Communicating One Bit over a Delay Constrained Gaussian MIMO Channel with Feedback

The energy-optimal scheme is found for communicating one bit over a memoryless Gaussian channel with an ideal feedback channel. It is assumed that the channel is allowed to be used at most N times before decoding. The optimal coding/decoding strategy is derived by dynamic programming. It is found that feedback gives a significant performance gain and that the optimal strategies are discontinuous. It is also shown that most of the performance increase can be obtained even with a one-bit feedback channel. The optimal scheme is compared with the strategy by Kailath-Schalkwijk and is found to be significantly more effective. For the case of a diagonal MIMO channel where measurement noise variances are equal along the sub channels we also show that the problem can be reduced to the previous case of transmitting one bit over a scalar feedback channel. 1