University of Malaya

Dr.HarikrishnanDepartment of Electrical Engineering

e-mail: hrkhari@um.edu.my

KEEE 2142Introduction to Communication System

Angle Modulation

Considering the carrier voltage:

Angle modulation is said to occur when the angle (t) is caused to vary by an amountdetermined by the instantaneous amplitudes of the modulating signal m(t).

( ) ( ) ( )Cx t V t cos t= (1)where V(t) = Peak carrier amplitude

(t) = Instantaneous phase deviation (radians)

University of Malaya KEEE 2142 HRK 2/10

The commonly encountered forms of angle modulation are phase modulation and frequencymodulation.

Phase modulation (PM) is said to occur when the deviation of the angle (t) from itsunmodulated value is caused to vary by an amount that is determined by the instantaneousamplitudes of the modulating signal m(t) and the amplitude V(t) of the carrier remains constant.

Frequency modulation (FM) is said to occur when the deviation of the carrier frequency from itsunmodulated value fC is caused to vary by an amount that is determined by the instantaneousamplitudes of the modulating signal m(t) and the amplitude of the carrier V(t) remains constant.

The carrier voltage in phase modulation is represented by:

In PM, the instantaneous phase deviation of the carrier from its unmodulated phase (t), isdirectly proportional to the instantaneous amplitudes of the modulating signal m(t). Thus,

Phase Modulation

( ) ( ) ( )C C C Cx t A cos t A cos t t= = + (2)where (t)= instantaneous phase in radians:

( ) ( ) ( )Ct t t instantaneous angle radians = + =

University of Malaya KEEE 2142 HRK 3/10

( ) ( ) ( )Pt 0 k m t = + (3)where kP is a phase deviation constant in radians per unit amplitude of m(t) and its value isdetermined by the design of the phase modulator used. If m(t) is represented by its variousspectral components, then

( ) ( ) ( ) ( )1 n1 2 n 2 k nkk

m t A m t A m t A m t= + + = (4)where the values of Ak are those amplitudes required to normalize each spectral componentand the mnk(t) are the normalized spectral components. Then,

( ) ( ) ( ) ( )P 1 n1 P 2 n 2t 0 k A m t k A m t ... = + + + (5)

Phase Modulation (contd)

where 1=kPA1 and 2=kPA2, are the individual maximum phase deviations in radiansproduced in the PM modulator by the components of m(t). It is usual to select a time scale that(0)=0

( ) ( ) ( ) ( )1 n1 2 n 2t 0 m t m t ... = + + +

Assuming that the normalized components of m(t) are given by:( )( )

n1 1m t sin tm t sin t

=

=

University of Malaya KEEE 2142 HRK 4/10

Then, with (0)=0,

When this expression is substituted in the carrier voltage and expanded, the spectrum of thephase modulated signal is obtained. The accompanying change in the carrier frequency isobtained as:

( )n 2 2m t sin t : :

: :

=

( ) C 1 1 2 2t t sin t sin t ... = + + +

( ) ( ) C 1 1 1 2 2 2d tt cos t cos t ...dt

= = + + + (6)

Dividing the equation by 2pi:

Phase Modulation (contd)

( ) C 1 1 1 2 2 2C 1 1 2 2

f t f f cos t f cos t ...f f cos t f cos t ...

= + + += + + + (7)

where f1=f11, f2=f22, represent the individual maximum frequency deviation in hertzproduced in the PM modulator by the individual components m(t). Notice that each frequencydeviation varies directly with both the frequency and the amplitude of a modulating component

University of Malaya KEEE 2142 HRK 5/10

Phase Modulation (contd) Thus, the phase modulated signal can be expressed as:

( ) ( )PM C Px t A cos t k m t= + (8)Example 1The sinusoidal output from a 100 kHz crystal oscillator is phase modulated. The modulating signal is:

The phase deviation constant of the phase modulator is kP=0.05 radian per volt of m(t). Calculate (a) theindividual maximum phase deviations in degrees produced by the phase modulator, (b) the individual

( )m t 2sin1000t sin 5000t V= +

University of Malaya KEEE 2142 HRK 6/10

individual maximum phase deviations in degrees produced by the phase modulator, (b) the individualmaximum frequency deviations in hertz produced by the components of m(t), (c) the total maximumfrequency deviation f produced by both modulating components together.

