Ch5 Angle Modulation

8/10/2019 Ch5 Angle Modulation

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FM Illustration

The frequency ofthe carrier is variedaround c inrelation with themessa e si nal.

i(t )= c + k f m(t )

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Instantaneous Frequency

.The angle could be constant [cos(30 0)], or varying with

The instantaneous angular frequency (in rad/sec) is therate of change of the angle. That is: i (t ) = d (t )/dt .For cos( c t + ), i (t )= c as expected.

.)()(t

i d t



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Representation of Angle Modulation in Time Domain

or an s gna : i = c f m

.)()()( t

f c

t

i FM d mk t d t

( ) cos ( )t

FM c f g t A t k m d

For Phase Modulation PM , the hase of the carrier isvaried in relation to the message signal: (t ) = k p m(t )

( ) cos ( ) g t A t k m t



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Relation Between FM and PMt t k m t

( )( ) ( ).i c p c pdm t t k k m t dt

( )t

d ( )

t

m t d FM Modulator

m(t ) g FM (t )

m(t ) o u ator PM

( )d dt

( )dm t dt



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Which is Which?



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5 5 5 -1 -1

FM and PM Modulation

f

k p = 10 rad/Volt = 5 v -1

f c = 100 MHzFM:

f i = f c + k f m(t )

108 -10 5 < f i < 10 8 +10 599.9 < f i < 100.1 MHZPM: f i = f c + k p dm(t )/dt

108 -10 5 < f i < 10 8 -10 599.9 < f i < 100.1 MHZPower (FM or PM) = A 2/2



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Frequency ModulationTheimagecannotbedisplayed.Your computer may nothaveenough memory to open theimage,or theimagemay havebeen corrupted.Restartyour computer,and then open thefil eagain.Ifthered x stil lappears,you may haveto deletetheimageand then insertitagain.

The equation cos sinc s c c mm

f v t = V t + t f

may be expressed as Bessel

series Besse unctionsTheimagecannotbedisplayed.Your computer may nothaveenough memory to open theimage,or theimagemay havebeen corrupted.Restartyour computer,and then open thefileagain.Ifthered x stillappears,you may haveto deletetheimageand then insertitagain.

= mcnc s t n+ J V =t v coswhere J n ( ) are Bessel functions of the first kind. Expanding theequation for a few terms we have:

t V t J V

t V t V t V t vmcmcc f f

mcc

f f

mcc

f

cc s

Amp

1

Amp

1

Amp

0

2cos2cos

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

mcmc f f

mcc

f f

mcc

2Amp2Amp



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FM Signal Spectrum.

The amplitudes drawn are completely arbitrary, since we have not foundn .



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Spectrum of FM/PM ,

to relate the spectrum of the FM/PM modulated signalto that of the modulating signal m(t ). We can deal witht on a case- y-case as s.

We are, however, particularly interested in finding the.

For that purpose, we will make some assumptions andwork on simple modulating messages.

Because of the close relation between FM and PM, wewill do the analysis for FM and extend it to PM.



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FM Spectrum Bessel Coefficients.

The FM signal spectrum may be determined from

mcnc t n J V t v )cos()()( n

The values for the Bessel coefficients, J n ( ) may be

found from graphs or, preferably, tables of Besselfunctions of the first kind.



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FM Spectrum Bessel Coefficients.

Jn ( )

In the series for v s( t ), n = 0 is the carrier component, i.e. )cos()(0 t J V cc

= 2.4 = 5

,f c , varies in amplitude, with modulation index .



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FM Spectrum Bessel Coefficients. n

read off the graph and hence the component amplitudes ( V c J

n( )) may be

determined.A further way to interpret these curves is to imagine them in 3 dimensions



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gn cant e an s pectrum.As ma be seen from the table of Bessel functions for values of n above acertain value, the values of J n ( ) become progressively smaller. In FM thesidebands are considered to be significant if J n ( ) 0.01 (1%).

Although the bandwidth of an FM signal is infinite, components withamplitudes V c J n ( ), for which J n ( ) < 0.01 are deemed to be insignificantand may be ignored.

