A STEP FURTHER

IN
TELECOMMUNICATION

Engr. O.T. Adewunmi

© 2021 A STEP FURTHER IN TELECOMMUNICATION
Copyright Eng. O.T. Adewunmi

All rights reserved. No part of this book may be reproduced, stored

in a retrieval system, or transmitted in any form or by any means:
electronic, mechanical, photocopying, recording, or otherwise,
without prior written permission of the publishers.

ISBN 978-978-993-204-7

Published by:
Mirror Age Concepts
No 2 Raji Olaosebikan Doherty Street,
Hamdalat Avenue, Atari Area,
Offa, Kwara State.
07035627734, 08130095597, 09022295511
mirrorageconcepts@gmail.com, matinsola4luv@yahoo.com

ii
 Dedication
Dedicate to Almighty God.

iii
 Table of contents
Dedication iii
Table of contents iv
CHAPTER ONE
INTRODUCTION 1
1.1 Communication System 1
1.2 Communication Channel 2
1.3 Baseband Digital Modulation 3
1.4 Classification of Communication Systems 6
1.5 Digital and Analog Signals 8
1.6 Classification of Microphones Under Physcial Design 10
CHAPTER TWO
2.1 Modulation Processes in a Communication System 26
2.2 Amplitude Modulation 27
CHAPTER THREE
3.1 Radio Receivers 31
3.2 Superheterodyne Radio Receiver 31
CHAPTER FOUR
4.0 Meaning of Transmitter 36
4.1 Function of Transmitter 36
4.2 Block Diagram of AM 37
4.3 F.M Transmitter Fig. 1.3 shows the typical configuration for
an F.M transmitter the indirect method of F.M generation is used.38
4.4 Black & White of Television Transmitter (Monocrome) 41
4.5 Field Strength of TV Signal 42
4.6 A Television Receiver 44
4.7 Radio Wave Propagation 47
CHAPTER FIVE
5.0 Antenna 48
5.1 Polarization 48
5.2 Half Wave Di-Pole Antenna 49
5.3 Current Voltage Characteristic of /2 Dipole 50
5.4 Radiation Resistance 51
5.5 Antenna Gain 52
5.6 Radiation Pattern of An Antenna 55
5.7 Types of Antenna 55
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5.8 Application 58
CHAPTER SIX
6.0 Mechanism of Propagation of Radio Wave 59
6.1 Ionospheric Layers 62
6.2 Effect of Ionosphere on the Sky Wave 65
6.3 Critical Frequency (CF) 65
6.4 Maximum Usable Frequency (MUF) 66
6.5 The Optimum Working Frequency (OWF) 66
CHAPTER SEVEN
7.0 Introduction to Signals 67
7.1 General signal characteristics 68
7.2 Classification of signals 69
7.3 Basic Operations of Signals 74
7.4 Decimation and Expansion 76
7.5 Combination of Operations 77
CHAPTER EIGHT
8.0 Introduction to Systems 83
8.1 Properties of systems (Classification of systems) 83
8.2 Linear –time convolution system (LTI) 92
8.3 System Properties 96
REFERENCES 98

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 CHAPTER ONE
INTRODUCTION
1.1 Communication System
Communication systems refer to the transmission, reception
and processing of information or messages by electrical means. The
commonest forms are telegraphy. Telephony, radio and TV
broadcasting.
The transmission system is usually through metallic wires
and cables, optical fibers or through space when radio signals are
employed.
The block diagram of a basic communication system is as
shown figure 1.1:

Information Transmitter Channel Receiver Destination

source

Noise source

Figure 1.1: Basic communication System

a) Information Source
The information or message to be communicated may
originate as speech (or music), picture (moving or stationary),
printed material or data generated by a computer or obtained from
some physical phenomena, whatever the source, the message must
first be converted into electrical signals before transmission takes

1
place. Then signal is received by the receiver and reconverted to a
desired from at the destination. Suitable TRANSDUCERS are
required to perform the conversion processes at the transmitter and
receiver ends. The table below illustrates various types of
transducers that may be employed.

Block diagram of a digital communication system

Estimate of message
Message signal
Source of signal
Sink of
information Information
Transmitter

Source of Source of
encoder encoder
Source code Estimate of source

Receiver
word code word

Channel Channel
encoder decoder
Channel code Estimate of channel
word code word

Modulator Demodulator

Wave form Receiver signal

Channel

Fig. 1.2: Communication Channel

1.2 Communication Channel

Types of Communication Channel
1. Cable e.g. Twisted pair, coaxial cable –
telephone, LAN etc

2
2. RF/Microwave links e.g. Microwave antennas,
optical fibre, waveguide – GSM, CDMA, AM & FM
broadcasting etc
3. Satellite links e.g. VSAT – Internet, satellite
broadcasting
Channels, where the noise/interference resides, can be
roughly divided into two groups:
i. Guided propagation channels e.g., telephone channels
e.g., telephone channels, coaxial cables and optical fibres
ii. Free propagation channels e.g., broadcast channels,
mobile radio channels, and satellite channels

1.3 Baseband Digital Modulation

Introduction
The previous chapters have been largely concerned with so-
called baseband signaling where the channel band is assumed to
extend from 0 Hz upwards. In applications where contiguous band
with encompassing 0 Hz is not available, bandpass signaling is
required. Here, the task is to centre the symbol energy at a given
frequency of operation, for example, 900MHz for a typical cellular
telephone channel and 30000Hz, (1000nm) for an optical fibre link.
This process usually involves modulating the amplitude,
frequency and/or phase of a carrier sinewave. The carrier is

3
commonly written as cos (wct). We shall see that the choice of
modulation method crucially affects the ease of communication.

Table 1: Types of the Transducer that may be employed

Message Transmitter Receiver Message

(Input) (Transducer) (Transducer) (Output)
Speech

Sound
Music Microphone Loudspeaker Sound

TV

Picture
Fax (printed material) T camera TV camera Picture or text
Photo detector Picture or text

Physical Temperature Bimetallic strip Electrical signal

Phenomenon Thermocouple

Pressure Piezoelectric Recording instruments

Crystal or strain guage

NOTE: The form of the message output at the destination needs not
be the same as the form transmitted since an electrical signal can be
converted into any form of message using transducers

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b) Transmitter
Electrical signals generated by the transducer are processed
by the transmitter. Such signal processing techniques involve
amplification to boost the signal current or voltage amplitude or
power level: others are modulation, filtering and coding etc.
c) Receiver
The reverse process of converting the electrical signal back
into the form of message desired at the destination takes place in
the receiver. Filtering demodulation and decoding are signal
processing techniques involve in the receiver

d) Disturbing Influence: Electrical Noise

Electrical noise is the most important disturbing influence
that limits the performance of a communication system. There are
various types of noise depending on the type of transmission
medium.
i) Thermal Noise exists on conducting wires, resulting from
the statistical fluctuations of the electrons in the wire. It
increases with temperature and is prominent when the signal
level is low. Noise also occurs at the transmitter and
receiver ends, but this can be reasonably controlled by
appropriate circuit sign.

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 In radio communication, noise introduced in the
transmission channel can be grouped into two broad categories;
Man-made noise and natural noise.
 Man-made Noise: This noise is from
electric motors, switches, automobile ignition, electric
welding equipment or high-voltage power lines etc.
 Natural Noise: This is from lighting
discharges during thunderstorms usually referred to as
static. Other sources of natural noise are solar noise arising
from the radiation from the sun and disturbance occurring in
the sun, and cosmic noise coming from distant stars in the
galaxy.

1.4 Classification of Communication Systems

This can be done in a number of ways
a) By Types of Systems
i) Telegraphy: Transmission of written texts. Familiar
form of this is the telegram which is a written material
intended to be transmitted by telegraphy for delivering to an
address. Telex is another form, which is the message
through a network of telegraphy exchanges
ii) Telephony: A form o telecommunication set up for the
transmission of speech or other sounds

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iii) Broadcasting: A radio communication service in which
the transmission are intended for direct reception by the
general public. The most familiar forms are sound
Broadcasting and Television Broadcasting. We have
terrestrial broadcasting when the transmission from the
transmitter to the receiver is close to the earth surface.
iv) Satellite-Broadcasting occurs when satellites are used as
repeater stations between the transmitter and receiver.

b) Classification by Signal Types

There are two types: Analog and Digital
- Analog Message: A physical quantity that varies with time,
usually in a smooth and continuous version, e.g the output
signal from a microphone when speech (or an acoustic
signal) impinges on it. The light intensity variation at some
points in a T signal is also analogue in form
- Digital Message: This is an ordered sequence of symbols
selected from a finite set of discrete elements e.g. the keys
pressed on a computer.
This signal is periodic with period
T = 2п

A
T

A 7
This signal is periodic with Ts = 2 п
1.5 Digital and Analog Signals
A digital information source produces a finite set of possible
messages. A typewriter is a good example of a digital source. There
are a finite number of characters (messages) that can be emitted by
this source. The output voltage describes the information in the
sound, and it is distributed over a continuous range of values.
A digital communication system transfers information from
a digital source to the intended receiver (also called the sink).
An analog communication system transfers information
from an analog source to the sink.
Strictly speaking, a digital waveform is defined as a
function of time that can have only a discrete set of amplitude
values.
An analog waveform is a function of time that has a
continuous range of values.

Advantage of digital communication

1. Relatively inexpensive digital circuits may be used
2. Privacy is preserved by using data encryption

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3. Greater dynamic range the difference between the
largest and smaller values) is possible
4. Data from voice, video and data sources may be merged
and transmitted over a common digital transmission system.
5. In long-distance system, noise does not accumulate from
repeater to repeater
6. Errors in detected data may be small, even when there is
a large amount of noise on the received signal
7. Errors may be often be corrected by the use of coding.

