Multidimensional Signals and Systems: Theory and Foundations

✍ Scribed by Rudolf Rabenstein, Maximilian Schäfer

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 514
• Edition: 1st ed. 2023
• Category: Library

✦ Synopsis

This book covers the theory of multidimensional signals and systems and related practical aspects. It extends the properties and mathematical tools of one-dimensional signals and systems to multiple dimensions and covers relevant timeless topics including multidimensional transformations, multidimensional sampling as well as discrete multidimensional systems. A special emphasis is placed on physical systems described by partial differential equations, the construction of suitable integral transformations and the implementation of the corresponding discrete-time algorithms. To this end, signal spaces and functional transformations are introduced at a mathematical level provided by undergraduate programs in engineering and science.

The presentation takes a comprehensive, illustrative and educational approach without reference to a particular application field. Instead, the book builds a solid theoretical concept of multidimensional signals and systems and shows the application to various problems relevant for practical scenarios.

✦ Table of Contents

Preface
Contents
Acronyms
1 Introduction
References
2 Overview on Multidimensional Signals
2.1 One-Dimensional Signals
2.1.1 One-Dimensional Time-Dependent Signals
2.1.2 One-Dimensional Space-Dependent Signals
2.2 Two-Dimensional Signals
2.2.1 Two-Dimensional Space-Dependent Signals
2.2.2 Two-Dimensional Space- and Time-Dependent Signals
2.3 Three-Dimensional Signals
2.3.1 Three-Dimensional Space-Dependent Signals
2.3.2 Three-Dimensional Space- and Time-Dependent Signals
2.4 Four-Dimensional Signals
2.5 Higher Dimensional Signals
2.6 Properties of Multidimensional Signals
2.7 Multidimensional Systems
2.7.1 Autonomous Systems and Input-Output-Systems
2.7.2 Linear, Time-, and Shift-Invariant Systems
2.7.3 Mathematical Formulation of Multidimensional Systems
2.7.3.1 Difference Equations
2.7.3.2 Differential Equations
2.8 Overview on the Next Chapters
2.9 Problems
References
3 Elements from One-Dimensional Signals and Systems
3.1 Convolution and Impulse Response
3.1.1 An Introductory Example
3.1.1.1 Integer Numbers
3.1.1.2 Sequences of Numbers
3.1.1.3 Continuous Functions
3.1.2 Delta Impulse
3.1.2.1 Classical Functions and Generalized Functions
3.1.2.2 Definition
3.1.2.3 Properties
3.1.2.4 Summary of the Properties of the Delta Impulse
3.1.3 Convolution
3.1.3.1 Derivation of Continuous-Time Convolution
3.1.3.2 Properties of Continuous-Time and Discrete-Time Convolution
3.1.3.3 Summary of the Properties of Continuous-Time and Discrete-Time Convolution
3.2 Fourier Transformation
3.2.1 Eigenfunctions of Linear and Time-Invariant Systems
3.2.2 Definition of the Fourier Transformation
3.2.3 Correspondences
3.2.4 Properties
3.2.5 Summary of Correspondences and Properties of the Fourier Transformation
3.3 Sampling
3.3.1 Sampling of Continuous-Time Functions
3.3.2 Spectrum of a Sampled Signal
3.4 Differential Equations and Transfer Functions
3.4.1 A Light Example
3.4.1.1 Circuit Analysis
3.4.1.2 Solution of the Homogeneous Differential Equation
3.4.1.3 Solution of the Inhomogeneous Differential Equation
3.4.1.4 Solution of the Initial Value Problem
3.4.2 State Space Systems
3.4.2.1 State Space Representation
3.4.2.2 Solution of the Homogeneous State Equation
3.4.2.3 Definition of the Propagator P P P P(t)
3.4.2.4 Properties of the Propagator P P P P(t)
