ANGLE MODULATION SYSTEM
1. Introduction:
Angle modulation is the process by which the angle (frequency or Phase) of the carrier signal is changed in accordance with the instantaneous amplitude of modulating or message signal. It is also known as “ Exponential modulation ". The amplitudes of the carrier remain contact in this process.
This is an advantage over AM since, all natural internal and external noises consist of electrical amplitude variations. The receiver cannot distinguish between amplitude variations that represent noise and those that represent desired signal .So AM reception is generally noisy than FM reception. The angle modulation can be broadly classified into two types such as ( 1) “Frequency modulations” (2) “Phase modulations”
2. Representation of FM
“Frequency modulations” can be defined as process by which the frequency the carrier wave is altered in accordance with instances amplitude if modulating or message signal the mathematical presentation of frequency modulation is obtained as follows:
Let the message signal
 m
And the carrier signal
 c

Where

V c
V m sin[ cos
 c t

 m t

]
--1
-- 2
V = maximum amplitude of message or modulating signal m
V = c

= m

= c 
=
 maximum amplitude of carrier signal angular frequency of modulating signal angular frequency of carrier signal total instantaneous phase angle of carrier

= (  t
 c


)
-- 3 v c
 

V c sin

To find angular velocity , differentiate the equations (3) w.r.t ‘t’
i.e, d
 dt
  c
During the process of frequency modulations the frequency of carrier signal is chanced in accordance with the instantaneous amplitude of message signal .Therefore the frequency of carrier after modulation is writer as
 i
  c

Kv m
 
C

KV cos
 m t

4
Where K = constant of proportionality
To find the instantaneous phase angle of modulated signal, integrate equation (4)
 i

1
   i dt
 


C

KV m cos
 m t
 dt
 
C t

KV m
 m sin
 m t
= Integrations constant, it is neglected because it plays no role in modulations process
The instantaneous amplitude of the modulating signal is given by
 
1

v v
 
 
FM
Vc
Vc sin sin

1
1
V
C
V
C sin( sin(

C
C t t t t
KV m
 m m m sin sin
 m m t t
)
) t t ) ) 5
5 v
 
FM v
 
FM

Vc sin

V
C
 sin(
1


V
C
C t sin(

C t
 m f sin

KV m
 m
 m t ) sin
  
6
 m t )
  
5
 max m f

KV m
 m
Modulations index of FM
From equation (4) the instantaneous angular frequency of FM signal is
 i



C

C
 cos
 m t )


1
KV m
The maximum and minim value of cosine terms is
.Hence the maximum value of angular frequency is given by
KV m
The minimum value of angular frequency is given by
 min
 
C

KV m
Then frequency deviation is given by
 d
  max
 
C
 
C
  min

KV m
  
7
The” Modulations Index” of FM System can be defined as the ratio of maximum frequency deviation to the modulating frequency. i.e

m f

 d w m

KV m
 m
      
8
 d

KV m

Maximum frequency deviation
Equation (6) represents the frequency modulating signal. v
 
FM

V
C sin(
 t

C m f sin
 m t )

V
C sin

C t cos( m f sin
 m t )
 cos
 c t sin( m f sin
 m t )
  
9

The graphical and phasor representation of frequency modulation is shown in figure 1 &2.In order to simplify the equation 9 it involves the use of Bessel function. v
 
FM

V
C
[ J
0
( m f
) sin

C t

J
1
( m f
)
 sin(

C
  m
) t
 sin(

C
  m
) t


V
C
[ J
2
( m f
) sin(

C

2
 m
) t
 sin(

C

2
 m
) t

V
C
[ J
3
( m f
) sin(

C

3
 m
) t
 sin(

C

3
 m
) t ]

.......
9 ( a )

It is seen that each pair of side band is preceded by J coefficients. The order of the coefficient is denoted by subscript m. The Bessel function can be written as
J m
  f


 m
2 f

 n 


1 n !


1 !
m
 n f
/

2
1

2

!

 m
2 !
 f n
/

2

2

4
!



