I have a similar problem with a Samsung TV that I haven't been able to solve, although it is the BBC SD channels that I have trouble with.
........
So I can only assume in my case at least that it is the TV that has a signal strength/quality problem, but of course I can get nothing out of Samsung!
TV tuners range in sensitivity within an allowable range, so you might find one TV has better or worse reception than another. However, all should give a picture so long as the aerial signal is also within acceptable limits.
Problems tend to arise if the tuner or TV aerial socket is broken, or if the aerial system has issues such as being split too many times or over amplified. Where the aerial on the roof and its down lead are in good working order and appropriate for the area, then in cases such as yours excessive splitting and over amplification are possibly the most common causes of problems.
From your description I'd say there's another clue too. You're having problems with the BBC SD channels. These are part of the core range of channels that most households should be able to receive as part of the Public Service Broadcasting (PSB) remit. They are broadcast in the transmitters higher power muxes. In short, these are the signals you should be able to receive in all but the worst conditions.
In the absence of other information, the fact that the Samsung picks up the lower power channels (I presume) without issue points to there being too much signal via your aerial system rather than too little. The signal for that mux could be swamping the tuner. This will result in problems that look very similar to low signal quality on digital, and it often goes misdiagnosed as exactly that.
The answer in most cases is to fit an attenuator to reduce the signal level for that particular TV. It's a very useful and very inexpensive diagnosis tool. This one is useful because there's a level control on it to allow the signal level to be adjusted:
MA1620 Adjustable 0-20dB attenuator, coaxial connector version
When making the adjustment, use the TV's Signal
Quality bar when tuned to the BBC SD channels as a guide. When there's too little or too much signal this will go down. When the level is getting towards the correct range then the bar will show higher quality. Stop adjusting when the bar indicates the highest quality. Check the other stations to make sure you still have good reception there too.
Let us know how you get on.