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Performance Analysis of Massive MIMO for Cell-Boundary Users

Yeon-Geun Lim, Chan-Byoung Chae, Giuseppe Caire

arXiv:1309.7817v2cs.ITmath.IT

TL;DR

Cell-boundary users are a low-SNR performance focus in a coordinated massive-MIMO system. The paper derives rate approximations and mode-selection thresholds for downlink and uplink schemes. It finds different preferred downlink normalizations for ZF and MRT, and uses thresholds to select transceiver modes.

  • Problem

    The paper focuses on cell-boundary user performance, described as a system bottleneck for both downlink and uplink.

    Source: §I · p. 2

  • Method

    The paper derives simple low- and high-SNR ergodic sum-rate approximations and mode-selection thresholds, and compares downlink normalization methods.

    Source: §I · pp. 2–3

  • Results

    ZF performs better with vector normalization, while MRT performs better with matrix normalization; mode-switching thresholds depend on transmit power and active-user count.

    Source: §I · p. 3, §VII · p. 10

  • Takeaways & Limitations

    The base station can select a transceiver mode to increase sum rate in both downlink and uplink scenarios.

    Source: §VIII · p. 12

    The closed-form expressions assume infinite M, although the approximations are described as quite accurate for finite but large M.

    Source: §I · p. 3

Abstract

from arXiv · show

In this paper, we consider massive multiple-input multiple-output (MIMO) systems for both downlink and uplink scenarios, where three radio units (RUs) connected via one digital unit (DU) support multiple user equipments (UEs) at the cell-boundary through the same radio resource, i.e., the same time-frequency slot. For downlink transmitter options, the study considers zero-forcing (ZF) and maximum ratio transmission (MRT), while for uplink receiver options it considers ZF and maximum ratio combining (MRC). For the sum rate of each of these, we derive simple closed-form formulas. In the simple but practically relevant case where uniform power is allocated to all downlink data streams, we observe that, for the downlink, vector normalization is better for ZF while matrix normalization is better for MRT. For a given antenna and user configuration, we also derive analytically the signal-to-noise-ratio (SNR) level below which MRC should be used instead of ZF. Numerical simulations confirm our analytical results.

I. INTRODUCTION

The paper studies cell-boundary massive MIMO, where three connected radio units support edge users, and develops rate approximations and mode-selection criteria. It also compares downlink precoding normalizations and specifies the system and beamforming models.

I. INTRODUCTION

  • The paper develops simple ergodic sum-rate approximations for ZF and MRT/MRC at low and high SNR, using near-determinism to simplify analysis.The approximations are described as accurate and simpler to evaluate than prior asymptotic expressions based on coupled fixed-point equations.Source: §I · p. 2
  • With uniform downlink stream power, vector normalization is preferred for ZF, while matrix normalization is preferred for MRT.The comparison addresses finite-antenna settings and practical, suboptimal power allocation.Source: §I · p. 2
  • Mode-selection thresholds specify when ZF provides a higher sum rate than downlink MRT or uplink MRC, as functions of transmit power and active-user count.The paper proposes algorithms from both transmit-power and number-of-users perspectives.Source: §I · pp. 2–3

II. SYSTEM MODEL: MASSIVE MIMO

  • One DU controls three fiber-connected radio units and K single-antenna users, with M cloud-BS antennas; pilot contamination is excluded.The model assumes perfect CSI at the radio units.Source: §II · p. 3

A. Downlink

  • Downlink beamforming uses ZF or MRT, with vector normalization assigning equal power per stream and matrix normalization allowing different stream powers.The paper does not optimize power allocation.Source: A. Downlink · pp. 3–4

1) ZF/MRT with vector normalization:

The paper summarizes its downlink and uplink sum-rate analyses, compares downlink normalization choices, and proposes mode-selection thresholds. Its uplink model compares ZF and MRC without normalization.

1) ZF/MRT with vector normalization:

B. Uplink

  • Uplink receivers use ZF or MRC combining; normalization is omitted because it does not change uplink SNR.The combining matrix is formed from per-user combining-filter vectors.Source: B. Uplink · p. 4

1) Asymptotic downlink sum rate:

  • ZF with vector normalization outperforms ZF with matrix normalization, while MRT with matrix normalization outperforms MRT with vector normalization.The section describes these as results of the ergodic achievable sum-rate analysis.Source: 1) Asymptotic downlink sum rate · p. 4

2) Asymptotic uplink sum rate:

3) Transceiver mode selection algorithm:

  • Power and active-user thresholds identify when ZF provides a higher sum rate than MRT in downlink or MRC in uplink.The paper presents these as two transceiver mode-selection algorithms.Source: 3) Transceiver mode selection algorithm · p. 4

III. MATHEMATICAL MOTIVATIONS AND PRELIMINARIES

  • The section introduces mathematical preliminaries for the later asymptotic analyses of network massive MIMO.These preliminaries are used in the downlink and uplink analyses.Source: §III · p. 4

A. Achievable Rate Bound

The section develops rate bounds and approximations for massive MIMO using random-vector statistics and near-deterministic behavior. It distinguishes the asymptotic effective channel from signal and interference powers, whose randomness can persist.