Disadvantages of Digital Communication

1. Generally, more bandwidth is required than that for
analog system.
2. Synchronization is required.

c) Classification by Types of Services

There are two types: fixed service and Mobile service
- Fixed service: This occurs when the transmitter
and receiver as fixed
- Mobile Service: This occurs when either the
transmitter or receiver or both are moving at different
speeds. Mobile service can further be divided into
i) Land mobile service e.g. cellular mobile the GSM

9
ii) Aero-neurical mobile, used in communicating with
aircraft
iii) Maritime mobile, used for communicating with ships at
sea
Any of these services can involve the use of the satellite and
we then have fixed satellites, mobile – satellite or satellite-
broadcasting services.
d) Classification by Frequency of Operation
The radio frequency spectrum used for communication
purposes is divided into frequency bands. Different frequency
bands are better suited for different types of services and
transmission media that all microphones needs a diaphragms which
vibrates when sound waves produce difference of pressure between
it’s force. Functionally microphones are classified into 2 categories.
i. Entertainment/Broadcasting microphone
ii. Measurement microphones
(i) Entertainment microphones comprises of ribbons
microphone, moving coil microphone and electric
microphone which can for to the requirement of speech and
music and preferred directionally.
(ii) Measurement microphones have capability that
extend well beyond requirement both in frequency response
and in dynamic range microphones can also be classified
according to physical design a polar or directivity patter.

10
 Directing Pattern: has to do in the way air pressure is
converted to movement of diaphragm which is responsible for
different directivity characteristics.
1.6 Classification of Microphones Under Physcial Design
1. Moving Coil Microphones
This involves magnetic structure and a vibrant element
which generate voltage n response to sound wave. When a
conductor moves in a magnetic field, an e.m.f voltage would be
generated at the coil terminal, such type of microphones has a
permanent magnet on which pole places are attached.
Diagram holds coils in position pole place (soft iron) helps in
contracting the magnetic field produced across a narrow gap, the
electric signal is a replica of sound pressure falling on the
diaphragm.
Diaphtagm
Pole piece

Oil

Magnet

2. Ribbon Microphone:
This which has a thin electrical conducting ribbon which
vibrates within a magnetic field to initiate the audio signal. It used
the same electromagnetic principle as moving coil microphone.
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This type of microphone is sometimes called (velocity
microphones), the velocity from both are equal, hence this type is
used in broadcasting studios for interviews
Ribbon types of microphones are considered to have good transient
response although they tend to be overloaded by a very loud sound

3. Carbon Microphone:
This type of microphone is designed purposely for
telephones use. It has in it granules which are packed into a box
with a diaphragm on one side. When the diaphragm vibrates, the
granules are subjected to varying pressure so that the area of
contact between the granukes and the resistance of the device to
electric current will changer, although the produce a high output but
suffer from excessive noise, poor frequency and high distortion.
They cannot be used for serious recording.
4. Crystal Microphone:
This involves the use of a substance which when
mechanically deform produce a difference of voltage between its
faces. This phenomenon is known as piece-electric effect. The
material response usually use is called “rochette Salt” because it is
very sensitive and it produces a good signal when the diaphragm
vibrates. The crystal layers are trusted and a voltage in sympathy

12
with the sound produced. As with carbon microphone, crystal
microphone and are not normally used for serious recording.
5. Condenser Microphone
This type operates on electro-static principle, capable of
extremely high quality output and frequency response is relatively
more accurate at very high frequency. In this type, diaphragm is in
one plate of condenser (capacitor) and the other plate being fired as
shown in the diagram below.
Diaphragm
(Metal)

Rigid back plate

Fig. 1.3: A condenser Microphone

Movement of diaphragm creates a change in capacitance
that can be transformed into voltage as the diaphragms is inside
very light condenser microphone tends to have transient response. It
requires separate d.c power which is known as platform power
(Remote vibrate Power). Other design used a replaceable battery in
the microphone body. It possess wide frequency response, low
distortion, little internal noise an very good sensitivity.
6. Electric Microphone

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 This is a type of capacitor (condenser) microphone in which
the diaphragm has an electrostatic charge scale within it, during
manufacture. This diaphragm is a foil electret in which a permanent
static eliminates the power supply that would otherwise be required
to change the condenser. However, a small d.c battery or phatom
power in response to the sound. The distance between the plates
varies so the electrostatic filed varies producing a varying voltage at
the output terminal. Its capacitance power is about 3 times that of a
capacitor microphone of comparable dimension with low
impedance.

Resistor

Battery

Fig. 1.4: Pressure Falling on the diaphragm

7. Radio (Wireless) Microphone:
This is an improvement of a condenser microphone. A
condenser microphone is attached to a frequency modulated
miniaturized radio transmitter enclosed in the same housing. The
transmitter signal is picked up by a remote receiver, demodulated
and sent as a line signal to the mixer. The distance between the
transmitter and the receiver is about 50m or less since the
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transmitter power is small. The frequency of transmitter is between
1741 mHz and 125mHz.

Advantages of Wireless Microphone

a. Freedom from microphone cable
that gets tangled during frequent movement

Disadvantages of Wireless Microphone

a. Frequency drift which leads to
jamming of one or more microphone used or fading of
signal when one microphone is used.
b. Howling effect when a
unidirectional microphone is placed very close to a field
back loudspeaker in a public address system.

Directivity Patterns
Polar (directivity) pattern is the graphical representation of
microphones response to sounds coming from different angle. The
shape of pattern of sounds (directs/reflected) has a great bearing on
how a microphone should be placed so that an accepted balance
between direct and reflected sound is obtained to avoid “howling”
or feedback.

15
 Howling occurs when sound radiated by the loudspeaker is
placed up by a microphone, amplified and feedback into the
loudspeaker. It can be put under control by reducing the volume of
the sound going into the microphone and also by positioning a
microphone with a known polar pattern correctly.
a) Omnidirectional Pattern: This is a non - directional
pattern that responds equally to sound coming from all
directions. They pick up sound from all directions with
equal sensitivity. The diaphragm is open to the air on one
side only. Many designs of moving coil and electrostatic
mic possess this pattern they are mostly used in a situation
there many singers to microphone

90
120 60
30
150

180 0

330
210
300
240
270

Fig. 1.5: Omnidirectional Pattern

b) Bidirectional Pattern: This measure difference in pressure
(pressure gradient) between two successive points along the
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 path of the sound wave, the pattern is general by exposing
the diaphragm of a transducer to the sound field from both
sides the pattern is often referred to as a bidirectional
pattern. Sound reducing microphone from the rear and front
generate the same electrical output but exactly opposite in
phase. The live angle is generally about 1000. An example
of this is the ribbon microphone, which is used for a group
of two facing each other in discussion.

90

180 0

270

Fig. 1.6: Bidirectional Pattern

c) Cardioids Pattern
It has a heart stage and it is developed to reduced pick from
the back of the microphone. This type of microphone is commonly

17
used in the performances to cope with acoustic feedback problem
such as howling.

Fig. 1.7: Cardoid Colour Pattern

b) Super cardioid and Hyper Cardioid Pattern
They are modification and improvement of cardioid
response that are highly discriminating against indirect sound. If the
pressure and pressure gradient modes of operation are mixed, in
varying proportion a range of polar diaphragma is produced,
passing from omni directional through cardioid, then super
cardioid, to bi- directional. The design is good in reducing the
amount of reverberation received and for rejection of unwanted
noises.

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 450

Fig. 1.8: Hyper carotid polar plate

c) Switchable Polar Pattern
Microphone with switchable polar pattern is of great
advantage in professional work. They are used in maintaining
balance in a studio on some occasions without howling to move
their artistes or microphones.

Electrical Properties of Microphone

a. Sensitivity: It is a measurement of the strength of the signal
that is produced. It is expressed as output voltage per unit of
sound pressure with the microphone diaphragm, the units
are millivolt/microbe
b. Impedance: this I the combination of resistance, capacitance
and inductance values of which only the resistance
determines the basic impedance of the microphone. The unit
of this is ohms. For complete safety, it is advisable that the

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 impedance of mic should be less than a third of that of the
system which is plugged unto.

LOUDSPEAKER

ms cs Rs RG ma
µ Voice coil

Loudspeaker converts electromagnetic signal to sound operation in

frequency 20Hz -20kHz and pressure is 10-5 to 50 Pascal.
A
Rc

L2
2

Mechanical resistor

Re Le
ms A2L2 2 Rs 2 Ra
2

Electrical

Electrical Properties of Loudspeaker

1. Radiation Impedance: - This is defined as the complex
ratio of reacting force of the air on the diaphragm to the
velocity of the diaphragm (f/v) the real part ma/Az of the
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 direction reactance characteristics. A large radiation
resistance is required for efficient sound radiation.
2. Input Impedance: The electrical o/p impedance is
defined as the ratio of the applied sinusoidal voltage across
the terminal of loudspeaker to the resulting current input
impedance of a specific loudspeaker should not be too low
or too high before connecting to an amplifier. If it is too
low, you will damage the amplifier and too high enough
sound a/p will not be produced. It is very important to study
input impedance in frequency characteristics of loudspeaker
drive or system before use.
3. Directional Characteristics: Loudspeaker has
omnidirectional characteristic in low frequency regulator
where the wavelength is much larger than the diameter of
the L.S. However, radiation becomes more directional as the
frequency increases.
4. Distortion: if loudspeaker contains frequency
component other than those contain in the input signal, the
new component are called non linear distortion production.
When a sinusoidal input with frequency applied, the output
contains component which are integral multiple of this
frequency such as 2F, 3F, etc. if the cords are a1, 12,13 etc,
the nth harmonic distortion is an/a and total harmonic is (a 22
+ a23………….)1/2ei

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5. Diaphragm Material: The loudspeaker diaphragm is
expected to have a wide range of piston like motion or free
from resources as much as possible, such loudspeaker is
expected to produce any loud of audio sound without
coloration.
6. Enclosure: Diaphragm that vibrates in a free space,
produces both +ve and –ve sound pressure alternatively in
front of and behind the diaphragm. Cancellation between
the +ve abd –ve pressure occurs at low frequency since has
an omnidirectional characteristics. But when put in an
enclosure, the pressure radiated behind the diaphragm is
contained, such that the interference with one generated in
front is removed (RIDE)

LOUDSPEAKER
A loudspeaker is a combination of a loudspeaker and a box,
the system is called the driver and the system itself, the
loudspeaker.

