Lecture 3 BME452 Biomedical Signal Processing 2013 (copyright Ali Işın, 2013) 1 BME452 Biomedical Signal Processing Lecture 3 Signal conditioning

Lecture 3 BME452 Biomedical Signal Processing 2013 (copyright Ali Işın, 2013) 1

BMEBME445252 Bio Biomedimedical Signal cal Signal ProcessingProcessing

Lecture 3Lecture 3

Signal conditioningSignal conditioning


Lecture 3 Outline

In this lecture, we’ll study the following signal conditioning methods (specifically for noise reduction) Ensemble averaging Median filtering Moving average filtering Principal component analysis Independent component analysis (in brief)

Before we study these, an introduction to some mathematics will be given


Mean

The arithmetic mean is the "standard" average, often simply called the "mean"

where N is used to denote the data size (length) In MATLAB, n=1,….N but sometimes we use n=0,1,….N-1.

Example An experiment yields the following data: 34,27,45,55,22,34 To get the arithmetic mean

How many items? There are 6. Therefore N=6 What is the sum of all items? =217. To get the arithmetic mean divide sum by N, here 217/6=36.1667

Expectation What is expected value of X, E[X]? Simply said, it refer to the sum divided by the

quantity, i.e. mean of the value in the square brackets

Eg: E[x2]=

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Very often, we set the mean to zero before performing any signal analysis This is to remove the dc (0 Hz) noise

xm=x-mean(x)

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Mean removal for signals


Mean removal across channels/recordings

Sometimes, a noise corrupts all the signals in a multi-channel signal or across all the recordings of a single channel signal Since the noise is common to all the channels/recordings, the simplest

way of removing this noise is to remove mean across channels/recordings


Standard deviation ()

Measures how spread out are the values in a data set Suppose we are given a signal x1, ..., xN of real value numbers (all recorded signals are real

values) The arithmetic mean of this population is defined as

The standard deviation of this population is defined as

Given only a sample of values x1,...,xN from some larger population, many authors define the sample (or estimated) standard deviation by

This is known as an unbiased estimator for the actual standard deviation


Standard deviation example


Interpreting standard deviation

A large standard deviation indicates that the data points are far from the mean and a small standard deviation indicates that they are clustered closely around the mean

For example, each of the three samples (0, 0, 14, 14), (0, 6, 8, 14), and (6, 6, 8, 8) has an average of 7.

Their standard deviations are 7, 5 and 1, respectively.

The third set has a much smaller standard deviation than the other two because its values are all close to 7.


Normalisation Sometimes, we may wish to normalise a signal to mean=0 and set the standard

deviation to 1 For example, if we record the same signal but using different instruments with different

amplification factor, it will be difficult to analyse the signals together In this regard, we will normalise the signals using

)(

)][(][

xstd

xnxnxnor


Variance Variance is simply the square of standard deviation

Uncertainty measure Variance may be thought of as a measure of uncertainty

When deciding whether measurements agree with a theoretical prediction, variance could be used

If variance (using the predicted mean) is high, then the measurements contradict the prediction

Example: say we have predicted that x[1]=7, x[2]=6, x[3]=5 x is measured 3 times => (7.2 6.7 5.6); (4.2 6.8 5.2); (11.2 6.3 5.9)

Do this =>

Compute the variance using the predicted value as mean

var[1]=12.76, var[2]=0.610, var[3]=0.605

So, we know that x[1] measurements are contradicting the prediction and probably not x[2] and x[3] measurements

N

ii xx

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1var


Covariance If we have multi-channel/multi-trial recorded signals, we can have cross variance or simply

covariance

Covariance measure the variance between different signals (from different channels/recordings)

Covariance between two signals, X and Y with respective means, μ and ν,

The covariance sometimes is used as a measure of "linear dependence" between the two signals but correlation is a better measure


Correlation Correlation between two signals, X and Y is

It is simply normalised covariance

It measures linear dependence between X and Y

The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables

The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables


Application of correlation (example)

The diagram shows how the unknown signal can be identified A copy of a known reference signal is correlated with the unknown signal The correlation will be high if the reference is similar to the unknown signal The unknown signal is correlated with a number of known reference

functions A large value for correlation shows the degree of similarity to the reference The largest value for correlation is the most likely match


Application of correlation (another example)

Application to heart disease detection using ECG signals Cross correlation is one way in which different types of heart diseases can be identified

using ECG signals Each heart disease has a unique ECG signal Some example of ECG signals for different diseases are shown below

