HPLC Analysis EEG

Question

I want to conduct a frequency-frequency analysis of the EEG signal. I found a GSL wavelet function to calculate wavelet coefficients. How can I extract the actual frequency bands (e.g. 8-12 Hz) from these coefficients? The GSL manual says:

For direct conversion, the elements of the original array are replaced by a discrete wavelet transform f_i -> w_{j,k} in a packed triangular storage, where J is the level index j = 0 ... J-1 and K is the coefficient index in each level, k = 0 ... (2^j)-1 . The total number of levels is J = \log_2(n) .

The output is as follows: (s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ..., d_{j,k}, ..., d_{J-1,2^{J-1}-1})

If I understand correctly that the output array data[] contains in position 1 (for example, data[1] ) the amplitude of the frequency band 2 ^ 0 = 1 Hz and

 data[2] = 2^1 Hz data[3] = 2^1 Hz data[4] = 2^2 Hz until data[7] = 2^2 Hz data[8] = 2^3 Hz 

etc.

This means that I only have amplitudes for the frequencies 1 Hz, 2 Hz, 4 Hz, 8 Hz, 16 Hz, ... How can I get, for example, the amplitude of the frequency component oscillating at 5.3 Hz? How can I get the amplitude of the entire frequency range, for example. amplitude of 8-13 Hz? Any recommendations on how to get a good frequency distribution?

Answer 1

I'm not sure how familiar you are with general signal processing, so I’ll try to be clear, but I don’t chew food for you.

Wavelets essentially filter banks . Each filter splits a given signal into two non-overlapping independent high-frequency and low-frequency sub-bands, so that they can then be restored using the inverse transform. When such filters are constantly applied, you get a filter tree with the output from one filed to the next. The simplest and most intuitive way to build such a tree is as follows:

• Decompose the signal into low-frequency (approximate) and high-frequency (detailed) components
• Take the low frequency component and do the same processing on this
• Continue until you process the required number of levels.

The reason for this is that you can downsample to get the approximation signal received. For example, if your filter breaks a signal with a sampling frequency (Fs) of 48000 Hz, which gives a maximum frequency of 24000 Hz according to the Nyquist theorem - in a component with an approximation of 0 to 12000 Hz and a component of detail from 12001 to 24000 Hz, you can take every second sample approximating component without aliasing , significantly reducing the signal. It is widely used in zoom and image compression .

In accordance with this description, at the first level, you divide the frequency content in the middle and create two separate signals. Then you take your low-frequency component and again divide it into the middle. Now you get three components: from 0 to 6000 Hz, from 6001 to 12000 Hz and from 12001 to 24000 Hz. You see that two new components are half the strip width of the first component of the part. You get the following picture:

This correlates with the passband described above ( 2^1 Hz , 2^2 Hz , 2^3 Hz , etc.). However, using the broader definition of the filter bank, we can arrange the tree structure indicated above as we like, and it will still remain the filter bank. For example, we can combine both the approximation and the part of the part to be divided into two high-frequency and low-frequency signals, for example:

If you look closely at this, you will see that both the high-frequency and low-frequency components are in the middle at their frequencies, and as a result you get a single filter bank, whose frequency separation is more like this:

Please note that all ranges are the same size. By creating a single filter bank with N levels, you get answers from 2 ^ (N-1) bandpass filters. You can fine-tune your filter bank to ultimately give you the desired range (8-13 Hz).

In general, I would not advise you to do this with bursts. You can read the literature on creating good bandpass filters and just create a filter that would only pass 8-13 Hz of your EEG signals. This is what I did before, and it worked fine for me.