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Multilevel Sequences and Line Codes

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Multilevel Sequences and Line Codes COPYRIGHT AND CITATION CONSIDERATIONS FOR THIS THESIS/ DISSERTATION o Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. o NonCommercial — You may not use the material for commercial purposes. o ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. How to cite this thesis Surname, Initial(s). (2012) Title of the thesis or dissertation. PhD. (Chemistry)/ M.Sc. (Physics)/ M.A. (Philosophy)/M.Com. (Finance) etc. [Unpublished]: University of Johannesburg. Retrieved from: https://ujdigispace.uj.ac.za (Accessed: Date). MULTILEVEL SEQUENCES AND LINE CODES by LOUIS BOTHA Thesis submitted as partial fulfilment of the requirements for the degree MASTER OF ENGINEERING in ELECTRICAL AND ELECTRONIC ENGINEERING in the FACULTY OF ENGINEERING at the RAND AFRIKAANS UNIVERSITY SUPERVISOR: PROF HC FERREIRA MAY 1991 SUMMARY As the demand for high-speed data communications over conventional channels such as coaxial cables and twisted pairs grows, it becomes neccesary to optimize every aspect of the communication system at reasonable cost to meet this demand effectively. The choice of a line code is one of the most important aspects in the design of a communications system, as the line code determines the complexity, and thus also the cost, of several circuits in the system. It has become known in recent years that a multilevel line code is preferable to a binary code in cases where high-speed communications are desired. Apart from ternary codes, not many multilevel codes are available. Some of the existing line codes also suffer from serious drawbacks regarding a lack of complying to input restrictions, small values of efficiency, and great code complexity. In this study, Markov models and values of channel capacity are presented for several classes of restricted multilevel sequences which are thought to be of practical importance in view of the channel input restrictions that these codes satisfy. Different coding methods are used to construct low-complexity encoders and decoders for generating and decoding these sequences with high values of efficiency, good error behaviour and favourable power spectral densities. OPSOMMING Om effektief in die groeiende aanvraag na datakommunikasie teen hoe spoed met die gebruikmaking van konvensionele kanale soas koaksiale kabels en gedraaide lynpare te voldoen, het dit nodig geword om elke aspek van die kommunikasiestelsel teen redelike koste te optimiseer. Die keuse van 'n lynkode is een van die belangrikste aspekte in die ontwerp van 'n kommunikasiestelsel. Die lynkode bepaal die kompleksiteit, en dus oak die koste, van verskeie stroambane in die stelsel. Dit het gaandeweg duidelik geword dat 'n multivlak lynkode verkieslik is bo 'n binere kode in gevalle waar kommunikasie teen 'n hoe tempo verlang word. Mgesien van ternere kodes is daar huidiglik nie baie multivlak lynkodes beskikbaar nie. Sommige van die beskikbare kodes het oak emstige nadele wat betref swak voldoening aan insetbeperkings, lae waardes van effektiwiteit, en hoe vlakke van kompleksiteit. In hierdie studie word Markovmodelle en waardes van kanaalkapasiteit aangebied vir verskeie klasse van insetbeperkte multivlak sekwensies wat as tegnologies belangrik geag word in die lig van die insetbeperkings wat hierdie sekwensies bevredig. Verskillende koderingstegnieke word gebruik om eenvoudige enkodeerders en dekodeerders te konstrueer wat hierdie sekwensies opwek en dekodeer met hoe waardes van effektiwiteit, goeie foutgedrageienskappe en gunstige drywingspektraaldigthede. II Ii - I II Ii !I i II I I I i I II I CONTENTS I !1 COntents CONTENTS PAGE LIST OF ILLUSTRATIONS LIST OF TABLES iv LIST OF SYMBOLS Vll 1. INTRODUCTION 1 2. BASEBAND COMMUNICATIONS 4 2.1. The ISDN subscriber loop 5 2.2. Model of a line coding system 5 3. TRANSMISSION ASPECTS OF BASEBAND COMMUNICATIONS 8 3.1. Noise 9 3.2. Intersymbol interference 10 3.3. Equalizer 11 3.4. The hybrid 13 3.5. Echo canceller 14 3.6. Timing recovery 15 3.7. Baseline wander 16 3.8. Scramblers 17 4. MARKOV MODELS AND CHANNEL CAPACITY OF RESTRICTED MULTILEVEL SEQUENCES 20 4.1. Input restrictions 21 4.2. Information and channel capacity 22 4.3. Enumerating multilevel sequences 23 4.4. Recursions and the characteristic equation 26 4.4.1. Direct determination of the recursion formula 26 4.4.2. Capacity of CC q-nary sequences with q even 29 4.5. Markov models and values of channel capacity for restricted multilevel sequences 33 4.5.1. Charge-constrained sequences 34 4.5.2. Runlength-limited sequences 41 COntents APPENDIXES A. PROGRAMS FOR ENUMERATING QUATERNARY CC SEQUENCES 123 B. PROGRAMS TO DETERMINE