There might be a little bit of reading involved but please help! You may find my specs at the bottom of this post.
-- To my knowledge, the HD5870 and LG 32" 32LD550 is HDMI 1.3 compatible. I went to Best Buy and asked for an HDMI 1.3 cable and, after 5-10 minutes of research, he handed me a Rocket Fish HDMI cable (It does not say 1.3 on it anywhere. All it says is 10.2Gbps, 1080p, 120hz).
-- Here's where I'm having the problem. I plugged my LG 32" in with every HDMI cable I have plus the one I purchased at Best Buy. While connected through HDMI, The 32" LG displays POST and the "Windows is Starting" screen...but it goes to "NO SIGNAL" right before my desktop pops up. It is OKAY till it reaches my desktop.
So since the LG 32LD550 came with a VGA slot in the back, i plugged in a spare DVI - VGA cable into the TV and my HD5850 and VIOLA...everything works fine...this is actually how I am typing to the world as we speak.
-- I also tried plugging in the hdmi cable from the HD5870 to my 23" LG monitor (the one that I am replacing with the 32"). I booted up, opened up Catalyst Control Center and proceeded to plug in the LG 32LD550. BOTH DISPLAYS (23" and 32") SHOW UP IN CATALYST CONTROL CENTER. I tried switching displays and my 23" just went black while the 32" continued with its "NO SIGNAL" bullcrap.
Try lowering the desktop resolution for HDMI in Windows 7. This error usually occurs when the LCD HDTV cannot handle a certain desktop resolution.
Not all HDTV's are built equally the same and some have cheaper components than others. I've had a1080p LCD HDTV that can only display a resolution of 1366x768 for the PC through HDMI, but was able to display a resolution of 1920x1080 just fine on DVI and VGA. Vice versa on another HDTV. If it is a hardware limitation then there is nothing much you can do about it.
The reason it still shows the "Windows is starting" screen is because on default the "Windows is starting" screen is displayed on a very low resolution; probably at a resolution of 640x480 or lower.
P.S. Not all HDTVs are meant to be used as computer monitors; due to many limitations like: lower resolutions on certain interfaces, large noticable pixels, slower response times, etc.
I have the same TV running off an HD5750 and had the same problem. I can't recall every step done, since it was a while ago, but I'll type what I remember.
I opened the AMD Control Center and under My Digital Flat-Panels, selected HDTV Support. Then, I checked the box for "Add 1080p60 format to the Display Manager (NTSC)". Clicking Apply when needed. Once the desktop appeared on the TV, I went into Scaling Options and moved the slider to 0% Overscan to get rid of the "black frame".
Like I mentioned earlier, there are probably some steps missing in the beginning, but maybe something above might help you out.