Linear Coding for AWGN Channels With Noisy Output Feedback via Dynamic Programming

The optimal coding scheme for Additive White Gaussian noise (AWGN) channels with noisy output feedback has been unknown for several decades. The best-known linear scheme is by Chance and Love, where the coefficients of the linear scheme are numerically optimized based on unique observations. In this paper, we introduce a new class of linear coding schemes, called
sequential linear schemes
, where the encoder sequentially updates a linear state process based on feedback. We then derive the optimal scheme within this class, in a closed form, by formulating a novel Markov decision process and solving it via dynamic programming. We demonstrate that our scheme outperforms the Chance-Love scheme for channels with noisy feedback and coincides with the Shalkwijk-Kailath scheme for channels with noiseless feedback. This problem is an instance of decentralized control
without any common information
and, to the best of our knowledge, the first such scenario where we can derive analytical solutions using dynamic programming.