UNIT I

ANALOG MODULATION
Amplitude modulation & demodulation
 Session objectives
▪ To understand the concept of amplitude modulation (AM)
▪ To study the mathematical representation of AM
▪ To introduce time domain and frequency domain concepts
▪ To learn the importance of modulation index in AM.
▪ To understand AM modulators and demodulators
 Session outcomes
At the end of the session, students will be able to

▪ Understand the concept of amplitude modulation (AM)

▪ Understand time domain and frequency domain
concepts
▪ Learn the importance of modulation index
▪ AM modulation and demodulation techniques
 Outline

▪ Introduction – Amplitude modulation

▪ Mathematical representation of AM
▪ Time domain and frequency domain concepts
▪ Modulation index
▪ AM modulators and demodulators
 Amplitude Modulation

Amplitude modulation is defined as a process in which

the amplitude of the carrier wave is varied about a
mean value, linearly with the baseband signal.
AM Modulator and Demodulator

AM Modulator

AM Demodulator
Time Domain Representation of AM
 Mathematical Analysis
• Modulating signal is given as
em = Em cos 2f mt (1.1)
where em represents message signal. Em represents
amplitude of the message signal. f m is frequency of the
modulating signal.
• Similarly, the carrier signal is denoted by,
ec = Ec cos 2f ct (1.2)
where ec represents the carrier signal. Ec represents the
amplitude of carrier signal. f c is carrier frequency.
 Mathematical Analysis
• Then the modulated signal is given by
eAM = Ec + Em cos 2f mt cos 2f ct (1.3)

 Em 
= Ec 1 + E cos 2f mt  cos 2f ct (1.4)
 c 

• Modulation index (m): It is the ratio of the

amplitude of the message signal to the amplitude of
the carrier signal.
Em (1.5)
m=
Ec
 Mathematical Analysis
• Equation (1.4) becomes,
eAM = EC 1+ m cos 2f mt cos 2f ct
• Now multiplying Ec cos 2f ct

eAM = Ec cos 2f ct + mEc cos 2f ct cos 2f mt (1.6)

• Apply, Cos A Cos B =

1
Cos ( A + B ) + Cos ( A − B ) formula to
2
the second term in the summation.
e AM = Ec cos 2f c t +
mEc
2
cos 2 ( f c + f m )t + cos 2 ( f c − f m )t
(1.7)
 Mathematical Analysis
• This is a signal made up of three signal components
i) A carrier at frequency fc Hz
ii) Upper side frequency at f c + f m Hz
iii) Lower side frequency at f c − f m Hz
• The bandwidth of AM (the difference between the
highest and the lowest frequency) is
Bandwidth = ( f c + f m ) − ( f c − f m ) = 2 f m Hz (1.8)
where fm is the maximum frequency of the
modulating signal.
Frequency domain representation of AM
Ec

mE c mE c
2 2

Source : Electronic Communication Systems, Wayne Tomasi

Frequency domain representation of AM
Frequency spectrum consists of
i) Ec cos 2f ct - carrier frequency f c with the
amplitude of Ec

ii) Side band at ( fc + f m ) and ( fc − f m ) with the

amplitude of mE c
2
.
Time and Frequency domain of AM
Modulated signal in time domain is only the aggregate
of the modulating signal, the carrier signal, the upper
sideband signals and the lower sideband signal.
Time and Frequency domain of AM
AM wave is the algebraic sum of the carrier, the upper
and the lower sideband frequencies.

a) modulating signal
b) carrier signal
c) upper sideband signal
d) lower sideband signal
e) composite AM wave
Modulation of complex modulating signal
• In the previous analysis, simple cosine or sinusoidal
signal has been considered as the message signal. In real
life scenario, the modulating signal is like a complex
signal. In other words, the modulating signal is the sum
of two or more cosine or sinusoidal signals.
• Hence, the method of modulation of a complex signal
with high frequency carrier is discussed.
• What is the spectrum for modulated signal? What is the
bandwidth? What is the modulation index for complex
modulating signal?
 Mathematical Analysis
• Let us consider a modulating signal x(t ) as, sum of two signals, i.e.
x(t ) = x1 (t ) + x2 (t ) where
x1 (t ) = Em1 cos 2f m1t and x2 (t ) = Em2 cos 2f m2t where Em1 is
amplitude of x1 (t ), f m1 is frequency of x1 (t ) . Similarly Em 2 is amplitude
of x2 (t ) , f m 2 is frequency of x2 (t ).
• Now the carrier signal is represented as, ec = Ec cos 2f ct where ec
represents the carrier signal. Ec represents the amplitude of carrier signal
and f c is the carrier frequency.
• Then the modulated signal is given as, eAM = EAM cos 2f ct
,
 Mathematical Analysis
• Here the amplitude of the modulated signal E AM is given by,

EAM = Ec + Em1 cos 2f m1t + Em2 cos 2f m2t

Now substitute vale of E AM in above equation,

eAM = (Ec + Em1 cos 2f m1t + Em2 cos 2f m2t ) cos 2f ct

 Em1 Em 2 

= Ec 1 + cos 2f m1t + cos 2f m 2t  cos 2f c t
 Ec Ec 
E m1
m
• Modulation index 1 = 1 =
Ec
Em 2
• Modulation index 2 = m 2 =
Ec
 Mathematical Analysis
• Therefore, the total modulation index is m = m12 + m22
• Hence the modulated signal can be written as,
eAM = Ec (1 + m1 cos 2f m1t + m2 cos 2f m2t )cos 2f ct

eAM = (Ec cos 2f ct + m1Ec cos 2f ct cos 2f m1t + m2 Ec cos 2f ct cos 2f m2t )

