Download Introduction to communi. sys.Module

Wollo University
Kombolcha Institute of Technology (KIoT)
School of Electrical and Computer Engineering
Course Name: Introduction to Communication System
Engineering
This Course Module is compiled by
Jemal Hassen (Msc)
Email: [email protected]
February, 2023
Introduction to communication system
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Course Content
Chapter One: Analysis and Transmission of Signals
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Analog communication system
Electromagnetic wave propagation models
Analysis of deterministic signals in frequency domain
Signal transmission in base band
Linear distortion
Nonlinear distortion and companding
Frequency allocation
Chapter Two : Amplitude (linear) modulation
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Analysis of linear modulations
Modulation Index and Percentage of Modulation
AM, DSB,DSB-SC,SSB,VSB
Power Content of AM wave
Linear modulation and demodulation techniques
Chapter Three: Angle modulation
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Mathematical Analysis of Angle Modulation
PM Modulation
FM modulation and demodulation
Narrow band and Wide band Angle modulation
Frequency and Bandwidth Analysis of Angle-Modulated Waves
Chapter Four: Base band pulse signalling
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The sampling theorem
Quantizing and Encoding
Pulse Modulation
Digital Modulation Techniques
PAM,PCM,QAM,FSK, PSK and ASK
Chapter Five : Introduction to Data communication
 Introduction
 Model for data communication
 TDM and PCM frames
 Digital carrier systems and multiplexing
Introduction to communication system
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Chapter One
Analysis and Transmission of Signals
1.1 Analog Communication
Analog communication is made from two words analog and communication. Analog refers to
the continuous time-varying signal. Communication refers to exchanging information
between two or more than two sources. Whereas communication system is a collection of
elements which works together to establish a communication bridge between the sender
(transmitters) and receivers. Two or more people communicating with each other by using
sound signals is an example of a communication system.
Analog communication means communication with the help of analog signals. The analog
communication is communication from the sender to the receiver in the form of an analog
signal. The analog signal is a continuous time varying signal. The example of analog signal
is sound waves. The signals that continuously vary with time are the examples of an analog
signal, such as audio and video signals.
The essential concept of the analog communication is modulation. It helps in removing the
noise or external disturbances from the data, which may deteriorate the quality of the signal
being transmitted.
There are two types of signals which are analog signal and digital signal.
Analog signals
Analog signal is a continuous signal whose characteristics (amplitude, voltage or frequency)
changes continuously over a period of time. Analog signals are continuous time-varying
signals. It means that these signals are the function of time. The common shape of an analog
signal is the sinusoidal wave. It is shown below.
Examples of analog signals are electrical signals, light signals, speech signals, etc. Every
signal requires a medium to propagation. For example, Electrical signals require cables to
propagate from one place to another
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Digital Signal
Digital signals are the signal that represents the data in the form of discrete values. It takes
only two values 0 and 1, which is known as bits. The data is transmitted in the form of these
bits. For example, 01000110 is a digital signal.
Digital signal is a non-continuous electrical signal, which is used to convey (send, receive,
and process) information between the sender and receiver. In digital signal, the original
information (analog information) is converted into a string of bits (digital information) before
being transmitted.
For example, computers are digital in nature. They send, receive, store and process
information in binary form, i.e. in the combination of 0s and 1s.
An analog or digital signal which repeats itself after a specific interval of time is called
periodic signal. They are deterministic signals.
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Basic Elements of Communication Systems
A collection of elements (devices) which works together to establish a communication
between the sender and receiver is called a communication system. Some examples of
communication system include radio broadcasting, television broadcasting, radio telegraphy,
mobile communication, computer communication
The basic components of a communication system are information source, input transducer,
transmitter, communication channel, receiver, output transducer, and destination
.
Information Source:
The information generated by the source may be in the form of sound (human speech),
picture (image source) or plain text.
Information is obtained from real life signals through the use of transducers. For example,
speech is converted into a corresponding electrical signal by a microphone and moving
picture signals are converted into the appropriate electrical signals by various cameras. The
information so obtained is called a signal that becomes a function of time which is usually
analog in nature. Signals may be described in time domain or in frequency domain. The
frequency domain description of a signal is known as spectrum that would be covered
subsequently. Data generated by the keystroke of a computer become the information when
communication is made through e-mail.
Transmitter:
•
•
•
•
•
•
The transmitter is a device which converts the signal produced by the source into a form
that is suitable for transmission over a given channel or medium.
Transmitters use a technique called modulation to convert the electrical signal into a
form that is suitable for transmission over a given channel or medium.
Converts electrical signal into a form suitable for transmission through the channel
(physical medium)
The transducer output signal cannot, in most cases, be transmitted directly (does not
match the channel)
Transmitter converts message to a suitable form and onversion is made through
modulation
Amplitude (AM), frequency (FM) and phase (PM) are examples of this.
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Communication Channel:
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The communication channel is a medium through which the signal travels.
Communication channels are divided into two categories: wired and wireless.
Physical medium over which the information will be transmitted from the transmitter
to the receiver
Can be wired (telephone line) or wireless (radio)
Whatever the medium, the signal is corrupted by noise and interference
E.g., thermal noise, lightning discharge, automobile ignition noise, interference from
other users etc.
Channel may be highly non-stationary (i.e., fading)
Significant signal attenuation may be introduced
E.g., 100 – 200dB
Other types of signal distortions (i.e., spectrum distortion)
Receiver:
The receiver is a device that receives the electrical signal from the channel and converts it
back to its original form which is understandable by humans at the destination.
 Main function: to recover the message from the received signal
 Somewhat inverse of the transmitter function
Types of Communication Systems
I.
Classification based on direction of communication
• Based on whether the system communicates only in one direction or otherwise, the
communication systems are classified as:
➢ Simplex System
➢ Half duplex System
➢ Full duplex System
Simplex System:
• In these systems, the information is communicated in only one direction.
Examples: Radio and TV broadcasting system
Half duplex System:
• These systems are bidirectional (they can transmit as well as receive) but not
simultaneously. At a time, these systems can either transmit or receive.
Examples: Walkie-talkies and CB (citizens band) radios
Full duplex System:
• These systems allow communication to take place in both directions simultaneously.
Example: Telephone systems
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II.
Classification based on nature of information signal
•
Based on the technique used for the signal transmission, we can categorize the
electronic communication system as under :
➢ Baseband transmission system
➢ Bandpass transmission system using modulation
Baseband transmission:
In baseband transmission systems, the baseband signals (original information signals) are
directly transmitted.
• The baseband transmission cannot be used with certain mediums e.g., it cannot be
used for signal transmission where the medium is free space. This is because the voice
signal cannot travel long distance in air. It gets suppressed after a short distance .
Bandpass transmission system using modulation:
• In the modulation process, some parameter of the carrier wave (such as amplitude,
frequency or phase ) is varied in accordance with the modulating signal .
III.
Classification based on nature of information signal
•
Based the nature of information signal, communication systems are classified into
two broad categories as:
➢ Analog communication system
➢ Digital communication system
Analog Communication System:
• Uses analog signals for communication.
• Analog signals are continuous and take continuous values
Digital Communication System:
• Uses digital signals for communication
• digital signals have discrete states and take discrete values.
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Modulation and Multiplexing
•
Modulation and multiplexing are electronic techniques for transmitting information
efficiently from one place to another.
• Modulation makes the information signal more compatible with the medium, and
multiplexing allows more than one signal to be transmitted concurrently over a single
medium.
• Modulation and multiplexing techniques are basic to electronic communication
.
Modulation is the process of transforming a baseband voice, video, or digital signal into
another higher-frequency signal using the carrier signal.
Multiplexing is the process of allowing two or more signals to share the same medium or
channel.
Propagation of Electromagnetic Waves
Unguided signals can travel from the source to destination in several ways
Depending on the frequency, radio waves and micro waves travel in space in different
ways depending on the behavior of these waves with respect to
the earth and the
atmosphere. They are:
1. Ground wave propagation
2. Sky ( ionospheric) wave propagation
3. Space (or tropospheric) wave propagation
Ground wave propagation
In ground wave propagation, the radio waves (AM) travel along the surface of the earth.
These waves are called ground waves or surface waves.
In fact, these waves are not confined to surface of the earth but are guided along the
earth’s surface and they follow the curvature of the earth.
The energy of the radio waves decreases as they travel over the surface of the earth due to
the conductivity and permittivity of the earth’s surface. Attenuation increases with the
increase in frequency.
Therefore, the ground waves are limited to frequency of 2 MHz (2000 kHZ) or
wavelength of 200 m.AM radio are good example of ground wave propagation.
Ground waves progress along the surface of the earth and must be vertically polarised to
prevent from short-circuiting the electric component.
The maximum range of a transmitter depends on its frequency as well as its power.
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Sky wave propagation or Ionospheric wave propagation
Sky waves are the AM radio waves which are received after being reflected from
ionosphere. The propagation of radio wave signals from one point to another via
reflection from ionosphere is known as sky wave propagation.
The sky wave propagation is a consequence of the total internal reflection of radiowaves.
Higher we go in the ionosphere, free electron density increases and refractive index
decreases.
The UV and high energy radiations from the Sun are absorbed by the air molecules and
they get ionised to form the ionised layer or electrons and ions. Ionosphere extends from
80 km to 300 km in the atmosphere above the earth’s surface.
If the maximum electron density of the ionosphere is Nmax per m3, then the critical
frequency is given by:
𝟏
𝒇𝒄 = 𝟗(𝑵𝒎𝒂𝒙 )
𝟐
Short wave radios( SW) are good example and the frequency range are from 2 to 30 MHz
Space wave propagation or Tropospheric wave propagation: (AM Radio waves)
A space wave travels in a straight line from transmitting antenna to the receiving antenna.
They propagate very much like electromagnetic waves in free space. In space wave
propagation there is a concept of radio horizon.
The radio horizon for space waves is about four-thirds as far as the optical horizon.This
beneficial effect is caused by the varying density of the atmosphere, and because of
diffraction around the curvature of the earth.
Applications of Communication Systems
The various major applications of electronic communication systems include:
1. AM and FM radio broadcasting
➢ Stations broadcast music, news, weather reports, and programs for entertainment
and information. It includes digital radio.
2. TV broadcasting
➢ Stations broadcast entertainment, informational, and educational programs. It
includes digital television (DTV) and cable television (CATV).
3. Telephones
 One-to-one verbal communication is transmitted over the vast worldwide telephone
networks employing wire, fiber optics, radio, and satellites. It includes cordless
telephones, cellular phones, VoIP phones and satellite phones.
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4. The Internet
 Worldwide interconnections mainly via fiber optic networks. It includes WideArea Networks (WANs), Metropolitan-area Networks (MANs) and Local-area
Networks (LANs)
5. Wireless remote control
 This category includes a device that controls any remote item by radio or infrared.
Examples are missiles, satellites, robots, vehicles, remote plants, garage door
opener, and the remote control on your TV set.
Range of Radio Frequency Bands
S.
No.
Name of the frequency range (Band)
Short
Form
Frequency Range
1
Very Low Frequency
VLF
3 kHz to 30 kHz
2
Low Frequency
LF
30 kHz to 300 kHz
3
Medium Frequency or Medium
Wave
MF or
MW
300 kHz to 3 MHz
4
High Frequency or Short Wave
HF
SW
3 MHz to 30 MHz
5
Very High Frequency
VHF
30 MHz to 300 MHz
6
Ultra High Frequency
UHF
300 MHz to 3,000 MHz
7
Super High Frequency or Micro
Waves
SHF
3,000 MHz to 30,000 MHz
(3 GHz to 30 GHz)
8
Extremely High Frequency
EHF
30 GHz to 300 GHz
Introduction to communication system
or
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Analysis of deterministic signals in frequency domain
Frequency domain representations are useful for:
– The analysis of signals and
– The design of systems for processing signals.
A large class of sequences can be represented as a linear combination of complex
exponentials whose frequencies lie in the range [-π, π].
Fourier transform (FT) allows us to represent aperiodic (not periodic) signal in term of its
frequency ω.
The Fourier transform integrals is performed as follows:
X ( ) 

 j t
x
(
t
)
e
dt


X ( )  X ( ) e  X ( )
The Fourier Transform Spectrum is also analyzed as of ;
X ( )  X ( ) eX ( )
The Phase Spectrum
The Amplitude (Magnitude) Spectrum
The amplitude spectrum is an even function and the phase is an odd function.
The Inverse Fourier transform:

