Coherence

We have seen that Power Spectrum represents the power carried by each frequency. When we have two signals, we can check the degree at which they are similar. To evaluate this degree we use `coherence'. In practice to calculate the coherence of two signals we calculate the cross-power spectrum of both signals with `pspect'.

Since coherence is a normalized measurement, the cross-power spectrum is divided by the squared root of the product of the spectra of the signals. Magnitude squared coherence (MSC) is frequently used. This is defined as the squared value of the cross-power spectrum divided by the product of the power of the spectra of both signals. Coherence has values between `0' and `1'. A value '0' means that this band is completely uncorrelated, while a value `1' means that, in this frequency, signals are completely correlated.

That a signal is correlated with other not only means that some energy in a frequency is present in both signals but that the frequency present in both signals is phase related. The coherence of white noise tends to `0'. Let us consider some related signals coming from the file signals. The file contains one recording of sleep with different electroencephalographic channels. We are going to calculate the coherence between three pairs of traces. We choose the traces contained in channel 8, 9, 11 and 12, that contain EOG (channels 8 and 9) , EMG of linked legs (11) and ECG (12).

-->//stacksize(20000000)
 
-->//load signals
 
-->x1 = signals(8);    // EOG Pos8M1
 
-->x2 = signals(9);    // EOG Pos18M1
 
 
-->y1 = signals(11);   // EMG linked legs
 
-->y2 = signals(12);   // ECG

Now we are going to plot the signals with the following code

-->inct = 0.01;
 
-->incf = 1 / (100 * inct) ;
 
 
-->segment = 1:500;
 
-->tscale = segment * inct;
 
 
-->xsetech([0,0,1,1/4])
 
-->plot2d(tscale ,x1(segment));
 
 
-->xsetech([0,1/4,1,1/4])
 
-->plot2d(tscale,x2(segment));
 
 
-->xsetech([0,2/4,1,1/4])
 
-->plot2d(tscale,y1(segment));
 
 
-->xsetech([0,3/4,1,1/4])
 
-->plot2d(tscale,y2(segment));

Figure 12.10: This figure plots a sample of the signals at which we are going to apply coherence analysis: the first and second signal represent EOG activity, the third the EMG of linked legs and de forth the ECG

Now we are going to calculate the coherence between both EOG channels and the coherence between the EMG and the ECG signals. The next code calculates the coherence

-->psx1 = pspect(50,100,'hn',x1);
 
-->psx2 = pspect(50,100,'hn',x2);
 
-->psx12 = pspect(50,100,'hn',x1,x2);
 
 
 
-->psy1 = pspect(50,100,'hn',y1);
 
-->psy2 = pspect(50,100,'hn',y2);
 
-->psy12 = pspect(50,100,'hn',y1,y2);
 
 
-->psxy = pspect(50,100,'hn',x1,y2);
 
 
-->cohx = ( (abs(psx12) ^ 2) ./ (psx1 .* psx2) );
 
-->cohy = ( (abs(psy12) ^ 2) ./ (psy1 .* psy2) );
 
-->cohxy = ( (abs(psxy) ^ 2) ./ (psx1 .* psy2) );

We calculate the spectra and the cross-spectra with `pspect'. In the previous code we calculated the coherence of both EOG channels (`cohx') the coherence of EMG with ECG (`cohy') and the coherence of an EOG channel with ECG (`cohxy'). So we calculated three coherence values. We want to represent these vectors. We are going to represent the values with the spectra of the original signals.

 
-->frec = [0:49] * incf;
 
-->xsetech([0,0,1/3,1/3]);                       // first row
 
-->plot2d (frec,20*log10( psx1(1:50))) ;
 
 
-->xsetech([1/3,0,1/3,1/3]);
 
-->plot2d (frec, cohx(1:50),1,'011',' ', [0,0,50,1] ) ;
 
 
-->xsetech([2/3,0,1/3,1/3]);
 
-->plot2d (frec,20* log10( psx2(1:50))) ;

The previous code plots the coherence of both EOG channels. We can expect a high degree of coherence since they have a common reference and EOG tends to present a wide electric field.

. Now, we are going to represent the coherence of EMG of legs with ECG.

-->xsetech([0,1/3,1/3,1/3]);                     // second row
 
-->plot2d (frec,20*log10( psy1(1:50))) ;
 
 
-->xsetech([1/3,1/3,1/3,1/3]);
 
-->plot2d (frec, cohy(1:50),1,'011',' ', [0,0,50,1] ) ;
 
 
-->xsetech([2/3,1/3,1/3,1/3]);
 
-->plot2d (frec,20* log10( psy2(1:50))) ;

Finally we will represent the coherence of an electrooculographic channel with ECG. Notice that coherence is concentrated in the frequencies present in QRS complex.

-->xsetech([0,2/3,1/3,1/3]);                     // third row
 
-->plot2d (frec,20*log10( psx1(1:50))) ;
 
 
-->xsetech([1/3,2/3,1/3,1/3]);
 
-->plot2d (frec, cohxy(1:50),1,'011',' ', [0,0,50,1] ) ;
 
 
 
-->xsetech([2/3,2/3,1/3,1/3]);
 
-->plot2d (frec,20* log10( psy2(1:50))) ;

And here is the result

Figure 12.11: The three rows of frames represent the coherence of both EOG channels (upper traces), the coherence between an EMG signal with ECG (middle traces) and the coherence between EOG and ECG (lower traces). For each row, the spectra of both signals is represented in logarithmic scale at the right and at the left of coherence, that is represented in the middle. A sample of the original signals can be seen in the previous figure

We can see that coherence is very high when we consider both electrooculographic channels. When we consider the coherence between EMG and ECG we can see that it is very high among 5 and 25 Hz. As we could see in the previous chapter the activity of the QRS complex is located principally in this band. A similar coherence is present between EOG and ECG although the presence of QRS complexes in the EOG trace is less evident.