Signal Processing: A Mathematical Approach

Content: Preface    Introduction    Chapter Summary    Aims and Topics    Some Examples of Remote Sensing    A Role for Mathematics    Limited Data    The Emphasis in this Book    Topics Covered    Applications of Interest    Sensing Modalities    Active and Passive Sensing    A Variety of Modalities    Using Prior Knowledge    An Urn Model of Remote Sensing    An Urn Model    Some Mathematical Notation    An Application to SPECT Imaging    Hidden Markov Models    Fourier Series and Fourier Transforms    Chapter Summary    Fourier Series    Complex Exponential Functions    Fourier Transforms    Basic Properties of the Fourier Transform    Some Fourier-Transform Pairs    Dirac Deltas    Convolution Filters    A Discontinuous Function    Shannon\'s Sampling Theorem    What Shannon Does Not Say    Inverse Problems    Two-Dimensional Fourier Transforms    The Basic Formulas    Radial Functions    An Example    The Uncertainty Principle    Best Approximation    The Orthogonality Principle    An Example    The DFT as Best Approximation    The Modified DFT (MDFT)    The PDFT    Analysis of the MDFT    Eigenvector Analysis of the MDFT    The Eigenfunctions of Sr    Remote Sensing    Chapter Summary    Fourier Series and Fourier Coefficients    The Unknown Strength Problem    Measurement in the Far Field    Limited Data    Can We Get More Data?    Measuring the Fourier Transform    Over-Sampling    The Modified DFT    Other Forms of Prior Knowledge    One-Dimensional Arrays    Measuring Fourier Coefficients    Over-Sampling    Under-Sampling    Using Matched Filtering    A Single Source    Multiple Sources    An Example: The Solar-Emission Problem    Estimating the Size of Distant Objects    The Transmission Problem    Directionality    The Case of Uniform Strength    The Laplace Transform and the Ozone Layer    The Laplace Transform    Scattering of Ultraviolet Radiation    Measuring the Scattered Intensity    The Laplace Transform Data    The Laplace Transform and Energy Spectral Estimation    The Attenuation Coefficient Function    The Absorption Function as a Laplace Transform    Finite-Parameter Models    Chapter Summary    Finite Fourier Series    The DFT and the Finite Fourier Series    The Vector DFT    The Vector DFT in Two Dimensions    The Issue of Units    Approximation, Models, or Truth?    Modeling the Data    Extrapolation    Filtering the Data    More on Coherent Summation    Uses in Quantum Electrodynamics    Using Coherence and Incoherence    The Discrete Fourier Transform    Complications    Multiple Signal Components    Resolution    Unequal Amplitudes and Complex Amplitudes    Phase Errors    Undetermined Exponential Models    Prony\'s Problem    Prony\'s Method    Transmission and Remote Sensing    Chapter Summary    Directional Transmission    Multiple-Antenna Arrays    The Array of Equi-Spaced Antennas    The Far-Field Strength Pattern    Can the Strength Be Zero?    Diffraction Gratings    Phase and Amplitude Modulation    Steering the Array    Maximal Concentration in a Sector    Scattering in Crystallography    The Fourier Transform and Convolution Filtering    Chapter Summary    Linear Filters    Shift-Invariant Filters    Some Properties of a SILO    The Dirac Delta    The Impulse Response Function    Using the Impulse-Response Function    The Filter Transfer Function    The Multiplication Theorem for Convolution    Summing Up    A Question    Band-Limiting    Infinite Sequences and Discrete Filters    Chapter Summary    Shifting    Shift-Invariant Discrete Linear Systems    The Delta Sequence    The Discrete Impulse Response    The Discrete Transfer Function    Using Fourier Series    The Multiplication Theorem for Convolution    The Three-Point Moving Average    Autocorrelation    Stable Systems    Causal Filters    Convolution and the Vector DFT    Chapter Summary    Nonperiodic Convolution    The DFT as a Polynomial    The Vector DFT and Periodic Convolution    The Vector DFT    Periodic Convolution    The vDFT of Sampled Data    Superposition of Sinusoids    Rescaling    The Aliasing Problem    The Discrete Fourier Transform    Calculating Values of the DFT    Zero-Padding    What the vDFT Achieves    Terminology    Understanding the Vector DFT    The Fast Fourier Transform (FFT)    Evaluating a Polynomial    The DFT and Vector DFT    Exploiting