Frequency modulation (FM) is a type of angle modulation where the instantaneous frequency of the carrier signal varies linearly with the modulating signal. There are two types of FM: narrowband FM (NBFM) where the modulation index is less than 1, and wideband FM (WBFM) where the modulation index is greater than 1. The bandwidth of an FM signal can be estimated using Carson's rule, which states that nearly all the signal power lies within a bandwidth equal to twice the maximum frequency deviation plus the maximum modulating frequency. FM signals have constant amplitude but varying frequency, so their average power does not depend on the modulating signal and remains constant.

1. Frequency modulation
•. Types of Modulation(Analog)
•Phase-Frequency Relationships
•FM and PM basics
•Frequency deviation
•MODULATION INDEX
•Classification of FM
•Narrow Band FM (NBFM)
•generating a narrowband FM signal.
•Wide Band FM (WBFM).
•Carson’s Rule
•Generation of WBFM
•Average Power
•FM BANDWIDTH
•Comparing Frequency Modulation to Phase Modulation

2. Types of Modulation(Analog)
1. Angle Modulation:
In the angle modulation, again there are two
different types of modulations.
• Frequency modulation
• Phase modulation.
2. Amplitude Modulation:

3. 0
time t
i
i
C
dt tt
Slope:  (t) 
d(t)
 
Phase-Frequency Relationship When Frequency is Constant
(t)  Acos((t))
(t) is generalized angle
(t)  Acos(Ct0 )
(t)
Ct 0
3

4. Concept of Instantaneous Frequency
(t) is generalized angle
(t)  Acos((t))
0

(t)  Acos(Ct0 )
(t)
Ct 0
(t)
d(t)
dt tti
Slope: i (t)   C
time t
ti
4

5. Frequency Modulation (FM)
But in frequency modulation the instantaneous angular frequency
i varies linearly with the modulating signal m(t),
i  C  kf m(t)
t
)d


 




FM (t)  Acos Ct  kf  m(
5
 kf m())d  Ct  kf
t
 (t)  (C
t
 m()d
 
kf is frequency-deviation (sensitivity) constant. Units: radians/volt-sec.
Then
FM and PM are very much related to each other.
In PM the angle is directly proportional to m(t).
In FM the angle is directly proportional to the integral of m(t), i.e., m(t)dt

6. Frequency Modulation
0
( ) cos 2 2 ( )
t
c c f
s t A f t k m d
   
 
 
 
 

( ) cos(2 )
m m
m t A f t

 cos(2 )
i c f m m
f f k A f t

 
 
 
0
Let
2 cos(2 )
2
1 1 1
2 2 2
1
2 cos(2 )
2
t
f m m
c
i
c f m m t
d k A f d
d f t
d
f
dt dt dt
f k A f 
   


  
  
 
 
 
 
  
 

max | ( ) |
f m f
f k A k m t
  
single-tone ( ) case: cos(2 )
general case:
i c m
c i c
m t f f f f t
f f f f f

  
     
Frequency deviation Δf

7. MODULATION INDEX


Directly proportional to the amplitude of the modulating
signal and inversely proportional to the frequency of the
modulating signal
Ratio of the frequency deviation and the modulating frequency
  as modulation index :


f m
k A
fm
 

8. Classification of FM
• On the basis of modulation index, FM is classified in
two parts :
• 1) If  <1, then FM is said to be Narrow
Band FM (NBFM).
• 2) If  >1, then FM is said to be Wide Band
FM (WBFM).

