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. Les mer
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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.

Fakta

Innholdsfortegnelse

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 = 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

Om forfatteren

Charles L. Byrne