Main Digital Communication: Principles and System Modelling

# Digital Communication: Principles and System Modelling

"Digital Communication" presents the theory and application of the philosophy of Digital Communication systems in a unique but lucid form. The book inserts equal importance to the theory and application aspect of the subject whereby the author selected a wide class of problems. The salient features of the book are: 1. A natural flow or writing is maintained from the first to the last chapters to visualize the information flow in an elegant way with intermediate conversion, processing like modulation and filtering, coding for error detection, correction and security. 2. In some cases like spread spectrum modulation, hypothetical experiments have been designed for clarifying the "why" of the topic. 3. The application area is rich and resemblance to the present trend of research, as the author himself is attached with those areas professionally.. 4. Supplementary electronic material is available via http://extras.springer.com which includes codes in MATLAB, with illuminating uncommon and common applications along with a large number of image, signal and speech demos with flash animation for better understanding of the subject. 5. Elegant worked out exercise section is designed in such a way that, the readers can get the flavor of the subject and get attracted towards the future scopes of the subject. 6. Unparallel tabular, flow chart based and pictorial methodology is employed for description of the subject for sustained impression of the proposed designs/ algorithms in mind.

Year: 2010
Edition: 1
Language: english
Pages: 246
ISBN 10: 3642127428
ISBN 13: 9783642127427
Series: Signals and Communication Technology
File: PDF, 8.88 MB

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```SIGNALS AND COMMUNICATION TECHNOLOGY

For further volumes:
http://www.springer.com/series/4748

Apurba Das

Digital Communication
Principles and System Modelling

123

Apurba Das
Scientist, Image Processing Lab
CDAC, Kolkata
A Scientific Society
Ministry of Communication and IT
Government of India
West Bengal
India
apurba.das@cdackolkata.in
apurba_das1@hotmail.com

Additional material to this book can be downloaded from http://extra.springer.com

ISBN 978-3-642-12742-7
e-ISBN 978-3-642-12743-4
DOI 10.1007/978-3-642-12743-4
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2010929759
© Springer-Verlag Berlin Heidelberg 2010
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer. Violations
are liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective
laws and regulations and therefore free for general use.
Cover design: WMXDesign GmbH, Heidelberg
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

To You
Angana (Oli),
my wife

Preface

In today’s world of human-machine interaction, the necessity of custom communication is increasing day by day. The adjective ‘custom’ in communication or in
true sense, telecommunication signifies a lot of things like secured communication, communication with hidden data through any type of multimedia covers like
image, speech or video, communication in user suggested bandwidth, multi-channel
communication without cross-talk, internetworking in LAN, MAN, WAN and integrated service digital network (ISDN) of image, video and data, data transmission in
encrypted but non-perceptible mode and so on. The information exchange between
the source and the destination is generally done through modulated signal both in
analog and digital form. Digital design and processing can not only be easily realizable, but also digital signal processing can help one designer to meet the practical
needs of this era as discussed before.
Therefore, the digital communication has become an important subject of study
for students of electronics and communication, computer science and information
technology both in undergraduate and post-graduate levels.
The aim of the book is to represent the theory and application of the design
philosophy of the subject Digital Communication systems in a unique but lucid
form. There are resource books that are supreme pedagogical documents of this
subject. It is a bit difficult to consider them as the standard text for the entire student
population, because the said books do not visualize the design problem clearly. An
attempt has been made to bridge the gap between the design principle and system
modelling. In this book, I have tried to introduce the subject through its obvious flow.
Supporting MATLAB codes and outputs are also included for better understanding
and visualization. Essentially it is designed for the large class of students in the
standards of bachelors, masters or those who have started their research carrier in
science, engineering or technology.
The title of the proposed book is ‘Digital Communication- Principles and system
modelling’. By selecting this name I essentially took the task to insert equal importance to the theory and application aspect of the subject. The subject is introduced
considering absolutely zero prerequisite of the readers. The introductory chapter
created a space for round table discussion between the author and the reader in the
subject of signals, systems, types and choices of telecommunication methodology.
Basic building blocks play their roles to design one complete digital communication
vii

viii

Preface

system according to necessity of successful interaction of information. In second
chapter, the process of conversion of natural (analog) signal to digital signal is
discussed for letting the input acceptable to digital communication system. In the
subsequent chapters the processes of signal transmission with and without modulation are described. Process of security embedding in terms of spreading the
bandwidth of a signal is represented. The issues of inter-symbol interference (ISI)
are solved by using equalizer. Theory of information and channel capacity are also
discussed in the light of probability theory. As the book claims zero prerequisite,
one Appendix titled as Elementary Probability Theory is also appended to the main
chapters. The error detection and correction in one digital communication system
is understood physically and then suitable coding schemes are being employed to
solve particular types of bit errors due to channel noise.
In the application part I have put my best effort to select a wide class of problems. Some of them are referral in nature and found in most of the books but the
other group includes the applications, which I have collected from different resource
books and different research papers (all the time submitting courtesy to the authors).
Some applications are also represented as case studies on frequently used concepts
convolution and correlation in Appendix B.
The style of writing is kept in the lucid level to attract the interest of a large class
of readers. I always believe that, writing a text book is very much difficult in terms
of presentation. Text book is not only for concept sharing, but also for preparing a
slide show or movie which can just put the readers into a smooth and comfortable
path of understanding the subject. Similarly, teaching is not a delivery of some raw
data; conversely, it is a performance in front of the students/ audience. I think this
book can be a good teaching aid, too.

Salient Features
1. The application area is rich and resemblance to the present trend of research.
2. An online content is included with the title, which includes codes and MATLAB,
with illuminating uncommon and common applications along with some flash
movies for better understanding of the subject.
3. Elegant worked out exercise section is designed in such a way that, the readers
can get the flavour of the subject and get attracted towards the future scopes of
the subject.
4. Unparallel tabular, flow chart based and pictorial methodology description is
included for sustained impression of the proposed design/algorithms in mind.

The book is for everybody, the reader may be someone without the knowledge of
basic engineering or someone like a researcher who wants to understand the subject
in a different angle. Let me use the technical terminology to let you understand.
Ultimately the received signal to you transmitted by this book must be converted

Preface

ix

successfully to analog/ natural form; otherwise the knowledge cycle does not get
completed.
Kolkata, India

Apurba Das

Acknowledgements

Human being is not at all an island. A person can be successful only when he/she is
blessed and encircled by the relatives and community. This book would not be completed without the blessings of my seniors, the well wishes of my friends, colleagues
and students.
First of all I like to express my deep respect to my parents Sri Barun Das and
Smt. Nabanita Das for setting up a vision for me. I like to express my regards to my
uncle and aunt (Jethu and Mamoni) Sri Arun Kr. Das and Smt. Moli Das for their
continuous support. I am also thankful to my brother Anirban and sister Adwitia as
they considered me as their benchmark of living.
I am fortunate enough getting Prof. Amit Konar of Electronics and TeleCommunication Engineering department of Jadavpur University as my teacher
during my M-Tech. Sir has always inspired me about writing text books. He himself
is the inspiration for thirst of knowledge.
I am grateful to my friend Mr. Manoj Kr. Pain for everything. Not only he helped
me in preparing the flash files for online content, moreover he is the man behind all
of my actions particularly in this phase of my life.
I thank my students, especially Dipto Pandit, Subarno Shekhar Gain, Subhojit
Chatterjee for their help in content preparation for error control coding chapter and
some parts of Delta Modulation.
I am really grateful to my senior Mr. Debasis Mazumdar of CDAC, Kolkata.
I have learnt how nicely mathematical models can infer a physics and how a physics
can generate one nice mathematics. I am also thankful to my colleague Mr. Raja
Gupta for preparing the flash files for online content.
Last but not the least I like to express my thanks to the editorial team of SpringerVerlag, specially Christoph Baumann and Carmen Wolf for such a wonderful
production.
Kolkata, India

Apurba Das

xi

Contents

1 Preview and Introduction . . . . . . . . . . . . . . . . . . . . . .
1.1 Process of Communication . . . . . . . . . . . . . . . . . . .
1.2 General Definition of Signal . . . . . . . . . . . . . . . . . . .
1.3 Time-Value Definition of Signals–Analog and Digital . . . . .
1.3.1 Continuous Time Continuous Valued Signal . . . . . .
1.3.2 Discrete Time Continuous Valued Signal . . . . . . . .
1.3.3 Discrete Time Discrete Valued Signal . . . . . . . . . .
1.4 Analog and Digital Communication Systems . . . . . . . . . .
1.5 Elements of Digital Communication System . . . . . . . . . .
1.6 MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Time and Frequency Domain Representation of Signals
1.6.2 CTSV, DTCV, DTDV Signals . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Waveform Encoding . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Pulse Code Modulation (PCM) . . . . . . . . . . . . . . . . .
2.2.1 Process of Sampling . . . . . . . . . . . . . . . . . . .
2.2.2 Process of Quantization . . . . . . . . . . . . . . . . .
2.2.3 PCM Transmitter and Receiver . . . . . . . . . . . . .
2.2.4 Quantization Error . . . . . . . . . . . . . . . . . . . .
2.2.5 Signal to Noise Ratio (SNR) for Quantized Pulses . . .
2.2.6 Non-uniform Quantization: Companding . . . . . . . .
2.3 Differential Pulse Code Modulation (DPCM) . . . . . . . . . .
2.3.1 Cumulative Error in PCM . . . . . . . . . . . . . . . .
2.3.2 Prevention of Cumulative Error by Applying Feedback .
2.3.3 How We Can Predict the Future? . . . . . . . . . . . .
2.3.4 Analysis of DPCM . . . . . . . . . . . . . . . . . . . .
2.4 Delta Modulation . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Drawbacks of Delta Modulation . . . . . . . . . . . . .
2.5 Adaptive Delta Modulation . . . . . . . . . . . . . . . . . . .
2.5.1 Song Algorithm . . . . . . . . . . . . . . . . . . . . .
2.5.2 Space-Shuttle Algorithm . . . . . . . . . . . . . . . . .

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Contents

2.6 Sigma-Delta Modulation (SDM) . . . . .
2.6.1 Noise Performance . . . . . . . . .
2.7 Linear Predictive Coder (LPC) . . . . . .
2.7.1 Concept . . . . . . . . . . . . . .
2.7.2 Genetic Algorithm Based Approach
2.8 MATLAB Programs . . . . . . . . . . . .
2.8.1 Aliasing . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . .

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3 Digital Baseband Signal Receivers . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . .
3.2 Integrate and Dump Type Filter . . . . . . . .
3.2.1 Noise Power and Variance . . . . . . .
3.2.2 Figure of Merit . . . . . . . . . . . . .
3.2.3 Probability of Error . . . . . . . . . .
3.3 The Optimum Filter . . . . . . . . . . . . . .
3.4 The Matched Filter . . . . . . . . . . . . . . .
3.4.1 Impulse Response . . . . . . . . . . .
3.4.2 Probability of Error . . . . . . . . . .
3.4.3 Properties of Matched Filter . . . . . .
3.5 The Correlator . . . . . . . . . . . . . . . . .
3.6 Simulink Communication Block Set Example
References . . . . . . . . . . . . . . . . . . . . . .

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4 Digital Baseband Signal Transmitter . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Elements of Digital Baseband Communication System
4.2.1 Formatting . . . . . . . . . . . . . . . . . . .
4.2.2 Regenerative Repeater . . . . . . . . . . . . .
4.3 Properties and Choice of Digital Formats . . . . . . .
4.4 Line Coding . . . . . . . . . . . . . . . . . . . . . .
4.5 Power Spectrum Density of Different Digital Formats
4.5.1 Unipolar-NRZ . . . . . . . . . . . . . . . . .
4.5.2 Unipolar-RZ . . . . . . . . . . . . . . . . . .
4.5.3 Polar-NRZ . . . . . . . . . . . . . . . . . . .
4.5.4 Polar-RZ . . . . . . . . . . . . . . . . . . . .
4.5.5 Bipolar-NRZ . . . . . . . . . . . . . . . . . .
4.5.6 Split-Phase (Manchester) . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Inter-Symbol Interference (ISI) . . . . . . . . . . . . . . . .
5.2 Nyquist Criterion for Distortion Less Transmission (Zero ISI)
5.2.1 Criteria in Frequency Domain . . . . . . . . . . . . .
5.2.2 Concept of Ideal Nyquist Channel . . . . . . . . . . .
5.2.3 Limitations of Ideal Solution: Raised Cosine Spectrum

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Contents

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5.3 Eye Pattern . . . . . . . . . . . . . . . . . . .
5.3.1 Information Obtained from Eye Pattern
5.4 System Design for Known Channel . . . . . .
5.5 Linear Equalizer . . . . . . . . . . . . . . . .
5.5.1 Linear Transversal Filter . . . . . . . .
5.6 Adaptive Equalizer . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . .

