EEG Wavelet Analysis - signal-processing

Question

I want to do a time-frequency analysis of an EEG signal. I found the GSL wavelet function for computing wavelet coefficients. How can I extract actual frequency bands (e.g. 8 - 12 Hz) from that coefficients? The GSL manual says:
For the forward transform, the elements of the original array are replaced by the discrete wavelet transform f_i -> w_{j,k} in a packed triangular storage layout, where J is the index of the level j = 0 ... J-1 and K is the index of the coefficient within each level, k = 0 ... (2^j)-1. The total number of levels is J = \log_2(n).
The output data has the following form, (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 that right an output array data[] contains at position 1 (e.g. 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
and so on ...
That means I have only the amplitudes for the frequencies 1 Hz, 2 Hz, 4 Hz, 8 Hz, 16 Hz, ... How can I get e.g. the amplitude of a frequency component oscillating at 5.3 Hz? How can I get the amplitude of a whole frequency range, e.g. the amplitude of 8 - 13 Hz? Any recommendations how to get a good time-frequency distribution?

Answer 1

I'm not sure how familiar you are with general signal processing, so I'll try to be clear, but not chew the food for you.
Wavelets are essentially filter banks. Each filter splits a given signal into two non-overlapping independent high frequency and low frequency subbands such that it can then be reconstructed by the means of an inverse transform. When such filters are applied continually, you get a tree of filters with output of one fed into the next. The simplest, and the most intuitive way to build such tree is as follows:
Decompose a signal into low frequency (approximation) and high frequency (detail) components
Take the low frequency component, and perform the same processing on that
Keep going until you've processed the required number of levels
The reason for this is that you can then downsample the resulting approximation signal. For example, if your filter splits a signal with sampling frequency (Fs) 48000 Hz -- which yields maximum frequency of 24000 Hz by Nyquist Theorem -- into 0 to 12000 Hz approximation component and 12001 to 24000 Hz detail component, you can then take every second sample of the approximation component without aliasing, essentially decimating the signal. This is widely used in signal and image compression.
According to this description, at level one you split your frequency content down the middle and create two separate signals. Then you take your lower frequency component and split it down the middle again. You now get three components in total: 0 to 6000 Hz, 6001 to 12000 Hz, and 12001 to 24000 Hz. You see that the two newer components are both half the bandwidth of the first detail component. You get this sort of a picture:
This correlates with the bandwidths you describe above (2^1 Hz, 2^2 Hz, 2^3 Hz and so on). However, using a broader definition of a filter bank, we can arrange the above tree structure as we like, and it will still remain a filter bank. For example, we can feed both approximation and detail component to to split into two high-frequency and low-frequency signals like so
If you look at it carefully, you see that both high and low frequency components down the middle in their frequencies and as a result you get a uniform filter bank whose frequency separation looks more like this:
Notice that all bands are of the same size. By building a uniform filter bank with N levels, you end up with responses of 2^(N-1) band-bass filters. You can fine-tune your filter bank to eventually give you the desired band (8-13 Hz).
In general, I would not advise you to do this with wavelets. You can go through some literature on designing good band-pass filters and simply build a filter that would only let through 8-13 Hz of your EEG signals. That's what I've done before and it worked quite well for me.