Contents
List of Figures...........................................................................................................xi
List of Tables...........................................................................................................xix
Preface.....................................................................................................................xxi
About the Authors..................................................................................................xxv
Chapter 1 Fundamental Materials and Tools ........................................................1
1.1 Galois Fields ...............................................................................1
1.1.1 Primitive Elements......................................................... 5
1.2 Vector Space ...............................................................................7
1.2.1 Linear Operations in Vector Space over a Field ............9
1.3 Matrix Theory........................................................................... 11
1.3.1 Basic Definitions..........................................................12
1.3.2 Basic Operations and Properties ..................................1 3
1.3.3 Determinant.................................................................. 15
1.3.4 Eigenvalues and Eigenvectors...................................... 16
1.4 Hamming Distance and Weight ................................................ 18
1.5 Correlation Functions ...............................................................19
1.5.1 1-D Auto- and Cross-Correlation Functions................ 19
1.5.2 2-D Auto- and Cross-Correlation Functions................ 20
1.6 Cardinality Upper Bound.......................................................... 21
1.7 Markov Chain ........................................................................... 23
1.8 Algebraic Tools for Performance Analysis............................... 2 4
1.8.1 Gaussian Approximation for Unipolar Codes..............25
1.8.2 Gaussian Approximation for Bipolar Codes................ 29
1.8.3 Combinatorial Analysis for Unipolar Codes................31
1.8.4 Hard-Limiting Analysis for Unipolar Codes ............... 32
1.8.5 Soft-Limiting Analysis without Chip Synchron-
ization .......................................................................... 35
1.8.6 Hard-Limiting Analysis without Chip Synchron-
ization........................................................................... 42
1.8.7 Spectral Efficiency ....................................................... 47
1.9 Summary................................................................................... 47
vii
viii Contents
Chapter 2 Optical Coding Schemes .................................................................... 51
2.1 1-D Temporal Amplitude Coding............................................. 55
2.2 1-D Temporal Phase Coding.....................................................60
2.3 1-D Spectral Phase Coding.......................................................61
2.4 1-D Spectral Amplitude Coding ............................................... 63
2.5 2-D Spatial-Temporal Amplitude Coding ................................6 5
2.6 2-D Spectral-Temporal Amplitude Coding ..............................68
2.7 Three-Dimensional Coding ......................................................69
2.8 Multirate and Multiple-QoS Coding.........................................69
2.9 Multicode Keying and Shifted-Code Keying............................ 71
2.10 Enabling Hardware Technologies............................................. 75
2.10.1 Wavelength-Aware Hard-Limiting Detector ...............75
2.10.2 Fiber Bragg Gratings....................................................76
2.10.3 Arrayed Waveguide Gratings.......................................77
2.11 Potential Applications...............................................................81
2.12 Summary...................................................................................82
Chapter 3 1-D Asynchronous Prime Codes ........................................................89
3.1 Original Prime Codes ............................................................... 89
3.1.1 Performance Analysis .................................................. 92
3.2 Extended Prime Codes..............................................................96
3.2.1 Performance Analysis .................................................. 97
3.3 Generalized Prime Codes .......................................................100
3.3.1 Performance Analysis ................................................ 103
3.4 2
n
Prime Codes.......................................................................106
3.4.1 Performance Analysis ................................................ 116
3.5 Optical Orthogonal Codes ......................................................117
3.5.1 Constructions of (N,w,1, 1) OOC .............................1 17
3.5.2 Constructions of (N,w,1, 2) OOC .............................1 19
3.5.3 Constructions of (N,w,2, 1) OOC .............................1 20
3.5.4 Performance Analysis ................................................ 122
3.6 Summary.................................................................................124
Chapter 4 1-D Synchronous Prime Codes ........................................................127
4.1 Synchronous Prime Codes......................................................128
4.1.1 Performance Analysis ................................................ 130