1
x (t ) 
X ( ) e j t d 

2  
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Some useful Functions
Unit Gate Function
0
t  
rect     0 . 5
  
1
| t |  / 2
| t |  / 2
| t |  / 2
Unit Triangle Function
 t  0
   
   1  2 t / 
| t |  / 2
| t |  / 2
Interpolation Function
sin t
t
sinc ( t )  0 for t
sinc ( t )  1 for t
sinc ( t ) 
sinc(t)
  k
 0
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Example 1.1
Find the FT, the magnitude, and the phase spectrum of 𝑥(𝑡) = 𝑟𝑒𝑐𝑡(𝑡/).
Answer
X ( ) 
 /2

 rect ( t /  ) e
 j t
dt   sinc(  / 2 )
/2
What is the bandwidth of the above pulse?
The spectrum of a pulse extend from 0 to . However, much of the spectrum is
concentrated within the first lobe ( = 𝟎 𝒕𝒐 𝟐/ )
Find the FT of the unit impulse (t).
X ( ) 

  (t )e

j t
dt
 1
 
Find the inverse FT of ().
1
x (t ) 
2
so the

j t

(

)
e
d 


spectrum
of a constant
1
2
is an impulse
1  2  (  )
Properties of the Fourier Transform
• Linearity:
Let
and
then
• Time Scaling:
Compression in the time domain results in
Let
then
x t   X   expansion in the frequency domain
x at  
1
 
X 
a
 a 
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• Time Reversal:
x t   X  
Let
x( t) 
then
•
X ( )
Multiplication by a Complex Exponential (Frequency Shift Property):
x t   X  
Let
x ( t ) e j 0 t 
then
•
X (
  0 )
Multiplication by a Sinusoid (Amplitude Modulation):
x t  
Let
x t  cos
then
X
 
1
 X    0   X    0 
2
 0 t  
Here 𝑐𝑜𝑠𝜔 𝑡 is the carrier, 𝑥(𝑡) is the modulating signal (message), 𝑥(𝑡) cos 𝜔0 𝑡 is the
modulated signal.
•
•
Differentiation in the Frequency Domain:
Let
xt   X  
then
n
t
x (t ) 
( j )
n
d
d 
n
n
X
( )
Differentiation in the Time Domain:
x t   X  
Let
then
d n
d t n
x (t ) 
( j ) n X
( )
Integration in the Time Domain:
Let
Then
t

 
x t   X  
x ( ) d 

1
j
Introduction to communication system
X ( )   X ( 0 ) ( )
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Convolution and Multiplication in the Time Domain:
x t   X  
Let
y t   Y  
Then
X ( )Y ( )
x (t )  y (t ) 
x1 (t ) x2 (t ) 
1
X 1 ( )  X 2 ( )
2
Frequency convolution
• Parseval’s Theorem: since x(t) is non-periodic and has FT X(), then it is
an energy signals:

E 
 x t 
2

Real signal has even spectrum

1
2


dt 
X

d
2  
X()= X(-),
E 
1

X   d 
 
2
0
Example
Find the energy of signal x(t) = e-at u(t) Determine the frequency  so that
the energy contributed by the spectrum components of all frequencies below
 is 95% of the signal energy EX.
Answer:  = 12.7a rad/sec
• Duality ( Similarity) :
• Let
x t   X  
then
X (t ) 
2 x (  )
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Signal Transmission through a linear system
A Linear-Time Invariant (LTI) system used to characterize communication channels.
LTI system is characterized in time domain by its impulse response h(t).
y ( t ) = h( t ) When
System response for input signal
x(t)= 𝜹(t)
x(t): y(t)=h(t)*x(t)
The frequency domain relationships are
Then according to convolution,
Y(f)=H(f)×X(f)
H( f ) is generally referred as Frequency Response or Transfer Function of the LTI system.
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Signal Distortion During Transmission
Deformation of the signal is called Distortion.
Distortion less transmission means that the output signal has the same “shape” as the input
signal.The output spectrum is given by the input spectrum multiplied by the spectral response
of the system.
 An input signal spectral component of f is modified in amplitude by a factor |H(f)| and is
shifted in phase by an angle θh(f).
 During transmission through the system, some frequency components may be boosted in
amplitude, while others may be attenuated.
 The relative phases of the various components also change.
 In general, the output waveform will be different from the input waveform.
Distortion less Transmission
•
•
•
•
In applications such as message transmission over communication channel, the output
waveform is required to be a replica of the input waveform.
To achieve this, distortion due to amplification or communication channel must be
minimized. Distortion less transmission is desired.
Transmission is said to be distortion less if the input and the output have identical wave
shapes within a multiplicative constant.
Given input x(t) and output y(t), a distortionless transmission satisfies:
Hence the transfer function required for distorionless transmission is:
•
For distortionless transmission, amplitude response |H(f)| must be a constant and phase
response θh(f) must be linear function.
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Energy Spectral Density

The energy of a signal can be expressed in the frequency domain as follows:

E   | x ( t ) | 2 dt


x * ( t )   X ( f ) e j 2  ft df  dt
  


 


Reversing the order of integration, we obtain


E   X ( f )   x * (t )e j 2ft dt  df
  


*


  X ( f )   x (t )e  j 2ft dt  df   X ( f ) X * ( f ) df
  




Thus





 | X ( f ) | df
2

E   | x(t) |2 dt   | X ( f ) |2 df
This is referred to as Parseval’s theorem for Fourier transforms.
The units of |X(f)|2 is (volts-sec)2. Since we consider power on a per-ohm basis, (wattssec)/hertz=joules/hertz.
Thus, we see that |X(f)|2 has the units of energy density, and we define the energy spectral
density of a signal as
G( f ) | X ( f ) |2
By integrating G(f) over all frequency, we obtain the total energy.

The energy and power of a signal represent the energy or power delivered by the signal
when it is interpreted as a voltage or current source feeding a 1-ohm resistor.

The energy content of x(t):
Ex  



The power content of x(t):
1
T   T
P x  lim
Introduction to communication system
2
x (t ) dt

T /2
T /2
x (t )
2
dt
Page 18
Low pass and Band pass Signals

Low pass signal - the spectrum (frequency content) of the signal is located around the
zero frequency.

Band pass signal - the spectrum is located around a frequency fc far from the zero
frequency.
The bandwidth of the band pass signal is usually much less than the frequency fc (recall that
he bandwidth of a signal is the set of the range of all positive frequencies present in the
signal).
The bandwidth of the band pass signal is usually much less than the frequency fc (recall that
he bandwidth of a signal is the set of the range of all positive frequencies present in the
signal).

The extreme case of a bandpass signal is a single frequency signal whose frequency is
equal to fc (The bandwidth of this signal is zero)
𝑥(𝑡) = 𝐴 cos( 2𝜋𝑓 𝑡 + 𝜃)

This is a sinusoidal signal that can be represented by a phasor which rotates
counterclockwise with an angular velocity of ωc = 2πfc
x(t) can be expanded as:
x ( t )  A cos( 2  f c t   )
 A cos(  ) cos( 2  f c t )  A sin(  ) sin( 2  f c t )
 x c cos( 2  f c t )  x s sin( 2  f c t )
This is a sinusoidal signal that can be represented by a phasor 𝑥 = 𝐴𝑒
which rotates
counterclockwise with an angular velocity of ωc = 2πfc
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Chapter Two
Amplitude (linear) Modulation
A message signal cannot travel a long distance because of its low signal strength. In addition
to this, physical surroundings, the addition of external noise and travel distance will further
reduce the signal strength of a message signal. So in order to send the message signal to a
long distance, we need to increase the signal strength of a message signal. This can be
achieved by using a high frequency or high energy signal called carrier signal. A high energy
signal can travel to a larger distance without getting affected by external disturbances.
The low energy message signal is mixed with the high energy or high frequency carrier signal
to produce a new high energy signal which carries information to a larger distance.
he Message signal contains information whereas the carrier signal contains no information.
Carrier signal is used just to transmit the information to a long distance. At the destination,
the message signal is consumed whereas the carrier signal is wasted.
In modulation process, the characteristics of the carrier signal is changed but the message
signal characteristics will not be changed. The carrier signal does not contain any information
so even if we change the characteristics of the carrier signal, the information contained in it
will not be changed. However, the message signal contains information so if we change the
characteristics of the message signal, the information contained in it will also changes.
Therefore, we always changes the characteristics of the carrier signal but not the message
signal.
Modulation allows the transmission to occur at high frequency while it simultaneously allows
the carrying of the message signal.
Modulation is the process of mixing a low energy message signal with the high energy carrier
signal to produce a new high energy signal which carries information to a long distance.
Types of Signals in Modulation
In modulation process, three types of signals are used to transmit information from source to
destination. They are:
1) Message signal
2) Carrier signal
3) Modulated signal
1) Message signal
The signal which contains a message to be transmitted to the destination is called a message
signal. The message signal is also known as a modulating signal or baseband signal.
The original frequency range of a transmission signal is called baseband signal. The message
signal or baseband signal undergoes a process called modulation before it gets transmitted
over the communication channel. Hence, the message signal is also known as the modulating
signal.
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2) Carrier signal
The high energy or high frequency signal which has characteristics like amplitude, frequency,
and phase but contains no information is called a carrier signal. It is also simply referred to as
a carrier. Carrier signal is used to carry the message signal from transmitter to receiver. The
carrier signal is also sometimes referred to as an empty signal.
3) Modulated signal
When the message signal is mixed with the carrier signal, a new signal is produced. This new
signal is known as a modulated signal. The modulated signal is the combination of the carrier
signal and modulating signal.
Amplitude modulation is the process of changing the amplitude of a relatively high frequency
carrier signal in proportion with the instantaneous value of modulating signal (information).
It is also a process of translating information signal from low band frequency to high band
frequency.
Signals from information sources are usually of low frequency, are called baseband signals
and baseband signals are not suitable for transmission. A technique called modulation is used
to turn a baseband message signal in to another form which is suitable for transmission. In
other words, modulation translates a base-band message signal to a band-pass signal.
The process of impressing low frequency information signals onto a high frequency carrier
signal is called modulation.
Demodulation is the reverse process where the received signal is transformed to their
original form. Band-pass signals have relatively higher frequency and are suitable for
transmission. In the process of modulation, the message signal is known as modulating
signal.
In any modulation technique, a high frequency Sin or Cosine wave is used as a carrier
signal. High quality oscillators are used to generate carrier signal.
 Based on the attribute used, modulation can be classified as follows:
 Amplitude Modulation (AM):
 If the amplitude of the carrier signal is chosen to carry the message signal
during modulation
 Frequency Modulation (FM):
 If the frequency is used to carry the message signal.
 Phase Modulation (PM):
 If the phase is used to carry message signal.
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During modulation, one of the attributes of the carrier signal is made to vary continuously
according to the changes in amplitude of the message signal. In this way, the message signal
will be superimposed in to the carrier signal, hence the name carrier.
Benefits of Modulation:1. Easy radiation
 Antennas operate effectively when antenna size is comparable to
the
wavelength. (for half-wave dipole antenna: 𝑳 = 1 2 𝝀 )
2. Frequency matching
 Modulation shifts the spectral of a message signal so as to fit the frequency
band of the channel
3. Multiplexing
 Accommodation for simultaneous transmission of several baseband signals
4. Less vulnerability to noise or interference
 Modulation provides a mechanism for putting the information content of a
message signal into a form that may be less vulnerable to noise or interference
5. Narrow banding
 For an audio signal range of 50 to 104 Hz, the ratio of the highest audio
frequency to the lowest is 200

