BIOSIGNAL PROCESSING Hee Chan Kim, Ph.D. Department of Biomedical Engineering College of Medicine Seoul National University
INTRODUCTION Biosignals (biological signals) : space, time, or space-time records of a biological event. electrical, chemical, and mechanical activities contain useful information that can be used to understand the underlying physiological mechanisms, which may be useful for medical diagnosis Biosignal Processing : a process to retrieve the useful information Acquisition and Analysis : Acquisition : a variety of sensors Common analysis methods : amplification, filtering, digitization, processing, storage Sophisticated processing methods : signal averaging, wavelet analysis, and artificial intelligent techniques
Characteristics of Biosignal mathematical representation where f = s(x,y,z,t :a,b,c,...) x,y,z,t... : independent variables, a,b,c : signal parameter(constant) f : signal magnitude(dependent variable) s : function representing signal highly dynamic nonstationarity a, b,c,.. & s are time-varying highly complex - large variability complex s with so many x,y,z,.. & a,b,c, A sinusoidal waveform : f = s(t) = Asin( t+ ) where t : time, A : amplitude, : frequency, : phase angle - / A -A f = s(t) = Asin( t+ ) t T = 1/f = 2 /
Dimensions in Biosignals 1D : f=s(t) 2D : f=s(x,y) 3D : f=s(x,y,z) 3D : f=s(x,y,t) 4D : f=s(x,y,z,t)
PHYSIOLOGICAL ORIGINS OF Bioelectric Signals : BIOSIGNALS Action potentials generated by excitable cells using intracellular or extracellular electrodes ECG, EGG, EEG, EMG, etc Biomagnetic Signals : Biomagnetism is the measurement of the magnetic signals that are associated with specific physiological activity typically linked to an accompanying electric field from a specific tissue or organ. (electromagnetism) SQUID (Superconducting Quantum Interference Device) magnetometer MEG, MNG, MGG, MCG, etc.
PHYSIOLOGICAL ORIGINS OF BIOSIGNALS Biochemical Signals Concentration of various chemical agents in the body Ions, po 2, pco 2, Biomechanical Signals Produced by mechanical functions of biological systems include motion, displacement, tension, force, pressure, and flow Bioacoustic Signals a special subset of biomechanical signals that involve vibrations (motion). Heart sound and respiratory sound measured by using acoustic transducers such as microphones and accelerometers. Biooptical Signals generated by the optical, or light-induced, attributes of biological systems. Fluorescence characteristics of the amniotic fluid (fetus health monitoring), Dye dilution method to measure CO, Oxygen saturation
CHARACTERISTICS OF BIOSIGNALS Classification according to various characteristics of signal waveform shape, statistical structure, and temporal properties Continuous vs Discrete signals : x(t) vs x(n) Deterministic vs Random (Stochastic) Mathematical functions : Statistical techniques : Stationary vs Nonstationary Periodic and Transient : x(t) = x(t+kt) Example : HRV (random) in ECG (periodic) x(t)=sin( t) y(t)=e -0.75t sin( t)
Biosignal representations Two independent windows to see one signal Time axis Frequency axis 1(Hz) = 1 cyclic change per 1 second
Periodic Signal Representation: The Trigonometric Fourier Series : fundamental frequency : harmonics Joseph Fourier initiated the study of Fourier series in order to solve the heat equation.
Example Problem Fourier Series
MATLAB Implementation (a) MATLAB result showing the first 10 terms of Fourier series approximation for the periodic square wave of Fig. 10.7a. (b) The Fourier coefficients are shown as a function of the harmonic frequency.
Compact Fourier Series The sum of sinusoids and cosine can be rewritten by a single cosine term with the addition of a phase constant; Example Problem
Exponential Fourier Series Euler s formula : Relationship to trigonometry : Proofs : using Talyor series,
Exponential Fourier Series Complex exponential functions are directly related to sinusoids and cosines; Euler s identities: Introduction of the negative frequencies The coefficient is a complex number. It requires only one integration. Example Problem
Harmonic Analysis Harmonic Analysis : the representation of functions or signals as the superposition of basic waves (harmonics) Fundamental Harmonics
Harmonics Analysis Figure 10.10 Harmonic coefficients of the aortic pressure waveform Figure 10.11 Harmonic reconstruction of the aortic pressure waveform.
