Biomedical Signal Processing

Typical Biomedical Signal

EEG

• Dr. Has Berger firstly recorded EEG signal in 1926
• Clinical EEG

• EEG signal is generated by Electro Neurophysiologic Signal

  • Spike
  • LFP: Local Field Potential
    • sEEG: stero-electroencephalography
  • ECog/iEEG
  • EEG (Non Invasive)
  • ERP

• Scalp EEG

  • history

    • 1926, Hans Berger-EEG
    • 1933, Adrain verified EEG in Cambridge
    • 1958, Dawson designed EP University of London
    • 1967, Sutton-P300-ERP

  • Classification

    • Spontaneous EEG
    • Evoked Potentials: External Stimulations
    • Event-Related Potentials

  • EEG Lead System: International 10-20 System

    • Bipolar
    • Unipolar

• EEG

  • EEG Sensor
  • Amplifier
  • Filter
  • A/D
  • Artifacts Rejection
  • EEG
  • ERP (from ERP)
  • Grand Average
  • Data analysis

• Evoked Potentials

  • VEP(Visual EP)
    • F-VEP
    • P-VEP

• ERPs: Event-related potential

  • Potential change when voking and revoking stimulations

• Basic Principle of ERPs Extraction

  • EEG covers ERPs
  • Data Averaging
  • ERP characteristic
    • Constant waveform
    • Constant incubation period
  • The EEG obtained on several trials can be averaged together time-locked to the stimulus to form an event-related potential(ERP)

• ECG

  • Collecting/Recording
    • History:
      • Augustus D. Waller (1856-1922) recorded the first human “electric potential of the heartbeat” using a capillary electrometer (1887)
      • Willem Eintowen (1860-1927) developed and Improved the string galvanometer(1901); established the theory and application to heart disease; Named PQRST(U) waves; Invented ECG leads
  • Theory of the Electrocardiogram (ECG)
    • The electrical properties of our heat can be measured non-invasively
    • ECG Principle:
      • P wave: Depolarization of the Atria
      • QRS complex: Depolarization of the ventricles
      • T wave: Repolarization of the ventricles

• EMG

  • The electral source is the muscle membrane potential of about 90 mV.
  • Surface EMG

• Others

  • Ultrasonic
  • Heart Rate

• Signal

  • Message, Information, Signal
  • Classification

    • Continuous and discrete

      • Continuous time signal
      • Discrete time signal (Series)

    • Analog and digital

      • Analog: continuous on time and amplitude
      • Digital: discrete on time and amplitude

    • Determinate and random

      • Determinate Signal: $x(t)$ could be expressed in a time function
        • Periodic signal: $x(t)=x(nT+t)$
        • Aperiodic
      • Random Signal: a stochastic signal with whole uncertainty
        • Stationary Stochastic Signal
        • Non Stationary Stochastic Signal

    • Finite energy signal and finite power signal

    • fractal, chaotic signal

    • Classification of signals

      • Signal
        • Deterministic
          • Periodic
            • Sino-soicial
            • Complex
          • Nonperiodic
            • Almost Periodic
            • Transient
        • Stochastic
          • Stationary
            • Ergodic
            • Non-Ergodic
          • Nonstationary
            • Special type

• Characteristics of Biomedical Signal

  • Weak signal (mv/uV)
    • Biomedical signal measurement is weak signal measurement
    • Measuring Requirements: high sensitivity, high resolution, noise superession, anti-interference
  • Strong Interference
  • Low frequency, thin bandwidth
  • Complicated, random, non stationary

• Characteristics of Biomedical System

  • Organism
  • Openness
  • Time-Varient
  • Complexity
  • Non linearity
  • Varability

  In a nutshell, Biomedical System is an extremely complicated system

• Basic mission of Biomedical Signal Processing

  • Motivation

    • Filter useless signal because they contaminate signal of interest
    • Extract signal in a more obvious or useful method
    • Prediction

  • Cited from Mark van Gils, Fall 2011

    • Some objectives of applications that use biosignal processing
      • Information gathering
      • Diagnosis
      • Monitoring
      • Therapy and control
      • Evaluation
    • Example: Notch Filter
    • Problems in biosignal processing
      • Accessibility
      • Variance
      • Inter-relationships and interactions among physiological system
      • Acquisition interference
      • Lack of Gold Standards

• Fourier Transformation

  • One of the most common analysis techiniques that is used for biological signals isaimed at breaking down the signal into its different spectral (frequency) components

  • Signal Processing Alogrithm == Transformation Operator

  • Orthogonal function

    • inner products equals to 0
    • Orthogonal function set
      • continuous function meet $$\int_a^b\Phi_n(t)\Phi_m(t)dt=\begin{cases} C,&m=n\ 0,&m\ne n\ \end{cases}$$ on interval [a,b]

  • Inner Product

    • $<f(x),g(x)>=\int_a^bf(x)g(x)dx$

  • Fourier Transformation

    Transform	Time	Spectrum
    CTFT	Continuous, Aperiodic	Continuous, Aperiodic
    FS	Continuous, Periodic	Discrete, Aperiodic
    DTFT	Discrete, Aperiodic	Continuous, Periodic
    DFT	Discrete, Periodic	Discrete, Periodic

