CIRCUITS, SYSTEMS, AND SIGNALS FOR BIOENGINEERS: A MATLAB-BASED INTRODUCTION John L. Semmlow ELSEVIER ACAUEMIC PRFSS AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
CONTENTS 1 BIOENGINEERING SIGNALS AND SYSTEMS 1 1.1 Biological Systems 1 1.2 Biosignals 3 1.2.1 Signal Encoding 6 1.3 Linear Signal Analysis: Overview 9 1.3.1 Analysis of Linear Systems 11 1.3.2 Analog Analysis and Analog Models 11 1.3.3 Systems Analysis and Systems Models 15 1.3.4 Systems and Analog Analysis: Summary 22 1.4 Noise and Variability 23 1.4.1 Electronic Noise 24 1.4.2 Signal-to-Noise Ratio 27 1.5 Summary 27 Problems 29 BASIC SIGNAL PROCESSING 31 2.1 Basic Signals: The Sinusoidal Waveform 31 2.1.1 Sinusoidal Arithmetic 34 2.1.2 Complex Representation 36 2.2 Signal Properties: Basic Measurements 38 2.2.1 Decibels 42 2.3 Advanced Measurements: Correlations and Covariances 44 2.3.1 Standard Correlation and Covariance 45 2.3.2 Autocorrelation and Cross-Correlation 48 2.4 MATLAB Implementation 54 2.4.1 Mean, Variance, and Standard Deviation 54 2.4.2 Ensemble Averaging 57 2.4.3 Covariance and Correlation 58
X CONTENTS 2.4.4 Autocorrelation and Cross-Correlation 61 2.5 Summary 65 Problems 66 FREQUENCY TRANSFORMATIONS 69 3.1 Useful Properties of the Sinusoidal Signal 72 3.2 Fourier Series Analysis 75 3.2.1 Symmetry 79 3.3 Frequency Representation 80 3.4 Compiex Representation 82 3.5 The Continuous Fourier Transform 85 3.6 Discrete Data: The Discrete Fourier Transform 87 3.6.1 Data Sampling: Sampling Theorem 88 3.6.2 Amplitude Slicing: Quantization (Optional) 93 3.6.3 Data Length: Truncation 95 3.7 Power Spectrum 97 3.7.1 Spectral Averaging 99 3.8 Signal Bandwidth 101 3.9 MATLAB Implementation 102 3.10 Summary 115 Problems 116 CIRCUIT AND ANALOG ANALYSIS IN SINUSOIDAL STEADY STATE 121 4.1 Circuits and Analog Systems 121 4.2 System Variables and Elements 123 4.2.1 Electrical Variables 125 4.2.2 Electrical Elements 127 4.2.3 Active Elements 135 4.2.4 The Fluid Analogy 137 4.3 Phasor Analysis 139 4.3.1 Phasor Representation: Electrical Elements 144 4.4 Mechanical Elements 147 4.4.1 Passive Elements 148 4.4.2 Elasticity 151 4.4.3 Sources 153 4.5 Summary 157 Problems 158
6 ANALYSIS OF ANALOG MODELS AND PROCESSES 1 61 5.1 Conservation Laws: Kirchhoff's Voltage Law 161 5.1.1 Mesh Analysis: Single Loops 162 5.1.2 Mesh Analysis: Multiple Loops 168 5.1.3 Mesh Analysis: MATLAB Implementation 171 5.2 Conservation Laws: Kirchhoff's Current Law Nodal Analysis 173 5.3 Conservation Laws: Newton's Law Mechanical Systems 178 5.4 Summary 186 Problems 187 FREQUENCY CHARACTERISTICS OF CIRCUITS AND ANALOG PROCESSES: THE TRANSFER FUNCTION 193 6.1 The Circuit or Mechanical System as a Process 193 6.1.1 Superposition 195 6.1.2 The Transfer Function 196 6.1.3 Transfer Function Characteristics 197 6.2 Transfer Function Frequency Plots: The Bode Plot 198 6.2.1 Frequency Characteristics of Bode Plot Primitives 201 6.3 Filters 213 6.3.1 Filter Types 214 6.3.2 Filter Bandwidth 215 6.3.3 Filter Order 216 6.3.4 Filter Initial Sharpness 217 6.3.5 Evaluating Filter Frequency Characteristics 217 63.6 Filter Design 219 6.4 MATLAB Implementation 221 6.4.1 Transfer Function 221 6.4.2 System Identification 227 6.4.3 The Transfer Function and Fourier Series Decomposition 230 6.5 Summary 234 Problems 234 RELATIONSH1PS BETWEEN ANALOG ELEMENTS 239 7.1 System Simplifications: Passive Network Reduction 239 7.1.1 Series Electrica! Elements 240 7.1.2 Parallel Elements 242 7.1.3 Network Reduction: Passive Networks 244 7.2 Ideal and Real Sources 252 7.2.1 The Voltage-Current or v-i Plot 252
