Discrete Signals and Inverse Problems: An introduction for engineers and scientists

Description:

Discrete Signals and Inverse Problems examines fundamental concepts necessary to engineers and scientists working with discrete signal processing and inverse problem solving, and places emphasis on the clear understanding of algorithms within the context of application needs.

Based on the original ‘Introduction to Discrete Signals and Inverse Problems in Civil Engineering’, this expanded and enriched version:

• combines discrete signal processing and inverse problem solving in one book
• covers the most versatile tools that are needed to process engineering and scientific data
• presents step-by-step ‘implementation procedures’ for the most relevant algorithms
• provides instructive figures, solved examples and insightful exercises

Discrete Signals and Inverse Problems is essential reading for experimental researchers and practicing engineers in civil, mechanical and electrical engineering, non-destructive testing and instrumentation.  This book is also an excellent reference for advanced undergraduate students and graduate students in engineering and science.

 

Table of Contents:

Preface.

Brief Comments on Notation.

1. Introduction1.1 Signals, Systems, and Problems.

1.2 Signals and Signal Processing -- Application Examples.

1.2.1 Nondestructive Testing by Echolocation (Active).

1.2.2 Listening and Understanding Emissions (Passive).

1.2.3 Feedback and Self-calibration.

1.2.4 Digital Image Processing.

1.2.5 Signals and Noise.

1.3 Inverse Problems -- Application Examples.

1.3.1 Profilometry (Deconvolution).

1.3.2 Model Calibration (System Identification).

1.3.3 Tomographic Imaging (System Identification)

1.4 History -- Discrete Mathematical Representation.

1.5 Summary.

Solved Problems.

Additional Problems.

2. Mathematical Concepts.

2.1 Complex Numbers and Exponential Functions.

2.1.1 Complex Numbers.

2.1.2 Exponential Functions.

2.1.3 Example.

2.2 Matrix Algebra.

2.2.1 Definitions and Fundamental Operations.

2.2.2 Matrices as Transformations.

2.2.3 Eigenvalues and Eigenvectors.

2.2.4 Matrix Decomposition.

2.3 Derivatives -- Constrained Optimization

2.4 Summary.

Further Reading.

Solved Problems.

Additional Problems.

3. Signals and Systems.

3.1 Signals: Types and Characteristics.

3.1.1 Continuous and Discrete Signals.

3.1.2 One-dimensional (1D) and Multidimensional Signals.

3.1.3 Even and Odd Signals.

3.1.4 Periodic and Aperiodic Signals (and Transformations).

3.1.5 Stationary and Ergodic Signals.

3.1.6 Related Comments

3.2 Implications of Digitization -- Aliasing

3.3 Elemental Signals and Other Important Signals.

3.3.1 Impulse.

3.3.2 Step.

3.3.3 Sinusoid.

3.3.4 Exponential.

3.3.5 Wavelets.

3.3.6 Random Noise

3.4 Signal Analysis with Elemental Signals.

3.4.1 Signal Analysis with Impulses.

3.4.2 Signal Analysis with Sinusoids.

3.4.3 Summary of Decomposition Methods -- Domain of Analysis

3.5 Systems: Characteristics and Properties.

3.5.1 Causality.

3.5.2 Linearity.

3.5.3 Time Invariance.

3.5.4 Stability.

3.5.5 Invertibility.

3.5.6 Linear Time-invariant (LTI) Systems

3.6 Combination of Systems

3.7 Summary.

Further Reading.

Solved Problems.

Additional Problems.

4. Time Domain Analyses of Signals and Systems.

4.1 Signals and Noise.

4.1.1 Signal Detrending and Spike Removal.

4.1.2 Stacking: Improving SNR and Resolution.

4.1.3 Moving Kernels.

4.1.4 Nonlinear Signal Enhancement.

4.1.5 Recommendations on Data Gathering

4.2 Cross- and Autocorrelation: Identifying Similarities.

4.2.1 Examples and Observations.

4.2.2 Autocorrelation.

4.2.3 Digital Images -- 2D Signals.

4.2.4 Properties of the Cross-correlation and Autocorrelation.

4.3 The Impulse Response -- System Identification.

4.3.1 The Impulse Response of a Linear Oscillator.

4.3.2 Determination of the Impulse Response.

4.3.3 System Identification.

4.4 Convolution: Computing the Output Signal.

4.4.1 Properties of the Convolution Operator.

4.4.2 Computing the Convolution Sum.

4.4.3 Revisiting Moving Kernels and Cross-correlation.

4.5 Time Domain Operations in Matrix Form.

4.6 Summary.

Further Reading.