Types of Box in Loudspeaker Design

1. The Close Box: consisting of a rigid and completely
close enclosure except for a single aperture in the front side
in which the driver is mounted.

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2. The Vented Box: consisting of an enclosure with two
apertures, one of them accommodates the driver and the
other known as the “port or vent” allows air to move in and
out of the enclosure
3. Passive Radiator Box: consisting of the vent box that
has in place of the port a “slave” diaphragm analogous to
the loudspeaker driver.

Closed box Vented box Vented box with

passive radiance

LOUDSPEAKER
A device that converts an electrical signal from amplifier
into sound. They are designed for production of audio signal which
have a frequency range of 20-20kHz and pressure of 10 to 50
Pascal.
Driver is a transducer mechanism without a structural radiation and
such as horn, baffle or enclosure.
Loudspeaker are used in homes, car, stereo, V, radio receivers, toys,
electric musical instrument, broad casting stations recording studios
and concert hall etc.

Types of Loudspeaker
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Loudspeaker can be classified in different ways.
In terms of radiation;
a. Direct
b. Horn-loaded

In terms of driving element

a. Magnetic: (i) dynamic (moving coil, ribbon), etc (ii) moving
armature (iii) magnetic structure
b. Electro static (condenser)
c. Piezo electric (crystal, ceramic polymer)
d. Ionophone: This operates on interaction between ionic
plasma and the surrounding corona effect with moving
mechanism system. (2) Air jet
In terms of reproduction range:
i) Low-frequency (woofer, subwoofer)
ii) Mid frequency (midrange, squawker)
iii) High frequency (tweeter, super tweeter)
iv) Full range

In terms of diaphragm shape;

i) Cone (straight , parabolic, flared
ii) Dome
iii) Flat: the most commonly used are:
1. Dynamic (moving coil) direct – radiation

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2. Horn loudspeaker in which the horn is a curved tube usually
made or metal plastic or wood, with a gradually increasing.
Cross Sectional Area. It has two ends, one adjacent to
diaphragm is called the THROAT and other radiates sound
into the air is called MOUTH. Some Horn Loudspeaker has
phase plugs I front of diaphragm for effective focusing of
wave front of the radiated sound in different area of
diaphragm

Horn increases the radiation efficiency and changes the

directivity of the sound. This type of loudspeaker so mostly used
for sound reinforcement in large rooms and in open spaces because
of high efficiency which ranges from 10-100 tunes (10 to 20db) and
directly for high sound production. It should be design with cone so
as to avoid colouration. The sound distortion produced around the
throat

Phasing of Loudspeaker
It is essentially important that when more than one
loudspeaker is used, the Loudspeaker should be operated in phase
with each other. That is, if the cone of one speaker moves outward,
more or less cancellation of sound output will occur. Therefore, a
less of each speaker is wide at time of installation to make certain
that the system operate in phase. The simplest way to check speaker

25
housing is to connect a dry cell across the voice coil terminals and
observe whether the coil is more in or out. The acoustics of a
Loudspeaker or of an array of speaker in a public address system is
an extensive topic that cannot be covered presently. Speakers are
mounted in baffles (sound board) the tend to isolate sound radiation
from the rear is 1800 out of phrase. Initially, with the front
radiation more or less in phase with the front radiation. Special
speakers cabinet are often utilized to bring

26
 CHAPTER TWO
2.1 Modulation Processes in a Communication System
Modulation is one of the important signals processing
technique undertaking at the transmitter stage. To be able to
transmit the signal efficiently, the information bearing signal
(baseband signal) must be processed in some manner before
sending the signal into the transmission medium.
Most commonly, the base band signals must be shifted to
higher frequencies for efficient transmission. It is a well-known
theory of electromagnetic radiation and the efficient radiator of an
electric energy (that is the antenna or aerial) must have a dimension
of the order of a wavelength in size.
Take, for instance, an audio signal of f = 1 kHz (or п =
300km). This would require a radiator of about 100 – 200 km in
size if it were to radiate this signal efficiently. However, by
translating the baseband to a higher frequency of say, 1MHz, the
size of an efficient radiator is reduced by a factor of 1000, down to
about 100-200m, which is more practicable. The use of higher
frequencies also provides wider bandwidth which can
accommodate other baseband signal for increased information
transfer.
Modulation process involves varying the amplitude,
frequency or phase (or combination of these) of the high frequency
sine-wave carrier, in accordance with the information to be

27
transmitted. This is referred to as sine were or continuous wave (c-
w) modulation.
Digital modulation occurs when the carrier is a pulse of high
bit – rates instead of a high frequency continuous carrier wave. The
pulse parameters suitable for modulation include amplitude,
duration or width and pulse position referred to a PAM, PDM, (or
PWM) and PPM systems respectively.

2.2 Amplitude Modulation

Amplitude modulation is defined as a system of modulation
in which the amplitude of the carrier is made proportional to the
instantaneous amplitude of the modulating (baseband) signal
Let ec = Ee sin wct be the carrier signal (2.1)
ec = Ee sin wct be the modulating signal (2.2)

e = A sin wct be the modulated signal (2.3)

Then, by the definition above,

A = Ec + kem (2.4)

A = Ec (2.5)

We may put k =1, for simplicity, then

28
A = Ec (2.6)

A = Ec 1+ k msin wmt (2.7)

Where m Em/Ec modulation index (2.8)

From equation (2.3), we have,

e = A sin wct
e = Ec (1 + m sin wmt) sin wct (2.9)
mE c cos Wc - Wm t mE c cos w c  w m t
e = Ec sinwct + - (2.10)
2 2

Using the trigonometric relation

Sin x sin y = ½[(cos(x-y)cos(x + y)]

(2.11)

R2/R1

Terminologies Regarding
Amplitude Modulation
29
(1) Modulation index (Ma): It is defined as the ratio of the
amplitude of the modulating signal to the amplitude of the

carrier i.e. ma =

Where Vm = Amplitude of the modulating signal (volt)

Vc = amplitude of the carrier (volt)

(2) Bandwidth: Is the difference between the uppr sideband to

the lower sideband.
i.e. Bw = FUsb – FLSB
It can also be defined as twice the frequency of the
modulating signal.
Bw = fc + fm – (Fe – fm)
Bw = fc + fm – (Fe – fm)
= 2fm

Example 2.2: An AM wave is represented by the expressions Vm =

10 sm 103 t and Vc = 30 cos 10 6 t. = Find (i) Amplitude (ii).
Modulation (iii) Index Frequency of the sidebands (iv) Bandwidth

Solution
Vm = 10sm 103t Vc = 30 cos 106
(i) Vm = 10v and Vc = 30v

(ii) Modulation index (ma) =

30
(iii) Frequency of the sidebands
Wm = 103 Wc = 106
2μfm = 103 2μfc = 106

Fm =

(iv) Bandwidth = 2fm

= 2 x 159
= 318Hz

31
 CHAPTER THREE
3.1 Radio Receivers
The primary requirement for any communication receiver is
that it has the ability to pick the desired signal from among (1000)
thousands of others present in free space and to provide sufficient
amplification to recover the modulating signal.

- Selectivity: Refers to the ability of a receiver to

pick a signal of a desired frequency while rejecting those on
closely adjacent frequencies.
- Sensitivity: Of a receiver refers to the receiver
ability to pick weak signals, sensitivity is primarily a
function of the overall receiver gain.
- Gain: Is a factor by which an input signal is
multiplied to produce an output signal. In general, the
higher the gain of a receiver, the better its sensitivity.

3.2 Superheterodyne Radio Receiver

Fig. 3.2 Shows a general block diagram of a superhet receiver

32
 Fig. 3.2: Block diagram of a suprhet receiver
The basic process of superhet receiver is to convert all
incoming signals to a lower frequency called the Intermediate
Frequency (I.F) where a single set of amplifier can be used to
provide a fixed level of sensitivity and selectivity. Most of the gain
and selectivity of superhet receiver is obtained in the I.F amplifier.
The heart of a superhet receiver is the mixer which produces the
frequency translation of the incoming signal down to the I.F
intermediate frequency. The incoming signal is mixed with the
local oscillator signal to produce this conversion.
The antennas pick up the weak radio signal and feed into the
radio frequency amplifier.
The purpose of R.F amplifier is to provide some initial gain
and selectivity. Some receivers do not used R.F amplifier but
instead the antenna is connected directly to a tuned circuit at the
input of the mixer. The input tuned circuit (tuned circuit provide
selectivity) will provide the desired initial selectivity.

