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Forum for Interdisciplinary Mathematics
K. Deergha Rao
Channel Coding
Techniques
for Wireless
Communications
SecondEdition

Editor-in-Chief
P. V. Subrahmanyam, Department of Mathematics, Indian Institute of Technology
Madras, Chennai, Tamil Nadu, India
Editorial Board
Yogendra Prasad Chaubey, Department of Mathematics and Statistics, Concordia
University, Montreal, QC, Canada
Jorge Cuellar, Principal Researcher, Siemens AG, München, Bayern, Germany
Janusz Matkowski, Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Góra, Zielona Góra, Poland
Thiruvenkatachari Parthasarathy, Chennai Mathematical Institute, Kelambakkam,
Tamil Nadu, India
Mathieu Dutour Sikirić, Institute Rudjer Boúsković, Zagreb, Croatia
Bhu Dev Sharma, Forum for Interdisciplinary Mathematics, Meerut, Uttar Pradesh,
India

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More information about this series at http://www.springer.com/series/13386

K. Deergha Rao
Channel Coding Techniques
for Wireless Communications
Second Edition
123

K. Deergha Rao
Department of ECE
Vasavi College of Engineering
(Autonomous college affiliated
to Osmania University)
Hyderabad, Telangana, India
ISSN 2364-6748 ISSN 2364-6756 (electronic)
ISBN 978-981-15-0560-7 ISBN 978-981-15-0561-4 (eBook)
https://doi.org/10.1007/978-981-15-0561-4
1st
edition: © Springer India 2015
2nd
edition: © Springer Nature Singapore Pte Ltd. 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, 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.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
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This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore

To
My Parents Boddu and Dalamma,
My Beloved Wife Sarojini,
and My Mentor Prof. M.N.S. Swamy

Preface
Lives of people have tremendously changed in view of the rapid growth of mobile
and wireless communication. Channel coding is the heart of digital communication
and data storage. Traditional block codes and conventional codes are commonly
used in digital communications. To approach the theoretical limit for Shannon’s
channel capacity, the length of a linear block code or constant lengths of convo-
lutional codes have to be increased, which in turn makes the decoder complexity
higher and may render it physically unrealizable. The powerful turbo and LDPC
codes approach the theoretical limit for Shannon’s channel capacity with feasible
complexity for decoding. MIMO communications is a multiple antenna technology
which is an effective way for high-speed or high-reliability communications.
The MIMO can be implemented by space-time coding. Recently, a new channel
coding technique, namely polar codes, has emerged as one of the channel coding
techniques for fifth-generation (5G) wireless communications, and it has been
recommended by third-generation partnership project (3GPP) as a channel coding
scheme for enhanced mobile broadband (eMBB) in 5G systems. However, the
market lacks a book which can serve as a textbook for graduate and undergraduate
students on channel coding techniques.
This book includes illustrative examples in each chapter for easy understanding
of coding techniques. An attractive feature of this book is the inclusion of
MATLAB-based examples with codes encouraging readers to implement them on
their personal computers and become confident of the fundamentals by gaining
more insight into coding theory. In addition to the problems that require analytical
solutions, MATLAB exercises are introduced to the reader at the end of each
chapter.
This book is divided into 13 chapters. Chapter 1 introduces the basic elements of
a digital communication system, statistical models for wireless channels, capacity
of a fading channel, Shannon’s noisy channel coding theorem, and the basic idea of
coding gain. Chapter 2 gives an overview of the performance analysis of different
modulation techniques and also deals with the performance of different diversity
combining techniques in a multichannel receiver. Chapter 3 introduces Galois fields
and polynomials over Galois fields. Chapter 4 covers linear block codes including
vii

RS codes because of their popularity in burst error correction in wireless networks.
Chapter 5 discusses the design of a convolutional encoder and Viterbi decoding
algorithm for the decoding of convolutional codes, as well as the performance
analysis of convolutional codes over AWGN and Rayleigh fading channels. In this
chapter, punctured convolutional codes, tail-biting convolutional codes, and their
performance analysis are also discussed. Chapter 6 provides a treatment of the
design of turbo codes, BCJR algorithm for iterative decoding of turbo codes, and
performance analysis of turbo codes. In this chapter, enhanced turbo codes,
enhanced list turbo decoding, and their performance evaluation are also described.
Chapter 7 focuses on the design and analysis of trellis-coded modulation
schemes using both the conventional and turbo codes. Chapter 8 describes the
design of low parity check codes (LDPC), quasi-cyclic (QC)-LDPC codes,
decoding algorithms, and performance analysis of LDPC and QC-LDPC codes. The
erasure correcting codes like Luby transform (LT) codes and Raptor codes are
described in Chap. 9. The design of polar encoder and successive cancelation
decoding (SCD), successive cancelation list decoding (SCLD), and multiple bit
decision successive cancelation list decoding algorithms and their performance
evaluation are provided in Chap. 10.
Chapter 11 provides an in-depth study of multiple-input multiple-output
(MIMO) systems in which multiple antennas are used both at the transmitter and
at the receiver. The advanced techniques for MIMO OFDM channel estimation are
also described in this chapter. The design of space-time codes and implementations
of MIMO systems are discussed in Chap. 12. Chapter 13 deals with the evolution of
channel codes for 5G wireless communications.
The motivation in writing this book is to include modern topics of increasing
importance such as turbo codes, LDPC codes, polar codes, LT and Raptor codes,
and space-time coding in detail, in addition to the traditional RS codes and con-
volutional codes, and also to provide a comprehensive exposition of all aspects of
coding for wireless channels. The text is integrated with MATLAB-based programs
to enhance the understanding of the underlying theories of the subject.
This book is written at a level suitable for undergraduate and master students in
electronics and communication engineering, electrical and computer engineering,
computer science, and applied physics as well as for self-study by researchers,
practicing engineers, and scientists. Depending on the chapters chosen, this text can
be used for teaching a one- or two-semester course on coding for wireless channels.
The prerequisite knowledge of the readers in principles of digital communication is
expected.
Hyderabad, India K. Deergha Rao
viii Preface

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Digital Communication System . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Wireless Communication Channels . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Binary Erasure Channel (BEC) . . . . . . . . . . . . . . . . . . 2
1.2.2 Binary Symmetric Channel (BSC) . . . . . . . . . . . . . . . . 3
1.2.3 Additive White Gaussian Noise Channel . . . . . . . . . . . 3
1.2.4 Gilbert–Elliott Channel . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.5 Fading Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.6 Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Statistical Models for Fading Channels. . . . . . . . . . . . . . . . . . . 8
1.3.1 Probability Density Function of Rician Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Probability Density Function of Rayleigh Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Probability Density Function of Nakagami Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Channel Capacity of Binary Erasure Channel . . . . . . . . 11
1.4.2 Channel Capacity of Binary Symmetric Channel . . . . . 11
1.4.3 Capacity of AWGN Channel. . . . . . . . . . . . . . . . . . . . 11
1.4.4 Channel Capacity of Gilbert–Elliott Channels . . . . . . . 13
1.4.5 Ergodic Capacity of Fading Channels . . . . . . . . . . . . . 13
1.4.6 Outage Probability of a Fading Channel . . . . . . . . . . . 15
1.4.7 Outage Capacity of Fading Channels . . . . . . . . . . . . . . 16
1.4.8 Capacity of Fading Channels with CSI
at the Transmitter and Receiver . . . . . . . . . . . . . . . . . . 17
1.5 Channel Coding for Improving the Performance
of Communication System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Shannon’s Noisy Channel Coding Theorem . . . . . . . . . 17
ix

1.5.2 Channel Coding Principle . . . . . . . . . . . . . . . . . . . . . . 18
1.5.3 Channel Coding Gain . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Some Application Examples of Channel Coding . . . . . . . . . . . . 19
1.6.1 Error Correction Coding in GSM . . . . . . . . . . . . . . . . 19
1.6.2 Error Correction Coding in W-CDMA . . . . . . . . . . . . . 20
1.6.3 Digital Video Broadcasting Channel Coding . . . . . . . . 20
1.6.4 Error Correction Coding in GPS L5 Signal . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Overview of the Performance of Digital Communication
Over Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 BER Performance of Different Modulation Schemes in
AWGN, Rayleigh, and Rician Fading Channels . . . . . . . . . . . . 23
2.1.1 BER of BPSK Modulation in AWGN Channel . . . . . . 24
2.1.2 BER of BPSK Modulation in Rayleigh Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 BER of BPSK Modulation in Rician Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.4 BER Performance of BFSK in AWGN, Rayleigh,
and Rician Fading Channels . . . . . . . . . . . . . . . . . . . . 27
2.1.5 Comparison of BER Performance of BPSK, QPSK,
and 16-QAM in AWGN and Rayleigh Fading
Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Wireless Communication Techniques . . . . . . . . . . . . . . . . . . . . 30
2.2.1 DS-CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 FH-CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.3 OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.4 MC-CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Diversity Reception. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 Receive Diversity with N Receive Antennas
in AWGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 Diversity Combining Techniques . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Selection Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.2 Equal Gain Combining (EGC) . . . . . . . . . . . . . . . . . . 47
2.4.3 Maximum Ratio Combining (MRC) . . . . . . . . . . . . . . 48
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6 MATLAB Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Galois Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
x Contents

3.5 Elementary Properties of Galois Fields . . . . . . . . . . . . . . . . . . . 59
3.6 Galois Field Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6.1 Addition and Subtraction of Polynomials . . . . . . . . . . . 59
3.6.2 Multiplication of Polynomials . . . . . . . . . . . . . . . . . . . 60
3.6.3 Multiplication of Polynomials Using MATLAB . . . . . . 60
3.6.4 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . 60
3.6.5 Division of Polynomials Using MATLAB . . . . . . . . . . 61
3.7 Polynomials Over Galois Fields . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7.1 Irreducible Polynomial . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7.2 Primitive Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7.3 Checking of Polynomials for Primitiveness Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7.4 Generation of Primitive Polynomials Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.8 Construction of Galois Field GF 2m
ð Þ from GF(2) . . . . . . . . . . . 64
3.8.1 Construction of GF 2m
ð Þ, Using MATLAB . . . . . . . . . . 70
3.9 Minimal Polynomials and Conjugacy Classes of GF 2m
ð Þ . . . . . 70
3.9.1 Minimal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.9.2 Conjugates of GF Elements . . . . . . . . . . . . . . . . . . . . 73
3.9.3 Properties of Minimal Polynomial . . . . . . . . . . . . . . . . 73
3.9.4 Construction of Minimal Polynomials . . . . . . . . . . . . . 74
3.9.5 Construction of Conjugacy Classes Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.9.6 Construction of Minimal Polynomials Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Linear Block Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1 Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Linear Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 Linear Block Code Properties . . . . . . . . . . . . . . . . . . . 81
4.2.2 Generator and Parity Check Matrices. . . . . . . . . . . . . . 82
4.2.3 Weight Distribution of Linear Block Codes . . . . . . . . . 84
4.2.4 Hamming Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.5 Syndrome Table Decoding . . . . . . . . . . . . . . . . . . . . . 87
4.2.6 Hamming Codes Decoding . . . . . . . . . . . . . . . . . . . . . 88
4.3 Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.1 The Basic Properties of Cyclic Codes . . . . . . . . . . . . . 90
4.3.2 Encoding Algorithm for an (n, k) Cyclic Codes . . . . . . 91
4.3.3 Encoder for Cyclic Codes Using Shift Registers. . . . . . 93
4.3.4 Shift Register Encoders for Cyclic Codes . . . . . . . . . . 94
4.3.5 Cyclic Redundancy Check Codes . . . . . . . . . . . . . . . . 96
Contents xi