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6 Digital Modulation Techniques . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
6.2 Amplitude Shift Keying (ASK) . . . . . . . . . . . .
6.2.1 Mathematical Model . . . . . . . . . . . . . .
6.2.2 ASK Modulator . . . . . . . . . . . . . . . .
6.2.3 Binary ASK Demodulator . . . . . . . . . . .
6.3 Frequency Shift Keying (FSK) . . . . . . . . . . . .
6.3.1 Mathematical Model . . . . . . . . . . . . . .
6.3.2 BFSK Modulator . . . . . . . . . . . . . . . .
6.3.3 FSK Demodulator . . . . . . . . . . . . . . .
6.4 Binary Phase Shift Keying (BPSK) . . . . . . . . . .
6.4.1 Mathematical Model . . . . . . . . . . . . . .
6.4.2 BPSK Modulator . . . . . . . . . . . . . . . .
6.4.3 BPSK Demodulator . . . . . . . . . . . . . .
6.5 Differential Phase Shift Keying (DPSK) . . . . . . .
6.5.1 DPSK Modulator . . . . . . . . . . . . . . .
6.5.2 DPSK Demodulator . . . . . . . . . . . . . .
6.6 Quadrature Phase Shift Keying (QPSK) . . . . . . . .
6.6.1 Mathematical Model . . . . . . . . . . . . . .
6.6.2 QPSK Modulator . . . . . . . . . . . . . . .
6.6.3 QPSK Demodulator . . . . . . . . . . . . . .
6.6.4 Offset QPSK (OQPSK) . . . . . . . . . . . .
6.7 Minimum Shift Keying (MSK) . . . . . . . . . . . .
6.8 Probability of Error for Different Modulation Schemes
6.8.1 Probability of Error in ASK . . . . . . . . . .
6.8.2 Probability of Error in FSK . . . . . . . . . .
6.8.3 Probability of Error in PSK . . . . . . . . . .
6.9 MATLAB Programs . . . . . . . . . . . . . . . . . .
6.9.1 QPSK Waveform . . . . . . . . . . . . . . . .
6.9.2 MSK Waveform . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Spread Spectrum Modulation . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . .
7.2 Processing Gain . . . . . . . . . . . . . . .
7.3 Pseudo-Noise (PN) Sequence . . . . . . . .
7.3.1 Concept: A Hypothetical Experiment
7.3.2 Generation of PN Sequence . . . . .

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xvi

Contents

7.3.3 Properties of PN Sequence . . . . .
7.4 Direct Sequence Spread Spectrum (DSSS)
7.4.1 Concept . . . . . . . . . . . . . .
7.4.2 DSSS with Coherent BPSK . . . .
7.4.3 Probability of Error Calculation . .
7.5 Frequency-Hopped Spread Spectrum . . .
7.5.1 Concept . . . . . . . . . . . . . .
7.5.2 FHSS with FSK . . . . . . . . . .
7.5.3 Rate of Hopping: Fast and Slow . .
7.6 Application of Spread Spectrum . . . . . .
7.6.1 GPS (Global Positioning System) .
7.7 CDMA (Code Division Multiple Access) .
7.7.1 Orthogonal Chip Sequence . . . .
7.7.2 Gold Sequence . . . . . . . . . . .
7.7.3 Principle of Operation . . . . . . .
References . . . . . . . . . . . . . . . . . . . .

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147
149
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151
152
155
155
157
159
159
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163
163
165
166
166

8 Information Theory . . . . . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . .
8.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Rate of Information . . . . . . . . . . . . . . . . .
8.4 Information Sources . . . . . . . . . . . . . . . . .
8.5 Discrete Memoryless Channel (DMC) . . . . . . .
8.5.1 Channel Representation . . . . . . . . . . .
8.5.2 The Channel Matrix . . . . . . . . . . . . .
8.6 Special Channels . . . . . . . . . . . . . . . . . . .
8.6.1 Lossless Channel . . . . . . . . . . . . . . .
8.6.2 Deterministic Channel . . . . . . . . . . . .
8.6.3 Noise-Less Channel . . . . . . . . . . . . .
8.6.4 Binary Symmetric Channel (BSC) . . . . . .
8.7 Mutual Information . . . . . . . . . . . . . . . . .
8.8 Channel Capacity . . . . . . . . . . . . . . . . . .
8.8.1 Gaussian Channel: Shanon-Hartley Theorem
8.9 Entropy Coding . . . . . . . . . . . . . . . . . . .
8.9.1 Shanon-Fano Coding . . . . . . . . . . . .
8.9.2 Huffman Coding . . . . . . . . . . . . . . .
8.10 MATLAB Code . . . . . . . . . . . . . . . . . . .
8.10.1 Convergence of Pe in Cascaded BSC . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .

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169
169
171
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176
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177
177
178
179
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182
183
183
185
186
187
188
188
189

9 Error Control Coding . . . . . . . . . . . . . . . . .
9.1 Introduction . . . . . . . . . . . . . . . . . . . .
9.2 Scope of Coding . . . . . . . . . . . . . . . . . .
9.3 Linear Block Code . . . . . . . . . . . . . . . . .
9.3.1 Coding Technique Using Generator Matrix
9.3.2 Syndrome Decoding . . . . . . . . . . . .

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191
191
192
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Contents

9.4 Convolutional Code . . . . . . . . . . . . . . . . .
9.4.1 Encoder . . . . . . . . . . . . . . . . . . .
9.4.2 State Diagram . . . . . . . . . . . . . . . .
9.4.3 Code Tree . . . . . . . . . . . . . . . . . .
9.4.4 Trellis Diagram . . . . . . . . . . . . . . .
9.4.5 Decoding of Convolutional Code by Viterbi
9.5 Cyclic Code . . . . . . . . . . . . . . . . . . . . .
9.5.1 Concept and Properties . . . . . . . . . . .
9.5.2 Encoder and Decoder . . . . . . . . . . . .
9.5.3 Meggitt Decoder . . . . . . . . . . . . . . .
9.6 BCH Code . . . . . . . . . . . . . . . . . . . . . .
9.6.1 Simplified BCH Codes . . . . . . . . . . . .
9.6.2 General BCH Codes . . . . . . . . . . . . .
9.6.3 Properties . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

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196
196
199
200
200
202
204
204
206
207
207
208
210
210
211

Appendix A: Elementary Probability Theory . . . . . . . . . . . . . . .

213

Appendix B: Convolution and Correlation – Some Case Studies . . . . .

225

Appendix C: Frequently Used MATLAB Functions . . . . . . . . . . .

237

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

Chapter 1

Preview and Introduction

1.1 Process of Communication

Transducer
Transducer

MIC

Audio
Amplifier

Electrical signal
(Audio)

Amplified Audio

Louder Speech

Speech

A man is discussing about a cricket game with his friend sitting just in front of
him. If this is the scenario, this process of interaction is interpersonal communication. In this case, no help of electronics is needed for communication. Let’s now
change the scenario. Say, a political meeting is going on at a large play ground.
In front of 1,000,000 people one speaker is giving his/her lecture. It’s not easy to
communicate now, as because (i) the distance between the communicating nodes
has been increased and (ii) the audience is not deaf and dumb therefore all of them
are responsible for generating unwanted signal (noise). Here we require some extra
circuits for successful communication. First the speech signal has to be converted
to electrical equivalent signal known as audio by a transducer namely, microphone.
Next, the electrical signal should be amplified by an audio amplifier. Finally, that
amplified audio signal is re-converted by another transducer namely loud speaker
to speech signal just as the amplified source signal to make it audible. The block
diagram representation is shown in Fig. 1.1.

Loud Speaker

Fig. 1.1 Speech signal amplification with the help of transducers and electrical amplifier

Therefore we have started using the knowledge of electrical and electronics engineering in communication. Now, let’s make the distance longer. Communication is
required between two persons; one in USA is trying to communicate other persons in Germany. Here, microphone and loud speaker cannot also solve the present
problem. Here two ways are there.
A. Das, Digital Communication, Signals and Communication Technology,
C Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-12743-4_1, 

1

2

1 Preview and Introduction

1. There must be a long connecting cable namely transmission line between the
persons. If this is the mode of communication, then that is known as Line
Communication.
2. Otherwise, we can take the help of an antenna to transmit and receive the unmodulated (base-band) or modulated audio signal, this is known as Radio wave
communication.
Combining the above two, the process of long distance communication is named
as Tele-Communication. The meaning of the word ‘tele’ means far. Obviously, we
are now convinced that, electronics must be needed for Tele-Communication. Our
study through out the title should be based on Tele-Communication only.
A communication system can also be classified into three categories in respect
of direction of communication  as:
1. Simplex
2. Half Duplex
3. Full Duplex
Simplex communication is a unidirectional communication system i.e., communication is possible in one direction only. Typically, the transmitter (the one talking
or sending information in any ways) sends a signal and it’s up to the other receiving
device (the listener) to figure out what was sent. This type of communication is most
efficient when there is a lot of data flow in one direction, and no traffic is needed
in the other direction. Broadcast systems like the T.V and radio signals, fire alarm
systems etc. are good examples of Simplex communication system.
In Half Duplex communication system, bi-directional communication is possible, but only in one direction at a time. That means one can either transmit or receive
a signal at a particular instant of time in this system of communication. One cannot
transmit and receive a signal simultaneously. Say, within a specific time interval t1 ≤
t < t2 , node A is transmitting a signal to another node B. At time t = t2 , node A will
command ‘over’ and both of the nodes A and B will change their mode from transmitter to receiver and receiver to transmitter, respectively by switching the duplexer.
Next, B can start sending information to A via the same link. The walky-talky
used in defense and by police is a good example of Half Duplex communication
system.
In Full Duplex communication system simultaneous two-way communication
is achieved. Unlike half duplex communication system, one can both transmit and
receive a signal simultaneously. Telephone conversation is an appropriate example
of Full Duplex communication system (Fig. 1.2).
When single transmitter is sending the same information to a plenty of receivers,
the special mode is called as broadcasting.

1.2

General Definition of Signal

3

Fig. 1.2 Modes of
directional communication:
(a) Simplex, (b) Half-duplex,
(c) Full-duplex
Fire alarm

(a)
over

over
or
(b)

and
(c)

1.2 General Definition of Signal
In a single phrase, signal is the physical quantity by which information is carried
between at least two entities. In other words by which physical quantity communication is done, is signal. Signal is not necessarily always represented as some
electrical form of energy, it may be some gait, may be an image, may be a video.
To construct a complete definition let us try to have a close look at the general
properties of a signal.
Property 1
Property 2

Property 3

Representation of a physical quantity.
Signal must carry some information. Meaning of a signal is not
important in the property discussion. If a signal does not have
meaning with respect to a particular communication, it may be in
coded form or meaningful for another arena of communication. It
means the physical quantity is unwanted with respect to that specific
application. Therefore, this is unwanted signal or noise.
To measure or predict the response of a physical phenomena or system, mathematical modeling is important. Electrical current signal is
generally characterized as
i(t) = I0 cos (ω0 t + φ)

(1.1)

If we observe a small time window of that signal, the waveform obtained would
be as shown in Fig. 1.3a.
Referring the Fig. 1.3, the time domain and frequency domain representations
are shown. Obviously, the spectrum of the single tone signal is showing an impulse

4

1 Preview and Introduction
Waveform

amplitude-->

2
1
0
–1
–2

0

0.5

1

1.5

amplitude-->

4000

2

2.5
3
3.5
time-->
Amplitude Spectrum

4

4.5

5

3000
2000
1000
0
0

1000

2000

3000
frequency-->

4000

5000

6000

(a)
Waveform

amplitude-->

2
1
0
–1
–2

0

0.5

1

amplitude-->

2000

1.5

2

2.5
3
3.5
time-->
Amplitude Spectrum

4

4.5

5

1500
1000
500
0

0

1000

2000

3000
frequency-->

4000

5000

6000

(b)
Fig. 1.3 (a) Time and frequency representation of a single tone signal, (b) Time and frequency
domain representation of a two tone signal

1.2

General Definition of Signal

5

only, whereas the two tone (DSB-SC type) signal generates a pair of impulses. These
representations of math models are satisfying the definition of signal but can only
be considered as an example of a special class of signal, which is modeled as function of time or frequency as f(t) or F(ω). Let’s consider now an image shown in the
following Fig. 1.4. To represent the image as a mathematical function, we have to
first divide the entire image into a number of unit squares of homogeneous intensity
value to each square. These smallest picture elements are called as pixels. Intensity
or color of each smallest square block is now can be represented as function of ordinate and abscissa, i.e., function of space where the entire image plane is considered
as a Cartesian co-ordinate system.

