lecture 8 angle modulation - university of ottawa

9-Jan-18

Angle Modulation

Angle modulation: frequency modulation (FM) or phase

modulation (PM).

Basic idea: vary frequency (FM) or phase (PM) according

to the message signal.

While AM is (almost) linear, FM or PM is highly nonlinear.

FM/PM provide many advantages (main – noise

immunity) over AM, at a cost of larger bandwidth.

Demodulation may be complex, but modern ICs allow

cost-effective implementation.

Example: FM radio (high quality, not expensive

receivers).

Lecture 8

Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 1(28)

9-Jan-18

Angle Modulation: Basic Definitions

• Angle-modulated signal (PM or FM) can be expressed as:

• Phase modulation:

• Frequency modulation:

( ) ( )( )cosc

x t A t= ψ

( ) ( ), ( ) ( )c

t t t t m tψ = ω +ϕ ϕ = ∆ϕ ⋅

( ) ( )0

, ( ) ( )

t

ct t d t m tψ = ω + Ω τ τ Ω = ∆Ω⋅∫

• Max phase deviation:

• Max frequency deviation:

• Normalized message signal:

Max ( ) Max ( )c

t t t∆ϕ = ϕ = ψ −ω

Max ( ) Max ( )c

t t∆Ω = Ω = ω −ω

( ) 1m t ≤

Note: deviation is w.r.t. unmodulated value.

[ ] 0( ) (0)

ct tψ ≈ ω +Ω +ϕfor a short period of time (small t):


9-Jan-18

Angle Modulation: Parameters

• Instantaneous frequency:

• Instantaneous phase:

• Effect of mod. signal amplitude:

( )( )

( ) ( )

( ) ( )

,

,

c c

c c

d t dm td t PM

t dt dtdt

t m t FM

ϕψ ω + = ω + ∆ϕ

ω = = ω +Ω = ω + ∆Ω⋅

( ) ( )

( )

( ) ( )0

0 0

( ),

,

c ct

t t

c c

t t t m t PM

t dt d t m d FM

ω +ϕ = ω + ∆ϕ⋅

ψ = ω τ τ = ω + Ω τ τ = ω + ∆Ω τ τ

∫∫ ∫

( ) ( ), max ( ) 1M t A m t m t= ⋅ =

,

,

p

f

k A PM

k A FM

∆ϕ =∆Ω =

, - modulation constants,

Hz/V & rad./V

f pk k

measured in lab 3.


9-Jan-18

Angle Modulation: Examplesrectangular pulse sawtooth pulse


9-Jan-18

Example: Sinusoidal Modulating Signal

• Assume that

• Instantaneous phase:

• Modulated signal:

• Modulation indices:

( )( ) cosm

m t t= ω

( )( )

( )

cos ,

sin ,

c m

c m

m

t t PM

tt t FM

ω + ∆ϕ⋅ ωψ = ∆Ωω + ω ω

( )

( )

( )

cos cos ,

cos sin ,

c c m

c c m

m

A t t PM

x t

A t t FM

ω + ∆ϕ⋅ ω = ∆Ω

ω + ω ω

,

,

p

fm

PM

FM

β = ∆ϕ

∆Ωβ = ω

Valid in general case

as well, with

-> max.

modulating frequencym

ω


9-Jan-18

Spectrum of Angle-Modulated Signal

• Consider sinusoidal modulating signal:

• Complex envelope is expanded in Fourier series:

• Expansion coefficients are

• Finally,

( ) ( ) ( )sincos sin Re m c

j t j tc c m cx t A t t A e e

β⋅ ω ω = ω +β⋅ ω =

( ) ( )sinm m

j t jn t

c c n

n

C t A e A c e

∞β⋅ ω ω

=−∞

= = ∑

( )

0

2 sinsin

0

1 1( )

2

mT m

m m

u tj u nuj t jn t

n n

m

c e e dt e du JT

=ω π β −β ω − ω= = = β

π∫ ∫

( ) ( ) ( )cosc n c m

n

x t A J n t

∞

=−∞

= β ω + ω ∑

- Bessel function of 1st kind & n-th order,( )n

J β ( ) ( )( 1)nn n

J J−

β = − β


9-Jan-18

Spectrum of Angle Modulation:

Increasing n moves the peak to the right!