Example 2A phase modulated signal is described by:

Considering xPM(t) as a PM signal with kP=10, find m(t)( ) ( ) ( )6 3PMx t 10cos 2 10 t 0.1sin 10 t = pi + pi

Example 3Consider a phase modulated signal

Find the maximum phase deviation.( ) ( ) ( )8 3PMx t 10cos 10 t 5sin 2 10 t = pi + pi

In frequency modulation (FM), the instantaneous frequency deviation of the carrier from itsunmodulated value fC is directly proportional to the instantaneous amplitude of the modulatingsignal m(t). Thus:

Frequency Modulation

( ) ( )C ff t f k m t= + (9)where kf is a frequency deviation constant in hertz per unit amplitude of m(t) and its value isdetermined by the design of the frequency modulator used. If m(t) is represented by its spectralcomponents, then:

( ) ( ) ( )f t f k A m t k A m t ...= + + +

University of Malaya KEEE 2142 HRK 7/10

If the normalized components of m(t) for FM can be given by:

( ) ( ) ( )C f 1 n1 f 2 n 2f t f k A m t k A m t ...= + + +or, multiplying by 2pi

( ) ( ) ( )C 1 n1 2 n 2t 2 f m t 2 f m t ... = + pi + pi +where f1=kfA1, f2=kfA2, represent the maximum frequency deviations product produced in theFM modulator by the individual components of m(t).

( ) ( ) ( ) ( )n1 1 n 2 2m t cos t ,m t cos t ,...= =

Even though this is frequency modulation, there is an accompanying change in the carrierphase that is obtained from:

Frequency Modulation (contd)or, ( ) C 1 1 2 2t cos t cos t ... = + + +

( ) ( )t t dt = where the constant of integration has been chosen equal to zero for simplicity. Thus:

( ) ( ) ( )C 1 n1 2 n 2t 2 f 2 f m t 2 f m t ... dt = pi + pi + pi +

University of Malaya KEEE 2142 HRK 8/10

( ) ( ) ( )( ) ( )

C 1 n1 2 n 2

C 1 n1 2 n1

t 2 f 2 f m t 2 f m t ... dt

2 f t 2 f m t dt 2 f m t dt ...

= pi + pi + pi + = pi + pi + pi + (10)

which for mn1(t) =cos 1t and mn2(t) = cos 2t , becomes

( ) 1 2C 1 21 2

1 2C 1 2

1 2

C 1 1 2 2

2 f 2 ft 2 f t sin t sin t

f ft sin t sin t

f ft sin t sin t

pi pi = pi + + +

= + + +

= + + +

Frequency Modulation (contd)where 1=f1/f1, 2=f2/f2, represent the maximum phase deviations produced in the FMmodulator by the individual components of m(t).

Thus, the frequency modulated signal can be expressed as:

( ) ( )tFM C fx t A cos t 2 k m t dt

= + pi (11)

University of Malaya KEEE 2142 HRK 9/10

Frequency Modulation (contd)Example 4The sinusoidal output of an 88.5 MHz oscillator is frequency modulated. The modulating voltage is:

the frequency deviation constant of the frequency modulator is 25 kHz per volt of m(t). Calculate.a. The individual maximum frequency deviations in kHz produced by the frequency modulator.b. The total maximum frequency deviation fmax in kHz produced by both modulating components

togetherc. The individual maximum phase deviations produced by the components of m(t).d. The total maximum phase deviation produced by both modulating components together.

( )m t 2cos1000t cos5000t V= +

University of Malaya KEEE 2142 HRK 10/10

d. The total maximum phase deviation produced by both modulating components together.

Example 5A frequency modulated signal is described by:

Considering xFM(t) as a FM signal with kf=10pi, find m(t)

Example 6Consider a frequency modulated signal

Find the maximum frequency deviation.

( ) ( ) ( )6 3FMx t 10cos 2 10 t 0.1sin 10 t = pi + pi

( ) ( ) ( )8 3FMx t 10cos 10 t 5sin 2 10 t = pi + pi