Ch t 5

Chapter

Angle Modulation

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FM Illustration
The frequency of
the carrier is varied
around c in
relation with the
g signal.
g
message

i(t)= c + kf m(t)

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Instantaneous Frequency
The argument of a cosine function represents an angle.
angle
The angle could be constant [cos(300)], or varying with
time, cos [(t)]
The instantaneous angular frequency (in rad/sec) is the
rate of change of the angle. That is:
i (t) = d (t)/dt .
For cos(c t +), i (t)= c as expected.
t

(t ) i ( )d .

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Representation of Angle Modulation in Time Domain

For
F an FM signal:
i l i (t) = c + kf m(t)
(t)
t

FM (t ) i ( )d ct k f m( )d .

g FM (t ) A cos c t k f

m ( )d
t

For Phase Modulation ((PM),

), the phase
p
of the carrier is
varied in relation to the message signal: (t) = kp m(t)
g PM (t ) A cos c t k p m (t )
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Relation Between FM and PM

PM (t ) c t k p m (t ),
dm (t )
i (t ) c k p
c k p m (t ).
dt
m(t)
m(t)
()

FM Modulator

gFM(t)

()d

m (t )d

PM M
Modulator
d l

gPM(t)

d ()
dt

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dm (t )
dt

Which is Which?

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FM and PM Modulation
kf = 2105 rad/sec/volt =105 Hz/Volt = 105 V-11sec-11
kp = 10 rad/Volt = 5 v-1
fc = 100 MHz
FM:
fi = fc + kf m(t)
108 -105 < fi < 108 +105
99.9 < fi < 100.1 MHZ
PM:
fi = fc + kp dm(t)/dt
108 -105 < fi < 108 -105
99.9 < fi < 100.1 MHZ
Power ((FM or PM)) = A2/2
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Frequency Modulation
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f c
sin mt may be expressed as Bessel
The equationvs t = Vc cos c t +
fm

series (Bessel
(
l ffunctions))
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v s t = Vc

J cos

n=

+ nm t

where Jn() are Bessel functions of the first kind. Expanding the
equation for a few terms we have:

v s (t ) Vc J 0 ( ) cos( c )t Vc J 1 ( ) cos( c m )t Vc J 1 ( ) cos( c m )t

Amp

fc

fc fm

Amp

Amp

fc fm

Vc J 2 ( ) cos( c 2 m )t Vc J 2 ( ) cos( c 2 m )t

Amp

fc 2 fm

Amp

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fc 2 f m

FM Signal Spectrum.

The amplitudes drawn are completely arbitrary, since we have not found
any value for Jn() this sketch is only to illustrate the spectrum.
spectrum
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Spectrum of FM/PM
Unlike Amplitude Modulation,
Modulation it is not straightforward
to relate the spectrum of the FM/PM modulated signal
to that of the modulating signal m(t). We can deal with
i on a case-by-case
it
b
b i
basis.
We are, however, particularly interested in finding the
bandwidth occupied by an FM/PM signal.
For that purpose, we will make some assumptions and
work on simple modulating messages.
Because of the close relation between FM and PM, we
will do the analysis for FM and extend it to PM.
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FM Spectrum Bessel Coefficients.

The FM signal spectrum may be determined from

v s (t ) Vc

( ) cos( c n m )t

The values for the Bessel coefficients, Jn() may be

found from graphs or, preferably, tables of Bessel
functions of the first kind.
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FM Spectrum Bessel Coefficients.

Jn()

= 2.4

=5

In the series for vs(t), n = 0 is the carrier component, i.e.Vc J 0 ( ) cos( c t )

hence the n = 0 curve shows how the component at the carrier frequency,
frequency
fc, varies in amplitude, with modulation index .
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FM Spectrum Bessel Coefficients.

Hence for a given value of modulation index , the values of Jn() may be
read off the graph and hence the component amplitudes (VcJn()) may be
determined.
A further way to interpret these curves is to imagine them in 3 dimensions

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Examples from the graph

= 0: When = 0 the carrier is unmodulated and J0(0) = 1,
1 all
other Jn(0) = 0, i.e.