4.4 BCH Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.1 BCH Code Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.2 Berlekamp’s Algorithm for Binary BCH Codes
Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4.3 Chien Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Reed–Solomon Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5.1 Reed–Solomon Encoder . . . . . . . . . . . . . . . . . . . . . . . 110
4.5.2 Decoding of Reed–Solomon Codes . . . . . . . . . . . . . . . 112
4.5.3 Binary Erasure Decoding . . . . . . . . . . . . . . . . . . . . . . 123
4.5.4 Non-binary Erasure Decoding . . . . . . . . . . . . . . . . . . . 123
4.6 Performance Analysis of RS Codes . . . . . . . . . . . . . . . . . . . . . 127
4.6.1 BER Performance of RS Codes for BPSK Modulation
in AWGN and Rayleigh Fading Channels . . . . . . . . . . 127
4.6.2 BER Performance of RS Codes for Non-coherent
BFSK Modulation in AWGN and Rayleigh Fading
Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5 Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Structure of Non-systematic Convolutional Encoder . . . . . . . . . 137
5.1.1 Impulse Response of Convolutional Codes . . . . . . . . . 139
5.1.2 Constraint Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.3 Convolutional Encoding Using MATLAB . . . . . . . . . . 141
5.2 Structure of Systematic Convolutional Encoder. . . . . . . . . . . . . 142
5.3 The Structural Properties of Convolutional Codes . . . . . . . . . . . 142
5.3.1 State Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3.2 Catastrophic Convolutional Codes . . . . . . . . . . . . . . . . 142
5.3.3 Transfer Function of a Convolutional Encoder . . . . . . . 144
5.3.4 Distance Properties of Convolutional Codes . . . . . . . . . 148
5.3.5 Trellis Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.4 Punctured Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . 154
5.5 The Viterbi Decoding Algorithm . . . . . . . . . . . . . . . . . . . . . . . 155
5.5.1 Hard Decision Decoding . . . . . . . . . . . . . . . . . . . . . . . 156
5.5.2 Soft Decision Decoding . . . . . . . . . . . . . . . . . . . . . . . 158
5.6 Performance Analysis of Convolutional Codes . . . . . . . . . . . . . 163
5.6.1 Binary Symmetric Channel . . . . . . . . . . . . . . . . . . . . . 163
5.6.2 AWGN Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6.3 Rayleigh Fading Channel . . . . . . . . . . . . . . . . . . . . . . 167
5.7 Tail-Biting Convolutional Code . . . . . . . . . . . . . . . . . . . . . . . . 169
5.7.1 Tail-Biting Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 170
xii Contents

5.7.2 Tail-Biting Encoding Using MATLAB . . . . . . . . . . . . 171
5.7.3 Tail-Biting Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.8 Performance Analysis of Tail-Biting Convolutional Codes . . . . 173
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6 Turbo Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.1 Non-recursive and Recursive Systematic Convolutional
Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.1.1 Recursive Systematic Convolutional (RSC) Encoder. . . 182
6.2 Turbo Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2.1 Different Types of Interleavers . . . . . . . . . . . . . . . . . . 184
6.2.2 Turbo Coding Illustration . . . . . . . . . . . . . . . . . . . . . . 185
6.2.3 Turbo Coding Using MATLAB . . . . . . . . . . . . . . . . . 189
6.2.4 Encoding Tail-Biting Codes with RSC (Feedback)
Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.3 Turbo Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.3.1 The BCJR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.3.2 Turbo Decoding Illustration . . . . . . . . . . . . . . . . . . . . 205
6.3.3 Convergence Behavior of the Turbo Codes . . . . . . . . . 213
6.3.4 EXIT Analysis of Turbo Codes . . . . . . . . . . . . . . . . . . 214
6.4 Performance Analysis of the Turbo Codes . . . . . . . . . . . . . . . . 217
6.4.1 Upper Bound for the Turbo Codes in AWGN
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4.2 Upper Bound for Turbo Codes in Rayleigh Fading
Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.4.3 Effect of Free Distance on the Performance
of the Turbo Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.4.4 Effect of Number of Iterations on the Performance
of the Turbo Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.4.5 Effect of Puncturing on the Performance of the Turbo
Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.5 Enhanced Turbo Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.5.1 Enhanced Turbo Encoder . . . . . . . . . . . . . . . . . . . . . . 226
6.5.2 Enhanced List Turbo Decoder . . . . . . . . . . . . . . . . . . . 226
6.6 Performance Analysis of Enhanced Turbo Codes . . . . . . . . . . . 226
6.6.1 Performance of Enhanced Tail-Biting Turbo Codes
Over AWGN and Rayleigh Fading Channels . . . . . . . . 226
6.6.2 Performance of Enhanced Turbo Codes
with Tail-Biting List Decoding in AWGN Channels . . . 228
Contents xiii

6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7 Bandwidth Efﬁcient Coded Modulation . . . . . . . . . . . . . . . . . . . . . 233
7.1 Set Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.2 Design of the TCM Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.3 Decoding TCM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.4 TCM Performance Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.4.1 Asymptotic Coding Gain . . . . . . . . . . . . . . . . . . . . . . 241
7.4.2 Bit Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.4.3 Simulation of the BER Performance of a 8-State
8-PSK TCM in the AWGN and Rayleigh Fading
Channels Using MATLAB . . . . . . . . . . . . . . . . . . . . . 250
7.5 Turbo Trellis-Coded Modulation (TTCM) . . . . . . . . . . . . . . . . 254
7.5.1 TTCM Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
7.5.2 TTCM Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.5.3 Simulation of the BER Performance of the 8-State
8-PSK TTCM in AWGN and Rayleigh Fading
Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.6 Bit-Interleaved Coded Modulation . . . . . . . . . . . . . . . . . . . . . . 258
7.6.1 BICM Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.6.2 BICM Decoder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.7 Bit-Interleaved Coded Modulation Using Iterative Decoding . . . 263
7.7.1 BICM-ID Encoder and Decoder . . . . . . . . . . . . . . . . . 263
7.7.2 Simulation of the BER Performance of 8-State 8-PSK
BICM and BICM-ID in AWGN and Rayleigh
Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
8 Low Density Parity Check Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.1 LDPC Code Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.2 Construction of Parity Check Matrix H . . . . . . . . . . . . . . . . . . 274
8.2.1 Gallager Method for Random Construction
of H for Regular Codes . . . . . . . . . . . . . . . . . . . . . . . 274
8.2.2 Algebraic Construction of H for Regular Codes . . . . . . 275
8.2.3 Random Construction of H for Irregular Codes . . . . . . 276
8.3 Representation of Parity Check Matrix Using Tanner
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.3.1 Cycles of Tanner Graph . . . . . . . . . . . . . . . . . . . . . . . 278
8.3.2 Detection and Removal of Girth 4 of a Parity
Check Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
xiv Contents

8.4 LDPC Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.4.1 Preprocessing Method. . . . . . . . . . . . . . . . . . . . . . . . . 282
8.5 Efﬁcient Encoding of LDPC Codes . . . . . . . . . . . . . . . . . . . . . 288
8.5.1 Efﬁcient Encoding of LDPC Codes Using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.6 LDPC Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.6.1 LDPC Decoding on Binary Erasure Channel
Using Message Passing Algorithm . . . . . . . . . . . . . . . 292
8.6.2 LDPC Decoding on Binary Erasure Channel
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
8.6.3 Bit-Flipping Decoding Algorithm . . . . . . . . . . . . . . . . 297
8.6.4 Bit-Flipping Decoding Using MATLAB . . . . . . . . . . . 299
8.7 Sum Product Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8.7.1 Log Domain Sum-Product Algorithm (SPA) . . . . . . . . 305
8.7.2 The Min-Sum Algorithm . . . . . . . . . . . . . . . . . . . . . . 306
8.7.3 Sum Product and Min-Sum Algorithms for Decoding
of Rate 1/2 LDPC Codes Using MATLAB . . . . . . . . . 309
8.8 EXIT Analysis of LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . 312
8.8.1 Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 312
8.8.2 Ensemble Decoding Thresholds. . . . . . . . . . . . . . . . . . 313
8.8.3 EXIT Charts for Irregular LDPC Codes in Binary
Input AWGN Channels. . . . . . . . . . . . . . . . . . . . . . . . 314
8.9 Performance Analysis of LDPC Codes . . . . . . . . . . . . . . . . . . . 316
8.9.1 Performance Comparison of Sum-Product
and Min-Sum Algorithms for Decoding of Regular
LDPC Codes in AWGN Channel . . . . . . . . . . . . . . . . 316
8.9.2 BER Performance Comparison of Regular and
Irregular LDPC Codes in AWGN Channel. . . . . . . . . . 317
8.9.3 Effect of Block Length on the BER Performance
of LDPC Codes in AWGN Channel . . . . . . . . . . . . . . 317
8.9.4 Error Floor Comparison of Irregular LDPC Codes
of Different Degree Distribution in AWGN Channel. . . 319
8.10 Quasi Cyclic (QC)-LDPC CODES . . . . . . . . . . . . . . . . . . . . . . 320
8.10.1 Brief Description of QC-LDPC Codes . . . . . . . . . . . . . 320
8.10.2 Base Matrix and Expansion . . . . . . . . . . . . . . . . . . . . 321
8.10.3 Performance Analysis of QC-LDPC Codes Over
AWGN Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Contents xv

9 LT and Raptor Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
9.1 LT Codes Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
9.1.1 LT Degree Distributions . . . . . . . . . . . . . . . . . . . . . . . 332
9.1.2 Important Properties of the Robust Soliton
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
9.1.3 LT Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
9.1.4 Tanner Graph of LT Codes . . . . . . . . . . . . . . . . . . . . . 335
9.1.5 LT Decoding with Hard Decision . . . . . . . . . . . . . . . . 335
9.1.6 Hard Decision LT Decoding Using MATLAB . . . . . . . 338
9.1.7 BER Performance of LT Decoding Over BEC
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
9.2 Systematic LT Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
9.2.1 Systematic LT Codes Decoding . . . . . . . . . . . . . . . . . 342
9.2.2 BER Performance Analysis of Systematic
LT Codes Using MATLAB . . . . . . . . . . . . . . . . . . . . 342
9.3 Raptor Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
10 Polar Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.1 Channel Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.1.1 Channel Combining Phase . . . . . . . . . . . . . . . . . . . . . 352
10.1.2 Channel Splitting Phase . . . . . . . . . . . . . . . . . . . . . . . 357
10.1.3 Polarization of Binary Erasure Channels . . . . . . . . . . . 358
10.1.4 Polarization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 361
10.1.5 Polarization of AWGN Channels . . . . . . . . . . . . . . . . . 362
10.2 Polar Encoder Structures and Encoding . . . . . . . . . . . . . . . . . . 362
10.2.1 Polar Encoder for N = 2 . . . . . . . . . . . . . . . . . . . . . . . 363
10.2.3 Polar Encoder for N = 4 with Input Data Vector
Permuted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
10.2.5 Polar Encoder for N = 8 with Input Data Vector
Permuted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
10.2.6 Non-systematic Polar Encoding Using MATLAB . . . . . 367
10.2.7 Systematic Polar Encoding . . . . . . . . . . . . . . . . . . . . . 368
10.2.8 Efﬁcient Systematic Polar Encoding Algorithm . . . . . . 369
10.3 Polar Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
10.3.1 Successive Cancelation Decoding . . . . . . . . . . . . . . . . 371
10.3.2 SC Decoding Algorithm . . . . . . . . . . . . . . . . . . . . . . . 374
10.3.3 Successive Cancelation List Decoding . . . . . . . . . . . . . 381
xvi Contents

