Chapter 4. Angle Modulation

4.7 Generation of FM Waves

• Direct Method– A sinusoidal oscillator, with one of the reactive elements

in the tank circuit of the oscillator being directly controllable by the message signal

– The tendency for the carrier frequency to drift, which is usually unacceptable for commercial radio applications.

– To overcome this limitation, frequency stabilization of the FM generator is required, which is realized through the use of feed-back around the oscillator

• Indirect Method : Armstrong Modulator– The message signal is first used to produce a narrow-

band FM, which is followed by frequency multiplication to increase the frequency deviation to the desired level.

– Armstrong wide-band frequency modulator• The carrier-frequency stability problem is alleviated by

using a highly stable oscillator

– A Frequency multiplier • A memoryless nonlinear device• The input-output relation of such a device is

• A new FM wave is

)40.4()(...)()()( 2

11 tsatsatsatv n

n

)41.4()(22cos)(0

t

fcc dmktfAts

)42.4()()( tmkftf fci

(4.43))(22cos)(0

'''

dmktfAts

t

fcc

)44.4()()(' tmnknftf fci

4.8 Demodulation of FM Signals

• Frequency Discriminator– The FM signal is

• We can motivate the formulation of a receiver for doing this recovery by nothing that if we take the derivative of Eq. (4.44) with respect to time

• A typical transfer characteristic that satisfies this requirement is

dmktfAts

t

fcc0

)(22cos)(

)45.4()(22sin)]([2)(

0

dmktftmkfA

dt

tds t

fcfcc

)46.4(2 fjdt

d

)47.4(otherwise ,0

)2/()2/()],2/([2)(1

TcTcTc BffBfBffjfH

– The slope circuit• The circuit is also not required to have zero response

outside the transmission bandwidth

– The complex envelope of the FM signal s(t) is

)48.4()(2exp)(0

~

t

fc dmkjAtS

)49.4(otherwise ,0

2/2/)],2/([2)(1

~

TTT BfBBfjfH

)50.4(

elsewhere ,0

2

1

2

1),(

2

1

)()(2

1)(

~

~~

1

~

1

TTT BfBfSBfj

fSfHfS

1. Multiplication of the Fourier transform by j2πf is equivalent to differentiating the inverse Fourier transform

2. Application of the linearity property to the nonzero part of yields

the actual response of the slope circuit due to the FM wave s(t) is given by

)52.4()(2exp)(2

12

1)(

0

~

1

t

f

T

f

Tc dmkjtmB

kBAjts

)51.4()(2

1)(

2

1)(

~~~

1 tsBjtsdt

dts T

)(2)(~~

fSfjtsdt

d

)53.4(2

)(22cos)(2

12

1

)2exp()(Re)(

0

1

~

1

t

fc

T

f

Tc

c

dmktftmB

kBA

tfjtsts

• The envelope detector

• Under ideal conditions, the output of the envelope detector is

• The overall output that is bias-free

ttmB

k

T

f allfor ,1)(2

max

)54.4()(2

12

1)(1

tm

B

kBAtv

T

f

Tc

)55.4()(2

12

1)(2

tm

B

kBAtv

T

f

Tc

(4.56) )(

)()()( 21

tcm

tvtvtv

• Phase-Locked Loop– A feedback system whose operation is closely linked to

frequency modulation– Three major components

• Voltage-controlled oscillator (VCO)• Multiplier• Loop filter of a low-pass kind

– Fig. 4.14, a closed-loop feedback system

– VCO has bee adjusted so that when the control signal is zero, two conditions are satisfied

1.The frequency of the VCO is set precisely at the unmodulated carrier frequency fc of the incoming FM wave s(t)

2.The VCO output has a 90◦-degree phase-shift with respect to the unmodulated carrier wave.

– Suppose the incoming FM wave is

– The FM wave produced by the VCO as

– The multiplication of the incoming FM wave by the locally generated FM wave produces two components• A high-frequency component

• A low-frequency component

)57.4()](2sin[)( 1 ttfAts cc

)58.4()(2)(0

1 dmktt

f

)59.4()](2cos[)( 2 ttfAtr cv

)60.4()(2)(0

2 dvktt

v

)]()(4sin[ 21 tttfAAk cvcm

)]()(sin[ 21 ttAAk vcm

– Discard the double-frequency term, we may reduce the signal applied to the loop filter to

– The phase error is

– Eq. (4.62), (4.63), (4.65), and (4.60)constitute a linearized feedback model of the phase-locked loop

)61.4()](sin[)( tAAkte evcm

)62.4()(2)(

)()()(

01

21

t

v

e

dvkt

ttt

)()](sin[ tt ee

)63.4( )(

)()(

0 tk

K

tAAkte

e

v

evcm

)64.4(0 vcvm AAkkK

)65.4()()()(

dthetv

Loop-gain parameter of the phase lock loop

1. The inverse of this feedback path is described in the time domain by the scaled differentiator

2. The closed-loop time-domain behavior of the phase-locked loop is described by the overall output v(t) produced in response to the angle Φ1(t) in the incoming FM wave s(t)

3. The magnitude of the open-loop transfer function of the phase-locked loop is controlled by the loop-gain parameter K0

When the open-loop transfer function of a linear feedback system has a large magnitude compared with unity for all frequencies, the closed-loop transfer function of the system is effectively determined

by the inverse of the transfer function of the feedback path.

)66.4()(

2

1)( 2

dt

td

ktv

v

– We may relate the overall output v(t) to the input angle Φ1(t) by

)67.4()(

2

1)( 1

dt

td

ktv

v

)68.4( )(

)(22

1)(

0

tmk

k

dmkdt

d

ktv

v

f

t

f

v

4.10 Summary and Discussion

• Two kinds of angle modulation– Phase modulation (PM), where the instantaneous phase of the

sinusoidal carrier wave is varied linearly with the message signal– Frequency modulation (FM), where the instantaneous frequency

of the sinusoidal carrier wave is varied linearly with the message signal

• Frequency modulation is typified by the equation

• FM is a nonlinear modulation process• In FM, the carrier amplitude and therefore the transmitted

average power is constant• Frequency modulation provides a practical method for the

tradeoff of channel bandwidth for improved noise performance.

)70.4()(22cos)(0

t

fcc dmktfAts