Chances are your modem has a fax capability. You may even already have fax software on your hard drive, and if not, there are plenty of fax programs you can download for free. Then you can just leave the program running at night, let it pick up and receive the fax, and voila, you'll know who's calling. Then you can file a complaint with the FCC.
Do you have a couple favorite fax programs that one can download for free, that you'd recommend?