BMEN90035 Biosignal Processing

Final Exam, Semester 2, 2024

Number of questions: 2          Total Marks: 120 (1 mark per minute of writing time)

• Question 1: 50 marks

• Question 2: 70 marks

Time allowed:

• 40 minutes reading time - you MAY begin writing during this time

• 120 minutes writing time

• 30 minutes upload time - you must NOT write during this time

The exam is openbook. You may use code from Class Exercises and Workshops as templates to modify to complete this exam. You may also consult resources on the internet, excluding artificial intelligence tools, such a Chat GPT or Wolfram Alpha. Additionally, a table of Fourier transforms available for download from the LMS Assignments page for this exam may be useful.

A copy of this exam is available in pdf format from the LMS Assignments page in case any equations are difficult to read in the .mlx file.

You may NOT share code or answers with other students during the exam.

Data provided

The following data are provided to be used in answering ALL Questions:

• e1, e2 - two Matlab variables corresponding to two electromyogram  (EMG) envelope signals elnl  and ,  respectively, in units of volts. The two EMG signals were recorded simultaneously from different electrodes, on channels 1 and 2 (respectively), on the forearm of a subject while typing.

• f_s - a Matlab variable corresponding the sampling frequency f = 1000 Hz used for elnl  and eal ml .

• cmap - a Matlab variable corresponding to a colour map that is useful for Q1f, Q2c, Q2d and Q2g. It is a 4x3 matrix, where rows 1 to 4 correspond to the colours cyan, blue, green, and red, respectively.

Please note that answers to some questions are used in subsequent questions as part of a multi-step signal processing pipeline. Consequently, the following supplementary variables are provided to allow you to complete these subsequent questions in case you are unable to calculate these variables in the original question. These   variables are all stored in a Matlab structure soln.

• soln.P1_volts and  soln.P2_volts - corresponding to key-press segments p1, m[k]  and p2, m[k] calculated in Q1b. These variables are stored in the 1000x80 matrices, pi , as described in Q1b.

• soln.p1_bar_volts and  soln.p2_bar_volts - corresponding to the mean key-press segments, and  , calculated in Q1c. These variables are 1000x1  vectors, .

• soln.Q1_volts and  soln.Q2_volts - corresponding to mean-subtracted key-press segments:  q1, m[k] and q2, m[k] , calculated in Q1c. These variables are stored in  1000x80 matrices, ,  in the same format to the matrices pi described in Q1b.

• soln.h1 and  soln.h2 - corresponding to first two principal components, and , calculated in Q1d.

• soln.ym1_volts and  soln.ym2_volts - corresponding the projections Y1,m and )2m calculated in Q1f. These variables are in 1x80 vectors giving the projections to the key-presses m=1,.,80 .

• soln.y1_volts and  soln.y2_volts - corresponding to filter outputs y[n]  and calculated in Q2c.

• soln.y1_3rd_volts and  soln.y2_3rd_volts - corresponding to filter outputs and for the third key press of each finger, as calculated in Q2d. These variables are 1000x4 matrices, where each column corresponds to a key-press for fingers 1 to 4, respectively.

• soln.r_volts - corresponding to the distance, r[n] , calculated in Q2e.

These variables can be loaded into your workspace by running the following lines of code.

clear all load ExamData2024.mat

Overview

Electromyograms (EMG) measure the electrical activity of muscles during their contraction using electrodes placed on the surface of the skin. Most useful information in the EMG appears in the envelope of the signal (this is a positive, relatively slow signal that modulates a faster, noisy carrier signal). In this exam you will analyse EMG envelope signals obtained from recordings made during typing, for potential application in control of a robotic hand for typing. These EMG envelopes, eu[n]  and , were obtained from simultaneous recordings on channels 1 and 2 (respectively), using from electrodes at different locations on the forearm. During the recordings, the subject pressed keys on a keyboard with one finger at a time. There were 80 key presses in all, made at a rate of 1 press per second as follows:

• key presses 1-20: finger 1 = index finger,

• key presses 21-40: finger 2 = middle finger,

• key presses 41-60: finger 3 = ring finger,

• key presses 61-80: finger 4 = little finger.

The goal of the application is to use the EMG envelopes, and , to determine when a key press has taken place and which finger (1 to 4) this corresponds to.