9. Narrow Band Angle Modulation
1
)
( 
t
a
kf
 
t
w
t
a
k
t
w
A
t c
f
c sin
)
(
cos
)
( 


Definition
Equation
Comparison with AM
Only phase difference of Pi/2
Frequency: similar
Time: AM: frequency constant
FM: amplitude constant
Conclusion: NBFM signal is similar to AM
signal
NBFM has also bandwidth 2W. (twice
message signal bandwidth)

10. Block diagram of a method for generating a narrowband FM signal.

11. Wide Band FM
 Wideband FM signal
 Fourier series representation
 
( ) cos(2 )
( ) cos 2 sin(2 )
m m
c c m
m t A f t
s t A f t f t

  

 
 
 
( ) ( )cos 2 ( )
( ) ( ) ( ) ( )
2
c n c m
n
c
n c m c m
n
s t A J f nf t
A
S f J f f nf f f nf
 
  




 
     


( ): -th order Bessel function of the first kind
n
J n


12. Bessel Function of First Kind
0
1
2
1. ( ) ( 1) ( )
2. If is small, then ( ) 1,
( ) ,
2
( ) 0 for all 2
3. ( ) 1
n
n n
n
n
n
J J
J
J
J n
J
 
 







 


 



13. Spectrum of WBFM (Chapter 5.2)
 Spectrum when m(t) is single-tone
 Example 2.2
   
 
( ) cos 2 sin(2 ) ( )cos 2 ( )
( ) ( ) ( ) ( )
2
c c m c n c m
n
c
n c m c m
n
s t A f t f t A J f nf t
A
S f J f f nf f f nf
    
  




   
     



14. Bandwidth of FM
• Facts
– FM has side frequencies extending to infinite frequency 
theoretically infinite bandwidth
– But side frequencies become negligibly small beyond a
point  practically finite bandwidth
– FM signal bandwidth equals the required transmission
(channel) bandwidth
• Bandwidth of FM signal is approximately by
– Carson’s Rule (which gives lower-bound)

15. Carson’s Rule
 Nearly all power lies within a bandwidth of
– For single-tone message signal with frequency fm
– For general message signal m(t) with bandwidth (or highest
frequency) W
2 2 2( 1)
T m m
B f f f

    
2 2 2( 1)
T
B f W D W
    
where is deviation ratio (equivalent to ),
max ( )
f
f
D
W
f k m t



 
   

16. NBFM and WBFM
• In NBFM the maximum modulating frequency is
3KHz and maximum frequency deviation is 75KHz.
• In WBFM the maximum modulating frequency is
30Hz to 15KHz and maximum frequency deviation is
75KHz.
• Bandwidth of WBFM is 15 times of that of NBFM.

17. Pre-emphasis and De-emphasis
• If freq. ↑ then amplitude ↓.
• Due to this signal decreases hence decreasing the
signal to noise ratio(SNR).
• Therefore, the high frequency component which
have low SNR are boosted or emphasized prior to
the transmission of the signal.
• This is done by using a pre-emphasis circuit and is
used at the transmitting end just before the
modulation takes place.

18. Pre-emphasis and De-emphasis
• Now, since the relative SNR for various
frequency components has been disturbed
then those frequency components which were
initially boosted or emphasized are now
brought down to the same level to keep the
same quality of the signal.
• This is done by using a de-emphasis circuit
and is used at the receiving end.

19. Generation of NBFM
NBFM Modulator:

20. Generation of WBFM
• There are two basic methods for generating FM
signals known as direct and indirect methods. The
direct method makes use of a device called voltage
controlled oscillator (VCO) whose oscillation
frequency depends linearly on the modulation
voltage.

21. Generation of WBFM
 Indirect Method (Armstrong’s Method):

22. Average Power of a FM or PM Wave
The amplitude A is constant in a phase modulated or a frequency
modulated signal. RF power does not depend upon the frequency
or the phase of the waveform.
FM or PM (t)  AcosCt  f (k,m(t))
Average Power 
A2
(always)
2
This is a result of FM and PM signals being constantamplitude.
22

23. Indirect Generation of an FM Signal Using Multiplication
In this method, a narrowband frequency-modulated
signal is first generated and then a frequency
multiplier is used to increase the modulation index.
The concept is shown below:
FM
NB
(t) FM
WB
(t)
NBFM
Frequency
Multiplier
m(t)
23
A frequency multiplier is used to increase both the
carrier frequency and the modulation index by integer N.