33
 The mixer receiver an input from local oscillator and the
output of R.F amplifier. The output of mixer will be the input
signal, the local oscillator signal as well as there sum and difference
frequencies. Usually a fined circuit at the output of the mixer select
the difference frequency which is called the intermediate frequency.
Fif = fi – f0, if fi >f0.
The output of the mixer is a signal at the I.F containing the
same modulation that appears on the input of I.F signal. This signal
is amplified by one or more I.F amplifier stages, and most of the
receiver gain is obtained in these I.F stages.
Another important circuit in the superhet receiver is the
A.G.C the output of the demodulator is the original modulating
signal whose amplitude is directly proportional to the amplitude of
the received signal. The recover signal which is usually A.C is
rectified and filtered into a D.C voltage by the A.G.C circuit. The
D.C voltage is fed back to the I.F amplifiers to control the gain of
the receiver. The purpose of the A.G.C circuit is to help maintain a
constant output voltage level over a wide range of R.F input signal
levels.
The output of the demodulator is passed into an audio
amplifier which is used to drive the loudspeaker. A speaker convert
incoming A.C signal back to speech.

34
Example
A parallel L.C Tuned circuit has a coil of 3µH and
capacitance of 75 pF the coil resistance is 10 Calculate (a) the
oscillating frequency of the tuned circuit (b) the quality factor of
circuit and hence what is the bandwidth of the selectivity curve.

Q=

BW =

Fr =

Solution

Parameters
Inductance 3H = 3 x 10–6H
Capacitance = 75pf = 75 x 10–12 F
Resistance = 10
(a) The oscillating frequency which is the same thing as
resonant frequency (fr):

Fr =

35
= 1.06 x 107 Hz

(b) Quality factor Q = X/R

XL = 2fL
= 2x 3.142 x 1.06 x 107 x 3 x 10– 6
= 200
Q = XL/R
= 200/10 = 20
BW = Fr/Q

= 0.53MHz

Exercise
Determine the intermediate frequency of a superhet if the
incoming signal is 100 MHz and local oscillator frequency is 110.7
MHz.

36
 CHAPTER FOUR
4.0 Meaning of Transmitter
Radio transmitter
The transmitter is an electronic unit that access the
information to be transmitted and converts it into a radio frequency
signal capable of be transmitted over long distances.
4.1 Function of Transmitter
Every transmitter has three basic functions
1. The transmitter must generate a signal of correct
frequency at desire p in spectrum e.g. radio Kwara operates
at 99.00MHZ while A.M station operates at 612KHZ.
2. It must provide some form of modulation that causes the
information signal to modify the carrier signal e.g. A.M. or
F.M.
3. It must provide sufficient power amplification to ensure
that the signal level is high enough so that it will carry over
the desired distance.

37
4.2 Block Diagram of AM

Fig 1.2 shows an A.M transmitter with an oscillator generating

the final carrier frequency
In most applications, the oscillator will be crystal
transmitter operate on an assigned frequency and the crystal
provide the best way to obtain the desire frequency with good
stability (we used crystal because of frequency stability).
The carrier signal is then fed to a buffer amplifier whose
primary purpose is to isolate the oscillator from the remaining
power amplifier stages.
The buffer amplifier usually operates CLASS A and the
main purpose of this amplifier is to prevent load changes from
causing frequency variation in the oscillator.
The signal from the buffer amplifier is applied to a driver
amplifier which is a CLASS C amplifier designed to provides an
intermediate level of power amplification.
The purpose of this circuit is to generate sufficient output
power to drive the final power amplifier stage.

38
 The final power amplifier stage also operates CLASS C at
very high power. The actual amount of power depends upon the
application e.g. A.M radio stations operate at higher power of 250
Watt, 500 Watts, IKW, 5Kwatts or 50Kwatts.
The voice signal is then fed to some form of speech-
processing unit, circuit, this speech processing unit is used for
filtering and amplitude control, this helps to minimize the
bandwidth occupied by the signal e.g. the voice frequency is
between 20Hz – 20KHz. After the speech processor a driver
amplifier is used. This stage increases the power level of the signal
so that it is capable of driving the high power modulation amplified.
In the A.M transmitter of (Fig. 1.0) high level modulation is used
because the power output of the modulation amplifier is usually
operate CLASS AB or CLASS B push pull to achieve such power
level.
F.M Operate between (88-108) MHZ
A.M Operate between (525-1605) KHZ
NBC (National Broadcasting Commission) they were the one give
out license.

4.3 F.M Transmitter Fig. 1.3 shows the typical configuration

for an F.M transmitter the indirect method of F.M
generation is used.

39
 Carrier
Antenna
oscillator

Buffer Phase Frequency Driver Final/power

oscillator Multiplier Amplifier Amplifier

Crystal

Audio Speech
Amplifier Processing

Microphone

Fig. 4.3 A typical FM transmitter using indirect FM with a phase

modulator. A stable crystal oscillator is used to generate the
carrier signal and buffer amplifier is used to isolate the carrier
oscillator from the remainder of the circuit.

The carrier signal is then applied to a phase modulator as

shown in fig 4.2 the voice input is amplified and processed to limit
the frequency range and prevent over deviation. The output of the
modulator is the desired F.M signal.
Most F.M transmitters are used in the VHf (30 - 300) MHz
and UHF (300- 3000) MHz range and crystals are not available to
generate those frequencies directly.

40
 To achieve the desire output frequency, on or more
frequency multiplier stages are used. A frequency multiplier is
CLASS C amplifier whose output frequency is some integral
multiple of the input frequency e.g. output frequency of a multiplier
is to increase the frequency by factor of 2, 3, 4, or 5.
The frequency multiplier do not only increases the carrier
frequency to the desired output frequency but it also multiplies the
frequency deviation produced by the modulator.
After frequency multipliers a CLASS C driver amplifier is
used to increase the power level sufficiently to operate the final
power amplifier. The final power amplifier also operates in CLASS
C.
Most F.M transmitters operate at less than 100watts. All
F.M transmitters in VHF and UHF range operate several 100watts
power level and make use of vacuum tubes.
The antenna is used to radiate the signal from the final
power amplifier into the free space inform of electromagnetic
waves.

41
4.4 Black & White of Television Transmitter (Monocrome)
The block diagram of typical TV transmitter is illustrated in
Fig. 4.6
Antenna radiate electromagnetic wave into the space
The video signal is generated from the video camera or TV
camera and the output is amplified by pre amplifier stage. The
output from this stage is then passed to the video processing unit
where frequency responds compensation, camera cable
equalization, synchronizing pulses, D.C restoration, gamma
correction take place. This video signal is then again amplified to a
suitable level become it is fed to the video modulating amplifier.
The video modulation amplifier is directly coupled to the picture
modulated amplifier. The audio signal from the microphone is
42
amplified in a pre-amplifier stage before it frequency modulates the
sound carrier generated in a crystal control oscillator.
The output of modulated amplifier is fed to a chain of
frequency multipliers to raise the carrier frequency to the desired
value. This is follow by a radio frequency amplifier to produce the
desired carrier power to feed to the duplexer.
The duplexer combines the sound carrier with the
modulated picture carrier after it has been passed through Vestigial
Sideband (VSB) filter, the job of the duplexer is to feed the separate
sound and vision signals in a common output feeder line in such a
way that there is no risk of the vision signal getting down back into
the sound transmitter, or of the sound signal getting back down the
line of vision transmitter; this enables a single antenna to be used in
radiating both sound energy and video energy.
Note that, the sound transmitter is usually frequency modulated, but
at times some station uses A.M. The vision transmitter is by
amplitude modulation technique.

4.5 Field Strength of TV Signal

The Effective Isotropic Radiated Power (EIRP) of a TV
transmitter is equal to the transmitter power multiplied by the loss
factor for the transmission lines and coupling networks and antenna
directive gain.
EIRP = Radiated Power x Loss factor x Antenna gain

43
 The net electric field strength of a TV signal using space
120hrht P
wave communication is given by E = v/m
d2
Where P is the EIRP wd2 in watt
ht is the height of transmitting antenna in meter
hr is the height of receiving antenna in meter
is wave length in meter
d is the distance in meter
E is the electric field strength
Example
Calculate electric field strength at a receiving point of a TV
transmitter if the radiating power is 100KW the height of
transmitting antenna is 300m, the height of receiving antenna is
10m and the distance between the transmitting and receiver is
50km. If the transmitting frequency is 100kHz.

Solution
d = 3.0 x 103m
hr = 10m
ht = 300m
p = 100kw = 100 x 103 watts
d = 50km or 50, 000m:. d2 = 2.5 109
To calculate the electric field strength (E) is given as

E=

44
E=

E=

E=

= 4.8 10–7 v/m

4.6 A Television Receiver
ATV receiver operates on the principle of superhet. Its main
requirement is to receive signal over a wideband width.
Fig. 1.8 Shows a simplified block diagram of a black & white
(monochrome) receiver.

Fig. 4.6 Black & White (Monochrome) Receiver

- The tuner, which commonly uses varactor diodes,
converts the incoming radio frequency signal to a constant

45
 Intermediate Frequency (I.F) for the vision and sound
carriers e.g. 39.5 MHz for vision and 33.5 MHz for sound, it
also matches the antenna to the input of the receiver in other
to achieve the best performance.
- The I.F signal containing both sound and picture is then
amplified to raise the signal level to a detectable value.
- The Automatic Frequency Control (AFC) provide a
sufficiently large pull in voltage range to ensure that any
radio frequency detuning, arising from in accuracies of
initial setting, e.g. temperature effect, aging etc, do not
affect the oscillator frequency beyond a particular value.
- The vision detector extracts the video signal from the
modulated carrier.
- The video output detector extracts the video signal from
the modulated carrier signal into a voltage level large
enough to drive the picture tube.
- The AGC circuit ensures the receiver provides a drive
voltage of the correct amplitude for the picture tube over the
range of antenna input voltage.
- The sound channel comprises of the sound I.F amplifier,
FM detector and the Audio frequency amplifier which
drives the loudspeaker.
- The synchronizing pulses are removed from the video
signal in the sync separate stage.