Redundancy    The Two-Dimensional Case    Plane-Wave Propagation    Chapter Summary    The Bobbing Boats    Transmission and Remote Sensing    The Transmission Problem    Reciprocity    Remote Sensing    The Wave Equation    Plane-wave Solutions    Superposition and the Fourier Transform    The Spherical Model    Sensor Arrays    The Two-Dimensional Array    The One-Dimensional Array    Limited Aperture    Sampling    The Limited-Aperture Problem    Resolution    The Solar-Emission Problem Revisited    Other Limitations on Resolution    Discrete Data    Reconstruction from Samples    The Finite-Data Problem    Functions of Several Variables    A Two-Dimensional Far-Field Object    Limited Apertures in Two Dimensions    Broadband Signals    The Phase Problem    Chapter Summary    Reconstructing from Over-Sampled Complex FT Data    The Phase Problem    A Phase-Retrieval Algorithm    Fienup\'s Method    Does the Iteration Converge?    Transmission Tomography    Chapter Summary    X-Ray Transmission Tomography    The Exponential-Decay Model    Difficulties to be Overcome    Reconstruction from Line Integrals    The Radon Transform    The Central Slice Theorem    Inverting the Fourier Transform    Back Projection    Ramp Filter, then Back Project    Back Project, then Ramp Filter    Radon\'s Inversion Formula    From Theory to Practice    The Practical Problems    A Practical Solution: Filtered Back Projection    Some Practical Concerns    Summary    Random Sequences    Chapter Summary    What is a Random Variable?    The Coin-Flip Random Sequence    Correlation    Filtering Random Sequences    An Example    Correlation Functions and Power Spectra    The Dirac Delta in Frequency Space    Random Sinusoidal Sequences    Random Noise Sequences    Increasing the SNR    Colored Noise    Spread-Spectrum Communication    Stochastic Difference Equations    Random Vectors and Correlation Matrices    The Prediction Problem    Prediction Through Interpolation    Divided Differences    Linear Predictive Coding    Discrete Random Processes    Wide-Sense Stationary Processes    Autoregressive Processes    Linear Systems with Random Input    Stochastic Prediction    Prediction for an Autoregressive Process    Nonlinear Methods    Chapter Summary    The Classical Methods    Modern Signal Processing and Entropy    Related Methods    Entropy Maximization    Estimating Nonnegative Functions    Philosophical Issues    The Autocorrelation Sequence fr(n)g    Minimum-Phase Vectors    Burg\'s MEM    The Minimum-Phase Property    Solving Ra = delta Using Levinson\'s Algorithm    A Sufficient Condition for Positive-Definiteness    The IPDFT    The Need for Prior Information in Nonlinear Estimation    What Wiener Filtering Suggests    Using a Prior Estimate    Properties of the IPDFT    Illustrations    Fourier Series and Analytic Functions    An Example    Hyperfunctions    Fejer-Riesz Factorization    Burg Entropy    Some Eigenvector Methods    The Sinusoids-in-Noise Model    Autocorrelation    Determining the Frequencies    The Case of Non-White Noise    Discrete Entropy Maximization    Chapter Summary    The Algebraic Reconstruction Technique    The Multiplicative Algebraic Reconstruction Technique    The Kullback-Leibler Distance    The EMART    Simultaneous Versions    The Landweber Algorithm    The SMART    The EMML Algorithm    Block-Iterative Versions    Convergence of the SMART    Analysis and Synthesis    Chapter Summary    The Basic Idea    Polynomial Approximation    Signal Analysis    Practical Considerations in Signal Analysis    The Finite-Data Problem    Frames    Bases, Riesz Bases, and Orthonormal Bases    Radar Problems    The Wideband Cross-Ambiguity Function    The Narrowband Cross-Ambiguity Function    Range Estimation    Time-Frequency Analysis    The Short-Time Fourier Transform    The Wigner-Ville Distribution    Wavelets    Chapter Summary    Background    A Simple Example    The Integral Wavelet Transform    Wavelet Series Expansions    Multiresolution Analysis    The Shannon Multiresolution Analysis    The Haar Multiresolution Analysis    Wavelets and Multiresolution Analysis    Signal Processing Using Wavelets    Decomposition and Reconstruction    Generating the Scaling Function    Generating the Two-Scale Sequence    Wavelets and Filter Banks    Using Wavelets    The BLUE and the Kalman Filter    Chapter Summary    The Simplest Case    A More General Case    Some Useful Matrix Identities    The BLUE with a Prior Estimate    Adaptive BLUE    The Kalman Filter    Kalman Filtering and the BLUE    Adaptive Kalman Filtering    Difficulties with the BLUE    Preliminaries from Linear Algebra    When are the BLUE and the LS Estimator the Same?    