Low-Complexity Statistically Robust Precoder/Detector Computation for Massive MIMO Systems

Massive MIMO is a variant of multiuser MIMO in which the number of antennas at the base station (BS) $M$ is very large and typically much larger than the number of served users (data streams) $K$. Recent research has illustrated the system-level advantages of such a system and in particular the beneficial effect of increasing the number of antennas $M$. These benefits, however, come at the cost of dramatic increase in hardware and computational complexity. This is partly due to the fact that the BS needs to compute suitable beamforming vectors in order to coherently transmit/receive data to/from each user, where the resulting complexity grows proportionally to the number of antennas $M$ and the number of served users $K$. Recently, different algorithms based on tools from random matrix theory in the asymptotic regime of $M,K \to \infty$ with $\frac{K}{M} \to \rho \in (0,1)$ have been proposed to reduce such complexity. The underlying assumption in all these techniques, however, is that the exact statistics (covariance matrix) of the channel vectors of the users is a priori known. This is far from being realistic, especially that in the high-dim regime of $M\to \infty$, estimation of the underlying covariance matrices is well known to be a very challenging problem. In this paper, we propose a novel technique for designing beamforming vectors in a massive MIMO system. Our method is based on the randomized Kaczmarz algorithm and does not require knowledge of the statistics of the users channel vectors. We analyze the performance of our proposed algorithm theoretically and compare its performance with that of other competitive techniques via numerical simulations. Our results indicate that our proposed technique has a comparable performance while it does not require the knowledge of the statistics of the users channel vectors.

💡 Research Summary

The paper addresses the computational bottleneck inherent in massive MIMO systems, where a base station equipped with hundreds of antennas must serve tens of single‑antenna users. Conventional linear precoders and detectors such as zero‑forcing (ZF), regularized ZF (RZF), and minimum‑mean‑square‑error (MMSE) require matrix inversions and matrix‑matrix multiplications whose complexities scale as O(K³) and O(MK²), respectively. Existing low‑complexity approaches—truncated polynomial expansion (TPE), approximate message passing (AMP), and other random‑matrix‑theory‑based methods—reduce this burden but rely on accurate knowledge of each user’s channel covariance matrix. In the high‑dimensional regime (M≫K) estimating and tracking these M×M covariance matrices is prohibitive in both sample complexity and storage.

The authors propose a “statistics‑free” solution based on the randomized Kaczmarz algorithm (KA). KA iteratively solves a linear system Ax = b by selecting rows of A at random (with probability proportional to the squared row norm) and projecting the current estimate onto the hyperplane defined by the selected equation. This simple update requires only a dot product and a scalar multiplication, i.e., O(K) or O(M) operations per iteration, and converges exponentially fast with a rate determined by the condition number of A.

Two novel extensions are introduced to adapt KA to massive MIMO’s specific needs:

1. Over‑determined, noisy uplink detection – The uplink signal model y = Hs + n leads to an inconsistent linear system when directly applying KA. The authors pre‑process the estimated channel matrix Q and the received vector y to construct a consistent system Ãw = b̃ (e.g., by forming Ã = Qᴴ and b̃ = Qᴴy, followed by appropriate scaling). The randomized KA then converges to the least‑squares solution, which is equivalent to the ZF or MMSE detector without explicitly computing (QᴴQ)⁻¹.

2. Under‑determined downlink precoding – Downlink precoding requires solving x = Q(QᴴQ + ξI)⁻¹s. By transposing the problem to Qᴴw = s, the authors apply KA to the under‑determined system, obtaining the minimum‑norm solution w* and finally forming the transmit vector x = Qw*. This avoids any matrix inversion and works for both ZF and regularized ZF precoding.

Theoretical analysis shows that, for a matrix A with rows aᵢ, the expected error after t iterations satisfies
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