46
- The line sync pulses are amplified to the line time base
circuit to lock it with the correct frequency. Its output it fed
to the detector coil constructed around the neck of the vision
tube to achieve horizontal scanning.
- The frequency of the time base oscillator can be
adjusted by horizontal hold circuit usually found at the back
of T.V receiver.
- The field time base is responsible for the vertical
deflection. It frequency can be adjusted by the vertical hold
control to prevent the picture from running up and down the
screen.
- The picture tube in a T.V receiver is essentially a CRT
that provide a display of the transmitter picture, the display
is produce by modulation. The beam current is produced by
the CRT, the screen of the tube consist of photo sensitive
plate having the same number of picture element (pixels) as
that of T.V camera tube hence, each picture element in the
camera is connected to the corresponding picture element in
the picture tube.
- The scanning of the photo sensitive plate in the camera
must, therefore, be synchronized to the scanning in the
picture tube.

4.7 Radio Wave Propagation

47
 Table 2.0 Frequency Bands within the Radio Spectrum
S/N NAME FREQUENCY APPLICATION
1 Extremely Low 30HZ – 3KHz Submarine
Frequency (ELF) comm.
2 Very Low Freq (VLF) 3HZ – 30KHz Radio telephone
3 Low Freq (LF) 30KHZ – 300KHz Radio telephone
4 Medium Freq (MF) 300 – 3000KHz Radio
5 High Freq (HF) 3 – 30MHz Radio
6 Very High Freq (VHF) 30 – 300MHz Television, radio
GSM
7 Ultra High Freq 300 – 3000GHz Television radio
military satellite
8 Super High Freq 3 – 30GHz Satellite comm.
Radar system
9 Extremely High Freq 30 – 300GHz Radar System,
(EHF) Satellite
10 Micro Wave Region Transmitting T.V
information

48
 CHAPTER FIVE
5.0 Antenna
An Antenna
Is a circuit element that provide radiated electromagnetic
wave when used as a transmitting antenna as a receiving antenna
electromagnetic wave cut through it for use by the receiver.
The receiving antenna transfer energy i.e. electromagnetic
wave from the atmosphere to its terminal with the same efficiency
with which it transfers energy from transmitter into the atmosphere.

5.1 Polarization
Is a measure of degree of orientation of an electric field in
an electromagnetic wave.

Fig. 5.1: Propagation direction for E & H fields

Electric field is always at angle 900 to magnetic field.

Electric field E & Magnetic field H.
49
 In electromagnetic wave the electric field E and the
magnetic field H are at right angles to one another. The
direction of propagation is perpendicular to the plane
containing both the E and H filed.
 If an electric field is vertical in an electromagnetic wave
then the wave is said to be vertically polarized thus, a
vertical antenna will transmit a vertically polarized wave.
This is because the direction of electric field is vertical. On
the other hand, if the antenna is horizontal then the
transmitted electromagnetic wave is horizontally polarized.

5.2 Half Wave Di-Pole Antenna

Any antenna having a physical length i.e. half wave length
of the applied frequency is called a half wave dipole antenna.
Example 1: If the frequency of the radiated electromagnetic wave
is 2MHz.
Calculate the appropriate half wave dipole antenna for this signal
C/f………...2.1
Where C = velocity of propagation (m/s)
F = is the frequency of the wave (HZ)
= wave length (m)
C = 3.0 x 108 m/s
F = 2 MHz = 2 x 106 Hz

= = 1.5 x 102m

50
 /2 which is half wave length dipole
= 0.7 x 102m = 75 or 1.5 x 102/2 = 0.75
= 0.75 x 102 m = 75m.

5.3 Current Voltage Characteristic of /2 Dipole

The antenna shown in fig 5.3 is composed of two quarter wave
(1/4) sections
Fig 5.3
λ/4

(v)

Fig 5.3 is the half wave antenna with excitation at the centre of /4
and open at both ends when the diagram of Fig 5.3 is developed
you generates a /2 dipole as
Illustrated in fig 5.4

IMAN
Current

Fig. 5.4 Basic half-wave

/2
dipole antenna
Fig 5.4 Illustrates both the current and voltage characteristic of a /2
dipole with the maximum current at the centre of the dipole and is
reduces to zero (0) at the extreme and however the voltage shows

51
that it is maximum at the two ends though alternating but zero at
the centre.
Example: Determine the physical length of a /2 dipole used in 150
MHz communication system.
= c/f
Where C = velocity propagation (m/s)
F = Frequency of the wave (Hz)
= Wave Length (m)
= 3.0 x 108 = 2m
150 x 106
/2 which is half –dipole length
= 2/2
= 1m

5.4 Radiation Resistance

The portion of an antenna input impedance that is (i.e.).
Result of power radiated into the space is called the radiation
resistance normally as Rr. Note that Rr is not the resistance of the
conductors that form antenna, it is simply an effective resistance
that is related to the power radiated by the antenna.
The relationship between the power radiated (p) by the
antenna and the antenna current (I) with the radiation resistance
(Rr).
Rr = p/12 in watts ……2.3

52
Where P = total power radiated from the antenna (w)
I = effective rms value of antenna current at the feed point (A)
Rr = radiation resistance ()
Irms = Ip
2
i = Asinwt
w = 2f
f = w/2
e.g i = 10sm 106 t
f = 106
2

5.5 Antenna Gain

This half wave dipole antenna has a gain with respect to the
theoretical isotropic radiation. And this isotropic radiator is an
hypothetical radiator that radiate energy equally in all direction.

Fig 5.5 Isotropic radiator

53
 The ratio of power per unit area of a given radiator to
the same amount of power per unit area of an isotropic
radiator is called the power gain of the antenna.
 The gain of an antenna is provided with respect to an
isotropic radiator often expressed in dBi
 Example: A half wave dipole antenna gain can be
express as 2.15 dBi if an antenna gain is give in dBi with
respect to a dipole it is express in dBd, th occur somewhat
less often than dBi in an antenna gain.
The amount of power received by an antenna through free
space can be predicted by the following equation.
Pr = Pt Gt Gr 2 ……………….2.4
16 2 d2
Where, Pr = power received (w)
Pt = Power transmitted (w)
Gt = transmitting antenna gain (not dB)
Gr = receiving antenna gain (not dB)
= wave length (m)
D = distance between antenna (m)

EXAMPLE: 2/2 dipole are separated by 50kw. They are aligned

for optimum reception. If the gain of the dipole is 2.15dB and the
transmitter fixed its antenna with 10watts at 144MHz. calculate the
power received.

54
Solution
Given Pt = 10w
F = 144 MHz = 144 x 106 Hz
d = 50km = 50 x 1000m
Gr = 2.15dB = 10 log102.15
G(dB) = 10log10k
2.15 = 10log10k
0.215 = 10log10k
K = 100.215 = 1.64
= 3.0 x 108 = 2.08m
144 x 106
Pr = = Pt Gt Gr 2
16 2 d2

= 10 x 1.64 x 1.6 x (2.08)2

16 x (3.142) 2 x (50, 000) 2
= 116.36
3.9 x 1011
= 2.98 x 10-10w
P (dB) = 10log10k
V (dB) = 20log10k1
P  V2

55
ASSIGNMENT
Design a 92 MHz dipole antenna if the radiated power is
10w and the antenna gain is 10dB. Determine the receiving power
if the distance between the transmitter and receiver is 100km.

5.6 Radiation Pattern of An Antenna

The radiation pattern of an antenna is a indication of
radiated field strength around the antenna.
The radiation pattern or polar diagram for the /2 dipole
antenna is illustrated in Fig 5.6a
Fig 5.8 Consist of a /2 dipole and a non-driven /2 element (not
electrically connected located /4 behind the dipole).
The non-driven element is also termed a parasitic element
because it is not electrically connected. The energy from parasitic
element travelling toward the driven element reaches it in phase and
causes a doubling of energy propagated in that direction. This effect
is shown by the radiation pattern in fig 5.8 (b)

5.7 Types of Antenna

5.7.1 Yagi-uda Antenna
The Yagi-uda antenna consists of a driven element and two
or more parasitic element and it is named after the two Japanese
scientist who were instrumental in its development, A two parasitic

56
elements consisting of a director and a reflector is illustrated in Fig
3.9.1

Fig 5.7.1 Yagi-Uda antenna

A director is a parasitic element that serves to direct
electromagnetic energy because it is in direction of the propagated
energy with respect to the driven element. The radiation pattern is
as shown in fig 5.7.1 (b). Note that two side lobes of the radiation
energy results in the radiated pattern. They are generally undesired
as is the small amount of the reverse propagation. The difference in
gain from the forward to the reverse direction is define as the front
to back ratio e.g. the pattern of fig 5.7.1 (b) has a forward gain of
12dB and 3dB in the reverse direction, therefore F/B ratio is 12dB -
-3dB = 15dB. They are often use as high frequency transmitting
antenna and as VHF/UHF television receiving antenna.

57
Exercise: Design a four element yagi-uda antenna operating at a
frequency of 100MHz if the velocity of propagation is assumed to
be 3.0 x 108 m/s. Draw the design antenna.

5.7.2 Rhombic Antenna

A rhombic antenna consists of a pair of wires in the form of
a horizontal rhombus, supported on poles, one end is energized and
the other is terminated in a resistor.

