outline - kmuttwebstaff.kmutt.ac.th/~sarawan.won/biosignal/biosignal2008.pdf · some matlab tools...

Introduction to Biosignal Processing

OutlineOverview of Signals

Measurement Systems

-Filtering-Acquisition Systems (Quantisation and Sampling)

Digital Filtering Design

Frequency Domain Characterisations

- Fourier Analysis- Power Spectral Density Analysis

Time-Frequency Analysis

- Spectrogram


What is a signal ?

Physical quantity, varying as a function of an independent variable, which carries information


Examples of Biomedical Signals

The electrocardiogram (ECG)

Source: http://medicalimages.allrefer.com/



The electromyogram (EMG)



The electroencephalogram (EEG)


Applications of Biosignal Processing


Uniform (periodic) sampling. Sampling frequency fs = 1/ T

Continuous-Time and Discrete-Time

Continuous Discrete)(tV )(][ ktVkV ≡

Types of Signals


Deterministic and Stochastic

Types of Signals

Deterministic

Stochastic


Typical Measurement Systems

Acquisition Signal processing

Information extractionInterpretation


Signal Acquisition System

Signal Conditioning

e.g. - Amplifier- Filter

A/D

Analog-to-digital converter

Sensors/Transducers

Sensors/transducers : to pick-up the physical quantity to an electrical signal

Amplifier : to amplify a small-amplitude electrical signal to range of several volts

Analog filter : to remove high frequencies (anti-aliasing)

A/D converter : to convert continuous signal to a discretised and quantised version


FilteringIdeal filters let frequency components over the passband pass through undistorted, while components at the stopband are completely cut off.

Filteru y

Filter Types


FilteringFilter properties are described in terms of frequency response and graphically in the form of a Bode plot

mmm

m

m

m

nnn

n

n

n

bdtdub

dtudb

dtudba

dtdya

dtyda

dtyd

++++=−−−− −−

−

−−

−

11

1

1011

1

1 ......

The Laplace Transform

ωjs =)( ωjH

LTI-Systemu y

Frequency Response Function


FilteringBode Plot

10-1

100

101

102

-90

-45

0

Pha

se (d

eg)

Bode Diagram

Frequency (rad/sec)

-50

-40

-30

-20

-10

0

Mag

nitu

de (d

B)

21)(+

=ω

ωj

jH

rad/sec 2=uω

o45H(j2)

0.3536)2(

−=∠

=jH

bode(tf(1,[1 2]))


FilteringFilter Bandwidth

The width of the positive frequency range in which the magnitude response is greater than or equal to 0.707 (3 dB attenuation) of the peak magnitude of a frequency plot

-3 dB


Filtering

Filter Order & Sharpness of the roll-off

Source: http://www.freqdev.com/guide/analog.html


Filtering

Analog Filter Implementation


Analog-to-Digital (A/D) Conversion

Analog signal

Anti-aliasingFilter

Sampling Quantisation Digital signal

A/D



0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Sample

x(t)

Analog signal

Anti-aliasingFilter


A/D



0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Sample

x(t)

Analog signal

Anti-aliasingFilter


A/D



0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Sample

x(t)

Analog signal

Anti-aliasingFilter


A/D


Sampling

An analog signal can be exactly recovered from its samples if the signal is sampled at a sampling rate of fs > 2fmax, where fmax is the highest frequencycontained in the signal

Example

)100cos()300sin(10)50cos(3)( ttttx πππ −+=If

what is the minimum sampling rate which guarantees no loss information ?

Sampling Theorem


Sampling

What if fs < fmax Aliasing

A 8 Hz-signal, sampled at 10 Hz, appears as 2 Hz


How to Prevent AliasingUse sufficiently high sampling frequency

Place an analog anti-aliasing filter in front of the sampler to reject 2/sFF >

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5(a) 1-Hz sine + 60-Hz noise

time (sec)0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5(c) prefiltered (a) with ωp = 20 rad/sec

time (sec)

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5(b) 28 Hz sampled (a)

time (sec)0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5(d) 28 Hz sampled (c)

time (sec)

p

pfilter s

sGω

ω+

=)(


Anti-Aliasing Filter

Avoid designing an anti-aliasingfilter ‘too tight’ against half the sampling frequency

In practice, the cut-off frequency (fc) of the filter may be selected a little above fmax. Then, select sampling rate fs > 3fc


Digital Filtering

)(...)1( ...)(...)2()1()(

10

21

mkubkubbnkyakyakyaky

m

n

−++−+++−−−−−−−=

)()(

...1...

