On precoding for constant K-User MIMO Gaussian interference channel with finite constellation inputs

Abstract

This paper considers linear precoding for constant channel-coefficient K-User MIMO Gaussian Interference Channel (K-MIMO GIC) where each transmitter-i (Tx-i) requires to send d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> independent complex symbols per channel use that take values from fixed finite constellations with uniform distribution to receiver-i (Rx-i), for i = 1, 2, ..., K. The maximum rate achieved by Tx-i as the signal to noise ratio (SNR) tends to infinity, using any linear precoder, when the interference channel-coefficients are zero is termed as Constellation Constrained Saturation Capacity (CCSC) for Tx-i. In this paper, we derive a high SNR approximation for the rate achieved by Tx-i when interference is treated as noise, which is given by the mutual information between Tx-i and Rx-i, denoted by I[X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ;Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ] where, X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> denotes the symbols generated at Tx-i before precoding and Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> denotes the symbols received at the antennas of Rx-i. Based on this high SNR approximation, we derive a set of necessary and sufficient conditions on the precoders under which I[X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ;Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ] tends to CCSC for Tx-i. Interestingly, the precoders that achieve interference alignment (IA) satisfy these necessary and sufficient conditions. However, finding precoders that achieve IA is known to be NP-hard in general whereas, the precoders that satisfy the derived necessary and sufficient conditions are easy to find for any given channel-coefficients. Further, we propose a gradient-ascent based algorithm to optimize the sum-rate achieved by precoding with finite constellation inputs and treating interference as noise. Simulation study for a 3-MIMO GIC with d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> = 1, for all i, equipped with two antennas at each node and QPSK inputs shows an improvement of 1.07 bits/sec/Hz in the ergodic sum-rate using the precoders obtained from the proposed algorithm over the precoders that achieve IA, at SNR = -2 dB.