Signals and systems analysis in biomedical engineering (1)

1557 Cover 2/10/03 1:38 PM Page C Composite M Y CM MY CY CMY K SIGNALS and SYSTEMS ANALYSIS in BIOMEDICAL ENGINEERING Biomedical Engineering Series Edited by Michael R Neuman Published Titles Electromagnetic Analysis and Design in Magnetic Resonance Imaging, Jianming Jin Endogenous and Exogenous Regulation and Control of Physiological Systems, Robert B Northrop Artificial Neural Networks in Cancer Diagnosis, Prognosis, and Treatment, Raouf N.G Naguib and Gajanan V Sherbet Medical Image Registration, Joseph V Hajnal, Derek Hill, and David J Hawkes Introduction to Dynamic Modeling of Neuro-Sensory Systems, Robert B Northrop Noninvasive Instrumentation and Measurement in Medical Diagnosis, Robert B Northrop Handbook of Neuroprosthetic Methods, Warren E Finn and Peter G LoPresti Signals and Systems Analysis in Biomedical Engineering, Robert B Northrop Forthcoming Titles Angiography and Plaque Imaging: Advanced Separation Techniques, Jasjit S Suri The BIOMEDICAL ENGINEERING Series Series Editor Michael Neuman SIGNALS and SYSTEMS ANALYSIS in BIOMEDICAL ENGINEERING Robert B Northrop CRC PR E S S Boca Raton London New York Washington, D.C 1557-discl Page Monday, February 10, 2003 5:05 PM Library of Congress Cataloging-in-Publication Data Northrop, Robert B Signals and systems analysis in biomedical engineering / Robert B Northrop p cm Includes bibliographical references and index ISBN 0-8493-1557-3 (alk paper) Biomedical engineering System analysis I Title R856.N58 2003 610′.28—dc21 2002191167 CIP This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S Government works International Standard Book Number 0-8493-1557-3 Library of Congress Card Number 2002191167 Printed in the United States of America Printed on acid-free paper Dedication I dedicate this text to my wife, Adelaide, whose encouragement catalyzes my inspiration Preface This text is intended for use in a classroom course on signals and systems analysis in biomedical engineering taken by undergraduate students specializing in biomedical engineering It will also serve as a reference book for biophysics and medical students interested in the topics Readers are assumed to have had introductory core courses up to the junior level in engineering mathematics, including complex algebra, calculus and introductory differential equations They also should have taken introductory human (medical) physiology and biomedical engineering After taking these courses, readers should be familiar with systems block diagrams, the concepts of frequency response and transfer functions, and should be able to solve simple, linear, ordinary differential equations and basic manipulations in linear algebra It is also important to have an understanding of how the physiological signals and systems being characterized figure in human health The interdisciplinary field of biomedical engineering is demanding in that it requires its followers to know and master not only certain engineering skills (electronic, materials, mechanical and photonic), but also a diversity of material in the biological sciences (anatomy, biochemistry, molecular biology, genomics, physiology etc.) Tying these diverse disciplines together is a common reticulum of mathematical skills characterized by both breadth and specialization This text was written to aid undergraduate biomedical engineering students by helping them to strengthen and understand this common network of applied mathematics, as well as to provide a ready source of information on the specialized mathematical tools and techniques most useful in describing and analyzing biomedical signals (including, but not limited to: ECG, EEG, EMG, ERG, heart sounds, breath sounds, blood pressure, tomographic images etc.) Of particular interest is the description of signals from nonstationary sources using the many algorithms for computing joint time-frequency spectrograms The text presents the traditional systems mathematics used to characterize linear, time-invariant (LTI) systems, and, given inputs, to find their outputs The relations between impulse response, real convolution, transfer functions and frequency response functions are explained Also, some specialized mathematical techniques used to characterize and model nonlinear systems are reviewed It is the very nature of living organisms that signals derived from them are noisy and nonstationary That is, the parameters of the nonlinear systems giving rise to the