If for any reason iTunes error code ‘1602’ comes up during your restore process then the key is not to have a power cable plugged in to the Apple TV. Disconnect your Apple TV power cable (while still keeping microUSB cable in) and try again.
To start over again you will need to put your Apple TV (2nd Gen black) into DFU mode for iTunes to recognise it again so you can restore it.
If your microUSB cable won’t power your Apple TV (without its power cable plugged in) then you don’t have the appropriate microUSB cable. I used my microUSB cable that comes with my Jawbone headset for my iPhone (it’s the short grey cable that connects my Jawbone to the power block). That little thing yes. That powers my Apple TV without a power cable and gets rid of the ‘1602’ iTunes error code.
Please let us know if you have other solutions that work as well.