Chapter 4. Angle Modulation
4.4 Narrow-Band Frequency Modulation
– We first consider the simple case of a single-tone modulation that
produces a narrow-band FM wave
– We next consider the more general case also involving a single-tone
modulation, but this time the FM wave is wide-band
– The two-stage spectral analysis described above provides us with
enough insight to propose a useful solution to the problem
– An FM signal is
m(t )  Am cos(2f mt ) (4.9)
f  k f A m
f i (t )  f c  k f Am cos(2f mt )
 f c  f cos(2f mt )
 i (t )  2f ct 
(4.10)
f
sin(2f mt ) (4.12)
f m Modulation index
i (t )  2f ct   sin(2f mt ) (4.14)
(4.11)
The frequency deviation

f
fm
( 4.13)
The phase deviation
– The FM wave is
s(t )  Ac cos2f ct   sin(2f mt ) (4.15)
cos( A  B)  cos A cos B  sin A sin B
s(t )  Ac cos(2f ct ) cos sin(2f mt )  Ac sin(2f ct ) sin  sin(2f mt ) (4.16)
– If the modulation index is small compared to one radian, the
approximate form of a narrow-band FM wave is
s(t )  Ac cos(2f ct )  Ac sin(2f ct ) sin(2f mt ) (4.17)
cos sin(2f mt )  1
sin  sin(2f mt )   sin(2f mt )
1. The envelope contains a residual amplitude modulation that varies
with time
2. The angel θi(t) contains harmonic distortion in the form of third- and
higher order harmonics of the modulation frequency fm
– We may expand the modulated wave further into three
frequency components
1
s(t )  Ac cos(2f c t )  Ac cos2 ( f c  f m )t   cos2 ( f c  f m )t  (4.18)
2
1
s AM (t )  Ac cos(2f c t )  Ac cos2 ( f c  f m )t   cos2 ( f c  f m )t  (4.19)
2
– The basic difference between and AM wave and a narrowband FM wave is that the algebraic sign of the lower sidefrequency in the narrow-band FM is reversed
– A narrow-band FM wave requires essentially the same
transmission bandwidth as the AM wave.
• Phasor Interpretation
– A resultant phasor representing the narrow-band FM wave
that is approximately of the same amplitude as the carrier
phasor, but out of phase with respect to it.
– The resultant phasor representing the AM wave has a different
amplitude from that of the carrier phasor, but always in phase
with it.
4.5 Wide-Band Frequency Modulation
– Assume that the carrier frequency fc is large enough to justify
rewriting Eq. 4.15) in the form
s (t )  Re Ac exp( j 2f c t  j sin( 2f mt ))
~

 Re s (t ) exp( j 2f c t )


(4.20)
– The complex envelope is
~
s(t )  Ac exp[ j sin( 2f mt ) ]
(4.21)
~
s (t )  Ac exp[ j sin( 2f m (t  k / f m )) ]
 Ac exp[ j sin( 2f mt  2k ) ]
 Ac exp[ j sin( 2f mt ) ]

s (t )   cn exp( j 2nf mt ) (4.22)
~
n  
•
The complex Fourier coefficient
1 /( 2 f m ) ~
cn  f m 
1 /( 2 f m )
 f m Ac 
s (t ) exp( j 2nf m t )dt
1 /( 2 f m )
1 /( 2 f m )
A
cn  c
2



exp[ j sin( 2f m t )  j 2nf mt ]dt (4.23)
x  2f mt (4.24)
exp[ j (  sin x  nx)]dx (4.25)
1
J n ( ) 
2



exp[ j (  sin x  nx)]dx (4.26)
cn  Ac J n ( ) (4.27)

s (t )  Ac  J n (  ) exp( j 2nf mt ) (4.28)
~
n  
 

s(t )  Re  Ac J n (  ) exp[ j 2 ( f c  nf m )t ] (4.29)
 n


• In the simplified form of Eq. (4.29)

s (t )  Ac  J n (  ) cos[2 ( f c  nf m )t ] (4.30)
n  
Ac 
S ( f )   J n (  )[ ( f  f c  nf m )   ( f  f c  nf m )] (4.31)
2 n 
•
Properties of single-tone FM for arbitrary modulation index β
1. For different integer values of n,
J n ( )  J n ( ), for n even (4.32)
J n ( )   J n ( ), for n odd (4.33)
2. For small values of the modulation index β
J 0 (  )  1,



J1 (  )  ,
 (4.34)
2

J n (  )  0, n  2
3. The equality holds exactly for arbitrary β

2
J
 n ( )  1 (4.35)
n  
1. The spectrum of an FM wave contains a carrier component and
an infinite set of side frequencies located symmetrically on either
side of the carrier at frequency separations of fm,2fm, 3fm….
2. The FM wave is effectively composed of a carrier and a single
pair of side-frequencies at fc±fm
3. The amplitude of the carrier component of an FM wave is
dependent on the modulation index β
The average power of such a signal developed across a 1-ohm
resistor is also constant.
Pav 
1 2
Ac
2
• The average power of an FM wave may also be determined form
1 2  2
P  Ac  J n (  ) (4.36)
2 
4.6 Transmission Bandwidth of FM waves
•
Carson’s Rule
–
–
The FM wave is effectively limited to a finite number of significant
side-frequencies compatible with a specified amount of distortion
Two limiting cases
1. For large values of the modulation index β, the bandwidth approaches,
and is only slightly greater than the total frequency excursion 2∆f,
2. For small values of the modulation index β, the spectrum of the FM wave
is effectively limited to the carrier frequency fc and one pair of sidefrequencies at fc±fm, so that the bandwidth approaches 2fm
–
An approximate rule for the transmission bandwidth of an FM wave
 1
BT  2f  2 f m  2f 1   (4.37)
 
• Universal Curve for FM Transmission Bandwidth
– A definition based on retaining the maximum number of
significant side frequencies whose amplitudes are all greater
than some selected value.
– A convenient choice for this value is one percent of the
unmodulated carrier amplitude
– The transmission bandwidth of an FM waves
• The separation between the two frequencies beyond which none
of the side frequencies is greater than one percent of the carrier
amplitude obtained when the modulation is removed.
• As the modulation index β is increased, the bandwidth occupied
by the significant side-frequencies drops toward that value over
which the carrier frequency actually deviates.
• Arbitrary Modulating Wave
– The bandwidth required to transmit an FM wave generated by
an arbitrary modulating wave is based on a worst-case tonemodulation analysis
– The deviation ratio D
D
f
W
(4.38)
– The generalized Carson rule is
BT  2(f  W ) (4.39)