Fig. 5.7.2: Rhombic Anternna

The arrangement may be regarded as a transmission lines

which has been open out to allow the system to radiate. The value
of the terminating resistance is such as to effectively match the line,
so that current distribution along the wires approximates to
traveling wave. The radiation pattern of a rhombic antenna can be
58
obtain by finding the resultant of a radiation pattern of four (4)
elementary dipoles each dipole producing its own lobe as illustrated
in fig 3.9.2

5.8 Application
1. Wildly use for T.V transmission and it can be cascaded
either serially of parallel.
2. It can be used as a multi steerable antenna
3. It can be used for transatlantic telephone system.

59
 CHAPTER SIX
6.0 Mechanism of Propagation of Radio Wave
There are four basic methods of getting a radio wave from
the transmitting end to the receiving end. These are;
i. Ground Wave
ii. Space Wave
iii. Sky Wave
iv. Satellite Communication

6.0.1 Ground Wave Propagation

A ground wave is a radio wave that travels along the earth
surface. The ground wave is sometimes referred to as a surface
wave. The ground wave. The ground wave must be vertically
polarized (electric field vertical) fig 6.0.1 illustrates the surface
wave mechanism.

The wave must be vertically polarized because the earth

would short out the electric field if horizontally polarized. If the

60
earth surface is highly conductive the absorption of wave energy
will take place and it is referred to as ATTENNATION (loss).
Ground wave propagation is much better over water
(especially salt water) than a very dry (poor conductivity) desert
TERRAIN. Ground waves are not very effective at frequencies
about 2MHz, but they are very reliable communication link than
sky wave propagation. Ground wave propagation is the only way to
communicate into the ocean with sub-marines.
ELF propagation utilized ground wave mechanism.

6.0.2 Space Wave Propagation

Occurs in the region of about 16km above the earth surface
and there are two types of space wave propagation.
i. The direct wave
ii. The ground reflected wave
This mechanism of propagation is illustrated in fig 6.0.2

61
6.0.3 Space Wave Propagation
The direct wave is by far the most widely mode of antenna
communications. The propagated wave is direct from transmitting
to the receiving antenna and it does not attenuate rapidly.

The direct space waves thus have one severe limitation and
this limitation is called line of sight (LOS) transmission distances
thus, the antenna high and the curvature of the are the limiting
factors.
The reflected wave shown in fig. 6.0.2 can cause reception
problem. If two received components are not in phase the resultant
signal will fade out. This can also result when both a direct and
ground wave are received or when any two or more signal paths
exist. A special case is Ghosting T.V reception.

6.0.4 Sky Wave Propagation

One of the most frequently used method of long distance
transmission is by the use of the sky wave. Sky waves are those
waves radiated from the transmitting antenna in a direction that
produces a large angle with reference to the earth surface. The sky
wave has the ability to strike the ionosphere, be refracted from it to
the ground, strike the ground and be reflected back toward the
ionosphere and so on.

62
Fig. 6.0.3 illustrates the sky wave propagation mechanism.

Fig. 6.0.3 Sky Wave Propagation Mechanism

The transmitted wave leave the antenna at point A is

refracted from the ionosphere to point B, is reflected from the
ground at point C, and again refracted from the ionosphere at point
D, before finally arriving at the receiving antenna at point E.

6.1 Ionospheric Layers

The ionosphere is composed of three layers designated from
the lowest level to the highest level as D, E and F respectively.
Fig. 6.1

Fig. 4.1 Layers of the ionosphere

63
6.1.1 The D – Layer
This layer ranges from about 40km to about 88km above the
earth surface.
Ionization in the D-layer is low because it is lowest region
of the ionosphere and most far from the sun.
The layer has the ability to refract signals of lower
frequencies. High frequencies pass right through it but are partially
attenuated in doing so. After sunset, the D-layer disappears because
of the rapid recombination of it’s ions.
6.1.2 The E-Layer
This layer limits are from approximately 88km to 144km
above the earth surface the rate of ionic recombination in this layer
is rather rapid after sunset and is almost complete by midnight. This
layer has the ability to refract signals of a higher frequency than
those refracted by the D-layer. The E-layer can refract signals with
frequencies as high as 20MHz.

6.1.3 The F-Layer

The F-layer exist from about 144km to 400km. During the
day time the F-layer separates into Two (2) layers namely
i. The F1-layer
ii. The F2-layer

64
 The ionization level in these layers is quite high and varies
widely during the cause of a day. At noon, this portion of
atmosphere is closet to the sun, and degree of ionization is
maximum. A fairly constant ionized layer is present at all times.
The F-layer are reasonable for high frequency, long distance
transmission due to refraction for frequencies up to 30MHz. In the
night F1 & F2 layers merges.
The relative distribution of the ionosphere layers with
reference to radiation from sun is illustrated in Fig 4.1.3

Fig. 6.1.3 Layers of the ionosphere with reference to the sun

D – layer -40-88km
E – layer -88-144km
F1 layer 144-248km
F2 layers 400km
With the disappearance of the D and E layers at night
signals normally refracted by these layers are refracted by the much
higher resulting in greater skip distances at night.

65
 The layers that form the ionosphere undergo considerable
variation in altitude, ionization density and thickness, due primarily
to very the degrees of solar activities. The unit of ionization is
electron per M3.

6.2 Effect of Ionosphere on the Sky Wave

The ability of the ionosphere to return a radio wave to the
earth depends on the ionization density, the frequency of the radio
wave and the angle of the transmission. The refractive ability of the
ionosphere increases with the degree of ionization.

The degree of ionization is greater during the day time than at

night.

6.3 Critical Frequency (CF)

If the frequency of a radio wave being transmitted vertically
is gradually increased a point is reached where the wave is not
refracted sufficiently to curve its path back to earth. Instead, these
waves continuing upward to the next layer, where refraction

66
continues. If the frequency is sufficiently high, the wave penetrates
all layers of the ionosphere and continues out into space. The
highest frequency that will be returned to earth when transmitted
vertically under given ionospheric conditions is called the
CRITICAL FREQUENCY.

6.4 Maximum Usable Frequency (MUF)

There is a best frequency for optimum communication
between any points at any specific condition of the ionosphere. The
highest frequency that is returned earth at a given distance is called
the maximum used from the relationship between the maximum
usable frequency and cal illustrate in figure 6.4.

Fig. 6.4 Relationship of frequency to critical angle

6.5 The Optimum Working Frequency (OWF)
Is the one that provides the most consistent communication
and therefore the best one to use.
Optimum working frequency is defined as 85% of the
maximum usable frequency.
67
 CHAPTER SEVEN
7.0 Introduction to Signals
What is a Signal?
A signal is formally defined as a function of one or more
variables that conveys information on the nature of a physical
phenomenon.
When the function depends on a single variable, the signal
is said to be one dimensional. E.g.; Speech signal (Amplitude varies
with respect to time)
When the function depends on two or more variables, the
signal is said to be multidimensional. E.g.; Image – 2D (Horizontal
& vertical coordinates of the images are two dimensional)

What is a System?
A system is formally defined as an entity that manipulates
one or more signals to accomplish a function, thereby yielding new
signals.
i/p signal System o/p signal

e.g.; In a communication system the input signal could be a speech

signal or computer data. The system itself is made up of the
combination of a transmitter, channel and a receiver. The output
signal is an estimate of the information contain in the original
message.

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 Message Transmitted Received Estimate of
Signal signal Transmitter signal Channel signal Receiver message

The examples of other systems are control systems,

biomedical signal processing system, audio system, remote sensing
system, microelectro mechanical system etc.

7.1 General signal characteristics

(a) Multichannel & multidimensional signals:
A signal is described by a function of one or more
independent variables.
The value of the function (dependent variable) can be real
valued scalar quantity, a complex valued quantity or perhaps a
vector.

Real valued signal x1 (A) = A sin3πt

Complex valued signal x2(A) = Ae j3 πt = A cos3πt + jAsin3πt
In some applications, signals are generated by multiple
sources or multiple sensors. Such signals can be represented in
vector form and we refer such a vector of signal as a multichannel
signal.

E.g.; In electrocardiography, 3-lead & 12-lead electrocardiograms

(ECG) are often used, which result in 3-channel & 12-channel
signals.

69
One dimensional: If the signal is a function of a single independent
variable, the signal is called 1-D signal.
Amp e.g.; Speech signal

Time

(b) Multidimensional signal:

Signals can be functions of more than one variable, e.g.,
image signals (2D), Colour image (3D), etc.

7.2 Classification of signals

Broadly we classify signals as:
1. Continuous-time signal: A signal x(t), is said to be
continuous-time signal if it is defined for all time t,
where t is a real-valued variable denoting time.
Ex: x(t) = e-3tu(t)

2. Discrete-time signal: A signal x(n), is said to be discrete-

time signal; if it is defined only at discrete instant of time,

70
 where n is an integer-valued variable denoting the discrete
samples of time. We use square brackets [·] to denote a
discrete-time signal.
Ex: x[n] = e-3nu[n]

X[n] e-3
e-6

e-9

3. Even and odd signal: A continuous-time signal x(t) is

even, if x(-t) = x(t) and it is odd if x(-t) = -x(t). A
discrete-time signal x[n] is even if x[-n] = x[n] and is odd
if x[-n] = -x[n].

Example 1: x(t) = t2-40 is even.

Example 2: x(t) = 0.1t3 is odd.

Example 3: x(t) = e0.4t is neither even nor odd.

Fig. 7.1: Illustrations of odd and even functions. (a) Even; (b)
Odd; (c) Neither.