The system has a library of pre-recorded ECG signals (known as templates) An unknown ECG signal is correlated with all the ECG templates in this library The largest correlation is the most likely match of the heart disease

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Signal-to-noise ratio (SNR)

Before we move into the noise reduction methods, we need a measure of noise in the signals This is important to gauge the performance of the noise reduction

techniques

For this purpose, we use SNR SNR=10log10[(signal energy)/(noise energy)]

The original noise x(noise) = x(original signal) – x(noisy signal)

After using some noise reduction method, x(noise) = x(original signal) – x(noise reduced signal)

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Ensemble averaging If we have many recordings, we can use ensemble averaging to reduce noise that is not

correlated between the recordings

Ensemble averaging to reduce noise from Evoked Potential (EP) EEG

Repeated different recordings are known as trials EP EEG signals from trial to another are about the same (high correlation) But noise will be different from one trial to another (low correlation) Hence, it would be possible to use ensemble averaging to reduce noise

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EP EEG

EP EEG+noise (trial 1)

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EP EEG after ensemble averaging

EP EEG+noise (trial 2) EP EEG+noise (trial 20)


Worked example 1 - Ensemble averaging

Assume we have 3 signals corrupted with noise. Assume we have the original also (for SNR computation). 1. Set the mean to zero first

2. The ensemble average is (the average is done for each sample point n)

3. The noises in the signals are(original signal – noise corrupted signal)

n 0 1 2 3

Noisy signal 1 -2.2 -0.2 0.2 2.2

Noisy signal 2 -2.1 0.1 0.1 1.9

Noisy signal 3 -1.9 -0.2 0.1 2.0

Original -2.0 0.0 0.0 2.0

n 0 1 2 3

Ensemble average -2.1 -0.1 0.1 2.0

n 0 1 2 3

Noisy signal 1 2.9 4.9 5.3 7.3

Noisy signal 2 2.9 5.1 5.1 6.9

Noisy signal 3 3.1 4.8 5.1 7

Original 3 5 5 7

n 0 1 2 3

Signal 1 noise 0.2 0.2 -0.2 -0.2

Signal 2 noise 0.1 -0.1 -0.1 0.1

Signal 3 noise -0.1 0.2 -0.1 0.0

Ensemble average noise 0.1 0.1 -0.1 0.0

4. SNR=10log10(e(signal)/e(noise))

Original signal energy 8

signal 1 signal 2 signal 3ensemble average

noise energy 0.16 0.04 0.06 0.03

SNR 16.99 23.01 21.25 24.26


Median filtering Similar to ensemble averaging, if we have many recordings, we can use median filtering to

reduce noise that is not correlated between the recordings

What is median filtering?

If we have x[1] as [3 2 1 0 6 7 9 3 2] from 9 trials, we sort the numbers from small to big, then the centre value (i.e. 5th) as the median

Sorted x[1] is [0 1 2 2 3 3 6 7 9], so median x[1]= 3

Median filtering is advantageous as compared to ensemble averaging if there is one trial containing a lot of noise AND if the number of trials/recordings are small

This is because the one heavily noise corrupted signal will distort the ensemble average values but will less likely affect the median values

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n=1,2,…………………………………………….,Nm

123....M

Number of trials

obtain median values

Data length


Worked example 2 – median filtering

Assume we have 3 signals corrupted with noise, one heavily corrupted

(assume the mean has been set to zero)

2. The noises in the signals are (original signal – noise corrupted signal)

4. SNR=10log10(e(signal)/e(noise))

1. The ensemble average and median filtered signals

Which technique gave better noise reduction using SNR – ensemble averaging or median filtering?

Why?

n 0 1 2 3

Noisy signal 1 -2.15 0.95 -0.25 1.45

Noisy signal 2 -9 0 5 4

Noisy signal 3 -2.25 -0.45 0.85 1.85

Original -2.0 0.0 0.0 2.0

n 0 1 2 3

Ensemble average -4.47 0.17 1.87 2.43

Median filter -2.25 0 0.85 1.85

n 0 1 2 3

Noise in signal 1 0.15 -0.95 0.25 0.55

Noise in signal 2 7 0 -5 -2

Noise in signal 3 0.25 0.45 -0.85 0.15

Noise in ensemble averaging 2.47 -0.17 -1.87 -0.43

Noise in median filter signal 0.25 0 -0.85 0.15

Original signal energy 8

signal 1 signal 2 signal 3ensemble average

Median filter

noise energy 1.29 78 1.01 9.81 0.81

SNR 7.93 -9.89 8.99 0.88 9.95


Moving average filtering How do we reduce noise if we have only one signal from one recording/ trial?