Fig. 1.4 Image
representation as a function
of 2D space

Origin

x
f(x,y)

y

Similarly, video images can be represented functionally as a multivariate function, where image signal is represented as I(t,x,y), i.e., function of both time and
space.
As per the discussion above, now we can characterize a signal mathematically as
a function of independent variable/variables.
Now from the above three properties, we can define signal as, A physical quantity used for carrying information from one entity to another and mathematically
modeled as a function of independent variable/ variables.
Our discussion would be mainly focused on the signals which can be modeled as
a function of time. In the next section, we’ll discuss the time-value definition and
classification of signals. Hence, we’ll clearly understand the time-value concept of
analog and digital signals. Also we can find the step-by-step procedure of conversion
of analog signal to digital signal.

6

1 Preview and Introduction

1.3 Time-Value Definition of Signals–Analog and Digital
The word ‘analog’ comes from the word ‘analogy’ which means similarity. Here the
similarity implies similarity with the nature. Directly we can say, any natural signal
is analog signal. As nature, the starting time of analog signal is also t = −∞ and
the ending time is t = ∞.
Let’s consider the waveform as shown in the Fig. 1.5. We have taken one time
window of the signal though the existence of the signal is beyond the time range
t = [t1 t2 ]. Within this time range, the signal x(t) has maximum and minimum values
Xmax and Xmin respectively. Now depending on the countability of the number of
possible amplitude and possible time values, we can classify signal and this is in
turn the step by step approach towards the procedure of ADC (analog to digital
conversion).

CTCV
amplitude-->

10

0

–10

0

1

2

3

4

5

3

4

5

time-->

amplitude-->

DTCV

X=1
Y=5.7063

10

0

–10

0

1

2
time(nTS)-->
DTDV

amplitude-->

10

Sampled signal
Quantized signal

0

–10

0

1

2

3
time(nTS)-->

Fig. 1.5 Time valued definition of signal

4

5

1.3

Time-Value Definition of Signals–Analog and Digital

7

1.3.1 Continuous Time Continuous Valued Signal
As shown in the Fig. 1.5, the signal x(t) is considered within the time interval t =
[t1 t2 ], say within time 10:10 a.m. and 10:11 a.m. This technique of watching a
signal in a specific interval is called as windowing. This is easily understandable
that, we can take any time value within the specified interval and any amplitude can
also be obtained at any specific time instant ti , i.e., the number of possible valid
time instants is infinite. From the concept of set theory, it is considered that, from a
set of finite number of time values, any valid time instant ti must be the element of
the predefined set. In the present case we cannot define any set of finite number of
elements. Because, we can have any instant like 10 h 10 min 1 s 1.3 ns, so possible
time values can be infinite. In the case of amplitude value also we can have infinite
number of possible amplitude values within the range Xmax and Xmin . This type of
signal is therefore called as continuous time continuous valued (CTCV) signal.

1.3.2 Discrete Time Continuous Valued Signal
Generally, the analog signals do have this property of continuous time and amplitude
values. But no digital memory or processor can deal with infinite long words. Digital
signals are represented using digits only. First, to express the CTCV signal, let’s take
the help of the equally samples of the signal. The equal time spacing between two
adjacent samples is called as sampling interval, Ts . Then the 2nd signal shown in the
figure must have finite number of time values but, we can have infinite number of
possible voltages values yet. This signal is called as discrete time continuous valued
signal (DTCV).

1.3.3 Discrete Time Discrete Valued Signal
As shown in Fig. 1.5, at t=1, the amplitude of the signal is 5.7063. It may be anything between Xmax and Xmin . Now, to get finite number of voltage values, a set
of finite number of amplitude values setting the Xmax and Xmin as upper and lower
limit, has to be defined. In the present case, within the amplitude range 10(Xmax )
and −10(Xmin ), 8 values have been taken to form that closed set. The elements are,
±1.25, ±3.75, ±6.25, ±8.75. Now, sampled ales can be approximated (i.e., quantized) to the closest defined element of the amplitude. As shown in Fig. 1.5, at time
instant t=1, the sampled value 5.7063 has been quantized to 6.25. The staircase
form of the signal is having finite number of time and amplitude values. This signal
is called as discrete time discrete valued (DTDV) signal. Now, that defined amplitude values can easily be encoded to get the digital bit stream. The entire process is
elaborated in the next chapter (Chap. 2).

8

1 Preview and Introduction

1.4 Analog and Digital Communication Systems
In the process of telecommunication, one may be interested to transmit directly the
natural signal (analog signal) with or without modulation (base band transmission).
There, all the processors/systems used for transmission process are analog processors designed using discrete analog elements like op-amps, transistors, resistors,
inductors, capacitors, etc. This is analog communication system.
On the other hand, the system which specifically deals with digital data and
digitally pre-processed signals is a digital communication system. Here, digital processors and accessories like stored program controlled processors, software codes,
digital memory etc. takes important role in the process of signal transmission and
reception. But, as we know, all the natural understandable signals are analog, overhead hardware for analog to digital conversion and digital to analog conversion are
needed at the transmitter and receiver end, respectively.
Although, the necessity of extra blocks for analog to digital conversion (ADC)
and digital to analog conversion (DAC), digital communication system is preferred
over analog communication system for some particular reasons as follows:
1. Noise immunity: While transmitting digital information, formatting is very essential, i.e., two analog correspondence amplitude values are to be assigned for logic
zero and logic one respectively in case of binary transmission. Noise immunity
into such formatted digital information is greater than that of an analog signal.
An Additive White Gaussian Noise (AWGN), (additive in nature, constant PSD
(Power Spectral Density) throughout all frequency components and Gaussian
PDF (Probability Density Function)) is added to both analog and digital signals.
From the Figure 1.6, it’s clearly seen that the actual analog signal after being
attacked by the noise is extremely distorted. Similarly the actual digital signal is
also distorted but in case of digital signal the original signal can be easily reconstructed by simply taking the average amplitude at required time intervals. In the
above example, the average value within time intervals (bit width) TB , 2TB , 3TB
& 4TB are somewhat nearer to +1 V, +1 V, 0 V & +1 V. Hence the original signal
can be easily reconstructed.
2. Memory: Analog signals are generally stored in devices like magnetic tapes,
floppy disks, etc. It requires many magnetic tapes to store the analog signals.
Moreover these are easily affected by the magnetic and other mechanical and
physical phenomenon. On the other hand, digital information is stored in devices
like CDs and registers. For example, in a D-flip-flop, the output can be retained
for many years without any external power. When the output is needed, the trigger clock pulse is supplied to the flip-flop & it will give the output. Lifetime of
digital memory is also higher than that of analog memory.
3. System Re-configurability: One of the most significant advantages of digital systems is their ease of system re-configurability. As for an example, let us consider
an analog low pass filter. To convert it to a high pass filter, we will have to remove
the components from the circuit and replace them with other appropriate components. This is a tiresome job if the system is a complex one. On the other

1.4

Analog and Digital Communication Systems

v(t)-->

d(t)-->

1

0.5

0

2

t-->

4

1
n(t)-->

0

2

4

0
0

6

2
n(t)+v(t)-->

n(t)+d(t)-->

2
1

0

2

4

6

4

6

t-->

AWGN Noise

0.5

t-->
Noisy d(t) at Rx

0

2
t-->

0.5
0

0
-1
0

6

AWGN Noise

1
n(t)-->

Analog signal

Digital data: 10110

1

0

9

2

4
t-->
Noisy v(t) at Rx

6

1
0
–1

0

2

4

6

t-->

Fig. 1.6 AWGN attacked digital and analog signal

hand of we want to convert a digital low pass filter into a high pass filter, we
can easily change the digital filter transfer function [H(z)] just by changing some
coefficients (a0 , a1 , a2 . . . ., b0 , b1 , b2 , . . . .. ).
Where
H(z) =

a0 + a1 z−1 + a2 z−2 + . . . ..
b0 + b1 z−1 + b2 z−2 + . . . ..

Just taking the example of any multimedia audio player, we can get a clear
view of the system. When the graphic equalizer is set to a ‘high treble’ preset, the
entire system of reconstruction process responds as a highpass filter. Just at the
very next instance if anyone chose to listen ‘high bass’ music, the reconstruction
system responds like a lowpass filter. It’s done only by dynamically change of
coefficients values.
4. Aging: Aging signifies growing older of the system. It is obviously less effective
in case of digital systems. In case of analog systems, the output may change after
a few years due to aging of the discrete analog components like diode. As time
goes, the cut-in voltage of the diode increases slowly and unsteadily. This causes
fluctuation in the system performance. In digital systems, the system error due
to the problem of aging is totally absent.

10

1 Preview and Introduction

1.5 Elements of Digital Communication System
The system which specifically deals with digital data and digitally pre-processed
signals is a digital communication system. In the Fig. 1.7, the basic elements of a
digital communication system is shown. As we know, objective of transmitting natural information in terms of digital data from one end to another can be achieved
by incorporating ADC at the transmitter side and DAC at the receiver side. Here,
the message generated by the information source is to be converted to binary bit
stream, first. At the time of this conversion, one should try to represent the message
through as few binary digits as possible, i.e., an efficient information representation of the source message has been obtained, which results in very low or zero
redundancy. This process of efficiently converting analog or digital message generated from source, into a sequence of binary digits is called as source encoding. This
process includes digital data compression.
Information Source

Input
transducer

Source
Encoder

Digital
modulator

Channel
decoder

Digital
demodulator

Communication
Channel

Channel
Encoder

Recovered Information

Output
transducer

Source
Decoder

Fig. 1.7 Basic elements of a Digital Communication system

The sequence of bits is then passed through the channel encoder. The purpose
of channel encoding is to introduce intentional custom redundancy into the information sequence to overcome the effects of noise encountered in the transmission
channel at the time of communication. Let’s consider a (trivial) form of channel
encoding where each bit of the information sequence is simply repeated by m times.
A better encoding (non-trivial) takes k information bits at the time of mapping each
k-bit sequence into a unique n-bit sequence, called a code word. The amount of
redundancy introduced by encoding the data in this manner is measured by the ratio
n/k. the reciprocal of the ratio k/n is called the code rate.
The channel encoded bits are then passed to a digital modulator. Digital modulation here signifies assigning an equivalent set of wave forms for channel encoded
binary digits. To understand in brief, we can set S0 (t) waveform is transmitted to

1.6

MATLAB Programs

11

transmit logic 0 and S1 (t) waveform is transmitted to transmit logic 1. This technique
of modulation can also be called as binary modulation.
Communication channel is the physical medium that is used to communicate
the digital modulated/un-modulated signal from transmitter to the receiver end.
In wireless communication, free space is considered as the channel. On the other
hand, telephone channels employ a variety of physical media, including wired
transmission line, optical fiber cables and microwave. The characteristic of the channel is randomly corrupting the transmitted signal in a variety of possible manner,
such as additive thermal noise generated by electronic deices, human made noise
like automobile ignition noise, atmospheric noise like lightning discharges due to
thunderstorm.
The digital demodulator processes the corrupted digital waveform and generates output bit stream. The bit stream is passed then through the channel decoder
to reconstruct correct information after removing the redundant bits from the
knowledge of encoding scheme used in the channel encoder end.
To get natural (analog) signal at the output, the last and final stage employed
is source decoder. Like channel decoding, source decoder also decodes the input
signal according to the source encoding scheme applied at the transmitter end.
The schemes may be anything out of PCM, Delta Modulation, Adaptive Delta
Modulation (ADM), sigma delta modulation (SDM). The performance of the channel decoder-source decoder pair depends upon the statistical prediction of the
amount of error introduced per unit time, i.e., bit error rate. The parameter bit error
rate (BER) is highly dependent upon the waveform used at the time of source encoding, the modulation scheme applied, etc. When the two extreme end signals (input
analog signal at the transmitter and recovered analog signal at the receiver) are compared, the difference is the measure of the total distortion introduced by the digital
communication system.