A. Achievable Rate Bound

  • For ZF, Jensen-based bounds are used to evaluate achievable sum rate and are reported as more accurate than prior work,,.The bounds use signal, interference, and noise powers.Source: A. Achievable Rate Bound · p. 4

B. Expectation and Variance of Random Vectors

C. Effective Channel

  • With perfect CSI, transmit energy can scale as 1/M, and the normalized effective channel matrix tends almost surely to the identity as M grows.Source: C. Effective Channel · p. 5

D. Signal and Interference Power

  • Unlike the effective channel limit, the normalized desired-signal and interference-power terms do not converge to fixed values because their variances do not vanish.Source: D. Signal and Interference Power · p. 5

E. Chebyshev’s Inequality

  • Chebyshev’s inequality is used to treat a scaled random variable as near deterministic when its variance is small relative to the squared scaling factor, supporting SINR approximations.Source: E. Chebyshev’s Inequality · p. 5

F. Ergodic Achievable Sum Rate of Massive MIMO Systems at the low/high SNR regime

The paper approximates ergodic sum rates in low- and high-SNR regimes using near-deterministic signal or interference terms. It applies these results to distinguish cell-boundary and cell-center operating regimes.

F. Ergodic Achievable Sum Rate of Massive MIMO Systems at the low/high SNR regime

  • As M grows in the low- or high-SNR regime, the ergodic sum-rate lower and upper bounds coincide, yielding an approximation based on expected signal power over expected interference plus noise.Source: F. Ergodic Achievable Sum Rate of… · p. 5

G. Arithmetic-Geometric Inequality

IV. ASYMPTOTIC DOWNLINK SUM RATE FOR CELL-BOUNDARY USERS

  • The downlink analysis assumes cell-boundary users operate at low SNR and cell-center users at high SNR, with desired or interference power becoming near deterministic in the relevant regime.Source: §IV · p. 6

1) Achievable rate bounds for ZF precoding:

The section derives achievable-rate bounds for ZF and low- and high-SNR rate expressions for MRT under vector and matrix normalization.

1) Achievable rate bounds for ZF precoding:

2) Achievable rate for MRT precoding:

B. Comparison between Vector and Matrix Normalizations

The comparison favors vector normalization for ZF and matrix normalization for MRT at low SNR. For cell-boundary users, the analysis then uses those preferred normalizations to evaluate low-SNR rates.

1) Performance comparison of ZF:

  • For ZF, vector normalization is always better than matrix normalization because normalization affects per-user power when desired power is one and interference power is zero.Source: 1) Performance comparison of ZF · pp. 6–7

2) Performance comparison of MRT:

  • For MRT, matrix normalization is better at low SNR, while the performance gap between the two normalizations is marginal at high SNR.The paper attributes the low-SNR difference to how each normalization scales desired power across users.Source: 2) Performance comparison of MRT · p. 7

C. Ergodic Achievable Sum Rate for Cell-boundary Users with the Best Normalization Method

V. ASYMPTOTIC UPLINK SUM RATE FOR CELL-BOUNDARY USERS

This section considers uplink ergodic sum rate under per-user power constraints and derives rates for MRC and ZF. It approximates MRC performance in high- and low-SNR regimes.

V. ASYMPTOTIC UPLINK SUM RATE FOR CELL-BOUNDARY USERS

  • The section derives the ergodic achievable uplink sum rate for MRC.Source: §V · p. 7

RMRC, UL

  • MRC sum rate is approximated separately in the high- and low-SNR regimes.Source: RMRC, UL · p. 7
  • The section also gives the low-SNR ergodic sum rate for uplink ZF.Source: RMRC, UL · p. 7

VI. TRANSCEIVER MODE SELECTION

The section proposes transceiver-mode selection rules based on transmit power and active-user count. Power and user-count thresholds identify when ZF or MRT/MRC has the better sum rate.

