[Paper Review] Linear Precoding Based on Polynomial Expansion: Reducing Complexity in Massive MIMO

Axel Müller, Abla Kammoun+2 more|arXiv (Cornell University)|Oct 7, 2013

Advanced MIMO Systems Optimization33 references13 citations

TL;DR

This paper proposes Truncated Polynomial Expansion (TPE) precoding as a low-complexity alternative to regularized zero-forcing (RZF) in massive MIMO systems, replacing matrix inversion with a polynomial approximation of the precoding matrix. The method achieves near-optimal spectral efficiency with significantly reduced computational latency, and the optimal polynomial coefficients are derived in closed form using random matrix theory to maximize the asymptotic SINR.

ABSTRACT

Large-scale multi-user multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements for future communication systems. Counter-intuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more-or-less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission (MRT) and more advanced RZF. By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in large-scale MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio (SNR).

Motivation & Objective

• Address the high computational complexity of optimal linear precoding schemes like RZF in massive MIMO, which require expensive matrix inversions in every coherence block.
• Develop a hardware-friendly precoding scheme suitable for real-time implementation by replacing matrix inversion with a truncated polynomial expansion (TPE).
• Derive a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding using random matrix theory.
• Provide a closed-form expression for the polynomial coefficients that maximize the asymptotic SINR, enabling performance optimization without iterative computation.
• Characterize the required polynomial degree as a function of channel quality and SNR, showing it does not scale with system size to maintain a fixed rate loss relative to RZF.

Proposed method

• Replace the matrix inversion in RZF precoding with a truncated polynomial expansion (TPE) of the form $ \mathbf{W} = \sum_{j=0}^{J-1} c_j \left( \frac{1}{K} \mathbf{H} \mathbf{H}^H \right)^j $, where $ \mathbf{H} $ is the channel matrix.
• Use random matrix theory to derive a deterministic equivalent for the asymptotic SINR, enabling analysis in the large-system limit ($ M, K \to \infty $ with $ M/K \to \beta $).
• Formulate the optimization of polynomial coefficients as a convex problem by maximizing the deterministic SINR approximation, leading to a closed-form solution.
• Introduce an iterative algorithm to compute the polynomial coefficients efficiently, based on recursive computation of traces and matrix products.
• Ensure hardware feasibility by designing the TPE scheme to allow smooth transition from simple maximum ratio transmission (MRT) to advanced RZF as the polynomial degree $ J $ increases.
• Validate the asymptotic SINR approximation through simulations and theoretical convergence analysis, showing tightness even at moderate system dimensions.

Experimental results

Research questions

• RQ1 Can a truncated polynomial expansion (TPE) effectively approximate the optimal RZF precoder while drastically reducing computational complexity?
• RQ2 What is the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems, and can it be expressed deterministically using random matrix theory?
• RQ3 What is the optimal set of polynomial coefficients for TPE that maximizes the asymptotic SINR, and can they be computed in closed form?
• RQ4 How does the required polynomial degree $ J $ scale with system parameters such as channel quality and SNR to maintain a fixed rate loss relative to RZF?
• RQ5 Does the TPE precoding scheme maintain robust performance across varying channel conditions and system dimensions, particularly in the large-$ M, K $ regime?

Key findings

• The TPE precoding scheme achieves asymptotically optimal spectral efficiency with a computational complexity that scales linearly with the number of users $ K $, compared to $ O(K^2M) $ for RZF.
• The asymptotic SINR of TPE precoding is accurately approximated by a deterministic expression derived via random matrix theory, which holds even for moderate system dimensions.
• The optimal polynomial coefficients for TPE are derived in closed form, maximizing the asymptotic SINR without requiring iterative or complex optimization.
• The required polynomial degree $ J $ does not need to scale with the number of base station antennas $ M $ to maintain a fixed rate loss relative to RZF, but should increase with channel quality and SNR.
• The TPE method enables a smooth transition from MRT to RZF performance by adjusting $ J $, making it adaptable to varying hardware capabilities.
• The proposed iterative algorithm efficiently computes the TPE coefficients with low overhead, supporting real-time implementation in massive MIMO systems.