An antenna suitable for use at one end of the range would be entirely too short or too long
for the other end
 If the audio spectrum were translated from (106 + 50) to (106 + 104), then the
ratio would be only 1.01
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Amplitude Modulation ( AM ):
In the modulation process, the voice, video, or digital signal modifies another signal called
the carrier. In amplitude modulation (AM) the information signal varies the amplitude of
the carrier sine wave. The instantaneous value of the carrier amplitude changes in accordance
with the amplitude and frequency variations of the modulating signal .An imaginary line
called the envelope connects the positive and negative peaks of the carrier waveform.
The AM wave is the algebraic sum of the carrier and upper and lower sideband sine waves.
Figure 2-1: Amplitude modulation. The carrier signal and message signal
In Amplitude Modulation (AM), it is particularly important that the peak value of the
modulating signal be less than the peak value of the carrier. Vm < Vc
Distortion occurs when the amplitude of the modulating signal is greater than the amplitude
of the carrier. A modulator is a circuit used to produce AM. Amplitude modulators compute
the product of the carrier and modulating signals.
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The circuit used for producing AM is called a modulator. Its two inputs, the carrier and the
modulating signal, and the resulting outputs are shown in Fig. below. Amplitude modulators
compute the product of the carrier and modulating signals. Circuits that compute the product
of two analog signals are also known as analog multipliers, mixers, converters, product
detectors, and phase detectors. A circuit that changes a lower-frequency baseband or
intelligence signal to a higher-frequency signal is usually called a modulator. A circuit used
to recover the original intelligence signal from an AM wave is known as a detector or
demodulator.
Figure 1-3: Amplitude modulator showing input and output signals.
where v2 is the instantaneous value of the AM wave (or vAM), Vc sin 2wfct is the carrier
waveform, and (Vm sin 2wfmt) (sin 2wfct) is the carrier waveform multiplied by the modulating signal waveform. It is the second part of the expression that is characteristic of AM.
A circuit must be able to produce mathematical multiplication of the carrier and modulating
signals in order for AM to occur. The AM wave is the product of the carrier and modulating
signals
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AM Frequency Sidebands and Bandwidth
Whenever a carrier is modulated by an information signal, new signals at different
frequencies are generated as part of the process. These new frequencies, which are called
side frequencies, or sidebands, occur in the frequency spectrum directly above and directly
below the carrier frequency. More specifically, the sidebands occur at frequencies that are
the sum and difference of the carrier and modulating frequencies.
When signals of more than one frequency make up a waveform, it is often better to show the
AM signal in the frequency domain rather than in the time domain Output envelope contains
of dc voltage, carrier frequency, the sum (fc + fm) and difference (fc – fm) frequencies.
When only a single-frequency sine wave modulating signal is used, the modulation process
generates two sidebands. If the modulating signal is a complex wave, such as voice or video,
a whole range of frequencies modulate the carrier, and thus a whole range of sidebands are
generated.
Bandwidth, B = difference between highest USB and lowest LSB
i.e B = 2fm(max).
Figure below shows the frequency spectrum for an AM waveform
The upper sideband fUSB and lower sideband fLSB are computed as
fUSB = fc + fm and
fLSB = fc — fm
where fc is the carrier frequency and fm is the modulating frequency
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Modulation Index and Percentage of Modulation
The modulation index (m) is a value that describes the relationship between the amplitude of
the modulating signal and the amplitude of the carrier signal. This index is also known as the
modulating factor or coefficient, or the degree of modulation. It is a measures of the depth
of the modulation
m 
Em
Ec
Percentage modulation (% m) is simply the modulation index (m) stated as a percentage.
More specifically percent modulation gives the percentage change in the amplitude of the
output wave when the carrier is acted on by a modulating signal.
𝑚(𝑖𝑛 %) =
𝐸
𝑥 100 %
𝐸
m = modulation index
Em = peak change in the amplitude output waveform (sum of voltages from upper and
lower side frequencies) and Ec = peak amplitude of the unmodulated carrier
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The maximum amplitude of the message signal must be less than (or equal to) the maximum
amplitude of the carrier signal to avoid any distortion in the modulated signal. For example, if
the carrier signal amplitude is 5 volts then the message signal amplitude must be less than (or
equal to) 5 volts. Hence, the maximum value of the modulation index will be less than one or
equal to one (Mi<=1) when Am <= Ac. The minimum value of the modulation index will be
zero.
Based on this, there are three types of modulation:
1. Perfect-Modulation
2. Under-Modulation
3. Over-Modulation
Perfect-Modulation:
Perfect-modulation occurs when the maximum amplitude of the message signal or
modulating signal is exactly equal to the maximum amplitude of the carrier signal (Am = Ac).
The modulation index is the ratio of the maximum amplitude of the message signal to the
maximum amplitude of carrier signal. For example, if the message signal maximum
amplitude is 4 volts and carrier signal maximum amplitude is also 4 volts, then the ratio of
modulating signal amplitude (4 volts) to the carrier signal amplitude (4 volts) is equal to 1.
Therefore, the modulation index in perfect-modulation is equal to one (Mi = 1).
The modulation index is also known as the modulation depth. The perfect-modulation has a
modulation depth of 100%. In perfect-modulation, the carrier level falls to zero. Perfectmodulation causes no distortion.
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Under-Modulation:
Under-modulation occurs when the maximum amplitude of the message signal or modulating
signal is less than the maximum amplitude of the carrier signal (Am < Ac).
The modulation index is the ratio of the maximum amplitude of the message signal to the
maximum amplitude of carrier signal. For example, if the message signal maximum
amplitude is 2 volts and carrier signal maximum amplitude is 4 volts, then the ratio of
modulating signal amplitude (2 volts) to the carrier signal amplitude (4 volts) is equal to 0.5.
Therefore, the modulation index in under-modulation is less than one (Mi < 1). The undermodulation has a modulation depth of less than 100%. In under-modulation, the carrier level
falls above zero. Under-modulation causes no distortion.
Over-Modulation:
Over-modulation occurs when the maximum amplitude of the message signal or modulating
signal is greater than the maximum amplitude of the carrier signal (Am > Ac).
The modulation index is the ratio of the maximum amplitude of the message signal to the
maximum amplitude of carrier signal. For example, if the message signal maximum
amplitude is 6 volts and carrier signal maximum amplitude is 4 volts, then the ratio of
modulating signal amplitude (6 volts) to the carrier signal amplitude (4 volts) is equal to 1.5.
Therefore, the modulation index in over-modulation is greater than one (Mi > 1).
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The over-modulation has a modulation depth of greater than 100%. In over-modulation, the
carrier wave experiences 180° phase reversals where the carrier level falls below the zero
point.
Over-modulation causes severe distortion of the waveform of the message signal which
results in data loss. Over-modulation is one of the reasons why amplitude modulation is no
longer used to transmit high-quality sound signals. At the transmitter, limiters are included
which prevent more than 100% modulation.
Determining modulation index from Vmax and Vmin
If the modulating signal is a pure, single-frequency sine wave and the process is
symmetrical then the modulation index can be derived as follows:
1
(V max  V min )
2
1
Ec 
(V max  V min )
2
Em 
1
(Vmax  Vmin )
(V  Vmin )
m 2
 max
1
(Vmax  Vmin ) (Vmax  Vmin )
2
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Since the peak change of modulated output wave Em is the sum of the usf and lsf voltages
hence,
E m  E usf  E lsf
where
E usf  E lsf
1
(V max  V min )
E
E usf  E lsf  m  2
2
2
1
 (V max  V min )
4
Eusf = peak amplitude of the upperside frequency (volts)
Elsf = peak amplitude of the lower side frequency (volts)
From the modulated wave displayed in the previous equation, the maximum and
minimum values of the envelope occurs at
+Vmax = Ec + Eusb + Elsb
+Vmin = Ec – Eusb – Elsb
-Vmax = -Ec - Eusb - Elsb
-Vmin = -Ec + Eusb + Elsb
For proper AM operation, Ec > Em means that 0≤ m ≤ 1.
 If Ec < Em means that m > 1 leads to severe distortion of the modulate wave.
 If Vc = Vm the percentage of modulation index goes to 100%, means the maximum
information signal is transmitted. In this case, Vmax = 2Vc and Vmin = 0.
In a certain AM radio transmitter, an audio signal 𝟏𝟎𝑺𝒊n(𝟐𝝅 ×𝟓𝟎𝟎 t ) is used to modulate a
carrier wave 𝟓𝟎𝑺𝒊𝒏 𝟐𝝅 × 𝟏𝟎𝟓𝒕 .Calculate the modulation index.
 Solution:
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Example
For an AM DSBFC modulator with a carrier frequency, fc = 100 kHz and a
maximum modulating signal frequency, fm(max) = 6 kHz, find
a) Frequency limit for upper and lower sideband.
b) Bandwidth.
c) Upper and lower side frequencies produced when the modulating signal is a
single frequency 4 kHz tone.
d) Draw the output frequency spectrum.
The Mathematical Representation and Analysis of AM
Representing both the modulating signal Vm(t) and the carrier signal Vc(t) in
trigonometric functions.
The AM DSBFC modulator must be able to produce mathematical multiplication of these
two analog signals
v m (t )  V m sin ( 2f m t )
v am (t )  [Vc  V m sin ( 2f m t )] sin ( 2f c t )
v c (t )  Vc sin ( 2f c t )
Substituting Vm = mVc gives:
v am ( t )  [V c  mV c sin ( 2  f m t )] sin ( 2  f c t )
 [1  m sin ( 2  f m t )] V c sin ( 2  f c t )
Constant + mod. signal
Introduction to communication system
Un-modulated carrier
Page 31
The constant in the first term produces the carrier frequency while the sinusoidal
component in the first term produces side bands frequencies
v am ( t )  V c sin ( 2 f c t )  [ mV c sin ( 2 f m t )] [sin ( 2 f c t )]
 V c sin ( 2 f c t ) 
m Vc
cos [ 2 ( f c  f m ) t ]
2
m Vc
cos [ 2 ( f c  f m ) t ]
2

The constant in the first term produces the carrier frequency while the sinusoidal
component in the first term produces side bands frequencies.
v am ( t )  V c sin ( 2 f c t )  [ mV c sin ( 2 f m t )] [sin ( 2 f c t )]
 V c sin ( 2 f c t ) 

Carrier frequency
signal (volts)
m Vc
cos [ 2 ( f c  f m ) t ]
2
m Vc
cos [ 2 ( f c  f m ) t ]
2
Upper side frequency signal (volts)
Lower side frequency signal (volts)
From the equation it is obvious that the amplitude of the carrier is unaffected by the
modulation process. The amplitude of the side frequencies depend on the both the carrier
amplitude and modulation index. At 100% modulation the amplitudes of side frequencies
are each equal to one-half the amplitude of the carrier.
Power Content of AM wave
In any electrical circuit, the power dissipated is equal to the voltage squared (rms) divided
by the resistance.
In radio transmission, the AM signal is amplified by a power amplifier and fed to the
antenna with a characteristic impedance that is ideally, but not necessarily, almost pure
resistance. The AM signal is really a composite of several signal voltages, namely,
the carrier and the two sidebands, and each of these signals produces power in the
antenna. The total transmitted power PT is simply the sum of the carrier power Pc and the
power in the two sidebands PUSB and PLSB:
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Recall that the AM signal contains three components.
 Carrier
 Lower side band
 Upper side band
The total power of the AM signal is the sum of the
powers contained in these
components
𝑃
=𝑃 +𝑃
+𝑃
Where: Pc is the power of the carrier
 Plsb / Pusb are the powers of lower & upper side bands respectively.
Mathematically power in unmodulated carrier is given by
Pc = carrier power (watts)
2
(V c / 2 ) 2 V c
Pc 

R
2R
Vc = peak carrier voltage (volts)
R = load resistance i.e antenna (ohms)
The upper and lower sideband powers will be
( mVc / 2) 2 m 2Vc
Pus b  Plsb 

2R
8R
2
where mVc/2 is the peak voltage of usf and lsf.
Rearranging in terms of Pc,
2
m 2  Vc  m 2


Pus b  Plsb 
Pc
4  2 R 
4
The total power in an AM wave is
Pt  Pc  Pusb  Plsb
Substituting the sidebands powers in terms of PC yields
m2
m2
Pc 
Pc
4
4
m2
m2
 Pc 
Pc  Pc [1 
]
2
2
Pt  Pc 
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Since carrier power in modulated wave is the same as unmodulated wave, obviously power of
the carrier is unaffected by modulation process. So the total power in an AM
envelopeincrease with modulation.
 m2 
m2
m2
m2

Pt  Pc  Pusb  Plsb  Pc 
Pc 
Pc  Pc 
Pc  Pc 1 
4
4
2
2 

Efficiency, E is defined as the percentage of total power that conveys information i.e it is the
percentage of total transmitted power that is in the sidebands.
Efficiency , E 
PSBs
Pusb  Plsb

PT
Pc  Pusb  Plsb
Example
For an AM DSBFC wave with a peak unmodulated carrier voltage Vc = 10Vp, frequency
of 100kHz, a load resistor of RL = 10 , frequency of modulating signal of 10kHz and m
= 1, determine the following
i) Powers of the carrier and the upper and lower sidebands.
ii) Total power of the modulated wave.
iii) Bandwidth of the transmitted wave.
iv) Draw the power and frequency spectrum.
2
(Vc / 2 ) 2 Vc
(10) 2
Pc 


 5W
R
2 R 2 10
m 2 Pc
Pusb  Plsb 
 1.25W
4
Introduction to communication system
Page 34
m2
m2
Pt  Pc 
Pc 
Pc
4
4
12
12
 5  (5)  (5)  7.5W
4
4
ii)
Bandwidth= 2xfmmax=2(10kHz)=20kHz
AM Current Calculations
Modulation index can be calculated by measuring the current of the carrier and the
modulated wave.
The measurement is simply by metering the transmit antenna current with and without the
presence of the modulating signal.
The relationship between the carrier current and the current of the modulated wave is
Pt It 2 R It 2
m2


 1
Pc Ic 2 R Ic 2
2
It
m2
 1
Ic
2
m2
2
Pt = total transmit power (watts)
Pc = carrier power (watts)
It = total transmit current (ampere)
Ic = carrier current (ampere)
R = antenna resistance (ohm)
It  Ic 1 
Where
Example
For an AM DSBFC transmitter with an unmodulated carrier power Pc = 100W that is
modulated simultaneously by 3 modulating signals with coefficient of modulation m1 =
0.2, m2 = 0.4 and m3 = 0.5, determine
a. Total coefficient of modulation
b. Upper and lower sideband power
c. Total transmitted power
Introduction to communication system
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One way of comparing communication systems is based on how they use the two primary
resources of communication, which are: Channel bandwidth, and
 Transmitted power
In this regard, conventional amplitude modulation (AM) modulation suffers two main
drawbacks as listed below.
i.
Inefficient use of transmission power:
Conventional AM signal spectrum includes a carrier component which consumes
high power for transmission but do not contain any message element.

