 2 is the pk-pk phase...

• 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

)(

)(

)(

)(

)()()(

)(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

+==

==

==

:signal modulated-angle dTransmitte

constantis power constant ais envelope real the

EnvelopeComplex

FM and PM

[ ] [ ]

voltHzD

dmDt

tmDtttAetgts

f

f

t

f

p

p

cctj c

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):

dmDD

tm

dttdm

DD

tm

t

fp

fp

p

f

pf

=

=

)()(

)()(

Figure 58 Angle modulator circuits. RFC = radio-frequency choke.

• 4

Figure 58 Angle modulator circuits. RFC = radio-frequency choke.

Instantaneous Frequency

2)()(

21)(

21)(

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

tfdt

tdttf

ttAPMorFMtA

ttRts

cii

cc

c

+=

==

+===

:is Hz in frequencyous instantane The

• 5

FM and PM differences

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

Modulation Constant Modulation sensitivity Phase sensitivity

2)(

2)(

2)()(

)()()(

tmDftf

dttdtf

tmDtttt

pcci

pcc

+=+==

+=+=

: Hz in frequencyous Instantane

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

f = )()(

2)(

2)(

2)()(

)()()(

tmDftf

dttdtf

dmDtttt

fcci

t

fcc

+=+==

+=+=

: Hz in frequencyous instantane The

• 6

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

)(21)(

21)()( tmDtftftf fcid

==

voltHz

ff

pp

2sec

=

=

=Modulation Constants

FM

[ ]

m(t) of bandwidth the is indexmodulationfrequency

deviationfrequencypeak

deviationfrequency

BBF

tmV

VDdt

tdF

dttdftftf

f

p

pf

cid

=

=

=

=

==

)(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

=

=

=

==

2

)(max

)(max)(

h

tmV

VDtt

fp

p

pp

Figure 59 FM with a sinusoidal baseband modulating signal.

• 8

Figure 59 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(2fmLet

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()(

tftfjc

mcc

mceA

tftfAtu

+=

+=

)2sin()( tft m =

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()(

tftfjc

mcc

mceA

tftfAtu

+=

+=

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 =

Jn Bessel Function

• 13

Jn Bessel Function

TABLE 52 FOUR-PLACE VALUES OF THE BESSEL FUNCTIONS Jn ()

• 14

TABLE 53 ZEROS OF BESSEL FUNCTIONS: VALUES FOR WHEN Jn() = 0

Figure 511 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

==

+=

• 15

Figure 511 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.

30

NBFM- Narrowband Frequency ModulationWBFM - Wideband FrequencyModulation Carsons Bandwidth Rule

EELE445-14Lecture 31

• 16

Narrowband FMOnly the Jo and J1 terms are significantSame Bandwidth as AMUsing Eulers identity, and (t)

• 17

Frequency Multiplication:Wideband FM from Narrowband FM

(s(t))nsi(t)cFM

so(t)n x cn x FM

The Output Carrier frequency = n x fcThe output modulation index = n x fmThe output bandwidth increases according to Carsons Rule

min

2)(2)(

)()(

)Re()Re()(

ffmout

t

f

tnfjtjnntfjtjo

n

dmnDtn

eeeets cc

=

=

==

Effective Bandwidth- Carsons Rule for Sine Wave ModulationWhere is the modulation

index fm is the sinusoidalmodulation frequency

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

• 18

Figure 511 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.

Figure 511 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.

• 19

Figure 511 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: f 2.4KHz:=

Modulation index: ffm

:= 2.4=

Reference equation: x t( )

n

Ac Jn n ,( ) cos c n m+( ) t =

Power in the signal: PcAc

2

2 1 := Pc 0.5 W=

Carsons rule bandwidth: BW 2 1+( ) fm:= BW 6.8 103 1s

=

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

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

k

k

n

Ac Jn n ,( )( )22 1

=

:=

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 =2.4

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

n 3=

P0 2

1

n

j

P j=

+ 0.991=

J

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-33.36710-45.92710-59.07610-51.2310-61.49610

= P

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-66.288100.2710.1860.039

-34.13510-42.63810-51.13410-73.51310-98.23710

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