It's actually NOT impossible to find the problem, it's just a matter of using logic to decide on a course of action. First step, think about the frequency range where this happens- since it's a booming sound, unplug the audio cables the subs and if it continues, the test proves the problem is in the subs. If it doesn't return, I would ask if you hear a low frequency hum when the subs are powered and connected to the rest of the system- this would usually indicate a ground loop and it must be eliminated. If you don't hear low frequency hum, connect some kind of audio recording device to the cable that sends signal to the subs and if the problem continues, it's somewhere ahead of the subs and it will be a matter of elimination to find it.
What kind of mics are used in this system? If they're hand-held, mounted on a podium and are powered by the board, unplug the cables from the board- if it stops, it may be a problem with the phantom power supply. If the mics are wireless, make sure the muting threshold isn't set too low. In this case, small sounds will cause the transmitter to send signal to the board and if could produce this sound. I would also look at the board to make sure it doesn't have its own muting threshold- they may be in conflict.
Make sure you were absolutely alone when you test the system- if someone else was handling a mic, switching the power on and off can cause this noise, too. Some wireless handheld mics should be switched on before the rest of the system. If the mics are the lavalier type (the small ones hung near the speaker's chest, with a small box that's worn on a belt), the transmitter may have been switched off. Also, if the batteries are close to depleted, they can cause intermittent operation in any wireless mic.
Bypass the signal processor- if it continues, the DSP isn't at fault.
What source was being used when the sound occurred? If it was used every time, substitute another device and see if it returns. If not, repair or replace it.