So, it is wasted power!
ii.
Spectral inefficiency:
The two side bands in conventional AM signal spectrum carry similar
information.

So, the bandwidth could have been reduced by half if this redundancy is
removed.
Thus, an AM or simply Double Side-Band Full Carrier (DSBFC) is both power and
bandwidth inefficient. To overcome the above drawbacks, some variants of AM modulation
have been developed.
These include: DSB-SC (Double Side-Band Suppressed Carrier) modulation
 SSB (Single Side-Band) modulation
 VSB (Vestigial Side-Band) modulation
DSB –SC (Double Side Band – Suppressed Carrier Modulation)
In DSB-SC, as the name implies, the carrier is removed (suppressed) from AM signal
spectrum. Only two sidebands are available for transmission and this is achieved by using
product modulator, also known as balanced modulator. Balanced modulator simply
multiplies the message signal with carrier signal.
 Let, the modulating (message) signal be:
And, the carrier signal be:
 Then, using product modulator, the DSB-SC signal can be developed as:-
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 DSB-SC signal equation can be expanded using the following trigonometric identity:-
Bandwidth of DSBSC Wave
We know the formula for bandwidth (BW) is
Consider the equation of DSBSC modulated wave in the above equation
The DSBSC modulated wave has only two frequencies. So, the maximum and minimum
frequencies are 𝒇𝒄 + 𝒇𝒎 and 𝒇𝒄 − 𝒇𝒎 respectively.
Substitute, 𝑓𝑚𝑎𝑥 and 𝑓𝑚𝑖𝑛 values in the bandwidth formula
Thus, the bandwidth of DSBSC wave is same as that of AM wave and it is equal to twice
the frequency of the modulating signal.
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Power of DSBSC wave is equal to the sum of powers of upper sideband and lower
sideband frequency components.
But powers of upper sideband and lower sideband are given by
Now, let us add these two sideband powers in order to get the power of DSBSC wave.
Therefore, the power required for transmitting DSBSC wave is equal to the power of both
the sidebands
Single-Side band Modulation
Single-sideband (SSB) is a form of AM where the carrier is suppressed and one sideband
is eliminated. The process of suppressing one of the sidebands along with the carrier and
transmitting a single sideband is called as Single Sideband Suppressed Carrier system
or simply SSBSC.
The carrier is transmitted at full power but only one sideband is transmitted
– requires half the bandwidth of DSBFC AM
– Carrier power constitutes 80% of total transmitted power, while sideband
power consumes 20%
– SSBFC requires less total power but utilizes a smaller percentage of the power
to carry the information
This SSBSC system, which transmits a single sideband has high power, as the power
allotted for both the carrier and the other sideband is utilized in transmitting this Single
Sideband.
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In DSB transmission, since the sidebands are the sum and difference of the carrier and
modulating signals, the information is contained in both sidebands. As it turns out, there
is no reason to transmit both sidebands in order to convey the information. One sideband can be suppressed; the remaining sideband is called a single-sideband suppressed
carrier (SSSC or SSB) signal. SSB signals offer four major benefits.
1. The primary benefit of an SSB signal is that the spectrum space it occupies is only onehalf that of AM and DSB signals. This greatly conserves spectrum space and allows
more signals to be transmitted in the same frequency range.
2. All the power previously devoted to the carrier and the other sideband can be chan- neled
into the single sideband, producing a stronger signal that should carry farther and be
more reliably received at greater distances. Alternatively, SSB transmitters can be made
smaller and lighter than an equivalent AM or DSB transmitter because less circuitry and
power are used.
3. Because SSB signals occupy a narrower bandwidth, the amount of noise in the sig- nal is
reduced.
4. There is less selective fading of an SSB signal over long distances. An AM signal is
really multiple signals, at least a carrier and two sidebands.
These are on different frequencies, so they are affected in slightly different ways by the
ionosphere and upper atmosphere, which have a great influence on radio signals of less than
about 50 MHz. The carrier and sidebands may arrive at the receiver at slightly different
times, causing a phase shift that can, in turn, cause them to add in such a way as to cancel
one another rather than add up to the original AM signal. Such cancellation, or selective
fading, is not a problem with SSB since only one sideband is being transmitted.
An SSB signal has some unusual characteristics. First, when no information or modulating
signal is present, no RF signal is transmitted. In a standard AM transmitter, the carrier is still
transmitted even though it may not be modulated. This is the condition that might occur
during a voice pause on an AM broadcast. But since there is no carrier transmitted in an SSB
system, no signals are present if the information signal is zero. Sidebands are generated only
during the modulation process, e.g., when someone speaks into a microphone. This explains
why SSB is so much more efficient than AM.
The main disadvantage of DSB and SSB signals is that they are harder to recover, or
demodulate, at the receiver. Demodulation depends upon the carrier being present. If the
carrier is not present, then it must be regenerated at the receiver and reinserted into the
signal. To faithfully recover the intelligence signal, the reinserted carrier must have the same
phase and frequency as those of the original carrier. This is a difficult requirement. When
SSB is used for voice transmission, the reinserted carrier can be made variable in frequency
so that it can be adjusted manually while listening to recover an intelligible signal. This is
not possible with some kinds of data signals
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Page 39
VESTIGIAL SIDEBAND (VSB)
•
•
•
•
•
•
•
•
•
VSB is similar to SSB but it retains a small portion (a vestige) of the undesired sideband
to reduce DC distortion.
VSB signals are generated using standard AM or DSBSC modulation, then passing
modulated signal through a sideband shaping filter.
Demodulation uses either standard AM or DSBSC demodulation.
Also called asymmetric sideband system.
Compromise between DSB & SSB and Easy to generate.
Bandwidth is only ~ 25% greater than SSB signals.
Derived by filtering DSB, one pass band is passed almost completely while just a trace
AM wave is applied to a vestigial sideband filter, producing a modulation scheme – VSB
+C
Mainly used for television video transmission and VSB Frequency Spectrum
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Chapter 3
Angle Modulation
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.
It results whenever the phase angle θ of a sinusoidal wave is varied with respect to time
An angle modulation results whenever the phase angle, θ of a sinusoidal wave is varied
with respect to time and can be expressed as
(1)
m ( t )  V c cos  ct   (t ) 
Where m(t) = angle-modulated wave
Vc = peak carrier amplitude
ωc = carrier radian frequency
θ(t) = instantaneous phase deviation
where θ(t) is a function of the modulating signal given by
 ( t )  F V m sin(
 mt)
(2)
where ωc = modulating signal radian frequency
Vm = peak amplitude of the modulating signal
Two types of Angle Modulation
There two basic types of angle modulation
•
•
Frequency Modulation: The frequency of the carrier signal is varied in proportion
to the message signal. In FM, the carrier amplitude remains constant and the carrier
frequency is changed by the modulating signal. As the amplitude of the information
signal varies, the carrier frequency shifts proportionately. As the modulating signal
amplitude increases, the carrier frequency increases. With no modulation the carrier is
at its normal center or resting frequency
Phase Modulation: The phase of the carrier signal is varied in proportion to the
message signal. Phase modulators produce a phase shift which is a time separation
between two sine waves of the same frequency. The greater the amplitude of the
modulating signal, the greater the phase shift.
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Mathematical Analysis of Angle Modulation
To differentiate between FM and PM, the following terms need to be defined:
1. Instantaneous Phase Deviation
The instantaneous change in the phase of the carrier at a given instant of time.
Instantaneous phase deviation = θ(t) rad
(3)
2. Instantaneous phase
The precise phase of the carrier at a given instant of time.
Instantaneous phase = ωct + θ(t) rad
(4)
3. Instantaneous frequency deviation
 the instantaneous change in the frequency of the carrier and is defined as the
first time derivative of the instantaneous phase deviation.
Instantaneous frequency deviation = θ’(t) rad/s
(5)
4. Instantaneous frequency
 the precise frequency of the carrier at a given instant of time and is defined
as the first time derivative of the instantaneous phase.
Instantaneous frequency = ωi = ωc + θ’(t) rad/s
Then from the previous 4 terms, (3) ~ (5), PM and FM can be defined as :
• PM : an angle modulation in which θ(t) is proportional to the amplitude of the
modulating signal.
• FM : an angle modulation in which θ’(t) is proportional to the amplitude of the
modulating signal.
For a modulating signal vm(t),
θ(t) = Kvm(t) rad
θ’(t) = K1vm(t) rad/s
(6)
(7)
where K and K1 are constants and are the deviation sensitivities of the phase and
frequency modulators, respectively.
Then substituting a modulating signal vm(t) = Vmcos(ωmt), equation (7) and (8) into
equation (1) yields
m (t )  V c cos  ct   (t )
PM :


 V c cos  c t  KV m cos(  m t ) 
FM : as
 (t )    ' (t )

m (t )  Vc cos  ct    ' (t )


 Vc cos ct  K 1 Vm cos(mt ) dt

K 1Vm


 V c cos  c t 
sin(  m t ) 
m


Introduction to communication system
(9)
( 10)
Page 42
FM and PM Waveforms
Waveforms of FM and PM of a sinusoidal carrier by a single-frequency modulating
signal.
•
•
•
FM and PM waveforms are identical except for their time relationship.
for FM, the maximum frequency deviation occurs during the maximum positive and
negative peaks of the modulating signal.
for PM, the maximum frequency deviation occurs during the zero crossings of the
modulating signal (i.e. the frequency deviation is proportional to the slope of first
derivative of the modulating signal).
Modulation Index and Percent Modulation
Comparing equation (9) and (10), equation (1) can be rewritten in general form as
m (t )  Vc cos ct  m cos(mt ) 
(11)
where m is called the modulation index.
Modulation Index and Percent Modulation for PM
For PM, the modulation index is also known as peak phase deviation Δθ, and is
proportional to the amplitude of the modulating signal and is expressed as
m     KV m (radians )
(12)
where m = modulation index
K = deviation sensitivity (radians/volt)
Vm = peak modulating signal amplitude (volt)
Introduction to communication system
Page 43
For PM we can use
m(t )  Vc cosct  KVm cos(mt )
 Vc cosct   cos(mt )
 Vc cosct  m cos(mt )
(13)
Modulation Index and Percent Modulation for FM
The modulation index is the maximum value of phase deviation for both PM and FM and
is dimensionless (unitless).
For FM, the modulation index is directly proportional to the amplitude of the modulating
signal and inversely proportional to the frequency of the modulating signal.
(14)
m
K 1V m
m

K 1V m
(unitless )
fm
Where K1 = deviation sensitivities (radians/second per volt or cycles/second per vol
Vm = peak modulating signal amplitude (volt)
ωm = radian frequency (radians/second)
fm = cyclic frequency (cycles/second or hertz)
Also for FM, the peak frequency deviation Δf is simply the product of the deviation
sensitivity and the peak modulating signal voltage. I.e.
f  K 1Vm  m 
f
(unitless)
fm
(15)
Therefore, for FM, equation (10) can be rewritten as