A Recursive Approach    Signal Detection and Estimation    Chapter Summary    The Model of Signal in Additive Noise    Optimal Linear Filtering for Detection    The Case of White Noise    Constant Signal    Sinusoidal Signal, Frequency Known    Sinusoidal Signal, Frequency Unknown    The Case of Correlated Noise    Constant Signal with Unequal-Variance Uncorrelated Noise    Sinusoidal signal, Frequency Known, in Correlated Noise    Sinusoidal Signal, Frequency Unknown, in Correlated Noise    Capon\'s Data-Adaptive Method    Appendix: Inner Products    Chapter Summary    Cauchy\'s Inequality    The Complex Vector Dot Product    Orthogonality    Generalizing the Dot Product: Inner Products    Another View of Orthogonality    Examples of Inner Products    An Inner Product for Infinite Sequences    An Inner Product for Functions    An Inner Product for Random Variables    An Inner Product for Complex Matrices    A Weighted Inner Product for Complex Vectors    A Weighted Inner Product for Functions    The Orthogonality Principle    Appendix: Wiener Filtering    Chapter Summary    The Vector Wiener Filter in Estimation    The Simplest Case    A More General Case    The Stochastic Case    The VWF and the BLUE    Wiener Filtering of Functions    Wiener Filter Approximation: The Discrete Stationary Case    Approximating the Wiener Filter    Adaptive Wiener Filters    An Adaptive Least-Mean-Square Approach    Adaptive Interference Cancellation (AIC)    Recursive Least Squares (RLS)    Appendix: Matrix Theory    Chapter Summary    Matrix Inverses    Basic Linear Algebra    Bases and Dimension    Systems of Linear Equations    Real and Complex Systems of Linear Equations    Solutions of Under-determined Systems of Linear Equations    Eigenvalues and Eigenvectors    Vectorization of a Matrix    The Singular Value Decomposition (SVD)    The SVD    An Application in Space Exploration    Pseudo-Inversion    Singular Values of Sparse Matrices    Matrix and Vector Differentiation    Differentiation with Respect to a Vector    Differentiation with Respect to a Matrix    Eigenvectors and Optimization    Appendix: Compressed Sensing    Chapter Summary    An Overview    Compressed Sensing    Sparse Solutions    Maximally Sparse Solutions    Minimum One-Norm Solutions    Minimum One-Norm as an LP Problem    Why the One-Norm?    Comparison with the PDFT    Iterative Reweighting    Why Sparseness?    Signal Analysis    Locally Constant Signals    Tomographic Imaging    Compressed Sampling    Appendix: Probability    Chapter Summary    Independent Random Variables    Maximum Likelihood Parameter Estimation    An Example: The Bias of a Coin    Estimating a Poisson Mean    Independent Poisson Random Variables    The Multinomial Distribution    Characteristic Functions    Gaussian Random Variables    Gaussian Random Vectors    Complex Gaussian Random Variables    Using A Priori Information    Conditional Probabilities and Bayes\' Rule    An Example of Bayes\' Rule    Using Prior Probabilities    Maximum A Posteriori Estimation    MAP Reconstruction of Images    Penalty-Function Methods    Basic Notions    Generating Correlated Noise Vectors    Covariance Matrices    Principal Component Analysis    Appendix: Using the Wave Equation    Chapter Summary    The Wave Equation    The Shallow-Water Case    The Homogeneous-Layer Model    The Pekeris Waveguide    The General Normal-Mode Model    Matched-Field Processing    Appendix: Reconstruction in Hilbert Space    Chapter Summary    The Basic Problem    Fourier-Transform Data    The General Case    Some Examples    Choosing the Inner Product    Choosing the Hilbert Space    Summary    Appendix: Some Theory of Fourier Analysis    Chapter Summary    Fourier Series    Fourier Transforms    Functions in the Schwartz Class    Generalized Fourier Series    Wiener Theory    Appendix: Reverberation and Echo Cancellation    Chapter Summary    The Echo Model    Finding the Inverse Filter    Using the Fourier Transform    The Teleconferencing Problem    Bibliography    Index