= 2.4: From the graph (approximately)

J0(2.4) = 0, J1(2.4) = 0.5, J2(2.4) = 0.45 and J3(2.4) = 0.2

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Si ifi t Sidebands
Significant
Sid b d Spectrum.
S t
As may
y be seen from the table of Bessel functions,, for values of n above a
certain value, the values of Jn() become progressively smaller. In FM the
sidebands are considered to be significant if Jn() 0.01 (1%).
Although
g the bandwidth of an FM signal
g
is infinite,, components
p
with
amplitudes VcJn(), for which Jn() < 0.01 are deemed to be insignificant
and may be ignored.
Example: A message signal with a frequency fm Hz modulates a carrier fc
to produce FM with a modulation index = 1. Sketch the spectrum.
n
0
1
2
3
4
5

Jn(1)
0.7652
0 4400
0.4400
0.1149
0.0196
0.0025
0.0002

Amplitude
0.7652Vc
0 44Vc
0.44V
0.1149Vc
0.0196Vc
Insignificant
Insignificant

Frequency
fc
fc+fm fc - fm
fc+2fm fc - 2fm
fc+3fm fc -3 fm

15

Significant Sidebands Spectrum.

As shown, the bandwidth of the spectrum containing

significant components is 6fm, for = 1.
1
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Significant Sidebands Spectrum.

The table below shows the number of significant sidebands for various
modulation indices () and the associated spectral bandwidth.

0.1
0.3
0.5
1.0
2.0
5.0
10.0

No of sidebands 1% of
unmodulated carrier
2
4
4
6
8
16
28

Bandwidth
2fm
4fm
4fm
6fm
8fm
16fm
28fm

e.g.
g for = 5,,
16 sidebands
(8 pairs).

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Carsons Rule for FM Bandwidth.

An approximation for the bandwidth of an FM signal
is given by BW = 2(Maximum frequency deviation +
hi h t modulated
highest
d l t d ffrequency))

Bandwidth 2(f c f m )

Carsons Rule

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Narrowband and Wideband FM

Narrowband FM NBFM
From the graph/table of Bessel functions it may be seen that for small , (
0.3) there is only the carrier and 2 significant sidebands, i.e. BW = 2fm.
FM with 0.3 is referred to as narrowband FM (NBFM) (Note, the
bandwidth is the same as DSBSC).

Wideband FM WBFM
For > 0.3
F
0 3 there
th
are more than
th
2 significant
i ifi
t sidebands.
id b d As
A increases
i
the number of sidebands increases. This is referred to as wideband FM
(WBFM).

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VHF/FM
VHF/FM (Very High Frequency band = 30MHz 300MHz) radio
transmissions, in the band 88MHz to 108MHz have the following
parameters:
Max frequency input (e.g. music) fm 15kHz
Deviation

75kHz

Modulation Index

f c Vm
f
c
fm

For = 5 there are 16 sidebands and the FM signal bandwidth is 16fm =

16 x 15kHz= 240kHz. Applying Carsons Rule BW = 2(75+15) = 180kHz.

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Comments FM
The FM spectrum contains a carrier component and an infinite number
of sidebands at frequencies fc nfm (n = 0, 1, 2, )
FM signal
signal,v s (t ) Vc

( ) cos( c n m )t

In FM we refer to sideband pairs not upper and lower sidebands.

C i or other
Carrier
th
components
t may nott be
b suppressed
d in
i FM.
FM
The relative amplitudes of components in FM depend on the values Jn(),
where

V m
fm

thus the component at the carrier frequency depends on

m(t), as do all the other components and none may be suppressed

.
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Comments FM
Components are significant if Jn() 0.01.
0 01 For <<1 ( 0.3
0 3 or less)
only J0() and J1() are significant, i.e. only a carrier and 2 sidebands.
Bandwidth is 2fm, similar to DSBSC in terms of bandwidth - called NBFM.
Large modulation index

f c
fm

means that a large bandwidth is required called WBFM

The FM process is non-linear. The principle of superposition does not
apply. When m(t) is a band of signals, e.g. speech or music the analysis
is very difficult (impossible?).
(impossible?) Calculations usually assume a single tone
frequency equal to the maximum input frequency. E.g. m(t) band 20Hz
15kHz, fm = 15kHz is used.

