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Spectral Efficiency Analysis of Cell-Free Massive MIMO Systems with Zero-Forcing Detector

Pei Liu, Kai Luo, Da Chen, Tao Jiang

arXiv:1805.10621v2cs.ITmath.IT

TL;DR

Cell-free massive MIMO presents a distinct rate-analysis problem because distributed antennas have different large-scale fading, unlike co-located arrays. The paper derives two uplink-rate approximations for perfect and imperfect CSI, finding asymptotic lower-bound scaling of (α/2) log2 L for cell-free systems when α > 2, except in free space, versus log2 L for conventional systems.

  • Problem

    Unlike conventional massive MIMO, cell-free systems have different large-scale fading across distributed antennas, making their channel-correlation statistics and spectral-efficiency analysis distinct.

    Source: §I · p. 2

  • Method

    The paper derives two approximate achievable uplink-rate expressions for cell-free massive MIMO with perfect and imperfect CSI and relates them to conventional-system bounds.

    Source: §VI · p. 21

  • Results

    For α > 2 except free space, cell-free approximations have asymptotic lower bound (α/2) log2 L, versus conventional bounds’ asymptotically tight log2 L.

    Source: §VI · p. 21

  • Takeaways & Limitations

    The asymptotic results indicate cell-free massive MIMO has potential for improving achievable uplink rate over conventional massive MIMO when α > 2, except in free space.

    Source: §VI · p. 21

Abstract

from arXiv · show

In this paper, we firstly derive two approximations of the achievable uplink rate with the perfect/imperfect channel state information (CSI) in cell-free massive multi-input multi-output (MIMO) systems, and all these approximations are not only in the simple, but also converge into the classical bounds achieved in conventional massive MIMO systems where the base-station (BS) antennas are co-located. It is worth noting that the obtained two approximations with perfect CSI could be regarded as the special cases of the obtained two approximations with imperfect CSI when the pilot sequence power becomes infinite, respectively. Moreover, the theory analysis shows that all obtained approximations with perfect/imperfect CSI have an asymptotic lower bound $\frac{\alpha}{2}\log_2 L$ thanks to the extra {\emph {distance diversity}} offered by massively distributed antennas, where $L$ is the number of BS antennas and the path-loss factor $\alpha>2$, except for the free space environment. Obviously, these results indicate that the cell-free massive MIMO system has huge potential of spectral efficiency than the conventional massive MIMO system with the asymptotically tight bound $\log_2 L$.

I. INTRODUCTION … A. Channel Model

The paper analyzes uplink spectral efficiency for cell-free massive MIMO with zero-forcing detection, deriving rate approximations for perfect and imperfect CSI. It attributes an asymptotic rate advantage over co-located antennas to distance diversity from distributed BS antennas.

I. INTRODUCTION

  • Cell-free massive MIMO distributes BS antennas to reduce average user-to-antenna distance, while unequal large-scale fading makes its SE analysis more complex than in co-located systems.The introduction also identifies coverage improvement, power savings, and additional spatial resources as benefits of distributed antennas.Source: §I · p. 2

A. Related Works

  • Prior MRC/MRT work extensively analyzes cell-free SE, whereas ZF studies leave relatively few insightful analytical uplink-rate expressions and parameter-impact analyses for perfect and imperfect CSI.The cited ZF work includes rate approximations, bounds, asymptotic rates, and SINR analyses.Source: A. Related Works · p. 3

B. Our Contributions

  • For perfect and imperfect CSI, the paper derives two tractable uplink-rate approximations; the imperfect-CSI approximations approach the perfect-CSI ones at high pilot power.When BS antennas are co-located, the approximations reduce to the conventional massive-MIMO bounds.Source: B. Our Contributions · p. 4
  • The approximations have an asymptotic lower bound α/2 log2 L for path-loss factor α > 2, while conventional massive MIMO has a tight bound log2 L.The paper attributes the difference to distance diversity from distributed antennas; the stated cell-free bound excludes free-space propagation.Source: B. Our Contributions · pp. 4–5

C. Organization and Notation

  • The paper proceeds from system modeling to perfect-CSI and imperfect-CSI SE analyses, then simulations and conclusions.Its notation section defines vector and matrix conventions and related mathematical symbols.Source: C. Organization and Notation · p. 5

II. SYSTEM MODEL

  • The model places L BS antennas and K single-antenna users in a circular area, with all users served simultaneously by all antennas and the antennas connected to a CPU.It assumes L ≫ K, TDD, perfect synchronization, and a unit-radius area.Source: §II · p. 6

A. Channel Model

  • Each user's channel has independent small-scale fading and distance-dependent, unequal large-scale fading across antennas; large-scale fading is known at the BS and remains constant over many coherence intervals.The channel is block-fading and frequency-flat during a coherence block, and channels are independent across users.Source: A. Channel Model · p. 6

B. Uplink Data Transmission … B. Upper Bound

The paper derives a closed-form approximation to the perfect-CSI ZF uplink-rate upper bound from large-scale fading. Averaging over fading yields growth logarithmic in L and linear in path-loss factor α, attributed to distance diversity from distributed antennas.

