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On the Matrix Inversion Approximation Based on Neumann Series in Massive MIMO Systems

Dengkui Zhu, Boyu Li, Ping Liang

arXiv:1503.05241v1cs.ITmath.IT

TL;DR

Massive MIMO makes exact ZF matrix inversion costly, while the convergence and accuracy conditions for practical Neumann Series approximations remain important questions. The paper derives antenna-to-user ratio conditions and approximation-error estimates; it reports a high-convergence condition of α > 5.83 and discusses how term count and diagonal dominance relate to accuracy and complexity.

  • Problem

    Exact ZF inversion becomes costly as the user count grows, while finite-size convergence, diagonal-dominance conditions, and accurate error estimates for practical NS term counts remain unresolved.

    Source: §I · p. 2

  • Method

    The paper analyzes how the antenna-to-user ratio affects NS-based MIA and derives ratio conditions and approximation-error estimates for practical term counts.

    Source: §I · p. 2, §VII · p. 18

  • Results

    α > 5.83 is given as a condition for very high convergence probability, and massive MIMO systems are described as generally satisfying it.

    Source: §VI · p. 15, §VII · p. 18

  • Takeaways & Limitations

    The error-floor estimates can guide performance assessment and selection of the smallest NS term count that meets a target performance with lower implementation complexity.

    Source: §VI · p. 17

    Good accuracy with quick convergence requires the diagonal-dominance condition, which the discussion says requires very small α.

    Source: §VI · p. 16

Abstract

from arXiv · show

Zero-Forcing (ZF) has been considered as one of the potential practical precoding and detection method for massive MIMO systems. One of the most important advantages of massive MIMO is the capability of supporting a large number of users in the same time-frequency resource, which requires much larger dimensions of matrix inversion for ZF than conventional multi-user MIMO systems. In this case, Neumann Series (NS) has been considered for the Matrix Inversion Approximation (MIA), because of its suitability for massive MIMO systems and its advantages in hardware implementation. The performance-complexity trade-off and the hardware implementation of NS-based MIA in massive MIMO systems have been discussed. In this paper, we analyze the effects of the ratio of the number of massive MIMO antennas to the number of users on the performance of NS-based MIA. In addition, we derive the approximation error estimation formulas for different practical numbers of terms of NS-based MIA. These results could offer useful guidelines for practical massive MIMO systems.

I. INTRODUCTION

Massive MIMO supports many users but makes the K × K inversion in ZF hardware costly, motivating Neumann Series matrix inversion approximation. The paper addresses uncertainty about convergence, diagonal dominance, and approximation error at practical term counts.

I. INTRODUCTION

  • When M is much larger than K, ZF can approach channel-capacity performance and is considered a potential practical method for massive MIMO.Source: §I · p. 2
  • The main hardware complexity of ZF is inverting a K × K matrix, and exact inversion can cause high complexity and processing delay in massive MIMO.The delay may prevent meeting channel-coherence-time demands.Source: §I · p. 2
  • For finite M/K, NS may fail to converge, and the ratio needed for high convergence or diagonal-dominance probability was unclear.More terms improve approximation but increase processing cycles, limiting practical hardware implementations.Source: §I · p. 2
  • The paper derives M/K ratio conditions for high convergence and diagonal-dominance probabilities, plus accurate approximation-error analysis for practical NS term counts.Source: §I · p. 2

II. BASIS OF NS-BASED MIA IN MASSIVE MIMO SYSTEMS

The section sets out the channel model and the NS-based approximation used to reduce ZF matrix-inversion complexity. It identifies convergence and diagonal-dominance conditions as key to obtaining accurate approximations with few terms.

II. BASIS OF NS-BASED MIA IN MASSIVE MIMO SYSTEMS

  • The analysis assumes i.i.d. zero-mean, unit-variance complex Gaussian channel coefficients under uncorrelated Rayleigh flat fading.In TDD, channel reciprocity gives the same H for uplink and downlink when transmission remains within channel coherence time.Source: §II · p. 3
  • ZF requires the pseudo-inverse of H; the hardware bottleneck is the K × K inverse G^-1, which NS approximates for massive MIMO.The system has M BS antennas serving K single-antenna users, with an M × K uplink channel matrix H.Source: §II · p. 3
  • The NS approximation uses N terms and a diagonal K × K matrix Θ, and its series requires a convergence condition.Source: §II · pp. 3–4
  • With choice (5), the stated convergence condition is satisfied when α ≥ 1, and the convergence rate improves as α/(1 + α) tends to zero.Source: §II · p. 4
  • A small number of NS terms can approach exact-inverse performance, but choosing attenuation factor δ is difficult: too large risks nonconvergence, while too small slows convergence.A smaller δ can require a very large N for good approximation, increasing hardware burden.Source: §II · p. 4
  • The analysis adopts choice (9), which requires G to be diagonally dominant for good approximation with quick convergence; the needed α conditions remained open.Prior work discussed its performance-complexity trade-off and hardware implementation for downlink and uplink.Source: §II · p. 5

III. CONVERGENCE AND α

The section derives an antenna-to-user ratio condition for high-probability convergence of Neumann-series matrix inversion approximation. Simulations show that the approximate condition tracks the accurate condition closely and identifies high-convergence user counts across antenna sizes.

