Signal Processing A Mathematical Approach, Second Edition

Signal Processing: A Mathematical Approach is designed to show how many of the mathematical tools the reader knows can be used to understand and employ signal processing techniques in an applied environment. Assuming an advanced undergraduate- or graduate-level understanding of mathematics—including familiarity with Fourier series, matrices, probability, and statistics—this Second Edition: Contains new chapters on convolution and the vector DFT, plane-wave propagation, and the BLUE and Kalman filters Expands the material on Fourier analysis to three new chapters to provide additional background information Presents real-world examples of applications that demonstrate how mathematics is used in remote sensing Featuring problems for use in the classroom or practice, Signal Processing: A Mathematical Approach, Second Edition covers topics such as Fourier series and transforms in one and several variables; applications to acoustic and electro-magnetic propagation models, transmission and emission tomography, and image reconstruction; sampling and the limited data problem; matrix methods, singular value decomposition, and data compression; optimization techniques in signal and image reconstruction from projections; autocorrelations and power spectra; high-resolution methods; detection and optimal filtering; and eigenvector-based methods for array processing and statistical filtering, time-frequency analysis, and wavelets.

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 = d 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 Fejér-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

Charles L. Byrne