We can’t use ensemble averaging and median filtering

Normally, in any signal, the few points before and after a certain point n are correlated (i.e. related)

But generally the noise is not correlated

So, we can use moving average (MA) filtering

It is defined as

where S is the filter order

Example, for S=3, y[5]=(x[5]+x[6]+x[7])/3

For signals x and y to remain of same sample length: We have to pad (S-1) zeros to the signal x to get the last (S-1) points of the signal y

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ni

nxS

ny

Signal – correlation is

high

Noise – correlation is

lown n+S-1


Moving average filtering –zero padding

If zero padding

is NOT allowed

If zero padding is allowed

x[n], N=256

y[n], N=254 if S=3

x[1] x[256]

Length y is S-1 less than x

Because y[254]=(x[254]+x[255]+x[256])/3Moving averaged signal

x[n], N=256

y[n], N=256 - no matter what value of S

x[1] x[256]

Length y is same as x

Because y[254]=(x[254]+x[255]+x[256])/3 y[255]=(x[255]+x[256]+0])/2 y[256]=(x[256]+0+0])

Moving averaged signal


Example - moving average filtering

Assume we have a EEG signal corrupted with noise

Set the mean to zero

Apply moving average filter to the noisy signal (use filter order=3 and 5)

The higher filter order will remove more noise, but it will also distort the signal more (i.e. remove the signal parts also)

So, a compromise has to be found for the value of S (normally by trial and error)

load eeg;N=length(eeg);for i=1:N-3,eegMA1(i)=(eeg(i)+eeg(i+1)+eeg(i+2))/3;endeegMA1(255)=(eeg(255)+eeg(256))/2;eegMA1(256)=eeg(256)/1;for i=1:N-5,eegMA2(i)=(eeg(i)+eeg(i+1)+eeg(i+2)+eeg(i+3)+eeg(i+4))/5;endeegMA2(253)= (eeg(253)+eeg(254)+eeg(255)+eeg(256))/4;eegMA2(254)=(eeg(254)+eeg(255)+eeg(256))/3;eegMA2(255)=(eeg(255)+eeg(256))/2;eegMA2(256)=eeg(256)/1;subplot(3,1,1), plot(eeg, 'g ');subplot(3,1,2), plot(eegMA1,'r');subplot(3,1,3), plot(eegMA2,‘b');

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Median filter for noisy images

Consider applying median filtering to some noisy images In computer, these grayscale images are stored as 2D arrays

x(i,j) where I and j are the coordinates and x is the grayscale values (in general from 0 (black) 255 (white))

After applying median filter

Mean (averaging) filter could be applied in similar manner though for images, median filter normally gives better results


Principal component analysis

PCA can be used to reduce noise from signals provided we have repeated recordings or signals from a number of trials or multi-channel signals

Principal components (PCs) are obtained from PCA, which are orthogonal signals, i.e. signals that are uncorrelated to each other

Since noise is less correlated between the trials as compared to the signals, the first few PCs will account for the signals while the last few PCs will account for the noise

By discarding the last few PCs before reconstruction, we’ll get the signals without noise/with less noise


Principal component analysis -algorithm

PCA algorithm Organise the data, X in M x N matrix Set mean to zero Compute CX=covariance of matrix, X Compute eigenvalue, eigenvector of CX Sort eigenvectors (i.e. principal components) in descending order Compute Zscores Decide how many PCs to keep using some criteria Reconstruct the noise reduced signals using the first few PCs and

Zscores


Eigenvector, eigenvalue – a brief review

The steps of setting mean to zero and computing covariance have been covered earlier, so let us move to the step of computing eigenvector, eigenvalue

Let us assume that A=cov(X), where X is the mean zero data

In MATLAB, [V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A

It is obtained from A.*V = D.*V Note: A has to be a square matrix

Eg:

So is the eigenvector and 4 is the eigenvalue

can be assumed to be the vector direction

And eigenvalue=4 is the weight of this vector

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Eigenvector, eigenvalue (cont.)