Solved Problems.

Additional Problems.

5. Frequency Domain Analysis of Signals (Discrete Fourier Transform).

5.1 Orthogonal Functions -- Fourier Series.

5.1.1 Fourier Series.

5.1.2 An Intuitive Preview of the Fourier Transform.

5.2 Discrete Fourier Analysis and Synthesis.

5.2.1 Synthesis: the Fourier Series Rewritten.

5.2.2 Analysis: Computing the Fourier Coefficients.

5.2.3 Selected Fourier Pair.

5.2.4 Computation -- Example.

5.3 Characteristics of the Discrete Fourier Transform.

5.3.1 Linearity.

5.3.2 Symmetry.

5.3.3 Periodicity.

5.3.4 Convergence -- Number of Unknown Fourier Coefficients.

5.3.5 One-sided and Two-sided Definitions.

5.3.6 Energy.

5.3.7 Time Shift.

5.3.8 Differentiation.

5.3.9 Duality.

5.3.10 Time and Frequency Resolution.

5.3.11 Time and Frequency Scaling.

5.4 Computation in Matrix Form.

5.5 Truncation, Leakage, and Windows.

5.6 Padding.

5.7 Plots.

5.8 The Two-dimensional Discrete Fourier Transform.

5.9 Procedure for Signal Recording.

5.10 Summary.

Further Reading and References.

Solved Problems.

Additional Problems.

6. Frequency Domain Analysis of Systems.

6.1 Sinusoids and Systems – Eigenfunctions.

6.2 Frequency Response.

6.2.1 Example: a Single Degree-of-freedom Oscillator.

6.2.2 Frequency Response and Impulse Response.

6.3 Convolution.

6.3.1 Computation.

6.3.2 Circularity.

6.3.3 Convolution in Matrix Form.

6.4 Cross-spectral and Autospectral Densities.

6.4.1 Important Relations.

6.5 Filters in the Frequency Domain -- Noise Control.

6.5.1 Filters.

6.5.2 Frequency and Time.

6.5.3 Computation.

6.5.4 Revisiting Windows in the Time Domain.

6.5.5 Filters in Two Dimensions (Frequency-wavenumber Filtering).

6.6 Determining H with Noiseless Signals (Phase Unwrapping).

6.6.1 Amplitude and Phase -- Phase Unwrapping.

6.7 Determining H with Noisy Signals (Coherence).

6.7.1 Measures of Noise -- Coherence.

6.7.2 Statistical Interpretation.

6.7.3 Number of Records -- Accuracy in the Frequency Response.

6.7.4 Experimental Determination of H in Noisy Conditions.

6.8 Summary.

Further Reading and References.

Solved Problems.

Additional Problems.

7. Time Variation and Nonlinearity.

7.1 Nonstationary Signals: Implications.

7.2 Nonstationary Signals: Instantaneous Parameters.

7.2.1 The Hilbert Transform.

7.2.2 The Analytic Signal.

7.2.3 Instantaneous Parameters.

7.3 Nonstationary Signals: Time Windows.

7.3.1 Time and Frequency Resolutions.

7.3.2 Procedure -- Example.

7.4 Nonstationary Signals: Frequency Windows.

7.4.1 Resolution.

7.4.2 Procedure -- Example.

7.5 Nonstationary Signals: Wavelet Analysis.

7.5.1 Wavelet Transform.

7.5.2 The Morlet Wavelet.

7.5.3 Resolution.

7.5.4 Procedure -- Example.

7.6 Nonlinear Systems: Detecting Nonlinearity.

7.6.1 Nonlinear Oscillator.

7.6.2 Multiples.

7.6.3 Detecting Nonlinearity.

7.7 Nonlinear Systems: Response to Different Excitations.

7.7.1 Input: Single-frequency, Constant-amplitude Sinusoid.

7.7.2 Input: Random Signal.

7.7.3 Input: Single-frequency Sinusoid -- Output: Constant Amplitude.

7.8 Time-varying Systems.

7.8.1 ARMA Model.

7.8.2 A Physically Based Example.

7.8.3 Time-varying System Analysis.

7.9 Summary.

7.9.1 Nonstationary Signals.

7.9.2 Nonlinear Time-varying Systems.

Further Reading and References

Solved Problems.

Additional Problems.