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22

Power in FM Signals.
From the equation for FM v s (t ) Vc

( ) cos( c n m )t

we see that the p

peak value of the components
p
is VcJn() for the nth
component.
2

V pk
(V RMS ) 2
Then the nth component Single normalised average power is=
2

V J ( )
Vc J n ( )
c n

2
2

Hence, the total power in the infinite spectrum is

(Vc J n ( )) 2
Total power PT
2

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Power in FM Signals.
By this method we would need to carry out an infinite number of
calculations to find PT. But, considering the waveform, the peak value is
V-c, which is constant.

V pk Vc
Since we know that the RMS value of a sine wave is

2
2

Vc J n ( )
V
V

c
c
2
P

and power = (VRMS) then we may deduce that T

2
2
n
2
2

Hence, if we know Vc for the FM signal, we can find the total power PT for
the infinite spectrum with a simple
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Power in FM Signals.
Now consider if we generate an FM signal, it will contain an infinite
number of sidebands. However, if we wish to transfer this signal, e.g. over a
radio or cable, this implies that we require an infinite bandwidth channel.
Even if there was an infinite channel bandwidth it would not all be
allocated to one user.
user Only a limited bandwidth is available for any
particular signal. Thus we have to make the signal spectrum fit into the
available channel bandwidth. We can think of the signal spectrum as a
train and the channel bandwidth as a tunnel obviously we make the
train slightly
l h l less
l
wider
d
than
h
the
h tunnell iff we can.

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Power in FM Signals.
However,, many
y signals
g
((e.g.
g FM,, square
q
waves,, digital
g
signals)
g
) contain an
infinite number of components. If we transfer such a signal via a limited
channel bandwidth, we will lose some of the components and the output
signal will be distorted. If we put an infinitely wide train through a tunnel,
the train would come out distorted,
distorted the question is how much distortion
can be tolerated?
Generally speaking, spectral components decrease in amplitude as we
move away
y from the spectrum
p
centre.

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Power in FM Signals.
In general distortion may be defined as

Power in total spectrum - Power in Bandlimited spectrum

Power in total spectrum

PT PBL
D
PT
With reference to FM the minimum channel bandwidth required would be
just wide enough to pass the spectrum of significant components. For a
bandlimited FM spectrum,
p
, let a = the number of sideband p
pairs,, e.g.
g for
= 5, a = 8 pairs (16 components). Hence, power in the bandlimited
spectrum PBL is

(Vc J n ( )) 2
PBL
= carrier power + sideband powers.
2
n a
a

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Power in FM Signals.
Vc2
Since PT
2

Vc2 Vc2 a

( J n ( )) 2

a
2
2 n a
2

(
J
(

))
Distortion D

n
Vc2
n a
2

Also, it is easily seen that the ratio

a
Power in Bandlimited spectrum PBL
D

( J n ( )) 2 = 1 Distortion
PT
Power in total spectrum
n a

i.e. proportion pf power in bandlimited spectrum to total power =

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(J

n a

28

( )) 2

Example
Consider NBFM, with = 0.2. Let Vc = 10 volts. The total power in the infinite
a
Vc2
2
spectrum
= 50 Watts, i.e. ( J n ( )) = 50 Watts.
2
n a

From the table the significant components are

n

Jn(0.2)
(0 2)

Amp = VcJn(0.2)
(0 2)

0
1

0.9900
0.0995

9.90
0.995

i e the carrier + 2 sidebands contain

i.e.

( Amp
A )2
Power =
2
49.005
0.4950125
PBL = 49.5 Watts

49.5
0.99or 99% of the total power
50

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36

Example
PT PBL 50 49.5

0.01 or 1%.
Distortion =
50
PT
A t ll we d
Actually,
dont
t need
d tto know
k
Vc, i.e.
i
alternatively
lt
ti l
Distortion = 1

(J

n 1

(0.2)) 2(a = 1)

D =1 (0.99) (0.0995) 0.01

2

1
PBL
( J n ( )) 2 1 D 0.99
Ratio
PT
n 1
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What is NOT the bandwidth of FM!

O
One may ttendd to
t believe
b li
th
thatt since
i
th
the modulated
d l t d
signal instantaneous frequency is varying between by
ff around fc, then the bandwidth of the FM signal
g is
2f. False!
In fact, the motivation behind introducing FM was to
reduce the bandwidth compared to that of Amplitude
Modulation, which turns out to be wrong.
What
Wh t was missing
i i from
f
the
th picture
i t
off bandwidth?
b d idth?