Decomposition Theorem
Every continuous-time signal x(t) can be expressed as:

71
x(t) = 𝑦 𝑡 + 𝑧 𝑡
where 𝑦 𝑡 is even, and 𝑧 𝑡 is odd.
𝑦 𝑡 = 𝑥(−𝑡) 2

𝑧 𝑡 m = 𝑥 𝑡 − 𝑥(−𝑡) 2
and

4. Periodic & non-periodic signals: A continuous time

signal x(t) is periodic if there is a constant T > 0, such
that x(t) = x(t + T ), for all t. A discrete time signal x[n] is
periodic if there is an integer constant N > 0, such that
x[n] = x[n + N ], for all n. Signals do not satisfy the
periodicity conditions are called non-periodic signals.

Note: The smallest value of T (N) that satisfies the above equations
is called fundamental period
Example: Determine the fundamental period of the following
signals:

(a) ej3πt/5
(b) ej3πn/5

Solution:
(a) Let x(t) = ej3πt/5. If x(t) is a periodic signal, then there
exists T > 0 such that x(t) = x(t + T ).
Therefore,
x(t) = x(t + T )
j3πt/5 j3π(t+T)/5
· e =e
j3πT/5
· 1=e
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 j2kπ j3πT/5
· e =e

=T=

(b) Let x[n] = ej3πn/5. If x[n] is a periodic signal, then

there exists an integer N > 0 such that x[n] = x[n + N ].
So,
x[n] = x[n + N ]

j3πn/5 j3π(n+N)/5
· e =e
j3πN/5
· 1=e
j3πN/5
· ej2kπ = e
T = 10 (k = 3)
·

5. Energy signals and power signals: In electrical systems,

a signal may represent a voltage or a current. Consider a
voltage v(t)developed across a resistor R, producing a
current i(t).
The instantaneous power dissipated in this resistor is
defined by the total energy of the continuous-time signal
x(t) as T/2
𝐸 = lim

= l

and its time-averaged, or average, power as

73
 1
P = lim

From above equation, we readily see that the time-averaged power

of a periodic signal
x(t) of fundamental period T is given by

1
P=
T

The square root of the average power P is called the root mean-
square (rms) value of the periodic signal x(t). In the case of a
discrete-time signal x[n], the integrals in above equations are
replaced by corresponding sums.

Thus, the total energy of x[ n] is defined by

A signal is referred to an energy signal if and only if the total

energy is finite. i.e., 0 < E < ∞

A signal is referred to a power signal if and only if the average

power is finite. i.e., 0 < P < ∞
Note: Energy signal has zero time average power and power signal
has infinite energy.

Example: x(n) = (- 0.5) nu[n]

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Solution
E=

P = lim

We got power zero and finite energy. Hence it is an energy signal.

6. Deterministic signals and random signals: The

deterministic signal is a signal about which there is no
uncertainty with respect to its value at any time. The
deterministic signals may be modeled as completely
specified function of time.
Example: x(t) = cos2(2πt)
A random signal is a signal about which there is uncertainty before
it occurs.
Example: The electrical noise generated in the amplifier of a radio
or television receiver.

7.3 Basic Operations of Signals

Operation performed on independent variable:
Time Shift
For any t0 and n0, time shift is an operation defined as
x(t)→ x(t - t0 )

x[n]→ x[n -n ].
If t0 > 0, the time shift is known as “delay”. If t0< 00, the

75
time shift is known as “advance”.
Example. In Fig. given below, the left image shows a continuous-
time signal x(t). A time- shifted version x(t-2) is shown in the
right image.

Figure: An example of time shift.

Time Reversal
Time reversal is defined as x(t) → x(-t) x[n] → x[-n],
which can be interpreted as the “flip over the y-axis”.
Example:

Figure: An example of time reversal.

Time Scaling
Time scaling is the operation where the time variable t is
multiplied by a constant a: x(t) → x(at), a > 0.
If a > 1, the time scale of the resultant signal is
“decimated” (speed up).

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 If 0 < a < 1, the time scale of the resultant signal is
“expanded” (slowed down).

Figure: An example of time scaling.

7.4 Decimation and Expansion

Decimation and expansion are standard discrete-time signal
processing operations.
Decimation is defined as yD [n] = x[Mn], for some integers M.
Where, M is the decimation factor

Expansion is defined as

Where L is the expansion factor.

77
Figure 7.8: Examples of decimation and expansion for M = 2 and
L=2
7.5 Combination of Operations
Generally, linear operation (in time) on a signal x(t) can
be expressed as y(t) = x(at-b). The recommended method is
“Shift, then Scale”.
Example: The signal x(t) shown in Figure of sketch x(3t –
5).

Figure: An example of Shift, then Scale Operation performed

on dependent variable:

Amplitude scaling:
Let x(t) denote a continuous time signal By amplitude
scaling, we get y(t) = cx(t) Where, c is the scaling factor.
Example: An electronic amplifier, a device that performs amplitude
scaling. For discrete time signal y[n] = cx[n]

Amplitude addition:

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 Let x1(t) and x2(t) is a pair of continuous time signal
By adding these two signals, we get y(t) = x1(t) + x2(t)
Example: An audio mixture
For discrete time signal, y[n] = x1[n] + x2[n]

Amplitude multiplication:
Let x1(t) and x2(t) is a pair of continuous time signal
By multiplying these two signals, we get y(t) = x1(t) x2(t)
Example: An AM radio signal, in which x 1(t) is an audio
signal x2(t) is an sinusoidal carrier wave For discrete time
signal, y[n] = x1[n] x2[n]

Differentiation:

Example: Voltage across an inductor L 

Integration:

Example: Voltage across a capacitor C 

Elementary Signals
Several elementary signals feature prominently in the study
of signals and systems. These are exponential and sinusoidal
signals, the step function, the impulse function, and the ramp

79
function, all of which serve as building blocks for the construction
of more complex signals
Exponential Signals
A real exponential signal, in its most general form, is
written as x(t) = Beat, where both B and a are real parameters. The
parameter B is the amplitude of the exponential signal measured at
time t = 0. Depending on whether the other parameter a is positive
or negative, we may identify two special cases:

Fig: Growing exponential, for a > 0 Decaying exponential, f o r a < 0

In discrete time, it is common practice to write a real exponential
signal as x[n] = Brn

Fig: Growing exponential for r > 1 Decaying exponential for 0 < r <1

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Impulse functions
The discrete-time version of the unit impulse is defined by

Fig: Discrete time form of unit impulse

The continuous-time version of the unit impulse is defined
by the following pair of relations:
δ(t) = 0 for t ≠ 0

and

δ (t)

0 t

Fig: Continuous time form of unit impulse

Above equation says that the impulse δ (t) is zero
everywhere except at the origin. Equation says that the total area
under the unit impulse is unity. The impulse δ (t) is also referred
to as the Dirac delta function.

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 Step function:

The discrete-time version of the unit- step function is defined by:

The continuous-time version of the unit- step function is

defined by:

Ramp function:
The integral of the step function u(t) is a ramp function of
unit slope.

82
 Fig: Ramp function of unit slope

The discrete-time version of the unit- ramp function is defined by:

83
 CHAPTER EIGHT
8.0 Introduction to Systems
Systems are used to process signals to allow
modification or extraction of additional information from
the signal.
A system may consist of physical components
(hardware realization) or an algorithm (operator) that
computes the output signal from the input signal.
A physical system consists of inter-connected
components which are characterized by their input-output
relationships.
Figure 2.1: Continuous-time and discrete-time systems: Here H & T are
operators.

Hx(t)
H =

Tx(n)
T =

8.1 Properties of systems (Classification of systems)

1. Static (Memoryless) & Dynamic (with memory):
- Static: A system is static if the output at time t (or n)

84
 depends only on the input at time t (or n).
Examples:

1. y(t) = (2x(t) - x2(t))2 is memoryless, because y(t)

depends on x(t) only. There is no x(t – 1), or x(t +
1) terms, for example.
2. y[n] = x2[n] is memoryless. In fact, this system is
passing the input to output directly, without any
processing.
3. Current flowing through a resistor i.e., i(t) = v(t)
- Dynamic: A system is said to possess memory
if its output signal depends on past or future
values of input.
Example:
1. Inductor and capacitor, since the current flowing
through the inductor at time „t‟ depends on the all

past values of the voltage v(t) i.e., i(t) =

and

2. The moving average system given by y(n) = (x[n]

+x[n-1]+x[n-2])

2. Stable & unstable system:

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 A system is said to be bounded-input, bounded-
output (BIBO) stable if and only if every bounded input
results in a bounded output, otherwise it is said to be
unstable.
If for |x(t)| ≤ Mx < ∞ for all t, output is |y(t)| ≤ M y
< ∞ for all t; where Mx & My are some finite positive
number.
Example:
1. y(t) = x (t-3) is a stable system.
2. y(t) = t x(t) is an unstable system.
3. y[n] = ex[n] is a stable system.

Assume that |x(n)| ≤ Mx< ∞, for all „t‟

y[n] = ex[n] = eMx = finite  Stable
4. y[n] = rn x[n], where r > 1
Assume that |x(n)| ≤ Mx < ∞, for all „t‟, then
|y[n]| = |rn x[n]| = |rn| | x[n]|
as „n‟  ∞ |rn| ∞
so y[n]∞ hence unstable.

3. Causal and non-Causal system:

- Causal: A system is said to be causal if the
present value of output signal depends only on the
present or past values of the input signal. A causal

86
 system is also known as physical or non-
anticipative system.
Example:

1. The moving average system given by y(n)= (x[n]

+x[n1]+x[n-2])
2. y(t) = x(t)cos 6t

Note:

i) Any practical system that operates in real time

must necessarily be causal.

ii) All static systems are causal.