)()(

11

110

zAzB

zazazbzbb

zUzY

nn

mm =

++++++

= −−

−−

H(z) - Transfer function

General form

z-transform


Digital Filtering

IIR and FIR filters

Infinite Impulse Response (IIR) filter:Response is a function of both current and past inputs and past outputs

Finite Impulse Response (FIR) filter: Response is a function of current and past inputs only

∑=

−=m

ii ikubky

1

)()(

∑∑==

−+−=m

ii

n

ii ikubikyaky

11

)()()(


Digital Filtering

IIR vs FIR filters

Pros Cons

IIR

FIR

Requiring a much lower filter order than FIR filters to achieve a specific frequency criterion

Nonlinear phase characteristics

Always being stableHaving linear phase shifts

less efficient in terms of computer time and memory than IIR because of their nonrecursive nature


Digital Filtering Design

Passband and stopband attenuation

Transition width(s)

ωp – Passband edge frequency

ωs – Stopband edge frequency


Some Digital Filters

Source: http://en.wikipedia.org/wiki/Digital_filter


Some MATLAB Tools for filter design

Frequency Response/Filter Analysis: freqz, fvtool

FIR: fir1, fir2, firls, remez

IIR: butter, cheby1, cheby2, ellip, yulewalk

Filter Design & Analysis GUI: fdatool


FDATool


Frequency Domain Characterisation

In physiological systems information is often more readily expressed in the frequency domain

Why frequency domain?

Noise and information are often more easily separated in frequency domainthan in time domain

Some algorithms are more efficient when implemented in the frequency domain, e.g. computation of convolution in frequency domain


The Fourier Transform (FT) method

Any signal may be presented as a series of sinusoids of different frequencies, each with an amplitude and phase.

Frequency Domain Analysis

Continuous Time Fourier Transform

∫∞

∞−

−⋅= dtetxfX ftj π2)()(

∫∞

∞−

⋅= dfefXtx ftj π

π2)(

21)(


Example

ttttx ⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−=

25cos0162.0

23cos0450.0

2cos4052.0)( πππ

0 1 2 3 4 5 6 7 8-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

x(t)

2/π 2/3π 2/5πfrequency (rad/sec)

Mag

nitu

de


Harmonics 1, 3 and 5


Example


Source: D. Simpson, Tutorial on Biomedical Signal Processing

EEG signal


Discrete Fourier Transform (DFT)


The frequency spectrum of a signal is numerically computed using its time samples, providing a discrete spectrum

∑−

=

−⋅=

1

0

2

][)(N

n

Nnkj

enxkXπ

∑−

=

⋅=1

0

2

)(1][N

n

Nnkj

ekXN

nxπ

frequencies 1,...,1,0, −=⋅= NkNfkf s

k

In practice, DFTs are computed using a high-speed algorithm, known as the Fast Fourier Transform (FFT)


Example: FFT with N=1024, fs=10 Hz


Ts=0.1; %sampling time Tst=[0:499]*Ts; %time vectorx=cos(2*pi*t/10)+0.5*sin(2*pi*t/5); %signal xN=1024; % no. of computational pointsX=abs(fft(x,N)); %magnitude of FFT of xX=X(1:N/2+1);F=[0:N/2]/N/Ts;figure;subplot(211);plot(t,x); xlabel('Time(sec)')subplot(212);plot(F,X); xlabel('Frequency (Hz)'); ylabel('Amplitude')

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2

Time(sec)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

Frequency (Hz)

Am

plitu

de

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

50

100

150

200

250

Frequency (Hz)

Am

plitu

de Nff s /=∆


Power Spectral Analysis


Frequency (Hz)

Pow

er/H

z

describes how the power of a signal is distributed with frequency

Parseval’s theorem

dffXdttxE ∫∫∞

∞−

∞

∞−

== 22 )()(

where E – the average energy of x(t)

x[n]offunction ation autocorrel theof )( FTfPx =

Power Spectrum


Estimate PS using FFT

Power Spectral Estimation

s

kkx fN

fXfP

⋅=

2)()(ˆ

at x[n]of DFT theis )( kfXwhere

The periodogram method

1,...,1,0, −=⋅= NkNfkf s

k


WindowingPower Spectral Estimation

Extracting a segment of a signal in time is the same as multiplying the signal with a rectangular window

Source: http://www.ee.ic.ac.uk/pcheung/teaching/ee2_signals/


Spectral Leakage


Original Signal

Window

WindowedSignal

0 0.1 0.2 0.3 0.4 0.5-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency (kHz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Sine frequency100 Hz + noise