1.6 MATLAB Programs
1.6.1 Time and Frequency Domain Representation of Signals
% In CD: ch1_1.m
% Time & frequency domain representation of signals
% Output: Fig. 1.3
% Programed by Apurba Das (Aug,’09)
clc;
clear all;
close all;
t=0:.001:5;
T=2;
f=1/T;

12

phi=pi/4;
a=2∗ cos(2∗ pi∗ f∗ t+phi);
subplot(2,1,1);
plot(t,a);
xlabel(’time-->’);
ylabel(’amplitude-->’);
title(’Waveform’);
f_dom=fft(a);
f_dom=fftshift(f_dom);
% figure;
subplot(2,1,2);
plot(abs(f_dom));
xlabel(’frequency-->’);
ylabel(’amplitude-->’);
title(’Amplitude Spectrum’);
figure;
f=20;
a=a.∗ sin(2∗ pi∗ f∗ t);
subplot(2,1,1);
plot(t,a);
xlabel(’time-->’);
ylabel(’amplitude-->’);
title(’Waveform’);
f_dom=fft(a);
f_dom=fftshift(f_dom);
% figure;
subplot(2,1,2);
plot(abs(f_dom));
xlabel(’frequency-->’);
ylabel(’amplitude-->’);
title(’Amplitude Spectrum’);

1.6.2 CTSV, DTCV, DTDV Signals
% In CD: ch1_2.m
% CTCV, DTCV, DTDV signals
% Output: Fig. 1.5
% Programed by Apurba Das (Aug,’09)
clc;

1 Preview and Introduction

References

clear all;
close all;
t=0:.001:5;
T=5;
f=1/T;
a=1.5∗ (5∗ sin(2∗ pi∗ f∗ t)+sin(7∗ pi∗ f∗ t));%+4∗ cos(4∗ pi∗ t);
subplot(3,1,1);
plot(t,a,’k’);
xlabel(’time-->’);
ylabel(’amplitude-->’);
title(’CTCV’);
subplot(3,1,2);
t1=0:.5:5;
a1=1.5∗ (5∗ sin(2∗ pi∗ f∗ t1)+sin(7∗ pi∗ f∗ t1));
stem(t1,a1,’k’);
xlabel(’time(nT_S)-->’);
ylabel(’amplitude-->’);
title(’DTCV’);
subplot(3,1,3);
res=20/8; % resolution
[r c]=size(a1);
s=zeros(1,9);
s(1)=-10;
for i=2:9
s(i)=s(i-1)+2.5;
end;
for i=1:c
for j=2:9
if (a1(i)<=s(j) & a1(i)>=s(j-1))
a2(i)=(s(j-1)+s(j))/2;
end;
end;
end;
stem(t1,a1,’k’);hold on;
stairs(t1,a2,’k--’);
xlabel(’time(nT_S)-->’);
ylabel(’amplitude-->’);
title(’DTDV’);
legend(’Sampled signal’,’Quantized signal’);

References
1. Das, A., “Line Communication System”, New Age International, 2006
2. Proakis, J.,“Digital Communication”, McGraw-Hill, 2000

13

Chapter 2

Waveform Encoding

2.1 Introduction
‘Any natural signal is in analog form’. To respect the said statement and to meet
the basic requirement of any type of digital signal processing and digital communication, the essential and prior step is converting the electrical form (through
transducer) of the natural analog signal into digital form, as digital modulator or any
type of digital signal processor does not accept analog signal as its input. Therefore,
to consider the digital transmission of analog signal, it is very important to encode
the waveform 1st. This process of waveform encoding is done through sampling,
quantization and encoding; finally the analog information is converted to digital
data.
The digitally coded analog signal produces a rugged signal with high immunity to
distortion, interference and noise. This source coding also allows the uses of regenerative repeater for long distance communication. In the process of quantization,
the approximation results in quantization noise and with a target of removing the
noise, the bandwidth becomes comparable to the analog signal. Hence, a trade-off
between the noise and bandwidth is to be established.
According to the specification in terms of accuracy-bandwidth trade off different
schemes of source encoding are employed. The basic procedure of digital transmission of analog signal is coded pulse modulation or PCM. Quantization can be
made non-uniformly  to catch up intimate details. To reduce the bandwidth in
an efficient way, the process of differential pulse code modulation (DPCM) and
delta-modulation (DM)  are used. In the prediction process of DM, reduction of
hardware can be ensured by a nice engineering called as delta-sigma modulation
(DSM). At the end of the chapter, the linear prediction is discussed with the help of
Genetic Algorithm (GA).

2.2 Pulse Code Modulation (PCM)
Pulse code modulation (PCM) is a digital transmission system of analog signal having an
analog to digital converter (ADC) at the input and a digital to analog converter (DAC) at the
output.

A. Das, Digital Communication, Signals and Communication Technology,
C Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-12743-4_2, 

15

16

2

Waveform Encoding

From the previous discussion it is granted that, the analog signals found in nature
are essentially continuous time continuous valued signal. To make the signal
compatible to feed into digital communication system, the signal needs to be converted to another form which is essentially represented using digits only. This
analog-to-digital conversion follows principally three chronological steps namely,
(a) Sampling, (b) Quantization and (c) Encoding. By the process of sampling, the
time continuity is broken taking care of sufficiently large (almost 1) probability
of regeneration. Quantization reshapes each sample height in voltage or in current
mode. Here we get defined levels of voltage/current with respect to defined instants
of time. That’s why this signal obtained after quantization is called as discrete time
discrete valued signal. Now, the remaining task is the representation of the defined
set of finite number of probable time and voltage/current values. That can be done by
mapping each of the elements of the set to a digital code. Automatically an encoder
can map the entire signal into digit space.
When the digits are received un-erroneously (for the simplicity of understanding,
it’s assumed that no errors occur into the communication channel; if error comes,
the problem can be taken care of by error detection and correction in channel coding
to be discussed in the Chap. 9) at the destination end, formation of the word is
essential. Each word is the tag of each quantization level. Next, a synchronized clock
oscillating in sampling frequency can segregate the levels, define the time instants
respective of the each level and then a discrete time discrete valued signal is obtained
here again merely identical to the input signal of the encoder of the transmitter. The
signal is obviously a multi-frequency signal where high frequency component arises
due to the process of sampling. Now a low pass filter can reconstruct the original
signal with some quantization error, which can be minimized by just increasing the
number of quantization levels, i.e., by increasing the number of bits to represent
each level.
Here a question of bandwidth noise trade-off arises. The objective of communication must play an important role here. There may a priority on accuracy,
may be a priority of speed, or there may exist a compromise between these two.
Depending upon the user or system requirement, a lot of variations of PCM are
designed.

2.2.1 Process of Sampling
The word ‘sample’ means a small portion of physical element/phenomena which
singularly or collectively has all the characteristics of the original physical element/phenomena. As discussed in Chap.1, this is the 1st step towards analog to
digital signal conversion. Let’s consider the signal m(t) of bandwidth B Hz. When
the signal is multiplied by a train of impulses, the resultant signal is obviously the
sampled version of m(t). The impulses are equi-spaced by an amount of time TS .
After sampling, the signal g(t) becomes discrete in time domain (Fig. 2.1).

2.2

Pulse Code Modulation (PCM)

17
M(ω)

m(t)

t→

– 2πB

0

2πB ω →

ΔnTs (t)

G(ω)

Ts
g(t)

Fig. 2.1 The process of sampling to get discrete time signal

To establish the relationship between the analog signal and discrete (sampled)
signal, let’s take the example of sampling of a sinusoid s(t), sampled at a rate of FS ,
i.e., sampling interval TS .
The analog signal is given by,
S(t) = A sin (t + φ)

(2.1)

where,
A is the amplitude,
 is the analog angular frequency = 2π F,
F is the analog frequency,
φ is the initial phase
After sampling, we are taking the signal at t = 0, TS , 2TS .,. . .NTS .
Or, t=nTS , (n=1,2,3,. . .N), N=number of samples.
Therefore, from Eq. (2.3),
S(nTS ) = A sin (2π FnTS + φ)


F
⇒ S(n) = A sin 2π ×
+ φ as TS = 1/FS
FS
⇒ S(n) = A sin (2π × f + φ)
⇒ S(n) = A sin (ωn + φ)

(2.2)

18

2

Waveform Encoding

taking
f =

F
FS

From Eq. (4.1) and (4.2), the correspondence of sampling is shown. The sequence
formed as S(n) and the input signal S(t) to the sampler, are correspondent.
Looking closely regarding the units of the different variations of frequencies,
generated as a bi-product of the process of sampling, we have the following derived
units.
Unit of F= cycles/s
Unit of FS = samples/s
Therefore,
Unit of digital frequency, f=cycles/sample
Unit of analog angular frequency, , i.e., 2π F = rad.cycles/s
Unit of digital angular frequency ω=rad/samples.
Here, if one decrease the spacing between the samples (TS ), i.e., increase the
rate of sampling FS , the reconstruction would be easier at the receiver side. But,
the bandwidth of transmission would be increased thereafter, it will affect the processing time of the sampled signal. So, there is a trade-off between noise and
transmission bandwidth. However, it’s really important to know the lower limit of
the choice of the sampling frequency for successful reconstruction.
2.2.1.1 Sampling Theorem
An analog signal can be reconstructed from its sampled values un-erroneously, if the
sampling frequency is at least twice the bandwidth of the analog signal.

Say an analog signal m(t) has three different frequency components f1 , f2 , and f3 , or
combination of all of them, where f1 < f2 < f3 then the bandwidth of that m(t) signal
must be f3 . For ease of calculation, we are taking bandwidth of that signal B Hz, in
general.
Therefore we can say the spectrum M(ω) is band limited by 2πB in ω scale,
and band limited by B Hz in f scale. Now, sampled signal is nothing but the signal
obtained by multiplying m(t) by unit impulse train nTs (t). From the figure the
sampled signal is g(t) = m(t) × nTs (t). Time period of the impulse train is Ts .
therefore frequency is fs = 1/ Ts . Let’s now expand the nTs (t) signal in Fourier
series so that we can study the spectrum of g(t) signal. As nTS (t) is even function
of time, By Fourier series,
nTS (t) = a0 +

∞

n=1

(an cos nωs t + bn sin nωs t)

(2.3)

2.2

Pulse Code Modulation (PCM)

19

Here,

1
1
nTS (t) dt =
TS TS
TS

2
2
an =
nTS (t) cos nω0 tdt =
TS TS
TS

a0 =

bn = 0, for even function

Here, TS signifies integration within upper limit to lower limit difference TS .
Therefore,
1
(1 + 2 cos ωs t + 2 cos 2ωs t + 2 cos 3ωs t + ............)
TS
⇒ g(t) =m(t) × nTS (t)
1
= (m(t) + 2m(t) cos ωs t + 2m(t) cos 2ωs t + 2m(t) cos 3ωs t + ........)
TS

1
e jωs t + e−jωs t
e2jωs t + e−2jωs t
=
m(t) + 2m(t) ×
+ 2m(t) ×
TS
2
2

3jω
t
−3jω
t
s
s
+e
e
+ .....
+2m(t) ×
2


1 
=
m(t) + m(t) e jωs t + e−jωs t + m(t) e2jωs t + e−2jωs t
TS

+m(t) e3jωs t + e−3jωs t + .....
(2.4)

nTS (t) =

From the 1st term of the equation above, it is very clear that spectrum of g(t) will be
similar to M(ω) except the amplitude. Amplitude will be 1/ Ts times of that of G(ω).
2nd term indicates M(ω) shifted in both sides by the amount of ωs (using negative
frequency concept). Similarly 3rd term indicates M(ω) shifted in both sides by the
amount of 2ωs and so on.
Now if we can extract the central spectrum by low pass filter from the assembly
of spectrum, then we can easily reconstruct the m(t) signal.

