Mutual Information and Minimum Mean-square Error in Gaussian Channels
<|reference_start|>Mutual Information and Minimum Mean-square Error in Gaussian Channels: This paper deals with arbitrarily distributed finite-power input signals observed through an additive Gaussian noise channel. It shows a new formula that connects the input-output mutual information and the minimum mean-square error (MMSE) achievable by optimal estimation of the input given the output. That is, the derivative of the mutual information (nats) with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input statistics. This relationship holds for both scalar and vector signals, as well as for discrete-time and continuous-time noncausal MMSE estimation. This fundamental information-theoretic result has an unexpected consequence in continuous-time nonlinear estimation: For any input signal with finite power, the causal filtering MMSE achieved at SNR is equal to the average value of the noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is chosen uniformly distributed between 0 and SNR.<|reference_end|>
@article{guo2004mutual,
    title={Mutual Information and Minimum Mean-square Error in Gaussian Channels},
    author={Dongning Guo, Shlomo Shamai (Shitz), and Sergio Verdu},
    journal={arXiv preprint arXiv:cs/0412108},
    year={2004},
    archivePrefix={arXiv},
    eprint={cs/0412108},
    primaryClass={cs.IT math.IT}
}