When does the source-channel separation theorem hold?

The meeting point of the two main branches of the Shannon (1948) theory is the joint source-channel coding theorem. This theorem has two parts: a direct part and a converse part. It follows that either reliable transmission is possible by separate source-channel coding or it is not possible at all. This is the reason why the joint source-channel coding theorem is commonly referred to as the separation theorem. We characterize those channels for which the classical statement of the separation theorem holds for every source. We also characterize those sources for which the separation theorem holds for every channel. A conclusion to be drawn from our results is that when dealing with nonstationary probabilistic models, care should be exercised before applying the separation theorem