EE 370: Dr. Wajih Abu-Al-Saud (Modified)
Chapter V: Angle Modulation
Lecture 23
Effect of Non–Linearity on AM and FM signals
Sometimes, the modulated signal after transmission gets distorted due to non–linearities
in the channel, for example. In general, when the transmitted modulated signal is affected
by non–linearity in the channel, the demodulated signal becomes a distorted version of
the message signal. We can easily show that the effect non–linearities on different types
of amplitude–modulated signals is devastating, while frequency–modulated signals are
immune to non–linearities. In fact, the effect of non–linearities on FM signals can be used
for generating wideband FM signals from narrowband FM signals, which is an important
feature.
Non–Linearity in AM: Consider the channel shown below with a DSBSC input signal.
The channel is a non–linear channel in which the output signal of the channel is the sum
of the input signal and other powers of the input signal.
gDSBSC(t)
f(t)
a1(.) + a2(.)2 + a3(.)3
Let gDSBSC(t) be
g DSBSC (t )  m (t )  cos ct  .
The output signal of the channel f(t) becomes
f (t )  a1m (t )  cos c t   a2  m (t )  cos c t    a3  m (t )  cos c t  
2
3
 a1m (t )  cos c t   a2 m 2 (t )  cos 2 c t   a3m 3 (t )  cos 3 c t 
a m 3 (t )
a2 m 2 (t )
1  cos  2c t    3
 cos c t  1  cos  2c t  
2
2
a m 3 (t ) 
a3 m 3 (t )
a m 2 (t ) 
a2 m 2 (t )
 2
 a1m (t )  3

c
os

t

cos
2

t

 cos c t   cos  3c t  




c
c

2
2 
2
4

 a1m (t )  cos c t  

3a m 3 (t ) 
a3m 3 (t )
a2 m 2 (t ) 
a2 m 2 (t )
 a1m (t )  3

cos

t

cos
2

t

 cos  3c t 




c
c

2
4
2
4


Around 0
Around c
Around 2c
Around 3c
So, unless a3 is zero, it is clear that the original modulated signal gDSBSC(t) given above
cannot be extracted from the received signal f(t) because the terms with frequency around
c does not contain only m(t) but also m3(t). So, DSBSC (and AM modulation in
general) is vulnerable to non–linearities.
Non–Linearity in FM: Again, consider the same channel shown given above with an FM
input signal.
EE 370: Dr. Wajih Abu-Al-Saud (Modified)
Chapter V: Angle Modulation
gFM(t)
Lecture 23
q(t)
a1(.) + a2(.)2 + a3(.)3
Let gFM(t) be

g FM (t )  A cos  c t  k f



 m ( )d   .

The output signal of the channel q(t) becomes

q (t )  a1A cos  c t  k f




 m ( )d    a2 A cos  ct  k f




 a3  A cos  c t  k f



m
(

)
d




 


 m ( )d   


2
3




1  cos  2c t  2k f  m ( )d   



 






a3A 

1  cos  2c t  2k f  m ( )d    cos  c t  k f  m ( )d  
2 



 



 a1A cos  c t  k f

 aA
 m ( )d    22


3a A 
aA 
 2  a1A  3  cos  c t  k f
2
4 


DC
 aA

 m ( )d    22 cos  2ct  2k f

Around c with k f  k f

aA
 3 cos  3c t  3k f
4



 m ( )d  

Around 2c with k f  2 k f

m
(

)
d






Around 3c with k f 3 k f
In this case, it is clear that all the coefficients of the different terms are simply constants.
The terms are in fact different FM signals with different frequencies and different values
of the parameter kf. But, the important conclusion is that the original FM signal can
easily be extracted from the output signal of the non–linear channel using a filter centered
at the carrier frequency and bandwidth equal to the bandwidth of the FM signal. So, FM
signals are IMMUNE to (do not get damaged by) non–linearities in the channel or in the
components.
Use of Non–linearity to Manipulate FM Signals
The fact that passing an FM signal through a non–linear device results in a set of FM
signals with different carrier frequencies and different parameter kf (and therefore
different frequency variation parameter ), we can use this process for manipulating the
EE 370: Dr. Wajih Abu-Al-Saud (Modified)
Chapter V: Angle Modulation
Lecture 23
carrier frequency c and/or the frequency variation . In general, passing the FM
signal through a non–linear device with a maximum non–linearity power of P will give P
different FM signals as shown in the block diagram below.
BPF
P
g FM (t ) 

A cos   c t  k f



 m ( ) d  


(.)
q (t )
contains the follow ing

cos   c t  k f



m ( ) d  




cos  2  c t  2 k f


cos  P  c t  Pk f



 m ( ) d  


C.F. = PC
BW = 2[P kf mp+Bm]
g FM
( output )
(t ) 

B cos   P  c  t   Pk f



m ( ) d  





m ( ) d  



Passing the output of the P–power non–linear device through a BPF with center
frequency Pc and bandwidth equal to the bandwidth of the last FM signal will extract
that signal and reject the other FM signal. This process can be used to obtain wideband
FM signals from narrowband FM signals as will be described through a set of examples
next. We will assume that the non–linear devices that we use in the following examples
have built-in BPFs to eliminate the undesired FM components.