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FM Visualization
Think of holding the frequency knob of a signal
generator, and wiggling it back and forth to
modulate the carries in response to some
message.
There are two wiggling
gg g parameters:
p
How far you deviate from the center frequency (f)
How fast you wiggle (related to Bm)

The rate of change of the instantaneous

frequency was missing!
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Carsons Rule
BFM 2(f+Bm)
where f = frequency deviation = kf |m(t)|max
Bm = bandwidth
b d id h off m(t)
()
Define the deviation ratio = f / Bm.
BFM 2(( +1)) Bm
The same rule applies to PM bandwidth,
BPM 2(f+Bm) = 2( +1) Bm
where (f )PM = kp |dm(t)/dt|max
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Narrow Band and Wide Band FM

Wh
When f << Bm (or
( <<1),
<<1) the
th scheme
h
is
i
called Narrow Band (NBFM, NBPM).
BNBFM 2Bm (same
(
f NBPM)
for
Therefore, no matter how small we make the
d i i aroundd fc , the
deviation
h bandwidth
b d id h off the
h
modulated signal does not get smaller than 2Bm.

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Estimate BFM and BPM

kf = 2105 rad/sec/volt =10
105
Hz/Volt = 105 V-1sec-1
kp = 5 rad/Volt = 2.5 v-1
fc = 1000 MHz
First estimate the Bm.
Cn = 8/2n2 for n odd, 0 n even
The 5th harmonic onward can
be neglected.
Bm = 15 kHz
For FM:
f = 100 kHz; BFM =230 KHz
For PM:
f = 50 kHz; BFM =130 KHz
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Repeat if m(t) is Doubled

kf = 2105 rad/sec/volt =105
Hz/Volt = 105 V-1sec-1
kp = 5 rad/Volt = 2.5 v-1
fc = 1000 MH
MHz
For FM:
f = 200 kHz; BFM = 430 KHz
For PM:
f = 100 kHz; BFM = 230 KHz
Doubling the signal peak has
significant effect on both FM
and PM bandwidth

-2
2

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40,000

-40,000
,

36

Repeat if the period of m(t) is Doubled

kf = 2105 rad/sec/volt =105
Hz/Volt = 105 V-1sec-1
kp = 5 rad/Volt = 2.5 v-1
fc = 1000 MH
MHz
Bm = 7.5 kHz
For FM:
f = 100 kHz; BFM = 215 KHz
For PM:
f = 25 kHz; BFM = 65 KHz
Expanding the signal varies its
spectrum. This has significant
effect on PM.

4x10-44

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10,000

-10,000
,

37

Spectrum of NBFM (1/2)

g FM (t ) A cos c t k f a (t ) where
h

a (t )

m ( )d

j t k a (t )
g FM (t ) A e c f A cos c t k f a (t ) jA sin c t k f a (t )

g FM (t ) A e

j c t

j 2 k f2a 2 (t ) j 3 k f3a 3 (t ) j 4 k f4a 4 (t )

1 jk f a (t )

2!
3!
4!

2 2
3 3
4 4
j t

k
a
(
t
)
jk
a
(
t
)
k
a (t ) j c t
j c t
j c t
j c t
f
f
f
c
A e
jk f a (t )e

e
e
e
2!
3!
4!

g FM (t ) Re g FM (t )

k f2a 2 (t )
k f3a 3 (t )
k f4a 4 (t )
A cos(c t ) k f a (t ) sin(c t )
cos(c t )
sin(c t )
cos(c t )
2!
3!
4!

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Spectrum of NBFM (2/2)

For NBFM,
NBFM |kf a(t)|<< 1

g FM ( Narrowband ) (t ) A cos(c t ) k f a (t ) sin(c t )

Bandwidth of a(t) is equal to the bandwidth of m(t), Bm.

