Introduction to Analog And Introduction to Analog And Digital CommunicationsDigital Communications

Second EditionSecond Edition

Simon Haykin, Michael MoherSimon Haykin, Michael Moher

Chapter 4 Angle ModulationChapter 4 Angle Modulation

4.1 Basic Definitions4.1 Basic Definitions

4.2 Properties of Angle-Modulated Waves4.2 Properties of Angle-Modulated Waves

4.3 Relationship between PM and FM waves4.3 Relationship between PM and FM waves

4.4 Narrow-Band Frequency Modulation4.4 Narrow-Band Frequency Modulation

4.5 Wide-Band Frequency Modulation4.5 Wide-Band Frequency Modulation

4.6 Transmission Bandwidth of FM waves4.6 Transmission Bandwidth of FM waves

4.7 Generation of FM waves4.7 Generation of FM waves

4.8 Demodulation of FM signals4.8 Demodulation of FM signals

4.9 Theme Example 4.9 Theme Example

: FM Stereo Multiplexing: FM Stereo Multiplexing

4.10 Summary and Discussion 4.10 Summary and Discussion

33

Angel modulation The angle of the carrier wave is varied according to the information-bearing

signal

Lesson 1 : Angle modulation is a nonlinear process, which testifies to its sophisticated nature. In the context of analog communications, this distinctive property of angle modulation has two implications : In analytic terms, the spectral analysis of angle modulation is complicated. In practical terms, the implementation of angle modulation is demanding

Lesson 2 : Whereas the transmission bandwidth of an amplitude-modulated wave is of limited extent, the transmission bandwidth of an angle-modulated wave may an infinite extent, at least in theory.

Lesson 3 : Given that the amplitude of the carrier wave is maintained constant, we would intuitively expect that additive noise would affect the performance of angle modulation to a lesser extent than amplitude modulation.

44

4.1 Basic Definitions4.1 Basic Definitions

Angle-modulated wave

the average frequency in hertz

The instantaneous frequency of the angle-modulated signal

)1.4()](cos[)( tAts ic

t

ttttf it

t

2

)()()(

(4.2) )(

2

1

2

)()(lim

)(lim)(

0

0

dt

td

t

ttt

tftf

i

it

t

tti

0)(for ,2)( tmtft cci

55

1. Phase modulation (PM) is that form of angle modulation in which the instantaneous angle is varied linearly with the message signal

2. Frequency modulation (FM) is that form of angle modulation in which the instantaneous frequency is varied linearly with the message signal

)3.4()(2)( tmktft pci

)4.4()(2cos)( tmktfAts pcc

)5.4()()( tmkftf fci

)6.4()(22

)(2)(

0

0

dmktf

dft

t

fc

t

ii

)7.4()(22cos)(0

t

fcc dmktfAts

Table. 4.1

66

Table.4.1Table.4.1 Back Next

77

4.2 Properties of Angle-Modulated Waves4.2 Properties of Angle-Modulated Waves

Property 1 : Constancy of transmitted power The amplitude of PM and FM waves is maintained at a constant value equal

to the carrier amplitude for all time. The average transmitted power of angle-modulated waves is a constant

Property 2 : Nonlinearity of the modulation process Its nonlinear character

)8.4(2

1 2

cav AP

)()()( 21 tmtmtm

)(2cos)(

))()((2cos)(

11

21

tmktfAts

tmtmktfAts

pcc

pcc

)(2cos)( 22 tmktfAts pcc

)()()( 21 tststs Fig. 4.1

88

Fig.4.1Fig.4.1 Back Next

99

Property 3 : Irregularity of zero-crossings Zero-crossings are defined as the instants of time at which a waveform

changes its amplitude from a positive to negative value or the other way around.

The irregularity of zero-crossings in angle-modulation waves is also attributed to the nonlinear character of the modulation process.

The message signal m(t) increases or decreases linearly with time t, in which case the instantaneous frequency fi(t) of the PM wave changes form the unmodulated carrier frequency fc to a new constant value dependent on the slope of m(t)

The message signal m(t) is maintained at some constant value, positive or negative, in which case the instantaneous frequency fi(t) of the FM wave changes from the unmodulated carrier frequency fc to a new constant value dependent on the constant value of m(t)

1010

Property 4 : Visualization difficulty of message waveform The difficulty in visualizing the message waveform in angle-modulated

waves is also attributed to the nonlinear character of angle-modulated waves.

