| |
| |
| Preface | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Signals | |
| |
| |
| |
| Systems | |
| |
| |
| |
| Overview of the Book | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Time-Domain Models of Continuous LTI-Systems | |
| |
| |
| |
| Differential Equations | |
| |
| |
| |
| Block Diagrams | |
| |
| |
| |
| State-Space Description of LTI-Systems | |
| |
| |
| |
| Higher-order Differential Equations, Block Diagrams and the State Model | |
| |
| |
| |
| Equivalent State-Space Representations | |
| |
| |
| |
| Controllable and Observable Systems | |
| |
| |
| |
| Summary | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Modelling LTI-Systems in the Frequency-Domain | |
| |
| |
| |
| Complex Frequencies | |
| |
| |
| |
| Eigenfunctions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Laplace Transform | |
| |
| |
| |
| The Eigenfunction Formulation | |
| |
| |
| |
| Definition of the Laplace Transform | |
| |
| |
| |
| Unilateral and Bilateral Laplace Transforms | |
| |
| |
| |
| Examples of Laplace Transforms | |
| |
| |
| |
| Region of Convergence of the Laplace Transform | |
| |
| |
| |
| Existence and Uniqueness of the Laplace Transform | |
| |
| |
| |
| Properties of the Laplace Transform | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Complex Analysis and the Inverse Laplace Transform | |
| |
| |
| |
| Path Integrals in the Complex Plane | |
| |
| |
| |
| The Main Principle of Complex Analysis | |
| |
| |
| |
| Circular Integrals that Enclose Singularities | |
| |
| |
| |
| Cauchy Integrals | |
| |
| |
| |
| Inverse Laplace Transform | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Analysis of Continuous-Time LTI-Systems with the Laplace Transform | |
| |
| |
| |
| System Response to Bilateral Input Signals | |
| |
| |
| |
| Finding the System Function | |
| |
| |
| |
| Poles and Zeros of the System Function | |
| |
| |
| |
| Determining the System Function from Differential Equations | |
| |
| |
| |
| Summarising Example | |
| |
| |
| |
| Combining Simple LTI-Systems | |
| |
| |
| |
| Combining LTI-Systems with Multiple Inputs and Outputs | |
| |
| |
| |
| Analysis of State-Space Descriptions | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Solving Initial Condition Problems with the Laplace Transform | |
| |
| |
| |
| First-Order Systems | |
| |
| |
| |
| Second-Order Systems | |
| |
| |
| |
| Higher-Order Systems | |
| |
| |
| |
| Assessment of the Procedures for Solving Initial Condition Problems | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Convolution and Impulse Response | |
| |
| |
| |
| Motivation | |
| |
| |
| |
| Time Behaviour of an RC-Circuit | |
| |
| |
| |
| The Delta Impulse | |
| |
| |
| |
| Convolution | |
| |
| |
| |
| Applications | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| The Fourier Transform | |
| |
| |
| |
| Review of the Laplace Transform | |
| |
| |
| |
| Definition of the Fourier Transform | |
| |
| |
| |
| Similarities and Differences between Fourier and Laplace Transforms | |
| |
| |
| |
| Examples of the Fourier Transform | |
| |
| |
| |
| Symmetries of the Fourier Transform | |
| |
| |
| |
| Inverse Fourier Transform | |
| |
| |
| |
| Properties of the Fourier Transform | |
| |
| |
| |
| Parseval's Theorem | |
| |
| |
| |
| Correlation of Deterministic Signals | |
| |
| |
| |
| Time-Bandwidth Product | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Bode Plots | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Contribution of Individual Poles and Zeros | |
| |
| |
| |
| Bode Plots for Multiple Poles and Zeros | |
| |
| |
| |
| Rules for Bode Plots | |
| |
| |
| |
| Complex Pairs of Poles and Zeros | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Sampling and Periodic Signals | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Delta Impulse Train and Periodic Functions | |
| |
| |
| |
| Sampling | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| The Spectrum of Discrete Signals | |
| |
| |
| |
| Discrete-Time Signals | |
| |
| |
| |
| Some Simple Sequences | |
| |
| |
| |
| Discrete-Time Fourier Transform | |
| |
| |
| |
| Sampling Continuous Signals | |
| |
| |
| |
| Properties of the F[subscript *] Transform | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| The z-Transform | |
| |
| |
| |
| Definition and Examples | |
| |
| |
| |
| Region of Convergence of the z-Transform | |
| |
| |
| |
| Relationships to Other Transformations | |
| |
| |
| |
| Theorems of the z-Transform | |
| |
| |
| |
| Inverse z-Transform | |
| |
| |
| |
| Pole-Zero Diagrams in the z-Plane | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Discrete-Time LTI-Systems | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Linearity and Time-Invariance | |
| |
| |
| |
| Linear Difference Equations with Constant Coefficients | |
| |
| |
| |
| Characteristic Sequences and System Functions of Discrete LTI-Systems | |
| |
| |
| |
| Block Diagrams and State-Space | |
| |
| |
| |
| Discrete Convolution and Impulse Response | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Causality and the Hilbert Transform | |
| |
| |
| |
| Causal Systems | |
| |
| |
| |
| Causal Signals | |
| |
| |
| |
| Signals with a One-Sided Spectrum | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Stability and Feedback Systems | |
| |
| |
| |
| BIBO, Impulse Response and Frequency Response Curve | |
| |
| |
| |
| Causal Stable LTI-Systems | |
| |
| |
| |
| Feedback Systems | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Describing Random Signals | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Expected Values | |
| |
| |
| |
| Stationary Random Processes | |
| |
| |
| |
| Correlation Functions | |
| |
| |
| |
| Power Density Spectra | |
| |
| |
| |
| Describing Discrete Random Signals | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Random Signals and LTI-Systems | |
| |
| |
| |
| Combining Random Signals | |
| |
| |
| |
| Response of LTI-Systems to Random Signals | |
| |
| |
| |
| Signal Estimation Using the Wiener Filter | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Solutions to the Exercises | |
| |
| |
| |
| Tables of Transformations | |
| |
| |
| |
| Bilateral Laplace Transform Pairs | |
| |
| |
| |
| Properties of the Bilateral Laplace Transform | |
| |
| |
| |
| Fourier Transform Pairs | |
| |
| |
| |
| Properties of the Fourier Transform | |
| |
| |
| |
| Two-sided z-Transform Pairs | |
| |
| |
| |
| Properties of the z-Transform | |
| |
| |
| |
| Discrete-Time Fourier Transform Pairs | |
| |
| |
| |
| Properties of the Discrete-Time Fourier Transform | |
| |
| |
| Bibliography | |
| |
| |
| Index | |