Performance Analysis and Modeling of Digital Transmission Systems

Preface.- Notation.- 1. Error Source Models.- 1.1 Description of Error Sources by Hidden Markov Models.- 1.1.1 Finite State Channel.- 1.1.2 Hidden Markov Models.- 1.1.3 Discrete-Time Finite State Systems.- 1.1.4 Error Source HMM.- 1.1.5 Gilbert’s Model.- 1.1.6 Equivalent Models.- 1.1.7 The HMM Model Generality.- 1.1.8 Satellite Channel Model.- 1.1.9 Fading Wireless Channels.- 1.1.10 Concatenated Channel Error Sources.- 1.2 Binary Symmetric Stationary Channel.- 1.2.1 Model Description.- 1.2.2 Diagonalization of the Matrices.- 1.2.3 Models with Two Sets of States.- 1.2.4 Block Matrix Representation.- 1.3 Error Source Description by Matrix Processes.- 1.3.1 Matrix Process Definition.- 1.3.2 Block Matrix Representation.- 1.3.3 Matrix Processes and Difference Equations.- 1.3.4 Matrix Processes and Markov Functions.- 1.4 Error Source Description by Semi-Markov Processes.- 1.4.1 Semi-Markov Processes.- 1.4.2 Semi-Markov Lumping of Markov Chain.- 1.5 Some Particular Error Source Models.- 1.5.1 Single-Error-State Models.- 1.5.2 Alternating Renewal Process Models.- 1.5.3 Alternating State HMM.- 1.5.4 Fading Channel Errors.- 1.6 Conclusion.- References.- 2. Matrix Probabilities.- 2.1 Matrix Probabilities and Their Properties.- 2.1.1 Basic Definitions.- 2.1.2 Properties of Matrix Probabilities.- 2.1.3 Random Variables.- 2.2 Matrix Transforms.- 2.2.1 Matrix Generating Functions.- 2.2.2 Matrix z-Transforms.- 2.2.3 Matrix Fourier Transform.- 2.2.4 Matrix Discrete Fourier Transform.- 2.2.5 Matrix Transforms and Difference Equations.- 2.3 Matrix Distributions.- 2.3.1 Matrix Multinomial Distribution.- 2.3.2 Matrix Binomial Distribution.- 2.3.3 Matrix Pascal Distribution.- 2.4 Markov Functions.- 2.4.1 Block Matrix Probabilities.- 2.4.2 Matrix Multinomial Distribution.- 2.4.3 Interval Distributions of Markov Functions.- 2.4.4 Signal Flow Graph Applications.- 2.5 Monte Carlo Method.- 2.5.1 Error Source Simulation.- 2.5.2 Performance Characteristic Calculation.- 2.6 Computing Scalar Probabilities.- 2.6.1 The Forward and Backward Algorithms.- 2.6.2 Scalar Generating Functions.- 2.7 Conclusion.- References.- 3. Model Parameter Estimation.- 3.1 The Em Algorithm.- 3.1.1 Kullback-Leibler Divergence.- 3.1.2 Minimization Algorithms.- 3.1.3 The EM Algorithm.- 3.1.4 Maximization of a Function.- 3.1.5 EM Algorithm Acceleration.- 3.1.6 Statistical EM Algorithm.- 3.2 Baum-Welch Algorithm.- 3.2.1 Phase-Type Distribution Approximation.- 3.2.2 HMM Approximation.- 3.2.3 Fitting HMM to Experimental Data.- 3.2.4 Matrix Form of the BWA.- 3.2.5 Scaling.- 3.3 Markov Renewal Process.- 3.3.1 Fast Forward and Backward Algorithms.- 3.3.2 Fitting MRP.- 3.4 Matrix-Geometric Distribution Parameter Estimation.- 3.4.1 Distribution Simplification.- 3.4.2 ML Parameter Estimation.- 3.4.3 Sequential Least Mean Square Method.- 3.4.4 Piecewise Logarithmic Transformation.- 3.4.5 Prony’s Method.- 3.4.6 Matrix Echelon Parametrization.- 3.4.7 Utilization of the Cumulative Distribution Function.- 3.4.8 Method of Moments.- 3.4.9 Transform Domain Approximation.- 3.5 Matrix Process Parameter Estimation.- 3.5.1 ML Parameter Estimation.- 3.5.2 Interval Distribution Estimation.- 3.5.3 Utilization of Two-Dimensional Distributions.- 3.5.4 Polygeometric Distributions.- 3.5.5 Error Bursts.- 3.6 Hmm Parameter Estimation.- 3.6.1 Multilayered Error Clusters.- 3.6.2 Nested Markov Chains.- 3.6.3 Single-Error-State Models.- 3.7 Monte Carlo Method of Model Building.- 3.8 Error Source Model in Several Channels.- 3.9 Conclusion.- References.- 4. Performance of Forward Error-Correction Systems.- 4.1 Basic Characteristics of One-Way Systems.- 4.2 Elements of Error-Correcting Coding.- 4.2.1 Field Galois Arithmetic.- 4.2.2 Linear Codes.- 4.2.3 Cyclic Codes.- 4.2.4 Bose-Chaudhuri-Hocquenghem Codes.- 4.2.5 Reed-Solomon Codes.- 4.2.6 Convolutional Codes.- 4.2.6 Trellis-Code Modulation.- 4.3 Maximum A Posteriori Decoding.- 4.3.1 MAP Symbol Estimation.- 4.3.2 MAP Sequence Decoding.- 4.4 Block Code Performance Characterization.- 4.4.1 The Probability of Undetected Error.