4.2 Synchronous Multilevel Prime Codes ....................................134
4.2.1 Performance Analysis ................................................ 138
4.3 Synchronous Coding Applications ......................................... 143
4.4 Summary.................................................................................146
Contents ix
Chapter 5 2-D Asynchronous Prime Codes ......................................................149
5.1 Carrier-Hopping Prime Codes ................................................150
5.1.1 Performance Analysis ................................................ 153
5.2 Multilevel Carrier-Hopping Prime Codes...............................156
5.2.1 Performance Analysis ................................................ 160
5.3 Shifted Carrier-Hopping Prime Codes.................................... 164
5.3.1 Construction 1: Time Shifts.......................................166
5.3.2 Construction 2: Wavelength Shifts ............................166
5.3.3 Performance Analysis ................................................ 168
5.3.4 Spectral Efficiency Study...........................................171
5.4 Extended Carrier-Hopping Prime Codes ................................176
5.4.1 Performance Analysis ................................................ 182
5.5 Expanded Carrier-Hopping Prime Codes ............................... 184
5.5.1 Performance Analysis ................................................ 187
5.6 Quadratic-Congruence Carrier-Hopping Prime Codes ........... 191
5.6.1 Performance Analysis ................................................ 193
5.6.2 Multicode and Shifted-Code Keying .........................195
5.6.3 Spectral Efficiency Study...........................................200
5.7 Prime-Permuted Codes with Unipolar Codes.........................202
5.7.1 Performance Analysis ................................................ 209
5.8 Prime-Permuted Codes with Bipolar Codes...........................218
5.8.1 Performance Analysis ................................................ 226
5.9 Quadratic-Congruence-Permuted Codes ................................234
5.9.1 Performance Analysis ................................................ 237
5.10 2-D Optical Orthogonal Codes ............................................... 239
5.10.1 Construction 1: From 1-D (N,w,1,1) OOC ............. . 240
5.10.2 Construction 2: From Reed–Solomon Code..............242
5.10.3 Performance Analysis ................................................244
5.11 Summary.................................................................................244
Chapter 6 2-D Synchronous Prime Codes ........................................................249
6.1 Synchronous Original, Expanded, and Quadratic-Congruence
Carrier-Hopping Prime Codes ................................................249
6.1.1 Performance Analysis ................................................ 253
6.2 Synchronous Multilevel Carrier-Hopping Prime Codes......... 256
6.2.1 Performance Analysis ................................................ 259
6.3 Synchronous Prime-Permuted Codes .....................................264
6.3.1 Performance Analysis ................................................ 267
6.4 Summary................................................................................. 270
x Contents
Chapter 7 Multilength Prime Codes .................................................................273
7.1 Multilength Carrier-Hopping Prime Codes ............................274
7.1.1 Performance Analysis ................................................ 280
7.2 Multilength Expanded Carrier-Hopping Prime Codes ...........285
7.2.1 Performance Analysis ................................................ 288
7.3 Multilength Quadratic-Congruence Carrier-Hopping Prime
Codes ......................................................................................291
7.3.1 Performance Analysis ................................................ 294
7.3.2 Multicode Keying ......................................................2 9 7
7.3.3 Spectral Efficiency Study...........................................301
7.4 2-D Multilength Prime-Permuted Codes................................303
7.4.1 Performance Analysis ................................................ 306
7.5 Variable-Weight Coding with Same Bit Power ......................309
7.6 Multilength 1-D Optical Orthogonal Codes ...........................3 19
7.6.1 Construction 1: Cross-Correlation of One .................319
7.6.2 Construction 2: Cross-Correlation of Two ................ 321
7.6.3 Performance Analysis ................................................ 324
7.7 Summary.................................................................................328
Chapter 8 3-D Prime Codes ..............................................................................333
8.1 Concatenated Prime Codes .....................................................333
8.1.1 Performance Analysis ................................................ 336
8.2 Multicarrier Prime Codes ....................................................... 338
8.2.1 Performance Analysis ................................................ 341
8.3 Summary.................................................................................344
Index......................................................................................................................345
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