Multiuser I-MMSE

TL;DR

This paper extends the fundamental relation between mutual information and MMSE to multiuser channels, deriving new formulas and insights for multiuser interference, especially in low-SNR regimes and for BPSK modulation.

Contribution

It introduces a generalized relation between mutual information derivative and MMSE in multiuser settings, including interference effects and closed-form expressions for specific channels.

Findings

Derived the relation between mutual information derivative and MMSE with interference terms.

Obtained closed-form mutual information expressions for two-user Gaussian MAC with BPSK.

Analyzed the low-SNR regime for multiuser MMSE and mutual information.

Abstract

In this paper, we generalize the fundamental relation between the derivative of the mutual information and the minimum mean squared error (MMSE) to multiuser setups. We prove that the derivative of the mutual information with respect to the signal to noise ratio (SNR) is equal to the MMSE plus a covariance induced due to the interference, quantified by a term with respect to the cross correlation of the multiuser input estimates, the channels and the precoding matrices. We also derive new relations for the gradient of the conditional and non-conditional mutual information with respect to the MMSE. Capitalizing on the new fundamental relations, we derive closed form expressions of the mutual information for the multiuser channels, particularly the two user multiple access Gaussian channel driven by binary phase shift keying (BPSK) to illustrate and shed light on methods to derive similar expressions for higher level constellations. We capitalize on the new unveiled relation to derive the multiuser MMSE and mutual information in the low-SNR regime.