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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

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1.6 Phase and Frequency Deviation:

Let a single-tone angle modulated signal is represented by

( ) cos[ sin ]c c ms t A ω t ω t 

PM: Suppose this equation represents the Phase Modulation, then  is the peak amplitude

of the phase information. In this case  is the maximum phase deviation 

( max{ sin }mω t ), usually referred to as modulation index p .

FM: Suppose this equation represents the Frequency Modulation, then  is the modulation

index ( f ). In this case the maximum frequency deviation mf f  ( mf f ).

max{ sin }m m
d

f f ω t
dt

    

In this case of the angle modulated signal the generalized angle is

( ) sinc mt ω t ω t  

The instantaneous frequency ( ) ( ) cosc m mi
d

ω t t ω ω ω t
dt
    or cosc m mif f f ω t 

Then the maximum frequency deviation mf f  .

1.6.1 Units for Phase / Frequency deviation and Modulation indices:

In Phase Modulation, the phase deviation is given by ( )pk m t  radians. Similarly in

Frequency Modulation the frequency deviation is given by

( )f
d

ω k m t
dt

    radians/sec. or ( )ff k m t  Hz.

where pk and fk are constants and are deviation sensitivities of the Phase and Frequency

Modulations respectively. The deviation sensitivities are the output versus input transfer

function for the modulation, which gives the relationship between the parameter changes in

respect to specified changes in the input signal.

 For a phase modulation, changes would occur in the phase of the output frequency in

respect to changes in the amplitude of the input modulating signal voltage. Therefore

the deviation sensitivity for a phase modulator is

p
Radians

k
V V

 
  

 


 For a frequency modulation, changes would occur in the output frequency in respect

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

7



to changes in the input modulating signal voltage. Therefore the deviation

sensitivity for a frequency modulator is


/ sec

f
ω Radians

k
V V

 
  

 
or

/ secf Cycles

V V

 
 
 

Hz / V

 Modulation index for PM is defined as p p mk A  for a single-tone modulating

signal ( ) cos2m mm t A f t . Then the units for modulation index p is defined as

Radians
V Radians

V
 .

 Modulation index for FM is defined as
mf

f
m

k A

ω
  for a single-tone modulating

signal ( ) cos2m mm t A f t . Then the units for modulation index f is defined as

/ sec

/ sec

Radians V

V Radians
or

/ sec

/ sec

Cycles V

V Cycles
(unit less).

1.6.1: Summary of Phase Modulation and Frequency Modulation

Phase Modulation Frequency Modulation

A Generalized PM equation


( ) cos ( )PM ps t A t k m tc c   

where pk


rad/sec is phase modulation

sensitivity constant.

A Generalized FM equation

( ) cos 2 ( )FM
t

fs t A t k m dc c   
  
  


where fk Hz/Volt is frequency modulation

sensitivity constant



Single-Tone case ( ) cos2m mm t A f t

( ) cos cos 2

cos cos 2

PM p m m

p m

s t A t k A f tc c

A t f tc c

 

  

   

   

Phase modulation index


radiansp p mk A 

Phase deviation radiansp mk A 


In Phase modulation, p  



Single-Tone case ( ) cos2m mm t A f t

( ) cos sin 2

cos sin 2

FM

mf
m

m

mf

k A
s t A t f tc c

f

A t f tc c

 

  

 
  

 

  
 



Frequency Modulation index

(unit less)
mf

f
m m

k A f

f f



 



Frequency

deviation


mff k A  Hz.