• Applying Cos A and Cos B formula to cos 2f ct cos 2f m1t and
cos 2f ct cos 2f m2t
 
eAM =  Ec cos 2f ct + 1 c cos 2 ( f c + f m1 )t + cos 2 ( f c − f m1 )t  + 2 c cos 2 ( f c + f m 2 )t + cos 2 ( f c − f m 2 )t 
mE mE
 2 2 
 Frequency domain representation
• Carrier signal Ec cos 2f ct
• Upper sideband at f c + f m1 and f c + f m2
• Lower sideband at f c − f m1 and f c − f m2
Bandwidth of the modulated signal

Bandwidth = 2  f m(max )

where f m(max ) is the maximum frequency of modulating

signal
 Importance of Modulation Index
Modulation index (m): It is the ratio of the amplitude
of the message signal to the amplitude of the carrier
signal.
Em
m=
Ec

Depth of modulation or % Modulation = m x 100

 Importance of Modulation Index
Modulation index should be in the range of 0  m  1.
Based on the values of m, the AM modulation can be
classified into two types.
• Linear modulation:
The type of modulation is linear modulation when m  1 .For
proper signal reception linear modulation is preferred
• Over modulation:
The type of modulation is over modulation when m  1. Upper
and lower envelopes are combined with each other and it may
lead to phase reversal in the modulated signal. Perfect
reconstruction is therefore not possible in over modulation
Importance of Modulation Index
Em
m=
Ec
Vm = Vmax − Vmin
Vc = Vmax + Vmin
Vmax − Vmin
m=
Vmax + Vmin
AM Modulator
 Generation of AM-DSBFC-
Square Law modulator
• In general any device operated in non-linear region, its
output characteristics is capable of producing amplitude
modulated waves when the carrier and the modulating
signals are fed as inputs. Thus, a transistor or a triode
tube or a diode may be used as a square law modulator.
• The square law modulator circuit consist of the
following:
– i) A non-linear device
– ii) A BPF (band pass filter)
– iii) A carrier and modulating signal sources
Square law modulator
• The modulating and carrier signals are connected in series with
each other and their sum e1 (t ) is applied at the input of a non-
linear devices, such as diode or transistor,
e1 (t ) = em (t ) + ec (t )

e1 (t ) = Em cos m t + Ec cos c t (1)

• The input-output relation for non-linear device is as follows,

e2 (t ) = a e1 (t ) + b e1 (t )
2 (2)

Where a and b are constants

 Substitute value of e1 (t ) from eq. (1) to eq. (2), we get,
e2 (t ) = a(E m cos  m t + E c cos  c t ) + b(E m cos  m t + E c cos  c t )
2

e2 (t ) = aEm cos  m t + aEc cos  c t + bEm cos 2  m t + bEc cos2  c t + 2bEm E c cos  m t cos  c t
2 2

The terms in the expression e2 (t ) are as follows,

i) Term I- Modulating signal
ii) Term II- Carrier signal
iii) Term III- Squared modulating signal
iv) Term IV-Squared carrier signal
v) Term V- AM wave only side bands
 cos (c − m )t + cos (c + m )t 
 2 bE E
m c cos  m t cos  c t = 2bE E
m c 
 2 
• The LC tuned circuit act as a BPF. This circuit is tuned to
carrier frequency f c and its bandwidth is equal to 2 f m
Therefore, it allows only c + m , andc − m neglecting all
other higher terms

 e2 (t ) = aEc cos c t + bEm Ec (cos(c − m )t + cos(c + m )t )

Generation of AM-DSBSC-Balanced
Modulator
Consists of two identical AM modulators. These two
modulators are arranged in a balanced configuration in
order to suppress the carrier signal. Hence, called as
Balanced modulator.
 Balanced Modulator

• The same carrier signal c(t)=Ac cos(2πfct) is applied as

one of the inputs to these two AM modulators.

• The modulating signal m(t) is applied as another input to

the upper AM modulator.

• Whereas, the modulating signal m(t) with opposite

polarity, i.e., −m(t) is applied as another input to the
lower AM modulator.
Balanced Modulator
AM Demodulator
 Demodulation of AM-
DSBFC/Detection of AM signal
• The process of recovering the message signal
from the received modulated signal is known
as demodulation
Envelope Detector
Diode Envelope Detector
 Envelope Detector
• During the positive half-cycle of the input signal, the
diode conducts and the capacitor charges up to the peak
value of the input signal.
• When the input falls below the voltage on the capacitor,
the diode becomes reverse-biased and the input
disconnects from the output
• During this period, the capacitor discharges slowly
through the load resistor R
• During the next cycle of the carrier, the process is
repeated
Envelope Detector
 Envelope Detector
• The time constant RC must be selected to follow the
variations in the envelope of the carrier-modulated
signal.
• If RC is too small, then the output of the filter falls
very rapidly after each peak and will not follow the
envelope of the modulated signal closely.
 Envelope Detector
• If RC is too large, then the discharge of the capacitor
is too slow and again the output will not follow the
envelope of the modulated signal
 Envelope Detector
• For an effective envelope detector,
1 1
 RC 
fc W

• In such a case, the capacitor discharges slowly

through the resistor; thus, the output of the envelope
detector, closely follows the message signal.
Comparison of AM, FM and PM
 Summary
• Concepts of Amplitude Modulation (AM)
• Mathematical analysis of AM
• Frequency and Time domain representation of AM
• Importance of modulation index
• AM Modulators and Demodulators
 Test your understanding
• Define Amplitude Modulation
• State the importance of modulation index
• Elaborate the mathematical representation of AM
wave
 References
1. H Taub, D L Schilling, G Saha, “Principles of
Communication Systems” 3/e, TMH 2007
2. S. Haykin, “Digital Communications”, John Wiley,
2005.