A. Algorithm

  • Above their respective power thresholds, ZF provides the better sum rate for downlink precoding and uplink combining.Source: A. Algorithm · pp. 7–8
  • The power threshold depends on the user count and increases monotonically with it; its minimum occurs at K = 2.Source: A. Algorithm · p. 8
  • Below the power cross points, MRT or MRC is always better than ZF in sum rate.The cross points are Pcross,DL = 4/(M−1) and Pcross,UL = 1/(M−1).Source: A. Algorithm · p. 8
  • When RU power exceeds Pcross,DL, MRT provides the better downlink sum rate if the active-user count exceeds Kcross,DL.Source: A. Algorithm · p. 8
  • When RU power exceeds Pcross,UL, MRC provides the better uplink sum rate if the user count exceeds Kcross,UL.Source: A. Algorithm · p. 8

B. Performance Comparison for a Large Number of Users Case

For cell-boundary users, the analysis compares rate behavior as user count and SNR vary, including mode-selection thresholds. The numerical results support the proposed rate approximations and show how normalization and user count affect the preferred technique.

B. Performance Comparison for a Large Number of Users Case

VII. NUMERICAL RESULTS

  • At low SNR, ZF with vector normalization is better, while MRT with matrix normalization achieves higher sum rates.The proposed MRT rate form is accurate when SNR is lower than 1/M; a gap appears for matrix normalization outside that condition.Source: §VII · p. 10
  • For MRC, the approximations in (20) and (21) are approximately the simulated uplink sum rate when Pu ≥ M and Pu ≤ 1/M, respectively.The paper notes a large gap between the finite-M MRC lower bound in and the ergodic achievable sum rate.Source: §VII · p. 10

B. Performance Comparison for a Large Number of Users Case

VII. NUMERICAL RESULTS

  • The rate-curve crosspoint is the power threshold: MRT/MRC performs better than ZF at low SNR, while the proposed threshold equation identifies the boundary.For M = 24, the reported thresholds are 6.2 dB downlink and −7 dB uplink; at −7.6 dB downlink and −13.6 dB uplink, MRT/MRC is better regardless of K.Source: Figure 6 · p. 10, §VII · p. 11, Figure 7 · p. 11
  • MRT/MRC performance increases with K while ZF performance decreases, because ZF’s downlink sum rate approaches a very small constant at M = K.The uplink ZF sum rate can likewise approach a very small value, since its lower bound is zero.Source: §VII · pp. 11–12
  • The proposed ergodic sum-rate analysis is tighter and simpler than prior work, with the same computation complexity as cited lower bounds.At larger antenna counts, the closed form in is less tight than the analysis in the low-SNR regime and uplink scenario.Source: §VII · p. 12, Figure 9 · p. 12

VIII. CONCLUSIONS

The conclusion summarizes rate analyses and mode-selection results for massive MIMO systems supporting multiple cell-boundary UEs.

VIII. CONCLUSIONS

  • The paper derives ZF sum-rate bounds and MRT approximations, confirms the preferred downlink normalizations, and studies mode-switching thresholds for downlink and uplink.It identifies limited RU cooperation and cooperation delay as more practical scenarios for future work.Source: §VIII · p. 12

APPENDIX

The appendix develops moment calculations and uses variance assumptions to support the MRT rate derivations.

A. Proof of Lemma 1

  • For Lemma 1, the appendix uses the complex Gaussian channel model to derive moments of channel terms and their variances and covariances.The squared magnitude of each channel element is treated as exponential with unit parameter.Source: A. Proof of Lemma 1 · pp. 12–13

B. Proof of Ergodic Sum Rate of MRT

  • The MRT proof assumes selected variance terms converge to zero in their respective SNR regimes to evaluate the ergodic rate under vector and matrix normalization.The vector-normalization argument considers high SNR; the matrix-normalization argument considers high-SNR convergence of cross-channel terms and related variance terms.Source: B. Proof of Ergodic Sum Rate of MRT · p. 13

1) Proof of (11):

  • The proof of (11) applies approximation (b) and equality (c) using Lemmas 4 and 1, after relation (32).Source: 1) Proof of (11) · p. 13

2) Proof of (12):

The appendix continues the rate proofs by invoking variance convergence and lemma-based substitutions across SNR regimes and uplink cases.

2) Proof of (12):

  • The proof of (12) relies on (33) and Lemmas 1 and 4.It notes that the variance of the channel norm is smaller than that of the cross-channel term, so its scaled variance converges faster at low SNR.Source: 2) Proof of (12) · p. 13

3) Proof of (13):

  • In the low-SNR regime, the derivation gives 1 + Pt(M^2 + M); step (f) follows directly from (34), while step (g) uses Lemma 4 and E(||H||_F^2) = MK.Source: 3) Proof of (13) · p. 14

4) Proof of (13):

  • In the high SNR regime, step (h) follows from (35), whereas step (i) can also be obtained using Lemmas 1 and 4.Source: 4) Proof of (13) · p. 14

RMRC, ULL

  • The uplink low-SNR MRC expression is derived using the same methods as (36).Source: RMRC, ULL · p. 14
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