10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
11 MIMO System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
11.1 What Is MIMO? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
11.2 MIMO Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
11.2.1 The Frequency Flat MIMO Channel . . . . . . . . . . . . . . 385
11.2.2 The Frequency-Selective MIMO Channel . . . . . . . . . . 387
11.2.3 MIMO-OFDM System . . . . . . . . . . . . . . . . . . . . . . . . 387
11.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
11.3.1 LS Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . 389
11.3.2 DFT-Based Channel Estimation . . . . . . . . . . . . . . . . . 389
11.3.3 MIMO-OFDM Channel Estimation Using LS . . . . . . . 390
11.3.4 Channel Estimation Using MATLAB . . . . . . . . . . . . . 390
11.4 MIMO Channel Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 392
11.5 MIMO Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
11.5.1 Capacity of Deterministic MIMO Channel When CSI
Is Known to the Transmitter . . . . . . . . . . . . . . . . . . . . 399
11.5.2 Deterministic MIMO Channel Capacity When CSI Is
Unknown at the Transmitter . . . . . . . . . . . . . . . . . . . . 401
11.5.3 Random MIMO Channel Capacity . . . . . . . . . . . . . . . 403
11.6 MIMO-OFDM Channel Estimation Using OMP Algorithm . . . . 411
11.6.1 OMP Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
11.7 MIMO Channel Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . 416
11.7.1 Zero Forcing (ZF) Equalization . . . . . . . . . . . . . . . . . . 417
11.7.2 Minimum Mean Square Error (MMSE)
Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
11.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
12 Space-Time Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
12.1 Space-Time-Coded MIMO System. . . . . . . . . . . . . . . . . . . . . . 423
12.2 Space-Time Block Code (STBC) . . . . . . . . . . . . . . . . . . . . . . . 424
12.2.1 Rate Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
12.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
12.2.3 Diversity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
12.2.4 Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 426
12.2.5 Decoding STBCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
12.3 Alamouti Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
12.3.1 2-Transmit, 1-Receive Alamouti STBC Coding . . . . . . 428
12.3.2 2-Transmit, 2-Receive Alamouti STBC Coding . . . . . . 429
12.3.3 Theoretical BER Performance of BPSK Alamouti
Codes Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . 433
Contents xvii

12.4 Higher-Order STBCs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
12.4.1 3-Transmit, 4-Receive STBC Coding. . . . . . . . . . . . . . 435
12.4.2 Simulation of BER Performance of STBCs
Using MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
12.5 Space-Time Trellis Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
12.5.1 Space-Time Trellis Encoder . . . . . . . . . . . . . . . . . . . . 443
12.5.2 Simulation of BER Performance of 4-state
QPSK STTC Using MATLAB . . . . . . . . . . . . . . . . . . 451
12.6 MIMO-OFDM Implementation . . . . . . . . . . . . . . . . . . . . . . . . 457
12.6.1 Space-Time-Coded OFDM . . . . . . . . . . . . . . . . . . . . . 459
12.6.2 Space-Frequency Coded OFDM . . . . . . . . . . . . . . . . . 459
12.6.3 Space-Time-Frequency Coded OFDM . . . . . . . . . . . . . 460
12.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
13 Channel Codes Evolution for 5G . . . . . . . . . . . . . . . . . . . . . . . . . . 465
13.1 5G Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
13.2 QC-LDPC and Polar Codes for eMBB . . . . . . . . . . . . . . . . . . . 466
13.2.1 Performance Evaluation of QC-LDPC Codes for
eMBB Data Channel . . . . . . . . . . . . . . . . . . . . . . . . . 467
13.2.2 Performance Evaluation of Polar Codes for eMBB
Control Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
13.3 Evaluation of Enhanced Turbo Codes and Polar Codes for
URLLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
13.3.1 Decoding Latency . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
13.3.2 Decoding Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 474
13.4 Channel Codes for mMTC . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
xviii Contents

About the Author
K. Deergha Rao is Professor at the Department of Electronics and Communication
Engineering, Vasavi College of Engineering, Osmania University, Hyderabad,
India. He is former Director and Professor of Research and Training Unit for
Navigational Electronics (NERTU), Osmania University. He was a postdoctoral
fellow and part-time professor for four years at the Department of Electrical and
Computer Engineering, Concordia University, Montreal, Canada. His teaching
areas are signals and systems, digital signal processing, channel coding techniques,
and MIMO communication systems. Professor Rao has executed research projects
for premium Indian organizations such as DRDO, HAL, and BEL and is interested
in research areas of wireless channel coding, MIMO-OFDM communications,
image processing, cryptosystems, and VLSI signal processing. He has also served
as the founder chairman of the joint chapter of IEEE Communications Society and
IEEE Signal Processing Society in Hyderabad from 2010–2012 and communica-
tions track chair for IEEE INDICON 2011 held in Hyderabad. Five students have
been so far awarded Ph.D. degrees under Professor Rao, while three are currently
working towards their Ph.D.
An awardee of the IETE K. S. Krishnan Memorial Award for the best system
oriented paper in 2013, Prof. Rao has presented papers at IEEE International
conferences several times in the USA, Switzerland, Russia, and Thailand. He has
more than 100 publications to his credit including more than 60 publications in the
IEEE journals and conference proceedings. Professor Rao is the author of two
books—Channel Coding Techniques for Wireless Communications (Springer,
2015) and Signals and Systems (Springer, 2018)—and has co-authored Digital
Signal Processing (Jaico Publishing House, 2012) and Digital Signal Processing:
Theory and Practice (Springer, 2018). He is an editorial board member for the
International Journal of Sustainable Aviation.
xix

Chapter 1
Introduction
In this chapter, a digital communication system with coding is first described. Second,
various wireless communication channels, their probability density functions, and
capacities are discussed. Further, Shannon’s noisy channel coding theorem, channel
coding principle and channel coding gain are explained. Finally, some application
examples of channel coding are included.
1.1 Digital Communication System
A communication system is a means of conveying information from one user to
another user. The digital communication system is one in which the data is trans-
mitted in digital form. A digital communication system schematic diagram is shown
in Fig. 1.1. The source coding is used to remove redundancy from source informa-
tion for efficient transmission. The transmitted signal power and channel bandwidth
are the key parameters in the design of digital communication system. Using these
parameters, the signal energy per bit (Eb) to noise power spectral density (N0) ratio
is determined. This ratio is unique in determining the probability of bit error, often
referred to as bit error rate (BER). In practice, for a fixed Eb/N0, acceptable BER
is possible with channel coding. This can be achieved by adding additional digits
to the transmitted information stream. These additional digits do not have any new
information, but they make it possible for the receiver to detect and correct errors,
thereby reducing the overall probability of error.
© Springer Nature Singapore Pte Ltd. 2019
K. D. Rao, Channel Coding Techniques for Wireless Communications,
Forum for Interdisciplinary Mathematics,
https://doi.org/10.1007/978-981-15-0561-4_1
1

2 1 Introduction
Fig. 1.1 Digital communication system with coding
1.2 Wireless Communication Channels
1.2.1 Binary Erasure Channel (BEC)
Erasure is a special type of error with known location. The binary erasure channel
(BEC) transmits one of the two binary bits 0 and 1. However, an erasure “e” is
produced when the receiver receives an unreliable bit. The BEC output consists of
0, 1, and e as shown in Fig. 1.2. The BEC erases a bit with probability ε, called the
erasure probability of the channel. Thus, the channel transition probabilities for the
BEC are:
Fig. 1.2 Binary erasure
channel
1
1
0
0
Transmit Receive

1.2 Wireless Communication Channels 3
Fig. 1.3 Binary symmetric
channel
1
1
0
0
P(y = 0|x = 0) = 1 − ε,
P(y = e|x = 0) = ε,
P(y = 1|x = 0) = 0,
P(y = 0|x = 1) = 0,
P(y = e|x = 1) = ε,
P(y = 1|x = 1) = 1 − ε.
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(1.1)
1.2.2 Binary Symmetric Channel (BSC)
The binary symmetric channel (BSC) is a discrete memoryless channel that has
binary symbols both in the input and output. It is symmetric because the probability
for receiving 0 when 1 is transmitted is same as the probability for receiving 1 when
0 is transmitted. This probability is called the crossover probability of the channel
denoted by P as shown in Fig. 1.3. The probability for no error, i.e., receiving the
same as transmitted is 1−P. Hence, the channel transition probabilities for the BSC
are:
P(y = 0|x = 0) = 1 − P,
P(y = 0|x = 1) = P,
P(y = 1|x = 0) = P,
P(y = 1|x = 1) = 1 − P,
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(1.2)
1.2.3 Additive White Gaussian Noise Channel
In an AWGN channel, the signal is degraded by white noise η which has a constant
spectral density and a Gaussian distribution of amplitude. The Gaussian distribution
has a Probability Density Function (PDF) given by

4 1 Introduction
Pdf (η) =
1
√
2πσ2
exp

−
η2
2σ2

(1.3)
where σ2
is the variance of a Gaussian random process.
1.2.4 Gilbert–Elliott Channel
For bursty wireless channels, the Gilbert–Elliott (GE) channel [1, 2] is one of the
simplest and practical models. The GE channel is a discrete-time stationary model
as shown in Fig. 1.4 with two states: one bad state or burst state “2” wherein a BSC
resides with high-error probabilities (1 − P2) and the other state is a good state “1”
wherein a BSC resides with low-error probabilities (1 − P1).
Another common GE Example is that the BEC resides in a bad state with ε close
to unity and assigns erasures to all of the bits transmitted during the high-error-rate
(bad) state.
1 2
Bad Channel
1
1
0
0
Good Channel
1
1
0
0
Fig. 1.4 Two-state channel

1.2.5 Fading Channel
In the radio channel, the received power is affected by the attenuations due to the
combinations of the following effects
1. The Path loss: It is the signal attenuation. The power received by the receiving
antenna decreases when the distance between transmitter and receiver increases.
The power attenuation is proportional to (distance)α
, where α values range from
2 to 4. When the distance varies with time, the path loss also varies.
2. The Shadowing loss: It is due to the absorption of the radiated signal by scattering
structure. It is derived from a random variable with log-normal distribution.
3. The Fading loss: The combination of multipath propagation and the Doppler
frequency shift produces the random fluctuations in the received power which
gives the fading losses.
1.2.6 Fading
Fading gives the variations of the received power along with the time. It is due to
the combination of multipath propagation and the Doppler frequency shift which
gives the time-varying attenuations and delays that may degrade the communication
system performance. The received signal is a distorted version of the transmitted
signal which is a sum of the signal components from the various paths with different
delays due to multipath and motion.
Let Ts be the duration of a transmitted signal and Bx be the signal bandwidth. The
fading channel can be classified based on coherence time and coherence bandwidth
of the channel. The coherence time and coherence bandwidth of a channel are defined
as
Doppler spread:
Doppler shift is defined as the perceived change in the frequency of the electromag-
netic wave due to the mobility of the mobile user.
If the mobile station is moving with a velocity v at an angle θ with the line joining
mobile user and base station, the Doppler shift fd is expressed by
fd =
v
c
cos θ fc (1.4a)
where fc is the carrier frequency, and c is the velocity of light.
From the above equation, it can be observed that the Doppler shift is maximum
when θ = 0 and the Doppler shift is zero when θ = 0 = π
2
that is, when the mobile
motion is perpendicular to the receive direction.

6 1 Introduction
Thus, the maximum Doppler shift fdmax is given by
fdmax =
v
c
fc (1.4b)
Then, the delay spread BD = 2fdmax
The coherence time of the channel Tc is
Tc
1
2BD
=
1
4fdmax
(1.4c)
The coherence time implies that the channel changes at every coherence time
duration, and the channel is to be estimated at least once in every coherence time
interval.
Example 1.1 Consider a mobile user is moving at 40 kmph with fc = 2.4 GHz.
Compute the coherence time.
Solution
40 kmph = 40 ×
5
18
=
2000
18
= 11.1111 m/s
fdmax =
11.1111
3 × 108
× 2400 × 106
= 88.8888 Hz
Tc =
1
4fdmax
=
1
4 × 88.8888
= 22.5 ms
Delay spread:
Themaximumamongthepathdelaydifferences,asignificantchangeoccurswhenthe
frequency change exceeds the inverse of TD, called the delay spread of the channel.
The channel bandwidth of the channel Bc is
Bc
1
TD
(1.5)
For example, if a 4-multipath channel with the delays corresponding to the first
and the last arriving are 0 μs and 6 μs, the maximum delay spread
TD = 6 μs − 0 μs = 6 μs
RMS Delay spread:
In a typical wireless channel, the later arriving paths are with lower power due to
larger propagation distances and weaker reflections. In such a scenario, maximum
delay spread metric is not reliable.

