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It is possible to send a signal weaker than the noise floor. The maximum information you can send is defined by shannon's channel capacity formula, which clearly says a signal can be weaker than the noise.

If you use a self-clocking signal, then effectively the information you are sending consists of both the clock timings and the information you're really trying to send. If your channel capacity is very low, you will find there is no channel capacity for even the clocking information, let alone the information you really want to send.



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