How to reduce leakage


Sine frequency100 Hz + noise

0 0.1 0.2 0.3 0.4 0.5-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency (kHz)

Pow

er/fr

eque

ncy

(dB/

Hz)

Using a window type which smoothes the edges of the signal

Modified periodogram methodCommonly used windows:• Hamming windows• Hanning windows• Barlett windows• Blackman windows• Kaiser windows


Estimate PS by Welch’s method


It reduces noise in the estimated power spectra in exchange for reducing the frequency resolution

Divides the data several segments, possibly overlapping, performs an FFTN each segment, computes the power spectrum, then averages these spectra

Averaging Overlap Averaging

Source: http://www.ave.kth.se/education/msce/TSOVM/courses/SD2130/


ExamplePower Spectral Estimation

0 50 100 150 200 250 300 350 400 450 500-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Periodogram Power Spectral Density Estimate

0 50 100 150 200 250 300 350 400 450 500-25

-20

-15

-10

-5

0

5

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Welch Power Spectral Density Estimate

% Generate datafs = 1000; % Sampling frequencyt = (0:0.3*fs)./fs; % 301 samplesA = [2 8]; % Sinusoid amplitudesf = [150;140]; % Sinusoid frequenciesxn = A*sin(2*pi*f*t) + 5*randn(size(t));

% PeriodogramHs = spectrum.periodogram('rectangular');psd(Hs,xn,'Fs',fs,'NFFT',1024);

% Welch’s spectral estimate for 3 segments with % 50% overlapHs = spectrum.welch('rectangular',150,50);figure;psd(Hs,xn,'Fs',fs,'NFFT',1024);


Time-Frequency Analysis

Fourier analysis assumes stationarity of the signal, e.g. its statistical properties do not change

Many signals - particularly those of biological origin - are not stationary.

Extract both time and frequency information from the signal


Short-Term Fourier Transform (STFT) – The Spectrogram

Divide signal into ‘stationary’ segments.

Perform Fourier transform on a window of data that slides along the time axis→ time-frequency function that describes the frequency distribution near time τ

))()((),( ττ −⋅= twtxFTfSTFT

The spectrogram is given by the magnitude of the STFT of the function

{ } 2),()( τfSTFTtxmSpectrogra =


Short-Term Fourier Transform (STFT) – The Spectrogram

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

A linear chirp, starting @ DC and crossing 150 Hz at t=1 sec


Short-Term Fourier Transform (STFT) – The SpectrogramThe time-frequency tradeoff: we cannot simultaneously obtain arbitrarily high resolution along both the time and frequency axes. Shortening the data length (N) to improve time resolution will reduce frequency resolution (≈ fs/N)

Time

Freq

uenc

y

N = 50

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

120

140

160

180

200

Time

Freq

uenc

y

N = 800

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

120

140

160

180

200

N=50 N=800


MATLAB Code

Ts=0.0025; %Sampling timefs=1/Ts; %sampling frequencyM=5/Ts; %M = Data length over 5 sec

% Generate Datat1=0:Ts:(M-1)*Ts; y=[cos(2*pi*10*t1) cos(2*pi*25*t1) cos(2*pi*50*t1) cos(2*pi*100*t1)]; t=[0:length(y)-1]*Ts; %Time vectornfft=[50 800]; %Window widths

%Plot the spectrogram for window widths of 50 and 800for n=1:2

N=nfft(n); %Tested window width (N)figure; %Call a new figurespecgram(y,N,fs,N,[]); %Plot the spectrogram using current Ntitle(['N = ' num2str(N)]) %Add title

end


More methodsNoise reduction- Optimal filters- Adaptive noise cancellation

Time-Frequency- Wigner-Ville- Wavelets

Signals and Systems- Correlation and coherence- Modelling (black box vs. physiological)

Pattern recognition and classification- Rule-based- Statistical methods- Learning


Suggested Reading

Oppenheim, A.V. and Schafer, R.W. Discrete-Time signal Processing. Prentice Hall: Engle-wood Cliffs, NJ, 1989.

Semmlow, J.L. Biosignal and Biomedical Image Processing: MATLAB-Based Applications. CRC, 2004.

Rangayyan, R.M. Biomedical Signal Analysis: A Case-Study Approach. Wiley-IEEE Press, 2001.

Bruce, E.N. Biomedical Signal Processing and Signal Modeling. Wiley-Interscience, 2000.

outline - kmuttwebstaff.kmutt.ac.th/~sarawan.won/biosignal/biosignal2008.pdf · some matlab tools...

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