How is your DirecTivo connected to that TV? Have you tried different cables and/or different AV connections?
Go look at wikipedia's article on "capacitor plague" and go to badcaps.net and educate yourself about how to visually spot caps gone bad, and then carefully inspect the ones in your DirecTiVo's power supply.I have the Samsung SIR-S4040R and have owned it for about 7 years now. It has worked great until the last few months. Now on the picture I have has a bunch of static horizontal lines on it. It's not actually static but covers the entire screen. It also seems random noises coming from it every once and awhile. No reboots or anything. I have reset the Tivo once to see if a reboot would help and there was a gray screen at bootup that eventually went away and everything loaded up normally.
I verified it was not the tv by hooking up my Directv HD DVR to the tv upstairs. The pictures is crystal clear. This also rules out a cable line issue. Also my tv is a TV/DVD combo and when playing a DVD the picture is crystal clear. So with that said I can only conclude that it is the Tivo.
I have the Directv protection plan and according to the terms of the plan since I own this Tivo Directv would give me a replacement DVR which I would own and not lease.
However, my wife really likes the Tivo interface and doesn't care much for the Directv interface. Is it worth fixing if in fact it is the Tivo?