Theory and application of the z-transform method

Title page
Preface
1 z-TRANSFORM DEFINITION AND THEOREMS
1.1 Discrete Time Function and z-Transform Definitions
1.2 Properties of z-Transforms
1.3 Inverse z-Transform and Branch Points
1.4 The Modified z-Transform
1.5 Relationship between Laplace and z-Transforms
1.6 Application to Sampled-Data Systems
1.7 Mean Square Value Theorem
1.8 Equivalence between Inverse Laplace and Modified z-Transforms
1.9 Other Transform Methods
Appendix. A Method of Determining the Coefficients of the z-Transform Expansion
2 z-TRANSFORM METHOD OF SOLUTION OF LINEAR DIFFERENCE EQUATIONS
2.1 Linear Difference Equations with Constant Coefficients
2.2 Solution of Difference Equations Whose Coefficients Are Periodic Functions
2.3 Linear Difference-Differential Equations
2.4 Difference Equations with Periodic Coefficients
2.5 Time-Varying Difference Equations
2.6 Time-Varying z-Transform and the System Function
2.7 Double z-Transformation and Solution of Partial Difference Equations
3 STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS
3.1 Definition of Stability
3.2 Stability Condition for Linear Time-Varying Discrete Systems
3.3 Tests for Stability
3.4 Stability Test Directly Applied in the z-Plane
3.5 Determinant Method
3.6 Critical Stability Constraints for System Design
3.7 Number of Roots of a Real Polynomial Inside the Unit Circle
3.8 Relationship between the Determinant Method and Hurwitz Criterion
3.9 Table Form
3.10 Division Method
3.11 Aperiodicity Criterion for Linear Discrete Systems
3.12 Theorems Related to Stability and Number of Roots
Appendices
1. Derivation of the Table Form of Stability
2. Singular Cases in Determinant and Table Forms
3. Summary of the Stability Criteria
4 CONVOLUTION z-TRANSFORM
4.1 Complex Convolution Theorem
4.2 Complex Convolution Theorem for the Modified z-Transform
4.3 Applications of the Convolution Modified or z-Transform Method
Appendices
1. Proof of Complex Convolution Formula
2. Derivation of Total Square Integrais Formula
5 CONVOLUTION z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS
5.1 Assumptions
5.2 Convolution z-Transforms of Certain Functions
5.3 Method of Solution for Second- and Higher-Order Equations
5.4 Illustrative Examples
6 PERIODIC MODES OF OSCILLATION IN NONLINEAR DISCRETE SYSTEMS
6.1 Limit Cycle Analysis of Nonlinear Discrete Systems
6.2 Application of the Fundamental Equation to Specific Examples
6.3 Limitation on the Period of Limit Cycles of Relay Mode Oscillations
6.4 Stability Study of Limit Cycles
6.5 Forced Oscillations in Nonlinear Discrete Systems
6.6 Direct z-Transform for Determining True Oscillation
6.7 Periodic Solution of Certain Nonlinear Difference Equations
7 z-TRANSFORM METHOD lN APPROXIMATION TECHNIQUES
7.1 Approximation Methods
7.2 Initial Conditions Nonzero
7.3 Integrating Operators
7.4 z-Forms and Modified z-Forms
7.5 The Choice of the Sampling Period
7.6 Analysis of the Error
7.7 Low-Pass Transformation for z-Transforms
7.8 Applications to Time-Varying Differential Équations
7.9 Application to Nonlinear Differential Equations
7.10 Other Numerical Techniques
8 APPLICATIONS TO VARIOUS AREAS OF SYSTEM THEORY
8.1 Nonlinear Sampled-Data Feedback Systems
8.2 Analysis of Discrete Antenna Array by z-Transform Method
8.3 Application to Information and Filtering Theory
8.4 z-Transform Method Applied to Problems of Economics
8.5 Linear Sequential Circuits
8.6 Application to Discrete Markov Processes
APPENDIX
Table I z-Transform Pairs
Table II Pairs of Modified z-Transforms
Table III Total Square IntegraIs
Table IV Closed Forms of the Function Σ₀^∞ n^r x^n (x<1)
PROBLEMS
INDEX