Sampling in Digital Signal Processing and Control

Undoubtably one of the key factors influencing recent technology has been the advent of high speed computational tools. Virtually every advanced engi­ neering system we come in contact with these days depends upon some form of sampling and digital signal processing. Well known examples are digital tele­ phone systems, digital recording of audio signals and computer control. These developments have been matched by the appearance of a plethora of books which explain a variety of analysis, synthesis and design tools applica­ ble to sampled-data systems. The reader might therefore wonder what is distinc­ tive about the current book. Our observation of the existing literature is that the underlying continuous-time system is usually forgotten once the samples are tak­ en. The alternative point of view, adopted in this book, is to formulate the analy­ sis in such a way that the user is constantly reminded of the presence of the under­ lying continuous-time signals. We thus give emphasis to two aspects of sampled-data analysis: Firstly, we formulate the various algorithms so that the appropriate contin­ uous-time case is approached as the sampling rate increases. Secondly we place emphasis on the continuous-time output response rath­ er than simply focusing on the sampled response.

1 Fourier Analysis.- 1.1 Introduction.- 1.2 The Basic Transforms.- 1.2.1 Fourier Series.- 1.2.2 The Continuous-Time Fourier Transform.- 1.2.3 The Discrete-Time Fourier Transform (DTFT).- 1.2.4 The Discrete Fourier Transform (DFT).- 1.3 Properties of Continuous-Time Fourier Transforms.- 1.4 Properties of Discrete-Time Fourier Transforms.- 1.5 The ? ― Impulse Stream.- 1.6 Inter-relating the Various Transforms.- 1.6.1 Fourier Series Revisited.- 1.6.2 The Discrete-Time Fourier Transform Revisited.- 1.6.3 The Discrete Fourier Transform Revisited.- 1.6.4 Summary.- 1.7 Special Topics.- 1.7.1 Sampling of Sequences.- 1.7.2 Irregular Sampling in Periodic Patterns.- 1.8 Further Reading and Discussion.- 1.9 Problems.- 2 Sampling and Reconstruction.- 2.1 Introduction.- 2.2 Sampled Data Sequences ― A Representation of Continuous Signals.- 2.3 Continuous Signal Reconstruction from a Sampled Data Sequence.- 2.4 Shannon’ s Reconstruction Theorem.- 2.5 Practical Methods of Reconstruction.- 2.5.1 Zero-Order-Hold (ZOH).- 2.5.2 First-Order-Hold (FOH).- 2.6 Signal Reconstruction from Periodic Samples.- 2.7 Further Reading and Discussion.- 2.8 Problems.- 3 Analysis of Discrete-Time Systems.- 3.1 Introduction.- 3.2 Shift Operator Models.- 3.3 z-Transforms.- 3.4 The Delta Operator.- 3.5 Difference Equations in Delta Operator Form.- 3.6 Discrete Delta Transform.- 3.7 Use of Discrete Delta Transforms to Solve Difference Equations.- 3.8 The Discrete Transfer Function.- 3.9 Summary of Delta Transform Properties.- 3.10 Stability of Discrete Systems.- 3.11 Discrete Frequency Response.- 3.12 Frequency Domain Stability Criteria for Discrete-Time Systems.- 3.13 Digital Filter Implementation.- 3.14 Further Reading and Discussion.- 3.15 Problems.- 4 Discrete-Time Models of Continuous Deterministic Systems.- 4.1 Introduction.- 4.2 State-Space Development.- 4.3 Transform Development.- 4.4 Continuous-Time and Discrete-Time Poles and Zeros.- 4.4.1 Poles.- 4.4.2 Zeros.- 4.5 Numerical Issues.- 4.6 Frequency Domain Development.- 4.7 Further Reading and Discussion.- 4.8 Problems.- 5 Optimal Linear Estimation with Finite Impulse Response Filters.- 5.1 Introduction.- 5.2 Problem Description.