analog communicationusers.okan.edu.tr/didem.kivanc/courses/eee307_2014_fall/eee307_… · 11/5/2013...

11/5/2013

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ANALOG COMMUNICATION (5) Fall 2013

Original slides by Yrd. Doç. Dr. Burak Kelleci

Modified by Yrd. Doç. Dr. Didem Kivanc Tureli

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING

OUTLINE

Phase and Frequency Modulation

Introduction

Basic Definitions

Properties of Angle-Modulated Waves

Spectrum of Frequency Modulation

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TYPES OF MODULATION

Modulation

Angle Modulation

Frequency Modulation

Phase Modulation

Amplitude Modulation

SSB DSB SSB-SC

Slides 3-4 Slides 5-6

TYPES OF MODULATION

ttfta c 2cos

If this changes, it is

frequency modulation

(FM) or phase

modulation (PM)

If this changes, it

is amplitude

modulation (AM)

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INTRODUCTION

Angle Modulation:

The angle of the carrier wave is varied according to

the baseband (message) signal.

The amplitude of the carrier wave is maintained

constant.

There are two forms of angle modulation

Phase modulation

Frequency modulation

Angle modulation gives better performance in the

presence of noise and interference at the expense of

increased bandwidth.

ttfta c 2cos

BASIC DEFINITIONS

qi(t) is the angle of a modulated sinusoidal carrier

at time t.

The angle modulated wave

where Ac is the carrier amplitude.

qi(t) changes 2 complete oscillation occurs

qi(t) increases monotonically with time

average frequency from t to t+Dt

tAts ic qcos

t

ttttf ii

tD

DD

qq

2

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BASIC DEFINITIONS

The instantaneous frequency of the angle-modulated

signal

Frequency is the derivative of the angle with

respect to time t

dt

td

t

ttt

tftf

i

ii

t

tt

i

q

qq

2

1

2lim

lim

0

0

D

D

D

DD

BASIC DEFINITIONS

Angle modulated signal s(t) may be interpreted

as a rotating phasor of length Ac and angle qi(t).

The angular velocity of this phasor is dqi(t)/dt.

As a simple case the angle of an unmodulated

carrier

The phasor velocity is 2fc

The constant c is the value of qi(t) at t=0

cci tft q 2

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ROTATING PHASORS??

ASIDE: why a rotating phasor? Didn’t we remove

the 2πfct term when we made it a phasor? Yes, usually,

in circuits class we do.

Yes we write it as above – but we always remember the

ej2πfct term, and remember that the phasor effectively

turns in time, as shown in the wikipedia:

http://en.wikipedia.org/wiki/Phasor

2 Phasor representationcos 2 c c cj f t j

c cA f t Ae Ae

BASIC DEFINITIONS

Phase Modulation

Instantaneous angle qi(t) is varied linearly with the

message signal

2fct represents the angle of unmodulated carrier

kp is the phase sensitivity of the modulator in radians

per volt (m(t) is assumed a voltage waveform)

For convenience the angle of unmodulated carrier at

t=0 is assumed 0

tmktft pci q 2

tmktfAts pcc 2cos

http://en.wikipedia.org/wiki/Phasor

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BASIC DEFINITIONS

Frequency Modulation

Instantaneous frequency fi(t) is varied linearly with

the message signal m(t)

fc is the frequency of the unmodulated carrier

kf is frequency sensitivity in Hertz per volt

The instantaneous phase is

Assuming the angle at t=0 is 0

tmkftf fci

t

fci dmktft0

22 q

t

fcc dmktfAts0

22cos

BASIC DEFINITIONS

Carrier Wave

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BASIC DEFINITIONS

Message Signal

Amplitude

Modulated Wave

Frequency

Modulated Wave

Phase Modulated

Wave

PROPERTIES OF ANGLE-MODULATED

WAVES

Property 1: Constancy of Transmitted Power

The amplitude of PM and FM waves equals to Ac

which is a constant value.

The average transmitted power of angle modulated

waves is constant.

2

2

1cav AP

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WAVES

Property 2: Nonlinearity of the modulation

process

Angle modulation is a nonlinear process violates

the principle of superposition

This nonlinearity property complicates the spectral

analysis of PM and FM waves.

tststs

tmktfAts

tmktfAts

tmtmktfAts

tmtmtm

pcc

pcc

pcc

21

22

11

21

21

2cos

2cos

2cos


WAVES

Property 3: Irregularity of Zero-Crossings

Since instantaneous angle depends on the message

signal or integral of message signal, the zero-

crossings of PM and FM wave have no perfect

regularity in their spacing across time-scale.

Zero-crossings: the instant time at which a waveform

changes its amplitude from positive to negative or the

other way around.

The information content of the message signal m(t)

resides in the zero-crossings of the modulated wave.

Due to the this property PM and FM waves shows

better performance in the presence of interferers.

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WAVES

Property 4: Visualization Difficulty of Message

Waveform

In AM the message signal is seen as the envelope of

the modulated wave. This is not so in angle

modulated waves.

Property 5: Trade-Off of Increased Transmission

Bandwidth for Improved Noise Performance

The improvement in noise performance is at the

expense of increasing the transmission bandwidth.

This trade-off is not possible with amplitude

modulation.