Example: A message signal with a frequency f m Hz modulates a carrier f c to produce FM with a modulation index = 1. Sketch the spectrum.

n J n(1) Amplitude Frequency0 0.7652 0.7652 V c f c

. . c c m c - m2 0.1149 0.1149 V c f c+2 f m f c - 2 f m 3 0.0196 0.0196 V c f c+3 f m f c -3 f m 4 0.0025 Insignificant5 0.0002 Insignificant

15


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Significant Sidebands Spectrum.

modulation indices ( ) and the associated spectral bandwidth.

No of sidebands 1% ofunmodulated carrier

Bandwidth

0.1 2 2 f m 0.3 4 4 f m 0.5 4 4 f m 1.0 6 6 f m 2.0 8 8 f m5.0 16 16 f m

10.0 28 28 f m

e. . for = 516 sidebands(8 pairs).



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Carsons Rule for FM Bandwidth.

An approximation for the bandwidth of an FM signalis given by BW = 2(Maximum frequency deviation +

g es mo u a e requency

)(2Bandwidth mc f f Carsons Rule



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Narrowband and Wideband FM

From the graph/table of Bessel functions it may be seen that for small , (0.3) there is only the carrier and 2 significant sidebands, i.e. BW = 2 fm .

FM with 0.3 is referred to as narrowband FM (NBFM) (Note, the

bandwidth is the same as DSBSC).

Wideband FM WBFM

or > . ere are more an s gn can s e an s. s ncreasesthe number of sidebands increases. This is referred to as wideband FM(WBFM).



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VHF/FM=

transmissions, in the band 88MHz to 108MHz have the followingparameters:Max frequency input ( e.g. music) f m 15kHz

mc V f Deviation 75kHz

Modulation Index 5 c f m

For = 5 there are 16 sidebands and the FM signal bandwidth is 16 fm =16 x 15kHz= 240kHz. Applying Carsons Rule BW = 2(75+15) = 180kHz.



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Comments FM

of sidebands at frequencies f c nf m ( n = 0, 1, 2, )

,n

mcnc s

In FM we refer to sideband pairs not upper and lower sidebands.arr er or o er componen s may no e suppresse n .

The relative amplitudes of components in FM depend on the values J n ( ),

where thus the component at the carrier frequency depends on

m ( t ), as do all the other components and none may be suppressedm

m

f

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Comments FM n . . .

only J 0 ( ) and J 1 ( ) are significant, i.e. only a carrier and 2 sidebands.Bandwidth is 2 f m , similar to DSBSC in terms of bandwidth - called NBFM.

Large modulation indexm

c

f f

The FM process is non-linear. The principle of superposition does notapply. When m ( t ) is a band of signals, e.g. speech or music the analysis

.frequency equal to the maximum input frequency. E.g. m ( t ) band 20Hz

15kHz, fm = 15kHz is used.



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Power in FM Signals.

From the equation for FM n mcnc s t n J V t v )cos()()( we see that the peak value of the components is V J ( ) for the n thcomponent.

Then the nth component Single normalised average power is=2

2

)( RMS pk

V V

2

)(

2

)( 22

ncnc J V J V

Hence, the total power in the infinite spectrum is

2

Total powern

ncT 2



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calculations to find P T . But, considering the waveform, the peak value is

V -c , which is constant.

V V Since we know that the RMS value of a sine wave is

2 2c

and power = ( V RMS ) 2 then we may deduce that

n

ncccT

J V V V P

2)(

22

222

Hence, if we know V c for the FM signal, we can find the total power P T for

the infinite spectrum with a simple calculation. 24EEEF311 (2014-2015)NITIN Sharma


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Power in FM Signals.Now consider if we generate an FM signal, it will contain an infinitenumber of sidebands. However, if we wish to transfer this signal, e.g. over a

radio or cable, this implies that we require an infinite bandwidth channel.Even if there was an infinite channel bandwidth it would not all be

.particular signal. Thus we have to make the signal spectrum fit into theavailable channel bandwidth. We can think of the signal spectrum as a

train and the channel bandwidth as a tunnel obviously we make thetrain s ig t y ess wi er t an t e tunne i we can.



