Using a Gaussian channel Twice

The problem of communicating one bit over a memoryless Gaussian channel with an energy constraint is discussed. It is assumed that the channel is allowed to be used only two times. An ideal feedback channel is also supposed available. The optimal feedback strategy and the bit-error probability are derived. It is shown that feedback gives a significant performance gain and that the optimal strategy is discontinuous. It is also shown that most of the performance increase can be obtained even with a one-bit feedback channel.