xii CONTENTS 8 7.2.2 Real Voltage Sources: The Thevenin Source 255 7.2.3 Real Current Sources: The Norton Source 258 7.2.4 Thevenin and Norton Circuit Conversion 261 7.3 Thevenin and Norton Theorems: Network Reduction with Sources 264 7.4 Measurement Loading 269 7.4.1 Jdeal and Real Measurement Devices 270 7.4.2 Maximum Power Transfer 273 7.5 Mechanical Systems 275 7.6 Multiple Sources: Revisited 281 7.7 Summary 283 Problems 283 THE ANALYSIS OF TRANSIENTS: THE LAPLACE TRANSFORM 289 8.1 The Laplace Transform 289 8.1.1 Definition of the Laplace Transform 290 8.1.2 Laplace Representation of Elements: Calculus Operations in the Laplace Domain 292 8.1.3 Initial Conditions 293 8.1.4 Voltage-Current and Force-Velocity Relationships in the Laplace Domain 294 8.1.5 Sources: Common Signals in the Laplace Domain 296 8.1.6 Converting the Laplace Transform to the Frequency Domain 298 8.1.7 The Time-Delay Element 299 8.1.8 The Inversc Laplace Transform 300 8.2 Laplace Analysis: The Laplace Transfer Function 300 8.2.1 First-Order Processes 302 8.2.2 Second-Order Processes 306 8.3 Nonzero Initial Conditions 316 8.4 Initial and Final Value Theorems 320 8.5 The Laplace Domain and the Frequency Domain 321 8.6 Summary 330 Problems 330 SYSTEM MODELS AND BEHAVIOR 335 9.1 The System Model 335 9.1.1 Feedback 337 9.2 The Convolution Integral 340 9.2.1 M.ATLAB Implementation 343 9.3 Resonance 354 9.3.1 Resonant Frequency 355 9.3.2 Resonant Bandwidth, Q 355
CONTENTS xiii 9.4 Summary 365 Problems 367 10 BASIC ANALOG ELECTRONICS: OPERATIONAL AMPLIFIERS 371 10.1 The Amplifier 372 10.2 The Operational Amplifier 374 10.3 The Noninverting Amplifier 376 10.4 The Inverting Amplifier 379 10.5 Practical Operational Amplifiers 381 10.5.1 Limitations in Transfer Characteristics of Real Operational Amplifiers 382 10.5.2 Input Characteristics 389 10.5.3 Output Characteristics 396 10.6 Power Supply 398 10.7 Operational Amplifier Circuits, or 101 Things to Do with an Operational Amplifier 399 10.7.1 The Differential Amplifier 400 10.7.2 The Adder 401 10.7.3 The Buffer Amplifier 402 10.7.4 The Transconductance Amplifier 403 10.7.5 Analog Filters 405 10.7.6 Instrumentation Amplifier 407 10.8 Summary 411 Problems 411 APPENDIX A 415 A.l Derivation of Euler's Formula 415 A.2 Confirmation of the Fourier Series 416 A.3 Derivation of the Transfer Function of a Second-Order Op Amp Filter 417 A.4 Derivation of the Transfer Function of an Instrumentation Amplifier 41? APPENDIX B Laplace Transforms 421 APPENDIX C Trigonometrie and Other Formulas 423
xiv CONTENTS APPENDIX D Units 425 APPENDIX E Complex Arithmetic 429 E.l.l Addition and Subtraction 430 E.1.2 Multiplication and Division 430 APPENDIX F LF 356 Specifications 433 APPENDIX G Determinants and Cramer's Ruie 435 BIBLIOCRAPHY 437 INDEX 439