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Power in FM Signals.However, many signals ( e.g. FM, square waves, digital signals) contain aninfinite number of components. If we transfer such a signal via a limitedchannel bandwidth, we will lose some of the components and the outputsignal will be distorted. If we put an infinitely wide train through a tunnel,

,can be tolerated?Generally speaking, spectral components decrease in amplitude as wemove awa from the s ectrum centre.



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In general distortion may be defined as

spectrumdBandlimiteinPower-spectrumin totalPower D

BLT P P D

With reference to FM the minimum channel bandwidth required would be just wide enough to pass the spectrum of significant components. For abandlimited FM s ectrum let a = the number of sideband airs e. . for

= 5, a = 8 pairs (16 components). Hence, power in the bandlimitedspectrum P BL is

a 2

an

nc BL P 2

= carrier power + sideband powers.



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Since 2c

T

V P

acc V V 222

a

ann

c

ann

J V

D 22 ))((1

2

22 Distortion

Also, it is easily seen that the ratio

a

ann

T P D ))((spectrumin totalPower = 1 Distortion

a

i.e. proportion p f power in bandlimited spectrum to total power = an

n ))((



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Example

Distortion = 01.0505.4950

T

BLT

P P P

or 1%.

c ua y, we on nee o now c , .e. a erna ve y

Distortion =

12

))2.0((1 n J ( a = 1)n

D = 01.0)0995.0()99.0(1 22

Ratio 99.01))((1

2 D P BL

1nT



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What is NOT the bandwidth of FM!

ne may en o e eve a s nce e mo u a e

signal instantaneous frequency is varying between byaround then the bandwidth of the FM si nal is

2 f . False!

In fact, the motivation behind introducing FM was toreduce the bandwidth compared to that of AmplitudeModulation, which turns out to be wrong.

a was m ss ng rom e p c ure o an w



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FM Visualization

u ygenerator, and wiggling it back and forth tom l h rri in r n m message.

There are two wi lin arameters:How far you deviate from the center frequency ( f)How fast you wiggle (related to Bm)

The rate of change of the instantaneous frequency was missing!



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Carsons Rule

FM m

where f = frequency deviation = k f |m(t )|maxm = an w t o m t

Define the deviation ratio = f / Bm. B FM 2( +1) BmThe same rule applies to PM bandwidth,

B PM 2( f + Bm) = 2( +1) Bmwhere ( f )PM = k p |dm(t )/dt| max



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Narrow Band and Wide Band FM

en m or , e sc eme scalled Narrow Band (NBFM, NBPM).

NBFM m same or

Therefore, no matter how small we make theev at on aroun c , t e an w t o t emodulated signal does not get smaller than 2 Bm.



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Estimate B FM and B PM = 5 = 5 f

Hz/Volt = 10 5 V-1sec -1

k p = 5 rad/Volt = 2.5 v -1

=c First estimate the Bm.Cn = 8/ 2n2 for n odd, 0 n evenThe 5 th harmonic onward can

be neglected. Bm = 15 kHzFor FM:

FMFor PM: f = 50 kHz; B FM =130 KHz



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Repeat if m(t ) is Doubled5 5

f

Hz/Volt = 105

V-1

sec-1

k p = 5 rad/Volt = 2.5 v -1

2

-

40,000

-40,000

c = zFor FM:

f = 200 kHz; B FM = 430 KHzFor PM: f = 100 kHz; B FM = 230 KHz

Doubling the signal peak has

significant effect on both FMand PM bandwidth



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Repeat if the period of m(t ) is Doubled

5 5 - f

Hz/Volt = 105

V-1

sec-1

k p = 5 rad/Volt = 2.5 v -1

10,000

-10,000

c = z Bm = 7.5 kHz

For FM: f = 100 kHz; B FM = 215 KHz

For PM: f = 25 kHz; B FM = 65 KHz

Expanding the signal varies itsspectrum. This has significanteffect on PM.



