Synthesis of signals with convolution

The second example illustrates the convolution with the use of two signals; one of them represents a shape similar to a motor unit potential and the other one something similar to the discharge rate of this unit. Our aim is to produce a trace analogous to an electromyographic signal at weak effort.

We are going to sample the signal at 20 kHz, so the sampling interval is 0.05 ms. We are going to represent 500 ms (0.5 s). We build a vector containing the time and a vector containing a shape similar to a motor unit potential.

-->t = 0:.05:500;
 
-->mup =  - (sin(%pi*[0:127]/128).^4) .* (sin(2*%pi*[0:127]/128));

To build the discharge rate we create a vector of 500 ms (it takes the characteristics of `t') and then introduce some values at the point where the motor unit potential will appear.

-->discharges = 500 : 1000 : 9500;

-->rate = zeros(1,length(t));
 
-->rate(discharges)=1;

Now we make the convolution between the 'motor unit potential' and the 'discharge' and plot the result

-->emg = convol (rate,mup);
 
-->xsetech([0,0,1/2,1/2]);
 
-->plot2d(t(1:length(mup)),mup);
 
-->xsetech([1/2,0,1/2 1/2]);
 
-->plot2d(t(1:length(rate)),rate,1,"011","a ",[0,0,500,3],[5,5,0,0]);
 
-->xsetech([0,1/2,1,1/2]);
 
-->plot2d(t(1:10000),emg(1:10000));

And here is the result

Figure 8.3: Generation of an EMG signal. The lower trace plots the convolution between a shape similar to a motor unit potential and a rhythmical discharge. Notice the different scale in x-axis. The upper frames plot the `motor unit potential' and the `motor unit potential discharge'.

Notice the different scale of the signals (both in ms). At the top line the left frame represents the `motor unit potential' (which lasts about 6 ms) and the right frame plots the discharge rate covering 0.5 s. The lower trace represents the convoluted signal. Each value of `1' in the discharge frame becomes a motor unit potential in the EMG signal. If we had added several units with different forms and discharge rates a more realistic electromyographic signal would have been got.

The inverse of the above process is called `deconvolution' and implies the extraction of the shape of the different MUP participating in the electromyographic signal. The motor unit potential contains information on the structure of the motor unit while the discharge pattern contains information on the organization of the muscular activity. Both aspects are mixed in real electromyographic traces and contain information used in neuromuscular diagnosis.