An electroencephalographic (EEG) signal x(t) is recorded from the surface of the head (scalp) via an EEG recording machine. The EEG machine has a bank of 4 ideal band-pass (BP) analog filters, H1(jω), H2(jω), H3(jω) and H4(jω), which are in parallel with each other. The bandwidths of these filters are 0-4Hz, 4-8Hz, 8-12Hz and 12-16Hz (plus the respective ones in the negative portion of the spectrum). Let's assume that the operator has full control of the magnitude of the spectrum in the pass-band of each of those filters.
a) Sketch a block diagram of the recording system described above
b) Select the magnitude of the frequency response at the pass-bands (GAIN) of each of the four filters if you need to filter out all EEG components except the alpha rhythm, which you would additionally like to amplify 1,000 times before you sample and send it to further digitally analyze it on the computer (see the assigned Project for those subsequent steps). [The alpha rhythm in the EEG is about 10Hz and signifies the level of alertness of a subject.]
c) Derive the global frequency response H(jω) of the EEG machine in terms of H1(jω), H2(jω), H3(jω) and H4(jω).
d) Derive the frequency response H(jω) of the EEG machine in terms of only H1(ω).