BEC/ECE/TSR

Angle Modulation

BEC/ECE/TSR

Contents

Properties of Angle (exponential) Modulation Types

– Phase Modulation– Frequency Modulation

Line Spectrum & Phase Diagram Implementation Power

BEC/ECE/TSR

Properties

Linear CW Modulation (AM):– Modulated spectrum is translated message spectrum– Bandwidth message bandwidth

– SNRo at the output can be improved only by increasing the transmitted power

Angle Modulation: A non-linear process:-– Modulated spectrum is not simply related to the

message spectrum– Bandwidth >>message bandwidth. This results in

improved SNRo without increasing the transmitted power

BEC/ECE/TSR

Basic Concept

First introduced in 1931

A sinusoidal carrier signal is defined as: )]([cos)( ttEtc ccc

For un-modulated carrier signal the total instantaneous angle is:

)()( ttt ccc

Thus one can express c(t) as:

][Re)(cos)( )(tjccc

ceEtEtc

Thus: • Varying the frequency fc Frequency modulation• Varying the phase c Phase modulation

BEC/ECE/TSR

Basic Concept - Cont’d.

In angle modulation: Amplitude is constant, but angle varies (increases linearly) with time

t

Amplitude Ec

Initial phase c

Unmodulated carrier

Slope = c/t

t = 0

t(ms)

Unmodulated carrier

0

c(t) (red)

-/2

11/2

23/235/247/2

1 2 3 4

Phase-modulatedangle

Frequency-modulatedangle

2

0-1

m(t)

BEC/ECE/TSR

Phase Modulation (PM)

PM is defined If 0180)()()( ppc KtmKtmt

Thus )]([cos)( tmKtEtc pccPM

Where Kp is known as the phase modulation index

Ec

c(t)

c(t)

c(t)

i(t)

Instantaneous frequency

Rotating Phasor diagram

)()(

)( tdt

tdt cc

ci

Instantaneous phase )()( tmKt pi

BEC/ECE/TSR

Frequency Modulation (FM)

The instantaneous frequency is;

Where Kf is known as the frequency deviation (or frequency modulation index). Note: Kf < fc to make sure that f(t) >0.

Note that )()( tmKt fc

)()( tmKt fci

Integrating )()( tt cci

0)()(0

t

fcc dttmKtt

Substituting c(t) in c(t) results in: ])([cos)(0t

fccFM dttmKtEtc

Instantaneous phase

BEC/ECE/TSR

Waveforms

BEC/ECE/TSR

Important Terms

Carrier Frequency Deviation (peak)

Frequency swing pmpfpp EKf

Rated System Deviation (i.e. maximum deviation allowed)

FD =

75 kHz, FM Radio, (88-108 MHz band)25 kHz, TV sound broadcast5 kHz, 2-way mobile radio2.5 kHz, 2-way mobile radio

Percent Modulation

mfdc EKff

%100D

d

F

fm

Modulation Index m

d

f

f

BEC/ECE/TSR

FM Spectral Analysis

Let modulating signal m(t) = Em cos mt

Substituting it in c(t)FM expression and integrating it results in:

]sin[cos])([cos)(0

tEK

tEdttmKtEtc mmm

fcc

t

fccFM

Since m

d

f

f mfdc EKff and

)sin(sinsin)sin(coscos]sin[cos)( ttEttEttEtc mccmccmccFM

the terms cos ( sin mt) and sin ( sin mt) are defined in trigonometric series, which gives Bessel Function Coefficient as:

BEC/ECE/TSR

Bessel Function Coefficients

cos ( sin x) = J0 () + 2 [J2() cos 2x + J4() cos 4x + ....]

And sin ( sin x) = 2 [J1() sin x + J3() sin 3x + ....]

where Jn() are the coefficient of Bessel function of the 1st kind, of the order n and argument of .

BEC/ECE/TSR

FM Spectral Analysis - Cont’d.

.........]3sin)(2sin)(2[sin

........]4cos)(22cos)(2)([cos/)(

31

420

tJtJt

tJtJJtEtc

mmc

mmccFM

Substituting the Bessel coefficient results in:

Expanding it results in:

.........}....................

)])3(cos)3()[cos({

)])2(cos)2()[cos({

)])(cos)()[cos({

cos)()(

3

2

1

0

ttJE

ttJE

ttJE

tJEtc

mcmcc

mcmcc

mcmcc

ccFM Carrier signal

Side-bands signal(infinite sets)

Since )()1))( nn

n JJ Then tnJEtc mn

cncFM )(cos)()(

tnJEtc mn

cncFM )(cos)()(

BEC/ECE/TSR

FM Spectrum

J0()

c

J1()

c+ m

c+ 2m

c+ 3m

c+ 4m

J2()J3()

J4()

Side bands

Side bands

Bandwidth (?)

c- 3m

J2()

J3()

J4()

c- 2mc- 4m

BEC/ECE/TSR

FM Spectrum - cont’d.

• The number of side bands with significant amplitude depend on see below

c

= 0.5

c

= 1.0

c

= 2.5

c

= 4

Bandwidth

Generation and transmission of pure FM requires infinite bandwidth, whether or not the modulating signal is bandlimited. However practical FM systems do have a finite bandwidth with quite well pwerformance.

Most practical FM systems have 2 < < 10

BEC/ECE/TSR

FM Bandwidth BFM

The commonly rule used to determine the bandwidth is:– Sideband amplitudes < 1% of the un-modulated carrier can be

ignored. Thus Jn()> 0.01

For large values of , BFM = 2nfm= 2fm=2 (fc/ fm).fm = 2 fcBFM = 2nfm= 2fm=2 (fc/ fm).fm = 2 fc

For small values of , BFM = 2fmBFM = 2fm For limited cases

General case: use Carson equation BFM 2(fc + fm)

BFM 2 fm (1 + )BFM 2 fm (1 + )