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Spectrum of NBFM (1/2)t

FM c f

( ) ( ) cos ( ) sin ( )c f j t k a t FM c f c f g t A e A t k a t jA t k a t

w ere

2 2 2 3 3 3 4 4 4( ) ( ) ( ) ( ) 1 ( )

2! 3! 4!

f f f c j t FM f

j k a t j k a t j k a t g t A e jk a t

2 2 3 3 4 4( ) ( ) ( )

( )2! 3! 4!

f f f c c c c c j t j t j t j t j t f

k a t jk a t k a t A e jk a t e e e e

2 2 3 3 4 4

( ) Re ( )

( ) ( ) ( )cos( ) ( ) sin( ) cos( ) sin( ) cos( ) f f f

FM FM

c f c c c c

g t g t

k a t k a t k a t A t k a t t t t t



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Spectrum of NBFM (2/2)

, f

( ) ( ) cos( ) ( ) sin( ) FM Narrowband c f c g t A t k a t t

Bandwidth of a(t ) is equal to the bandwidth of m(t ), Bm. NBFM m .

Similarly for PM ( |k p m(t )|


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NBFM Modulator

( ) ( ) cos( ) ( ) sin( ) FM Narrowband c f ct A t k a t t

( )t

d



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NBPM Modulator

( ) ( ) cos( ) ( ) sin( ) PM Narrowband c p c g t A t k m t t



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Immunity of FM to Non-linearities

2

1 2

3

3

( ) cos ( ) cos ( )

cos ( )

c f c f

c f

q t a A t k m d a A t k m d

a A t k m d

21 cos ( ) 1 cos 2 2 ( )2c f c f

a Aa A t k m d t k m d

3 1 cos 2 2 (2 c f

a At k m

32 21

) cos ( )

3cos ( ) cos 2 2 ( )

c f

c f c f

d t k m d

a Aa A a Aa A t k m d t k m d

Around with Around 2 with 2c f f c f f DC k k k k

3 cos 3 3 ( )4 c f

a At k m d

Around 3 with 3c f f k k



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Frequency Multipliers

( ) FM g t ( ) contains the followingq t

cos ( )c f A t k m d cos ( )c f t k m d

( )( )

cos ( )

FM output

c f

g t

B P t Pk m d

cos 2 2 ( )c f t k m d

cos ( )c f

P t Pk m d



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Generation of WBFM: Indirect Method,

interested in generatingan FM signal of certain bandwidth (or f or )

c.In the indirect method,we generate a NBFM

with small then use ascale to the requiredvalue.This way, f c will also

e sca e y e samefactor. We may need afrequency mixer toadjust f c.



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Example: From NBFM to WBFM

mo u a or s mo u a ng a message s gna

m(t ) with bandwidth 5 kHz and is producing an FMsi nal with the followin s ecifications

f c1 = 300 kHz, f 1 = 35 Hz.

We would like to use this si nal to enerate a WBFMsignal with the following specifications

f c2 = 135 MHz, f 2 = 77 kHz.3

2

1

77*102200

35 f f

6

23

1

135*10450

300*10c

c

f f



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Generation of WBFM: Direct Method

as poor requency sta ty. equ resfeedback to stabilize it.



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FM Demodulation: Signal Differentiation

( ) cos ( ) FM c f

g t A t k m d

( ) sin ( ) FM c f c f g

A k m t t k m d dt



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FM Demodulation: Signal Differentiation

( )d ( ) sin ( )c f c f A k m t t k m d

dt

( )sin ( )

dm t k t k m t

t ( )dm t

dt t ( )d



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Frequency Discriminatorsny sys em w a

transfer function of theform = a + bover the band of the FMsignal can be used for

emo u a onThe differentiator is just

.