( )n

J β


9-Jan-18

n 0.1β = 0.2β = 0.5β = 1β = 2β = 5β = 8β = 10β = n

0 0.998 0.990 0.938 0.765 0.224 -0.178 0.172 -0.246 0

1 0.050 0.100 0.242 0.440 0.577 -0.328 0.235 0.043 1

2 0.001 0.005 0.031 0.115 0.353 0.047 -0.113 0.255 2

3 0.020 0.129 0.365 -0.291 0.058 3

4 0.002 0.034 0.391 -0.105 -0.220 4

5 0.007 0.261 0.186 -0.234 5

6 0.001 0.131 0.338 -0.014 6

7 0.053 0.321 0.217 7

8 0.018 0.223 0.318 8

9 0.006 0.126 0.292 9

10 0.001 0.061 0.207 10

11 0.026 0.123 11

12 0.010 0.063 12

13 0.003 0.029 13

14 0.001 0.012 14

15 0.004 15

16 0.001 16

the last significant

spectral component:

[ ]1n = β+

( ) ( ) ( )cosc n c m

n

x t A J n t

∞

=−∞

= β ω + ω ∑

Spectrum of Angle Modulation: ( )n

J β


9-Jan-18

Spectrum: Examples

100 150 200 250 3000

0.5

1

.

0.3β =

100 150 200 250 3000

0.5

1

.

1β =

5β =

0 100 200 300 4000

0.1

0.2

0.3

0.4

.

0 100 200 300 4000

0.2

0.4

.

10β =


9-Jan-18

Bandwidth of Angle-Modulated Signal

• Power bandwidth (98% of the power) of angle-modulated

signal (Carson’s rule):

• Power bandwidth of PM and FM signals:

• These expressions hold for a general modulating signal

as well, - the max. modulating frequency.

• Angle modulation with large index expands spectrum!

( )2 1m

∆ω ≈ β+ ω

( )( )

( )

2 1 ,2 1

2 ,

m

m

m

PM

FM

∆ϕ+ ω∆ω≈ β+ ω = ∆Ω+ω

mω


9-Jan-18

Narrowband Angle Modulation

• Modulation index is low,

• Modulated signal can be expressed as:

• Similar to AM signal, the bandwidth is (both, PM & FM)

( ) ( )

( ) ( )

cos sin

cos cos cos2 2

c c m

c c

c c c m c m

x t A t t

A AA t t t

= ω +β⋅ ω = β β

= ω + ω +ω − ω −ω

1β<<

2 ( )x

S f

f

cf( )c

f F− ( )cf F+

0,180

2

cA β 0, 0

2

cA β

cA

2m

∆ω≈ ω


9-Jan-18

Wideband Angle Modulation

• Modulation index is high,

• The signal bandwidth is:

• Different for PM and FM!

• Wideband FM: the bandwidth is twice the frequency

deviation. Does not depend on the modulating

frequency.

• Wideband PM: the bandwidth depends on modulating

frequency.

• Modulation index bandwidth expansion factor.

1β >>

2 ,2

2 ,

m

m

PM

FM

∆ϕ⋅ω∆ω≈ βω =

∆Ω

β =


9-Jan-18

PM Modulator

variable phase

shifter

( )m t

cos( )c

A tω

~

( ) ( )cos ( )c c

x t A t m t= ω +β⋅

RF oscillator

General

principle:

Practical

implementation:

L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.

0

0

,

modulation constant

resonant frequency

ck f f f f

k

f

∆ϕ ≈ ∆ ∆ = −

=

=

Q.: sketch ( )f∆ϕ ∆


9-Jan-18

FM Modulator

RF oscillator

( )m t

( ) ( ) 0

0

cos

t

c cx t A t d

= ω + Ω τ τ + ϕ

∫variable

resonant

circuit

General

principle:

Practical

implementation:

Difficulty: frequency

stability.

Suitable for

narrowband FM only.



9-Jan-18

Narrowband Angle Modulator

( ) [ ]cos ( )

cos sin ( )

c c

c c c c

x t A t m t

A t A t m t

= ω +β⋅ ≈

≈ ω − ω ⋅β⋅

( )pm tβ

( )fm tβ

Small modulation index: 1β≪

sin ( )c c

A t m tω ⋅β⋅

Message signal

Out

J.Proakis, M.Salehi, Communications Systems Engineering, Prentice Hall, 2002

Q.: can an AM modulator

be used instead?


