Extensions of the I-MMSE relation

Unveiling a fundamental link between information theory and estimation theory, the I-MMSE relation by Guo, Shamai and Verdú [4] has great theoretical significance and numerous practical applications. On the other hand, its influences to date have been restricted to channels without feedback or memory, due to the lack of extensions of the I-MMSE relation to such channels. In this paper, we propose extensions of the I-MMSE relation for discrete and continuous-time Gaussian channels with feedback 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 of the classical de Bruijn´s identity.