Property 5 : Tradeoff of increased transmission bandwidth for improved noise performance The transmission of a message signal by modulating the angle of a

sinusoidal carrier wave is less sensitive to the presence of additive noise

1111

Fig. 4.2

1212

Fig.4.2Fig.4.2 Back Next

1313

1414

1515

1616

4.3 Relationship Between PM and 4.3 Relationship Between PM and FM wavesFM waves

Fig. 4.3(a) An FM wave can be generated by first integrating the message signal m

(t) with respect to time t and then using the resulting signal as the input to a phase modulation

Fig. 4.3(b) A PM wave can be generated by first differentiating m(t) with respect t

o time t and then using the resulting signal as the input to a frequency modulator

We may deduce the properties of phase modulation from those of frequency modulation and vice versa

Fig. 4.3

1717

Fig.4.3Fig.4.3 Back Next

1818

4.4 Narrow-Band Frequency Modulation4.4 Narrow-Band Frequency Modulation

We first consider the simple case of a single-tone modulation that produces a narrow-band FM wave

We next consider the more general case also involving a single-tone modulation, but this time the FM wave is wide-band

The two-stage spectral analysis described above provides us with enough insight to propose a useful solution to the problem

A FM signal is )9.4()2cos()( tfAtm mm

(4.10) )2cos(

)2cos()(

tfff

tfAkftf

mc

mmfci

(4.11) Amfkf

The frequency deviation

)12.4()2sin(2)( tff

ftft m

m

ci

)13.4(mf

f

Modulation index of the FM wave

The phase deviation of the FM wave

)14.4()2sin(2)( tftft mci

1919

The FM wave is

If the modulation index is small compared to one radian, the approximate form of a narrow-band FM wave is

1. The envelope contains a residual amplitude modulation that varies with time

2. The angel θi(t) contains harmonic distortion in the form of third- and higher order harmonics of the modulation frequency fm

)15.4()2sin(2cos)( tftfAts mcc

BABABA sinsincoscos)cos(

)16.4()2sin(sin)2sin()2sin(cos)2cos()( tftfAtftfAts mccmcc

1)2sin(cos tfm )2sin()2sin(sin tftf mm

)17.4()2sin()2sin()2cos()( tftfAtfAts mcccc

Fig. 4.4

2020

Fig.4.4Fig.4.4 Back Next

2121

We may expand the modulated wave further into three frequency components

The basic difference between and AM wave and a narrow-band FM wave is that the algebraic sign of the lower side-frequency in the narrow-band FM is reversed

A narrow-band FM wave requires essentially the same transmission bandwidth as the AM wave.

)18.4()(2cos)(2cos2

1)2cos()( tfftffAtfAts mcmcccc

)19.4()(2cos)(2cos2

1)2cos()( tfftffAtfAts mcmccccAM

2222

Phasor Interpretation A resultant phasor representing the narrow-band FM wave that is appro

ximately of the same amplitude as the carrier phasor, but out of phase with respect to it.

The resultant phasor representing the AM wave has a different amplitude from that of the carrier phasor, but always in phase with it.

Fig. 4.5

2323

Fig.4.5Fig.4.5 Back Next

2424

4.5 Wide-Band Frequency 4.5 Wide-Band Frequency ModulationModulation

Assume that the carrier frequency fc is large enough to justify rewriting Eq. 4.15) in the form

The complex envelope is

(4.20) )2exp()(Re

))2sin(2exp(Re)(~

tfjts

tfjtfjAts

c

mcc

(4.21) ] )2sin(exp[)(~

tfjAts mc

] )2sin(exp[

] )22sin(exp[

] ))/(2sin(exp[)(~

tfjA

ktfjA

fktfjAts

mc

mc

mmc

(4.22))2exp()(~

tnfjcts mn

n

2525

The complex Fourier coefficient

(4.23)]2)2sin(exp[

)2exp()(

)2/(1

)2/(1

)2/(1

)2/(1

~

m

m

m

m

f

fmmcm

m

f

fmn

dttnfjtfjAf

dttnfjtsfc

)24.4(2 tfx m

)25.4()]sin(exp[2

dxnxxj

Ac c

n

)26.4()]sin(exp[2

1)(

dxnxxjJ n

)27.4()(ncn JAc

(4.28))2exp()()(~

tnfjJAts mn

nc

(4.29)])(2exp[)(Re)(

tnffjJAts mcn

nc

2626

In the simplified form of Eq. (4.29)

(4.30)])(2cos[)()( tnffJAts mcn

nc

(4.31))]()()[(2

)( mcmcn

nc nfffnfffJ

AfS

2727

Properties of single-tone FM for arbitrary modulation index β

1. For different integer values of n,

2. For small values of the modulation index β

3. The equality holds exactly for arbitrary β

)32.4(even for ),()( nJJ nn

)33.4(odd for ),()( nJJ nn

)34.4(

2,0)(

,2

)(

,1)(

1

0

nJ

J

J

n

)35.4(1)(2

nnJ

Fig. 4.6

2828

Fig.4.6Fig.4.6 Back Next

2929

1. The spectrum of an FM wave contains a carrier component and an infinite set of side frequencies located symmetrically on either side of the carrier at frequency separations of fm,2fm, 3fm….