signals change with time There are many causes for nonstationary signals in biomedical systems: One is circadian rhythm, another is the action of drugs, another involves inherent periodic rhythms such as those associated with breathing or the heart’s beating, and still other nonstationarity can be associated with natural processes such as the digestion of food or locomotion Because nature has implemented many physiological systems with parallel architectures for redundancy and reliability, when recording from one “channel” of one system, one is likely to pick up the “cross-talk” from other channels as noise (e.g., in EMG recording) Also, many bioelectric signals are in the microvolt range, so electrode, amplifier and environmental noises are often significant compared with the signal level This text introduces the basic mathematical tools used to describe noise and how it propagates through LTI and NLTI systems It also describes at a basic level how signal-to-noise ratio can be improved by signal averaging and linear and nonlinear filtering Bandwidths associated with endogenous (natural) biomedical signals range from dc (e.g., hormone concentrations or dc potentials on the body surface) to hundreds of kilohertz (bat ultrasound) Exogenous signals associated with certain noninvasive imaging modalities (e.g., ultrasound, MRI) can reach into the 10s of MHz It is axiomatic that the large physiological systems are nonlinear and nonstationary, although early workers avoided their complexity by characterizing them as linear and stationary Nonstationarity can generally be ignored if it is slow compared with the time epoch over which data is acquired Nonlinearity can arise from the concatenated chemical reactions underlying physiological system function (there are no negative concentrations) The coupled ODEs of mass-action kinetics are generally nonlinear, which makes system characterization a challenge Other nonlinearities arise in the signal processing properties of the nervous system By considering the system behavior in a limited parameter space around an operating point, some systems can be linearized Such piecewise linearization is often an over-simplification that obscures the detailed understanding of the system It is important to eschew reductionism when analyzing and describing physiological and biochemical systems The text was written based on both the author’s experience in teaching EE 202 Signals and Systems, EE 232 Systems Analysis, EE 271 Physiological Control Systems, and EE 372, Communications and Control in Physiological Systems for over 30 years in the Electrical and Computer Engineering Department at the University of Connecticut, and on his personal research in biomedical instrumentation and on certain neurosensory systems Signals and Systems Analysis in Biomedical Engineering is organized into 10 chapters, plus an Index, a wide-ranging Bibliography and four Appendices Extensive chapter examples based on problems in biomedical engineering are given The chapter contents are summarized below: • Chapter 1, Introduction to Biomedical Signals and Systems, sets forth the general characteristics of biomedical signals and the general properties of physiological systems, including nonlinearity and nonstationarity, are examined Also reviewed are the various means of modulating (and demodulating) signals from physiological systems Discrete signals and systems are also introduced • Chapter 2, Review of Linear Systems Theory, formally presents the concepts of linearity, causality and stationarity Linear time-invariant (LTI) dynamic analog systems are introduced and shown to be described by sets of ordinary differential equations (ODEs) General solutions of first- and second-order linear ODEs are covered The basics of linear algebra are introduced and the solution of sets of simultaneous ODEs by the state variable method is presented In characterizing LTI systems, the concepts of system impulse response, real convolution, general transient response, and steady-state sinusoidal frequency response are covered, including Bode and Nyquist plots Chapter also treats discrete systems and signals, including difference equations and the use of the z-transform and discrete state equations Finally, the factors that affect the stability of systems and review certain stability tests are described • In Chapter 3, The Laplace Transform and Its Applications, the Laplace transform is defined and its mathematical properties are presented Many