   
10
In order to find the amplitude of a sideband and the amplitude of the carrier, it is necessary to know the value of the corresponding Bessel function. The plot of Bessel function is shown in figure 3
Conclusion: i) From the equation 9(a) we observe that FM has infinite number of side bands as well as carrier and they are separated from carrier by ….. but in AM there are only three terms (carrier ,LSB,USB). ii) The modulation index determines how many side band components have significant amplitude ie if ‘’ ,is large more number of significance side bands iii) are present ,if ‘’ is small ,less number of side bands exist.
The side bands at equal distance from ‘f
C
‘ have equals amplitudes so that ,the side band distribution is symmetrical about the carrier frequency. iv) Theoretically infinite number of side bands is produced and the amplitude of each side band is deiced by Bessel function.

v) For small values of m f
(i.e m f <
1) only the amplitudes of J o
(m f
) and J
1
(m f
) are significant and other terms are neglected .Thus FM carrier term has only one pair of sidebands. This is equivalent to Narrow Band FM. vi) The amplitude of FM remains unchanged ,hence the power of FM is same as that of the unmodulated carrier. vii) The total power of FM signal depends upon the power of the unmodulated carrier, whereas in AM the total power depends on the modulations index. viii) In AM the increased modulations index increased the sideband power. In FM the total power remains constant with increased modulation index and only the bandwidth is increased.
3. Multitone Modulation
Modualtion done with more than one message signal is called
“multitone modulation” .
Let us consider the message signal as v

V m 1 cos

1 t

V m 2 cos

2 t
   
11
Let the carrier signal be v
C

V
C sin(

C t
 
)

V
C sin

Let

)
; and the angular velocity
 
C d
 dt
During frequency modulation the frequency of carrier is changed in accordance with the amplitude of modulating signal. Hence the modulated signal is

1
  c

Kv m

1
 
C

KV m 1 cos

1 t

KV m 2 cos

2 t
  
12


 
C t

2
  f
1

1 sin

1 t

2
  f
2

2 sin

2 t

1
 
C

K [ V m 1 cos

1 t

V m 2 cos

2 t ]
The frequency deviation will be maximum when .The frequency deviation is proportional to the amplitude of modulating signal ie, hence equation 4.12 can be written as i
  c
 i

  c
2
  f
1

Kv m 1
 cos

1 t

Kv m
2

2
 f
2 cos

2 t .....
13
The instantaneous phase is given bye
After frequency modulation the instaneouis amplitude of the modulated signal is
 i



C
 t
(
 c

2



2

 f
1
 f
1 sin cos

1 t
1

1 t

2



2
2

 f
 f
2
2 sin cos

2 t

2 t ) dt
 i
 
C t

 f
1 f
1 sin

1 t

 f
2 f
2 sin

2 t .......
14


V
C sin
 i v fm v fm

V
C sin[

C t
 sin[
If

1

C t

 m f 1 sin[

C t
 sin

 f
1

1 t f
1
 m f 1 sin

1 t sin m f 2

1 t sin

 f
2 sin

2 t ] f
2

2 t ].........
.....
15

2 sin m f

2 t ]
Then v fm

V
C sin[

C t

(

1
 
)]



V
C
[sin
V
C
{sin cos

C

C

C t .
t .[sin cos( t .[cos

1


1 cos
1


2

2 cos


)

 cos sin
2 cos


C
 sin t .
sin( sin

2

2

]
1

]}.....
16

2
)]
In order to simplify the above equation the Bessel function can be used hence the resultant equation given by

 v fm

V
C m




J n
(

1
).
J m
(

2
)(cos

C t
 n

1 t
 n

2
)...
17
From the above equation we observe that, it has four frequency terms i) A carrier freque3ncy component with amplitude
[ J o
(

1
).
J
0
(

2
).
E
C
] ii) A set of sidebands corresponding to first tone .The sidebands have
J n
(

1
).
J
0
(

2
).
E
C amplitude and frequencies .where n=1, 2, 3….
Phase modulation is defied as the process by which changing the phase of carrier signal in accordance with the instantaneous of message signal .The amplitude and frequency remains constant after the modulation process.
Let the modulating signal is given by v m
( t )

V m cos
 m t
The carrier signal v
C
( t )

V
C sin(

C t
 
)
Where

=phase angle of carrier signal .It is changed in accordance with the amplitude of the message signal v m
( t ) ;
Ie.,
 
KV m
( t )

KV m cos

C t ..........
18
After phase modulation the instantaneous voltage will be v pm
( t )

V
C sin(

C t
 
)
ANGLE MODULATION SYSTEM


V
C v pm sin(
( t )


C t
V
C
 sin(
KV m

C t cos
 m t )....
19
 m p cos
 m t )....
20
Where mp = Modulation index of phase modulation let
5 Conversion of PM to FM
Frequency modulated wage can be obtained from PM. This is done by integrating the modulating signal before applying it to the phase modulator it is shown fig 4. v m
( t )

V m cos
 m t v m
( t )
 
V m cos
 m tdt

V
 m m sin
 m t ...
21
After Integrat6ion

v m
( t )

V m cos
 m t
After phase modulation;
 v m
( t )
 