Finding the eigenvalues and eigenvectors for bigger than 3 x 3 matrix is extremely difficult, so we will skip the algorithms and just use MATLAB function eig

Example, for the following square matrix:

Decide which, if any, of the following vectors are eigenvectors of that matrix and give the corresponding eigenvalue

Answer: The eigenvector is because = 1.

The eigenvalue is 1

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Sort the eigenvectors

Sort the eigenvectors from big to small using eigenvalues Let’s use the example we saw earlier for ensemble averaging and median filtering X=[2.9 4.9 5.3 7.3; 2.9 5.1 5.1 6.9; 3.1 4.8 5.1 7] Xm=[-2.2 -0.2 0.2 2.2; -2.1 0.1 0.1 1.9; -1.9 -0.2 0.1 2.0]

A=Cov(Xm’)

The eigenvectors are, [V,D]=eig(A)

The corresponding eigenvalues are 0.0017, 0.0272, 8.4578

So now sort the eigenvectors in the order of eigenvalues: 8.4578, 0.0272, 0.0017

So the eigenvectors are

3.2533 2.9333 2.8800 2.9333 2.6800 2.5933 2.8800 2.5933 2.5533

V =

0.7103 0.3335 0.6198 -0.1039 -0.8213 0.5610 -0.6962 0.4629 0.5487

D =

0.0017 0 0 0 0.0272 0 0 0 8.4578


Zscores

Zscores=Vsort’*Xm where V is the sorted eigenvectors and Xm is the mean zero data

matrix

In the previous example, the size of A=3

So, we will have 3 Zscores

Zscores will have the same dimensions as Xm

Zscore 1

Zscore 2

-3.5843 -0.1776 0.2349 3.5270 0.1115 -0.2414 0.0309 0.0991 -0.0218 -0.0132 0.0621 -0.0270


How to select the number of PCs to keep

The PCs with higher eigenvalues represent the signals while the PCs with lower eigenvalues represent the noise

So we keep the first few PCs and discard the rest

But how many PCs do we keep?

Using certain percentage of variance to retain, normally 95% or 99%

Eigenvalues represent the weight of the PCs i.e. some sort of variance (power) measure of the PCs

So, we can use sum(D1:Dq)/sum(D1:Dlast)>0.99, where D represents the eigenvalues [D1,D2,D3,…Dlast]

In our example, say we wish to retain 99% variance eigenvalues are 8.4578, 0.0272, 0.0017

Sum(D1:Dlast)= 8.4867 Sum(D1:D1)=8.4578; sum(D1:D1)/sum(D1:Dlast)=0.9996 Sum(D1:D2)=8.4849; sum(D1:D1)/sum(D1:Dlast)=0.9998 Sum(D1:Dlast)=8.4867; sum(D1:Dlast)/sum(D1:Dlast)=1.0

Since the first eigenvalue accounted for 99.96% variance (which is more than 99%) and we can discard the second and third PC

If we wish to retain 99.97%, how many PCs do we retain? Answer=2


Reconstruct using the selected PCs To get back the original signals without noise, we need to reconstruct using the selected PCs

Xnonoise=Vselected*Zscoreselected

In our example, only 1 PC was selected, so the first eigenvector and the first Zscore will be used to get back the 3 noise reduced signals

Xnonoise=Vsort(:,1)*Zscore(1,:)

noise=Xm-Xnonoise

Energy (noise) =

Original signal, x=[-2 0 0 2]; this is the actual original mean removed signal - from the earlier slide

Energy (original signal)=8;

SNR=

SNR using PCA is generally higher than ensemble averaging or median filtering and we do get 3 signal outputs unlike one signal output from ensemble averaging or median filtering

-2.2217 -0.1101 0.1456 2.1861 -2.0107 -0.0996 0.1318 1.9786 -1.9668 -0.0975 0.1289 1.9353

0.0217 -0.0899 0.0544 0.0139 -0.0893 0.1996 -0.0318 -0.0786 0.0668 -0.1025 -0.0289 0.0647

0.0117 0.0550 0.0200

28.3483 21.6265 26.0222


Principal component analysis – an example of application

Consider the following 3 noise corrupted signals

Obtain the principal components (in descending order of eigenvalue magnitude)

Obtain the Zscores

Decide how many PCs to retain - assume that we retain only the first PC

By retaining the first PC only for reconstruction, we will have 3 noise reduced EP

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EP signal (trial 1)

EP signal (trial 2)

EP signal (trial 3)

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Noisy EP signal (trial 1)




