Chapter 5

Chapter 5
ANGLE MODULATION:
FREQUENCY and PHASE
MODULATIONS(FM,PM)

Outlines
• Introduction
• Concepts of instantaneous frequency
• Bandwidth of angle modulated signals
• Narrow-band and wide-band frequency
modulations
• Generation of FM signals
• Demodulation of FM signals
• superhetrodyne FM radio

Introduction
• Angle modulation: either frequency modulation
(FM) or phase modulation (PM).
• Basic idea: vary the carrier frequency (FM) or
phase (PM) according to the message signal.

• While AM is linear process, FM and PM are
highly nonlinear.
• FM/PM provide many advantages (main –
noise immunity, interference, exchange of
power with bandwidth ) over AM, at a cost of
larger transmission bandwidth.
• Demodulation may be complex, but modern
ICs allow cost-effective implementation.
Example: FM radio (high quality, not
expensive receivers).

Concepts of Instantaneous
Frequency
• A general form of an angle modulated signal is
given by
is the instantaneous angle
is the instantaneous phase deviation.
• The instantaneous angular frequency of
( ) cos ( ) cos(2 ( ))EM i c iS t A t A f t tθ π φ= = +
( ) ( )
( ) i i
i c
d t d t
t
dt dt
θ φ
ω ω= = +
( )i tθ
( )i tφ
( )EMS t

• The instantaneous frequency of
• The instantaneous frequency deviation
( ) ( )1 1
( )
2 2
i i
i c
d t d t
f t f
dt dt
θ φ
π π
= = +
( )1
( )
2
i
i
d t
f t
dt
φ
π
∆ =
( )EMS t

Example
• for the signal below find the
instantaneous frequency and maximum
frequency deviation.
2
( ) cos(10 )x t A t tπ π= +

• For phase modulation (PM), the instantaneous
phase deviation is
•
kp is the phase sensitivity of the PM modulator
expressed in (rad/ V) if m(t) is in Volts
( )
( )i c p
dm t
f t f k
dt
= +
Phase modulation (PM)
( ) ( )i t kp m tφ =
( ) cos [2 ( )]PM c pS t A f t k m tπ= +
( )PMS t

• For Frequency Modulation (FM), the
instantaneous phase deviation is
• kf is the frequency sensitivity of the FM
modulator expressed in rad/ V s if m(t) in Volts.
( ) cos 2 ( )
t
FM c fS t A f t k m dπ α α
−∞
 
= + 
 
∫
Frequency Modulation (FM)
( ) ( )
t
i ft k m dφ α α
−∞
= ∫
( )FMS t
( ) ( )
2
f
i c
k
f t f m t
π
= +

Angle modulation viewed as FM or
PM

Phase
Modulator
Frequency
Modulator
Phase
Modulator∫
Frequency
Modulator
( )m t ( )PMS t
( )m t
( )FMS t
( )m t ( )FMS t
( )PMS t( )m t d
dt

• A PM/FM modulator may be used to
generate an FM/PM waveform
• FM is much more frequently used than PM
• All the properties of a PM signal may be
deduced from that of an FM signal
• In the remaining part of the chapter we
deal mainly with FM signals.

Example 5.1
• Sketch FM and PM waves for the modulating
signal m(t) shown in Fig. 5.4a. The constants kf
and kp are 2πx105
and 10π, respectively, and the
carrier frequency fc is 100 MHz..

Bandwidth of Angle Modulated
Signals
1) FM signals
[ ]
2 3
2 3
( ) cos(2 ) ( )sin(2 )
( )cos(2 ) ( )sin(2 ) ...
2! 3!
FM c f c
f f
c c
S t A f t k a t f t
k k
A a t f t a t f t
π π
π π
= −
 
+ − + + 
 
where ( ) ( )
t
a t m dα α
−∞
= ∫

• Narrow-Band Frequency Modulation
(NBFM):
• Narrow-Band Phase Modulation (NBPM):
[ ]( ) cos(2 ) ( )sin(2 )NBFM c f cS t A f t k a t f tπ π≈ −
( ) cos(2 ) ( )sin(2 )NBPM c p cS t A f t k m t f tπ π ≈ − 
BBNBFM 2=
| ( ) | 1fk a t <<
2NBPMB B=
| ( ) | 1Pk m t <<