K 1Vm
m (t )  Vc cos  ct 
sin( wmt ) 
fm




f
 V c cos  ct  sin( w mt ) 
fm


 Vc cosct  m sin(wmt )
•
•
(16)
Percent modulation for angle modulation is determined in different manner than for
amplitude modulation.
with angle modulation, percent modulation is the ratio of frequency deviation actually
produced to the maximum frequency deviation allowed, stated in percent form
Percent modulation

 f ( actual )
 100 %
 f (max)
Introduction to communication system
( 17)
Page 44
Frequency and Bandwidth Analysis of Angle-Modulated Waves
•
•
frequency analysis of the angle-modulated wave is much more complex compared to the
amplitude modulation analysis.
in phase/frequency modulator, a modulating signal produces an infinite number of side
frequencies pairs (i.e. it has infinite bandwidth), where each side frequency is displaced
from the carrier by an integral multiple of the modulating frequency.
Bessel Function
•
from equation (11), the angle-modulated wave is expressed as
m(t )  Vc cosct  m cos(mt )
•
•
•
based on the above equation, the individual frequency components of the anglemodulated wave is not obvious.
The equation that expresses the phase angle in terms of the sine wave modulating signal
is solved with a complex mathematical process known as Bessel functions.
Bessel function identities can be used to determine the side frequencies components

cos(  m cos  )   Jn (m) cos(  n 
n  
n
)
2
(18)
where Jn(m) is the Bessel function of the first kind. Then applying equation (18) to equation
(11) yields,

m (t )  Vc  Jn ( m ) cos(ct  n mt 
n  
n
)
2
(19)
expanding equation (19),

 

m ( t )  V c  J 0 ( m ) cos(  c t )  J 1 ( m ) cos  (  c   m ) t  
2


 

 J 1 ( m ) cos  (  c   m ) t    J 2 ( m ) cos (  c  2  m ) t 
2

  Jn ( m )  
Where m(t) = angle modulated wave
m = modulation index
Vc = peak carrier ampitude
J0(m) = carrier component
J1(m) = first set of side frequencies displaced from carrier by ωm
J2(m) = second set of side frequencies displaced from carrier by 2ωm
Jn(m) = nth set of side frequencies displaced from carrier by n ωm
Introduction to communication system
Page 45
In other words, angle modulation produces infinite number of sidebands, called as firstorder sidebands, second-order sidebands, and so on. Also their magnitude are determined
by the coefficients J1(m), J2(m),...Jn(m).
Bessel function of the first kind for several values of modulation index are shown below.
Curves for the relative amplitudes of the carrier and several sets of side frequencies for
values of m up to 10.
Conclusion from the table & graph can be summarized as follows :
modulation index m of 0 produces zero side frequencies.
the larger the m, the more sets of side frequencies are produced.
values shown for Jn are relative to the amplitude of the unmodulated carrier.
as the m decreases below unity, the amplitude of the higher-order side frequencies
rapidly becomes insignificant.
Introduction to communication system
Page 46
as the m increases from 0, the magnitude of the carrier J0(m) decreases.
the negative values for Jn simply indicate the relative phase of that side frequency
set
a side frequency is not considered significant unless its amplitude is equal or
greater that 1% of the unmodulated carrier amplitude (Jn ≥
0.01).
as m increases, the number of significant side frequencies increases. I.e. the
bandwidth of an angle-modulated wave is a function of the modulation index.
Bandwidth Requirement of Angle modulation
Angle-modulated wave consumes larger bandwidth than an amplitude-modulated wave.
Bandwidth of an angle-modulated wave is a function of the modulating signal and the
modulation index. Theoretically, the generation and transmission of FM requires infinite
bandwidth. Practically, FM system have finite bandwidth and they perform well.
However only those sidebands with significant power are considered and by Carson’s
rule; The minimum bandwidth for angle-modulated is given as;
BWFM  2( f  fm )  2(   1) f m Hz
(20)
Depending on the value of the modulation index 𝛽; frequency modulation (FM) wave is
classified as (i) Narrowband FM (NBFM) and
(ii) Wideband FM (WBFM)
(a) In narrow band FM, the modulation index 𝛽 is small as compared to one radian and the
bandwidth of the FM wave is small.
(b) In Wide band FM, the bandwidth is much larger and value of 𝛽 is very high. For larger
values of 𝛽, the FM wave ideally contains carrier and an infinite number of sidebands
located symmetrically around the carrier. Hence the BW approaches to infinity and
hence it is called wideband FM.
Characteristic
1. Modulation index
NBFM
<1
WBFM
>1
2. Maximum deviation ∆𝑓
5 kHz
75 kHz
3. Maximum modulation
index (mf)
4. Modulating frequency
range
5. BW
Slightly > 1
5 to 2500
30 Hz – 3 kHz
30 Hz – 15 kHz
6. Applications
Small


FM mobile

communications like: 
police wireless,
used for speech
transmission
Introduction to communication system
LARGE approximately 15
times of AM
Entertainment broadcasting
can be used for high quality
music transmission
Page 47
NOTE:
 Many of the advantages obtained with WBFM such as – noise reduction are not
available with NBFM.
 BW of NBFM is almost same as that of AM (amplitude modulation)
 BW of WBFM ≅ 15 times that of AM.
Carson’s Rule
o It is a general rule to estimate the bandwidth for all angle-modulated systems
regardless of the modulation index.
o The Carson’s Rule states that the bandwidth necessary to transmit an anglemodulated wave as twice the sum of the peak frequency deviation and the highest
modulating signal frequency.
Carson’s Rule
B  2(  f  fm ) Hz
(21)
o for a low modulation index (narrow band) fm is much larger than Δf ,
B  2 fm( Hz )
(22)
o For a high modulation index (wideband) Δf is much larger than fm or
B  2f ( Hz )
(23)
o Carson’s Rule approximate and gives a narrower bandwidth than the bandwidth
determined using Bessel function. Therefore, a system designed using Carson’s
Rule would have a narrower bandwidth but a poorer performance than system
designed using the Bessel table.
o for modulation index above 5, Carson’s Rule is a close approximation to the actual
bandwidth required.
o For FM, the bandwidth varies with both deviation and modulating frequency.
o Increasing modulating frequency reduces modulation index so it reduces the number
of sidebands with significant amplitude.
o On the other hand, increasing modulating frequency increases the frequency
separation between sidebands.
o Bandwidth increases with modulation frequency but is not directly proportional to it.
Introduction to communication system
Page 48
Deviation ratio
Deviation ratio DR is the worst case modulation index and is equal to the maximum
peak frequency deviation divided by the maximum modulating-signal frequency –
producing the widest frequency spectrum.
DR

 f (max)
f m (max)
Where DR = deviation ratio (unitless)
Δf(max) = maximum peak frequency deviation (Hertz)
fm(max) = maximum modulating-signal frequency (Hertz)
Average Power of a FM or PM Wave
⚫
⚫
⚫
⚫
⚫
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 power of a sinusoidal signal depends only on the square of the amplitude and not
on the frequency. The amplitude of an FM wave is constant and therefore the total
power of an FM signal is independent of the modulation index. This contrasts with
the AM where the power of the modulated signal is a function of the modulation
index.
The average power in undulated carrier
Vc 2
P ave 
2
An angle-modulated signal is described by the equation:
𝑠(𝑡) = 5𝑐𝑜𝑠(2𝜋 𝑥 10 𝑡 + 20𝑠𝑖𝑛1000 𝜋𝑡 + 10𝑠𝑖𝑛 2000𝜋𝑡)
Determine:
a. the power of the angle-modulated signal
b. the frequency deviation, ∆𝑓
c. The phase deviation,∆𝜙
d. The bandwidth of the angle-modulated signal using Carson’s rule.
Introduction to communication system
Page 49
FM/ PM Modulators
A phase modulator is a circuit in which the carrier instantaneous phase is proportional to
the modulating signal.
A frequency modulator is a circuit in which the carrier instantaneous phase is proportional
to the integral of the modulating signal.
PM modulator :
 (t )  v (t )
FM modulator :
 (t )   v(t )
Considering the FM modulator, if the modulating signal is v(t) is differentiate before being
applied to the FM modulator, the instantaneous phase is now proportional to the modulating
signal (i.e. PM modulator).
Differentiator + FM modulator =  (t ) 
dv (t )
 dt   (t )  v (t ) = PM modulator
Meanwhile, if the modulating signal is integrated before being applied to the PM modulator,
the instantaneous phase is now proportional to the integral of the modulating signal (i.e. FM
modulator).
Integrator + PM modulator =  ( t ) 
 v (t )
= FM modulator
The most common circuits used for FM signal demodulation are slope detector, balanced
slope detector and PLL demodulator.
The slope detector and balanced slope detector are categorized as tuned-circuit frequency
discriminator.
Advantages of Angle Modulation
 Noise immunity – most noise results in unwanted amplitude variations in the modulated
wave (i.e. AM noise). FM and PM receivers include limiters that remove most of the Am
noise from the received signal before the final demodulation process occurs – a process
that cannot be used with AM receivers because the information is also contained in
amplitude variations, and removing the noise would also remove the information.
 Noise performance and S/N improvement – with the use of limiters, FM and PM actually
reduce the noise level and improve the S/N ratio during the demodulation process.
 Capture effect - with FM and PM, a phenomenon of capture effect allows a receiver to
differentiate between two signals received with the same frequency by capturing the
stronger signal and eliminate the weaker one. With AM, both signals will be demodulated
and produce audio signals.
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Page 50
Disadvantages of Angle Modulation
 Bandwidth
- high quality angle modulation produces many side frequencies, thus
necessitating a much wider bandwidth than is necessary for AM transmission.
 Circuit complexity and cost
- PM and FM modulators, demodulators, transmitters, and receivers are
more complex to design and build than their AM counterparts.
- At one time, more complex means more expensive.
- However with the advent of inexpensive, large-scale integration ICs, the
cost is comparable to their AM counterparts.
Noise and Angle Modulation
When a constant density of thermal noise is added to an angle-modulated signal, unwanted
deviation of carrier frequency is expected. Magnitude of the unwanted deviation depends on
the relative amplitude of the noise with respect to the carrier amplitude.
Consider a noise signal with amplitude Vn and frequency fn :
 for PM, the unwanted peak phase deviation due to this interfering noise signal is
given by
  peak 