This will be done in two stages:

• Question 1: Performing an off-line analysis using principal components analysis of the EMG envelopes. This will identify a two-dimensional projection space in which it is possible to classify which finger was     used to press the key for each segment of the EMG envelope.

• Question 2: Designing and implementing an on-line processing algorithm to determine when a key-press has taken place and which finger (1 to 4) was used. This is based on the principal components analysis    from Question 1.

Question 1: Performing an off-line analysis using principal components analysis of the EMG envelopes. This    will identify a two-dimensional projection space in which it is possible to classify which finger was used to press the key for each segment of the EMG envelope.

Q1a. [4 marks] Figure 1:  Plot the two EMG envelopes, elnl  and e a[n] , over the duration of the recording. Plot elnl in subplot(2,1,1) and ea[n] in subplot(2,1,2). Label your axes showing time in seconds and signal    amplitude in volts. Give the figure a title of Fig. 1. (Note here and throughout the exam, when giving a title to a   figure with subplots, you may apply title to just the first subplot.)

%% Q1a code figure(1), clf subplot(2,1,1) plot(t,e1,'k') title('Fig.1)') subplot(2,1,2) plot(t,e2,'k') xlabel('time [s]') ylabel('amplitude [V]')

Q1b. [6 marks] The first step in performing principal components analysis is to partition each of the envelopes elnl and e a[n] into 80 one-second segments, each corresponding to a key-press. Call these key-press segments p1, m[k] and p2, m[k] ,  respectively, form=1,.,80 key-presses, and k=1,…,1000 samples.

Extract each segment pj. m[k] , for each electrode. For the first segment of each electrode use samples n= l to 1000 (corresponding to the 1st to 1000th samples of eu[n]  and eznl ); for the second segment use samples    n= 1001 up to 2000, and so on for the 80 key-presses.  (Hint: consider using the Matlab command reshape to produce a 1000x80 matrix, pi , for each electrode with pi. m[k] , k=1,…,1000 , as the columns).

In Figure 2  plot all 80 key-press segments overlaid, plotting those for the first electrode, p1, m[n] , in subplot(2,1,1) and those for the second electrode, p2, m[n] , in subplot(2,1,2). Label your axes showing time in seconds and signal amplitude in volts. Give the figure a title of Fig. 2.

%% Q1b code figure(2), clf

Q1c. [8 marks] The next step in performing principal components analysis is to calculate the mean key-press segments,  and  , over the 80 key-presses for electrodes 1 and 2, respectively. Then use these to

calculate the mean-subtracted key-press segments:   and  , for each key-press m=1,.,80 .

Calculate the mean-subtracted key-press segments:  q1, m[k]  and q2, m[k] , m=1,.,80 . Plot them overlaid in

Figure 3, plotting those for the first electrode, q1, m[k] , in subplot(2,1,1) and those for the second electrode, q2, m[k] , in subplot(2,1,2). Label your axes showing time in seconds and signal amplitude in volts. Give the figure a title of Fig. 3.

(Note: if you were not able to calculate the key-press segments p1, m[k]  and p2, m[k]   you may use a version

stored in soln.P1_volts and  soln.P2_volts to complete this question. These variables are the 1000x80 matrices, pi , described in Q1a.)

%% Q1c code figure(3), clf

Q1d. [10 marks] Perform. principal component analysis on the combined mean-subtracted key-press

segments. To combine them, concatenate the segments together for channels 1 and 2  for each key-press,

m. I.e. if  1,m and q2m are the column vectors corresponding to segments q1, m[k]  and q2, m[k] , respectively, make a vector of twice their length by appending 92m under 1,m :  qm=Iq1, m;q2, m] .

Plot the first two principal components, hi  and hc in Figure 4, using sample numbers for the timeaxis. Label    your axes showing time in samples and component amplitude (unitless). Give the figure a legend and a title of Fig. 4.

(Note: if you were not able to calculate the mean-subtracted key-press segments,  q1, m[k]  and q2, m[k] ,  you

may use a version stored in soln.Q1_volts and  soln.Q2_volts to complete this question. These variables are 1000x80 matrices, ,  in the same format to the matrices pi described in Q1b.)

%% Q1d code figure(4), clf

Q1e. [6 marks] Interpret the two principal components in terms of the relative amplitude and polarity of the signals on each of the two electrodes.