examples are given of finding the Laplace transforms of transient signals, including causal LTI system impulse responses Examples of the use of the Laplace transform to find the transient output of a causal LTI system given a transient input are given and the inverse Laplace transform is introduced Real convolution of a system’s impulse response with its input to find its output, y(t), in the time domain is shown to be equivalent to the Laplace transform of the output, Y(s), being equal to the product of the Laplace transforms of the input and the impulse response The partial fraction expansion is shown to be an effective method for finding y(t), given Y(s) Solution of state equations in the frequency domain using the Laplace transform method is given • Chapter 4, Fourier Series Analysis of Periodic Signals, defines the real and complex forms of the Fourier series (FS) and the mathematical properties of the FS are presented Gibbs phenomena are shown to persist even as the number of harmonic terms → ∞, but their area → Several examples of finding the FS of periodic waveforms are given • The Continuous Fourier Transform is derived from the FS in Chapter The (CFT) is seen to be equivalent to the Laplace transform for many applications, but the radian frequency ω is real, while s is complex The properties of the CFT are presented and the IFT is introduced Several applications of the CFT are given; the periodic spectrum of a sampled analog signal is derived in the Poisson sum form, and the sampling theorem is presented Next, the generation of the analytical signal is derived using the Hilbert transform and applications are given Finally, the modulation transfer function (MTF) is defined as the normalized spatial frequency response of an imaging system Properties of the MTF are explored, as well as its significance in image resolution The relation of the contrast transfer function (CTF) for a 1-D square-wave object to the MTF is discussed In addition, Section 5.4 describes the analytical signal and the Hilbert transform and some of its biomedical applications • In Chapter 6, The Discrete Fourier Transform, the DFT and IDFT are compared with the CFT and the ICFT and their properties are described Data window functions for finite sampled data sets are introduced and how they affect spectral resolution is demonstrated Finally, the computational advantages of the FFT are described and several examples are given of FFT implementation The Mathematics of Tomographic Imaging 389 MatlabTM offers an enormous collection of object programs or executable *.m files, which enable an wide range of system simulation and signal analysis of LTI systems and signals *.m files are organized by application into collections known as “toolboxes.” To facilitate system analysis, the 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Coding 270, Tables 8.5.1 & 8.5.2 Analog-to-Digital Conversion 139 Analytical Signal 142 Autocorrelation Function, Continuous 229–231 Autocorrelation Function, Discrete 255 Auto-Power Density Spectrum 234 B Bayes’ Theorem 277 Bessel Functions (in FM) Binomial Distribution 278 Bioinformatics 265 Bode Plots: 56–60, App C Cross-Correlation Function 231–234 Cross-Power Density Spectrum 235 D Data Scrubbing 249– Deconvolution 51 Delta Function (aka unit impulse) 45 Describing Functions: 322– Table 326, 327 Diffusion: 303– Carrier-mediated 303 Inhomogeneous equation for 305 Ligand-gated 304 Voltage-gated 304 DNA (structure): 266– Codon 270, Tables 8.5.1, 8.5.2 Exon 270 Intron 270 E C Characteristic Functions 240–242 Cis-Regulatory Elements (in intron) 270 Codon 270 Coherence Function 333 Combinations (equation) 277 Compartment 35, 308 Contrast 155 Contrast Transfer Function 156–158 Convolution: Discrete 68 Complex 76, 137 Real 46–51 Ensemble Averaging 257 Ergodic (noise) 226, 237 Error Function 227 Expected Value (probability average) 228 F Fast Fourier Transform (FFT) 179–186 Fick’s Law 303–305 Filtering, Continuous 261–264 Filtered Back-Projection Algorithm (FBPA) 361–364 399 400 Fourier Transform, Continuous: 135– Properties: Table 5.1 Fourier Transform, Discrete: 165– Examples 167–171 Fast (FFT) 182–187 Properties: 166–167, Table 6.1 Fourier Series: 123– Gibbs phenomena 124 Examples 126–130 Properties of: 125, Table 4.1 Fourier Slice Theorem 358–361 Index J Genome 256 Genomics 265 Joint Time-Frequency Analysis: 191– Applications in Biomedicine 211–221 Choi-Williams 202–203 Cohen’s Class 201–204 Cone-shaped 201 Gabor & Adaptive