A Representation Scheme

I had an idea once for vastly improving the quality of representation in a representative democracy. As far as I can recall I came up with it independently, although someone else must have suggested it first because it's so completely obvious. It's possible I read it somewhere and I'm just blanking out where.

The idea is this: instead of having any sort of election, you have people announce that they are willing to serve as representatives, and citizens choose one. A representative's vote in congress has a weight proportional to the number of constituents he represents. Perhaps a representative needs a minimum number of constituents (say, 5000 or so) to be seated. Representatives could be chosen from anywhere in the country, although it's likely that some of them would announce that they are particularly devoted to the interests of some particular area, and so citizens who live in that area who are particularly concerned with local issues would be likely to choose that representative. Perhaps changing representatives could be done at any time, or perhaps only once a year or so. Who represents whom would be a matter of public record, so if a representatives "constituents" were largely dead or fictitious persons this would quickly be discovered and he would be prosecuted.

It seems to me that something along those lines would enormously increase the degree to which a "representative" really did represents his constituents, would eliminate all concerns about districting/gerrymandering, and would substantially reduce voter fraud.

I mention the idea because the idea itself and its advantages (assuming one thinks accurate representation is a good thing) seem so obvious to me that I wonder why no nation as implemented it (although I think the German system of electing Bundestag members has some similarities). I can't recommend it as stated, because it would still seem to allow a majority to arbitrarily impose its will on a minority. But I think it may be useful as a mental stepping-stone.