B. Uplink Data Transmission

  • Users transmit uplink data synchronously; their symbols have zero mean and identity covariance, while Gaussian received noise is uncorrelated with user channels and symbols.Source: B. Uplink Data Transmission · p. 7

A. General Achievable Uplink Rate

B. Upper Bound

  • The rate depends on large-scale fading, and averaging it is difficult because the user-to-antenna access-distance distribution is complicated.The analysis defines an averaged rate metric to study average and asymptotic performance.Source: A. General Achievable Uplink Rate · p. 8, B. Upper Bound · p. 9
  • Theorem 1 gives a closed-form approximation to user k’s upper bound for large L, remains valid with not-so-large antenna counts, and exceeds the simulated rate in Section V.The approximation depends on large-scale fading rather than small-scale fading, so it need only be recalculated when large-scale coefficients change.Source: B. Upper Bound · pp. 8–9
  • ZF projects the received signal onto the null space of other users’ interference, making at most K −1 large-scale fading coefficients useless for approximating user k’s rate.Source: B. Upper Bound · p. 8
  • Under suitable conditions, the approximation reduces to the exact conventional massive-MIMO ZF upper bound with perfect CSI when intercell interference is ignored [36, Theorem 3].Source: B. Upper Bound · p. 9
  • The averaged-rate approximation grows logarithmically with L and linearly with α because distributed antennas provide extra distance diversity.The passage contrasts different access distances to distributed antennas with placing all massive BS antennas together.Source: B. Upper Bound · pp. 9–10

C. Lower Bound

For perfect-CSI ZF detection, the section derives approximations to the achievable uplink rate and shows that cell-free systems gain distance-diversity scaling unavailable to co-located antennas. The two approximations share an asymptotic lower bound, subject to the path-loss condition stated below.

C. Lower Bound

  • Theorem 2 gives an approximate closed-form lower bound for the achievable uplink rate when L is large, for the case of distributed BS antennas.The theorem is described as a simple analytical investigation of the rate approximation.Source: C. Lower Bound · p. 10
  • Theorem 2’s expression is the exact lower bound for perfect-CSI ZF uplink rate in conventional massive MIMO [16, Proposition 3], and provides a formula for assessing the general communication system.The section states that the derived lower-bound approximation is smaller than the corresponding upper-bound approximation, supporting Theorem 2’s validity.Source: C. Lower Bound · p. 10
  • Corollary 2 gives an approximate lower bound for the average achievable uplink rate after averaging out large-scale fading.This average-rate approximation is considered to examine the scaling behavior of the obtained approximation.Source: C. Lower Bound · p. 11
  • The two perfect-CSI achievable-uplink-rate approximations share the same asymptotic lower bound, while conventional massive MIMO with perfect CSI and ZF has rate Θ(log2 L), growing no faster than log2 L.The path-loss factor α is greater than 2 except in the case of free-space propagation.Source: C. Lower Bound · p. 11

IV. SE ANALYSIS WITH IMPERFECT CSI · A. General Achievable Uplink Rate · B. Upper Bound

The section analyzes imperfect-CSI uplink rates using MMSE channel estimates and derives upper-bound approximations. With fixed system parameters, their scaling behavior as the antenna count grows matches the perfect-CSI case.

A. General Achievable Uplink Rate

  • For imperfect CSI, the BS estimates user channels from uplink pilots using MMSE estimation and orthogonal pilot sequences, then forms the detector from the estimated channel matrix.The channel-estimation overhead within the coherence interval is excluded when comparing imperfect and perfect CSI rates [16].Source: A. General Achievable Uplink Rate · pp. 11–12
  • As pilot sequence power becomes high, channel-estimation error vanishes and the imperfect-CSI rate expression reduces to the perfect-CSI case.Source: A. General Achievable Uplink Rate · p. 12

B. Upper Bound

  • Theorem 3 derives an approximate upper bound on the achievable uplink rate with ZF detection and imperfect CSI.The approximation is evaluable from large-scale fading coefficients, uplink data power, a minimum involving channel-estimation-error variance, and a base-2 logarithm.Source: B. Upper Bound · p. 13
  • The imperfect-CSI upper bound is always below the perfect-CSI upper bound..Source: B. Upper Bound · p. 13
  • When all BS antennas are co-located and out-of-cell interference is ignored, the imperfect-CSI upper bound is exact for conventional massive MIMO and is a special case of the cell-free bound [38, Theorem 1].Source: B. Upper Bound · p. 13
  • With fixed uplink data power, pilot power, user count, and path-loss factor, the imperfect-CSI upper-bound approximation has the same scaling behavior as the perfect-CSI case as L increases.The gap between the large-scale-fading-averaged perfect- and imperfect-CSI approximations grows when pilot sequence power is reduced.Source: B. Upper Bound · p. 14

C. Lower Bound

The section presents tractable lower-bound approximations for uplink rate with imperfect CSI and shows that their asymptotic growth is independent of pilot power. In cell-free systems, the lower bounds capture a higher growth rate than the conventional massive-MIMO bound.