• If
∆f: maximum carrier frequency deviation
β: deviation ratio or modulation index
• Wide- Band Frequency Modulation (WBFM)
|kf a(t)|>>1 or β>100 fBWBFM ∆= 2
π2
pf mk
f =∆
)1(2)(2 +=+∆= βBBfBFM
B
f∆
=β
| ( ) | 1fk a t ?
max ( )Pm m t=

• For phase modulation: if
π2
'
ppmk
f =∆
| ( ) | 1Pk m t ?
2( ) 2 ( 1)PMB f B B β= ∆ + = +
' '
max ( )Pm m t=
2WBPMB f= ∆

Single tone modulation
• Let
[ ])2sin(2cos)( tftfAtx mcFM πβπ +=
[ ]∑
∞
−∞=
+=
n
mcnFM tfnfJAtx )(2cos)()( πβ
( ) cos2 mm t f tα π=

• The results is valid only for sinusoidal signal
• The single tone method can be used for
finding the spectrum of an FM wave when
m(t) is any periodic signal.
2 ( 1)
2
FM m
f
m
B f
k
f
f
f
β
α
π
β
= +
∆ =
∆
=

Example 1
• A single tone FM signal is
Determine
a) the carrier frequency fc
b) the modulation index β
c) the peak frequency deviation
d) the bandwidth of xFM(t)
6 3
FMx (t)=10 cos[ 2 (10 )t+ 8 sin(2 (10 )t)]π π

Example 2
• A 10 MHz carrier is frequency modulated by
a sinusoidal signal such that the peak
frequency deviation is ∆f=50 KHz. Determine
the approximate bandwidth of the FM signal if
the frequency of the modulating sinusoid fm is
a) 500 kHz, b) 500 Hz, c) 10 kHz.

Example 3
• An angle modulated signal with carrier
frequency 100kHz is
Find
a) the power of xFM(t)
b) the frequency deviation ∆f
c) The deviation ratio β
d) the phase deviation ∆φ
e) the bandwidth of xFM(t).
EM cx (t)=10 cos[ 2 f t+ 5 sin(3000 t)+10 sin(2000 t) ]π π π

Example 5.3 (Txt book)
a) Estimate BFM and BPM for m(t) when
kf= 2πx105
rad/sV and kp= 5πrad/V
b) Repeat the problem if the amplitude of m(t)
is doubled.

Features of Angle Modulation
• Channel bandwidth may be exchanged for
improved noise performance. Such trade-off
is not possible with AM
• Angle modulation is less vulnerable than AM
to small signal interference from adjacent
channels and more resistant to noise.
• Immunity of angle modulation to
nonlinearities thus used for high power
systems as microwave radio.

• FM is used for: radio broadcasting, sound
signal in TV, two-way fixed and mobile
radio systems, cellular telephone systems,
and satellite communications.
• PM is used extensively in data
communications and for indirect FM.
• WBFM is used widely in space and
satellite communication systems.
• WBFM is also used for high fidelity radio
transmission over rather limited areas.

Generation of FM Signals
• There are two ways of generating FM
waves:
–Indirect generation
–Direct generation

Indirect Generation of NBFM
m(t)

Indirect Generation of
Wideband FM
• In this method, a narrowband frequency-
modulated signal is first generated and then a
frequency multiplier is used to increase the
modulation index.
m(t)
NBFM
xFM(t)
Frequency
Multiplier

m(t)
N fc
NBFM
Frequency
Multiplier
BPF
Local Oscillator
(fLo)
xFM(t)
fc
Frequency Converter

m(t)
NBFM
Frequency
Multiplier
x64
Power
Amplifier
Crystal
Oscillator
10.9 MHz
fc1=200 kHz
∆f1= 25 Hz
Frequency
Multiplier
x48
fc2=12.8MHz
∆f2= 1.6 kHz
fc3=1.9 MHz
∆f3= 1.6 kHz
fc4= 91.2MHz
∆f4= 76.8 kHz
Armstrong Indirect FM TransmitterArmstrong Indirect FM Transmitter
BPF