The RMS delay spread is defined by
TDRMS =
L−1
i=0 gi(τi − τ̃)2
L−1
i=0 gi
where
τi is the delay of the ith path.
gi is the power corresponding to the ith path.
τ̃ is the average delay given by
τ̃ =
L−1
i=0 giτi
L−1
i=0 gi
Example 1.2 Consider a wireless channel consists of L = 4 multipath with the delays
and power tabulated as follows
i τi (μs) gi
0 0 1
1 1 0.1
2 3 0.1
3 5 0.01
Solution
τ̃ =
1 × 0 + 0.1 × 1 + 0.1 × 3 + 0.01 × 5
1 + 0.1 + 0.1 + 0.01
μs = 0.3719 μs
TDRMS =
1 × (0 − 0.3719)2 + 0.1 × (1 − 0.3719)2 + 0.1 × (3 − 0.3719)2 + 0.01 × (5 − 0.3719)2
1 + 0.1 + 0.1 + 0.01
μs
= 0.9459 μs
1.2.6.1 Fading Channels Classification
The classification of fading channels is shown in Fig. 1.5.
The fast fading causes short burst errors which are easy to correct. The slow
fading will affect many successive symbols leading to long burst errors. Due to
energy absorption and scattering in physical channel propagation media, the trans-
mitted signal is attenuated and becomes noisy. The attenuation will vary in mobile

8 1 Introduction
Slow fading Fast fading Frequency flat Frequency selective
Fading Channel Classification
Fig. 1.5 Classification of fading channels
communications based on the vehicle speed, surrounding trees, buildings, moun-
tains, and terrain. Based on the receiver location, moving receiver signals interfere
with one another and take several different paths. As such, the wireless channels are
called multipath fading channels. Hence, the additive white Gaussian noise (AWGN)
assumption for wireless channels is not realistic. Thus, the amplitudes in the wireless
channels are often modeled using Rayleigh or Rician probability density function.
The most common fading channel models are
1. Flat independent fading channel
2. Block fading channel.
In flat independent fading channel, the attenuation remains constant for one sym-
bol period and varies from symbol to symbol. Whereas in block fading channel, the
attenuation is constant over a block of symbols and varies from block to block.
1.3 Statistical Models for Fading Channels
1.3.1 Probability Density Function of Rician Fading Channel
When the received signal is made up of multiple reflective rays plus a significant
line-of-sight (non-faded) component, the received envelope amplitude has a Rician
Probability Density Function (PDF) as given in Eq. (1.6), and the fading is referred
to as Rician fading.
Pdf (x) = x
σ2 exp

−(x2
+A2
)
2σ2 I0
xA
σ2 ; for x ≥ 0, A ≥ 0
= 0 ; Otherwise
(1.6)
where x is the amplitude of the received faded signal, I0 is the zero-order modified
Bessel function of the first kind, and A denotes the peak magnitude of the non-faded
signal component called the specular component. The Rician PDF for different values
of sigma and A = 1 is shown in Fig. 1.6.

1.3 Statistical Models for Fading Channels 9
Fig. 1.6 Probability density
of Rician fading channel
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x
probability
density
sigma=0.25
sigma=0.6
sigma=1
1.3.2 Probability Density Function of Rayleigh Fading
Channel
Rayleigh fading occurs when there are multiple indirect paths between the transmit-
ter and the receiver and no direct non-fading or line-of-sight (LOS) path. It represents
the worst case scenario for the transmission channel. Rayleigh fading assumes that a
received multipath signal consists of a large number of reflected waves with indepen-
dent and identically distributed phase and amplitude. The envelope of the received
carrier signal is Rayleigh distributed in wireless communications [3].
As the magnitude of the specular component approaches zero, the Rician PDF
approaches a Rayleigh PDF expressed as follows:
Pdf (x) = x
σ2 exp

− x2
2σ2 for x ≥ 0
= 0 Otherwise
(1.7)
The Rayleigh PDF for different values of sigma is shown in Fig. 1.7.
Additive white Gaussian noise and Rician channels provide fairly good perfor-
mance corresponding to an open country environment, while Rayleigh channel,
which best describes the urban environment fading, provides relatively worse per-
formance.
1.3.3 Probability Density Function of Nakagami Fading
Channel
The Nakagami model is another very popular empirical fading model [4]

10 1 Introduction
Fig. 1.7 Probability density
of Rayleigh fading channel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3
3.5
x
Probability
density
sigma=0.2
sigma=0.6
sigma=1.0
Pdf (r) =
2
Γ (m)
m
2σ2
m
r2m−1
e−m r2
2σ2
(1.8)
where
σ2
= 1
2
E

r2

, Γ (.) is the gamma function.
m ≥ 1
2
is the fading figure.
The received instantaneous power r2
satisfies a gamma distribution. The phase of
the signal is uniformly distributed in [0, 2π). The Nakagami distribution is a general
model obtained from experimental data fitting and its shape very similar to that of
the Rice distribution. The shape parameter “m” measures the severity of fading.
when
m = 1, it is Rayleigh fading.
m → ∞, it is AWGN channel, that is, there is no fading.
m 1, it is close to Rician fading.
However, due to lack of physical basis, the Nakagami distribution is not as popular
as the Rician and Rayleigh fading models in mobile communications. Many other
fading channel models are discussed in [5].
1.4 Channel Capacity
Channel capacity can be defined as the maximum rate at which the information can
be transmitted over a reliable channel.

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Fig. 67., a larger teaser, which is introduced at the top of the furnace, for
keeping a complete supply of charcoal around the muffle.
Fig. 68., the tongs used for charging the assays into the cups.
Fig. 69. represents a board of wood used as a register, and is divided into 45
equal compartments, upon which the assays are placed previously to their being
introduced into the furnace. When the operation is performed, the cupels are
placed in the furnace in situations corresponding to these assays on the board.
By these means all confusion is avoided, and without this regularity it would be
impossible to preserve the accuracy which the delicate operations of the assayer
require.

I shall now proceed to a description of a small assay furnace, invented by
Messrs. Anfrye and d’Arcet, of Paris. They term it, Le Petit Fourneau à Coupelle.
Fig. 70. represents this furnace, and it is composed of a chimney or pipe of
wrought iron a, and of the furnace B. It is 171⁄2 inches high, and 71⁄4 inches
wide. The furnace is formed of three pieces; of a dome A; the body of the
furnace B; and the ash-pit C, which is used as the base of the furnace, fig. 70.
and 71. The principal piece, or body of the furnace, B, has the form of a hollow
tower, or of a hollow cylinder, flattened equally at the two opposite sides parallel
to the axis, in such a manner that the horizontal section is elliptical. The foot
which supports it is a hollow truncated cone, flattened in like manner upon the
two opposite sides, and having consequently for its basis two ellipses of
different diameters; the smallest ought to be equal to that of the furnace, so
that the bottom of the latter may exactly fit it. The dome, which forms an arch
above the furnace, has also its base elliptical, whilst that of the superior orifice
by which the smoke goes out preserves the cylindrical form. The tube of
wrought iron is 18 inches long and 21⁄2 inches diameter, having one of its ends a
little enlarged, and slightly conical, that it may be exactly fitted or jointed upon

the upper part of the furnace dome d, fig. 70. At the union of the conical and
cylindrical parts of the tube, there is placed a small gallery of iron, e, fig. 70, 71.
See also a plan of it, fig. 72. This gallery is both ingenious and useful. Upon it
are placed the cupels, which are thus annealed during the ordinary work of the
furnace, that they may be introduced into the muffle, when it is brought into its
proper degree of heat. A little above this gallery is a door f, by which, if thought
proper, the charcoal could be introduced into the furnace; above that there is
placed at g a throttle valve, which is used for regulating the draught of the
furnace at pleasure. Messrs. Anfrye and d’Arcet say, that, to give the furnace the
necessary degree of heat so as to work the assays of gold, the tube must be
about 18 inches above the gallery, for annealing or heating the cupels. The
circular opening h, in the dome, fig. 70., and as seen in the section, fig. 71., is
used to introduce the charcoal into the furnace: it is also used to inspect the
interior of the furnace, and to arrange the charcoal round the muffle. This
opening is kept shut during the working of the furnace, with the mouth-piece, of
which the face is seen at n, fig. 71.
The section of the furnace, fig. 71., presents several openings, the principal of
which is that of the muffle; it is placed at i; it is shut with the semicircular door
m, fig. 70., and seen in the section m, fig. 71. In front of this opening, is the
table or shelf, upon which the door of the muffle is made to advance or recede;
the letter q, fig. 71., shows the face, side, and cross section of the shelf, which
makes part of the furnace. Immediately under the shelf, is a horizontal slit, l,
which is pierced at the level of the upper part of the grate, and used for the
introduction of a slender rod of iron, that the grate may be easily kept clean.
This opening is shut at pleasure, by the wedge represented at k, fig. 70. and 71.
Upon the back of the furnace is a horizontal slit p, fig. 71, which supports the
fire-brick, s, and upon which the end of the muffle, if necessary, may rest; u,
fig. 71., is the opening in the furnace where the muffle is placed.
The plan of the grate of the furnace is an ellipse: fig. 73. is a horizontal view of
it. The dimensions of that ellipsis determine the general form of the furnace,
and thickness of the grate. To give strength and solidity to the grate, it is
encircled by a bar or hoop of iron. There is a groove in which the hoop of iron is
fixed. The holes of the grate are truncated cones, having the greater base
below, that the ashes may more easily fall into the ash-pit. The letter v, fig. 71.,

shows the form of these holes. The grate is supported by a small bank or shelf,
making part of the furnace, as seen at a, fig. 71.
The ash-pit, C, has an opening y in front, fig. 71.; and is shut when necessary
by the mouth-piece r, fig. 70. and 71.
To give strength and solidity to the furnace, it is bound with hoops of iron, at b,
b, b, b, fig. 70.
Figs. 74. 75. 76. are views of the muffle.
Fig. 77. is a view of a crucible for annealing gold.
Figs. 78. 79. 80. are cupels of various sizes, to be used in the furnace. They are
the same as those used by assayers in their ordinary furnaces.

Figs. 81. and 82. are views of the hand-shovels, used for filling the furnace with
charcoal; they should be made of such size and form as to fit the opening h, in
figs. 70. and 71.
The smaller pincers or tongs, by which the assays are charged into the cupels,
and by which the latter are withdrawn from the furnace, as well as the teaser
for cleaning the grate of the furnace, are similar to those used in the British
Mint.
In the furnace of the Mint above described, the number of assays that can be
made at one time, is 45. The same number of cupels are put into the muffle.
The furnace is then filled with charcoal to the top, and upon this are laid a few
pieces already ignited. In the course of three hours, a little more or less,
according to circumstances, the whole is ignited; during which period, the
muffle, which is made of fire-clay, is gradually heated to redness, and is
prevented from cracking; which a less regular or more sudden increase of
temperature would not fail to do: the cupels, also, become properly annealed.
All moisture being dispelled, they are in a fit state to receive the piece of silver
or gold to be assayed.
The greater care that is exercised in this operation, the less liable is the assayer
to accidents from the breaking of the muffle; which it is both expensive and
troublesome to fit properly into the furnace.
The cupels used in the assay process, are made of the ashes of burnt bones
(phosphate of lime). In the Royal Mint, the cores of ox-horn are selected for this
purpose; and the ashes produced are about four times the expense of the bone-
ash, used in the process of cupellation upon the large scale. So much depends
upon the accuracy of an assay of gold or silver, where a mass of 15lbs. troy in
the first, and 60lbs. troy in the second instance, is determined by the analysis of
a portion not exceeding 20 troy grains, that every precaution which the longest
experience has suggested, is used to obtain an accurate result. Hence the
attention paid to the selection of the most proper materials for making the
cupels.
The cupels are formed in a circular mould made of cast steel, very nicely turned,
by which means they are easily freed from the mould when struck. The bone-
ash is used moistened with a quantity of water, sufficient to make the particles
adhere firmly together. The circular mould is filled, and pressed level with its
surface; after which, a pestle or rammer, having its end nicely turned, of a
globular or convex shape, and of a size equal to the degree of concavity wished
to be made in the cupel for the reception of the assay, is placed upon the ashes
in the mould, and struck with a hammer until the cupel is properly formed.