(Note: if you were not able to calculate the first two principal components, and ,  you may use a version stored in soln.h1 and  soln.h2 to complete this question.)

Q1f. [8 marks] Project the segments qm onto the 2-dimensional projection space defined by the first two

principal components and : i.e. onto points (ym,yz m)=(hf qm, h⃞qm)    (Eq.1)

Plot the results in Figure 5 as a scatterplot of 2m versus Y1,m . Colour code the points in the figure so that "finger 1" = cyan, "finger 2" = blue, "finger 3" = green and "finger 4" = red. (The colour map cmap is provided for you.

It is a 4x3 matrix, where rows 1 to 4 correspond to the colours cyan, blue, green, and red, respectively.) Label your axes, including showing the projection amplitude scale in volts. Give the figure a title of Fig. 5.

(Note: if you were not able to calculate the first two principal components, and ,  you may use a version

stored in soln.h1 and  soln.h2 to complete this question. Similarly, if you were not able to calculate the

mean-subtracted key-press segments:  q1, m[k]  and q2, m[k] ,  you may use a version stored in soln.Q1_volts

and  soln.Q2_volts to complete this question. These variables are 1000x80 matrices, Q; ,  in the same format to the matrices pi described in Q1b.)

%% Q1f code figure(5), clf

Q1g. [8 marks] Propose a criteria based on the projections y1,m  and Y2,m  in Figure 5 to classify each key-press as corresponding to one of the fingers 1 to 4 (the criteria need not have perfect accuracy).  Explain the merits    and deficiencies of your criteria.

(Note: If you were not able to calculate the projections Y1,m  and Y2,m , you may use a version stored in

soln.ym1_volts and  soln.ym2_volts to complete this question. These variables are 1x80 vectors giving the projections to the key-presses m=1,.,80).

Question 2. Designing and implementing an on-line processing algorithm to determine when a key-press has  taken place and which finger (1 to 4) this corresponds to. This is based on principal components analysis from Question 1.

Consider the decomposition of the first principal component, h1=[h11;h12] ,  where

• hn is the subcomponent of hi  corresponding to signals from electrode 1, and

• is the subcomponent of corresponding to signals from electrode 2.

Similarly for the second principal component, h2= [h21;h22] , where

• is the subcomponent of corresponding to signals from electrode 1, and

• h22 is the subcomponent of hc corresponding to signals from electrode 2.

Q2a. [14 marks] [This question requires handwritten answers to be scanned and uploaded]. Considering, applying these subcomponents, hj ,  as templates for matched filters, sol kl  (k=0,…,999), to their

corresponding envelopes eynl so as to correctly transform. them into the principal component projection space (y[n], ya[n]) defined by Eq.1 in Q1f. That is, at each time step, n, xnl  is calculated by applying     the steps of mean subtraction and projection described in Q1b, Q1c and Q1f to the two vectors

e,in l=le, in-999),e,in-998],…e, in ll', (j=1,2) that contain the last 1000 samples of einl . Show that

(y[n], ya[n]) are given by

y[n]=(g11xe)[n]+(g12xe)[n]-h fp1-hp2     (Eq.2)

(Eq.3)

where (j=1,2) are the column vectors corresponding to the mean key-press segments,

.

Q2b. [10 marks] [This question requires handwritten answers to be scanned and uploaded]. Consider the pair of signals:

• dI[n]=hu[n] for n=0,…,999 ,  and d[n]=0 otherwise, on channel 1,

• dz[n]=h2[n] for n=0,…,999 ,  and d[n]=0 otherwise, on channel 2.

That is,dnl is the subcomponent of the first principal component corresponding to channel j, starting at time zero.

1. Prove that the value of delay, na , from the onset of the signal at n=0 that will lead to a maximal output y[n] for the matched-filter is n a=999 samples?

2. Theoretically, what will the value of be at this delay time, ? Explain your answer.

Q2c. [8 marks] Apply the formulae in Eqs. 2 & 3 to calculate the filter outputs (y[n], ya[n]) . Plot the trajectories of versus in the principal component projection space, in Figure 6.