Gabor 196–197 Reduced Interference 203 Short-term Fourier 194–196, 200 Smoothed Wigner 203 Software 221–224 Spectrogram 191 Wavelet-based JTFTs 204–210 Wigner-Ville & Pseudo-Wigner 197–202 H L Hempel Filter 250–252 Hidden Markov Model (in genomics) 284–287 Hilbert Transform: 142– Application 148–151 Properties 144 Table of: 145 Hill Function 302 Hydrogen Bonds (in DNA & RNA) 269, Fig 8.15 Laplace Transforms: 95– Application to continuous state systems 107 Convergence 95 Delay operator 63 Inverse 100–101 Partial fractions in finding inverse 103–107 Properties 97–98 Table of: 102 Limit Cycles in Nonlinear Feedback Systems 302, 332 G I Implicit Summing Point 299 Impulse Modulation & Sampling 139–140 Impulse Response 45 Information Theory in Genomics 279–284 Intensity Irradiance M Mass-Action Kinetics (examples) 306–310 Matrix: 36– Addition 38 Adjoint 37 Cofactor 38 Conformable Determinant 37 Index Matrix (Continued) Diagonal 37 Identity 39 Inverse 39–40 Multiplication 38 Singular 37 State transition 41 Time-variable 40 Transpose 37 Michaelis-Menten Reaction Architecture 307 Modulation & Demodulation: Adaptive delta modulation 15 AM AMSSB Angle Delta 12 DSBCM FM 6, IPFM 17 Narrow-band FM (NBFM) Phase modulation (PhM) 6, 11 RPFM 18 Modulation Transfer Function (MTF) 153–156 N Noise-based system characterization 332–335 Noise Factor (of averaging process) 259–261 Noise Figure 259 Nonlinear Systems (properties) 301–303 Nonstationary Nucleoside 268 Nucleotide 268 Nyquist Plots 60–62 Nyquist Stability Criterion 311–322 O Operating Point 300 Operator Notation (for derivatives) 31 401 Ordinary Differential Equation (ODE) 20, 28–35 P Parametric Control 298 Partial Fractions 101, 103–107 Permutations (equation) 278 Physiological Systems (properties) 20, 297–299 Point-spread Function 152 Poisson Sum 139–141 Power Density Spectrum 234 Price’s Theorem 242–244 Probability Density Functions: 226– Gaussian 226 Maxwell 227 Rectangular 226 Rayleigh 227 Propagation of Noise Through Stationary, Causal, Continuous, LTI Systems 236 Propagation of Noise Through Stationary, Causal, Discrete, LTI Systems 237–240 Purines Fig 8.17 Pyrimidines Fig 8.17 Q Quadratic Characteristic Equation: 31 Types of Roots 31–33 Quantization Noise 245–249 Quantizer, 8-bit: Fig 8.7 R Radon Transform 353–358 Rectangular Integrator 66 RNA: 271– Messenger 272–275 Ribosomal 272–275 Transfer 272–275, Fig 8.18 402 Index S T Sample Mean 228, 254 Sample Mean Square 254 Sample Mean Variance 254 Sampling Theorem 141–142 Scale-Invariant Filter 250 Shannon Measure of Average Information 280 Signal Averaging 257–261, Fig 8.11 Signal-Dependent, Rank-Order Mean Data Scrubbing Filter 252–253 Signal Flow Graphs 108–112, App B Simpson’s Rule Integration 68 Sinogram 356–360 Sinusoidal Steady-State 52 Spliceosome 270 State Equations, continuous 41–45 State Equations, discrete 82–83 Static Nonlinearity (example) 302 Stationary (noise) 226 Stem Cells 266 Suprisal 280 Systems: Causal 28 Discrete 62–74 Impulse response 45 Linear time-invariant (LTI) 28 Multiple-input, multiple-output (MIMO) 28 Nonlinear 301 Single-input, single output (SISO) 28 Stability 83–85 Steady-state, sinusoidal frequency response 52–55, App C Transient response 51 Time-Frequency Analysis (see Joint TFA) Tomography: 347– Algebraic Reconstruction Technique (ART) 351–353 Directional Sensitivity Function 348 Emission 347 Line of Response (LOR) 348 MRI 347 Scintillation Camera Fig 10.2 Transmission 347 Trapezoidal Integration 67 Trits 281 Type Closed-Loop System 298 V Variance 228 W Weighting Function 46, 233 White Noise Used in LTIS Identification 333 Wiener Kernel Method of NLS Identification 333–336 Window Functions 172–179, Table 6.2, Fig 6.3 Z z -Transform Pair: 64–68, 74–81, Table 2.2 Pairs & Theorems Table 2.2 Properties Table 2.3 Solution of discrete state equations 82–83 1557 Cover 2/10/03 1:38 PM Page C Composite M Y CM MY CY CMY K ... B Signals and systems analysis in biomedical engineering / Robert B Northrop p cm Includes bibliographical references and index ISBN 0-8493-1557-3 (alk paper) Biomedical engineering System analysis. . .SIGNALS and SYSTEMS ANALYSIS in BIOMEDICAL ENGINEERING Biomedical Engineering Series Edited by Michael R Neuman Published Titles Electromagnetic Analysis and Design in Magnetic Resonance... inspiration Preface This text is intended for use in a classroom course on signals and systems analysis in biomedical engineering taken by undergraduate students specializing in biomedical engineering