These cupels are allowed to dry in the air for some time before they are used. If
the weather is fine, a fortnight will be sufficient.
An assay may prove defective for several reasons. Sometimes the button or
bead sends forth crystalline vegetations on its surface with such force, as to
make one suppose a portion of the silver may be thrown out of the cupel. When
the surface of the bead is dull and flat, the assay is considered to have been too
hot, and it indicates a loss of silver in fumes. When the tint of the bead is not
uniform, when its inferior surface is bubbly, when yellow scales of oxide of lead
remain on the bottom of the cupel, and the bead adheres strongly to it, by
these signs it is judged that the assay has been too cold, and that the silver
retains some lead.
Lastly, the assay is thought to be good if the bead is of a round form, if its
upper surface is brilliant, if its lower surface is granular and of a dead white, and
if it separates readily from the cupel.
After the lead is put into the cupel, it gets immediately covered with a coat of
oxide, which resists the admission of the silver to be assayed into the melted
metal; so that the alloy cannot form. When a bit of silver is laid on a lead bath
in this predicament, we see it swim about for a long time without dissolving. In
order to avoid this result, the silver is wrapped up in a bit of paper; and the
carburetted hydrogen generated by its combustion, reduces the film of the lead
oxide, gives the bath immediately a bright metallic lustre, and enables the two
metals readily to combine.
As the heat rises, the oxide of lead flows round about over the surface, till it is
absorbed by the cupel. When the lead is wasted to a certain degree, a very thin
film of it only remains on the silver, which causes the iridescent appearance, like
the colours of soap-bubbles; a phenomenon, called by the old chemists,
fulguration.
When the cupel cools in the progress of the assay, the oxygenation of the lead
ceases; and, instead of a very liquid vitreous oxide, an imperfectly melted oxide
is formed, which the cupel cannot absorb. To correct a cold assay, the
temperature of the furnace ought to be raised, and pieces of paper ought to be
put into the cupel, till the oxide of lead which adheres to it, be reduced. On
keeping up the heat, the assay will resume its ordinary train.
Pure silver almost always vegetates. Some traces of copper destroy this
property, which is obviously due to the oxygen which the silver can absorb while
it is in fusion, and which is disengaged the moment it solidifies. An excess of
lead, by removing all the copper at an early stage, tends to cause the
vegetation.

The brightening is caused by the heat evolved, when the button passes from
the liquid to the solid state. Many other substances present the same
phenomenon.
In the above operation it is necessary to employ lead which is very pure, or at
least free from silver. That kind is called poor lead.
It has been observed at all times, that the oxide of lead carries off with it, into
the cupel, a little silver in the state of an oxide. This effect becomes less, or
even disappears, when there is some copper remaining; and the more copper,
the less chance there is of any silver being lost. The loss of silver increases, on
the other hand, with the dose of lead. Hence the reason why it is so important
to proportion the lead with a precision which, at first sight, would appear to be
superfluous. Hence, also, the reason of the attempts which have, of late years,
been made to change the whole system of silver assays, and to have recourse
to a method exempt from the above causes of error.
M. d’Arcet, charged by the Commission of the Mint in Paris, to examine into the
justice of the reclamations made by the French silversmiths against the public
assays, ascertained that they were well founded; and that the results of
cupellation gave for the alloys between 897 and 903 thousandths (the limits of
their standard coin) an inferior standard, by from 4 to 5 thousandth parts, from
the standard or title which should result from the absolute or actual alloy.
The mode of assay shows, in fact, that an ingot, experimentally composed of
900 thousandths of fine silver, and 100 thousandths of copper, appears, by
cupellation, to be only, at the utmost, 896 or 897 thousandths; whereas fine
silver, of 1000 thousandths, comes out nearly of its real standard. Consequently
a director of the Mint, who should compound his alloy with fine silver, would be
obliged to employ 903 or 904 thousandths, in order that, by the assay in the
laboratory of the Mint, it should appear to have the standard of 900
thousandths. These 3 or 4 thousandths would be lost to him, since they would
be disguised by the mode of assay, the definitive criterion of the quantity of
silver, of which the government keeps count from the coiner of the money.
From experiments subsequently made by M. d’Arcet, it appears that silver
assays always suffer a loss of the precious metal, which varies, however, with
the standard of the alloy. It is 1 thousandth for fine silver,
4·3 thousandths for silver of 900 thousandths,
4·9 — for — of 800 —
4·2 — for — of 500 —

and diminishes thereafter, progressively, till the alloy contains only 100
thousandths of silver, at which point the loss is only 0·4.
Assays requested by the Commission of the Paris Mint, from the assayers of the
principal Royal Mints in Europe, to which the same alloys, synthetically
compounded, were sent, afforded the results inscribed in the following table.
Names of the Assayers. Cities where
they reside.
Standards found for
the Mathematical Alloys.
950 mill. 900 mill. 800 mill.
F. de Castenhole, Mint Assayer Vienna 946·20 898·40 795·10
A. R. Vervaëz, Ditto Madrid 944·40 893·70 789·20
D. M. Cabrera, Assayer in Spain Ditto 944·40 893·70 788·60
Assayer Amsterdam 947·00 895·00 795·00
Mr. Bingley, Assay Master London 946·25 896·25 794·25
Mr. Johnson, Assayer Ditto 933·33 883·50 783·33
Inspector of the Mint Utrecht 945·00 896·50 799·00
Assayer of the Mint Naples 945·00 891·00 787·00
Assayer of Trade Ditto 945·00 891·00 787·00
Assayer of the Mint Hamburgh 946·13⁄72 897·41⁄72 798·44⁄72
Ditto Altona 942·1⁄4 894·00 790
These results, as well as those in still greater numbers, obtained from the ablest
Parisian assayers, upon identical alloys of silver and copper, prove that the mode
of assay applied to them brings out the standard too low; and further, that the
quantity of silver masked or disguised, is not uniform for these different eminent
assay masters. An alloy, for example, at the standard of 900 thousandths is
judged at
M.
the Mint of Paris to have a standard of 895·6
At that of Vienna — 898·4
— Madrid — 893·7
— Naples — 891·0
The fact thus so clearly made out of a loss in the standard of silver bullion and
coin, merits the most serious attention; and it will appear astonishing, perhaps,
that a thing recurring every day, should have remained for so long a time in the
dark. In reality, however, the fact is not new; as the very numerous and well-

made experiments of Tillet from 1760 to 1763, which are related in the memoirs
of the Academy of Sciences, show, in the silver assays, a loss still greater than
that which was experienced lately in the laboratory of the Commission of the
French Mint. But he thought that, as the error was common to the nations in
general, it was not worth while or prudent to introduce any innovation.
A mode of assaying, to give, with certainty, the standard of silver bullion, should
be entirely independent of the variable circumstances of temperature, and the
unknown proportions of copper, so difficult to regulate by the mere judgment of
the senses. The process by the humid way, recommended by me to the Royal
Mint in 1829, and exhibited as to its principles before the Right Honourable John
Herries, then Master, in 1830, has all the precision and certainty we could wish.
It is founded on the well-known property which silver has, when dissolved in
nitric acid, to be precipitated in a chloride of silver quite insoluble, by a solution
of sea salt, or by muriatic acid; but, instead of determining the weight of the
chloride of silver, which would be somewhat uncertain and rather tedious, on
account of the difficulty of drying it, we take the quantity of the solution of sea
salt which has been necessary for the precipitation of the silver. To put the
process in execution, a liquor is prepared, composed of water and sea salt in
such proportions that 1000 measures of this liquor may precipitate, completely,
12 grains of silver, perfectly pure, or of the standard 1000, previously dissolved
in nitric acid. The liquor thus prepared, gives, immediately, the true standard of
any alloy whatever, of silver and copper, by the weight of it which may be
necessary to precipitate 12 grains of this alloy. If, for example 905 measures
have been required to precipitate the 12 grains of alloy, its standard would be
905 thousandths.
The process by the humid way is, so to speak, independent of the operator. The
manipulations are so easy; and the term of the operation is very distinctly
announced by the absence of any sensible nebulosities on the affusion of sea
salt into the silver solution, while there remains in it 1⁄2 thousandth of metal.
The process is not tedious, and in experienced hands it may rival the cupel in
rapidity; it has the advantage over the cupel of being more within the reach of
ordinary operators, and of not requiring a long apprenticeship. It is particularly
useful to such assayers as have only a few assays to make daily, as it will cost
them very little time and expense.
By agitating briskly during two minutes, or thereby, the liquid rendered milky by
the precipitation of the chloride of silver, it may be sufficiently clarified to enable
us to appreciate, after a few moments of repose, the disturbance that can be
produced in it by the addition of 1000 of a grain of silver. Filtration is more
efficacious than agitation, especially when it is employed afterwards; it may be

sometimes used; but agitation, which is much more prompt, is generally
sufficient. The presence of lead and copper, or any other metal, except mercury,
has no perceptible influence on the quantity of sea salt necessary to precipitate
the silver; that is to say, the same quantity of silver, pure or alloyed, requires for
its precipitation a constant quantity of the solution of sea salt.
Supposing that we operate upon a gramme of pure silver, the solution of sea
salt ought to be such that 100 centimetres cube may precipitate exactly the
whole silver. The standard of an alloy is given by the number of thousandths of
solution of sea salt necessary to precipitate the silver contained in a gramme of
the alloy.
When any mercury is accidentally present, which is, however, a rare occurrence,
it is made obvious by the precipitated chloride remaining white when exposed to
daylight, whereas when there is no mercury present, it becomes speedily first
grey and then purple. Silver so contaminated must be strongly ignited in fusion
before being assayed, and its loss of weight noted. In this case, a cupel assay
must be had recourse to.
Preparation of the Normal Solution of Sea Salt, when it is measured by Weight.
—Supposing the sea salt pure as well as the water, we have only to take these
two bodies in the proportion of 0·5427 k. of salt to 99·4573 k. of water, to have
100 k. of solution, of which 100 grammes will precipitate exactly one gramme of
silver. But instead of pure salt, which is to be procured with difficulty, and which
besides may be altered readily by absorbing the humidity of the air, a
concentrated solution of the sea salt of commerce is to be preferred, of which a
large quantity may be prepared at a time, to be kept in reserve for use, as it is
wanted. Instruction de Gay Lussac.
Preparation of the Normal Solution of Sea Salt, when measured by Volume.—
The measure by weight has the advantage of being independent of
temperature, of having the same degree of precision as the balance, and of
standing in need of no correction. The measure by volume has not all these
advantages; but, by giving it sufficient precision, it is more rapid, and is quite
sufficient for the numerous daily assays of the mint. This normal solution is so
made, that a volume equal to that of 100 grammes of water, or 100 centimetres
cube, at a determinate temperature, may precipitate exactly one gramme of
silver. The solution may be kept at a constant temperature, and in this case the
assay stands in want of no correction; or if its temperature be variable, the
assay must be corrected according to its influence. These two circumstances
make no change in the principle of the process, but they are sufficiently
important to occasion some modifications in the apparatus. Experience has