Vn
rad
Vc
for FM, when Vc > Vn, the unwanted instantaneous phase deviation is
approximately,
 (t ) 
taking derivative,
  (t ) 
Vn
sin(  n t   n ) rad
Vc
Vn
 n cos(  n t   n ) rad / s
Vc
Therefore, the unwanted peak frequency deviation is
 peak 
Vn
nrad / s
Vc
Introduction to communication system
 f peak 
Vn
f n Hz
Vc
Page 51
Chapter Four
Base band pulse signalling
4.1 The Sampling Theorem
Sampling theorem is based on the fixed sampling rate, called Nyquist rate. Hence, sampling
theorem is also known as Nyquist theorem. It is based on the theory of the bandlimited
signals. Let's discuss the sampling theorem of the bandpass signals and baseband signals.
According to the sampling theory of the bandpass signals, a signal can be successfully
reconstructed if its sampling rate is not greater than the maximum frequency W. The samples
are spaced at sampling time 'Ts' seconds apart without zero mean square error.
𝑇𝑠 = 1/2W
According to the sampling theory of the baseband signals, a signal can be successfully
reconstructed if the samples are separated with a uniform intervals less than or equal to
1/2Fm.
It can be represented as:
𝑇𝑠 ≤ 1/2𝐹𝑚
Sampling rate
Sampling rate is defined as the number of samples taken per second from a continuous signal
for a finite set of values. We can also define it as a sampling frequency, which is the
reciprocal of the sampling time.
𝐹𝑠 = 1/𝑇𝑠
Where, Fs is the sampling frequency
Ts
is the sampling time
As discussed, sampling rate is an essential period for the sampler to perform sampling
process. It helps in the successful recovery of the digital signal at the receiving end. Hence, a
fixed parameter was defined for the sampling rate, known as Nyquist rate.
Nyquist rate
Suppose H is the highest selected frequency. A bandlimited signal is transmitted at the
frequency components lower than W Hz. Thus, for the replication of the original signal, the
sampling rate should be twice the highest frequency.
It is given by:
𝐹𝑠 = 2𝑊
Where, Fs is the sampling rate
Introduction to communication system
W is the highest frequency
Page 52
Such rate of sampling is known as Nyquist rate. The sampling at the Nyquist rate does not
introduce any distortion.
Nyquist rate is also known as the minimum sampling rate and is represented by the
condition:
𝐹𝑠 = 2𝐹𝑚
Where, Fs is the sampling frequency or sampling rate
Fm is the maximum frequency of the input signal or the message signal
Nyquist Interval
Nyquist interval is the reciprocal of the Nyquist rate. It is given by:
𝑇𝑠 = 1/2𝑊
Where, TS is the Nyquist Interval W is the highest frequency
Methods of sampling
The methods of sampling are classified as follows:
o
Ideal Sampling
o
Natural Sampling
o
Flat-top sampling
Ideal Sampling
Ideal sampling is also known as instantaneous sampling or impulse sampling. The sampling
process multiplies the input signal and the carrier signal, which is present in the form of train
of pulses.
The above diagram shows the waveforms of the message signal, sampling signal in the form
of train of pulses, and the sampled signal.
Introduction to communication system
Page 53
Natural Sampling
Natural Sampling is considered an efficient multiplexing method in Pulse Amplitude
Modulation. Here, the analog signal is multiplied by the uniformly spaced rectangular pulses.
The above diagram shows the waveforms of the message signal, sampling signal, and the
sampled signal.
Flat-top sampling
The design and reconstruction of flat-top sampling is easy than the natural sampling process.
The pulses in the flat-top sampling method are in the flat shape at the top and are held at a
constant height. It means that the samples are flat and have constant amplitude.
Introduction to communication system
Page 54
Anti-aliasing filter
Aliasing is the common effect that can arise post the sampling process. In the aliasing
process, the high frequency components in the signal override the low frequency components.
The aliasing occurs when the signal frequency exceeds half the sampling frequency (Fs/2).
It can be represented as:
𝐹𝑚 > 𝐹𝑠/2
2𝐹𝑚 > 𝐹𝑠
Where,
Fs is the sampling frequency or sampling rate
Fm is the maximum frequency of the input signal or the message signal
Similarly, the aliasing effect can be reduced when the sampling frequency exceeds twice the
signal's frequency.
It is represented as:
𝐹𝑠 > 2𝐹𝑚
The anti-aliasing filters are used to prevent the aliasing effect during the transmission
process. The cut-off frequency of such filters is equal to half the sampling rate (𝐹𝑠/2). The
function of aliasing filters is to remove and filter the high-frequency components from the
signal. It is inserted before the sampler. It is also known as low-pass anti-aliasing filter. The
signal after passing through the anti-aliasing filter is sampled at a rate higher than the Nyquist
rate. It helps in the easy recovery of the signal.
Why sampling is required?
We know that the sampling process helps in the conversion of an analog signal to the digital
signal. The data transmission in the form of digital signal offers various advantages, such as
high efficiency, fast speed, low cost, low interference, low distortion, and high security.
Hence, sampling is essential to improve the quality and transmission ability of the signals
over the communication channel.
Introduction to communication system
Page 55
Advantages of sampling
The major advantages of the sampling process are due to the conversion of the transmission
to the digital form, which has various advantages as discussed above. It converts an analog
signal to the discrete values.
The advantages of sampling are as follows:
o
Low cost
o
High accuracy
o
Easy to implement
o
Less time consuming
o
Low signal loss
o
High scope
It prevents the signal loss or any information loss by converting the incoming data to the
suitable rate for transmission. For example, if a signal contains high frequency components,
the sampling process will convert it into high rates for effective transmission. Generally, the
input signal is sampled at the frequency rate twice that the incoming signal. It is done to
preserve the full information in the signal.
Applications of sampling
Sampling describes the number of possible digital values that are used to represent a sample.
Sampling is essential because it prevents any information loss during the transmission loss. It
also increases the accuracy of the system. Sampling is used in various processes, such as
PAM, PCM, and TDM. The major applications of sampling will be discussed in detail.
Pulse Modulation
The process of transmitting signals in the form of pulses (discontinuous signals) by using
special techniques. In pulse modulation, some parameter of a pulse train is varied in
accordance with the message signal.
We may distinguish two families of pulse modulation:
1. Analog pulse modulation and
2. Digital pulse modulation.
In analog pulse modulation, a periodic pulse train is used as the carrier wave, and some
characteristic feature of each pulse (e.g., amplitude, duration, or position) is varied in a
continuous manner in accordance with the corresponding sample value of the message
signal.
Introduction to communication system
Page 56
Thus in analog pulse modulation, information is transmitted basically in analog form, but the
transmission takes place at discrete times.
In digital pulse modulation, on the other hand, the message signal is represented in a
form that is discrete in both time and amplitude, thereby permitting its transmission in digital
form as a sequence of coded pulses; this form of signal transmission has no CW counterpart.
The use of coded pulses for the transmission of analog information-bearing signals represents
a basic ingredient in the application of digital communications
In pulse modulation a periodic pulse train is used as a carrier. The following parameters of
the pulse are modified in accordance with the message signal. Signal is transmitted at discrete
intervals of time.
 Pulse amplitude modulation
 Pulse width modulation
 Pulse position modulation
Pulse-amplitude modulation (PAM)
PAM is the simplest and most basic form of analog pulse modulation.In pulse-amplitude
modulation (PAM), the amplitudes of regularly spaced pulses are varied in proportion to the
corresponding sample values of a continuous message signal. In PAM: width and position
are fixed but amplitude varies.
Pulses can be of a rectangular form or some other appropriate shape.
Pulse-amplitude modulation is similar to natural sampling, where the message signal is
multiplied by a periodic train of rectangular pulses. In natural sampling the top of each
modulated rectangular pulse varies with the message signal, whereas in PAM it is maintained
flat. For minimum distortion, the sampling rate should be more than twice the signal
frequency.
There are two operations involved in the generation of PAM signal:
1. Instantaneous sampling of the message signal m(t) every Ts seconds, where the
sampling rate fs = 1/Ts is chosen.
2. Lengthening the duration of each sample so obtained to some constant value T.
Bandwidth required for transmitter of PAM signal will be equal to maximum frequency
B T  f max

1
2
Introduction to communication system
Page 57
The following graph represents natural sampling of PAM
PAM Modulators:
The PAM modulator is a simple “Emitter Follower” circuit. The modulating signal is
applied at the input. At the base, a CLOCK signal is applied. The frequency of the clock
signal is made equal to the frequency of carrier pulse train.
When the CLOCK signal is “high”, the circuit behaves as “Emitter follower” and the output
follows the input (modulating) signal, when the CLOCK is “low”, the transistor is “cut off”
and the output is zero. In this way, at the output we get PAM signal.
Introduction to communication system
Page 58
Advantages & Disadvantages of PAM
Disadvantages of PAM :
It has simple transmitter and receiver designs.
It allows multiplexing, so that the sharing of the same transmission media by different
sources (or users). This is because a PAM signal only occurs in slots of time, leaving the idle
time for the transmission of other PAM signals.
It is used to carry information as well as to generate other pulse modulations.
Disadvantages of PAM :
Amplitude keeps varying so there is noise associated with it.
Due to amplitude variation peak power of receiver also varies with it.
It requires a larger transmission bandwidth (very large compare to its maximum frequency)
Interference of noise is maximum and also needed for varies transmission power.
Pulse Density Modulation (PDM)
Sometimes called Pulse Duration Modulation/ Pulse Width Duration (PWM). In PDM the
width of pulses is varied in accordance to information signal but Amplitude & position
constant. PDM is used in a great number of applications Communications. The width of the
transmitted pulse corresponds to the encoded data value.
PDM is
• Immune to noise
• Power Delivery
– Reduce the total amount of power delivered to a load
– Applications: DC Motors, Light Dimmers, Anti-Lock Breaking System
Pulse Position Modulation (PPM)
•
•
•
•
•
•
•
Modulation in which the temporal positions of the pulses are varied in accordance with
some characteristic of the information signal.
Amplitude & width constant.
The higher the amplitude of the sample, the farther to the right the pulse is position within
the prescribed time slot.
The amplitude is held constant thus less noise interference and signal and noise separation
is very easy
Due to constant pulse widths and amplitudes, transmission power for each pulse is same.
It Require less power compare to PAM and PDM because of short duration pulses.
As a disadvantage it require very large bandwidth compare to PAM.
Introduction to communication system
Page 59
Transmission BW of PDM/PPM Signal
PPM and PDM need a sharp rise time and fall time for pulses in order to preserve the
message information.
• Lets rise time, tr
tr« Ts
BT 
1
2t r
From formula above, we know that transmission BW of PPM and PDM is higher than PAM.
Pulse duration (τ) supposed to be very small compare to the period, Ts between 2 samples
Lets max frequency of the signal, W
Fs >= 2 W
Ts =< 1/2W
T « Ts =< 1/2W
f max 
1
2
Example
For PAM transmission of voice signal with W = 3kHz. Calculate BT if fs = 8 kHz and τ = 0.1
Ts
1
1
Solution
4
Ts 
fs

