how to rectify/amplify voltage across current sensing resistor?
30mV (peak-peak?) is in the same order as the output from a microphone so a 100x voltage amplifier could easily take this up to 3V. A precision rectifier (active diode) simply uses the op amp (open loop) gain to overcome the turn on voltage of a diode.

https://en.wikipedia.org/wiki/Precision_rectifier