Angle Modulation (FM/PM)

by Erol Seke

For the course “Communications”

ESKİŞEHİR OSMANGAZİ UNIVERSITY

Changing the frequency and phase angle of the carrier signal with the message

signal are called “Frequency Modulation” (FM) and “Phase Modulation” (PM)

respectively. The class of such techniques is called “Angle Modulation”.

))()(cos()( tpttAtu c

phase deviation constant for PM

dt

tdtfi

)(

2

1)(

instantaneous frequency

))(cos()( ttAtu cc dt

tdt ci

)()(

so

Let )(tm be the message signal )()( tmkt p

dt

tdtmkt ci

)()()(

frequency deviation constant for FM

dt

tdti

)()(

))(cos()( tAtu c

center frequency deviation around center

dt

tdftf ci

)(

2

1)(

or

or

Phase is only meaningful with a particular given frequency, cannot be thought of independent entities

FMdmk

PMtmk

tt

f

p

,)(2

,)(

)(

FMtmk

PMtmdt

dk

tdt

d

f

p

,)(2

,)(

)(

or

FM modulator)(tm

PM modulator dt)(tm

PM modulator)(tm

FM modulatordt

d)(tm

)cos()( tAtx mExample : Tone signal is given. Find FM and PM waveforms.

)cos()()( tAktxkt mppPM

)sin()()( tAk

dxkt m

m

t

FM

(PM)

(FM)

therefore))cos(cos()( tAktAtu mpcc

))sin(cos()( tAk

tAtu m

m

cc

(PM)

(FM)

FMPM

Example :

Rectangular wave as message signal.

Note that neither FM nor PM changes the

amplitude of the carrier.

PM

FM

FM changes the frequency

PM changes the phase of the carrier

Remember that when there is only a finite

number of frequency values, it is FSK.

When there is only a finite number of

phase values, it is called PSK.

Spectral Characteristics of Angle Modulated Signals

Let ))2sin(2cos()( tftfAtu mcc

Modulation index

}Re{)()2sin(2 tfjtfj

mcc eeAtu

Since is periodic it can be represented by Fourier series)(tu

m

mm

f

tfjntfj

mn dteefC

/1

0

2)2sin(

2

0

))sin((

2

1dueC uuj

n

Let tfu m2

This integral is called the Bessel function

of the first kind of order n)(nJ

The Fourier series is

n

tnfj

n

tfj mm eJe 2)2sin(

)(

FM

f

tmk

PMtmk

m

f

p

2

|})(max{|

|})(max{|

Therefore

n

ctnfj

nc

tfjeeJAtu m

22

)(Re)(

n

mcnc tnffJAtu ))(2cos()()(

Actual bandwidth of FM/PM is infinite. So we usually define a finite “effective bandwidth”

Bessel function can be represented by a series expansion

0

2

2

)!(!

)()1()(

k

knk

nnkk

J

For small β!2

)(n

Jn

n

n

(approximation)

For 2n )(nJ is negligible.

Therefore only the first sideband corresponding to n=1 is of importance.

The effective bandwidth of an angle modulated signal is, in general, given by

mc fB )1(2 which contains the 98% of the signal power.

Modulation index Frequency of the sinusoidal message signal (or BW of m(t))

For the message signal )2cos()( tfatm m

FMff

ak

PMfak

fB

m

m

f

mp

mc

,)1(2

,)1(2

)1(2

increasing a increases the effective bandwidth

increasingmf increases the bw.

proportional increase in PM (large)

additive increase in FM (small)

Generation of Narrowband FM/PM (Phasor View)

re

im phase of the carrier

re

imsum

90° phase shifted carrier

with lower amplitude

re

sum

changing amplitude of this carrier deviates the phase angle of the sum

(do AM) (get PM)

Generation of Narrowband FM/PM

90phase delay

)(tx

)cos( tc)sin( tc

PM

90phase delay

)(tx

)sin( tc)cos( tc

FM dt

-

+

-

+

AM

AM'

AM demodulatorFM to AM convertorFM

AM

Message signal

Demodulation

This is usually an LTI system whose response curve is

approximately an increasing linear line in the related band.

Approximately linear region

Balanced Discriminator

VCO

DiscriminatorBPF LPFFM

Message signal

voltage Controlled Oscillator

AM

demod.

(for center freq stability)

General Commercial Receiver

RF Amplifier

RF

OSC

IF Demodulator Audio

AGC

MixerTune

Volume

FM band is selected to be 87.5 to 108.0 MHz (not in all countries)

with 200 kHz band for each station

f

M=L+R

Baseband Signal 15kHz 23kHz 53kHz

DSB-SC centered at 38kHz

S=L-R

19kHz pilot tone

(in phase with 38kHz carrier)

Mono FM receivers will only receive L+R

Stereo receivers will do: L=(M+S)/2 and R=(M-S)/2

Stereo FM

New baseband signal

ExampleBlock diagram of stereo AM/FM transmitter

right(t)

left(t)

M(t)

S(t)

-

38 kHz

19 kHz

freq. dup.

RF modulator Stereo AM/FM

RF carrier

ExampleBlock diagram of stereo AM/FM receiver

right(t)

left(t)

M(t)

S(t)

-

38 kHz

19 kHz

freq. dup.

RF

demodulatorStereo AM/FM

LPF

fc=15kHz

RF carrier

sharp-filter

19 kHz

M+S

LPF

fc=15kHz

M-S

Hmw : Add some blocks for Mono-transmitter reception

HomeworkBlock diagram to generate the following spectrum

f

|L(f)| |R(f)|

fc-15 fc+15fc

END