24. Generation of Narrowband Frequency Modulation (NBFM)
t

 
FM (t)  AcosCt  kf
 
 m()d
NBFM requires  << 1radian
DSB-SC
modulator
Lathi & Ding;
Figure 5.10
Page 276
-/2
NBFM

m(t)
kf

24
Acos(ct)
Asin(ct)
Carrier

25. Generation of Narrowband Phase Modulation (NBPM)
PM (t)  Acos(Ct  kpm(t))
-/2
NBPM

Acos(ct)
m(t)
kp
Asin(ct)
25

26. FREQUENCY MODULATION
(FM)
 Variation of d/dt produces Frequency
Modulation
 Frequency modulation implies that d/dt is
proportional to the modulating signal.
 This yields
c c
c c f m
c c f m m
f m
m
cos (t)

vFM (t) Vc sinct  (t)
 
 V sin  t   '(t)dt
 
 
 V sin  t  k v (t)dt
 
 
 V sin  t  k V sin (t)dt
 
k V 
 V sin

t 
c c m 
 




27. Specifications for transmission of FM signal
 Table 1 display the transmission band that use FM and the legal
frequency deviation limit for each category

28. FM BANDWIDTH
 The total BW of an FM signal can be determined by knowing
the modulation index and Bessel function.
N = number of significant sidebands
fm = modulating signal frequency (Hz)
Another way to determine the BW is use Carson’s rule

 This rule recognizes only the power in the most significant
sidebands with amplitude greater than 2% of the
carrier.
BW  2 fm N

29. CARSON’S RULE
fd (max)
fm (max)
= max. frequency deviation
= max. modulating frequency


• Carson’s rule always give a lower BW calculated with the
formula BW = 2fmN.
• Consider only the power in the most
significant sidebands whose amplitudes are
greater than 1% of the carrier.

Rule for the transmission bandwidth of an FM
signal
generated by a single of frequency fm as follows:
BW  2[ fd (max)  fm(max) ]
or
T m

B  BW  2f  2 f  2f (1  1)
= 2 fm 1  

30. DEVIATION RATIO (DR)
 Minimum bandwidth is greatest when maximum freq
deviation is obtained with the maximum modulating
signal frequency


Worst case modulation index and is equal to the
maximum peak frequency deviation divided by the
maximum modulating signal frequency
Worst case modulation index produces the widest

output frequency spectrum
Mathematically,
max mod signal freq fm(max)
DR 
max peak freq deviation

fmax

31. 
P = VC /2R W
2
 Thus the power contained in the FM signal is independent
of the message signal. This is an important difference
between FM and AM.
 The time-average power of an FM signal may also
be obtained from
vFM (t)  Vc cos(2 fct  (t))
POWER IN ANGLE-
MODULATED SIGNAL
The power in an angle-modulated signal is easily computed

32. FM SIGNAL GENERATION
 They are two basic methods of
generating frequency-Modulated
signals:
 Direct Method
 Indirect Method

33. 13
AC cos(Ct)
C  2 fC
Am cos(mt) A single tone frequency
m  2 fm (radians/sec)
kf
Carrier signal
Carrier frequency
Modulating wave m(t)
Modulating frequency
Deviation sensitivity
Frequency deviation f m


 
max min
22
f 
 m m
f  k A  k
 
f
fm
fi  fC  k f Am cos(mt)  fC  f cos(mt)
  


 
 t
FM (t)  ACcosCt  k f  m( )d  , generally
Modulation Index
Instantaneous frequency
Remember
Modulated wave FM C

 
 (t)  ACcos t  

 
f m
m
m
 k A
sin( t)
f
FM (t)  AC 
cosCt   sin(mt)

Handout
or
Equations for FM Wave with Single Tone Modulation