Case I
From the Fig. 2.2 above specific fs i.e. ωs is taken into account. It is very
clear that central spectrum can be extracted using LPF. Therefore it is success of
reconstruction.
Here ωs − 2πB = 2πB
Or, ωs = 4πB
Or, fs = 2B

20

2

Waveform Encoding

G(ω)

–fS–B –fS

–fS+B

0

fS–B

–B

0

B

fS

fS+B

2fS 2fS+B
f→

Fig. 2.2 Sampled spectra for fS =B

Case II
From the Fig. 2.3 above other fs i.e. ωs is taken into account. It is very clear that central spectrum cannot be extracted using LPF; because, there is an overlap region of
two spectra. The error due to this overlap region is called as aliasing error. Therefore
it is failureof reconstruction.
Here ωs − 2πB < 2πB
Or, ωs < 4πB
Or, fs < 2B
Fig. 2.3 Sampled spectra for
fS <B

G(ω)

–fS–B

–fS

–fS+B 0 fS–B
–B

0

fS fS+B
B

f

Case III
From the Fig. 2.4 above specific fs i.e. ωs is taken into account. It is very clear that
central spectrum can be extracted using LPF more easily than case I. Therefore it is
success of reconstruction.
Here ωs − 2πB > 2πB
Or, ωs > 4πB
Or, fs > 2B
Therefore collectively we can say fs > 2B i.e., sampling theorem is proved. The
minimum frequency of sampling for successful reconstruction is Nyquist sampling
rate (fs = 2B)

2.2

Pulse Code Modulation (PCM)

21

G(ω)

–fS–B

–fS –fS+B

fS–B

0
–B

0

fS

B

fS+B

2fS

f

Fig. 2.4 Sampled spectra for fS >B

Collectively, the condition for successful reconstruction of the message signal
can be derived therefore as,
fS ≥ 2B

(2.5)

2.2.1.2 Aliasing
Say, two sequences are there s1(n) and s2(n). If the digital angular frequency difference between these two sequences is integral multiple of 360◦ , the sequences would
be identical.
Say,
s1(n) = A sin (ω1 n + φ)
s2(n) = A sin (ω2 n + φ)

(2.6)

and
ω1 = ω2 + 2π
Then,
s1(n) = A sin (ω2 n + φ + 2π n) = A sin (ω2 n + φ)

(2.7)

It is a severe error. The above equation signifies, after sampling the uniqueness of
the signal destroys. Two different sinusoids map to a single sequence after sampling
(Fig. 2.5). This error is called as aliasing error and these two sequences are called
as alias of other.
To overcome the problem of aliasing, the choice of ω should be such that, any ω
must reside within ±π . Then only the difference between any two ω’s be restricted
within the range of 2π as shown in the Fig. 2.6.

22

2

Waveform Encoding

Simulated analog signals, x = dots y = solid

Amplitude

2
1
0
–1
–2

0

2

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Time
Simulated Sampled signals, x = x y = o

Samples

1
0
–1
–2

0

1

2

3

4

5

6

7

8

9

10

Time

Fig. 2.5 Illustration of the sampling theorem
Fig. 2.6 Solution to the error
of aliasing
−π

ω1

ω2

π

2π

Therefore, for 0 aliasing, the digital angular frequency should be as follows
|ω| ≤ π
F
≤π
⇒ 2π
Fs
Fs
⇒F≤
2
⇒ Fs ≤ 2F

(2.8)

Equation (2.8) is indeed the mathematical model for statement of the sampling theorem, and the process of sampled sinusoid signal correspondence is another way to
prove the sampling theorem.

2.2.2 Process of Quantization
After sampling what we get is a sequence of impulses with value continuity. The
impulses are separated by an amount of the sampling time TS , but one cannot infer

2.2

Pulse Code Modulation (PCM)

23

about the amplitude of the signal even after sampling in terms of discontinuity. It
means, a set can be defined with finite number of time elements but a set with finite
number of probable amplitude elements cannot be defined yet. The amplitude may
be anything between the Amax and Amin . As for an example, if the signal is within
voltage range +5 V and −5 V, at an instant of sampling the amplitude may be
3.0000000029 V. And we do not have the control over the accuracy permission, i.e.,
how many decimal digits we should allow, or truncate or round off and so on. This
signal is therefore a discrete time continuous valued (DTCV) signal after sampling.
Now, we need to make it properly a discrete time discrete valued (DTDV) signal.
The process by which a set of finite number of amplitude values are also defined
is called as quantization. Say, a set is defined with voltage values 1, 2, 3, 4, 5 and
their negative counterpart volts. Then any original value obtained by sampling will
be approximated to the closest defined value. For some cases, the sampled values
may be quantized high; sometimes they may be quantized low with respect to the
defined levels (Fig. 2.7).
To understand closely, let’s consider the signal of Fig. 2.8. The maximum and
minimum values are designated as Amax and Amin . Now, the entire range Amax

CTCV
amplitude-->

10
0
– 10

0

1

2

3

4

5

3

4

5

time-->

amplitude-->

DTCV

X=1
Y = 5.7063

10
0
– 10

0

1

2
time(nTS)-->
DTDV

amplitude-->

10

Sampled signal
Quantized signal

0

– 10

0

1

2

3
time(nTS)-->

Fig. 2.7 Time valued definition of signal

4

5

24
Fig. 2.8 Illustration of
uniform quantization and
encoding

2

Waveform Encoding

Δ3

q3 (11)

Δ2

q2 (10)

Δ1

q1 (01)

Δ0

q0 (00)

to Amin is equally divided into 4 regions of width S. The 4 regions are named as
0 to 3 . The middle of each of the -regions is our defined probable amplitude
level. The levels are called as q-levels or quantization levels. The algorithm of quantization states, ‘If the sampled signal amplitude lies within i region, the amplitude
will be quantized (approximated) to the amplitude qi ’. Figure 2.8 clearly depicts the
process.
The shaded signal in the above figure is the staircase representation of the quantized signal. Next, each q-level can easily be encoded to through 2 bit encoder or
MUX to get proper digital representation of the input analog signal. After quantization, the signal xq (kTS ) is represented as q2 , q3 , q2 , q2 , q3 , q0 , q0 , q0 , q0 , q1 . And
after encoding, each level is inferred by a pair of digits in parallel. Next, by employing a parallel to serial converter, the serial bit stream is obtained which is the
corresponding digital representation of the input analog signal.
There are two types of uniform quantizer namely,
(b) Midrise quantizer
Figure 2.9 shows the input-output characteristics of the two types of the uniform
quantizer. In midtread type quantizer, the origin lies in the middle of the tread of
the staircase like graph. In midrise type, the origin the graph lies in the middle of
a rising part of the staircase like graph. It can be noticed that, both midtread and
midrise graphs are symmetric about the origin.

2.2.3 PCM Transmitter and Receiver
2.2.3.1 PCM Transmitter
The entire principle of analog to digital conversion is now unveiled to us. Therefore,
we can just club up all the procedural blocks to construct the PCM transmitter as
shown in Fig. 2.10. We have already discussed the functioning of the blocks. First
the input analog signal x(t) is being passed through a low pass filter of bandwidth

2.2

Pulse Code Modulation (PCM)

25
Midrise quantizer

4
3
3
2
2
1
xq(kTs)-->

xq(kTs)-->

1
0

0
–1

–1

–2

–2

–3
–3
–4

–2

0
2
x(kTs)-->

4

–4
–4

–2

0
x(kTs)-->

2

4

Fig. 2.9 Midtread and midrise quantizer characteristics

x(t)

LPF
B

x(kTS)

S/H

q-level
quantizer

xq(kTS)

M-ary
Encoder
ν digits

fS ≥ 2B

Timer
P to S
converter
PCM

Fig. 2.10 PCM transmitter

B Hz to restrict the bandwidth of the signal for prevention of aliasing error. Next,
the three basic steps of analog to digital conversion (ADC) is being followed
i.e., sampling-quantization-encoding, as discussed before. But here the encoded bit
stream is essentially parallel in form, although the objective is to get serial bit stream
in response to serial analog signal input .
The encoder works with M-ary digits (if M=2, it becomes binary) and produces
a codeword consisting of ν digits in parallel, corresponding to each sample. Since
there are M ν (in binary, 2ν ) possible M-ary codewords, with ν digits per word,

26

2

Waveform Encoding

unique coding of the q different quantization levels requires that, M ν ≥ q. The
parameters should be chosen removing the inequality, such that,
q = M ν or,

ν = logM q

(2.9)

For binary PCM, number of bits per q-level can be calculated as ν = log2 q.
Finally, the successive code words are read out serially. The parallel to serial conversion also needs the clocking by sampling frequency. Otherwise, either one word
would superimpose or overwrite the other, or one code word would be converted
twice. In both the cases, we get improper conversion. Therefore, clocking in the
parallel to serial converter is essential.
As, each encoded sample is represented using ν digits, so the signaling rate at the
output becomes r = νfS . Therefore, for transmission of PCM signal, the bandwidth
needed is
B.W.tr ≥

1
1
r = νfS ≥ νB
2
2

(2.10)

The transmitted PCM signal is now passed through transmission channel towards
the receiver. Contamination of noise is almost obvious during the traversal of the
signal through transmission path. Though the received signal is noise accumulated,
the process of regeneration yields a nearly errorless signal as the SNR becomes
sufficiently large. In the receiver section ultimately we need to get the reconstructed
analog signal. Therefore, one digital to analog conversion (DAC) is to be designed.
As shown in the Fig. 2.11, the DAC operation is just the opposite, i.e., serial to
parallel conversion (for getting proper form of input to the next stage, i.e. decoder),
M-ary decoding and sample and hold. The output of the sample and hold circuit is
the staircase type waveform, xq (t).
The waveform shown in Fig. 2.12 may be regarded as a staircase approximation
x(t). A low pass filter is then employed for smoothening. The low pass filtering
produces the reconstructedanalog signal x̃(t), which differs from x(t) to the extent
that the quantized samples xq (kTS ) differ from the exact sampled values (x (kTS )).

DAC
PCM
+
Noise

PCM
Regen

fS ≥ 2B

Fig. 2.11 PCM receiver

S to P
converter

Timer

ν digits

M-ary
Decoder

S/H

~
x (t)

LPF
B

2.2

Pulse Code Modulation (PCM)

27
x (t) xq (t)

Fig. 2.12 Waveform
reconstruction through PCM
receiver as shown in Fig. 2.11

~
x (t )

Δ3
Δ2
Δ1
Δ0

2.2.4 Quantization Error
It is now understood that, the process of quantization is nothing but approximation to
nearer level of voltage/current. Due to this approximation, there a random amount of
difference occurs between actual and quantized value, named as quantization error.
As the error is like a random number and adds noise to modulation process, the error
can also be viewed as a noise called as quantization noise Fig. 2.13.