    • Fourier Transformation
      • $$\int_{-\infty}^\infty S(t)\cdot e^{-2\pi j \omega t}dt$$

  • Properties of DFT

    • Linearity $$DFT[ax(n)+by(n)]=aX(k)+bY(k)$$
    • Periodical $$x(n)=X(n+N), X(k)=X(k+N)$$
    • Time Reversal $$DFT[x(N-n)]=X(N-k)$$
    • Circular Time Shift $$DFT[x((n-l))_N]=X(k)e^{-j\frac{2\pi kl}{N}}$$
    • Circular Frequency Shift $$DFT[x(n)e^{-j\frac{2\pi nl}{N}}]=X((k-l))_N$$
    • Complex Conjugate $$DFT[x^{*}(n)]=X^*(N-k)$$
    • Circular Convolution $$DFT[x_1(n)\otimes x_2(n)]=X_1(k)X_2(k)$$
    • Multiplication $$DFT[x_1(n)x_2(n)]=X_1(k)\otimes X_2(k)$$
    • Parseval’s Theorem $$\sum_{n=0}^{N-1}x(n)x^*(n)=\cfrac{1}{2\pi}\sum_{n=0}^{N-1}X(k)X^*(k)$$

• Spectrum Analysis

  • Magnitude Spectrum: $|X(\omega)|$
  • Phase Spectrum: $\theta$

• Random/Stochastic Process

  • A Stochastic function of time $X(t,\xi)$.

• Types of Stochastic Process

  • based on time
    • Discrete
    • Continuous
  • based on sample function
    • Deterministic stochastic process
    • Uncertain stochastic process
  • based on distribution function / density function
    • stable, normal, rayleigh, markov, …

• Random Signal

  • Random signal could only be described within limited accuracy and confidence.

• Statistics of Random signal

  • Statistical average
  • Distribution
  • Statistical Features

• Probability Distribution Function

  • 1-D probability distribution: $F_X(x_1;t_1)=P{X(t_1)\le x_1}$
    • probability density function: $f_x(x_1;t_1)=\cfrac{\partial F_X(x_1;t_1)}{\partial x_1}$
  • 2-D probability distribution: $F_X(x_1,x_2;t_1,t_2)=P{X(t_1)\le x_1,X(t_2)\le x_2}$
    • $f_x(x_1,x_2;t_1,t_2)=\cfrac{\partial^2F_X(x_1,x_2;t_1,t_2)}{\partial x_1\partial x_2}$

• Statistical Features

  • Expectation(Mean; first moment around zero): $m_x(t)=E[X(t)]=\int_{-\infty}^\infty xf_X(x;t)dx$: DC component
  • Mean Square Value(second moment around zero): $\Psi_X(t)=E[X^2(t)]=\int_{-\infty}^\infty x^2f_X(x;t)dx$: Power
  • Variance(second moment around central): $\sigma^2_X(t)=D[X(t)]=E[X^2]=E[(X(t)-m_x(t))^2]$: Ac Power
  • $$\Psi_X(t)=m_X^2(t)+\sigma_X^2(t)$$
  • Correlation Function: $R_X(t_1,t_2)=E[X(t_1)X(t_2)]=\int_{-\infty}^\infty\int_{-\infty}^\infty x_1 x_2 f_X(x_1,x_2;t_1,t_2)dx_1dx_2$
    • When $t_1=t_2$, $\tau=0$, auto-correlation function $R_X(0)=E(X^2)$
  • Covariance function: $C_X(t_1,t_2)=E[(X(t_1)-m_X(t_1))(X(t_2)-m_X(t_2))]$
    • When $\tau=0$, $C_X(0)=\sigma^2(x)$

• Stationary random signal

  • First order stable process(m=1): mean of the signal not relate to t
  • Second order stable process(m=2; wide sense stationary): mean and mean square value not related to t

• Ergodic

  • relating to or denoting systems or processes with the property that, given sufficient time, they include or impinge on all points in a given space and can be represented statistically by a reasonably large selection of points.

• Some typical random process

  • Gaussian/Normal process
    • $f(x)=\cfrac{1}{\sqrt{2\pi}\sigma}\exp^{-\cfrac{(x-a)^2}{2\sigma^2}}$
  • Ideal White Noise
    • $P_\xi(\omega)=\cfrac{n_0}{2} (W/Hz)$
    • $R(\tau)=\cfrac{n_0}{2}\delta(\tau)$
  • Band-limit White Noise
    • $P_\xi=\begin{cases} \cfrac{n_0}{2},&|f|<f_0\ 0, &\text{otherwise} \end{cases}$
    • $R(\tau)=2f_0\cfrac{n_0}{2}\cfrac{\sin(w_0\tau)}{w_0\tau}$

• Correlation

  • Similarity
  • Correlation coefficient: Pearson correlation coeffient
    • $r(X,y)=\cfrac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$

• Linearity

  • $f(ax+by)=af(x)+bf(y)$

• Nonlinear models

  • linear models are linear in the parameters which have to be estimated

• Linear Correlation

  • Synchronism/Similarity/In-phase/Linear Relationship
  • $r_{xy}(m)=\sum_{n=-\infty}^\infty x(n)y(n+m)$
  • $r_{xy}(m)=x(n)\cdot y(n)$
  • $r_{xy}(m)=r_{yx}(-m)$