BNBFM 22 Bm (as expected).
expected)
Similarly for PM (|kp m(t)|<< 1 ):

g PM ( Narrowband ) (t ) A cos(c t ) k p m (t ) sin(c t )

BNBPM 2
2 Bm
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NBFM Modulator
g FM ( Narrowband ) (t ) A cos(c t ) k f a (t ) sin(c t )

()d

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NBPM Modulator
g PM ( Narrowband ) (t ) A cos(c t ) k p m (t ) sin(c t )

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Immunity of FM to Non-linearities

q (t ) a1A cos c t k f

m
(

)
d

a
A
cos

c t k f
2

a3 A cos c t k f

m
(

)
d

m ( )d

a2 A

a1A cos c t k f m ( )d
1 cos 2c t 2k f

aA
3
2

1 cos 2c t 2k f

)
d

cos
c t k f
m

m ( )d

m
(

)
d

a2 A

3a3 A
a2 A
t
k
m
d
t
k
m
d

a1A
cos

(
)

cos
2

2
(
)

c
f
c
f

2
4
2

DC
Around c with k f k f

Around 2c with k f 2 k f

aA
3 cos 3c t 3k f m ( )d
4

Around 3c with k f 3 k f

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Frequency Multipliers
g FM (t )

A cos c t k f

m
(

)
d

q (t ) contains the followingg

cos c t k f

m
(

)
d

cos 2c t 2k f

m
(

)
d

cos P c t Pk f

m
(

)
d

g FM (output ) (t )

B cos P c t Pk f

m ( )d

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Generation of WBFM: Indirect Method

Usually, we are
interested in generating
an FM signal of certain
bandwidth (or f or )
and certain fc.
In the indirect method,
we generate a NBFM
with small then use a
frequency multiplier to
scale to the required
value.
This way, fc will also
b scaled
be
l d by
b the
th same
factor. We may need a
frequency mixer to
adjust fc.
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Example: From NBFM to WBFM

A NBFM modulator
d l t iis modulating
d l ti a message signal
i l
m(t) with bandwidth 5 kHz and is producing an FM
signal
g with the following
g specifications
p
fc1 = 300 kHz,
f1 = 35 Hz.
g to ggenerate a WBFM
We would like to use this signal
signal with the following specifications
fc2 = 135 MHz,
f 2 = 77 kHz.
f 2 77 *103

2200
f 1
35

f c 2 135*106

450
3
f c 1 300*10

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From NBFM to WBFM: System 1

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From NBFM to WBFM: System 2

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Generation of WBFM: Direct Method

H
Has poor ffrequency stability.
bili Requires
R i
feedback to stabilize it.
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FM Demodulator
Four
F
primary
i
methods
th d
Differentiator with envelope detector/Slope detector
FM to AM conversion

Phase-shift discriminator/Ratio detector

Approximates the differentiator

Zero-crossing detector
Frequency feedback
Phase lock loops (PLL)

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FM Demodulation: Signal Differentiation

g FM (t ) A cos c t k f

m ( )d

dg FM (t )
A c k f m (t ) sin c t k f
dt

m ( )d

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FM Demodulation: Signal Differentiation

d ()
dt

d ()
dt

A c k f m (t ) sin c t k f

m
(

)
d

dm (t )

A c k f
sin c t k p m (t )

d
dt

dm (t )
dt

()d

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Frequency Discriminators
A
Any system
t with
ith a
transfer function of the
form ||H(()| = a + b
over the band of the FM
signal can be used for
FM ddemodulation
d l ti
The differentiator is just
one example
example.

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Slope Detectors (Demodulators)

A c Ck f m (t ) cos ct k f

m
(

)
d

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FM Slope Demodulator
Principle: use slope detector (slope circuit) as
frequency discriminator, which implements frequency
to voltage conversion (FVC)
Slope circuit: output voltage is proportional to the input
frequency. Example: filters, differentiator
s(t)

S(f)

x(t)

d
dt
H(f)=j2 f

|H(f)|
X(f)
output
voltage
range

X(f)

freqency in s(t)

voltage in x(t)

10 Hz
20 Hz

j 20
j 40

f
Input frequency
range in S(f)
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FM Slope Demodulator cont.

Block diagram
g
of direct method ((slope
p detector = slope
p
circuit + envelope detector)
s(t)

slope
circuit

s1(t)

(FM
AM)
(FVC)

envelope
detector

so(t)

(AM demodulator)

s (t ) Ac cos 2 f c t 2 k f m( )d ,

where fi (t ) f c k f m(t )

Let the slope circuit be simply differentiator:

t

s1 (t ) Ac 2 f c 2 k f m(t ) sin 2 f c t 2 k f m( )d

so (t ) Ac 2 f c 2 k f m(t )

so(t) linear with m(t)