C. Lower Bound

  • Theorem 4’s imperfect-CSI lower bound is readily evaluated from pilot and uplink data powers, large-scale fading coefficients, and the antenna index set.Its form is similar to the perfect-CSI approximation, with a difference due to channel-estimation error; infinite pilot power recovers the perfect-CSI special case.Source: C. Lower Bound · p. 15
  • The cell-free lower bound’s conventional massive-MIMO limit is the exact imperfect-CSI ZF lower bound in [16, Proposition 7].The section presents this expression as a tractable way to connect cell-free and conventional massive-MIMO rate analysis.Source: C. Lower Bound · p. 15
  • Corollary 4 provides an approximation to the lower bound after averaging over the large-scale fading coefficients.Source: C. Lower Bound · p. 15
  • The imperfect-CSI lower bounds grow asymptotically as (α/2)log2 L, independently of pilot power, while the conventional massive-MIMO rate grows no faster than log2 L.The asymptotic results match those for perfect CSI; channel-estimation error affects the deviation between imperfect- and perfect-CSI rates, not the asymptotic result.Source: C. Lower Bound · p. 16

V. NUMERICAL RESULTS

Numerical simulations support the rate approximations for perfect and imperfect CSI, show cell-free systems outperform conventional massive MIMO as antenna counts grow, and find the approximations accurate across the tested conditions.

V. NUMERICAL RESULTS

  • The simulations average over 900 large-scale fading realizations and 200 small-scale fading realizations per large-scale realization, using α = 4 and K = 10.The setup distributes antennas and users uniformly in a radius-one circular area and examines pilot-sequence power.Source: §V · p. 16
  • As L increases, the cell-free system’s rate advantage over conventional massive MIMO grows, while both systems’ simulated rates and analytical approximations support the predicted rate behavior.For imperfect CSI, the same observations hold, with rate loss in both systems; the conventional system’s bounds and simulated rate approach the asymptotically tight bound log2 L.Source: Figure 2 · p. 17, §V · p. 18

LIST OF THE AVERAGE PER USER RAE (LOWER BOUND)

  • The average per-user RAE indicates accurate upper- and lower-bound approximations, negligible path-loss-factor influence, and decreasing error as L grows, except for a small lower-bound gap at α = 4 and L = 150.The RAE comparison covers perfect and imperfect CSI with ρu = −10 dB, ρp = 0 dB, and α = 3, 4.Source: LIST OF THE AVERAGE PER USER RAE… · p. 19, Table I · p. 19, Table II · p. 19
  • At L = 300, the CDF results show cell-free massive MIMO outperforming conventional massive MIMO, while its approximations remain valid across the rate region for both CSI cases.For imperfect CSI, channel-estimation error causes an inevitable rate loss.Source: Figure 3 · p. 20, LIST OF THE AVERAGE PER USER RAE… · p. 20
  • Across pilot powers, cell-free approximations track simulated rates; imperfect-CSI performance stays below perfect-CSI performance, and cell-free rates remain far above conventional-system rates.As ρp → ∞, imperfect-CSI approximations and simulated rates approach the corresponding perfect-CSI values.Source: LIST OF THE AVERAGE PER USER RAE… · pp. 20–21, Figure 4 · p. 21

VI. CONCLUSION

The paper derives two approximate uplink-rate expressions for cell-free massive MIMO with perfect and imperfect CSI. Their asymptotic scaling highlights the potential of distributed antennas, while imperfect CSI retains the perfect-CSI conclusions with a slight rate loss.

VI. CONCLUSION

  • For perfect CSI, the paper derives two approximate uplink-rate expressions that converge to the exact bounds when all BS antennas are co-located.Source: §VI · p. 21
  • For path-loss factor α > 2, asymptotic results indicate potential for cell-free massive MIMO, except in free space; conventional massive MIMO has asymptotically tight bound log2 L.Source: §VI · p. 21
  • For imperfect CSI, the perfect-CSI conclusions also hold with a slight rate loss, and the approximations converge to the perfect-CSI approximations as pilot power ρ_p becomes infinite.Source: §VI · p. 21

APPENDIX A SEVERAL USEFUL RESULTS

Appendix A provides distance-distribution and Gamma-approximation results used in the analysis, along with an asymptotic result for general order index l. It also gives bounds relating inverse distance CDFs and the quantities Q_l1 and Q_l2.