9-Jan-18

Indirect Wideband Angle Modulator

Frequency multiplier:

( )2 1cos ( ) 1 cos 2 ( )

2t tψ = + ψ

( )1cos 2 ( )

2tψ

BPF

J.P

roakis

, M

.Sale

hi, C

om

munic

ations S

yste

ms E

ngin

eering, P

rentice H

all, 2002


9-Jan-18

Indirect Wideband FM Transmitter

(Amstrong)

B.P. Lathi, Modern Digital and Analog Communications Systems, Oxford University Press


9-Jan-18

Direct Wideband Angle Modulator


Explain how it operates

Hint: consider it without feedback first

Explain why feedback is required

Explain why frequency divider is required


9-Jan-18

FM Demodulators

• FM-to-AM conversion:

( ) ( )0for

2

c

c c

BH f H f f f f= + α − − <

( ) ( ) 0

0

cos

t

c cx t A t d

= ω + Ω τ τ +ϕ

∫ ( ) ( ) [ ]0

( ) cos *c

y t A H t= +αΩ

( )tΩ∼

• Possible candidate: (differentiator) ( ) 2H f f= π


9-Jan-18

FM Slope Detector• Circuit diagram:

• Magnitude frequency response:

fre

qu

en

cy b

an

d fo

r lin

ea

r F

M-

AM

co

nve

rsio

n


9-Jan-18

Balanced Discriminator: Block Diagram



9-Jan-18

Balanced Discriminator: Circuit Diagram



9-Jan-18

Phased Locked Loop (PLL) Detector

( ) [ ]sin ( )in in c inv t A t t= ω +ϕ

( ) [ ]0 0 0cos ( )

cv t A t t= ω +ϕ

( ) ( )0 2( ) ( )

VCO c c

dt t t v t

dtω = ω +ϕ = ω +α

Informally,

0( ) ( )

int tϕ ≈ ϕ

2

1( ) ( )

in

dv t t

dt≈ ϕα

( ) [ ]1 2

1 0sin ( ) ( ) (2 )term

2in c

A Av t t t= ϕ −ϕ + ω


9-Jan-18

PLL Detector: Linear Model

( )in

tϕ

0( )tϕ

( )t∆ϕ ( )1v t

0 2( ) ( )

dt v t

dtϕ = α

( ) [ ]

( )

1 2

2 0

1 2

0

sin ( ) ( )2

( ) ( ) ( )2

in

in d

A Av t t t

A At t K t

= ϕ −ϕ ≈

≈ ϕ −ϕ = ∆ϕ

[ ]2

1 1( ) ( ) ( ) ( )

in in

d dv t t t t

dt dt= ϕ − ∆ϕ ≈ ϕα α

( ) [ ]1 2

1 0sin ( ) ( )

2

(2 )term

in

c

A Av t t t= ϕ −ϕ +

+ ω


9-Jan-18

FM Demodulator: Lab 3

explain its operation !


9-Jan-18

Comparison of AM and FM/PM

• AM is simple (envelope detector) but no

noise/interference immunity (low quality).

• AM bandwidth is twice or the same as the modulating

signal (no bandwidth expansion).

• Power efficiency is low for conventional AM.

• DSB-SC & SSB – good power efficiency, but complex

circuitry.

• FM/PM – spectrum expansion & noise immunity. Good

quality.

• More complex circuitry. However, ICs allow for cost-

effective implementation.


9-Jan-18

Important Properties of Angle-Modulated

Signals: Summary

• FM/PM signal is a nonlinear function of the message.

• The signal’s bandwidth increases with the modulation

index.

• The carrier spectral level varies with the modulation

index, being 0 in some cases.

• Narrowband FM/PM: the signal’s bandwidth is twice that

of the message (the same as for AM).

• The amplitude of the FM/PM signal is constant (hence,

the power does not depend on the message).


9-Jan-18

Summary

Angle modulation: PM & FM

Spectra of angle-modulated signals. Modulation index.

Narrowband (low-index) & wideband (large-index) modulation.

Signal bandwidth.

Relation between PM and FM.

Generation of angle-modulated signals. Narrowband & wideband

modulators.

Demodulation of PM and FM signals. Slope detector & balanced

discriminator. PLL detector.

Comparison of AM and FM/PM.

Homework: Reading: Couch, 5.6, 4.13, 4.14. Study carefully all

the examples and make sure you understand them.

Lecture 8


lecture 8 angle modulation - university of ottawa

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