Biomedical Signal Processing and Signal Modeling

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1 Biomedical Signal Processing and Signal Modeling Eugene N. Bruce University of Kentucky A Wiley-lnterscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

Contents Preface Chapter 1 The Nature of Biomedical Signals 1 1.1 The Reasons for Studying Biomedical Signal Processing 1 1.2 What Is a Signal? 3 1.3 Some Typical Sources of Biomedical Signals 5 1.

2 Contents Preface Chapter 1 The Nature of Biomedical Signals The Reasons for Studying Biomedical Signal Processing What Is a Signal? Some Typical Sources of Biomedical Signals Continuous-Time and Discrete-Time Assessing the Relationships Between Two Signals Why Do We "Process" Signals? Types of Signals: Deterministic, Stochastic, Fractal and Chaotic Signal Modeling as a Framework for Signal Processing What Is Noise? Summary 22 Exercises 23 Chapter 2 Memory and Correlation Introduction Properties of Operators and Transformations Memory in a Physical System Energy and Power Signals The Concept of Autocorrelation Autocovariance and Autocorrelation for DT Signals 50 xi v

VI CONTENTS 2.7 Summary 55 Exercises 55 Chapter 3 The Impulse Response 63 3.1 Introduction 63 3.2 Thought Experiment and Computer Exercise: Glucose Control 65 3.

3 VI CONTENTS 2.7 Summary 55 Exercises 55 Chapter 3 The Impulse Response Introduction Thought Experiment and Computer Exercise: Glucose Control Convolution Form of an LSI System Convolution for Continuous-Time Systems Convolution as Signal Processing Relation of Impulse Response to Differential Equation Convolution as a Filtering Process Impulse Responses for Nonlinear Systems The Glucose Control Problem Revisited Summary 100 Exercises 101 Chapter 4 Frequency Response Introduction Biomedical Example (Transducers for Measuring Клее Angle) Sinusoidal Inputs to LTIC Systems Generalized Frequency Response Frequency Response of Discrete-Time Systems Series and Parallel Filter Cascades Ideal Filters Frequency Response and Nonlinear Systems Other Biomedical Examples Summary 138 Exercises 139 Chapter 5 Modeling Continuous-Time Signals as Sums of Sine Waves Introduction Introductory Example (Analysis of Circadian Rhythm) Orthogonal Functions Sinusoidal Basis Functions The Fourier Series 152

CONTENTS VII 5.6 The Frequency Response and Nonsinusoidal Periodic Inputs 161 5.7 Parseval's Relation for Periodic Signals 163 5.8 The Continuous-Time Fourier Transform (CTFT) 164 5.

4 CONTENTS VII 5.6 The Frequency Response and Nonsinusoidal Periodic Inputs Parseval's Relation for Periodic Signals The Continuous-Time Fourier Transform (CTFT) Relationship of Fourier Transform to Frequency Response Properties of the Fourier Transform The Generalized Fourier Transform Examples of Fourier Transform Calculations Parseval's Relation for Nonperiodic Signals Filtering Output Response via the Fourier Transform Summary 185 Exercises 186 Chapter 6 Responses of Linear Continuous-Time Filters to Arbitrary Inputs Introduction Introductory Example Conceptual Basis of the Laplace Transform Properties of (Unilateral) Laplace Transforms The Inverse (Unilateral) Laplace Transform Transfer Functions Feedback Systems Biomedical Applications of Laplace Transforms Summary 225 Exercises 226 Chapter 7 Modeling Signals as Sums of Discrete-Time Sine Waves Introduction Interactive Example: Periodic Oscillations in the Amplitude of Breathing The Discrete-Time Fourier Series Fourier Transform of Discrete-Time Signals Parseval's Relation for DT Nonperiodic Signals Output of an LSI System Relation of DFS and DTFT Windowing 250

VIII CONTENTS 7.9 Sampling 255 7.10 The Discrete Fourier Transform (DFT) 261 7.11 Biomedical Applications 267 7.12 Summary 271 Exercises 272 Chapter 8 Noise Removal and Signal Compensation 279 8.

5 VIII CONTENTS 7.9 Sampling The Discrete Fourier Transform (DFT) Biomedical Applications Summary 271 Exercises 272 Chapter 8 Noise Removal and Signal Compensation Introduction Introductory Example: Reducing the ECG Artifact in an EMG Recording Eigenfunctions of LSI Systems and the Z-Transform Properties of the Bilateral Z-Transform Poles and Zeros of Z-Transforms The Inverse Z-Transform Pole Locations and Time Responses The Unilateral Z-Transform Analyzing Digital Filters Using Z-Transforms (DT Transfer Functions) Biomedical Applications of DT Filters Overview: Design of Digital Filters IIR Filter Design by Approximating a CT Filter IIR Filter Design by Impulse Invariance IIR Filter Design by Bilinear Transformation Biomedical Examples of IIR Digital Filter Design IIR Filter Design by Minimization of an Error Function FIR Filter Design Frequency-Band Transformations Biomedical Applications of Digital Filtering Summary 338 Exercises 339 Chapter 9 Modeling Stochastic Signals as Filtered White Noise Introduction Introductory Exercise: EEG Analysis Random Processes Mean and Autocorrelation Function of a Random Process Stationarity and Ergodicity 356

CONTENTS IX 9.6 General Linear Processes 363 9.7 Yule-Walker Equations 369 9.8 Autoregressive (AR) Processes 373 9.9 Moving Average (MA) Processes 381 9.

6 CONTENTS IX 9.6 General Linear Processes Yule-Walker Equations Autoregressive (AR) Processes Moving Average (MA) Processes Autoregressive-Moving Average (ARMA) Processes Harmonic Processes Other Biomedical Examples Introductory Example Continued Summary 391 Exercises 393 Chapter 10 Scaling and Long-Term Memory Introduction Geometrical Scaling and Self-Similarity Measures of Dimension Self-Similarity and Functions of Time Theoretical Signals Having Statistical Similarity Measures of Statistical Similarity for Real Signals Generation of Synthetic Fractal Signals Fractional Differencing Models Biomedical Examples Summary 430 Exercises 432 Chapter 11 Nonlinear Models of Signals Introductory Exercise Nonlinear Signals and Systems: Basic Concepts Poincare Sections and Return Maps Chaos Measures of Nonlinear Signals and Systems Characteristic Multipliers and Lyapunov Exponents Estimating the Dimension of Real Data Tests of Null Hypotheses Based on Surrogate Data Other Biomedical Applications Summary 480 Exercises 481 Chapter 12 Assessing Stationarity and Reproducibility Introduction 485

X CONTENTS 12.2 Assessing Stationarity of a Random Process from a Sample Function 486 12.3 Statistical Properties of Autocovariance Estimators 490 12.

7 X CONTENTS 12.2 Assessing Stationarity of a Random Process from a Sample Function Statistical Properties of Autocovariance Estimators Statistical Properties of the Periodogram Analysis of Nonstationary Signals Nonstationary Second-Order Statistics Summary 508 Exercises 509 Bibliography 511 Index 517

Continuous and Discrete Time Signals and Systems

Continuous and Discrete Time Signals and Systems Mrinal Mandal University of Alberta, Edmonton, Canada and Amir Asif York University, Toronto, Canada CAMBRIDGE UNIVERSITY PRESS Contents Preface Parti Introduction