- 5.3 Sampled Model.- 5.4 The Discrete Lattice Filter.- 5.5 Continuous-Time Lattice Structure.- 5.6 Relationships between the Discrete and Continuous Lattice Filters.- 5.7 Further Reading and Discussion.- 5.8 Problems.- 6 Optimal Linear Estimation with State-Space Filters.- 6.1 Introduction.- 6.2 Signal Model.- 6.3 The Sampling Process.- 6.4 Discrete Stochastic Model.- 6.5 The Discrete Kalman Filter.- 6.5.1 Model Simplification.- 6.5.2 The Optimal Filter.- 6.5.3 Relationship to Finite Impulse Response Filters.- 6.6 Continuous-Time State Estimation.- 6.6.1 Continuous-Time Data.- 6.6.2 Sampled Data.- 6.6.3 Relationships to Shannon Reconstruction Theorem.- 6.7 Further Reading and Discussion.- 6.8 Problems.- 7 Periodic and Multirate Filtering.- 7.1 Introduction.- 7.2 Models for Periodic Linear Systems.- 7.3 The Raising Procedure.- 7.4 Frequency Domain Analysis of Periodic Filters.- 7.5 Models for Sampled Periodic Stochastic Systems.- 7.6 Periodic Optimal Filtering.- 7.7 Further Reading and Discussion.- 7.8 Problems.- 8 Discrete-Time Control.- 8.1 Introduction.- 8.2 Closed-Loop Stability and Pole Assignment.- 8.3 Some Special Discrete-Time Control Laws.- 8.3.1 Deadbeat Control.- 8.3.2 Model Reference Control.- 8.3.3 Minimal Prototype Control.- 8.4 Sensitivity and Complementary Sensitivity Functions.- 8.5 All Stabilizing Control Laws.- 8.5.1 Open Loop Stable Plants.- 8.5.2 Open-Loop Unstable Plants.- 8.6 State Estimate Feedback.- 8.7 Rapprochement Between State Estimate Feedback and Stabilizing Controllers.- 8.7.1 Implicit Disturbance Modelling.- 8.7.2 Explicit Disturbance Modelling.- 8.8 Linear Quadratic Optimal Regulator.- 8.9 Duality Relationships.- 8.9.1 Filtering as a Control Problem.- 8.9.2 Associations.- 8.10 Further Reading and Discussion.- 8.11 Problems.- 9 Sampled Data Control.- 9.1 Introduction.- 9.2 Mixing Continuous and Discrete Transfer Functions.- 9.3 Sensitivity Considerations.- 9.4 Modified Discrete Transforms.- 9.5 Examples.- 9.5.1 Servo System with Minimal Prototype Controller.- 9.5.2 Servo System with Dead-Beat Control.- 9.5.3 Resonant System #1.- 9.5.4 Resonant System #2 with Anti-aliasing Filter.- 9.6 Observations and Comments from the Examples.- 9.7 The Class of All Stabilizing Sampled-Data Controllers.- 9.8 Linear Quadratic Design of Sampled-Data Controllers.- 9.9 Duality Relationships for Hybrid Optimal Controller.- 9.9.1 Review of Hybrid Optimal Controller.- 9.9.2 Review of Hybrid Optimal Filter.- 9.9.3 Duality.- 9.10 Further Reading and Discussion.- 9.11 Problems.- 10 Generalized Sample-Hold Functions.- 10.1 Introduction.- 10.2 Generalized Sample-Hold Function: A Time Domain Perspective.- 10.3 Other Applications of Generalized Sample-Hold Functions.- 10.4 Frequency Domain Analysis of GSHF.- 10.5 Sensitivity Considerations.- 10.6 Further Reading and Discussion.- 10.7 Problems.- 11 Periodic Control of Linear Time-Invariant Systems.- 11.1 Introduction.- 11.2 Periodic Control of Linear Time-Invariant Systems.- 11.3 Time Domain Analysis.- 11.4 Frequency Domain Analysis.- 11.5 Further Reading and Discussion.- 11.6 Problems.- 12 Multirate Control.- 12.1 Introduction.- 12.2 A Unifying Approach.- 12.3 Slow Output Sampling, Fast Input Sampling.- 12.4 Fast Output Sampling, Slow Input Sampling.- 12.5 Further Reading and Discussion.- 12.6 Problems.- 13 Optimal Control of Periodic Systems.- 13.1 Introduction.- 13.2 Control of Linear Periodic Systems.- 13.3 Control based on State Estimate Feedback.- 13.4 Further Reading and Discussion.- 13.5 Problems.