Well before you worry about the software signal analysis method how are you handling the electrical interface from the microphone to the analog input pin? You know that audio is an AC voltage of variable amplitude and frequencies, right? An arduino cannot safely measure the negative portion of an AC signal, just DC voltages in the range of 0 to +5vdc. Also a microphone without additional amplification outputs a very small AC voltage range just some millivolts. So how is your electrical part set-up so far?
Lefty