I don't really understand FFTs, and filtering neither. I hope I can explain my confusion in a way that is not too confusing for you to understand.
consider the following example: I have an EEG signal, and am interested in the frequency bands from 1 to 40 Hz.
if I apply a LP filter at 40, and a HP at 1, my signal of interest would be distorted around the beginning and end because of the filter's transition band. right? why does it have a transition band at all, and how does it work? is a filter a sum of sine waves in the specified frequency range? (sorry, seems like a basic question, still I couldn't find an answer to it)
now to get the FFT of different frequency bands in the 1-40Hz range, I could either first bandpass filter the specific ranges and apply FFT to each (e.g. to 1-4,4-8,8-12,12-20). however, this would lead to a distorted result in each band, because of the transition bands, right?
if on the other hand, I do FFT over the whole 1-40 range, and then just 'extract' the results from 8-12 etc, the results would be more accurate? and how would the extraction mathematically work, is it just the mean of all values at the frequencies between 8 and 12?
I hope you can help me understand these things