Kv m
( t )

KV m
 m sin
The instantaneous value of modulated voltage is given by v fm
( t )

V
C sin(

C t
 
)
 m t .....
22 v fm
( t )

 v fm
( t )
V
C
 sin(
V
C
 sin
C t

.(

C t
KV
 m m
 m f sin
 m t )........
23 cos
 m t ........
24 wherem
 v fm
( t ) f


V
C
 f f m sin(

KV
 m m

C t
 m as [ 2
  f f cos
 m t

KV m
]
).......
24
This is the expression for FM wave.
Conversation of FM to PM
The PM wave can be obtained from FM by differentiating the modulating signal before applying it to the frequency modulator circuit it is shown in fig.5.

We know that v m
( t )

V m cos
 m t
After differentiation; d dt v m
( t )
   m
V m sin
 m t .....
25
After frequency modulation

 i
 
C
 dv m
( t ) dt

 i

C



C
K [
  m

K .
 m
.
V
.
V m sin m sin
 m t ]
 m t .....
26
We know that, the instantaneous phase angle of frequency modulated signal is
 i

 i


C t
  i dt
 

K

 m
V m m
(

C
 cos
K
 m
V m
 m t
 
C t

KV m sin cos
 m t ..........
27
 m t ) dt
The instantaneous voltage after modulation is given by v pm
( t )

V
C
Sin
 i

v pm
( t ) v pm
( t )

V
C
(

C t

V
C
(

C t

KV m
 m p cos
 m t ) cos
 m t )....
28
This is the expression for the phase modulated wave. The process of integration and differentiation are liner .Therefore now new frequencies are generated
6.
Generation of Narrow Band FM
The bandwidth of FM signal depends on the modulation index. if the modulation index is high, then the bandwidth is large and vice versa. Thus depending on the modulation index, FM could be divided into two types, namely
(1) Narrow band FM and 92) Wide band FM i) Narrowband FM for which modulation index is small compared to one radian ii) Wideband FM for which modulation index is large compared to one radian.
Let the message signal be represented as v m
( t )

V m cos
 m t
Let the carrier signal be given by v
C
( t )

V
C sin

C t
  
V
C sin

 
(

C t
Where
 
).........
....
29
Differentiating equation 94.29) w.r.t. ‘t’ we get

d
 dt
 
C

Angular frequency of carrier signal
After frequency modulation
 i
 
C

KV m
( t )

KV m cos
 m t
The frequency deviation is maximum, when cos
 t
 
1
Hence
 i
 
C

KV m
The frequency deviation is proportion al to the amplitude of modulating voltage ,hence it can be written as

2
  f
 i



C
KV m

2
  f .
cos
 m t
 i
   i dt
 
 f
(

C

 i
 
C v fm
( t ) t
 f m

V
C
.
sin sin

 m t m t
2
  f .
cos
 m t ) dt

V
C v sin( fm
( t )


C
V
C t
 sin
 f sin
 m t f m

C t .
cos( m
) f

V
C
.
sin sin(

C t
 m t )

V
C
 m f cos sin

 m t )
C t .
sin( m f
.
sin
 m t )...
30
For narrow band FM assume the modulation index ‘m f
’ is small compared to one radian, hence we may use the following approximation. cos( m f and sin( sin m f
 m t ) sin

1
 m t )
 m f sin
 m t (because sin
   if '

' small )
V fm
( t )

V
C sin

C t

V
C sin(

C t
 m f sin
 m t ).....
31

The equation (31) defines the approximate form of narrow band FM. From this representation we deduce the modulator shown in fig 4.6
Fig. 6. Generation of narrowband FM
This modulation involves splitting the carrier wave into two paths. One path is direct, the other path contains a -90
0 phase
shift network and a product modulator, the combination of which generates DSB-SC-AM signal. The difference between these two signals produces narrowband FM shown figure 6 differs from this ideal condition in two fundamental respects. i) ii)
The envelope contains residual amplitude and varies with time.
It produces some harmonic distortions.
It can be avoided by restricting the modulation index m <0.3 radians.
7.
Generation of Wide Band FM
Wide band FM for which modulation index is large compared to narrow band
AM. It can be obtained by multiplying narrow band FM signal by using suitable frequency multiplier.
Let the message signal be represented as v m
( t )

V m cos
 m t
Let the carrier signal be represented as

v c
( t )
 where

V c
 sin(
(

C

C t
 t

 
)

V c sin
)........
4 .
32

Differentiating equation (32) w.r.t ‘t’ d
 dt
 
C
= angular frequency of the carrier signal.
After frequency modulation the frequency of the modulated voltage is
 i
 