Quantization error characteristics

5
4
3

ε = xq(kTs) –x(kTs)-->

2
1
0
–1
–2
–3
–4
–5
–4

–3

–2

–1

Fig. 2.13 Quantization error characteristics

0
x(kTs)-->

1

2

3

4

28

2

Waveform Encoding

The noise is now to be calculated as mean-square quantization error, e2 where
e is the difference between the actual and approximated value of voltage/current.
The peak to peak amplitude of the sampled signal xs (t) is divided into M equal
levels each of width S. At the centre of each level of width the quantization levels
are located as x1 , x2 , . . . xM as shown the Fig. 2.14. Since, in the figure xs (t) happens
to be the closest to the level xk , the quantizer output will be xk and obviously, the
quantization error is e = xs (t) − xk .
Fig. 2.14 Interpretation of
quantization error

xS
xk
x3

x2
x1

Let, p(x) dx be the probability that xs (t) lies in the voltage/current range x + dx/2
to x − dx/2. Then the mean square quantization error  is
x1+S/2

e2

=

x2+S/2

p(x) (x − x1 ) dx+
2

x1 −S/2

p(x) (x − x2 )2 dx + . . .

(2.11)

x2 −S/2

Now, the PDF (probability density function) p(x) of the message signal will not
certainly be constant throughout each division. Assuming large M, i.e., with sufficiently small S, p(x) can be taken as constant throughout each division. Then, the 1st
term of the right hand side of the Eq. (2.11), p(x) = p(1) =constant. The 2nd term is
p(x) = p(2) , and so on. Hence, the constant terms may be taken out of the integration
sign. If we now substitute y ≡ x − xk , the expression in Eq. (2.11) becomes

e2


= p(1) + p(2) + . . .

S/2
y2 dy
−S/2


S3
= p(1) + p(2) + . . .
12

S2
= p(1) S + p(2) S + . . .
12

(2.12)

Now, according to definition, p(1) S be the probability that the signal lies within the
1st quantization range, p(2) S be the probability that the signal lies within the 2nd
quantization range, and so on. Hence, the sum of the terms in the bracket in the

2.2

Pulse Code Modulation (PCM)

29

Eq. (2.12) must have unity value, as signal must lie within the peak-to-peak value,
i.e., summation of all quantization range. Therefore, the mean square quantization
error is
e2 =

S2
12

(2.13)

2.2.5 Signal to Noise Ratio (SNR) for Quantized Pulses
A signal with maximum and minimum amplitudes as +A and –A is assumed. The
signal is quantized in M levels with each of width S. Then the peak to peak range of
amplitude (voltage or current) is App =A− (−A) =2A. As understood from the graph
shown in Fig. 2.13, the maximum value of quantization error is S/2 or –S/2 in either
direction. Therefore, the degradation of the signal is limited to the amount of S/2
in additive or subtractive manner. It is obvious that, the peak to peak amplitude can
also be expressed in terms of number and width or quantization levels as App =M×S.
The peak power of the analog signal (normalized to 1 ) can be expressed as

Ap 2 =

App
2

2


=

MS
2

2
=

M 2 S2
4

(2.14)

Now, we do have already the expression for quantization noise. Therefore, from
Eq. 2.13 and Eq. 2.14 we can derive the signal to noise ratio for uniformly quantized
pulses as 
M 2 S2
St
= 42 = 3 M 2
Nq
S
12

(2.15)

The above equation supports our common understanding that, SNR will be
improved as the number of levels to be squared is increased. As a limit if M → ∞,
the PCM signals will be converted to PAM signal, i.e., no quantization will be done.
Example 2.1 Consider an audio signal as given
s(t) = 3 cos 500π t V
(i) Find the SNR when s(t) is quantized using 10 bit PCM.
(ii) How many bits of quantization is needed for achieving SNR of 40 dB.
(i) SNR =

St
12PS
= 2 ,
N
S

30

2

Waveform Encoding

where PS is the RMS signal power defined as,
32
= 4.5 W
2
VH − VL
3 − (−3)
S=
= 5.86 mV
=
M
2 × 1010
St
12 × 4.5
∴ SNR =
= 1.56 × 106
=
N
(5.46 × 10−3 )2

St
⇒ SNR |dB =
= 10 log 1.57 × 106 ≈ 62 dB
N dB

PS =

(ii) We know, 40 dB = 4 × 10 log 10 = 10 log 104
Therefore, to find out the number of bits of quantization,
12 × 4.5

= 104
(6/2ν )2
22ν
104
⇒
=
2
12 × 4.5
(6)
36 × 104
⇒ 22ν =
12 × 4.5

Taking log2 of both sides, we get

⇒ 2ν = log2
∴ν=

12.7
2

36 × 104
12 × 4.5


= 12.7027

= 7 bits

2.2.6 Non-uniform Quantization: Companding
For most of the voice communication channels, very low speech volumes predominate for more than half of the times, the voltage characterizing detected speech
energy is 25% of the RMS value. Large amplitude values are relatively rare; only
10–15% of the time the voltage exceeds the RMS value. This is a peculiar statistics of speech signals. Therefore, the spacing at the high amplitude level becomes
an wastage and for incorporation of that unused q-levels, we do have to go for
high bandwidth transmission of low accuracy transmission, if we employ uniform
quantization.
In case of the design of uniform quantization, the quantization error varies
directly proportional to the square of the uniform width of quantization levels,
i.e., the quantization error is inversely proportional to the square of number of
quantization levels. Now, to reduce the error, if we plan to increase the number

2.2

Pulse Code Modulation (PCM)

31

of q-levels, it requires more number of bits to represent each level. Hence, the transmission bandwidth will be increased significantly. To bypass the trade-off between
these two (transmission bandwidth and quantization error), an overall reduction of
quantization error is achieved by judicially choosing the widths (non-uniformly) of
the levels keeping the total number of quantization levels unchanged [4, 5].
The ratio of voltage levels covered by voice signals from peaks of loud talk to the
weak passages of weak talk is of the order of 1000:1. Such variation is taken care of
by non-uniform quantization of the signal. The non-uniform quantization helps the
practical variability is such a way that, the step size automatically increases as the
separation from the origin of the input-output amplitude characteristics is increased.
In weak passages, the step size becomes smaller and the details are not missed at
all. On the other hand, by employing uniform quantization, either number of steps
should be made higher or the weaker signal details have to be compromised.
The procedure for getting non-uniformly quantized samples is compression followed by uniform quantization. At the receiver side also, the reverse operation, i.e.,
expanding followed by uniform quantization is done, as shown in Fig. 2.15. The
inclusion of COMpression and exPANDING renames the process of non-uniform
quantization as a contraction COMPANDING (Fig. 2.16).
It does can be easily understood that, the compression characteristic and
parameter of the characteristic is really an important factor to know for reliable
and successful communication. A typical compressor characteristic is shown in

Strong signal
Weak signal

5
4
3
2
1
0
−1
−2
−3
−4
−5

0

1

2

3

Uniform Quantization

4

5

6

7

8

9

Non-uniform Quantization

Fig. 2.15 Uniform quantization vs. non-uniform quantization

10

32

2

Compressor

Uniform
quantizer

Uniform dequantizer

Non-uniform quantization

Waveform Encoding

Expander

Non-uniform de-quantization

Fig. 2.16 The process of COMPANDING

Fig. 2.17 Compressor
characteristics

S2max

Compression
No compression

Output, S2
S1min

S1max
Input, S1
S2min

Fig. 2.17. A lot of compressor laws with different parameter orientation are there.
Among them A-law and μ law are accepted by CCITT1 and are discussed here.
μ law are mainly used in North America and Japan. A-law is used in Europe and
rest of the country.
2.2.6.1 µ-Law
The compression algorithm following μ-law is given by
|s2 | =

log (1 + μ|s1 |)
log (1 + μ)

(2.16)

where, s1 and s2 are normalized input and output voltage/current respectively. μ is
a non-negative parameter which determine the degree of compression. For μ = 0,
we obtain uniform quantization. The plot of the normalized output with normalized

1 CCITT is the contraction of Consultative Committee for International Telegraphy and Telephony.

2.2

Pulse Code Modulation (PCM)

33

μ = 255
1

s2 ↑
0.8

μ=5

0.6
0.4

μ=0

0.2
0

0

0.2

0.4

0.6

0.8

1

s1 →

Fig. 2.18 Compression by μ-law

input with different values of μ is shown in Fig. 2.18. The quantization step is
defined as the reciprocal of the slope of the compression curve given by
d |s1 |
log (1 + μ)
=
(1 + μ |s1 |)
d |s2 |
μ

(2.17)

For lower values of input, μ |s1 | << 1, therefore,
|s2 | ≈

μ |s1 |
log (1 + μ)

(2.18)

and
d |s1 |
log (1 + μ)
≈
d |s2 |
μ

(2.19)

For higher values of input, μ |s1 | >> 1, therefore,
|s2 | ≈

log (μ |s1 |)
log (1 + μ)

(2.20)

and
d |s1 |
≈ |s1 | log (1 + μ)
d |s2 |

(2.21)

Therefore, the μ-law is linear at low input levels and approximately logarithmic at
high input levels.

34

2

Waveform Encoding

2.2.6.2 A-Law
Another well known compression algorithm following A-law is given by
A |s1 |
;
1 + log A
1 + log (A |s1 |)
|s2 | =
;
1 + log A
|s2 | =

0 ≤ |s1 | ≤

1
A

(2.22)

1
≤ |s1 | ≤ 1
A

(2.23)

Here also, A is a non-negative parameter of compression. A=1 signifies uniformquantization. The plot of the normalized output with normalized input with different
values of A is shown in Fig. 2.19. The quantization step is defined as the reciprocal
of the slope of the compression curve given by
⎧
1 + log A
1
⎪
⎪
; 0 ≤ |s1 | ≤
d |s1 | ⎨
A
A
(2.24)
=
d |s2 | ⎪
1
⎪
⎩ (1 + log A) |s1 | ;
≤ |s1 | ≤ 1
A
Thus, the quantization steps over the central linear segment are diminished by factor A/(1+logA). These steps over the linear segment have a dominant effect on
small signals and are diminished practically by about 25 dB as compared to uniform
quantization.

A = 100
1

s2

↑
0.8
0.6

A=5

0.4

A=1

0.2
0

0

0.2

0.4

0.6

0.8

1

s1

→

Fig. 2.19 Compression by A-law

Example 2.2 When a 256 level quantization is employed for a sinusoid with peak
voltage 40 Volts, what is the voltage interval without compression?
For μ = 255, what is the maximum and minimum separation between the
levels?

2.3

Differential Pulse Code Modulation (DPCM)

35

(i) For uniform quantization, the step size, S can be determined easily
as
S=

2 × 40
2×V
=
= 312.5 mV
256
256

(ii) To apply μ-law for non-uniform quantization, for μ = 255
The smallest height h would be increased in GP as
h + 2 h + 4 h + 8 h + 16 h + 32 h + 64 h + 128 h = 2 V
⇒ 255 h = 2 V
⇒ h = 0.0078 V = 7.8 mV
⇒ 128 h = 1.0039 V
Therefore the maximum and minimum step size employing μ-law for nonuniform quantization, for μ = 255 are 1.0039 V and 7.8 mV respectively.

2.3 Differential Pulse Code Modulation (DPCM)
In analog messages we can make a good guess about a sampled value from the
knowledge of the past sampled values. In other words, the sampled values are
not independent, and generally there is a great deal of redundancy in the Nyquist
samples. Proper exploitation of this redundancy leads to encoding a signal with
lesser number of bits. Consider a sampling scheme where instead of transmitting
the sampled values, we transmit the difference between the successive samples. By
employing the technique of transmitting the quantized difference values of the successive samples we can efficiently use the bandwidth provided by the transmitting
channel.

2.3.1 Cumulative Error in PCM
In general PCM system, a quantization error eqi is added to the quantized output
while quantizing the ith sample. Now just take a look on how the quantization error
affects the DPCM output.
Here Zi is the sampled value at ith instant and Ẑi is the predicted sample (considered
as delayed sample) at ith instant.
Now, from the Fig. 2.20,
di = Zi − Ẑi
= Zi − Zi−1
Or recovered sample at the receiver is Z i = d̂i + Z i−1
For the 1st sample, i=1. So, Z 1 = d̂1 + Z 0

(2.25)

36

2
di

^

di

∑

Zi

Zi

∑

Quantizer

Waveform Encoding

–

Delay

^

Zi

Zi−1
Transmitter

Fig. 2.20 Basic DPCM communication system

= d̂1 + Z0 (As at 0th instant Z 0 = Z0 )
And the quantized difference
d̂i = di + eqi

(2.26)

So, from the diagram presented in Fig. 2.20,
Z 1 = d1 + eq1 + Z0
Or Z 1 = Z1 + eq1

(2.27)

Now, for i=2, we get from Eq. (2.25),
Z 2 = d̂2 + Z 1
Z 2 = d2 + eq2 + Z1 + eq1

(2.28)

Z 2 = Z2 + eq1 + eq2
Proceeding in this way we can see
Z n = Zn +

n


eqi

(2.29)

i=1

So, here we see that in case of basic configuration of DPCM, the output at nth instant
is affected not only by the quantization error of nth instant, but also by the errors
of previous instants. The error becomes severe as the term of samples increases. To
overcome the problem, a new configuration is designed.