3.4.2.5 Similarity Transformation
3.4.2.6 Interpretation of the State Transformation as a Signal Transformation
3.4.2.7 Propagator P P P P(t) in the Transform Domain
3.4.2.8 Laplace Transfer Function
3.4.2.9 Solution of the State Equations
3.4.2.10 Discrete-Time State Space Representation
3.4.3 Conclusion and Outlook
3.5 Problems
References
4 Signal Spaces
4.1 Foundations
4.1.1 Vectors, Functions, and Signals
4.1.2 Topics from Signal Processing
4.1.2.1 Superposition of Signals
4.1.2.2 Distance and Angle
4.1.2.3 Power and Energy
4.1.2.4 Square-Integrability and Square-Summability
4.2 Introduction to Signal Spaces
4.2.1 What Is a Signal Space?
4.2.1.1 Operations in a Signal Space
4.2.1.2 Basis of a Signal Space
4.2.2 Scalar Product
4.2.2.1 Definition
4.2.2.2 Schwarz Inequality
4.2.2.3 Square-Integrable Functions and Square-Summable Sequences
4.2.3 Norm, Distance, and Angle
4.2.3.1 Definition of a Norm
4.2.3.2 Lp-Norms
4.2.3.3 Norms in Inner Product Spaces
4.2.3.4 Norms for Square-Integrable Functions and Square-Summable Sequences
4.2.3.5 Distance and Metric
4.2.3.6 Angle
4.2.3.7 Law of Cosines
4.2.4 Completeness
4.2.5 Hilbert Spaces
4.2.6 Extension to Generalized Functions
4.2.6.1 Delta Impulse
4.2.6.2 Constant Value
4.2.6.3 Signal Spaces with Generalized Functions
4.3 Orthogonality
4.3.1 Definition
4.3.2 Perpendicularity in Two Dimensions
4.3.3 Expansion into Basis Vectors
4.3.3.1 Orthogonal and Non-orthogonal Basis Vectors
4.3.3.2 The Gramian Matrix
4.3.3.3 Examples for Different Kinds of Basis Vectors
4.3.4 Gram-Schmidt Orthogonalization
4.3.4.1 Projection
4.3.4.2 Gram-Schmidt-Orthogonalization Procedure
4.3.4.3 Expansion of a Function into Powers
4.4 Duality and Biorthogonality
4.4.1 Dual Spaces and Biorthogonal Signal Spaces
4.4.2 Sets of Biorthogonal Vectors
4.4.3 General Biorthogonal Signal Spaces
4.5 Signal Transformations
4.5.1 General Procedure
4.5.2 Discrete Fourier Transformation
4.5.3 Fourier Series
4.5.4 Discrete-Time Fourier Transformation
4.5.5 Fourier Transformation
4.5.6 Discrete Cosine Transformation
4.5.7 Summary of the Fourier-Type Transformations
4.5.8 A Review of the Notation
4.5.9 Signal Transformations with Complex FrequencyVariables
4.5.9.1 Laplace Transformation
4.5.9.2 z-Transformation
4.6 Problems
References
5 Multidimensional Signals
5.1 Properties
5.1.1 Basic Properties
5.1.2 Separable Signals
5.1.3 Symmetrical Signals
5.1.3.1 Axial Symmetry
5.1.3.2 Radial Symmetry
5.1.4 Coordinate Systems
5.1.4.1 Cartesian Coordinates
5.1.4.2 Polar Coordinates
5.1.4.3 Cylindrical Coordinates
5.1.4.4 Spherical Coordinates
5.1.5 Symmetry and Separability in Different CoordinateSystems
5.2 Convolution
5.2.1 Review of One-Dimensional Convolution
5.2.2 Definitions and Notation
5.2.2.1 2D Convolution of Continuous Signals
5.2.2.2 2D Convolution of Discrete Signals
5.2.2.3 Examples for 2D Convolution
5.2.3 Convolution of Separable Signals
5.2.4 2D Convolution and Imaging
5.3 Distributions
5.3.1 A Note on Dimensionality
5.3.2 Two-Dimensional Point Impulses
5.3.2.1 Definition
5.3.2.2 Properties of Delta Impulses
5.3.3 Two-Dimensional Line Impulses
5.3.4 Properties of Line Impulses
5.3.5 Ring Impulses
5.3.6 Combinations of Line Impulses
5.3.6.1 Product of Line Impulses
5.3.6.2 Periodic Sequences of Line Impulses
5.3.7 Applications of 2D Impulses
5.3.8 Point Impulses in Different Coordinate Systems
5.3.9 Review of Delta Impulses
5.4 Problems
References
6 Multidimensional Transformations
6.1 Fourier Transformation in Cartesian Coordinates
6.1.1 Definition of the 2D Fourier Transformation
6.1.1.1 Analysis and Synthesis Equation
