BIOSIGNAL PROCESSING

Hee Chan Kim, Ph.D.Department of Biomedical Engineering

College of MedicineSeoul 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

A

-A

T = 1/f = 2/

t

f = s(t) = Asin(t+)

- /

• mathematical representationf = s(x,y,z,t…:a,b,c,...)

where 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

1D : f=s(t) 2D : f=s(x,y) 3D : f=s(x,y,z)

Dimensions in Biosignals

4D : f=s(x,y,z,t) 3D : f=s(x,y,t)

PHYSIOLOGICAL ORIGINS OF

BIOSIGNALS• Bioelectric Signals :

– 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, pO2, pCO2,

• 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-0.75tsin(t)

Biosignal – representations

Frequency axis

Time axis

Two independent windows to see one signal

1(Hz) = 1 cyclic change per 1 second

Periodic Signal Representation:

The Trigonometric Fourier Series

Joseph Fourier initiated the study of Fourier series in order to solve the heat equation.

: fundamental frequency

: harmonics

Fourier Series

• Example Problem

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:

• Example Problem

It requires only one integration.

Introduction of the negative frequencies

The coefficient is a complex number.

Fundamental

Harmonics

Harmonic Analysis

Harmonic Analysis : the representation of functions or signals as the superposition of basic waves (harmonics)

Harmonics Analysis

Figure 10.11 Harmonic reconstruction of the aortic pressure waveform.

Figure 10.10 Harmonic coefficients of the aortic pressure waveform

Biosignal – high frequency component

Biosignal – time & freq : waveshape

Transition from Fourier Series to

Fourier Transform

T→, 0=2/T →0,

m0→

Fourier Series Fourier Transform

t

Fourier Series

t

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 cm 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 : 연속시간 비주기 신호의 주파수 변환 유한한 구간에서 0이 아닌 임의의 비주기신호를 한 주기로 하는

주기신호의 Fourier Series를 구한다. 푸리에급수로 표현된 주기신호의 주기를 무한대로 크게 할 때 해당

Fourier Series가 점근적으로 Fourier Transform으로 접근

( ) ( ) j tF f t e dt

1

( ) ( )2

j tf t F e dt

0

(t)

t

0

1

t 0

|F()|

0

|F()|1

0 t

|F()|

0

1

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*:

N - 1

• DFT (Discrete Frourier Transform) : Fourier series of a periodic extension of the digital samples of a continuous signal;

*which indicates that a periodic extension of function can be constructed from the samples of function

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

t t

Fourier Series

Discrete Time Fourier Transform Discrete Fourier Transform

t 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(N2) 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 thenoisy 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.

• All linear systems are characterized by the principles of superposition (or additivity) and scaling.

Block diagram representation of a system

Time-Domain Representation of

Linear System

• impulse response andconvolution

– The impulse response of a system is its output when presented with a very brief input signal, an impulse.

– Convolution

(Commutativity)Visual explanation of convolution.

Time-Domain Representation of

Linear System

SYSTEMt

SYSTEM

SYSTEM ???

Frequency-Domain Representation

of Linear System• Transfer Function :

– 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.

Amplifier&

Filter

Biosignal – processing

Stimulus/Measurand

Transducible

Property

Principle of

Transduction

DetectionMeans

ConversionPhenomenon

SENSOR

ElectricalOutput

InformationPC

Sensor Analog signalprocessing A/D conversion

Main Control UnitDigital Signal Processing

Data Input

User Interface

Calibration

FeedbackControl

Actuator

BioSystem

measurand

Display Storage Xmission

Signal Processing in an Embedded System

target

Processor /Algorithm

Diagnostic Instrument

Therapeutic Instrument

Electrical Signal

Information

sensor

Personal Computer

Signal

transduction

Signal

acquisition

Signal

transformation

Parameter

extraction

Signal

classification

digitizedsignal

transformedsignal

signalparameters

interpretedsignal

(digital data)

(information)

Patient/Biological

process

Four stages of biosignal processing.

Biosignal

Electrical biosignal

Signal Processing

(signal)

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.

bioinstrumentation system

Electrical signal

SIGNAL ACQUISITION

• 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 (analog-to-digital converter)

Frequency

( )T j

1f 2f 3f 4f 5f 6f

G

- f1 : HPF cut-off

- f2 : lower limit of biosignal freq.

- f3 : power-line freq.

- f4 : upper limit of biosignal freq.

- f5 : LPF cut-off & anti-aliasing

- f6 : A/D sampling freq

• Continuity :analog vs digital

아날로그신호

디지털신호

A/D Conversion

디지털

아날로그

(1) A/D 변환 : 아날로그 신호를 디지털 신호로 변환

- 표본화(sampling) : 시간축 상의 이산화- 양자화(quantization) : 진폭축 상의 이산화

(2) D/A 변환 : 디지털 신호를 아날로그 신호로 변환

Conversions between Analog & Digital

Sampling Theorem

시간축 주파수축

- Nyquist Rate : 신호에 포함된 최대 주파수 성분(fc)보다최소 2배 이상(2fc)의 빠르기로 표본화하면 정보의 손실없이 원래의 신호를 완벽하게 복원할 수 있다.

Sampling of Signals: How Often?

Aliasing-1

Spectrum Effect Waveform Effect

undersampling 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

Resolution of N-bit quantization = 2N-1 steps(ex: 8 bit quantization for 1(V) signal = 1/255

= 3.92(mV) Resolution)

Sampling of Signals: How Accurate?

A/D Conversion

QUIZ

외래 환자로부터 측정한 10초 동안의 근전도(EMG) 신호를 디지털 데이터

로 변환하여 760MByte의 CD Rom으로 저장하라”는 order에 대해 CD

Rom 1장에 최대 몇 명의 환자 데이터를 수록할 수 있을지 계산하시오.

(단, 근전도 신호의 최대 주파수 성분은 1.5(kHz)이며, 근전계의 출력은 최

대 1(V)의 크기 신호를 제공하는데 향후 분석 목적상 250(㎶) 까지의 분해

능을 요구한다.)