8kHz
 1.25x10 s
  0.1Ts  1.25x105 s
1
2W
1
BT 
 W
2
1
BT 
 40 kHz
2
 
For the same information as in example above, find minimum transmission BW needed for
PPM and PDM. Given tr= 1% of the width of the pulse
Solution
1
7
tr 
100
1
BT 
2t r
  1.25 x10 s
BT  4 MHz
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Pulse Width Modulation (PWM)
In PWM, the width of the carrier pulse is varied according to the instantaneous value of the
modulating signal, while the amplitude remains constant. This system is also called “Pulse
duration modulation” (PDM) or “Pulse length modulation” (PLM). PWM is more often used
for control than for communication.
Noise is less in PWM as the amplitude is kept constant. The signal and noise separation is
easy. The PWM does not require synchronization between transmitter and receiver
But as a disadvantage Large bandwidth is required for PWM communication as compared to
PAM.The transmitter should be able to handle more power (equal to the power of the
maximum width pulse).
Pulse Code Modulation (PCM)
A PCM stream is a digital representation of an analog signal, in which the magnitude of the
analog signal is sampled regularly at uniform intervals, with each sample being quantized to
the nearest value within a range of digital steps.
In PCM the sampling rate which is the number of times per second that samples are taken.
According to sampling theorem, number of Pulses per second should be twice of signal
frequency.
Let's discuss the function of a PCM system with the help of an example of an audio signal.
The audio signal is first applied to the low-pass filter, which rejects the higher range of
frequencies from the signal. The sampler performs the sampling of the left and right
channels of the audio signal based on the sampling rate of 44100 Hz or 44.1k Hz and 16/32bit resolution. The quantizer and encoder set the digital value based on the specified
resolution and bit rate and send it to the receiver. The digital signal passes through
the quantizer that generates the pulse according to the received positive or negative pulse.
The decoder converts the regenerated pulse back to the analog signal. Further, it sends to
the reconstruction filter, which helps in the smooth conversion of the digital signal back to
the original analog signal.
The bit depth which determines the number of possible digital values that each sample can
take.
PCM consists of three steps to digitize an analog signal:
1. Sampling
2. Quantization
3. Binary encoding
Sampling PAM:
The first step in pulse code modulation is sampling. The analog signal is sampled at equal
interval, every Ts s (sample interval). The inverse of sampling interval is sampling rate or
sampling frequency. fs= 1/Ts
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Sampling rate is the number of samples per second. The process of generating pulses of zero
width and of amplitude equal to the instantaneous amplitude of the analog signal.
Quantized PAM Signal
The process of dividing the maximum value of the analog signal into a fixed number of levels
in order to convert the PAM into a binary code. The levels obtained are called quantization
levels. The result of PAM is a series of pulses with amplitude values between the maximum
and minimum amplitudes of the signal with real values. Quantization is a method of
assigning integer values in a specific range to sampled instances.
When a signal is quantized, an error will be introduced - the coded signal is an approximation
of the actual amplitude value.
The more zones, the smaller ∆ which results in smaller errors. But, the more zones the more
bits required to encode the samples so that higher bit rate
Thus the quantization error is the sequence eq (n) defined as the difference between the
quantized value and the actual sample value eq (n) = xq (n) - x(n)
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The bit rate of a PCM signal can be calculated form the number of bits per sample x the
sampling rate
Bit rate = nb x fs
The bandwidth required to transmit this signal depends on the type of line encoding used
A digitized signal will always need more bandwidth than the original analog signal. Price we
pay for robustness and other features of digital transmission.
Encoding : Encoding maps the quantized values to digital words that are n bits long. The
mapping is one-to-one so there is no distortion introduced by encoding.
The output of the quantizer is one of L possible signal levels.
If we want to use a binary transmission system, then we need to map each quantized sample
into an n bit binary word.
L  2n
n  log 2 L
Encoding is the process of representing each quantized sample by n bit code word. The
mapping is one-to-one so there is no distortion introduced by encoding
The component of PCM shown in the following
Bandwidth of PCM Signals
The spectrum of the PCM signal is not directly related to the spectrum of the input signal.
The bandwidth of (serial) binary PCM waveforms depends on the bit rate R and the
waveform pulse shape used to represent the data.
The Bit Rate R is
R=nfs
Where n is the number of bits in the PCM word (M=2n) and fs is the sampling rate.
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For no aliasing case (fs≥ 2B), the MINIMUM Bandwidth of PCM Bpcm(Min) is:
Bpcm(Min) = R/2 = nfs//2
The Minimum Bandwidth of nfs//2 is obtained only when sin(x)/x pulse is used to generate
the PCM waveform.
For PCM waveform generated by rectangular pulses, the First-null Bandwidth is:
Bpcm = R = nfs
Various forms of PCM processes are used in coding and signal processing in communication.
The following are some of the most common coding processes related to the Pulse Code
Modulation.
o
LPCM
PCM converts the analog signal to the digital signal for fast and efficient transmission
by converting the analog data into binary digits 0 and 1. Linear Pulse Code
Modulation uses the linear quantization method. The data during the quantization
process is generally compressed for better transmission. But, in LPCM, the data is in
the uncompressed form. Examples include blue-ray discs, Red Book compact discs,
etc.
o
DPCM
Differential Pulse Code Modulation requires fewer bits to encode the input pulse
level. It requires less bandwidth, an increased number of quantization levels, and
decreased quantization noise compared to the Pulse Code Modulation method.
o
ADPCM
Adaptive Differential Pulse Code Modulation is a type of DPCM that allows the
reduction of bandwidth by varying the size of the quantization step.
o
DM
Delta Modulation is a simplest type of DPCM that can convert both analog and
digital signals. It works similar to the A/D and D/A converters. It is generally used to
transmit voice signals because such signals do not require high quality at the output.
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Pulse Code Modulation Advantages
1. Analog signal can be transmitted over a high speed digital communication system.
2. Probability of occurring error will reduce by the use of appropriate coding methods.
3. PCM is used in Telkom system, digital audio recording, digitized video special
effects, digital video, voice mail.
4. PCM is also used in Radio control units as transmitter and also receiver for remote
controlled cars, boats, planes.
5. The PCM signal is more resistant to interference than normal signal.
•
•
•
Pulse Code Demodulation: will be doing the same modulation process in reverse.
Demodulation starts with decoding process
During transmission the PCM signal will effected by the noise interference.
The following table summarizes the Pulse modulation techniques based on some basic
parameter.
Relation with
modulating
signal
BW of the
transmission
channel
Instantaneous
power
Noise
interference
Complexity
the system
PAM
PDM
PPM
Amplitude of
the pulse is
proportional to
amplitude
of
modulating
signal
Width of the pulse is
proportional
to
amplitude
of
modulating signal
Relative position of the
pulse is proportional to
amplitude of modulating
signal
Depends of rise time
of the pulse
Depends on rising time of
the pulse
varies
Remains constant
Minimum
Minimum
Simple
simple
depends
width of
pulse
on
the
varies
High
of
Complex
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Digital Modulation Techniques
Digital Modulation provides more information capacity, high data security, quicker system
availability with great quality communication. Hence, digital modulation techniques have a
greater demand, for their capacity to convey larger amounts of data than analog ones.
There are many types of digital modulation techniques which will be discussed in the
following.
Amplitude Shift Keying ( ASK)
The amplitude of the resultant output depends upon the input data whether it should be a zero
level or a variation of positive and negative, depending upon the carrier frequency.
A digital modulation technique in which the amplitude of the carrier wave is altered
according to the modulating signal (bitstream) is known as Amplitude Shift Keying (ASK).
It is the easiest and straightforward digital modulation scheme.
Amplitude Shift Keying (ASK) is a type of Amplitude Modulation which represents the
binary data in the form of variations in the amplitude of a signal.
In ASK, frequency and phase of the carrier wave is kept constant and only the amplitude is
varied according to the digitized modulating signal. It is also referred as Binary Amplitude
Shift Keying (BASK) as its usual operation is associated with only two levels.
Following is the diagram for ASK modulated waveform along with its input.
Any modulated signal has a high frequency carrier. The binary signal when ASK is
modulated, gives a zero value for LOW input and gives the carrier output for HIGH input.
In ASK, only the amplitude of the carrier signal is modified in modulation. The simplest
version is on–off keying (OOK).
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Advantage of Amplitude shift keying

Its generation and detection are easy thus facilitate simple transmitter and receiver
sections.
Disadvantages of Amplitude shift keying

ASK technique is not suitable for high bit rate data transmission.

Poor bandwidth efficiency.

Highly susceptible to noise and other external factors.
Applications of Amplitude shift keying
1. Digital data through an optical fiber is transmitted using ASK technique.
2. The technique was widely used in traditional telephone modems.
Frequency Shift Keying ( FSK)
The frequency of the output signal will be either high or low, depending upon the input data
applied.
Frequency Shift Keying (FSK) is the digital modulation technique in which the frequency of
the carrier signal varies according to the discrete digital changes. FSK is a scheme of
frequency modulation. It is the most straightforward and efficient digital signal transmission
scheme.
The simplest form of FSK is Binary frequency shift keying (BFSK). Here, the frequency of
the carrier wave changed between discrete binary values of the modulating signal. Thus, the
frequency of the carrier shows variation according to the binary message signal.
In frequency shift keying, the carrier is modulated in such a way that high-frequency signal is
achieved for high level i.e., 1 of binary data input. Similarly, the low-frequency signal is
obtained in case of low level i.e., 0 of the message signal.
Following is the diagram for FSK modulated waveform along with its input.
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Advantages of frequency shift keying

FSK provides better noise immunity.

The signal transmission through FSK is quite simple.

It is suitable for long-distance data transmission.

Bit error rate performance is better than ASK.
Disadvantages of frequency shift keying

It utilizes more bandwidth as compared to ASK and PSK thus is not bandwidth efficient.

Detection of the signal at the receiver is somewhat complex.
Frequency shift keying (FSK) is used in the high-frequency data transmission system. and
also extensively used in low-speed modems
Phase Shift Keying – PSK
Digital modulation technique that transmits data by varying the phase of the carrier wave in
accordance with the digital modulating signal, is called Phase Shift Keying (PSK).The easiest
form of PSK is BPSK i.e., binary phase shift keying. However, PSK can be extended to 4
level and 8 level PSK that totally depends on the need of the system.
Binary Phase Shift Keying (BPSK)
This is also called as 2-phase PSK (or) Phase Reversal Keying. In this technique, the sine
wave carrier takes two phase reversals such as 0° and 180°.
BPSK is basically a DSB-SC (Double Sideband Suppressed Carrier) modulation scheme, for
message being the digital information.
Following is the image of BPSK Modulated output wave along with its input.
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Chapter Five
Introduction to Data communication
Data communication is the combination of two terms Data and Communication
 Data- representation of information
 Communication - process of sharing data
Data communication is the exchange of data between two devices via some form of
transmission medium.
Data communication covers three general areas: data communications, networking, and
protocols.
● Data communications deals with the transmission of signals in a reliable and efficient
manner.
● Networking deals with the technology and architecture of the communications
networks used to interconnect communicating devices.
● Protocols deals with the protocol being used in communication and networking
Basic trends are
 traffic growth at a high & steady rate
 development of new services
 advances in technology
The significant change in requirements includes
 emergence of high-speed LANs
 corporate WAN needs
 digital electronics
A data communication system has five Basic components





Message – Information to be communicated
o text, picture, video, audio
Sender – the device that generates the information
o e.g., Computer, telephone
Receiver - the devices that receives the message
Transmission Medium – the physical path by which a message travels from sender to
receiver
o It can be a simple transmission line or a complex network
o Coaxial cable, optical fiber, radio wave
Protocol – sets of rules that governs data communication
o Defines what is communicated, how it is communicated, when it is
communicated
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Data Communications Model
Source - generates data to be transmitted
Transmitter - converts data into transmittable signals
Transmission System - carries data from source to destination
Receiver - converts received signal into data
Destination - takes incoming data
The performance of data communication systems can be measured by the following
criterion
 Delivery
 The data must be delivered to the intended device (receiver)
 Accuracy
 The data has to be delivered correctly
 Timeliness
 The system must deliver the data in a timely manner
 Jitter
 variation in the packet arrival time
Information in data communication can be in form of text, image, video, audio, etc. and
this information is represented in binary format (data). A binary digit (bit) has only two
values, 0 and 1. Text, image, audio, etc. are represented as a sequence of bits
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OSI Reference Model
o
OSI stands for Open System Interconnection is a reference model that describes
how information from a software application in one computer moves through a
physical medium to the software application in another computer.
o
OSI consists of seven layers, and each layer performs a particular network function
There are the seven OSI layers. Each layer has different functions. A list of seven layers are
given below:







Application (layer 7)
Presentation (layer 6)
Session (layer 5)
Transport (layer 4)
Network (layer 3)
Data link (layer 2)
Physical (layer 1)
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OSI-Application layer
o
Provides service to users


Implement the function needed by the users
Enable users to access a network
Application layer specific services



File transfer, access and management – allows a user to access, retrieve and
mange files in a remote computer
Mail service – email forwarding and storage
User authentication – logging to remote host
OSI-Presentation Layer
•
•
Deals with the presentation of data
Specific responsibility of Presentation layer:
 Translation – from sender dependent data representation (encoding) format
into common format or from common format into receiver dependent format
 Encryption/decryption – from original form to another form or vice verca
 Compression – reducing the number of bits
OSI-Session Layer
•
•
Coordinate the interaction between applications on communication devices
Provides service to
 Establishes, maintains and terminates session between applications
 Synchronize data flow (support orderly data exchange )
 Manage dialog
 Decides which device communicate first
 Session refers to a connection for ongoing data exchange
OSI-Transport Layer
•
•
•
Responsible for end-to-end (process to process) delivery of the entire message
Ensures that the whole message arrives intact and in order
Functions of transport layer
 Segmentation and reassembling
 Convert application data into smaller block of segments
 Each segment contains a sequence number
 Connection control and Flow and error control
 Ensures that the message arrive at the receiving transport layer without
error
 Service point addressing
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OSI-Network Layer
•
•
Network layer is responsible for the source-destination delivery of a packet
Responsibilities of network layer
 Logical addressing
 Used to distinguished the source and destination system
 Routing
 Moving the data across a series of interconnected networks
OSI-Data link layer
• Responsible to for moving frames from one hop to the next
Specific functions of data link layer
• Framing – divides the stream of bits received from the network layer into
manageable data unit
• Physical addressing – adds header that defines the sender and receiver of the frame
• Flow control - regulates the amount of data the sender sends
• Error control – detects and retransmit damaged or lost frames
• Access control – control access to the shared link
OSI-Physical Layer
•
•
•
Coordinates the functions required to carry a bit stream over a physical medium
Defines the procedures and functions that the physical devices and interfaces have to
perform for transmission to occur
Other functions
 Representation of bits – encodes bits into electrical or optical signals
 Synchronization of bits
- synchronizes the sender and the receiver clocks
 Transmission mode- The direction of transmission between two devices
(simplex, half- duplex, full-duplex)
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Multiplexing
Multiplexing is a technique used to combine and send the multiple data streams over a single
medium. The process of combining the data streams is known as multiplexing and hardware
used for multiplexing is known as a multiplexer.
Multiplexing is achieved by using a device called Multiplexer (MUX) that combines n input
lines to generate a single output line. Multiplexing follows many-to-one, i.e., n input lines
and one output line.
Demultiplexing is achieved by using a device called Demultiplexer (DEMUX) available at
the receiving end. DEMUX separates a signal into its component signals (one input and n
outputs). Therefore, we can say that demultiplexing follows the one-to-many approach.
Why Multiplexing is needed ?
o
The transmission medium is used to send the signal from sender to receiver. The medium
can only have one signal at a time.
o
If there are multiple signals to share one medium, then the medium must be divided in
such a way that each signal is given some portion of the available bandwidth. For
example: If there are 10 signals and bandwidth of medium is100 units, then the 10 unit is
shared by each signal.
o
When multiple signals share the common medium, there is a possibility of collision.
Multiplexing concept is used to avoid such collision.
o
Transmission services are very expensive
o
The 'n' input lines are transmitted through a multiplexer and multiplexer combines the
signals to form a composite signal.
o
The composite signal is passed through a Demultiplexer and demultiplexer separates a
signal to component signals and transfers them to their respective destinations.
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Multiplexing Techniques
Multiplexing techniques can be classified as
Frequency-division Multiplexing (FDM)
o
Frequency Division Multiplexing is a technique in which the available bandwidth of a
single transmission medium is subdivided into several channels. It is an analog technique.
In the above diagram, a single transmission medium is subdivided into several frequency
channels, and each frequency channel is given to different devices. Device 1 has a frequency
channel of range from 1 to 5

The input signals are translated into frequency bands by using modulation techniques, and
they are combined by a multiplexer to form a composite signal.