APPENDIX A SEVERAL USEFUL RESULTS

  • Lemma 1 characterizes access distances from a user to uniformly distributed antennas in a unit-radius circular area using their distance distribution [12, Eqs. (57-59)], [23, Eqs. (43-46)], and [39, Eq. (3.1)].It considers the ordered access distances among L−K+1 antennas.Source: Appendix A · p. 22
  • Lemma 2 approximates a sum of independent Gamma variables by a Gamma-distributed variable with shape and scale parameters specified in the lemma [10, Lemma 2] and [40, Lemma 6 & Proposition 8].Source: Appendix A · p. 22
  • For fixed l, Proposition 1 gives an asymptotic property as L→∞ under Lemma 1’s geometry, for path-loss α>2 except free space; it extends the prior l=1 result.The proposition presents the detailed result for the general case.Source: Appendix A · p. 23
  • For 0<x≤1, F^-1_1 and F^-1_2 satisfy 1−x < F^-1_1(z;x) < F^-1_2(z;x) ≤ 2F^-1_1(z;x)−(1−x); at x=0, Q_l2=0<Q_l1.The inverse-CDF relation is used to upper-bound Q_l2.Source: Appendix A · p. 24

APPENDIX B PROOF OF THEOREM 1

The proof of Theorem 1 upper-bounds (7) using Jensen’s inequality and derives (11) through a large-L approximation and an L ≫ K simplification.

APPENDIX B PROOF OF THEOREM 1

  • Jensen’s inequality provides an upper bound on (7).Source: Appendix B · p. 25
  • For large L, the proof applies the methodology of [23, Appendix D] to obtain an approximation.Source: Appendix B · p. 25
  • When L ≫ K, the proof identifies the relevant elements as set A_k, defined in (12), and obtains the resulting expression.Source: Appendix B · p. 25
  • The passage states that equation (11) is obtained by substituting equations (2), (63), and (64) into equation (62), then simplifying.Source: Appendix B · p. 25

APPENDIX C PROOF OF COROLLARY 1

The appendix derives approximation (14) from Theorem 1 and analyzes its asymptotic behavior using the large-L distribution of serving antennas and a bound on Q1(L).

APPENDIX C PROOF OF COROLLARY 1

  • Theorem 1 directly yields approximation (14), which is the starting point for the appendix’s asymptotic analysis.Source: Appendix C · p. 26
  • For large L, |Lk| approaches L−K +1, and the antennas in Lk are treated as uniformly distributed in a unit-radius circular area to obtain a lower bound.The bound uses equation (3) and Jensen’s inequality.Source: Appendix C · p. 26
  • Q1(L) is a special case of Proposition 1’s equation (51), and its asymptotically tight bound supports the asymptotic conclusion.The conclusion follows from the relationship among Q1(L) and equations (65) and (66).Source: Appendix C · p. 26

APPENDIX D PROOF OF THEOREM 2

The proof of Theorem 2 uses Jensen’s inequality to lower-bound (7), then approximates the relevant channel-norm terms with Gamma-distributed random variables to obtain (15).

APPENDIX D PROOF OF THEOREM 2

  • Substituting (63) and (64) into it approximates ||ak||_2^2; separately, Jensen’s inequality is cited as the basis for lower-bounding (7).Source: Appendix D · p. 26
  • The channel gains |g̃_l,k|^2 are Gamma distributed, and their sum is approximated by Λ_k ∼ Γ(Ψ_k, Φ_k), with parameters defined in (16) and (17).The proof uses these approximations in the subsequent algebraic derivation.Source: Appendix D · p. 27
  • The proof obtains (15) by substituting (69) into (67) and simplifying.The derivation begins by lower-bounding (7) using Jensen’s inequality.Source: Appendix D · pp. 26–27

APPENDIX E PROOF OF COROLLARY 2

Appendix E derives Corollary 2’s asymptotic performance by adapting the approach of Corollary 1, including its sorting method for large-scale fading coefficients.

APPENDIX E PROOF OF COROLLARY 2

  • The proof follows Corollary 1’s methodology and sorts the large-scale fading coefficients before deriving Corollary 2’s asymptotic performance.The appendix presents the asymptotic result as the outcome of these intermediate proof steps.Source: Appendix E · pp. 27–28
  • The derivation establishes the positivity relation in (73), then substitutes it into the expression for f̄Rlb.The appendix uses (73) in the subsequent derivation.Source: Appendix E · pp. 27–28
  • The proof next uses (3) and special cases from Appendix A’s Lemma 1 and Proposition 1 to obtain an intermediate bound before concluding Corollary 2’s asymptotic performance.The cited passages identify the special cases as those of (47) and (51), respectively.Source: Appendix E · p. 28
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