Colour-code the trace so that those parts of the signal corresponding fingers 1, 2, 3 and 4 are plotted in cyan, blue, green and red, respectively.  In doing so, account for the delay, na ,  due to the matched filter (defined

in Eqs. 2  & 3) that would be expected to result in an optimal match to the key-press segments, p1, m[k]  and

p2, m[k] , defined in Q1b. (Note: The colour map cmap is provided for you. It is a 4x3  matrix, where rows 1 to 4 correspond to  the colours cyan, blue, green, and red, respectively.)

Label your axes showing the projection amplitude in volts. Give the figure a legend and a title of Fig. 6.

(Note:  if you were not able to calculate the envelopes elnl  and eznl  you may use a version stored in

soln.e1_volts and  soln.e2_volts to complete this question. Similarly, if you were not able to calculate the   first two principal components, hi  and hc ,  you may use a version stored in soln.h1 and  soln.h2 to complete this question. Similarly, if you were not able to calculate the mean key-press segments,   and  ,  you

may use a version stored in soln.p1_bar_volts and  soln.p2_bar_volts to complete this question. These variables are 1000x1 vectors, .)

%% Q2c code figure(6), clf

Q2d. [8 marks] Using your calculation of the filter outputs (y ln], ya[n]) , plot the trajectories of versus for just the third key-press for each finger, 1 to 4. To do this use the segment of the signal (y[n], ya[n]) that:

• corresponds to 500 ms before the third key-press was initiated, to 500 ms after the third key-press was initiated, and

• accounts for the delay, na ,  due to the matched filter (defined in Eqs. 2  & 3) that would be expected to result in an optimal match to the key-press segments, p1, m[k]  and p2, m[k] , defined in Q1b.

Once again, colour-code the trace for each finger 1, 2, 3 and 4 so they are plotted with cyan, blue, green and

red, respectively. (The colour map cmap is provided for you. It is a 4x3 matrix, where rows 1 to 4 correspond to the colours cyan, blue, green, and red, respectively.)

Label your axes showing projection amplitude in volts. Give the figure a legend and a title of Fig. 7.

(Note:  if you were not able to calculate the envelopes and you may use a version stored in

soln.e1_volts and  soln.e2_volts to complete this question. Similarly, if you were not able to calculate the   first two principal components, and ,  you may use a version stored in soln.h1 and  soln.h2 to complete this question. Similarly, if you were not able to calculate the mean key-press segments,   and  ,  you

may use a version stored in soln.p1_bar_volts and  soln.p2_bar_volts to complete this question. These variables are 1000x1 vectors, .)

%% Q2d code figure(7), clf

Q2e. [8 marks] Identify by sight the approximate "return" position (yr1, yr,.2) ,  evident in Figure 7; i.e. the

approximate point that the trajectory (y[n], ya[n]) returns to between each key-press. In Figure 8 plot the

distance, r[n] ,  of (y[n], yz[n]) from the "return" position (yr,1,yr,2) , as a function of time. (In your answer, specify the "return" position by giving values (yr,1, yr:2) to a 2-dimensional Matlab variable y_r in your code).

(Note: if you were not able to calculate the segments of and corresponding to the third key-press

in Q2d, you may use a version stored in soln.y1_3rd_volts and  soln.y2_3rd_volts to complete this

question. These variables are 1000x4 matrices, where each column corresponds to a key-press for fingers 1 to 4, respectively. )

%% Q2e code figure(8), clf

Q2f. [8 marks] Propose (in words) a criterion based on the distance, r[n] , for first identifying when a key-press occurred, regardless of which finger was used. Justify your answer using Figures 6, 7 and 8.

(Note: if you were not able to calculate r[n]  in Q2e, you may use a version stored in soln.r_volts) Answer:

Q2g. [14 marks] Based on your answers to Q1gand Q2f devise a method to determine from the EMG

envelope data when a key-press has taken place and which finger (1 to 4) this corresponds to. Apply your method to the data in envelopes and .

Report the results by first plotting the two EMG envelopes, elnj and elnl , in black over the duration of the  recording in Figure 9 in subplot(2,1,1) and subplot(2,1,2), respectively. Then plot on top an asterisk   (*) along the timeaxis of both subplots at the estimated time of each key-press from Q2f. Colour code each asterisk with black, blue, green or red, corresponding fingers 1, 2, 3 and 4, respectively.

Label your axes showing time in seconds and envelope amplitude in volts. Give the figure a title of Fig. 9.

%% Q2g code figure(9), clf

Total marks: 120