2.3.2 Prevention of Cumulative Error by Applying Feedback
In the receiver section, a delay feedback path is used for proper recovery of the
message sample. To solve the problem of cumulative quantization error in the transmitted samples, we have just copied the feedback loop of the receiver section and
pasted to the transmitter section as shown in Fig. 2.21.

2.3

Differential Pulse Code Modulation (DPCM)

Fig. 2.21 DPCM
communication system
with error suppressor

di
∑

Zi

–

Zˆi

37
^

^

Quantizer di

di

Zi−1

–

∑

Zi

∑

Zi−1

Z−1

Z−1
Transmitter

feedback circuit has been added to the transmitter side. The feedback circuit is basically same as the receiver circuit. Now let us analyze the transmitter. Here the block
Z −1 signifies 1 unit delay (delay by an amount of TS ).
At ith instant
di = Zi − Z i−1 and d̂i + Z i−1 = Z i

(2.30)

d̂i = di + eqi

(2.31)

Also,

For the 1st sample, i.e. for i=1,
d̂1 = d1 + eq1
So, Z 1 = d̂1 + Z 0
= d1 + Z 0 + eq1
= Z1 + eq1

(2.32)

For i=2, from Eq. (2.31) we get
d̂2 = d2 + eq2

(2.33)

Now,
d̂2 + Z 1 = Z 2
So,
Z 2 = d2 + eq2 + Z 1
= Z2 + eq2

(2.34)

38

2

Waveform Encoding

Proceeding in this way, we can say reconstructed output at any instant
Z n = Zn + eqn

(2.35)

So, comparing Eqs. (2.29) and (2.35), we can say that in the present case, the reconstructed output depends only on the quantization error of that particular instant. So
it’s free from cumulative error.
Thus if m[k] is the kth sample instead of transmitting m[k] we transmit the difference d[k] = m[k] − m[k − 1]. At the receiver knowing d[k] and the previous
sample m[k − 1], we reconstruct m[k] iteratively at the receiver. Now, the differences between successive samples are generally much smaller than the sample
values. Thus the peak amplitude mp of the transmitted value is reduced considerably because the quantization interval Av = mp /l, for a given l (no of bits per
sample) we can reduce the quantization noise given by A2v /12. This also means that
for a given transmission bandwidth (which is related to number of bits per sample
as BW = l×samples/s
) and for a given SNR we can reduce BW.
2
We can improve upon this scheme by estimating (predicting) the value of kth
sample m̂[k] from a knowledge of the previous sample. If this estimate is m̂[k], then
we transmit the difference (prediction error) d[k] = m[k] − m̂[k]). At the receiver
also, we determine the estimated m̂[k] from the previous sample, and then generate
m[k] by adding the received d̂[k] to the estimated m̂[k].
Now if our prediction is quite worth, the difference between m̂[k] and m[k] will
be much smaller than the difference between two sample values m[k] and m[k − 1].
Thus the number of bits required will be even lower than the previous case and
hence the bandwidth will also be lower.
In the basic assumption, the prediction for a particular sample is taken granted
as the delayed sample. In the next section, we’ll try to discuss the reason and logic
behind this assumption.

2.3.3 How We Can Predict the Future?
The approach to signal prediction (estimation) is discussed here using the spirit of
Taylor .
Let us consider, a signal m(t), which have derivatives of all order at t. Using
Taylor series we can express m(t + TS ) as
m(t + TS ) = m(t) +

T2
TS
m(t) + S m(t) + .............
1!
2!

≈ m(t) + TS m(t) For small TS

(2.36a)
(2.36b)

Equation (2.36a) shows that from knowledge of the signal and its derivative at
instant t, we can predict a future signal at t = TS . In fact, even if we know
just the first derivative, we can still predict this value approximately, as shown in

2.3

Differential Pulse Code Modulation (DPCM)

39

Eq. (2.36b). Let us denote the kth sample of m(t) as m[k], that is m(kTS ) = m[k],
and m(kTS ± TS ) = m[k ± 1].
Now, the derivative
m(kTS ) ≈

m[kTS ] − m(kTS − TS )
.
TS

So form Eq. (2.36b), we obtain


m[k] − m[k − 1]
TS
= 2m[k] − m[k − 1].



m[k + 1] ≈ m[k] + TS

(2.37)

It shows that, we can find a crude prediction of the k + 1th sample from the two
previous samples. The approximation improves if we take more terms in Eq. (2.36a).
To determine the higher order derivatives in the services, we require more samples
from the past. Larger the number of past samples, we can get a better approximation.

m[k] ≈ a1 m[k − 1] + a2 m[k − 2] + · · · · · · + aN m[k − N]

(2.38)

The right hand side is m̂[k] the predicted value of m[k]. Thus m̂[k] = a1 m[k − 1] +
a2 m[k − 2] + · · · + aN m[k − N].
This is the equation of an Nth order prediction. Thus we can design one prediction
filter whose output is m̂[k] and input is m[k − 1], m[k − 2], . . . m[k − N]. But we can
get these by simply delaying m[k] by TS .
The parameters a1, a2, . . ., aN are called prediction coefficients. This is also
called linier prediction. It is actually a transversal filter (a tapped delay line), where
the tap gains are set equal to the prediction coefficients as shown in Fig. 2.22.

m[k]

Delay

Delay

Delay

Delay

TS

TS

TS

TS

a1

a3

a2

∑

Fig. 2.22 Basic idea of linear prediction

Output m̂[k]

aN

40

2

Waveform Encoding

2.3.4 Analysis of DPCM
DPCM transmit d[k], the difference between m[k] and m̂[k] and at receiver we generate m̂[k] from the past sample values to which the received d[k] is added to generate
m[k].
There is, however, one difficulty in this scheme. At the receiver, instead of past
sample m[k − 1], m[k − 2], . . . m[k − N] as well as d[k], we have there quantized
versions, mq [k − 2],. . . mq [k − N]. Hence, we cannot determine m̂[k]. We can only
determine m̂q [k] the predicted value of quantized sample from previous quantized
samples. This will increase the error in reconstruction (because the error is here
additive, error in each quantized signal is added together).
In such case, a better strategy is to determine m̂q [k], the estimate of mq [k] instead
of m[k] at the transmitter also from the quantized sample. The difference d[k] =
m[k] − m̂q [k] is now transmitted using PCM (Fig. 2.23). At the receiver, we can
generate m̂q [k], and from the received d[k], we can reconstruct mq [k] (Fig. 2.24).
Here dq [k] = d[k] + q[k], where q[k] is the quantization error. The predictor
output m̂q [k] is fed back to its input so that the predictor input mq [k] is mq [k] =
m̂q [k] + dq [k]
= m[k] − d[k] + dq [k]

(2.39)

= m[k] + q[k]

d[k]
mq[k−1]

∑

dq[k]
Quantizer

–

m^ q[k]

+
∑

Prediction filter

Fig. 2.23 DPCM transmitter

Accumulator

Output mq[k]

dq[k]
∑

m̂q[k]

mq[k]
Predictor

Fig. 2.24 DPCM receiver

2.4

Delta Modulation

41

This shows that mq [k] is a quantized form of m[k]. The predictor input is indeed
mq [k] as assumed.
The receiver shown in Fig. 2.24 is identical to the shaded portion of the transmitter. The inputs in both cases are also the same dq [k]. The predictor output must be
mq [k].

2.4 Delta Modulation
Sample correlation used in DPCM is further explained in Delta Modulation (DM)
by over sampling (typically 4 times the Nyquist rate) the baseband signal. This
increases the correlation between adjacent samples, which results in a small prediction error that can be encoded using only one bit (no. of levels = 2). Thus DM
is basically a one bit DPCM, that is, a DPCM that uses only two levels (L=2) for
quantization of the m[k] − m̂q [k]. For more simple scheme the difference between
m[k] and m[k − 1] is encoded by only one bit. That is if m[k] is higher than m[k]
then 1 is transmitted, otherwise 0.
In comparison to PCM (and also DPCM), it is very simple method of A/D conversion. In DM the word framing is unnecessary in transmitter and receiver. This
strategy allows us to use fewer bits per sample for encoding a base band signal.
In delta modulation, we use a first order predictor, which, as seen earlier, is just
a time delay of TS . Thus the DPCM transmitter and receiver can be reconfigured to
delta modulation as shown in the following Fig. 2.25.
Fig. 2.25 Delta modulation
as a special case of DPCM

m[k]

dq[k]

∑

Quantizer

mq[k−1]

∑

Delay TS

mq[k]
Accumulator

Here
mq [k] = mq [k − 1] + dq [k]
Hence,
mq [k − 1] = mq [k − 2] + dq [k − 1]

42

2

Waveform Encoding

So,
mq [k] = mq [k − 2] + dq [k − 1] + dq [k]

(2.40)

Proceeding iteratively in this manner and considering 0 initial condition, i.e.
mq  = 0, we have mq [k] =

k


dq [n]

(2.41)

n=0

This shows that the receiver is just an accumulator. If the output dq [k] is represented by impulses, then the accumulator (receiver) may be realized by an integrator
because its output is the sum of the strengths of input impulses (sum of the areas
under the impulses). We may replace the feedback portion of the modulator (which
is identical to the demodulator) by an integrator. The demodulator output is mq [k],
which when passed through an LPF gives us the retrieved signal. The Fig. 2.26
bellows shows a practical implementation of the delta modulator and demodulator.

Comparator
m(t)

Sampler frequency fs

E

∑

dq(t)

–

–E

m^ q(t)

∫
Integrator

(a)
dq[k]

∫

~
m(t)

^
m
q(t)

Low pass filter

(b)
~

m(t)

m(t)

t
Pi(t)

Fig. 2.26 (a) Delta
modulator, (b) Delta
demodulator, (c) DM
waveforms

t
Po(t)
t

(c)

2.4

Delta Modulation

43

2.4.1 Drawbacks of Delta Modulation
The waveform 
m(t) needs to closely follow the waveform of m(t) in order to the
recovered waveform 
m(t) resembles m(t). Taking a careful look at the waveform
shows situations where 
m(t) is unable to follow m(t) as the slope of m(t) is higher
than that of 
m(t). Also when m(t) is varying slowly, the change of 
m(t) is higher
enough to cause an error.
When, at any instant or any duration of time the slope of m(t) is higher then the slope
of 
m(t), i.e. step size/time period, 
m(t) is unable to follow the input message signal
m(t). So the recovered waveform will be distorted. Figure 2.27 shows the case.
error in delta modulation
(DM)

~

~
m(t)

m(t)

Step size Δ

Time period TS

The condition for slope over loading is then,
dm(t)

≥
dtmax
TS

(2.42)

Example 2.3 For the signal s(t) = A sin 2π ft calculate the sampling frequency to
avoid slope overload error, considering 256 quantization levels and 1 kHz signal
frequency.
ds(t)
dt

= A.2π f
max

1
2Aπ f
≤
Ts



2A
,
⇒ fs ≤ Mπ f
∵M=

∴ fs =

M = Number of levels,
2A = peak to peak amplitude.
Now, putting M=256, f=1,

44

2

Waveform Encoding

fS ≥ 256 × 3.14 × 1 kHz = 803.84 kHz
Therefore, to avoid the error of slope overload, the sampling frequency must be
greater than 803.84 kHz.
2.4.1.2 Granular Noise
In other hand, when the input message signal m(t) is varying slowly, the change in

m(t) is comparatively higher that causes error in reconstruction. This type of error
or distortion is called Granular noise. Figure 2.28 shows the case.