THE CONNECTION MATRIX 127 B.1. Charge-constrained sequences 128 B.2. CCMXRLL sequences 129 C. PROGRAMS TO SEARCH FOR THE APPROXIMATE EIGENVECTOR 134 D. PROGRAMS TO DETERMINE ERROR BEHAVIOUR 137 D.l. Programs to determine error propagation properties 138 D.2. Programs to determine the EMF 141 E. List of programs on the floppy disk 144 Contents 4.5.3. Charge-constrained, maximum runlength-limited sequences 44 5. PREVIOUS MULTILEVEL LINE CODES 47 5.1. Multilevel feedback balanced codes 48 5.2. Multilevel balanced codes 49 5.3. The 4L7-4 code 51 5.4. Eleftheriou and Cideciyan's quaternary codes 51 5.5. The 3B2Q code 51 5.6. The 16B9Q line code 52 5.7. The 2B1Q line code 52 5.8. Tunev's 8B4QI code 53 5.9. The BK-45 line code 53 5.10. The L742 code 54 6. SYNTHESIS OF NEW MULTILEVEL LINE CODES 55 6.1. Sequence-state coding methods 57 6.1.1. The 2B2T and 2B2TA line codes 57 6.1.2. The 3B2Q line code 65 6.1.3. The QRLL line code 67 6.1.4. The 1B1Q line code 70 6.1.5. The 3B2QI line code 74 6.1.6. The 2B1QI line code .. 76 6.1.7. The 4B2H line code 77 6.2. Sliding block code algorithm 80 6.3. Block coding techniques 85 7. ERROR BEHAVIOUR OF THE NEW LINE CODES 90 7.1. Channel model 91 7.2. Results on error propagation properties 91 8. POWER SPECTRAL DENSITIES OF THE NEW LINE CODES 97 8.1. PSD of the PST code 98 8.2. Power spectral densities for the new line codes 107 9. DISCUSSION AND OVERVIEW 114 REFERENCES 118 I II il I I . - LIST OF ILLUSTRATIONS i List of illustrations FIGURE No TITLE PAGE 2.1 Model of a coding system 6 3.1 ISDN V-Interface transceiver structure 9 3.2 NEXT and FEXT mechanisms 10 3.3 Equalizer structure 12 3.4 Hybrid diagram 13 3.5 Echo canceller with n taps 14 3.6 Deductive timing recovery 15 3.7 Inductive timing recovery 16 3.8 AC Coupling networks 16 3.9 A self-synchronizing scrambler 18 3.10 A reset scrambler 19 4.1 Markov model for the quaternary sequence with C =2 24 4.2 Graph of approximate values of H against l for quaternary sequences with C =2 26 4.3 Trellis diagram corresponding to the Markov model of Fig 4.1 27 4.4 Markov model for the 5-level sequence with C =1 and L~ax =1 31 4.5 Trellis diagram for 5-level sequences with C =1 and L~ax =1. 32 4.6 General Markov model for CC quaternary sequences 36 4.7 General Markov model for CC 5-level sequences 37 4.8 Markov model for (q.C) =(6,3) Sequences 39 4.9 General Markov model for CC seven-level sequences 40 4.10 General Markov model for RLL quinternary sequences 42 4.11 General Markov model for a CCMXRLL quinternary sequence 45 5.1 A BK-45 coding system 53 6.1 Markov model for ternary sequences with max C =1 and L =2 58 2' 6.2 The graph G 61 List of illustrations 6.3 Encoder for the 2B2T line code with q = 3, R = 1, T] = 83%, C = 1, and Lmax = 2 62 6.4 Encoder for the 2B2TA line code 63 6.5 Encoder for the 3B2Q code with q = 4, 3 max R = 2' fJ = 95.7%, C = 3, and L = 4 66 6.6 Encoder for the QRLL code with q = 4, 3 max 1 R = 2' T] = 94.6%, and L = 69 6.7 Markov model for the CCMXRLL quaternary sequence with Lmax = 1 and C = 2 71 6.8 Encoder for the 1BIQ code with q = 4, 1 max R = I' T] = 95%, C = 2, and L = 1 73 6.9 Encoder for the 3B2Q code with q = 5, 3 max 2 R = 2' T] = 97%, C = 1, and L = 75 6.10 Markov model for a CC ternary sequence with the symbol '0' prohibited when RDS :t 0 80 6.11 Markov model for a RLL ternary sequence with Lmax = 1 86 7.1 Channel transition diagram 92 7.2 Error propagation of the new ternary line codes 93 7.3 Error propagation properties for the new quaternary line codes 93 7.4 Error propagation properties of the new quinternary and six-level line codes 94 7.5 Error propagation properties for some HDBn codes 95 7.6 Error propagation properties for some BnZS codes 95 8.1 PSD for sequences corresponding to the PST code if p = 0.5 107 8.2 Continuous PSD for sequences of the 2B2T code 108 8.3 Continuous PSD for sequences of the 2B2TA code ] 08 8.4 Continuous PSD for sequences of the 4B5T code 109 8.5 Continuous PSD for sequences of the 3B2Q code 110 8.6 Continuous PSD for sequences of the QRLL code 110 8.7 Continuous PSD for sequences of the 3B2QI code 111 II List of illustrations 8.8 Continuous PSD for sequences of the 2B1QI code 111 8.9 Continuous PSD for sequences of the 4B2H code 112 III LIST OF TABLES ! I, I,il II II I! iiI ==============,.cc~,.:J lV List of tables TABLE No TITLE PAGE 4.1 Approximate values of H for quaternary charge limited sequences 25 4.2 Values of H for CC quaternary sequences 35 4.3 Values of channel capacity (bits/symbol) for charge-constrained quinternary sequences 38 4.4 Values of H for CC six-level sequences 38 4.5 Values of channel capacity (bits/symbol) for charge-constrained seven-level sequences 41 4.6 Values of channel
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