- Non-Causal: A system is said to be non-causal

if the present value of output signal depends on
one or more future values of the input signal.
Example:
1
1. The moving average system given by y(n)= (x[n]
+x[n-1]+x[n+2])

4. Time invariant and time variant system:

- Time invariant: A system is time-invariant if a
time-shift of the input signal results in the same
time-shift of the output signal.
That is, if

87
 x(t) → y(t),
then the system is time-invariant if
x(t – t0) → y(t – t0), for any t0.

Figure 8.1: Illustration of a time-invariant system.

Example1.
The system y(t) = sin[x(t)] is time-invariant
Proof. Let us consider a time-shifted signal x1(t) = x(t
- tO). Correspondingly, we let y1(t) be the output of
x1(t).
Therefore,
y1(t) = sin[x1(t)] = sin[x(t - tO)].
Now, we have to check whether y1(t) = y(t -
tO). To show this, we note that
y(t - tO) = sin[x(t - tO)],
which is the same as y1(t). Therefore, the system is
time-invariant.

- Time variant: A system is time-variant if its

input –output characteristic changes with time.
Example 2:
The system y[n] = nx[n] is time-variant.

88
Proof: Output for a time shifted input is
y[n] | x(n-k) = nx(n-k)
then the same time shifted output is
y(n-k) = (n-k)x(n-k)
the above two equations are not same. Hence it is
time variant
4 Linear and non-linear system:
- Linear system: A system is said to be linear if
it satisfies two properties i.e.; superposition &
homogeneity.
- Superposition: It states that the response of the
system to a weighted sum of signals be equal to
the corresponding weighted sum of responses
(Outputs of the system to each of the individual
input signal.
For an input x(t) = x1(t), the output y(t) = y1(t)
and input x(t) = x2(t), the output y(t) = y2(t)
then, the system is linear if & only if
T [a1x1(t) + a2x2(t)] = a1T [x1(t)] + a2T [x2(t)]

- Homogeneity: If the input x(t) is scaled by a

constant factor “a”, then the output y(t) is also
scaled by exactly the same constant factor „a‟.
For an input x(t)  output y(t)

89
and input x1(t) = ax (t)output y1(t) = ay(t)

Example 1:
The system y(t) = 2πx(t) is linear. To see this, let‟s
consider a signal
x(t) = ax1(t) + bx2(t),
where y1(t) = 2πx1(t) and y2(t) = 2πx2(t). Then
ay1(t) + by2(t) = a (2πx1(t)) + b (2πx2(t))
= 2π [ax1 (t) + bx2 (t)] = 2πx(t) = y(t).

Example 2.
The system y[n] = (x[2n])2 is not linear. To see this, let‟s
consider the signal
x[n] = ax1 [n] + bx2 [n],
where y1 [n] = (x1 [2n])2 and y2 [n] = (x2 [2n])2 .
We want to see whether y[n] = ay1[n] + by2[n]. It holds
that ay1 [n] + by2 [n] = a (x1 [2n])2 + b (x2 [2n])2
However,
y[n] = (x[2n])2 = (ax1 [2n] + bx2 [2n])2 = a2 (x1 [2n])2 +
b2 (x2 [2n])2 + 2abx1 [n]x2 [n].

5. Invertible and non-invertible system:

A system is said to be invertible if the input of the
system can be recovered from the output.
Let the set of operations needed to recover the input
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represents the second system, which is connected in
cascade with the given system such that the output
signal of the second system is equal to the input applied
to the given system.

Let H the continuous time system x(t)input signal

to the system y(t)output signal of the system
Hinvthe second continuous time system
x(t) y(t) x(t)
H Hinv

The output signal of the second system is given by

Hinv{y(t)} = Hinv{Hx(t)}
= HinvH{x(t)}
For the output signal to equal to the original input, we
require that

H Hinv = I

Where “I‟ denotes the identity operator.

The system whose output is equal to the input is an
identity system. The operator Hinv must satisfy the above
condition for H to be an invertible system. Cascading a
system, with its inverse system, result in an identity
system.
Example:
An inductor is described by the relation
91
y(t) = 𝑥(𝜏)𝑑𝜏 is an invertible system

because, by rearranging terms, we get

which is the inversion formula.

Note:
(i) A system is not invertible unless distinct inputs applied to
the system produce distinct outputs.
(ii) There must be a one to one mapping between input and
output signal for system to be invertible.

- Non-invertible System: When several different inputs

results in the same output, it is impossible to obtain the
input from output. Such system is called a non-invertible
system.

Example:
A square-law system described by the input output relation
y(t) = x2(t), is non-invertible,
because distinct inputs x(t) & -x(t) produce the same output y(t)
[not distinct output].

92
8.2 Linear –time convolution system (LTI)
Linear time invariant (LTI) systems are good models for many real-
life systems, and they have properties that lead to a very powerful
and effective theory for analyzing their behavior. The LTI systems
can be studied through its characteristic function, called the impulse
response. Further, any arbitrary input signal can be decomposed
and represented as a weighted sum of unit sample sequences. As a
consequence of the linearity and time invariance properties of the
system, the response of the system to any arbitrary input signal can
be expressed in terms of the unit sample response of the system.
The general form of the expression that relates the unit sample
response of the system and the arbitrary input signal to the output
signal, called the convolution sum, is also derived.

Resolution of a Discrete-time signal into impulses:

Any arbitrary sequence x(n) can be represented in terms of delayed
and scaled impulse sequence δ(n). Let x(n) is an infinite sequence
as shown in figure below.

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 ●● ●●
● ●

Figure 1.13: Representing of a signal x[n] using a train of impulses

δ[n - k].
The sample x(0) can be obtained by multiplying x(0), the
magnitude, with unit impulse δ(n)
i.e., x[n] δ[n] = x(0)
n= 0
0, n ≠ 0
Similarly, the sample x(-3) can be obtained as shown in the figure.
i.e., x[-3] δ[n+3] = x(-3)
n = -3
0, n ≠ -3
In the same way we can get the sequence x[n] by summing all the
shifted and scaled impulse function
i.e., x[n] = …. x[-3] δ[n+3] + x[-2] δ[n+2] + …. + x[0] δ[n] + ….+

94
x[4 ] δ[n-4]…

Impulse response and convolution sum:

- Impulse response: A discrete-time system performs an
operation on an input signal based on predefined criteria to
produce a modified output signal. The input signal x[n] is
the system excitation, and y[n] is the system response. The
transform operation is shown in the figure below

x[n] y[n]=T[x[n]]
T

If the input to the system is the unit impulse i.e., x[n] = δ[n],
then the output of the system is known as impulse response
represented by h[n] where
h[n] = T [δ[n]]
Response of LTI system to arbitrary inputs: The convolution sum

From the above discussion, we get the response of an LTI system to

an unit impulse as the impulse response h[n] i.e.,
δ[n] h[n]
δ[n-k] h[n-k], by time invariant property
x(k)δ[n-k] x(k)h[n-k], by homogeneity principle

𝑥(𝑘)𝛿[𝑛 − 𝑘] 𝑥(𝑘)ℎ[𝑛 −

𝑘], by super position

As we know the arbitrary input signal is a weighted sum of

95
impulse, the LHS = x[n] having a response in RHS = y[n] known as
convolution summation
i.e., x[n] y[n]

In other words, given a signal x[n] and the impulse response of an

LTI system h[n], the convolution between x[n] and h[n] is defined

as 𝑦[𝑛] = 𝑥(𝑘)ℎ[𝑛 − 𝑘]

We denote convolution as y[n] = x[n] ∗ h[n].

Equivalent form: Letting m = n - k, we can show that
𝑥(𝑘)ℎ[𝑛 − 𝑘] 𝑥(𝑛 − 𝑚)ℎ[𝑚] = 𝑥[𝑛 −
𝑘]ℎ[𝑘]

Properties of convolution:
The following “standard” properties can be proved easily:
1. Commutative: x[n] ∗ h[n] = h[n] ∗ x[n]
2. Associative: x[n] ∗ (h1 [n] ∗ h2 [n]) = (x[n] ∗ h1 [n])
∗ h2 [n]
3. Distributive: x[n] ∗ (h1 [n] + h2 [n]) = (x(t) ∗ h1 [n]) +
(x[n] ∗ h2 [n])
How to Evaluate Convolution?
To evaluate convolution, there are four basic steps:
1. Fold
2. Shift
3. Multiply

96
4. Summation

Example1: Consider the signal x[n] and the impulse response h[n]
shown below.

Let’s compute the output y[n] one by one. First, consider y[0]:

𝑦[𝑜] = 𝑥 𝑘 ℎ[0 − 𝑘] =

𝑥 𝑘 ℎ[1 − 𝑘] = 1 × 1 + 2 × 1 = 3

The calculation is shown in the figure below.

8.3 System Properties

With the notion of convolution, we can now proceed to
discuss the system properties in terms of impulse responses.
Memoryless
A system is memoryless if the output depends on the current
input only. An equivalent statement using the impulse response h[n]
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is that:
An LTI system is memoryless if and only if
h[n] = aδ[n], for some a.

Invertible
An LTI system is invertible if and only if there exist g[n] such that
h[n] * g[n] = δ[n].

Causal
An LTI system is causal if and only if
h[n] = 0, for all n < 0.

98
REFERENCES
1. Electronic device and circuit by Jacob Millman and
Christos C. Halkias
2. Electronics a course for Engineers by R.J Maddock &
D.M Calcutta.
3. Electronics fundaments and Application by John D.
Ryder.

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