 Essentials of Error-Control Coding
ISBN: 978-0-470-02920-6
Hardcover
388 pages
September 2006
US $90.00
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Symbols & Abbreviations.
1 INFORMATION AND CODING THEORY.
1.2 Entropy and Information Rate.
1.3 Extended Discrete Memoryless Source.
1.4 Channels and Mutual Information.
1.5 Channel Probability Relationships.
1.6 The a priori and a posteriori Entropies.
1.7 Mutual Information.
1.8 Capacity of a Discrete Channel.
1.9 Shannon’s Theorems.
1.10 Signal Spaces and the Channel Coding Theorem.
1.11 Error Control Coding.
1.12 Limits to Communication and their Consequences.
Bibliography and References.
Problems.
2. BLOCK CODES.
2.1 Error Control Coding.
2.2 Error Detection and Correction.
2.3 Block Codes: Introduction and Parameters.
2.4 The Vector Space over the Binary Field.
2.5 Linear Block Codes.
2.6 Syndrome Error Detection.
2.7 Minimum Distance of a Block Code.
2.8 Error Correction Capability of a Block Code.
2.9 Syndrome Detection and the Standard Array.
2.10 Hamming Codes.
2.11 Forward Error Correction (FEC) and Automatic Repeat ReQuest (ARQ).
Bibliography and References.
Problems.
3 CYCLIC CODES.
3.1 Description.
3.2 Polynomial Representation of Codewords.
3.3 Generator Polynomial of a Cyclic Code.
3.4 Cyclic Codes in Systematic Form.
3.5 Generator Matrix of a Cyclic Code.
3.6 Syndrome Calculation and Error Detection.
3.7 Decoding of Cyclic Codes.
3.8 An Application Example: CRC Code for the Ethernet Standard.
Bibliography and References.
Problems.
4 BCH CODES.
4.1 Introduction: The Minimal Polynomial.
4.2 Description of BCH Cyclic Codes.
4.3 Decoding of BCH Codes.
4.4 Error Location and Error Evaluation Polynomials.
4.5 The Key Equation.
4.6 Decoding of BCH Codes using the Euclidean Algorithm.
Bibliography and References.
Problems.
5 REED-SOLOMON CODES.
5.1 Introduction.
5.2 Error Correction Capability of RS Codes: The Vandermonde Determinant.
5.3 RS Codes in Systematic Form.
5.4 Syndrome Decoding of RS Codes.
5.5 The Euclidean Algorithm. Error Location and Evaluation Polynomials.
5.6 Decoding of RS Codes using the Euclidean Algorithm.
5.7 Decoding of RS and BCH Codes using the Berlekamp- Massey Algorithm.
5.8 A Practical Application: Error-Control Coding for the Compact Disc (CD).
5.9 Encoding for RS codes C(RS)(28,24), C(RS)(32,28) and C(RS)(255,251).
5.10 Decoding of RS Codes C(RS)(28,24) and C(RS)(32,28). .
5.11 Importance of Interleaving Bibliography and References Problems.
6 CONVOLUTIONAL CODES.
6.1 Linear Sequential Circuits.
6.2 Convolutional Codes and Encoders.
6.3 Description in the D-Transform Domain.
6.4 Convolutional Encoder Representations.
6.5 Convolutional Codes in Systematic Form.
6.6 General Structure of FIR and IIR Finite State Sequential Machines.
6.7 State Transfer Function Matrix: Calculation of the Transfer Function.
6.8 Relationship between the Systematic and Non-Systematic Forms.
6.9 Distance Properties of Convolutional Codes.
6.10 Minimum Free Distance of a Convolutional Code.
6.11 Maximum Likelihood Detection (MLD).
6.12 Decoding of Convolutional Codes: The Viterbi Algorithm.
6.13 Extended and Modified State Diagram.
6.14 Error Probability Analysis for Convolutional Codes.
6.15 Hard and soft Decisions.
6.16 Punctured Convolutional Codes and Rate Compatible Schemes.
Bibliography and References.
Problems.
7 TURBO CODES.
7.1 A Turbo Encoder.
7.2 Decoding of Turbo Codes.
7.3 Markov Sources and Discrete Channels.
7.4 The BCJR Algorithm: Trellis Coding and Discrete Memoryless Channels.
7.5 Iterative Coefficient Calculation.
7.6 MAP BCJR Algorithm and the Log Likelihood Ratio.
7.7 Turbo Decoding.
7.8 Construction Methods for Turbo Codes.
7.9 Other Decoding Algorithms for Turbo Codes.
7.10 EXIT Charts for Turbo Codes.
Bibliography and References.
Problems.
8 LOW-DENSITY PARITY-CHECK (LDPC) CODES.
8.1 Different Systematic Forms of a Block Code.
8.2 Description of LDPC Codes.
8.3 Construction of LDPC Codes.
8.4 The Sum-Product Algorithm.
8.5 Sum-Product Algorithm for LDPC Codes: An Example.
8.6 Simplifications of the Sum-Product Algorithm.
8.7 A Logarithmic LDPC Decoder.
8.8 EXIT Charts for LDPC Codes.
8.9 Fountain and LT Codes.
8.10 LDPC and Turbo Codes.
Bibliography and References.
Problems.
APPENDIX A: Error Probability in the Transmission of Digital Signals.
A.1 Digital Signalling.
A.2 Bit Error Rate (BER).
Bibliography.
APPENDIX B: Galois Fields GF(q).
B.1 Groups.
B.2 Addition and Multiplication modulo-2.
B.3 Fields.
B.4 Polynomials over Binary Fields.
B.5 Construction of a Galois Field GF(2m).
B.6 Properties of Extended Galois Fields GF(2m).
B.7 Minimal Polynomials.
Bibliography.
Answers to Problems.
Index.
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