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Slope Detectors (Demodulators)

( ) cos ( )

c f c f A Ck m t t k m d



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FM Slope Demodulator Principle: use slope detector (slope circuit) asfrequency discriminator, which implements frequencyto voltage conversion (FVC)

frequency. Example: filters, differentiator

s(t) x(t) |H f |

X(f)

dt

H(f)=j2 f S(f)

X(f)outputvoltage

f freqency in s(t) voltage in x(t)10 20

20 40

Hz j

Hz j

range in S(f)



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FM Slope Demodulator cont.Block dia ram of direct method slo e detector = slo ecircuit + envelope detector)

slope envelopes(t) s 1 (t) so (t)

(AM demodulator)

circuit detector

(FM AM)(FVC)

0( ) cos 2 2 ( ) , where ( ) ( )

t

c c f i c f s t A f t k m d f t f k m t

1 0

( ) 2 2 ( ) sin 2 2 ( )t

c c f c f s t A f k m t f t k m d

( ) 2 2 ( )o c c f s t A f k m t s o(t) linear with m(t) 55EEEF311 (2014-2015)NITIN Sharma


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Slope Detector

Magnitude frequencyresponse of

.



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Hard Limiter

filter that suppresses the unwanted products (harmonics) of thelimiting process.Input Signal

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

t

f ci daamk t wt At t At v

utput o ar m ter

)(5cos1)(3cos1)(cos4)( t t t t vo

Bandpass filter ))(cos(4)(

t

f co daamk t wt e

Remove the amplitude variations57EEEF311 (2014-2015)

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Ratio Detector Foster-Seele / hase shift discriminator

uses a double-tuned transformer to convert the instantaneous frequency

variations of the FM input signal to instantaneous amplitude variations. Theseamplitude variations are rectified to provide a DC output voltage which variesin amplitude and polarity with the input signal frequency.Example

Ratio detector o e os er- ee ey scr m na or, no response o , u



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Zero Crossing Detector



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Phased-Locked Loop (PLL)e mu p er o owe

by the filter estimatesthe error bewteen theangle of g FM (t ) and

g VCO (t ).LoopFilter

to adjust the angle.When the angles are VCO

locked, the output of thePLL would be followingm t attern.

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FM Demodulator PLL-

A closed-loop feedback control circuit, make a signal in fixed phase(and frequency) relation to a reference signal

Or, change frequency (or phase) according to inputsPLL can be used for both FM modulator and demodulator

modulations and demodulations

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PLL FMRemember the followin relations

Si=Acos(w ct+ 1(t)), Sv=A vcos(w ct+ c(t))Sp=0.5AA v[sin(2w ct+ 1+ c)+sin( 1- c)]

o= . vs n 1- c = v 1- cSection 2.14

s(t)+

+

freqencydevided

LP

FilterLoops(t) e(t) v(t)

y

ReferenceCarrier

r

VCO

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Superheterodyne Receiver Radio receivers main function

Demodulation get message signalCarrier frequency tuning select station

er ng remove no se n er erenceAmplification combat transmission power loss

Su erheterod ne receiver Heterodyne: mixing two signals for new frequencySuperheterodyne receiver: heterodyne RF signals with localtuner convert to common IF

Invented by E. Armstrong in 1918.AM: RF 0.535MHz-1.605 MHz, Midband 0.455MHz

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FM BroadcastingThe fre uenc of an FM broadcast station is usuall anexact multiple of 100 kHz from 87.5 to 108.5 MHz . Inmost of the Americas and Caribbean only oddmultiples are used.f m=15KHz, f=75KHz, =5, B=2(f m+ f)=180kHzPre-emphasis and de-emphasis

Random noise has a 'triangular' spectral distribution in an,

at the highest frequencies within the baseband . This can beoffset, to a limited extent, by boosting the high frequencies

before transmission and reducing them by a correspondingamount in the receiver. 65EEEF311 (2014-2015)

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FM Stereo MultiplexingFM Stereo Multiplexing


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FM Stereo MultiplexingFM Stereo Multiplexing

Fc=19KHz.(a) Multiplexer in transmitter

of FM stereo.(b) Demultiplexer in receivero FM stereo.

For non-stereo receiver

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TV FM broadcasting

f m=15KHz, f=25KHz, =5/3, B=2(f m+ f)=80kHzCenter f c+4.5MHz

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Ch5 Angle Modulation

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