To define and explain frequency
modulation (FM) and phase modulation
(PM)
 To analyze the FM in terms of
Mathematical analysis
 To analyze the Bessel function for FM and
PM
 To analyze the FM bandwidth and FM
power distribution

Frequency modulation (FM) and phase
modulation (PM)
 Analysis of FM
 Bessel function for FM and PM
 FM bandwidth
 Power distribution of FM

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.
 Also known as “Exponential modulation"


Angle modulation is classified into two types
such as
› Frequency modulation (FM)
› Phase modulation (PM)

Used for :
›
›
›
›
›
Commercial radio broadcasting
Television sound transmission
Two way mobile radio
Cellular radio
Microwave and satellite communication system
FM is the process of varying the
frequency of a carrier wave in
proportion to a modulating signal.
 The amplitude of the carrier wave is kept
constant while its frequency and a rate
of change are varied by the modulating
signal.


Fig 3.1 :
Frequency
Modulated signal

i.
ii.
iii.
iv.
The important features
waveforms are :
about
FM
The frequency varies.
The rate of change of carrier frequency changes
is the same as the frequency of the information
signal.
The amount of carrier frequency changes is
proportional to the amplitude of the information
signal.
The amplitude is constant.




The FM modulator receives two signals,the information signal
from an external source and the carrier signal from a built in
oscillator.
The modulator circuit combines the two signals producing a
FM signal which passed on to the transmission medium.
The demodulator receives the FM signal and separates it,
passing the information signal on and eliminating the carrier
signal.
Federal Communication Coporation (FCC) allocation for a
standard broadcast FM station is as shown in Fig.3.2
Mathematical analysis:
 Let the message signal:



 m t   Vm cos mt
(3.1)
And carrier signal:
 c t   Vc cos[ ct   ]
(3.2)
 Where carrier frequency is very much
higher than message frequency.



In FM, frequency changes with the change of the
amplitude of the information signal. So the
instantenous frequency of the FM wave is;
i  c  Kvm t   C  KVm cos mt
K is constant of proportionality
(3.3)

Thus, we get the FM wave as:
vFM (t )  Vc cos 1  VC cos(C t 
KVm
m
sin mt )
vFM (t )  VC cos(C t  m f sin mt ) (3.4)

Where modulation index for FM is given by
mf 
KVm
m

Frequency deviation: ∆f is the relative
placement of carrier frequency (Hz) with
respect to its unmodulated value. Given
as:
max  C  KVm
min  C  KVm
d  max  C  C  min  KVm
d KVm
f 

2
2

Therefore:
KVm
f 
2
or
f  KVm ( Hz )
f
mf 
fm
f  Vm ;

FM broadcast station is allowed to have
a frequency deviation of 75 kHz. If a 4
kHz (highest voice frequency) audio
signal causes full deviation (i.e. at
maximun
amplitude of information
signal) , calculate the modulation index.

Determine the peak frequency
deviation,  f , and the modulation
index, mf, for an FM modulator with
a deviation Kf = 10 kHz/V. The
modulating signal to be transmitted
is Vm(t) = 5 cos ( cos 10kπt).

Thus, for general equation:
vFM (t )  VC cos(C t  m f cos mt )
vFM (t )  VC sin( C t  m f sin mt )
 VC sin C t cos( m f sin mt )  cos c t sin( m f sin mt )
vt FM  VC [ J 0 (m f ) sin C t  J1 (m f )sin( C  m )t  sin( C  m )t 
 VC [ J 2 (m f ) sin( C  2m )t  sin( C  2m )t
 VC [ J 3 (m f ) sin( C  3m )t  sin( C  3m )t ]  ..

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
 mf
J m m f   
 2



n
 1 m f / 22 m f / 24


 ....
 
 n! 1!n  1! 2!n  2!

N=number of the side frequency
 M=modulation index




Theoretically, the generation and transmission of FM
requires infinite bandwidth. Practically, FM system have
finite bandwidth and they perform well.
The value of modulation index determine the number of
sidebands that have the significant relative amplitudes
If n is the number of sideband pairs, and line of frequency
spectrum are spaced by fm, thus, the bandwidth is:
B fm  2nf m

For n=>1



Estimation of transmission bandwidth;
Assume mf is large and n is approximate mf +2; thus
Bfm=2(mf +2)fm
=
f
2(
 2) f m
fm
B fm  2(f  f m )........(1)
(1) is called Carson’s rule

The worse case modulation index which produces
the widest output frequency spectrum.
DR 

f (max )
f m (max )
Where
› ∆f(max) = max. peak frequency deviation
› fm(max) = max. modulating signal frequency
An FM modulator is operating with a peak
frequency deviation
∆f = 20 kHz. The
modulating signal frequency, fm is 10 kHz,
and the 100 kHz carrier signal has an
amplitude of 10 V. Determine :
a)The minimum bandwidth using Bessel
Function table.
b)The minimum bandwidth using Carson’s
Rule.
c)Sketch the frequency spectrum for (a), with
actual amplitudes.
For an FM modulator with a modulation
index m=1, a modulating signal Vm(t)=Vm
sin(2π1000t), and an unmodulated carrier
Vc(t) = 10sin(2π500kt), determine :
a)Number of sets of significant side
frequencies
b)Their amplitudes
c)Draw the frequency spectrum showing their
relative amplitudes.

As seen in Bessel function table, it shows that as
the sideband relative amplitude increases, the
carrier amplitude,J0 decreases.

This is because, in FM, the total transmitted
power is always constant and the total average
power is equal to the unmodulated carrier power,
that is the amplitude of the FM remains constant
whether or not it is modulated.
In effect, in FM, the total power that is originally
in the carrier is redistributed between all
components of the spectrum, in an amount
determined by the modulation index, mf, and the
corresponding Bessel functions.
 At certain value of modulation index, the carrier
component goes to zero, where in this condition,
the power is carried by the sidebands only.


The average power in unmodulated carrier
Vc2
Pc 
2R

The total instantaneous power in the angle modulated carrier.
m(t ) 2 Vc2
Pt 

cos 2 [c t   (t )]
R
R
2
Vc2  1 1
 Vc
Pt 
  cos[ 2c t  2 (t )] 
R 2 2
 2R

The total modulated power
Vc2 2(V1 ) 2 2(V2 ) 2
2(Vn ) 2
Pt  P0  P1  P2  ..  Pn 


 .. 
2R
2R
2R
2R

a) Determine the unmodulated carrier
power for the FM modulator and
condition given in example 3.4, (assume
a load resistance RL = 50 Ώ)

b) Determine the total power in the
angle modulated wave.
To define and explain frequency
modulation (FM) and phase modulation
(PM)
 To analyze the FM in terms of
Mathematical analysis
 To analyze the Bessel function for FM and
PM
 To analyze the FM bandwidth and FM
power distribution

END OF CHAPTER 3
PART 1