The scale parameter in wavelet decomposition refers to the level of decomposition or the number of decomposition stages. It determines the number of sub-bands or frequency bands obtained after the decomposition.
In the context of the paper you mentioned, the scale of 2^3 corresponds to performing three levels of decomposition. Each level of decomposition generates sub-bands representing different frequency ranges. The specific frequency range for each sub-band depends on the chosen wavelet and the sampling rate of the signal.
Regarding the sampling rate, the choice of the scale or number of decomposition levels is not directly related to the sampling rate. However, it is essential to consider the Nyquist frequency, which is half the sampling rate. The highest frequency that can be accurately represented in the signal is limited by the Nyquist frequency. Therefore, it is advisable to choose the scale and wavelet basis according to the frequency range of interest and the Nyquist frequency.
In the case of the paper, where the raw EEG signal was sampled at 250Hz, it is not explicitly mentioned whether the signal was downsampled or not. If the signal is already sampled at the desired frequency range (47-63Hz), downsampling may not be necessary. However, if the signal contains higher frequency components, it might be beneficial to preprocess the signal by low-pass filtering or downsampling before applying the wavelet decomposition.
To reproduce the processing step in the paper, you can use the Coiflet-1 wavelet basis and perform three levels of decomposition using the wavelet package. Adjust the parameters according to your specific sampling rate and frequency range of interest.
Please note that replicating the exact results of the paper may also require considering other preprocessing steps or specific implementation details mentioned in the paper.
I hope this clarifies your doubts. If you have any further questions, feel free to ask!