I am studying for my signals and systems final in 3 hours and cannot seem to wrap my head around one of the homework/practice problems. We are given the following figure:
For each figures (1, 2, and 3) we are asked to prove that the 3 dB bandwidth of the system is:
I understand how to go about doing the first system. I first calculate |H(w)| and then get |H(0)| and |H(Wc)| and then run the results through 20log(|H(Wc)|/|H(0)|) (which comes out to -3 dB).
However I cannot figure out how to calculate the transfer function for the next two systems. The solutions provide me with this formula:
Which for the second system ends up being
However I would love to understand how they got to that point. I guess I do not understand how to translate the circuit diagram into a new equation.
Can anyone provide some hints in this regard? Thanks!