IPHONE 4 WONT STAY ON WHILE I RESTORE IT.

it just started yesterday.  the phone would shut off randomly and then the only way for it to come back up is to plug it in.  sometime it complete shut off where i would have to turn it back on or plugging it would just go straight to the home screen.   now the phone wont go past the apple sign at all.  i got to the home screen two times today and one time it it had no connection and lost service and then it came back and then it had no service again.   so i turned it off to try and get signal again.   so eventually it came back on and the phone worked for about 2 minutes and then just complete shut off while still plugged in.   I am now trying to restore it and it keeps shutting of mid restore and then I have to keep putting it back into DFU mode or however you spell it.  The phone has the most up to date firmware but do I need to start the download over if the phone cycled randomly while it the download was still downloading from itunes or do I just put it back into that mode and let it continue downloading?