Fig. 2.28 Granular noise
error in delta modulation

m(t)

m(t)

Re constructed signal

2.5 Adaptive Delta Modulation
The drawbacks of delta modulation are primarily due to the slope. These drawbacks
can be overcome by suitably changing the slop. Slope can be changed either by
changing the time period or the step size. Changing the time period will cause in
changing the frequency and bandwidth. So it is not possible to vary the time period
for a single message. Rather we can vary the step size instead to overcome the
limitation. One may question that, by increasing the step size we are increasing
the quantization error, because the quantization error is directly proportional to the
square of the step size. But here the basic assumption of squared error calculation is
missing and does not hold for the present case.
A Delta Modulation system with variable step size is known as the Adaptive
The block diagram of an ADM system is shown in the following Fig. 2.29.
In the concept discussed above, the squaring circuit is responsible for the size
of step () adaptation (tuning). The shaded region shown in the figure above is the
‘adaptation algorithm’ in generic. Two other well known adaptation algorithms are
discussed below.

2.5.1 Song Algorithm
In the year of 1971, C. L. Song et al.  proposed a nice algorithm for step size
adaptation by which the step size of the predicted waveform (
m (t)) can nicely be
adapted. According to the algorithm, positive slope of the prediction with respect
to time results in the next prediction equal to the previous step size added with the

2.5

Fig. 2.29 Adaptive delta
communication system.

45
m(t)

Pulse generator

Differenceamplifier

Pi(t)
Modulator

Δ(t)

~
m(t)

Po(t)

Integrator
Variable-gain
amplifier
Integrator
Square law
device

Feedback circuit

Po(t)

Variable-gain
amplifier

Quantizer

Integrator

Integrator
LPF
Square law
device

circuit

m̂(t)

step size at 0th prediction(0 ), similarly, for negative slope of the prediction with
respect to time results in the next prediction equal to the previous step size minus
the step size at 0th prediction (0 ).
If the nth prediction sample is (n), the algorithm prescribes the adaptation as
shown in the following two equations.
(n) = (n − 1) + 0

if

m(t) > 
m(t)

(2.43)

(n) = (n − 1) − 0

if

m(t) < 
m(t)

(2.44)

This addition and subtraction is called as accumulation in other terms. To combine
the Eqs. (2.43) and (2.44) let us assume another parameter called as discrepancy
parameter (dn ) as

dn =

+1
−1

if
if

m(t) > 
m(t)
m(t) < 
m(t)

(2.45)

Now, combining Eqs. (2.43) and (2.44), the nth sample can be calculated using
song algorithm of adaptation as

46

2

Waveform Encoding

(n) = |(n − 1) dn + 0 dn−1 |

(2.46)

The song algorithm can now be used for digitization of any type of analog signal shape. Let us take a typical step size analog signal for prediction through
accumulation. The ADM waveform through song algorithm is shown below.
The level of decrement is controlled by the new algorithm. In the previous section
(adaptation by squaring), the increment and decrement amount is only the step size
of 0th instant (0 ). But, as shown in the Fig. 2.30, the ADM predicted waveform
is showing undue oscillation for typically step like analog signal. For low slope and
high slope signals of other types can be tracked more efficiently by this method of
adaptation. The slope overload problem is solved totally by the song algorithm. The
undue oscillation can be viewed as a special type of granular noise.
Fig. 2.30 Adaptive delta
modulation by song
algorithm

4Δ 0

3Δ 0

3Δ 0
Undue oscillation
during constant
level of m(t)

2Δ 0
Δ0
~ by Delta modulation
m(t)

~ (t) by ADM (Song Algorithm)
m

2.5.2 Space-Shuttle Algorithm
Space-shuttle algorithm is a modification over song algorithm only. We have seen
that, the negative slope or maintenance of constant levels of the analog signal cannot
be properly tracked by the prediction through adaptive delta modulation by song
algorithm. The step size of the accumulated waveform is incremented and decremented by same amount (0 ). But, allowance of this linear decrement gives rise to
an undue oscillation during constant level of m(t). The new algorithm only changes
the rule of decrement. Instead of linear decrement space shuttle algorithm suggests
constant decrement of amount 0 .
(n) = (n − 1) + 0
(n) = 0

if

if

m(t) > 
m(t)

m(t) < 
m(t)

(2.47)
(2.48)

2.6

Sigma-Delta Modulation (SDM)

47

Otherwise, the algorithm is very much similar to the song algorithm. Here also,
the step size is incremented if the direction of the step at the current clock edge
is the same as the previous clock edge. If the direction is opposite, the step size is
decremented.

2.6 Sigma-Delta Modulation (SDM)
Delta modulation requires two integrators for modulation and demodulation processes as shown in Fig. 2.31 below. Since integration is a linear operation, the
second integrator can be moved before the modulator without altering the overall
input/output characteristics. Furthermore, the two integrators in Fig. 2.31a can be
combined into a single integrator by the linear operation property (Fig. 2.31b).
1 bit
quantizer
Analog
signal

Channel

∑

∫

LPF

Analog
signal

+_

∫
(a)

Analog
signal

∑

∫

Channel

LPF

Analog
signal

+_

∫
(b)
Fig. 2.31 (a) Delta modulator and demodulator, (b) Modification of (a) by integrator matching

The arrangement shown in Fig. 2.32 is the design diagram of a Sigma-Delta
(S-D) Modulation system. This structure, besides being simpler, can be considered
as being a ‘smoothed version’ of a 1-bit pulse code modulation (delta modulator).
Here, the smoothness signifies low pass filtering (i.e., integrated) of the comparator

Analog
signal

∑

∫

Channel

+_

Fig. 2.32 Sigma-Delta modulator with a single integrator

LPF

Analog
signal

48

2

Waveform Encoding

output as shown in Fig. 2.32 and 1 bit PCM signifies a hard limiter of 2 output levels
of saturation at the output (Fig. 2.26).
The name Sigma-Delta modulator comes from putting the integrator (sigma) in
front of the delta modulator. The use of the integrator in the manner described here
has also the following benefits:
1. The low frequency content of the input analog signal is pre-emphasized.
2. Correlation between adjacent samples of the delta modulator input is increased,
which improves the overall system performance by reducing the variance of the
error signal at the quantizer input.
3. Design of the receiver is simplified.
4. Noise performance is improved as discussed in the following section.

2.6.1 Noise Performance
Sometimes, the S-D modulator is referred to as an interpolative coder. The quantization noise characteristic (noise performance) of such a coder is frequency dependent
in contrast to delta modulation. As will be discussed further, this noise-shaping
property is well suited to signal processing applications such as digital audio and
communication. Like delta modulators, the S-D modulators use a simple course
quantizer (comparator). However, unlike delta modulators, these systems encode
the integral of the signal itself and thus their performance is insensitive to the rate
of change of the signal (Fig. 2.33).

Fig. 2.33 Delta-Sigma modulation in frequency domain representation

• Signal transfer function (when N(S)=0):
Y(s) = [X(s) − Y(s)] ×
Y(s)
=
X(s)

1
s
1
1+
s

Clearly, it signifies transfer function of LPF.

=

1
s

1
1+s

(2.49)

(2.50)

2.7

Linear Predictive Coder (LPC)

49

• Noise transfer function (when X(S)=0):
1
+ N(s)
s
Y(s)
1
s
=
=
1
N(s)
s+1
1+
s

Y(s) = −Y(s) ×

(2.51)
(2.52)

Clearly, it signifies transfer function of HPF.
Transfer function shown in Fig. 2.34 illustrates the modulator’s main action. As
the loop integrates the error between the sampled signal and the input signal, it low
pass filters the signal and high pass filters the noise. In other words, the signal is
left unchanged as long as its frequency content doesn’t exceed the filter’s cutoff frequency, but the S-D loop pushes the noise into a higher frequency band. Grossly
over-sampling the input causes the quantization noise to spread over a wide bandwidth and the noise density in the bandwidth of interest (baseband) to significantly
decrease.

Transfer function

Fig. 2.34 Noise performance
of signal delta modulation

Signal response

Noise response

Frequency

2.7 Linear Predictive Coder (LPC)
2.7.1 Concept
From the Sect. 2.3.3 (DPCM prediction), it is understood that, for prediction of the
next sample, only the delayed sample may not reduce the error of prediction to our
threshold level of satisfaction. For that, we have incorporated a series of delayed
samples with suitable scaling factors. The word ‘suitable’ is important in respect of
the design (Fig. 2.35)
The accuracy of the prediction is dependent on the proper choice of the scaling
factors. Here the predicted sample is expressible as linear combination of the series
of samples as under:

50

2
m[k]

Waveform Encoding

Delay

Delay

Delay

Delay

TS

TS

TS

TS

a1

a2

a3

aN

∑

Output m̂[k]
_

+
Error

()

2

Fitness function
that needs to be
minimized

Fig. 2.35 Linear predictive coding with optimization of fitness of prediction

m̂[k] = a1 m[k − 1] + a2 m[k − 2] + . . . aN m[k − N]
=

N


ai m[k − i]

(2.53)

i=1

Next, the error is calculated with respect to real m[k] and the squared value of error
(or we can use one squaring circuit to make it sign free) is minimized by any optimization algorithm to find out ‘suitable’ scaling factors ai s for proper prediction.
This is LSE (least square error) estimation algorithm. Once the scaling factors are
fixed for training samples, now our LPC (linear predictive coder) is ready to use
for the other query samples also. The optimization process needs to solve a huge
number of differential equations to find out proper ai s for good predictor design.
To overcome the hazard of solving the differential equations, we take help of soft
computing approaches like Genetic Algorithm (GA), to be discussed in the next
section.

2.7.2 Genetic Algorithm Based Approach
Genetic algorithm is a biologically inspired algorithm based on Darwinian evolution, and is a nice algorithm for object optimization. Here, we can parallely discuss
about basic GA and application of GA to find out suitable ai ’s for minimum squared
error.

2.7

Linear Predictive Coder (LPC)

51

Genetic algorithms are implemented in a computer simulation in which a population of abstract representations (called chromosomes or the genotype of the
genome) of candidate solutions (called individuals, creatures, or phenotypes) to an
optimization problem evolves toward better solutions. Traditionally, solutions are
represented in binary as strings of 0s and 1s, but other encodings are also possible.
Here, we will use real-coded GA. The evolution usually starts from a population of
randomly generated individuals and allowed for reproduction. In each generation,
the fitness of every individual in the population is evaluated, multiple individuals
are stochastically selected from the current population (based on their fitness to the
particular problem), and modified (recombined and possibly randomly mutated) to
form a new population. The new population is then used in the next iteration of the
algorithm. Commonly, the algorithm terminates when either a maximum number
of generations has been produced, or a satisfactory fitness level has been reached
for the population. If the algorithm has terminated due to a maximum number of
generations, a satisfactory solution may or may not have been reached.
A typical genetic algorithm requires:
1. A genetic representation of the solution domain,
2. A fitness function to evaluate the solution domain.
Here, the gene is represented as an array of solutions in real code. The fitness
function is defined over the genetic representation and measures the quality of
the represented solution. The fitness function is always problem dependent. In the
present case, the fitness function f is defined as the squared error as given below:

f = ε2 = m[k] − m̂[k]


2

= m[k] −

N


2
ai m[k − i]

(2.54)

i=1

After having the genetic representation and the fitness function defined, GA proceeds to initialize a population of solutions randomly, and then improve it through
repetitive application of mutation, crossover, inversion and selection operators.
Step1: Population
A genetic pool is created with 20 chromosomes each with N number of genes
with random values, covering the entire range of possible solutions. Here the
scaling factors must reside within 0 and 1. Therefore, the initialized random values must be fractions only. Say, the values of genes are the trial solutions of ai ’s
(∀i) for minimum squared error.
Step2: Selection
During each successive generation, a proportion of the existing population is
selected to breed a new generation. Individual solutions are selected through a
fitness-based process. It means, from that 20 strings of arrays (20 chromosomes),
the most fit genes will be selected. The ```
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