Independent component analysis –a brief study ICA is a new method that could be used to separate noise from signal Sometimes known as blind source separation Requires more than one signal recording (like PCA) ICA separates the signals into independent signals (signals and noises – we keep the signals,

discard the noises) Example: Assume, we have 3 observed (i.e. recorded signals): x1[n], x2[n] and x3[n] from 3 original signals

sources: s1[n], s2[n] and s3[n] x1[n]=a11.s1[n]+a12.s2[n]+a13.s3[n] x2[n]=a21.s1[n]+a22.s2[n]+a23.s3[n] x3[n]=a31.s1[n]+a32.s2[n]+a33.s3[n]

The matrix, is known as mixing matrix

ICA can be used to obtain the original signals by obtaining the unmixing matrix W W=A-1

The original signals can be obtained by using s1[n]=w11.x1[n]+w12.x2[n]+w13.x3[n] s2[n]=w21.x1[n]+w22.x2[n]+w23.x3[n] s3[n]=w31.x1[n]+w32.x2[n]+w33.x3[n]

333231

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www

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Independent component analysis – a pictorial example

Figures from Independent Component Analysis, Hyvarinen, Karhunen and Oja


Maximising non-gaussianity using kurtosis How ICA works?

The central limit theorem says that sums of non-gaussian random variables are closer to gaussian than the original ones

=> the independent signals are less gaussian than the combined signals So by maximising non-gaussian behaviour, we get closer to the original signals Kurtosis could be used to measure gaussian behaviour

BUT what is gaussian? See next slide

more gaussianless gaussian

Source (original signals)

Mixed (combined signals)


Gaussian and probability distributions

Gaussian (or normal) probability distribution is

BUT what is probability distribution?

Probability distribution for discrete-time signals is simply the number of occurences vs value

Eg: if x has values from 1 to 10

Gaussian distribution

Super-gaussian distribution The data close to mean have higher occurences

Sub-gaussian distribution Most the data have similar number of occurences

count(1:10)=0;for i=1:10,y=find(x==i);count(i)=length(y);endplot(y);x = [4 1 2 3 9 8 6 5 7 3 4 2 2 6 9 5 6 7] 0 2 4 6 8 10

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Probability distribution of x

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Kurtosis

Non-gaussianity can be measure using kurtosis

Gaussian signals have kurtosis=3 Sub-gaussian signals have lower kurtosis value Super-gaussian signals have higher kurtosis value

Examples

x = [4 1 2 3 9 8 6 5 7 3 4 2 2 6 9 5 6 7];y=kurtosis(x,0); %unbiased kurtosis using MATLAB

y=1.9509

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Gaussian distribution signalx = randn(1,100000); % gaussian signal with mean=0, std=1plot(x);y=kurtosis(x,0) %unbiased kurtosis using MATLAB

y=3.00


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X1= EP+noise, kurtosis=2.79

Example – Kurtosis for EP and noise

Original signals Recorded signals

Can you see that kurtosis is lower for combined signals, i.e. the actual independent signals (i.e. sources) have higher kurtosis


ICA tries to obtain EP and noise by estimating the unmixing matrix

The solution is In the beginning, we don’t know the unmixing matrix!

A simple ICA method is to randomly generate values in [0,1] for the unmixing matrix

Now, EP[n]=w11.X1[n]+w12.X2[n] and noise[n]=w21.X1[n]+w22.X2[n]

Kurtosis values are computed for these estimated EP and noise

Repeat with other random values for the unmixing matrix (say for a thousand times)

The unmixing matrix that gave the highest kurtosis values will denote the actual EP and noise

Actual ICA algorithms use complicated neural network learning algorithms, so we’ll skip them

It suffices to know that by using certain measures like kurtosis (representing non-gaussianity), we can separate the signals into independent components

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Simple ICA algorithm – an example using EP and noise

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Unmixing matrix


Study guide (Lecture 3)

From this week’s lecture, you should know

Basic mathematics– mean, standard deviation, variance, covariance, correlation, autocorrelation, SNR, etc.

Uses of these basic maths in signal analysis

Noise reduction methods like ensemble averaging, median filtering, moving average filtering, principal component analysis and basics of independent component analysis

End of lecture 3

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Lecture 3 BME452 Biomedical Signal Processing 2013 (copyright Ali Işın, 2013) 1 BME452 Biomedical Signal Processing Lecture 3 Signal conditioning