- 4.4.2 Performance of Forward Error-Correcting Codes.- 4.4.3 Symmetrically Dependent Errors.- 4.4.4 Upper and Lower Bounds.- 4.4.5 Postdecoding Error Probability.- 4.4.6 Soft Decision Decoding.- 4.4.7 Correcting Errors and Erasures.- 4.4.8 Product Codes.- 4.4.9 Concatenated Codes.- 4.5 Convolutional Code Performance.- 4.5.1 Viterbi Algorithm Performance.- 4.5.2 Syndrome Decoding Performance.- 4.6 Computer Simulation.- 4.7 Zero-Redundancy Codes.- 4.7.1 Interleaving.- 4.7.2 Encryptors and Scramblers.- 4.8 Conclusion.- References.- 5. Performance Analysis of Communication Protocol.- 5.1 Basic Characteristics of Two-Way Systems.- 5.2 Return-Channel Messages.- 5.3 Synchronization.- 5.4 Arq Performance Characteristics.- 5.4.1 Stop-and-Wait ARQ.- 5.4.2 Message Delay Distribution.- 5.4.3 Insertion and Loss Interval Distribution.- 5.4.4 Accepted Messages.- 5.4.5 Alternative Assumptions.- 5.4.6 Modified Stop-and-Wait ARQ.- 5.4.7 Go-Back-N ARQ.- 5.4.8 Multiframe Messages.- 5.5 Delay-Constained Systems.- 5.5.1 Queue-Length Probability Distribution.- 5.5.2 Packet Delay Probability.- 5.6 Conclusion.- References.- 6. Continuous Time Hmm.- 6.1 Continuous and Discrete Time Hmm.- 6.1.1 Markov Arrival Processes.- 6.1.2 MAP Discrete Skeleton.- 6.1.3 Matrix Poisson Distribution.- 6.2 Fitting Continuous Time Hmm.- 6.2.1 Fitting Phase-Type Distribution.- 6.2.2 HMM Approximation.- 6.3 Conclusion.- References.- 7. Continuous State Hmm.- 7.1 Continuous and Discrete State Hmm.- 7.2 Operator Probability.- 7.3 Filtering, Prediction, and Smoothing.- 7.4 Linear Systems.- 7.4.1 Autocovariance Function.- 7.4.2 Observation Sequence PDF.- 7.4.3 Kalman Filter PDF.- 7.4.4 The Innovation Representation PDF.- 7.4.5 The Backward Algorithm.- 7.4.6 The Forward-Backward Algorithm PDF.- 7.4.7 RTS Smoother.- 7.4.8 Viterbi Algorithm.- 7.5 Autoregressive Moving Average Processes.- 7.5.1 State Space Representation.- 7.5.2 Autocovariance Function.- 7.5.3 The Forward Algorithm.- 7.5.4 RTS Smoother.- 7.6 Parameter Estimation.- 7.6.1 HMM Approximation.- 7.6.2 HGMM Approximation.- 7.7 Arma Channel Modeling.- 7.7.1 Fading Channel.- 7.7.2 Ultra-Wide Bandwidth Channel.- 7.8 Conclusion.- References.- Appendix 1.- 1.1 Matrix Processes.- 1.1.1 Reduced Canonic Representation.- 1.2 Markov Lumpable Chains.- 1.3 Semi-Markov Lumpable Chains.- References.- Appendix 2.- 2.1 Asymptotic Expansion of Matrix Probabilities.- 2.2 Chernoff Bounds.- 2.3 Block Graphs.- References.- Appendix 3.- 3.1 Statistical Inference.- 3.1.1 Distribution Parameter Estimation.- 3.1.2 Hypothesis Testing.- 3.1.3 Goodness of Fit.- 3.1.4 Confidence Limits.- 3.2 Markov Chain Model Building.- 3.2.1 Statistical Tests for Simple Markov Chains.- 3.2.2 Multiple Markov Chains.- 3.3 Semi-Markov Process Hypothesis Testing.- 3.3.1 ML Parameter Estimation.- 3.3.2 Grouped Data.- 3.3.3 Underlying Markov Chain Parameter Estimation.- 3.3.4 Autonomous Semi-Markov Processes.- 3.4 Matrix Process Parameter Estimation.- References.- Appendix 4.- 4.1 Sums With Binomial Coefficients.- 4.2 Maximum-Distance-Separable Code Weight Generating Function.- 4.3 Union Bounds on Viterbi Algorithm Performance.- References.- Appendix 5.- 5.1 Matrices.- 5.1.1 Basic Definitions.- 5.1.2 Systems of Linear Equations.- 5.1.3 Calculating Powers of a Matrix.- 5.1.4 Eigenvectors and Eigenvalues.- 5.1.5 Similar Matrices.- 5.1.6 Matrix Series Summation.- 5.1.7 Product of Several Matrices.- 5.1.8 Fast Exponentiation.- 5.1.9 Matrix Derivatives.- 5.1.10 Quadratic Polynomials.- 5.1.11 Quadratic Forms.- References.- Appendix 6.- 6.1 Markov Chains and Graphs.- 6.1.1 Transition Probabilities.- 6.1.2 Stationary Markov Chains.- 6.1.3 Partitioning States of a Markov Chains.- 6.1.4 Signal Flow Graphs.- References.- Appendix 7.- 7.1 Markov Processes.- 7.1.1 Transition Probability Densities.- 7.1.2 Transition Probability Operators.- 7.2 Gauss-Markov Processes.- 7.2.1 Gaussian Random variables and Processes.- 7.2.2 Linear Systems.- 7.2.3 Operator Pm.- 7.2.4 Stationary Distribution.- 7.2.5 Autocovariance Function.- 7.2.6 Autoregressive Processes.- 7.2.7 Parameter Estimation.- References.