III. CONVERGENCE AND α

  • A good Neumann-series approximation requires the spectral radius of B to be less than 1, and the derived α condition gives a maximum K for a chosen M to achieve very high convergence probability.The probability comparison validates the condition as an acceptable approximation to the accurate condition.Source: §III · pp. 5–7
  • For large M, the diagonal matrix D is approximated by M I_K, reducing the convergence condition to an eigenvalue bound on G.Because G is positive definite, its eigenvalues are positive; the resulting bound is 0 < λ(G) < 2M.Source: §III · pp. 5–6
  • The exact probability calculation for condition (13) involves an infinite sum whose extreme complexity prevents a closed-form convergence condition in α.The sum uses a matrix hypergeometric function.Source: §III · p. 6
  • The α condition identifies maximum K values for M from 64 to 512, and the approximate and accurate convergence probabilities are close, with the accurate condition slightly better at large M.The condition offers high convergence probability but does not ensure fast convergence.Source: Figure 1 · p. 7, §III · p. 7, Figure 2 · p. 8

IV. DIAGONALLY DOMINANT AND α

The section derives an α-based high-probability condition for diagonal dominance of G, then validates its probability approximation and relates the condition to convergence.

IV. DIAGONALLY DOMINANT AND α

  • As M grows, diagonal entries g_ii approach M, so diagonal dominance is approximated through normalized correlations between user-channel vectors.The derivation uses the mean correlation magnitude and a high-probability bound because the finite-K sum can exceed (K−1)E(x).Source: §IV · pp. 8–9
  • The authors approximate (20) and derive a high-probability DDM condition in α, which gives the maximum K for a specified M.The condition is based on the bound for summed normalized correlations.Source: §IV · p. 9
  • For M=64–512, Fig. 3 plots maximum K satisfying (28); simulations find the approximated DDM probability close to the definition, especially at large M.The table also reports typical M values, associated maximum K values, and DDM probabilities.Source: §IV · pp. 9–10, Figure 3 · p. 10, Table II · p. 10
  • Condition (28) offers high DDM probability and is sufficient for convergence, leading to quicker convergence in practice.The section presents (28) as more useful in practice for this reason.Source: §IV · p. 10

V. ERROR ANALYSIS

This section replaces a loose SIR lower bound with more accurate residual-error estimates for practical Neumann Series term counts. The resulting estimated SIR values closely match the exact values across the reported antenna and user settings.

V. ERROR ANALYSIS

  • For N > 1, the lower bound from (33) differs substantially from exact SIR and offers insufficient insight into residual error.The comparison is reported for M = 128 and different K and N values.Source: §V · p. 12, Figure 5 · p. 13
  • Approximating the PDF of x = |zij| enables a more accurate approximation of γ_N from the residual error.The PDF approximation is derived for large M.Source: §V · pp. 12–13
  • For N = 1, the MSE expression is specialized using E(x^2) and the definition of B_a,M.The passage also states that convergence condition (3) is relevant to G^-1.Source: §V · pp. 11, 13
  • For N = 2, the MSE expansion groups polynomial terms by powers and index patterns, approximating the third-category sum as zero.The first and second category means are used to approximate ε_2.Source: §V · pp. 13–14
  • The same derivation approach provides MSE formulas for N = 3 and N = 4, while the uplink and downlink MSE expressions are treated as the same ε_N.The downlink result is stated to match the earlier expression.Source: §V · pp. 11, 12, 14
  • Estimated SIR values are very close to exact values for M = 128, 256, and 512 across the reported K and N settings.The comparisons use formulas (36), (42)–(44), with Fig. 6 showing the M = 128 case.Source: Figure 6 · p. 15, §V · p. 15

VI. DISCUSSIONS

The discussion finds that massive-MIMO operating ratios generally exceed the derived NS convergence threshold, while faster convergence and good accuracy require a stricter diagonal-dominance condition. It also explains how SIR error-floor estimates and an alternative series expression can guide practical choices of term count and implementation.

VI. DISCUSSIONS

  • α > 5.83 is the convergence condition; because massive MIMO commonly uses α > 10, convergence probabilities are at least the already near-one values reported at the threshold, and accuracy improves with N.The cited SIR figures compare exact and estimated values across K and N for M = 256 and M = 512.Source: §VI · pp. 15–16, Figure 7 · p. 16, Figure 8 · p. 17
  • The diagonal-dominance condition enables good accuracy with small N, but its strict α requirement can limit M = 512 to at most K = 17 users; slightly larger N can accommodate α just above the condition.With M = 128 and N = 3, K = 4 met the condition and approached exact-inverse performance, whereas K = 12 did not and suffered large performance loss.Source: §VI · p. 16
  • SIR estimates the NS-based MIA performance error floor, allowing performance to be assessed from M, K, and N and the smallest adequate N to be chosen for a target performance.The section describes larger N as increasing hardware implementation complexity.Source: §VI · p. 17
  • The alternative expression uses N = 2L and permits direct selection of N = 8 when N = 4 is inadequate; cascaded matched-filter implementation reduces complexity to O(K^2), retaining an advantage over exact inversion.The alternative expression corresponds to N = 2, 4, and 8 for L = 1, 2, and 3, respectively.Source: §VI · pp. 17–18

VII. CONCLUSIONS

The paper derives conditions for NS-based MIA convergence and diagonal dominance, then provides residual-error formulas for practical term counts. Together, these results are presented as guidelines for applying NS-based MIA in massive MIMO systems.

VII. CONCLUSIONS

  • α > 5.83 provides very high convergence probability and lets the maximum served-user count K be determined for a given antenna count M.This condition addresses whether NS-based MIA is a valid method.Source: §VII · p. 18
  • Condition (28) gives a high-probability diagonal-dominance bound for G, supporting good NS-based MIA performance and quick convergence with small N.For a given M, the condition can determine the maximum K for good performance and quick convergence in ZF decoding or detection.Source: §VII · p. 18
  • Residual-error formulas (36), (42)–(44) estimate the NS-based MIA error floor with very high accuracy for practical N values, enabling SIR estimation from M, K, and N.The paper presents these results as practical-application guidelines.Source: §VII · p. 18
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