C

Kv m
( t )
The frequency deviation is maximum when cos
 m t



1

C

KV m cos
 m t
   
C

KV m f
And the frequency deviation is propositional to the amplitude of the modulating
Voltage

2
  f
 i



C
KV m

2
  f .
cos
 m t
 i
   i dt
 
(

C

2
  f .
cos
 m t ) dt
 i
 
C t

 f sin
 m t f m v fm
( t )

V
C sin
 i

V
C sin(

C t

 f sin
 m t ) f m

V
C v fm
( t ) sin(

C t

V
C
 m f sin sin
 m t )

C t .
cos( m f sin
 m t )

V
C cos

C t .
sin( m f weknow  v fm
( t ) sin

V
C
 m t ) sin(

C t
 m f sin
 m t )
Hence
It can be rewritten by using exponetional form as follows:

v fm
( t )

V
Ce j (

C t
 m f sin
 m t )

Re[ V
Ce j (

C t
 m f sin
 m t )

Re[ v ( t ).
e j
 c t
]  33 when v ( t )

V
C
[ e jmSin
 c t
]  34
Thus, unlike the original FM signal v(t) ,the complex envelope v(t) is a periodic function of the time with a fundamental frequency equal to the modulation frequency f m
Hence, the equation (34) could be expanded by Fourier series. v m
( t )
 n



 
C n e jn
 m t  35
Where C n
= Fourier coefficient

C n

C n
 let f m f m


2 f m v ( t
2
) e
 jn
 m t .
d t
 36 f m f m x


2 f m
V
2
  m
C e jn sin
 m t
.
e
 t
 jn
 m t .
d t
2
 f m t
 37 and dx
  m

2
 f m or dx
  m dt
 dt
The integral on the right hand side of equation is recognized as the n th order Bessel
2
 f m dt  38 function of the first kind. This function is commonalty denoted by the symbol Jn(m f
) It is given by

J n
( m f
)

1
2

Hence C n
 

 e
 jm f
.
sin x
 nx
.
dx  39
J n
( m f
).
V
C v ( t )
 v m
( t )
 n



 
J ( m f
) V
Ce jn
 m t v mf
( t ) v mf
( t )

Re[ v ( t ).
e j
 c t
]

Re[ n



 
J n
( m f
) V
C
.
e jn
 m t

V
C n



 
J n
( m f
) cos n (

C t
 
C t )  41
].
e j

C t  40
This is the desired form for the Fourier series representation of the singe tone FM signal for an arbitrary value of m f
for wide band FM wide band FM the modulation index is greater than one radian
Transmission bandwidth of FM
The number of significant sidebands ‘n’ produced is an FM wave’s can obtained from the plot of Bessel function Jn(m f
).For n> m f
, the values of Jn(m f
) are negligible particularly when m f
>>1.Therefore ,the significant sidebands produced in wideband FM may be considered to be an integer approximately equal to m f
ie n- m f
if m f
>>1

The USB are separated by and form a frequency span of ‘’ .Similar span is produced by the LSB.
Therefore transmission band width of FM wave is defined as the separation between the frequencies beyond which none off side frequencies is greater than 1% of the carrier amplitude obtained when the modulation is removed. i.e ,B.W.= 2n
 m
rad/sec where n=number of sidebands if n= m f
then B.W= 2m f

or m m m



 f hence B.W.=

2
 
 m m f

2
  rad .

2 (
 f ) Hz
Thus the approximate bandwidth of a wide band FM system is given as twice the frequency deviation .This approximation holds true for m f
>>1.For smaller values of m f, the bandwidth may be more than
The approximate rule of transmission bandwidth of an FM signal generated by a single tone modulating signal is
We know that
B .
W

2 (
    m
)  m

 m
 f

2
 
( 1

1
) radian m f
This empirical relation is known as
“Carson’s rule”
Band width of PM
The PM bandwidth as per Carson’s rule ( BM ) pm

2
  
2 K p
V m
 m
Thus the B.W of the PM signal varies tremendously with a change in modulating frequency
 m
Comparison of WBFM and NBFM.

S.No WBFM
I.
II.
III. iV.
V.
Vi.
Modulating index is greater than1
Frequency deviation =75 KHz.
Modulating frequency range
From 30 Hz-15 Khz.
Bandwidth 15 times NBFM.
Noise is more suppressed.
Use: Entertainment and broadcasting
NBFM
Modulation index is less than 1
Frequency deviation 5 Khz.
Modulation frequency =3 Khz
Bandwidth =2 FM
Less suppressing of noise
Use: Mobile communication.