Of course you
can put up a LCD without upgrading your cable. I'm not sure "digital" cable would really make a difference, as it's still broadcasted in standard definition or 480i. What you really need to make that LCD shine is HD content! Usually this is in the form of Blu-ray (1080p) or HD Cable or Satellite Television (1080i or 720p). Some companies offer "cable cards" to put into televisions instead of using cable boxes if that is one of your concerns. You could also put the cable box in a closet or away from the display. Having an HDTV with no HD source is like driving a ferrari on a dirt road. It's possible, but isn't the optimum.
Typically fixed pixel displays (Plasma, LCD, DLP, etc.) have a hard time with SD content. Generally this is because the picture then has to be scaled to the display's native resolution (usually 1080p in the case of modern LCD's). Some do better than other, so I'd check online reviews before purchasing a specific model. Typically you can get an idea of how it will do with SD content.
Hope this long, rambling post helps. In anything you do, keep reading/researching, there's a ton of good info on this site if you use the search function or just browse around!!
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