decided the preference in favour of applying a correction to a variable
temperature.
We readily obtain a volume of 100 cubic centimetres by means of a pipette, fig.
83., so gauged that when filled with water up to the mark a, b, and well dried at
its point, it will run out, at a continuous efflux, 100 grammes of water at the
temperature of 15 C. (59 Fah.). We say purposely at one efflux, because after
the cessation of the jet, the pipette may still furnish two or three drops of liquid,
which must not be counted or reckoned upon. The weight of the volume of the
normal solution, taken in this manner with suitable precautions, will be uniform
from one extreme to another, upon two centimetres and a half, at most, or to a
quarter of a thousandth, and the difference from the mean will be obviously
twice less, or one half. Let us indicate the most simple manner of taking a
measure of the normal solution of sea salt.
After having immersed the beak c of the pipette in the solution, we apply
suction by the mouth, to the upper orifice, and thereby raise the liquid to d
above the circular line a b. We next apply neatly the forefinger of one hand to
this orifice, remove the pipette from the liquid, and seize it as represented in fig.
84. The mark a b being placed at the level of the eye, we make the surface of

the solution become exactly a tangent to the plane a b. At the instant it
becomes a tangent, we leave the beak c of the pipette open, by taking away the
finger that had been applied to it, and without changing any thing else in the
position of the hands, we empty it into the bottle which should receive the
solution, taking care to remove it whenever the efflux has run out.
If after filling the pipette by suction, any one should find a difficulty in applying
the forefinger fast enough to the upper orifice, without letting the liquid run
down below the mark a b, he should remove the pipette from the solution with
its top still closed with his tongue, then apply the middle finger of one of his
hands to the lower orifice; after which he may withdraw his tongue, and apply
the forefinger of the other hand to the orifice previously wiped. This mode of
obtaining a measure of normal solution of sea salt is very simple, and requires
no complex apparatus; but we shall indicate another manipulation still easier,
and also more exact.
In this new process the pipette is filled from the top like a bottle, instead of
being filled by suction, and it is moreover fixed. Fig. 85. represents the
apparatus. D and D′ are two sockets separated by a stop cock R. The upper one,
tapped interiorly, receives, by means of a cork stopper L, the tube T, which
admits the solution of sea salt. The lower socket is cemented on to the pipette;
it bears a small air-cock R′, and a screw plug V, which regulates a minute
opening intended to let the air enter very slowly into the pipette. Below the
stop-cock R′, a silver tube N, of narrow diameter, soldered to the socket, leads
the solution into the pipette, by allowing the air, which it displaces, to escape by
the stop-cock R′. The screw plug, with the milled head V′, replaces the ordinary
screw by which the key of the stop-cock may be made to press, with more or
less force, upon its conical seat.

Fig. 86. represents, in a side view, the apparatus just described. We here
remark an air-cock R, and an opening m. At the extremity Q of the same figure,
the conical pipe T enters, with friction. It is by this pipe that the air is sucked
into the pipette, when it is to be filled from its beak.

The pipette is supported by two horizontal arms H K (fig. 87.) moveable about a
common axis A A, and capable of being drawn out or shortened by the aid of
two longitudinal slits. They are fixed steadily by two screw nuts e e′, and their
distance may be varied by means of round bits of wood or cork interposed, or
even by opposite screw nuts o o′. The upper arm H is pierced with a hole, in
which is fixed, by the pressure of a wooden screw v, the socket of the pipette.
The corresponding hole of the lower arm is larger; and the beak of the pipette is
supported in it by a cork stopper L. The apparatus is fixed by its tail-piece P, by
means of a screw to the corner of a wall, or any other prop.
The manner of filling the pipette is very simple. We begin by applying the fore-
finger of the left hand to the lower aperture c; we then open the two stop-cocks
R and R′. Whenever the liquor approaches the neck of the pipette, we must
temper its influx, and when it has arrived at some millimetres above the mark a
b, we close the two stop-cocks, and remove our forefinger. We have now
nothing more to do than to regulate the pipette; for which purpose the liquid
must touch the line a b, and must simply adhere externally to the beak of the
pipette.

This last circumstance is easily adjusted. After taking away the finger which
closed the aperture c of the pipette, we apply to this orifice a moist sponge m,
fig. 88., wrapped up in a linen rag, to absorb the superfluous liquor as it drops
out. This sponge is called the handkerchief (mouchoir), by M. Gay Lussac. The
pipette is said to be wiped when there is no liquor adhering to its point
exteriorly.
For the convenience of operating, the handkerchief is fixed by friction in a tube
of tin plate, terminated by a cup, open at bottom to let the droppings flow off
into the cistern C, to which the tube is soldered. It may be easily removed for
the purpose of washing it; and, if necessary, a little wedge of wood, o, can raise
it towards the pipette.
To complete the adjustment of the pipette, the liquid must be made merely to
descend to the mark a, b. With this view, and whilst the handkerchief is applied
to the beak of the pipette, the air must be allowed to enter very slowly by
unscrewing the plug V, fig. 85.; and at the moment of the contact the
handkerchief must be removed, and the bottle F, destined to receive the
solution, must be placed below the orifice of the pipette, fig. 88. As the motion
must be made rapidly, and without hesitation, the bottle is placed in a cylinder
of tin-plate, of a diameter somewhat greater, and forming one body with the
cistern and the handkerchief. The whole of this apparatus has for a basis a plate
of tinned iron, moveable between two wooden rulers R R, one of which bears a
groove, under which the edge of the plate slips. Its traverses are fixed by two
abutments b b, placed so that when it is stopped by one of them, the beak of
the pipette corresponds to the centre of the neck of the bottle, or is a tangent
to the handkerchief. This arrangement, very convenient for wiping the pipette
and emptying it, gives the apparatus sufficient solidity, and allows of its being
taken away, and replaced without deranging any thing. It is obvious that it is of
advantage, when once the entry of the air into the pipette has been regulated

by the screw V, to leave it constantly open, because the motion from the
handkerchief to the bottle is performed with sufficient rapidity to prevent a drop
of the solution from collecting and falling down.
Temperature of the Solution.—After having described the manner of measuring
by volume the normal solution of the sea salt, we shall indicate the most
convenient means of taking the temperature. The thermometer is placed in a
tube of glass T, fig. 89., which the solution traverses to arrive at the pipette. It is
suspended in it by a piece of cork, grooved on the four sides to afford passage
to the liquid. The scale is engraved upon the tube itself, and is repeated at the
opposite side, to fix the eye by the coincidence of this double division at the
level of the thermometric column. The tube is joined below to another narrower
one, through which it is attached by means of a cork stopper B, in the socket of
the stop-cock of the pipette. At its upper part it is cemented into a brass socket,
screw-tapped in the inside, which is connected in its turn by a cock, with the
extremity, also tapped, of the tube above T, belonging to the reservoir of the
normal solution. The corks employed here as connecting links between the parts
of the apparatus, give them a certain flexibility, and allow of their being
dismounted and remounted in a very short time; but it is indispensable to make

them be traversed by a hollow tube of glass or metal, which will hinder them
from being crushed by the pressure they are exposed to. If the precaution be
taken to grease them with a little suet and to fill their pores, they will suffer no
leakage.
Preservation of the Normal Solution of Sea Salt in metallic Vessels.—M. Gay
Lussac uses for this purpose a cylindrical vessel or drum of copper, of a capacity
of about 110 litres, having its inside covered with a rosin and wax cement.
Preparation of the Normal Solution of Sea Salt, measuring it by Volume.—If the
drum contains 110 litres, we should put only 105 into it, in order that sufficient
space may be left for agitating the liquor without throwing it out. According to
the principle that 100 centimetres cube, or 1⁄10 of a litre of the solution should
contain enough of sea salt to precipitate a gramme of pure silver; and,
admitting moreover, 13·516 for the prime equivalent of silver, and 7·335 for that
of sea salt, we shall find the quantity of pure salt that should be dissolved in the
105 litres of water, and which corresponds to 105 × 10 = 1050 grammes of
silver, to be by the following proportion:—
13·516 : 7·335 ∷ 1050 gramm. : x = 569·83 gr.
And as the solution of the sea salt of commerce, formerly mentioned, contains
approximately 250 grammes per kilogramme, we must take 2279·3 grammes of
this solution to have 569·83 gram. of salt. The mixture being perfectly made,
the tubes and the pipette must be several times washed by running the solution
through them, and putting it into the drum. The standard of the solution must
be determined after it has been well agitated, supposing the temperature to
remain uniform.
To arrive more conveniently at this result, we begin by preparing two decimes
solutions; one of silver, and another of sea salt.
The decime solution of silver is obtained by dissolving 1 gramme of silver in
nitric acid, and diluting the solution with water till its volume become a litre.
The decime solution of sea salt may be obtained by dissolving 0·543 grammes
of pure sea salt in water, so that the solution shall occupy a litre; but we shall
prepare it even with the normal solution which we wish to test, by mixing a
measure of it with 9 measures of water; it being understood that this solution is
not rigorously equivalent to that of silver, and that it will become so, only when
the normal solution employed for its preparation shall be finally of the true
standard. Lastly, we prepare beforehand several stoppered phials, in each of
which we dissolve 1 gramme of silver in 8 or 10 grammes of nitric acid. For
brevity’s sake we shall call these tests.

Now to investigate the standard of the normal solution, we must transfer a
pipette of it into one of these test phials; and we must agitate the liquors briskly
to clarify them. After some instants of repose, we must pour in 2 thousandths of
the decime solution of sea salt, which, we suppose, will produce a precipitate.
The normal liquor is consequently too feeble; and we should expect this, since
the sea salt employed was not perfectly pure. We agitate and add 2 fresh
thousandths, which will also produce a precipitate. We continue thus by
successive additions of 2 thousandths, till the last produces no precipitation.
Suppose that we have added 16 thousandths: the last two should not be
reckoned, as they produced no precipitate; the preceding two were necessary,
but only in part; that is to say, the useful thousandths added are above 12 and
below 14, or otherwise they are on an average equal to 13.
Thus, in the condition of the normal solution, we require 1013 parts of it to
precipitate one gramme of silver, while we should require only 1000. We shall
find the quantity of concentrated solution of sea salt that we should add, by
noting that the quantity of solution of sea salt, at first employed, viz. 2279·3
grammes, produced a standard of only 987 thousandths = 1000 - 13; and by
using the following proportion:
987 : 2279·3 ∷ 13 : x = 30·02 grammes.
This quantity of the strong solution of salt, mixed with the normal solution in the
drum, will correct its standard, and we shall now see by how much.
After having washed the tubes and the pipette, with the new solution, we must
repeat the experiment upon a fresh gramme of silver. We shall find, for
example, in proceeding only by a thousandth at a time, that the first causes a
precipitate, but not the second. The standard of the solution is still too weak,
and is comprised between 1000 and 1001; that is to say, it may be equal to
10001⁄2, but we must make a closer approximation.
We pour into the test bottle 2 thousandths of the decime solution of silver,
which will destroy, perceptibly, two thousandths of sea salt, and the operation
will have retrograded by two thousandths; that is to say, it will be brought back
to the point at which it was first of all. If, after having cleared up the liquor, we
add half a thousandth of the decime solution, there will necessarily be a
precipitate, as we knew beforehand, but a second will cause no turbidity. The
standard of the normal liquor will be consequently comprehended between 1000
and 10001⁄2, or equal to 10001⁄4.
We should rest content with this standard, but if we wish to correct it, we may
remark that the two quantities of solution of salt added, viz. 2279·3 gr. + 30·02
gr. = 2309·32 gr. have produced only 999·75 thousandths, and that we must

