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 t

( ) ( )( ) i ii c

d t d tt

dt dt

( )i t( )i t

( )EMS t

• The instantaneous frequency of

• The instantaneous frequency deviation

( ) ( )1 1( )

2 2i i

i c

d t d tf t f

dt dt

( )1( )

2i

i

d tf t

dt

( )EMS t

Example

• for the signal below find the instantaneous frequency and maximum frequency deviation.

2( ) cos(10 ) x t A t t

• 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

• The instantaneous frequency of ( )

( )i c p

dm tf t f k

dt

Phase modulation (PM)

( ) ( )i t kp m t

( ) cos [2 ( )] PM c pS t A f t k m t

( )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.

• The instantaneous frequency of

( ) cos 2 ( )t

FM c fS t A f t k m d

Frequency Modulation (FM)

( ) ( )t

i ft k m d

( )FMS t

( ) ( )2f

i c

kf t f m t

Angle modulation viewed as FM or PM

Phase Modulator

FrequencyModulator

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 2x105 and 10respectively, and the carrier frequency fc is 100 MHz.

Example

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 fc c

S t A f t k a t f t

k kA a t f t a t f t

where ( ) ( )t

a t m d

• 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 t

( ) cos(2 ) ( )sin(2 )NBPM c p cS t A f t k m t f t

BBNBFM 2

| ( ) | 1fk a t

2NBPMB B

| ( ) | 1Pk m t

Generation of NBFM

m(t)

Generation of NBPM

m(t)

• If

f: maximum carrier frequency deviation : deviation ratio or modulation index

• Wide- Band Frequency Modulation (WBFM)

|kf a(t)|>>1 or >100 fBWBFM 2

2pfmkf

)1(2)(2 BBfBFM

B

f

| ( ) | 1fk a t

max ( )Pm m t

• For phase modulation: if

2

'ppmkf

| ( ) | 1Pk m t

2( ) 2 ( 1)PMB f B B

' 'max ( )Pm m t

2WBPMB f

Single tone modulation

• Let

)2sin(2cos)( tftfAtx mcFM

n

mcnFM tfnfJAtx )(2cos)()(

( ) cos 2 mm t f t

• 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

kf

f

f

Example 1• A single tone FM signal is

Determine

a) the carrier frequency fc

b) the modulation index c) the peak frequency deviation

d) the bandwidth of xFM(t)

6 3FMx (t)=10 cos[ 2 (10 )t+ 8 sin(2 (10 )t)]

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 fc) The deviation ratio d) the phase deviation e) the bandwidth of xFM(t).

EM cx (t)=10 cos[ 2 f t+ 5 sin(3000 t)+10 sin(2000 t) ]

Example 5.3 (Txt book)

a) Estimate BFM and BPM for m(t) when

kf= 2x105 rad/sV and kp= 5 rad/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

NBFMFrequency 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 kHzf1= 25 Hz

Frequency Multiplier

x48

fc2=12.8MHzf2= 1.6 kHz

fc3=1.9 MHzf3= 1.6 kHz

fc4= 91.2MHzf4= 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

02

)(1

C

tmkc

0

1

LCc

Varactor Modulator Circuit

• 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

dt

tdkty

)()(

• A frequency-selective network with a transfer function of the form |H()|= a +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 d

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)

FM Slope Detector

FM Slope Detector

FM Slope Detector

Balanced Discriminator

Balanced Discriminator (Cont.)

Balanced Discriminator (Cont.)

Quadrature Demodulator• FM is converted into PM

• PM detector is used to recover message signal

Quadrature Demodulator

Transfer function of Quadrature demodulator

Phase-Locked Loop (PLL)

)sin()( icin tAtv

)cos()( ocout tBtv

vout(t)

vin(t) e0(t)x(t) Loop Filter

H(s)

Voltage-Controlled Oscillator (VCO)

dt

tdkte i )(

)(0

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