On the use of training sequences for channel estimation

Suppose Q is a family of discrete memoryless channels. An unknown member of Q will be available with perfect (causal) feedback for communication. A recent result [9] shows the existence, for certain families of channels (e.g. Binary Symmetric Channels and Z channels), of coding schemes that achieve Burnashev's exponent universally over these families. In other words, in certain cases, there is no loss in the error exponent by ignoring the channel: transmitter and receiver can design optimal blind coding schemes that perform as well as the best feedback coding schemes tuned for the channel under use. Here we study the situation where communication is carried by first testing the channel by means of a training sequence, then coding the information according to the channel estimate. We provide an upper bound on the maximum achievable error exponent of any such scheme. If we consider Binary Symmetric Channels and Z channels this bound is much lower than Burnashev's exponent. This suggests that in terms of error exponent, a good universal feedback scheme entangles channel estimation with information delivery, rather than separating them.