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The optimal coding scheme for communicating a Gaussian message over anAdditive White Gaussian noise (AWGN) channel with AWGN output feedback, with alimited number of transmissions is unknown. Even if we restrict the scope ofthe coding scheme to linear schemes, still, deriving the optimal coding schemeis a challenging task. The state-of-the-art linear scheme for channels withnoisy feedback is by Chance and Love, where the coefficients of the linearscheme are numerically optimized based on unique observations [1]. In thispaper, we introduce a new class of sequential linear schemes for this channelby introducing a novel linear state process at the transmitter and derive theoptimal sequential scheme within this class of schemes in a closed-form byformulating a novel Dynamic Programming (DP). We empirically show that ourscheme outperforms the state-of-the-art linear scheme in [1] for noisy feedbackand coincides with the SK scheme for noiseless feedback. We also show that incommunicating message bits as opposed to a Gaussian message, a learning-basedapproach further improves the reliability of sequential linear schemes. Thisproblem is an instance of decentralized control without any common informationand to the best of our knowledge the first such scenario where we can deriveanalytical solutions using a DP.

Linear Coding for AWGN channels with Noisy Output Feedback via Dynamic Programming