Biosignal high frequency component
Biosignal time & freq : waveshape
Transition from Fourier Series to Fourier Transform Fourier Series Fourier Transform T, 0 =2 /T 0, m 0 t t Fourier Series Fourier Transform
Fourier Transform Fourier Integral or Fourier Transform; Used to decompose a continuous aperiodic signal into its constituent frequency components. X( ) is a complex valued function of the continuous frequency,. The coefficients c m of the exponential Fourier series approaches X( ) as T. Aperiodic function = a periodic function that repeats at infinity Example Problem
Properties of the Fourier Transform Linearity Time Shifting / Delay Frequency Shifting Convolution theorem Symmetry if f(t) is even and f(t) F( ), then F(t) f( )
Fourier Transform Pairs
Biosignal time & freq:equivalence Fourier Transform : 연속시간비주기신호의주파수변환 유한한구간에서 0 이아닌임의의비주기신호를한주기로하는주기신호의 Fourier Series 를구한다. 푸리에급수로표현된주기신호의주기를무한대로크게할때해당 Fourier Series 가점근적으로 Fourier Transform 으로접근 (t) F( ) 1 1 j t f ( t) F( ) e dt 2 0 1 t 0 F( ) j t F( ) f ( t) e dt 0 1 t 0 F( ) 0 0 t
Discrete Fourier Transform DTFT (Discrete Time Fourier Transform) : Fourier transform of the sampled version of a continuous signal; X( ) is a periodic extension of X ( ) - Fourier transform of a continuous signal x(t) ; Periodicity : Poisson summation formula*: *which indicates that a periodic extension of function samples of function can be constructed from the DFT (Discrete Frourier Transform) : Fourier series of a periodic extension of the digital samples of a continuous signal; N - 1
Discrete Fourier Transform Symmetry (or Duality) if the signal is even: x(t) = x(-t) then we have For example, the spectrum of an even square wave is a sinc function, and the spectrum of a sinc function is an even square wave. Extended Symmetry t Fourier Series t Fourier Transform t t Discrete Time Fourier Transform Discrete Fourier Transform
Discrete Fourier Transform fast Fourier transform (FFT) : an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the obvious way, using the definition, takes O(N 2 ) arithmetical operations, while an FFT can compute the same result in only O(NlogN) operations. Figure 10.12 (a) 100 Hz sine wave. (b) Fast Fourier transform (FFT) of 100 Hz sine wave. Figure 10.13 (a) 100 Hz sine wave corrupted with noise. (b) Fast Fourier transform (FFT) of the noisy 100 Hz sine wave.
LINEAR SYSTEMS A system is a process, machine, or a device that takes a signal as an input and manipulates it to produce an output that is related to, but is distinctly different from its input. Block diagram representation of a system All linear systems are characterized by the principles of superposition (or additivity) and scaling.
Time-Domain Representation of impulse response and convolution The impulse response of a system is its output when presented with a very brief input signal, an impulse. Convolution Linear System (Commutativity) Visual explanation of convolution.
Time-Domain Representation of Linear System t SYSTEM SYSTEM??? SYSTEM
Frequency-Domain Representation Transfer Function : of Linear System a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system.
Frequency-Domain Representation of Linear System Transfer Function : A Fourier transform of the impulse response.
Biosignal processing SENSOR Stimulus/ Measurand Transducible Property Principle of Transduction Electrical Output Detection Means Conversion Phenomenon Information PC Amplifier & Filter
Signal Processing in an Embedded System BioSystem measurand target Diagnostic Instrument Sensor Analog signal processing Electrical Signal A/D conversion Data Input Actuator Calibration Feedback Control Therapeutic Instrument Processor /Algorithm Information Main Control Unit Digital Signal Processing User Interface Display Storage Xmission
Signal Processing Patient/ Biological process Biosignal (signal) Electrical biosignal Signal transduction sensor interpreted signal (information) Signal acquisition digitized signal Signal transformation transformed signal (digital data) Parameter extraction Personal Computer signal parameters Signal classification Four stages of biosignal processing.
SIGNAL ACQUISITION Overview of Biosignal Data Acquisition Unwanted interference or noise : exogenous or endogenous High-precision low-noise equipment is often necessary to minimize the effects of unwanted noise the information and structure of the original biological signal of interest should be faithfully preserved. Electrical signal bioinstrumentation system
SIGNAL ACQUISITION T( j ) Sensors, Amplifiers, and Analog Filters Sensor should not adversely affect the properties and characteristics of the signal it is measuring Amplifier and Filter : OP Amp circuits To boost amplitude To compensate for distortion caused by the sensor To meet the specifications of the data acquisition system (analogto-digital converter) G f1 f2 f3 f4 f5 f6 Frequency - f 1 : HPF cut-off - f 2 : lower limit of biosignal freq. - f 3 : power-line freq. - f 4 : upper limit of biosignal freq. - f 5 : LPF cut-off & anti-aliasing - f 6 : A/D sampling freq
A/D Conversion Continuity :analog vs digital 아날로그신호 디지털신호
Conversions between Analog & Digital (1) A/D 변환 : 아날로그신호를디지털신호로변환 - 표본화 (sampling) : 시간축상의이산화 - 양자화 (quantization) : 진폭축상의이산화 (2) D/A 변환 : 디지털신호를아날로그신호로변환 디지털 아날로그
Sampling Theorem Sampling of Signals: How Often? - Nyquist Rate : 신호에포함된최대주파수성분 (f c ) 보다최소 2 배이상 (2f c ) 의빠르기로표본화하면정보의손실없이원래의신호를완벽하게복원할수있다. 시간축 주파수축
Aliasing-1 undersampling effect Spectrum Effect Waveform Effect
Aliasing - 2 Figure 10.6 A 360Hz sine wave is sampled every 5ms (i.e., at 200 samples/s). This sampling rate will adequately sample a 40Hz sine wave, but not a 360Hz sine wave.
Quantization Effect Sampling of Signals: How Accurate? Resolution of N-bit quantization = 2 N -1 steps (ex: 8 bit quantization for 1(V) signal = 1/255 = 3.92(mV) Resolution)
A/D Conversion QUIZ 외래환자로부터측정한 10초동안의근전도 (EMG) 신호를디지털데이터로변환하여 760MByte의 CD Rom으로저장하라 는 order에대해 CD Rom 1장에최대몇명의환자데이터를수록할수있을지계산하시오. ( 단, 근전도신호의최대주파수성분은 1.5(kHz) 이며, 근전계의출력은최대 1(V) 의크기신호를제공하는데향후분석목적상 250( μv ) 까지의분해능을요구한다.)