6.1.1.2 2D Basis Functions
6.1.1.3 Vector Notation
6.1.1.4 1D and 2D Fourier Transformations
6.1.2 Fourier Transforms of Frequently Used Functions
6.1.2.1 Delta Impulse
6.1.2.2 Gaussian Function
6.1.2.3 Rectangle Function
6.1.2.4 Impulse Grid
6.1.2.5 Line Impulses δ(x) and δ(y)
6.1.2.6 First Set of Correspondences
6.1.3 Basic Properties of the 2D Fourier Transformation
6.2 Affine Mappings
6.2.1 Motivation
6.2.2 Affine Mappings in Two Dimensions
6.2.3 Properties of Affine Mappings
6.2.4 Coordinate Transformation in Multiple Integrals
6.3 More Properties of the 2D Fourier Transformation
6.3.1 Affine Theorem
6.3.2 Special Cases
6.3.3 Projection Slice Theorem
6.3.4 Radon Transformation
6.3.5 Back-Projection
6.3.6 Differentiation
6.3.7 Summary of Correspondences and Properties
6.4 Fourier Transformation in Non-Cartesian Coordinates
6.5 Fourier Transformation in Polar Coordinates
6.5.1 Notation and Overview
6.5.2 Coordinate Transformation
6.5.3 Angular Series Expansion
6.5.4 Hankel Transformation
6.5.5 Summary of the Fourier Transformation in PolarCoordinates
6.6 Fourier Transformation in Spherical Coordinates
6.6.1 Notation
6.6.2 Coordinate Transformation
6.6.3 Angular Series Expansion
6.6.4 Summary of the Fourier Transformation in Spherical Coordinates
6.7 Other 2D Transformations
6.7.1 2D Discrete-Time Fourier Transformation
6.7.2 2D z-Transformation
6.7.2.1 Definition
6.7.2.2 Properties
6.8 Problems
References
7 Multidimensional Sampling
7.1 Rectangular Sampling of Two-Dimensional Signals
7.1.1 Rectangular Sampling of a Continuous 2DSignal
7.1.2 2D Rectangular Sampling in Vector Notation
7.2 2D Sampling on General Sampling Grids
7.2.1 A Glimpse of Lattice Theory
7.2.2 Definition of a General 2D Sampling Grid
7.2.3 Repetition Pattern for General 2D Sampling
7.2.4 Summary of Non-Rectangular Sampling and Spectral Repetition
7.2.5 An Extended Example for Non-Rectangular Sampling
7.3 Aliasing
7.3.1 Aliasing in One Dimension
7.3.2 Aliasing in Two Dimensions
7.4 Summary of 1D and 2D sampling
7.5 Frequently Used Sampling Lattices
7.5.1 Introduction
7.5.2 Sampling Density
7.5.3 2D Continuous Functions
7.5.4 Rectangular Sampling
7.5.5 Diagonal Sampling
7.5.6 Hexagonal Sampling
7.5.7 Summary and Outlook
7.6 Problems
References
8 Discrete Multidimensional Systems
8.1 Discrete Finite Impulse Response Systems
8.1.1 Discrete Convolution
8.1.2 Definition of 2D FIR Systems
8.1.3 Order of Computations
8.1.4 Typical 2D FIR Systems
8.1.5 Transfer Functions of 2D FIR Systems
8.1.6 Stability of 2D FIR Systems
8.2 Discrete Infinite Impulse Response Systems
8.2.1 Definition of 2D IIR Systems
8.2.2 Order of Computations
8.2.2.1 Quarter-Plane and Half-Plane Filters
8.2.2.2 Order of Computations for Quarter-Plane Filters
8.2.2.3 Order of Computations for Half-Plane Filters
8.2.3 Transfer Functions of 2D IIR Systems
8.2.4 Stability of 2D IIR Systems
8.2.5 Application of 2D IIR Systems
8.3 Discretization of Differential Equations
8.3.1 Continuous Poisson Equation
8.3.2 The Method of Weighted Residuals
8.3.2.1 Weighted Residuals and Basis Functions
8.3.2.2 Reduction to an Algebraic Problem
8.3.3 Numerical Methods
8.3.3.1 Global and Local Basis Functions
8.3.3.2 Collocation Methods
8.3.3.3 Subdomain Methods
8.3.3.4 Galerkin Methods
8.3.3.5 Examples of Weighted Residual Methods
8.3.3.6 Summary
8.4 Iterative Methods for the Solution of Systems of Linear Equations
8.4.1 Finite Difference Approximation
8.4.1.1 Derivation of Finite Difference Operators
8.4.1.2 Discrete Poisson Equation
8.4.2 Iterative Solution of Large Systems of Linear Equations