add a new quantity of it corresponding to 1⁄4 of a thousandth. We make,
therefore, the proportion
999·75 : 2309·32 ∷ 0·25 : x.
But since the first term differs very little from 1000, we may content ourselves
to have x by taking the 0·25⁄1000 of 2309·32, and we shall find 0·577 gr. for the
quantity of solution of sea salt to be added to the normal solution.
It is not convenient to take exactly so small a quantity of solution of sea salt by
the balance, but we shall succeed easily by the following process. We weigh 50
grammes of this solution, and we dilute it with water; so that it occupies exactly
half a litre, or 500 centimetres cube. A pipette of this solution, one centimetre
cube in volume, will give a decigramme of the primitive solution, and as such a
small pipette is divided into twenty drops, each drop, for example, will represent
5 milligrammes of the solution. We should arrive at quantities smaller still by
diluting the solution with a proper quantity of water; but greater precision would
be entirely needless.
The testing of the normal liquor just described, is, in reality, less tedious than
might be supposed. It deserves also to be remarked, that liquor has been
prepared for more than 1000 assays; and that, in preparing a fresh quantity, we
shall obtain directly its true standard, or nearly so, if we bear in mind the
quantities of water and solution of salt which had been employed.
Correction of the Standard of the Normal Solution of Sea Salt, when the
Temperature changes.—We have supposed, in determining the standard of the
normal solution of sea salt, that the temperature remained uniform. The assays
made in such circumstances, have no need of correction; but if the temperature
should change, the same measure of the solution will not contain the same
quantity of sea salt. Supposing that we have tested the solution of the salt at
the temperature of 15° C.; if, at the time of making the experiment, the
temperature is 18° C., for example, the solution will be too weak on account of
its expansion, and the pipette will contain less of it by weight; if, on the
contrary, the temperature has fallen to 12°, the solution will be thereby
concentrated and will prove too strong. It is therefore proper to determine the
correction necessary to be made, for any variation of temperature.
To ascertain this point, the temperature of the solution of sea salt was made
successively to be 0°, 5°, 10°, 15°, 20°, 25°, and 30° C.; and three pipettes of
the solution were weighed exactly at each of these temperatures. The third of
these weighings gave the mean weight of a pipette. The corresponding weights
of a pipette of the solution, were afterwards graphically interpolated from
degree to degree. These weights form the second column of the following table,

intitled, Table of Correction for the Variations in the Temperature of the Normal
Solution of the Sea Salt. They enable us to correct any temperature between 0
and 30 degrees centigrade (32° and 86° Fahr.) when the solution of sea salt has
been prepared in the same limits.
Let us suppose, for example, that the solution has been made standard at 15°,
and that at the time of using it, the temperature has become 18°. We see by
the second column of the table, that the weight of a measure of the solution is
100·099 gr. at 15°, and 100·065 at 18°; the difference 0·034 gr., is the quantity
of solution less which has been really taken; and of course we must add it to
the normal measure, in order to make it equal to one thousand millièmes. If the
temperature of the solution had fallen to 10 degrees, the difference of the
weight of a measure from 10 to 15 degrees would be 0·019 gr. which we must
on the contrary deduct from the measure, since it had been taken too large.
These differences of weight of a measure of solution at 15°, from that of a
measure at any other temperature, form the column 15° of the table, where
they are expressed in thousandths; they are inscribed on the same horizontal
lines as the temperatures to which each of them relates with the sign + plus,
when they must be added, and with the sign - minus, when they must be
subtracted. The columns 5°, 10°, 20°, 25°, 35°, have been calculated in the
same manner for the cases in which the normal solution may have been
graduated to each of these temperatures. Thus, to calculate the column 10, the
number 100·118 has been taken of the column of weights for a term of
departure, and its difference from all the numbers of the same column has been
sought.
Table of Correction for the Variations in the Temperature of the Normal Solution
of the Sea Salt.
Tem-
pera-
ture.
Weight. 5° 10° 15° 20° 25° 30°
gram. mill. mill. mill. mill. mill. mill.
4 100,109 0·0 - 0·1 + 0·1 + 0·7 + 1·7 + 2·7
5 100,113 0·0 - 0·1 + 0·1 + 0·7 + 1·7 + 2·8
6 100,115 0·0 0·0 + 0·2 + 0·8 + 1·7 + 2·8
7 110,118 + 0·1 0·0 + 0·2 + 0·8 + 1·7 + 2·8
8 100,120 + 0·1 0·0 + 0·2 + 0·8 + 1·8 + 2·8
9 100,120 + 0·1 0·0 + 0·2 + 0·8 + 1·8 + 2·8
10 100,118 + 0·1 0·0 + 0·2 + 0·8 + 1·7 + 2·8
11 100,116 0·0 0·0 + 0·2 + 0·8 + 1·7 + 2·8

12 100,114 0·0 0·0 + 0·2 + 0·8 + 1·7 + 2·8
13 100,110 0·0 - 0·1 + 0·1 + 0·7 + 1·7 + 2·7
14 100,106 - 0·1 - 0·1 + 0·1 + 0·7 + 1·6 + 2·7
15 100,099 - 0·1 - 0·2 - 0·0 + 0·6 + 1·6 + 2·6
16 100,090 - 0·2 - 0·3 - 0·1 + 0·5 + 1·5 + 2·5
17 100,078 - 0·4 - 0·4 - 0·2 + 0·4 + 1·3 + 2·4
18 100,065 - 0·5 - 0·5 - 0·3 + 0·3 + 1·2 + 2·3
19 100,053 - 0·6 - 0·7 - 0·5 + 0·1 + 1·1 + 2·2
20 100,039 - 0·7 - 0·8 - 0·6 0·0 + 1·0 + 2·0
21 100,021 - 0·9 - 1·0 - 0·8 - 0·2 + 0·8 + 1·9
22 100,001 - 1·1 - 1·2 - 1·0 - 0·4 + 0·6 + 1·7
23 99,983 - 1·3 - 1·4 - 1·2 - 0·6 + 0·4 + 1·5
24 99,964 - 1·5 - 1·5 - 1·4 - 0·8 + 0·2 + 1·3
25 99,944 - 1·7 - 1·7 - 1·6 - 1·0 0·0 + 1·1
26 99,924 - 1·9 - 1·9 - 1·8 - 1·2 - 0·2 + 0·9
27 99,902 - 2·1 - 2·2 - 2·0 - 1·4 - 0·4 + 0·7
28 99,879 - 2·3 - 2·4 - 2·2 - 1·6 - 0·7 + 0·4
29 99,858 - 2·6 - 2·6 - 2·4 - 1·8 - 0·9 + 0·2
30 99,836 - 2·8 - 2·8 - 2·6 - 2·0 - 1·1 0·0
Several expedients have been employed to facilitate and abridge the
manipulations. In the first place, the phials for testing or assaying the
specimens of silver should all be of the same height and of the same diameter.
They should be numbered at their top, as well as on their stoppers, in the order
1, 2, 3, c. They may be ranged successively in tens; the stoppers of the same
series being placed on a support in their proper order. Each two phials should, in
their turn, be placed in a japanned tin case (fig. 90.) with ten compartments
duly numbered. These compartments are cut out anteriorly to about half their
height, to allow the bottoms of the bottles to be seen. When each phial has
received its portion of alloy, through a wide-beaked funnel, there must be
poured into it about 10 grammes of nitric acid, of specific gravity 1·28, with a
pipette, containing that quantity; it is then exposed to the heat of a water bath,
in order to facilitate the solution of the alloy. The water bath is an oblong vessel
made of tin plate, intended to receive the phials. It has a moveable double
bottom, pierced with small holes, for the purpose of preventing the phials being
broken, as it insulates them from the bottom to which the heat is applied. The
solution is rapid; and, since it emits nitrous vapours in abundance, it ought to be
carried on under a chimney.

The agitator.—Fig. 91. gives a sufficiently exact idea of it, and may dispense
with a lengthened description. It has ten cylindrical compartments, numbered
from 1 to 10. The phials, after the solution of the alloy, are arranged in it in the
order of their numbers. The agitator is then placed within reach of the pipette,
intended to measure out the normal solution of sea salt, and a pipette full of
this solution is put into each phial. Each is then closed with its glass stopper,
previously dipped in pure water. They are fixed in the cells of the agitator by
wooden wedges. The agitator is then suspended to a spring R, and, seizing it
with the two hands, the operator gives an alternating rapid movement, which
agitates the solution, and makes it, in less than a minute, as limpid as water.
This movement is promoted by a spiral spring, B, fixed to the agitator and the
ground; but this is seldom made use of, because it is convenient to be able to
transport the agitator from one place to another. When the agitation is finished,
the wedges are to be taken out, and the phials are placed in order upon a table
furnished with round cells destined to receive them, and to screen them from
too free a light.
When we place the phials upon this table, we must give them a brisk circular
motion, to collect the chloride of silver scattered round their sides; we must lift
out their stoppers, and suspend them in wire rings, or pincers. We next pour a

thousandth of the decime solution into each phial; and before this operation is
terminated, there is formed in the first phials, when there should be a
precipitate, a nebulous stratum, very well marked, of about a centimetre in
thickness.
At the back of the table there is a black board divided into compartments
numbered from 1 to 10, upon each of which we mark, with chalk, the
thousandths of the decime liquor put into the correspondent phial. The
thousandths of sea salt, which indicate an augmentation of standard, are
preceded by the sign +, and the thousandths of nitrate of silver by the sign -.
When the assays are finished, the liquor of each phial is to be poured into a
large vessel, in which a slight excess of sea salt is kept; and when it is full, the
supernatant clear liquid must be run off with a syphon.
The chloride of silver may be reduced without any perceptible loss. After having
washed it well, we immerse pieces of iron or zinc into it, and add sulphuric acid
in sufficient quantity to keep up a feeble disengagement of hydrogen gas. The
mass must not be touched. In a few days the silver is completely reduced. This
is easily recognised by the colour and nature of the product; or by treating a
small quantity of it with water of ammonia, we shall see whether there be any
chloride unreduced; for it will be dissolved by the ammonia, and will afterwards
appear upon saturating the ammonia with an acid. The chlorine remains
associated with the iron or the zinc in a state of solution. The first washings of
the reduced silver must be made with an acidulous water, to dissolve the oxide
of iron which may have been formed, and the other washings with common
water. After decanting the water of the last washing, we dry the mass, and add
a little powdered borax to it. It must be now fused. The silver being in a bulky
powder is to be put in successive portions into a crucible as it sinks down. The
heat should be at first moderate; but towards the end of the operation it must
be pretty strong to bring into complete fusion the silver and the scoriæ, and to
effect their complete separation. In case it should be supposed that the whole
of the silver had not been reduced by the iron or zinc, a little carbonate of
potash should be added to the borax. The silver may also be reduced by
exposing the chloride to a strong heat, in contact with chalk and charcoal.
The following remarks by M. Gay Lussac, the author of the above method, upon
the effect of a little mercury in the humid assay, are important:—
It is well known that chloride of silver blackens the more readily as it is exposed
to an intense light, and that even in the diffused light of a room, it becomes
soon sensibly coloured. If it contains four to five thousandths of mercury, it does
not blacken; it remains of a dead white: with three thousandths of mercury,

