New Digital Signal Processing Methods : Applications to Measurement and Diagnostics

This book is intended as a manual on modern advanced statistical methods for signal processing. The objectives of signal processing are the analysis, synthesis, and modification of signals measured from different natural phenomena, including engineering applications as well. Often the measured signals are affected by noise, distortion and incompleteness, and this makes it difficult to extract significant signal information. The main topic of the book is the extraction of significant information from measured data, with the aim of reducing the data size while keeping the basic information/knowledge about the peculiarities and properties of the analyzed system; to this aim, advanced and recently developed methods in signal analysis and treatment are introduced and described in depth. More in details, the book covers the following new advanced topics (and the corresponding algorithms), including detailed descriptions and discussions: the Eigen-Coordinates (ECs) method, The statistics of the fractional moments, The quantitative "universal" label (QUL) and the universal distribution function for the relative fluctuations (UDFRF), the generalized Prony spectrum, the Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals (NAFASS), the discrete geometrical invariants (DGI) serving as the common platform for quantitative comparison of different random functions. Although advanced topics are discussed in signal analysis, each subject is introduced gradually, with the use of only the necessary mathematics, and avoiding unnecessary abstractions. Each chapter presents testing and verification examples on real data for each proposed method. In comparison with other books, here it is adopted a more practical approach with numerous real case studies. Preface......Page 6 Symbols......Page 10 Contents......Page 14 Abbreviations......Page 19 1.1 Classical Approach: Formulation of One-Dimensional Regression Problems......Page 21 1.1.2 Regression Model......Page 22 1.1.4 Assumption About the Regression Function and Its Possible Recognition......Page 23 1.1.5 Analysis of Remnants......Page 25 1.2 Procedure of Optimal Linear Smoothing of Noisy Data......Page 26 1.2.1 Possible Generalizations......Page 29 1.3 Description of the Eigen-Coordinates Method......Page 31 1.3.1 Determining the Basic Linear Relationship for Functions with Nonlinear Fitting Parameters......Page 32 1.3.2 Using the Orthogonal Variables......Page 35 1.3.3 Selection of the Most Suitable Hypothesis......Page 37 1.4 Generalizations and Recommendations for the Eigen-Coordinates Method......Page 45 1.4.1 Using a Priori Information......Page 47 1.4.2 The Problem of Elimination of Depending Constants......Page 52 1.5 Concluding Remarks......Page 61 1.7 Exercises......Page 65 References......Page 67 2.1 Introduction and Problem Formulation......Page 69 2.2 The Reduced Fractal Model and its Realisation in the Fractal-Branched Structures......Page 73 2.3 The Fitting Procedure of Functions Containing Log-Periodic Oscillations......Page 78 2.4 Application of the Original Procedure to Real Data......Page 81 2.4.1 Description of the Bronchial Asthma Disease......Page 82 2.4.2 Description of Fragments of the Queen Bee Acoustic Signals......Page 83 2.4.3 Description of Acoustic Data Recorded from Car Valves in the Idling Regime......Page 90 2.5 Concluding Remarks on Fundamental Results and Open Questions......Page 92 2.A Appendix 1: Evaluation of the Product (2.4) for the Case xib 1......Page 95 2.B Appendix 2: Synthesis of the Eigen-Coordinates Method and the Basic Linear Relationship......Page 97 2.C Appendix 3: Computation of the Power-Law Exponent nu in L0(t)......Page 101 2.D Appendix 4: Basic Linear Relationship for Inoculating Log-Periodic Functions......Page 102 2.F Questions for Self-Testing......Page 103 2.G Exercises......Page 104 References......Page 105 3.1 Introduction and Formulation of the Problem......Page 107 3.2 Evaluation of Statistical Stability of Random Sequences by Higher Moments......Page 109 3.3 The Approximate Expression for the Generalised Mean Value Function, Fractional and Complex Moments, and External Correlati.........Page 113 3.4 The Generalised Pearson Correlation Function, External and Internal Correlations, and Some Useful Inequalities......Page 119 3.5.1 Statistical Protection of Valuable Documents......Page 126 3.5.2 Detection of a Small Signal and Statistical Proximity......Page 131 3.5.4 Dielectric Data and Calibration Curve......Page 135 3.5.5 Fractional Exponential Reduction Moments Approach......Page 138 3.5.6 Integration and Differentiation Pre-processing......Page 142 3.5.7 Some Simulation Results......Page 143 3.6 Concluding Remarks and Open Problems......Page 156 References......Page 157 4.1 Introduction and Formulation of the Problem......Page 160 4.2 The Universal Distribution of Stable Points......Page 163 4.3 Representation of Data from Complex Systems Without an Explicit Model by Means of the Generalised