8.4.2.1 Formulation of the Problem
8.4.2.2 Description of the Approach
8.4.3 Classical Iteration Methods
8.4.3.1 Matrix Splitting
8.4.3.2 Jacobi Iteration
8.4.3.3 Gauss-Seidel Iteration
8.4.3.4 Successive Over-Relaxation
8.4.4 Convergence
8.4.4.1 Structure of the Iteration Procedure
8.4.4.2 Frequency Domain Analysis
8.4.4.3 Multigrid Methods
8.4.5 Review of Iterative Matrix Inversion
8.5 Multidimensional Systems and MIMO Systems
8.5.1 Multiple-Input Multiple-Output Systems
8.5.2 Relations Between Multidimensional Systems and MIMO Systems
8.5.3 Conclusions
8.6 Summary and Outlook
8.7 Problems
References
9 Continuous Multidimensional Systems
9.1 Distributed Parameter Systems
9.1.1 Lumped and Distributed Parameter Systems
9.1.2 Electrical Transmission Line
9.1.3 Telegraph Equation
9.1.4 Special Cases
9.1.5 Initial and Boundary Conditions
9.1.6 Summary
9.2 Scalar Linear Partial Differential Equations
9.2.1 Spatio-Temporal Domain
9.2.2 Initial-Boundary-Value Problems for a Scalar Variable
9.2.3 Partial Differential Equations
9.2.4 Initial Conditions
9.2.5 Boundary Conditions
9.2.6 Telegraph Equation
9.3 Vector-Valued Linear Partial Differential Equations
9.3.1 Coupled Partial Differential Equations
9.3.2 A Note on Analogies of Physical Variables
9.3.2.1 Initial Conditions
9.3.3 Boundary Conditions
9.4 Vector-Valued and Scalar Partial Differential Equations
9.4.1 Converting a Vector Representation into a Scalar Representation
9.4.1.1 Examples
9.4.2 Converting a Scalar Representation into a Vector Representation
9.4.3 Transformation of the Dependent Variables
9.5 General Solution
9.5.1 Review of One-Dimensional Systems
9.5.2 Solution of Multidimensional Systems
9.6 Problems
References
10 Sturm-Liouville Transformation
10.1 Introductory Example
10.1.1 Physical Problem
10.1.2 Laplace Transformation
10.1.3 Finite Fourier-Sine Transformation
10.1.4 Transfer Function Description
10.1.5 Inverse Fourier Sine Transformation
10.1.6 Inverse Laplace Transformation
10.1.7 Solution in the Space-Time Domain
10.1.8 Review
10.2 Spatial Differentiation Operators
10.2.1 Initial-Boundary-Value Problem
10.2.1.1 Time-Domain Formulation
10.2.1.2 Frequency-Domain Formulation
10.2.1.3 Objectives for a Spatial Transformation
10.2.2 Spatial Differentiation Operator and its Adjoint Operator
10.2.2.1 Scalar Product
10.2.2.2 Integration by Parts in One Dimension
10.2.2.3 Integration by Parts in n Dimensions
10.2.2.4 Adjoint Spatial Operator
10.2.3 Eigenfunctions of the Spatial Operators
10.2.3.1 Eigenvalue Problems
10.2.3.2 Orthogonality of the Eigenfunctions
10.2.3.3 Boundary Conditions for the Eigenfunctions
10.2.4 Recapitulation of the Eigenvalue Problems
10.3 Spatial Transformation
10.3.1 Eigenfunctions and Basisfunctions
10.3.2 Definition of the Sturm-Liouville Transformation
10.3.3 Differentiation Theorem
10.3.4 Application to the Initial-Boundary-Value Problem
10.3.5 Transfer Function Description
10.3.5.1 Solution of the Algebraic Equation
10.3.5.2 Relation to One-Dimensional Transfer Functions
10.4 Green's Functions
10.4.1 Green's Function for the Initial Value
10.4.2 Green's Function for the Excitation Function
10.4.3 Green's Function for the Boundary Value
10.4.4 Summary of the Green's Functions
10.4.5 Response to an Impulse
10.5 Propagator
10.5.1 Definition
10.5.2 Properties
10.6 Review of the Solution of Initial-Boundary-Value Problems
10.7 Continuous Multidimensional Systems with Space-Dependent Coefficients
10.7.1 Introduction
10.7.2 Formulation in the Time-Domain andFrequency-Domain
10.7.3 Adjoint Operator
10.7.4 Eigenvalue Problems
