Discrete Signals and Inverse Problems

J. Carlos Santamarina, Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355, USA Dante Fratta, 3418H CEBA, Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

• Preface xiBrief Comments on Notation xiii1 Introduction 11.1 Signals, Systems, and Problems 11.2 Signals and Signal Processing – Application Examples 31.3 Inverse Problems – Application Examples 81.4 History – Discrete Mathematical Representation 101.5 Summary 12Solved Problems 12Additional Problems 142 Mathematical Concepts 172.1 Complex Numbers and Exponential Functions 172.2 Matrix Algebra 212.3 Derivatives – Constrained Optimization 282.4 Summary 29Further Reading 29Solved Problems 30Additional Problems 333 Signals and Systems 353.1 Signals: Types and Characteristics 353.2 Implications of Digitization – Aliasing 403.3 Elemental Signals and Other Important Signals 453.4 Signal Analysis with Elemental Signals 493.5 Systems: Characteristics and Properties 533.6 Combination of Systems 573.7 Summary 59Further Reading 59Solved Problems 60Additional Problems 634 Time Domain Analyses of Signals and Systems 654.1 Signals and Noise 654.2 Cross- and Autocorrelation: Identifying Similarities 774.3 The Impulse Response – System Identification 854.4 Convolution: Computing the Output Signal 894.5 Time Domain Operations in Matrix Form 944.6 Summary 96Further Reading 96Solved Problems 97Additional Problems 995 Frequency Domain Analysis of Signals (Discrete Fourier Transform) 1035.1 Orthogonal Functions – Fourier Series 1035.2 Discrete Fourier Analysis and Synthesis 1075.3 Characteristics of the Discrete Fourier Transform 1125.4 Computation in Matrix Form 1195.5 Truncation, Leakage, and Windows 1215.6 Padding 1235.7 Plots 1255.8 The Two-Dimensional Discrete Fourier Transform 1275.9 Procedure for Signal Recording 1285.10 Summary 130Further Reading and References 131Solved Problems 131Additional Problems 1346 Frequency Domain Analysis of Systems 1376.1 Sinusoids and Systems – Eigenfunctions 1376.2 Frequency Response 1386.3 Convolution 1426.4 Cross-Spectral and Autospectral Densities 1476.5 Filters in the Frequency Domain – Noise Control 1516.6 Determining H with Noiseless Signals (Phase Unwrapping) 1566.7 Determining H with Noisy Signals (Coherence) 1606.8 Summary 168Further Reading and References 169Solved Problems 169Additional Problems 1727 Time Variation and Nonlinearity 1757.1 Nonstationary Signals: Implications 1757.2 Nonstationary Signals: Instantaneous Parameters 1797.3 Nonstationary Signals: Time Windows 1847.4 Nonstationary Signals: Frequency Windows 1887.5 Nonstationary Signals: Wavelet Analysis 1917.6 Nonlinear Systems: Detecting Nonlinearity 1977.7 Nonlinear Systems: Response to Different Excitations 2007.8 Time-Varying Systems 2047.9 Summary 207Further Reading and References 209Solved Problems 209Additional Problems 2128 Concepts in Discrete Inverse Problems 2158.1 Inverse Problems – Discrete Formulation 2158.2 Linearization of Nonlinear Problems 2278.3 Data-Driven Solution – Error Norms 2288.4 Model Selection – Ockham’s Razor 2348.5 Information 2388.6 Data and Model Errors 2408.7 Nonconvex Error Surfaces 2418.8 Discussion on Inverse Problems 2428.9 Summary 243Further Reading and References 244Solved Problems 244Additional Problems 2469 Solution by Matrix Inversion 2499.1 Pseudoinverse 2499.2 Classification of Inverse Problems 2509.3 Least Squares Solution (LSS) 2539.4 Regularized Least Squares Solution (RLSS) 2559.5 Incorporating Additional Information 2629.6 Solution Based on Singular Value Decomposition 2659.7 Nonlinearity 2679.8 Statistical Concepts – Error Propagation 2689.9 Experimental Design for Inverse Problems 2729.10 Methodology for the Solution of Inverse Problems 2749.11 Summary 275Further Reading 276Solved Problems 277Additional Problems 28210 Other Inversion Methods 28510.1 Transformed Problem Representation 28610.2 Iterative Solution of System of Equations 29310.3 Solution by Successive Forward Simulations 29810.4 Techniques from the Field of Artificial Intelligence 30110.5 Summary 308Further Reading 308Solved Problems 309Additional Problems 31211 Strategy for Inverse Problem Solving 31511.1 Step 1: Analyze the Problem 31511.2 Step 2: Pay Close Attention to Experimental Design 32011.3 Step 3: Gather High-quality Data 32111.4 Step 4: Preprocess the Data 32111.5 Step 5: Select an Adequate Physical Model 32711.6 Step 6: Explore Different Inversion Methods 33011.7 Step 7: Analyze the Final Solution 33811.8 Summary 338Solved Problems 339Additional Problems 342Index 347