8. Concepts in Discrete Inverse Problems.

8.1 Inverse Problems -- Discrete Formulation.

8.1.1 Continuous vs Discrete.

8.1.2 Revisiting Signal Processing: Inverse Operations.

8.1.3 Regression Analysis.

8.1.4 Travel Time Tomographic Imaging.

8.1.5 Determination of Source Location.

8.2 Linearization of Nonlinear Problems.

8.3 Data-driven Solution -- Error Norms.

8.3.1 Errors.

8.3.2 Error Norms.

8.4 Model Selection -- Ockham's Razor.

8.4.1 Favor Simplicity: Ockham's Razor.

8.4.2 Reducing the Number of Unknowns.

8.4.2 Dimensionless Ratios -- Buckingham's  Theorem.

8.5 Information.

8.5.1 Available Information.

8.5.2 Information Density -- Spatial Distribution.

8.6 Data and Model Errors.

8.7 Nonconvex Error Surfaces.

8.8 Discussion on Inverse Problems.

8.9 Summary.

Further Reading and References.

Solved Problems.

Additional Problems.

9. Solution by Matrix Inversion.

9.1 Pseudoinverse.

9.2 Classification of Inverse Problems.

9.2.1 Information: Rank Deficiency and Condition Number.

9.2.2 Errors -- Consistency.

9.2.3 Problem Classification.

9.3 Least Squares Solution (LSS).

9.4 Regularized Least Squares Solution (RLSS).

9.4.1 Special Cases.

9.4.2 The Regularization Matrix.

9.4.3 The Regularization Coefficient.

9.5 Incorporating Additional Information.

9.5.1 Weighted Measurements.

9.5.2 Initial Guess of the Solution.

9.5.3 Simple Model -- Ockham's Criterion.

9.5.4 Combined Solutions.

9.6 Solution Based on Singular Value Decomposition.

9.6.1 Selecting the Optimal Number of Meaningful Singular Values p.

9.6.2 SVD and Other Inverse Solutions.

9.7 Nonlinearity.

9.8 Statistical Concepts -- Error Propagation.

9.8.1 Least Squares Solution with Standard Errors.

9.8.2 Gaussian Statistics -- Outliers.

9.8.3 Accidental Errors.

9.8.4 Systematic and Proportional Errors.

9.8.5 Error Propagation -- Regularization and SVD Solutions9.

.9 Experimental Design for Inverse Problems.

9.10 Methodology for the Solution of Inverse Problems.

9.11 Summary.

Further Reading.

Solved Problems.

Additional Problems.

10. Other Inversion Methods.

10.1 Transformed Problem Representation.

10.1.1 Parametric Representation.

10.1.2 Flexible Narrow-band Representation.

10.1.3 Solution in the Frequency Domain.

10.2 Iterative Solution of System of Equations.

10.2.1 Algebraic Reconstruction Technique (ART).

10.2.2 Simultaneous Iterative Reconstruction Technique (SIRT).

10.2.3 Multiplicative Algebraic Reconstruction Technique (MART).

10.2.4 Convergence in Iterative Methods.

10.2.5 Nonlinear Problems.

10.2.6 Incorporating Additional Information in Iterative Solutions.

10.3 Solution by Successive Forward Simulations.

10.4 Techniques from the Field of Artificial Intelligence.

10.4.1 Artificial Neural Networks (Repetitive Problems).

10.4.2 Genetic Algorithms.

10.4.3 Heuristic Methods -- Fuzzy Logic.

10.5 Summary.

Further Reading.

Solved Problems.

Additional Problems.

11. Strategy for Inverse Problem Solving.

11.1 Step 1: Analyze the Problem.

11.1.1 Identify Physical Processes and Constraints.

11.1.2 Address Measurement and Transducer-related Difficulties.

11.1.3 Keep-in-mind Inversion-related Issues.

11.2 Step 2: Pay Close Attention to Experimental Design.

11.2.1 Design the Distribution Measurements to Attain Good Coverage.

11.2.2 Design the Experiment to Obtain High-quality Data.

11.3 Step 3: Gather High-quality Data.

11.4 Step 4: Pre-process the Data.

11.4.1 Evaluate the Measured Data y.

11.4.2 Infer Outstanding Characteristics of the Solution x.

11.4.3 Hypothesize Physical Models that Can Explain the Data.

11.5 Step 5: Select an Adequate Physical Model.

11.6 Step 6: Explore Different Inversion Methods.

11.6.1 Heuristic Methods.

11.6.2 Parametric Representation -- Successive Forward Simulations.

11.6.3 Matrix-based Inversion.

11.6.4 Investigate Other Inversion Methods.

11.7 Step 7: Analyze the Final Solution.

11.8 Summary.

Solved Problems.

Additional Problems.