Direct Generation
• The modulating signal m(t) directly controls
the carrier frequency. [ ]
• A common method is to vary the inductance
or capacitance of a voltage controlled
oscillator.
( ) ( )i c ff t f k m t= +

• In Hartley or Colpitt oscillator , the frequency is
given by
• We can show that for k m(t) << C0
LC
1
=ω






+=
02
)(
1
C
tmk
cωω
0
1
LC
c =ω

• Advantage - Large frequency deviations are
possible and thus less frequency multiplication
is needed.
• Disadvantage - The carrier frequency tends to
drift and additional circuitry is required for
frequency stabilization.
To stabilize the carrier frequency, a phase-
locked loop can be used.

Example 5.6
• Discuss the nature of distortion inherent in the
Armstrong FM generator
–Amplitude distortion
–Frequency distortion

Example
• A given angle modulated signal has a peak
frequency deviation of 20 Hz for an input
sinusoid of unit amplitude and a frequency of
50 Hz. Determine the required frequency
multiplication factor, N, to produce a peak
frequency deviation of 20 kHz when the input
sinusoid has unit amplitude and a frequency
of 100Hz, and the angle-modulation used is
(a) FM; (b) PM

Demodulation of FM Signals
• Demodulation of an FM signal requires a
system that produces an output proportional to
the instantaneous frequency deviation of the
input signal.
• Such system is called a frequency
discriminator.
FM
Demodulator
[ ])(cos)( ttAtx c φω +=
dt
td
kty
)(
)(
φ
=

• A frequency-selective network with a transfer
function of the form |H(ω)|= a ω+b over the
FM band would yield an output proportional
to the instantaneous frequency.
• There are several possible examples for
frequency discriminator, the simplest is the
FM demodulator by direct differentiation

FM demodulator by direct differentiation
• The basic idea is to convert FM into AM
and then use AM demodulator.
[ ]'
( ) 2 ( ) sin 2 ( )
t
c c f c fs t A f k m t f t k m dπ π α α
−∞
 
= − + + 
 
∫

Bandpass Limiter
• Input-output characteristic of a hard limiter
Hard
Limiter
BPF

• Any signal which exceeds the preset limits are
simply chopped off

Practical Frequency Demodulators
• There are several possible networks for
frequency discriminator
–FM slope detector
–Balanced discriminator
– Quadrature Demodulator
• Another superior technique for the
demodulation of the FM signal is to use the
Phased locked loop (PLL)

Balanced Discriminator (Cont.)

Quadrature Demodulator
• FM is converted into PM
• PM detector is used to recover message
signal

Transfer function of
Quadrature demodulator

Phase-Locked Loop (PLL)
)sin()( icin tAtv θω +=
)cos()( ocout tBtv θω +=
vout(t)
vin(t) e0(t)x(t) Loop Filter
H(s)
Voltage-Controlled
Oscillator (VCO)
dt
td
kte i )(
)(0
θ
=

Zero-Crossing Detectors
• Zero-Crossing Detectors are also used
because of advances in digital integrated
circuits.
• These are the frequency counters designed to
measure the instantaneous frequency by the
number of zero crossings.
• The rate of zero crossings is equal to the
instantaneous frequency of the input signal

Summary
• Concepts of instantaneous frequency
• FM and PM signals
• Bandwidth of angle modulated signals
NBFM and WBFM
• Generation of FM signals
– Direct and indirect generation
• Demodulation of FM signals
– frequency discriminator
– PLL

Suggested Problems
• 5.1-1 5.1-2 5.1-3 5.2-1 5.2-2
5.2-3 5.2-4 , 5.2-5 5.2-6 .
• 5.2-7 5.3-1 5.3-2 5.4-1 5.4-2

Chapter 5

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