there is no marked discolouring in diffused light; with two thousandths it is
slight; with one it is much more marked, but still it is much less intense than
with pure chloride. With half a thousandth of mercury the difference of colour is
not remarkable, and is perceived only in a very moderate light.
But when the quantity of mercury is so small that it cannot be detected by the
difference of colour in the chloride of silver, it may be rendered quite evident by
a very simple process of concentration. Dissolve one gramme of the silver
supposed to contain 1⁄4 of a thousandth of mercury, and let only 1⁄4 of it be
precipitated, by adding only 1⁄4 of the common salt necessary to precipitate it
entirely. In thus operating, the 1⁄4 thousandth of mercury is concentrated in a
quantity of chloride of silver four times smaller: it is as if the silver having been
entirely precipitated, four times as much mercury, equal to two thousandths,
had been precipitated with it.
In taking two grammes of silver, and precipitating only 1⁄4 by common salt, the
precipitate would be, with respect to the chloride of silver, as if it amounted to
four thousandths. By this process, which occupies only five minutes, because
exact weighing is not necessary, 1⁄10 of a thousandth of mercury may be
detected in silver.
It is not useless to observe, that in making those experiments the most exact
manner of introducing small quantities of mercury into a solution of silver, is to
weigh a minute globule of mercury, and to dissolve it in nitric acid, diluting the
solution so that it may contain as many cubic centimetres as the globule weighs
of centigrammes. Each cubic centimetre, taken by means of a pipette, will
contain one milligramme of mercury.
If the ingot of silver to be assayed is found to contain a greater quantity of
mercury, one thousandth for example, the humid process ought either to be
given up in this case, or to be compared with cupellation.
When the silver contains mercury, the solution from which the mixed chlorides
are precipitated, does not readily become clear.
Silver containing mercury, put into a small crucible and mixed with lamp black,
to prevent the volatilization of the silver, was heated for three quarters of an
hour in a muffle, but the silver increased sensibly in weight. This process for
separating the mercury, therefore, failed. It is to be observed, that mercury is
the only metal which has thus the power of disturbing the analysis by the humid
way.
Assaying of Gold.—In estimating or expressing the fineness of gold, the whole
mass spoken of is supposed to weigh 24 carats of 12 grains each, either real, or

merely proportional, like the assayer’s weights; and the pure gold is called fine.
Thus, if gold be said to be 23 carats fine, it is to be understood, that in a mass,
weighing 24 carats, the quantity of pure gold amounts to 23 carats.
In such small work as cannot be assayed by scraping off a part and cupelling it,
the assayers endeavour to ascertain its fineness or quality by the touch. This is
a method of comparing the colour and other properties, of a minute portion of
the metal, with those of small bars, the composition of which is known. These
bars are called touch needles, and they are rubbed upon a smooth piece of
black basaltes or pottery, which, for this reason, is called the touchstone. Black
flint slate will serve the same purpose. Sets of gold needles may consist of pure
gold; of pure gold, 231⁄2 carats with 1⁄2 carat of silver; 23 carats of gold with
one carat of silver; 221⁄2 carats of gold with 11⁄2 carat of silver; and so on, till
the silver amounts to four carats; after which the additions may proceed by
whole carats. Other needles may be made in the same manner, with copper
instead of silver; and other sets may have the addition, consisting either of
equal parts of silver and copper, or of such proportions as the occasions of
business require. The examination by the touch may be advantageously
employed previous to quartation, to indicate the quantity of silver necessary to
be added.
In foreign countries, where trinkets and small work are required to be submitted
to the assay of the touch, a variety of needles is necessary; but they are not
much used in England. They afford, however, a degree of information which is
more considerable than might at first be expected. The attentive assayer
compares not only the colour of the stroke made upon the touchstone by the
metal under examination, with that produced by his needle, but will likewise
attend to the sensation of roughness, dryness, smoothness, or greasiness,
which the texture of the rubbed metal excites, when abraded by the stone.
When two strokes perfectly alike in colour are made upon the stone, he may
then wet them with aquafortis, which will affect them very differently, if they be
not similar compositions; or the stone itself may be made red-hot by the fire, or
by the blowpipe, if thin black pottery be used; in which case the phenomena of
oxidation will differ, according to the nature and quantity of the alloy. Six
principal circumstances appear to affect the operation of parting; namely, the
quantity of acid used in parting, or in the first boiling; the concentration of this
acid; the time employed in its application; the quantity of acid made use of in
the reprise, or second operation; its concentration; and the time during which it
is applied. From experiment it has been shown, that each of these unfavourable
circumstances might easily occasion a loss of from the half of a thirty-second
part of a carat, to two thirty-second parts. The assayers explain their technical
language by observing, that in the whole mass consisting of twenty-four carats,

this thirty-second part denotes 1-768th part of the mass. It may easily be
conceived, therefore, that if the whole six circumstances were to exist, and be
productive of errors, falling the same way, the loss would be very considerable.
It is therefore indispensably necessary, that one uniform process should be
followed in the assays of gold; and it is a matter of astonishment, that such an
accurate process should not have been prescribed by government for assayers,
in an operation of such great commercial importance, instead of every one
being left to follow his own judgment. The process recommended in the old
French official report is as follows:—twelve grains of the gold intended to be
assayed must be mixed with thirty grains of fine silver, and cupelled with 108
grains of lead. The cupellation must be carefully attended to, and all the
imperfect buttons rejected. When the cupellation is ended, the button must be
reduced, by lamination, into a plate of 11⁄2 inches, or rather more, in length,
and four or five lines in breadth. This must be rolled up upon a quill, and placed
in a matrass capable of holding about three ounces of liquid, when filled up to
its narrow part. Two ounces and a half of very pure aquafortis, of the strength
of 20 degrees of Baumé’s areometer, must then be poured upon it; and the
matrass being placed upon hot ashes, or sand, the acid must be kept gently
boiling for a quarter of an hour: the acid must then be cautiously decanted, and
an additional quantity of 11⁄2 ounces must be poured upon the metal, and
slightly boiled for twelve minutes. This being likewise carefully decanted, the
small spiral piece of metal must be washed with filtered river water, or distilled
water, by filling the matrass with this fluid. The vessel is then to be reversed, by
applying the extremity of its neck against the bottom of a crucible of fine earth,
the internal surface of which is very smooth. The annealing must now be made,
after having separated the portion of water which had fallen into the crucible;
and, lastly, the annealed gold must be weighed. For the certainty of this
operation, two assays must be made in the same manner, together with a third
assay upon gold of twenty-four carats, or upon gold the fineness of which is
perfectly and generally known.
No conclusion must be drawn from this assay, unless the latter gold should
prove to be of the fineness of twenty-four carats exactly, or of its known degree
of fineness; for, if there be either loss or surplus, it may be inferred, that the
other two assays, having undergone the same operation, must be subject to the
same error. The operation being made according to this process by several
assayers, in circumstances of importance, such as those which relate to large
fabrications, the fineness of the gold must not be depended upon, nor
considered as accurately known, unless all the assayers have obtained an
uniform result, without communication with each other. This identity must be
considered as referring to the accuracy of half the thirty-second part of a carat.

For, notwithstanding every possible precaution or uniformity, it very seldom
happens that an absolute agreement is obtained between the different assays of
one and the same ingot; because the ingot itself may differ in its fineness in
different parts of its mass.
The phenomena of the cupellation of gold are the same as of silver, only the
operation is less delicate, for no gold is lost by evaporation or penetration into
the bone-ash, and therefore it bears safely the highest heat of the assay
furnace. The button of gold never vegetates, and need not therefore be drawn
out to the front of the muffle, but may be left at the further end till the assay is
complete. Copper is retained more strongly by gold than it is by silver; so that
with it 16 parts of lead are requisite to sweat out 1 of copper; or, in general,
twice as much lead must be taken for the copper alloys of gold, as for those of
silver. When the copper is alloyed with very small quantities of gold, cupellation
would afford very uncertain results; we must then have recourse to liquid
analysis.
M. Vauquelin recommends to boil 60 parts of nitric acid at 22° Baumé, on the
spiral slip or cornet of gold and silver alloy, for twenty-five minutes, and replace
the liquid afterwards by acid of 32°, which must be boiled on it for eight
minutes. This process is free from uncertainty when the assay is performed
upon an alloy containing a considerable quantity of copper. But this is not the
case in assaying finer gold; for then a little silver always remains in the gold.
The surcharge which occurs here is 2 or 3 thousandths; this is too much, and it
is an intolerable error when it becomes greater, which often happens. This evil
may be completely avoided by employing the following process of M. Chaudet.
He takes 0·500 of the fine gold to be assayed; cupels it with 1·500 of silver, and
1·000 of lead; forms, with the button from the cupel, a riband or strip three
inches long, which he rolls into a cornet. He puts this into a mattrass with acid
at 22° B., which he boils for 3 or 4 minutes. He replaces this by acid of 32° B.,
and boils for ten minutes; then decants off, and boils again with acid of 32°,
which must be finally boiled for 8 or 10 minutes.
Gold thus treated is very pure. He washes the cornet, and puts it entire into a
small crucible permeable to water; heats the crucible to dull redness under the
muffle, when the gold assumes the metallic lustre, and the cornet becomes
solid. It is now taken out of the crucible and weighed.
When the alloy contains platinum, the assay presents greater difficulties. In
general, to separate the platinum from the gold with accuracy, we must avail
ourselves of a peculiar property of platinum; when alloyed with silver, it
becomes soluble in nitric acid. Therefore, by a proper quartation of the alloy by
cupellation, and boiling the button with nitric acid, we may get a residuum of

pure gold. If we were to treat the button with sulphuric acid, however, we
should dissolve nothing but the silver. The copper is easily removed by
cupellation. Hence, supposing that we have a quaternary compound of copper,
silver, platinum, and gold, we first cupel it, and weigh the button obtained; the
loss denotes the copper. This button, treated by sulphuric acid, will suffer a loss
of weight equal to the amount of silver present. The residuum, by quartation
with silver and boiling with nitric acid, will part with its platinum, and the gold
will remain pure. For more detailed explanations, see Platinum.
ATOMIC WEIGHTS or ATOMS, are the primal quantities in which the different
objects of chemistry, simple or compound, combine with each other, referred to
a common body, taken as unity. Oxygen is assumed by some philosophers, and
hydrogen by others, as the standard of comparison. Every chemical
manufacturer should be thoroughly acquainted with the combining ratios which
are, for the same two substances, not only definite, but multiple; two great
truths, upon which are founded not merely the rationale of his operations, but
also the means of modifying them to useful purposes. The discussion of the
doctrine of atomic weights, or prime equivalents, belongs to pure chemistry; but
several of its happiest applications are to be found in the processes of art, as
pursued upon the greatest scale. For many instructive examples of this
proposition, the various chemical manufactures may be consulted in this
Dictionary.
ATTAR OF ROSES. See Oils, Volatile, and Perfumery.
AURUM MUSIVUM. Mosaic gold, a preparation of tin; which see.

AUTOMATIC, a term which I have employed to designate such
economic arts as are carried on by self-acting machinery. The word
“manufacture,” in its etymological sense, means any system, or objects
of industry, executed by the hands; but in the vicissitude of language,
it has now come to signify every extensive product of art which is
made by machinery, with little or no aid of the human hand, so that the
most perfect manufacture is that which dispenses entirely with manual
labour.[4] It is in our modern cotton and flax mills that automatic
operations are displayed to most advantage; for there the elemental
powers have been made to animate millions of complex organs,
infusing into forms of wood, iron, and brass, an intelligent agency. And
as the philosophy of the fine arts, poetry, painting, and music, may be
best studied in their individual master-pieces, so may the philosophy of
manufactures in these its noblest creations.[5]
[4] Philosophy of Manufactures, p. 1.
[5] Ibid., p. 2.
The constant aim and effect of these automatic improvements in the
arts are philanthropic, as they tend to relieve the workmen either from
niceties of adjustment, which exhaust his mind and fatigue his eyes, or
from painful repetition of effort, which distort and wear out his frame.
A well arranged power-mill combines the operation of many work-
people, adult and young, in tending with assiduous skill, a system of
productive machines continuously impelled by a central force. How
vastly conducive to the commercial greatness of a nation, and the
comforts of mankind, human industry can become, when no longer
proportioned in its results to muscular effort, which is by its nature
fitful and capricious, but when made to consist in the task of guiding
the work of mechanical fingers and arms regularly impelled, with equal
precision and velocity, by some indefatigable physical agent, is
apparent to every visitor of our cotton, flax, silk, wool, and machine
factories. This great era in the useful arts is mainly due to the genius of
Arkwright. Prior to the introduction of his system, manufactures were
every where feeble and fluctuating in their development; shooting forth
luxuriantly for a season, and again withering almost to the roots like

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