The main aim of the FDM is to subdivide the available bandwidth into different
frequency channels and allocate them to different devices.

Using the modulation technique, the input signals are transmitted into frequency bands
and then combined to form a composite signal.

The carriers which are used for modulating the signals are known as sub-carriers. They
are represented as f1,f2..fn.

FDM is mainly used in radio broadcasts and TV networks.
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Wavelength Division Multiplexing (WDM)
Wavelength Division Multiplexing is same as FDM except that the optical signals are
o
transmitted through the fibre optic cable.
o
WDM is used on fibre optics to increase the capacity of a single fibre.
o
It is used to utilize the high data rate capability of fibre optic cable.
o
It is an analog multiplexing technique.
o
Optical signals from different source are combined to form a wider band of light with the
help of multiplexer.
At the receiving end, demultiplexer separates the signals to transmit them to their
o
respective destinations
Time Division Multiplexing
o
It is a digital technique.
o
In Frequency Division Multiplexing Technique, all signals operate at the same time with
different frequency, but in case of Time Division Multiplexing technique, all signals
operate at the same frequency with different time.
o
In Time Division Multiplexing technique, the total time available in the channel is
distributed among different users. Therefore, each user is allocated with different time
interval known as a Time slot at which data is to be transmitted by the sender.
o
In Time Division Multiplexing technique, data is not transmitted simultaneously rather
the data is transmitted one-by-one.
o
In TDM, the signal is transmitted in the form of frames. Frames contain a cycle of time
slots in which each frame contains one or more slots dedicated to each user.
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Supplementary Multiple Choose Questions
Choose the correct answer from the given alternatives
1. Modulation has a number of advantages. Which of one of the following is not correct?
a) Efficient transmission
b) Reduction in noise and interference
c) Overcomes hardware limitations
d) Requires higher power transmitter
2. For message signal 𝑚(𝑡) = 𝑐𝑜𝑠 (2𝜋𝑓 𝑡) and carrier frequency fc , which one of the
following represents a SSB signal
a) 𝑐𝑜𝑠(2𝜋𝑓 𝑡) 𝑐𝑜𝑠(2𝜋𝑓 𝑡)
b) 𝑐𝑜𝑠(2𝜋𝑓 𝑡)
c) 𝑐𝑜𝑠[2𝜋(𝑓 + 𝑓 ) 𝑡]
d) 1 + 𝑐𝑜𝑠(2𝜋𝑓 𝑡) 𝑐𝑜𝑠(2𝜋𝑓 𝑡)
3.
4.
5.
6.
7.
Vestigial Sideband (VSB) modulation is preferred in TV because
a) It reduces the bandwidth requirement to half
b) It avoids phase distortion at low frequencies
c) it results in better reception
d) none of the above
For telegraphy the most commonly used modulation system is
a) FSK
b) two tone modulation
c) PCM
d) single tone modulation
In Pulse Code Modulation( PCM) system
a) large bandwidth is required
b) quantising noise can be overcome by companding
c) quantising noise can be reduced by decreasing the number of standard levels
d) suffers from the disadvantage of its incompatibly with TDM
_______determines the number of sideband components in frequency modulation.
a) carrier frequency
b) modulation frequency
c) modulation index
d) deviation ratio
Fourier analysis indicate that a square wave can be represented as
a) a fundamental sine wave and odd harmonics
b) a fundamental sine wave and even harmonics
c) a fundamental sine wave and harmonics
d) fundamental and subharmonic sine waves
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8. Gate function is defined as_________
a) 𝐺(𝑡) =
b) 𝐺(𝑡) =
c) 𝐺(𝑡) =
d) 𝐺(𝑡) =
1, |𝑡| <
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
1, |𝑡| >
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
1, |𝑡| ≤
0,
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
1, |𝑡| ≥
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
9. The disadvantage of frequency modulation over amplitude modulation is that
a) high output power is needed
b) high modulating power is needed
c) noise is very high for high frequency
d) large bandwidth is required
10. Frequency component of an AM wave are
a) Carrier frequency (ωc) with amplitude A
b) Lower side band (ωc + ωm) having amplitude mA 2
c) Upper side band (ωc – ωm) having amplitude mA 2
d) Carrier frequency (ωc / 2) with amplitude A
11. A frequency modulated voltage wave is given by the equation :
e = 12 cos (6 × 108t + 5 sin 1250 t)
What will be the maximum frequency deviation and power dissipated by the FM wave
in 10-ohm resistor respectively.
a). 212 Hz and 11.5 W
c) 755 Hz and 8.9 W
b) 995 Hz and 7.2 W
d) 677 Hz and 5.7 W
12. In an amplitude modulation ( AM) wave, useful power is carried by____________
a) carrier
b) sidebands
c) both sidebands and carrier
d) none of the above
13. Modulation is done in___________________
a) transmitter
b) radio receiver
c) between transmitter and radio receiver
d) none of the above
14. Over modulation results in________________
a) weakening of the signal
b) excessive carrier power
c) distortion
d) none of the above
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15. The amount of power and bandwidth necessary to be transmitted for a given amount of
information are reduced in _________
a) Single Sideband Modulation
b) Double Sideband Modulation
c) Vestigial Sideband Modulation
d) Amplitude Modulation
16. In FM, if we decreases modulating frequency then the modulation index ________
a) will increase, if the modulating voltage amplitude increases
b) will decrease, if the modulating voltage amplitude increases
c) will increase, if the modulating voltage amplitude remains constant
d) will decrease, if the modulating voltage amplitude remains constant
17. . For demodulation of PCM, it is first converted into __________
a) PDM
b) PWM
c) PPM
d) PAM
18. What is the equation for full-carrier amplitude modulation (AM)
a) V(t) = (Ec + Em) × (sin ωc t)
b) V(t) = (Ec + Em) × (sinωm t) + (sin ωc t)
c) V(t) = (Ec × Em) × (sin ωm t) × (sin ωc t)
d) V(t) = (Ec + Em sin ωm t) × (sin ωc t)
19. In a frequency modulated (FM) system, when the audio frequency is 500 Hz and audio
frequency voltage is 2.4 V, The frequency deviation δ is 4.8 kHz. If the audio frequency
voltage is now increased to 7.2 V then what is the new value of deviation?
a) 3.6 kHz
b) 0.6 kHz
c) 14.4 kHz
d) 12.4 kH
20. The equation V = A sin (ωct + m sin ωmt) is the expression for
a) Amplitude modulation
b) Phase modulated signal
c) Carrier signal used for modulation
d) None of the above
21. When the modulation frequency is doubled the modulation index is halved and the
modulating index is halved and the modulation voltage remains constant. This happens
when the modulating system is
a) AM
b) PM
c) FM
d) Delta Modulation
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22. An FM signal is represented by
V = 12 sin (6 x 108t + 5 sin 1250t)
The carrier frequency f and frequency deviation δ, respectively, are
a) 191 MHz and 665 Hz
b) 95.5 MHz and 995 Hz
c) 191 MHz and 995 Hz
d) 95.5 MHz and 665 H
23. Which of the following are the advantages of FM over AM?
I.
II.
III.
IV.
Better noise immunity is provided
Lower bandwidth is required
Transmitted power is more useful
Less modulating power is required
a) i, ii and iv
b) i, ii and iii
c) i, iii and iv
d) ii, iii and iv
24. Which one of the following statement is not correct?
a) FM has an infinite number of side-bands
b) Modulation index for FM is always greater than one
c) As modulation depth increases the BW increases
d) As modulation depth increases the sideband power increases
25. Consider the following statements about frequency modulation
1. Modulation index determines the number of significant sideband components.
2. Theoretical bandwidth is infinite.
3. Carrier suppression is not possible
4. Sidebands are not symmetric about carrier
Which of these statements is/are correct?
a.
b.
c.
d.
1,2,3 and 4
1 and 2 only
3 only
3 and 4 only
26. An angle modulated signal is described by the equation
xc(t) = 10 cos[2πfct + 10 sin(4000πt) + 5 sin 2000πt]
Then what is the bandwidth of this modulated signal?
a) 6 kHz
b) 45 kHz
c) 54 kHz
d) 63 kHz
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27. In TV transmission, the modulation schemes for Video and Audio are, respectively
a) FM and AM
b) FM and FM
c) AM and FM
d) AM and AM
28. In the frequency modulation if fm is modulating frequency, Δf is maximum frequency
deviation and B is bandwidth, then
a) B = Δf – fm
b) B = Δf + fm
c) B= 2(Δf – fm)
d) B=2(Δf + fm)
29. A binary channel with a capacity of 36 kbits/sec is available for PCM voice transmission.
If signal is band-limited to 3.2 kHz, then the appropriate values of quantizing level L and
the sampling frequency respectively are
a) 64 and 7.2 kHz
b) 32 and 3.6 kHz
c) 64 and 3.6 kHz
d) 32 and 7.2 kHz
30. Comparison of FDM and TDM systems shows that
a) FDM requires lower bandwidth, but TDM has greater noise immunity.
b) FDM has greater noise immunity and requires lower bandwidth the TDM.
c) FDM requires channel synchronization while TDM has greater noise immunity.
d) FDM requires more multiplexing, while TDM requires band-pass filter
31. Which one of the following is a disadvantage of digital transmission as compared to
analog transmission?
a) Digital signals cannot be multiplexed efficiently
b) Digital transmission is less immune to channel noise
c) Digital signals need to be coded before transmission
d) Digital transmission needs more bandwidth
32. What are the three steps in generating PCM in the correct sequence?
a) Sampling, quantizing and encoding
b) Encoding, sampling and quantizing
c) Sampling, encoding and quantizing
d) Quantizing, sampling and encoding
33. For a 10-bit PCM system, the signal to quantization noise ratio is 62 dB. If the number of
bits increased by 2, then how would the signal quantization noise ratio change?
a) Increase by 6 Db
b) Decrease by 6 dB
c) Increase by 12 dB
d) Decrease by 12 dB
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34. Quantization noise occurs in______________
a) Pulse amplitude modulation
b) Pulse width modulation
c) Pulse code modulation
d) Pulse position modulation
35. For 100% modulation, total power is
a) same as the power of unmodulated signal
b) twice as the power of unmodulated signal
c) four times as the power of unmodulated signal
d) one and half times as the power of unmodulated signal
36. The modulation technique that uses the minimum channel bandwidth and transmitted
power is
a) FM
b) DSB-SC
c) VSB
d) SSB
37. The function of multiplexing is
a) To reduce the bandwidth of the signal to be transmitted
b) To combine multiple data streams over a single data channel
c) To allow multiple data streams over multiple channels in a prescribed format
d) To match the frequencies of the signal at the transmitter as well as the receiver
38. Aliasing refers to
a) Sampling of signals less than at Nyquist rate
b) Sampling of signals greater than at Nyquist rate
c) Sampling of signals at Nyquist rate
d) None of the above
39. An AM signal has a total power of 48 Watts with 45% modulation. Calculate the power
in the carrier and the sidebands.
a) 39.59 W, 4.505 W
b) 40.59 W, 4.205 W
c) 43.59 W, 2.205 W
d) 31.59 W, 8.205 W
40. Analog communication indicates
a) Continuous signal with varying amplitude or phase
b) No numerical coding
c) AM or FM signal
d) All of the above
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Answer Key for Multiple Choose Questions
Question
number
Answer
Question
number
Answer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
C
C
A
A
A
C
A
A
D
A
B
B
A
C
A
C
D
D
C
B
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
C
B
C
B
B
C
C
D
D
C
D
A
C
C
D
D
B
A
C
D
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