10.7.5 Sturm-Liouville Transformation
10.7.6 Transfer Function Description
10.8 Classical Sturm-Liouville Problems
10.8.1 Derivation from a 22 Eigenvalue Problem
10.8.2 Properties
10.8.3 Solution of Classical Sturm-Liouville Problems
10.8.4 Legendre Polynomials
10.8.5 Associated Legendre Functions
10.8.6 Bessel Functions
10.9 Problems
References
11 Solution Methods
11.1 Solution of Eigenvalue Problems
11.1.1 An Introductory Example
11.1.1.1 Eigenfunctions
11.1.1.2 Boundary Conditions
11.1.1.3 Eigenvalues
11.1.1.4 Final Result for the Eigenvalues and Eigenfunctions
11.1.1.5 Dispersion Relation
11.1.1.6 Summary
11.1.2 Definition and Properties of the Matrix Exponential
11.1.2.1 Definition
11.1.2.2 Properties
11.1.2.3 Matrix Exponential and Propagator
11.1.3 Representations of the Matrix Exponential by FiniteSums
11.1.3.1 Exploiting the Cayley-Hamilton Theorem
11.1.3.2 Diagonalization
11.1.3.3 Power Series Expansion
11.1.4 Matrix Representation of a 1D Continuous-Time System
11.1.4.1 System Description
11.1.4.2 Solution in Terms of a Vandermonde Matrix
11.1.4.3 Solution by Laplace Transformation
11.1.4.4 A Comparison of Both Solutions
11.1.4.5 Conclusion
11.1.5 Calculation of the Matrix Exponential
11.1.5.1 Diagonalization
11.1.5.2 Comparing Boundary Values
11.1.5.3 Coefficient Matrix Representation
11.1.6 Summary and Examples
11.2 Calculation of Further Quantities
11.2.1 Matrix Exponential of the Adjoint Operator
11.2.1.1 Review of the Primal and the Adjoint Eigenvalue Problem
11.2.1.2 Calculation of the Matrix Exponential of the Matrix
11.2.1.3 An Identity for the Primal and the Adjoint Matrix Exponentials
11.2.2 Normalization Factor Nμ
11.2.2.1 Calculation from the Eigenfunctions
11.2.2.2 Calculation from the Matrix Exponentials
11.2.2.3 Review
11.2.3 Boundary Terms
11.2.4 Conclusions
11.3 Solve Initial-Boundary-Value Problems in Seven Steps
11.3.1 Seven Step Procedure
11.3.2 Example for the Solution of Sturm-Liouville Problems
11.3.2.1 Conclusions
11.4 Problems
References
12 Algorithmic Implementation
12.1 State-Space Representation of Continuous Multidimensional Systems
12.1.1 Infinite-Dimensional Linear Operators
12.1.1.1 Forward Transformation
12.1.1.2 Differentiation Theorem
12.1.1.3 Application to the Partial Differentiation Equation
12.1.1.4 Inverse Transformation
12.1.1.5 Interpretation as Biorthogonal Signal Spaces
12.1.2 State Space Representation
12.1.2.1 Infinite-Dimensional State Space Representation
12.1.2.2 Input Variables with Space-Time Separability
12.1.2.3 Finite-Dimensional Approximation
12.1.3 Relation to the Green's Functions and to the Propagator
12.1.3.1 Relation to the Green's Functions
12.1.3.2 Relation to the Propagator
12.1.4 Enumeration of the Eigenvalues and Eigenfunctions
12.2 Time Discretization
12.2.1 Bilinear Transformation
12.2.1.1 Fundamentals
12.2.1.2 Stability and Minimum Phase
12.2.1.3 Frequency Warping
12.2.1.4 Vector-Valued Bilinear Transformation
12.2.2 Impulse Invariant Transformation
12.2.2.1 Fundamentals
12.2.2.2 Classical Version
12.2.2.3 Corrected Version
12.2.2.4 Stability
12.2.2.5 Aliasing
12.2.2.6 Vector-Valued Impulse Invariant Transformation
12.2.3 Comparing Bilinear and Impulse InvariantTransformation
12.2.3.1 Applicability
12.2.3.2 Stability
12.2.3.3 Frequency Response
12.2.3.4 Impulse Response
12.2.3.5 Conclusion
12.2.4 Discrete-Time Algorithmic Structure
12.2.4.1 Computable Finite-Dimensional Formulation
12.2.4.2 Discrete-Time Formulation
12.2.4.3 Space-Time Separability
12.3 Outlook
12.4 Problems
References
Appendix A Solutions to the Problems
Index