Gaussian Distribution......Page 167 4.3.1 Detection of Few `Strange ́ Points Located in a Narrow Interval......Page 168 4.3.2 The Presence of Random Points That Disturb the Whole Interval l......Page 172 4.4.1 Similar Random Sequences Without Trend (Triple-Correlations of the Transcendental Numbers π and e)......Page 178 4.4.2 Random Sequences with a Trend: The `Forex ́ Currency Market Data......Page 184 4.4.3 Quantitative Classification of Earthquakes......Page 188 4.4.4 Coding Information with the Help of Quantum Dots......Page 196 4.5 Basic Results and Open Problems......Page 199 4.6.1 Introduction and Formulation of the Problem......Page 201 4.6.2 Scaling Properties of the Beta-Distribution and Description of the Treatment Procedure......Page 204 4.6.3 Treatment of Long-Times Membrane Current Series......Page 214 4.6.4 Clusterization of Final Parameters Based on the Generalised Pearson Correlation Function......Page 217 4.6.5 Results and Discussion......Page 222 4.8 Exercises......Page 223 References......Page 224 5.1 Introduction and Formulation of the Problem......Page 226 5.2 Description of a Quasi-Periodic Process in Terms of the Prony ́s Spectrum......Page 228 5.3 Description of the General Detection Algorithm......Page 231 5.4 Detection of Quasi-Periodic Processes from Real Data......Page 235 5.4.1 Detection from Raman Spectra Recorded for Pure Water......Page 236 5.4.2 Detection from Random Geophysical Acoustic Signals......Page 242 5.5 Results and Discussion......Page 245 5.A Appendix: Generalization of the Model for Quasi-Periodic Processes to Consider Incommensurable Periods......Page 249 References......Page 251 6.1 Introduction and Formulation of the Problem: The Reproducible Experiments and their Description......Page 253 6.2 The Basis of the General Theory of Reproducible Experiments. The Physical Meaning of the Prony Decomposition......Page 254 6.3 Description of the General Algorithm and Its Testing on Available Data......Page 262 6.3.1 The Raman Spectra of Distilled Water......Page 264 6.3.2 Two Wireless Sensor Nodes Exchanging Packets in a Noisy Wireless Channel......Page 270 6.4 Generalisation of Results for Quasi-Reproducible (Non-stationary) Measurements......Page 275 6.4.1 Self-Consistent Solutions of the Functional Equation (6.30)......Page 278 6.4.2 The Clusterization Procedure and Reduction to an ``Ideal Experiment ́ ́......Page 282 6.5 Validation of the General Theory on Experimental Data......Page 287 6.6 Final Results and Further Perspectives......Page 298 6.7 Questions for Self-Testing......Page 303 References......Page 304 7.1 Describing Complex Signals with Beatings......Page 306 7.2 Basics of the Non-orthogonal Amplitude Frequency Analysis of Smoothed Signals (NAFASS) Approach: Evaluation of the Initial.........Page 310 7.3.1 The Fitting Function for the Integer Case......Page 313 7.3.2 The Fitting Function for the Fractional Case......Page 315 7.4.1 Application to Economic Data......Page 318 7.4.2 Application to the Noise Created by Transcendental Numbers......Page 323 7.5 New Type of Fluctuation Spectroscopy Based on the NAFASS Approach......Page 332 7.6 The NAFASS Approach and Chaos......Page 340 7.7 Concluding Remarks and the Basic Principles of Fluctuation Metrology......Page 351 7.A Appendix......Page 354 7.B Questions for Self-Testing......Page 356 References......Page 357 Chapter 8: Applications of NIMRAD in Electrochemistry......Page 359 8.1.1 Formulation of the Problem......Page 360 8.1.2.1 Preliminary Considerations: The Second-Order DGIs......Page 361 8.1.2.2 The General Theory of the Discrete Geometrical Invariants Based on the Higher-Order Curves and the Fourth-Order GDI......Page 363 8.1.2.3 Application of the Statistics of the Fractional Moments (SFM) and Use of the Internal Correlation Factor......Page 366 8.1.3.2 Algorithm Description......Page 368 8.1.4 Results and Discussion......Page 373 8.2.1 Formulation of the Problem......Page 375 8.2.2 Experimental Set-Up and Preliminary Data Analysis......Page 379 8.2.3 The Mathematical Section of the PCA and the Modified F-Transform......Page 386 8.2.4 Application of the Modified Platform to Real Data......Page 391 8.2.5 Results and Discussion......Page 396 8.3.1 Formulation of the Problem......Page 398 8.3.2 Foundations of the General Theory of Percolation Currents......Page 400 8.3.3.2 Experimental Measurements......Page 406 8.3.4.2 Representation of Initial Data in the Uniform Logarithmic Scale......Page 407 8.3.4.3 Clusterization of all Measurements to the Averaged Triad Cluster......Page 409 8.3.4.4 The Final Fit......Page 413 8.A Appendix......Page 417 8.B Questions for Self-Testing......Page 420 References......Page 421 9.1 Compact Universal Parameters to Reduce Initial Data......Page 425 9.2 The ``Struggle ́ ́ Principle and Justification of the Chosen Set......Page 427 9.3 Verification by Model Data......Page 431 9.4 Finding Quantitative Differences when the Input Is in a Descriptive Form......Page 432 References......Page 444 Epilogue......Page 446 Index......Page 448