Gaussian Channels with Feedback: A Dynamic Programming Approach

In this paper, we consider a communication system where a sender sends messages over a memoryless Gaussian point-to-point channel to a receiver and receives the output feedback over another Gaussian channel with known variance and unit delay. The sender sequentially transmits the message over multiple times till a certain error performance is achieved. The aim of our work is to design a transmission strategy to process every transmission with the information that was received in the previous feedback and send a signal so that the estimation error drops as quickly as possible. The optimal code is unknown for channels with noisy output feedback when the block length is finite. Even within the family of linear codes, optimal codes are unknown in general. Bridging this gap, we propose a family of linear sequential codes and provide a dynamic programming algorithm to solve for a closed form expression for the optimal code within a class of sequential linear codes. The optimal code discovered via dynamic programming is a generalized version of which the Schalkwijk-Kailath (SK) scheme is one special case with noiseless feedback; our proposed code coincides with the celebrated SK scheme for noiseless feedback settings.