PHASE AND FREQUENCY MODULATION

Comparing PM and FM waveforms reveals that

FM signal can be considered as a PM signal in

which the message signal is integrated version of

m(t).

The properties of PM signals can be deduced from

the properties of FM signal

Therefore. we will concentrate on FM signals

t

fcc dmktfAts0

22cos

tmktfAts pcc 2cos

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FREQUENCY MODULATION

Since the FM signal is a nonlinear function of the

message signal m(t), it is not easy to analyze its

spectrum.

Let’s start the simplest case of the message

signal, which is a single sinusoidal tone.

tfAtm mm 2cos


The ratio of peak frequency deviation to the max modulation

frequency is called modulation index

tfjtfj

C

tftfj

C

mcC

m

m

f

mcC

t

mmfcC

mc

mc

eeA

eA

tftfA

tff

kAtfA

dfAktfAts

2sin2

2sin2

0

2sin2cos

2sin2cos

2cos22cos

tmff

kA

m

f

m

fm

offrequencymax

deviationfrequencypeak

D

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Expanding this signal to Fourier Series

The Fourier Series Coefficients

m

tfjfTe m /1 with periodic is 0

2sin

n

tnfj

n

tfj mm eCe 22sin

2/

2/

22sin

0

0

0

1 T

T

tnfjtfj

n dteeT

C mm


Jn() nth order Bessel function of the first kind

n

xnxj

T

T

tftnfj

T

T

tnfjtfj

n

J

dxe

dteT

dteeT

C

mm

mm

sin

2/

2/

2sin2

0

2/

2/

22sin

0

2

1

1

1

0

0

0

0

n

tnfj

n

tfj mm eJe 22sin

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Plots of Bessel functions of the first kind

Matlab command: besselj

nJJ n

n

n allfor 1


The spectrum of FM signal with sinusoidal input is

n

mCnC

n

tnffj

nC

n

tnfj

n

tfj

C

tnffJA

eJA

eJeAts

mC

mC

2cos

2

22

n

mCmCnC nfffnfffJ

AfS

2

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NARROWBAND FREQUENCY MODULATION

For small values of the modulation index

20

22

1

11

0

nJ

JJ

J

n

tfftfftfAts mCmCCC

2cos

22cos

22cos

mCmC

mCmCCCC

ffffff

ffffffffffA

fS

2

2

2

PROPERTIES OF FM SPECTRUM

The spectrum of an FM signal contains carrier

components and an infinite set of side

frequencies located symmetrically on either side

of the carrier at frequency separations of fm, 2fm,

3fm, …

For the special case of small compared with

unity, only the Bessel coefficients J0() and J1()

have significant values, so that the FM signal is

effectively composed of a carrier and a single pair

of side frequencies at fc ± fm.

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The amplitude of the carrier component varies with

according to J0(). The physical explanation for this

property is that the envelope of an FM signal is

constant, so that the average power across 1-ohm

resistor is also constant

When the carrier is modulated to generate FM signal,

the power in the side frequencies appear only at the

expense of the carrier power.

2

2

1cAP

n

n

n

nc JJAP 222 12

1


The integral of the power spectral density of s(t)

will give the total energy of the FM signal.

mC

n

nC

mC

n

nC nfffJA

nfffJA

fS

44

2222

2

2

2

44

2

2

2

22

2222

C

n

n

C

n

nC

n

nC

n

nC

A

JA

JA

JAJA

dffSP

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SPECTRA OF FM SIGNALS


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TRANSMISSION BANDWIDTH OF FM

SIGNALS

In theory, FM signals has infinite number of side

frequencies, so the required bandwidth is infinite.

In practice, FM signals are limited to a finite number

of significant side frequencies.

For large values of modulation index , the bandwidth

approaches the total frequency excursion 2Df.

For small values of modulation index , the bandwidth

approaches 2fm.

This empirical relation is known as Carson’s rule.

DD

11222 fffB mT

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SIGNALS

More accurate definition:

Transmission bandwidth of an FM wave as the

separation between the two frequencies beyond

which none of the side frequencies is greater than 1

percent of the carrier amplitude obtained when the

modulation is removed.

The bandwidth is 2nmaxfm, where fm is the frequency

modulation and nmax is the largest value of the

integer n that satisfies the requirement |Jn()|>0.01


SIGNALS

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SIGNALS

Let’s consider that m(t) is an arbitrary waveform.

The deviation ratio D, defined as the ratio of the

frequency deviation Df, which corresponds to the

maximum possible amplitude of the modulation

signal m(t), to the highest modulation frequency

W.

The deviation ratio D plays the same role for

nonsinusoidal modulation that modulation index

b plays for the case of sinusoidal modulation.

W

fD

D

EXAMPLE

The frequency deviation Df of FM broadcasting is

fixed at 75KHz. The maximum audio frequency of

interest is 15KHz. What is deviation ratio?

Calculate the bandwidth using Carson’s rule and

using the curve.

KHzfB

KHzKHzKHzB

KHz

KHzD

T

T

240752.32.3

18015752

515

75

D

Carson’s rule underestimates the bandwidth by 25% compared with

the result of using the universal curve

analog communicationusers.okan.edu.tr/didem.kivanc/courses/eee307_2014_fall/eee307_… · 11/5/2013...

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