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Unveiling a fundamental link between information theory and estimationtheory, the I-MMSE relationship by Guo, Shamai and Verdu~\cite{gu05}, togetherwith its numerous extensions, has great theoretical significance and variouspractical applications. On the other hand, its influences to date have beenrestricted to channels without feedback or memory, due to the absence of itsextensions to such channels. In this paper, we propose extensions of the I-MMSErelationship to discrete-time and continuous-time Gaussian channels withfeedback and/or memory. Our approach is based on a very simple observation,which can be applied to other scenarios, such as a simple and direct proof ofthe classical de Bruijn's identity. This submission corrects the mistakes inthe previous version.

Extensions of the I-MMSE Relationship to Gaussian Channels with Feedback and Memory