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Slope Detector

Magnitude frequency
response of
transformer BPF
BPF.
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Hard Limiter
A device that imposes hard limiting on a signal and contains a
filter that suppresses the unwanted products (harmonics) of the
limiting process.
Input Signal
t

vi (t ) A(t ) cos (t ) A(t ) cos( wc t k f m(a)da)

Output
O
off hhard
d li
limiter
i
vo (t )

4
1
1

cos
(
t
)
cos
3
(
t
)

cos 5 (t )

3
5

Bandpass filter e (t ) 4 cos( w t k t m(a )da )

o
c
f

Remove the amplitude EEEF311

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Ratio Detector
Foster
Foster-Seeley/phase
Seeley/phase shift discriminator

uses a double-tuned transformer to convert the instantaneous frequency

variations of the FM input signal to instantaneous amplitude variations. These
amplitude variations are rectified to provide a DC output voltage which varies
in amplitude and polarity with the input signal frequency.
Example

Ratio detector

M difi d Foster-Seeley
Modified
F t S l discriminator,
di i i t nott response to
t AM,
AM but
b t 50%
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Zero Crossing Detector

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Phased-Locked Loop (PLL)

Th
The multiplier
lti li followed
f ll
d
by the filter estimates
the error bewteen the
angle of gFM(t) and
gVCO(t).
The error is fed to VCO
to adjust the angle.
When the angles are
locked, the output of the
PLL would be following
m(t) pattern.
patte .
EEEF311 (2014-2015)
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Loop
Filter

VCO

60

FM Demodulator PLL
Phase
Phase-locked
locked loop (PLL)
A closed-loop feedback control circuit, make a signal in fixed phase
(and frequency) relation to a reference signal
Track frequency (or phase) variation of inputs
Or, change frequency (or phase) according to inputs
PLL can be used for both FM modulator and demodulator
Just as Balanced Modulator IC can be used for most amplitude
modulations and demodulations

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PLL FM
Remember the following relations
Si=Acos(wct+1(t)), Sv=Avcos(wct+c(t))
Sp=0.5AAv[sin(2wct+1+c)+sin(1-c)]
So=0.5AA
S 0 5AAvsin(
i (1-c)=AA
) AAv(1-c)
Section 2.14
m(t)

s(t)

VCO

s(t)
freqency
devided
by N

LP

e(t)
r(t)

v(t)

Loop
Filter
VCO

Reference
Carrier

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Superheterodyne Receiver
Radio receiver
receiverss main function

Demodulation get message signal

Carrier frequency tuning select station
Filt i remove noise/interference
Filtering
i /i t f
Amplification combat transmission power loss

Superheterodyne
p
y receiver
Heterodyne: mixing two signals for new frequency
Superheterodyne receiver: heterodyne RF signals with local
tuner, convert to common IF
Invented by E. Armstrong in 1918.
AM: RF 0.535MHz-1.605 MHz, Midband 0.455MHz
FM: RF 88M-108MHz,
88M 108MHz Midband 10.7MHz
10 7MHz
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Advantage of superheterodyne receiver

A signal block (of circuit) can hardly achieve all: selectivity,
selectivity signal
quality, and power amplification
Superheterodyne receiver deals them with different blocks
RF bl
blocks:
k selectivity
l ti it only
l
IF blocks: filter for high signal quality, and amplification, use circuits
that work in only a constant IF, not a large band

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FM Broadcasting
The frequency
q
y of an FM broadcast station is usuallyy an
exact multiple of 100 kHz from 87.5 to 108.5 MHz . In
most of the Americas and Caribbean only odd
multiples are used.
fm=15KHz, f=75KHz, =5, B=2(fm+f)=180kHz
Pre-emphasis and de-emphasis
Random noise has a 'triangular' spectral distribution in an
FM system,
system with the effect that noise occurs predominantly
at the highest frequencies within the baseband. This can be
offset, to a limited extent, by boosting the high frequencies
before transmission and reducing them by a corresponding
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amount in the receiver.
NITIN Sharma

FM Stereo Multiplexing
Fc=19KHz.
(a) Multiplexer in transmitter
of FM stereo.
(b) Demultiplexer in receiver
off FM stereo.
Backward compatible
For non-stereo receiver

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TV FM broadcasting
fm=15KHz, f=25KHz, =5/3, B=2(fm+f)=80kHz
Center fc+4.5MHz

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