2. The FM wave is effectively composed of a carrier and a single pair of side-frequencies at fc±fm

3. The amplitude of the carrier component of an FM wave is dependent on the modulation index β

The average power of such a signal developed across a 1-ohm resistor is also constant.

The average power of an FM wave may also be determined form

2

av 2

1cAP

)36.4()(2

1 22

nc JAP

3030

Fig. 4.7

Fig. 4.8

3131

Fig.4.7Fig.4.7 Back Next

3232

Fig.4.8Fig.4.8 Back Next

3333

4.6 Transmission Bandwidth of FM 4.6 Transmission Bandwidth of FM waveswaves

Carson’s Rule The FM wave is effectively limited to a finite number of significant side-freq

uencies compatible with a specified amount of distortion Two limiting cases

1. For large values of the modulation index β, the bandwidth approaches, and is only slightly greater than the total frequency excursion 2∆f,

2. For small values of the modulation index β, the spectrum of the FM wave is effectively limited to the carrier frequency fc and one pair of side-frequencies at fc±fm, so that the bandwidth approaches 2fm

An approximate rule for the transmission bandwidth of an FM wave

)37.4(1

1222

fffB mT

3434

Universal Curve for FM Transmission Bandwidth A definition based on retaining the maximum number of significant sid

e frequencies whose amplitudes are all greater than some selected value. A convenient choice for this value is one percent of the unmodulated ca

rrier amplitude The transmission bandwidth of an FM waves

The separation between the two frequencies beyond which none of the side frequencies is greater than one percent of the carrier amplitude obtained when the modulation is removed.

As the modulation index β is increased, the bandwidth occupied by the significant side-frequencies drops toward that value over which the carrier frequency actually deviates.

Fig. 4.9

Table. 4.2

3535

Table.4.2Table.4.2 Back Next

3636

Fig.4.9Fig.4.9 Back Next

3737

Arbitrary Modulating Wave The bandwidth required to transmit an FM wave generated by an

arbitrary modulating wave is based on a worst-case tone-modulation analysis

The deviation ratio D

The generalized Carson rule is

)38.4(W

fD

)39.4()W(2 fBT

Fig. 4.9

3838

3939

4.7 Generation of FM Waves4.7 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 Fig. 4.10

4040

Fig.4.10Fig.4.10 Back Next

4141

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

Fig. 4.11

)41.4()(22cos)(0

t

fcc dmktfAts

)42.4()()( tmkftf fci

(4.43))(22cos)(0

'''

dmktfAts

t

fcc

)44.4()()(' tmnknftf fci

4242

Fig.4.11Fig.4.11 Back Next

4343

4.8 Demodulation of FM Signals4.8 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

4444

The slope circuit The circuit is also not required to have zero response outside the transmissi

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

Fig. 4.12

)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

4545

Fig.4.12Fig.4.12 Back Next

4646

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

4747

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

Fig. 4.13

4848

Fig.4.13Fig.4.13 Back Next

4949

Phase-Locked Loop A feedback system whose operation is closely linked to frequency mod

ulation 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.

Fig. 4.14

5050

Fig.4.14Fig.4.14 Back Next

5151

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

5252

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

5353

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

5454

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

Fig. 4.15

5555

Fig.4.15Fig.4.15 Back Next

5656

4.9 Theme Example : FM Stereo 4.9 Theme Example : FM Stereo MultiplexingMultiplexing

The specification of standards for FM stereo transmission is influenced by two factors

1. The transmission has to operate within the allocated FM broadcast channels

2. It has to be compatible with monophonic radio receivers

The multiplied signal is recovered by frequency demodulating the incoming FM wave

)69.4()4cos()4cos()]()([)]()([)( tfKtftmtmtmtmtm ccrlrl

Fig. 4.16

5757

Fig.4.16Fig.4.16 Back Next

5858

4.10 Summary and Discussion4.10 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

5959

Fig.4.17Fig.4.17 Back Next

Fig. 4.17

6060

Fig.4.18Fig.4.18 Back Next

Fig. 4.18

6161

Fig.4.19Fig.4.19 Back Next

Fig. 4.19

6262

Fig.4.20Fig.4.20 Back Next

Fig. 4.20

6363

Fig.4.21Fig.4.21 Back Next

Fig. 4.21