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Sum-Rate and Power Scaling of Massive MIMO Systems with Channel Aging

Chuili Kong, Caijun Zhong, Anastasios K. Papazafeiropoulos, Michail Matthaiou, Zhaoyang Zhang

arXiv:1510.06827v1cs.ITmath.IT

TL;DR

Because channel aging has been little studied in massive MIMO and existing analyses seek bounds valid for finite antenna and user counts, this paper derives sum-rate lower bounds for MRC and ZF with aged CSI. It finds that aging degrades achievable rate but does not change the power-scaling law, while channel prediction does not improve that law.

  • Problem

    The impact of channel aging on massive MIMO has received little study, and tight sum-rate bounds for arbitrary finite M and K are sought.

    Source: §I · pp. 2–3

  • Method

    The paper derives tractable lower bounds for achievable rates with and without channel prediction and characterizes power scaling for aged CSI.

    Source: §VIII · p. 29

  • Results

    Aged CSI degrades achievable sum-rate, with more severe aging lowering the rate, while channel aging and prediction leave the power-scaling law unchanged.

    Source: §VIII · pp. 29–30

  • Takeaways & Limitations

    The transmit-power scaling law is retained under channel aging and is not improved by channel prediction.

    Source: §VIII · p. 30

Abstract

from arXiv · show

This paper investigates the achievable sum-rate of massive multiple-input multiple-output (MIMO) systems in the presence of channel aging. For the uplink, by assuming that the base station (BS) deploys maximum ratio combining (MRC) or zero-forcing (ZF) receivers, we present tight closed-form lower bounds on the achievable sum-rate for both receivers with aged channel state information (CSI). In addition, the benefit of implementing channel prediction methods on the sum-rate is examined, and closed-form sum rate lower bounds are derived. Moreover, the impact of channel aging and channel prediction on the power scaling law is characterized. Extension to the downlink scenario and multi-cell scenario are also considered. It is found that, for a system with/without channel prediction, the transmit power of each user can be scaled down at most by $1/\sqrt{M}$ (where $M$ is the number of BS antennas), which indicates that aged CSI does not degrade the power scaling law, and channel prediction does not enhance the power scaling law; instead, these phenomena affect the achievable sum-rate by degrading or enhancing the effective signal to interference and noise ratio, respectively.

I. INTRODUCTION

Channel aging changes the channel between estimation and use, yet its impact on massive MIMO has received little study. This paper develops finite-system sum-rate bounds and analyzes power scaling across receiver and deployment scenarios.

Motivation

  • The paper seeks tight sum-rate bounds for arbitrary finite M and K because existing deterministic-equivalent analyses rely on large-system limits and provide approximations.Source: §I · pp. 2–3

Contributions

  • Simple power-scaling laws are derived to clarify how channel aging affects achievable rate, alongside analysis of both single-cell and multi-cell scenarios.The paper motivates the single-cell analysis as a source of engineering insight and as an approach that can also be applied to multi-cell scenarios.Source: §I · p. 3
  • The paper derives tight uplink sum-rate lower bounds for MRC and ZF with aged CSI, valid for arbitrary antenna and user counts.It also derives bounds with channel prediction.Source: §I · p. 3
  • Aged CSI does not reduce the user power-scaling law, and channel prediction does not improve it.The paper also extends power-scaling analysis to single-cell downlink and multi-cell uplink, where pilot contamination produces a different law.Source: §I · p. 3

II. SINGLE-CELL UPLINK MODEL

The model considers a single-cell uplink with M base-station antennas and K single-antenna users. It estimates channels from uplink pilots and models channel aging between estimation and later use.

II. SINGLE-CELL UPLINK MODEL

  • The single-cell uplink model has M base-station antennas serving K noncooperative single-antenna users over flat-fading channels that vary between symbols.Large-scale channel effects are assumed constant across symbols.Source: §II · p. 4

A. Channel Estimation

  • The base station estimates channels using uplink pilots and an MMSE estimator; the channel estimate and estimation error are independent.The pilots satisfy ΦΦ† = I_K.Source: A. Channel Estimation · pp. 5–6

B. Channel Aging

  • Channel aging is modeled as a first-order autoregressive process, with temporal correlation α = J0(2πf_DT_s) under Jakes fading.Smaller |α| corresponds to more severe channel aging.Source: B. Channel Aging · p. 6

III. ACHIEVABLE UPLINK SUM-RATE WITH CHANNEL AGING

This section derives achievable uplink sum-rate lower bounds for MRC and ZF under aged CSI and examines their power-scaling behavior. For MRC, more severe aging lowers the achievable rate, while the power-scaling law remains unchanged.

III. ACHIEVABLE UPLINK SUM-RATE WITH CHANNEL AGING

  • The section derives closed-form lower bounds on achievable sum-rate with aged CSI for both MRC and ZF receivers.It also characterizes the effect of aged CSI on asymptotic power scaling.Source: §III · p. 7

A. MRC Receivers

  • With MRC, channel aging leaves the power-scaling law unchanged from the channel-estimation-error-only case; it reduces effective SINR instead.For α = 1, there is no channel-aging effect.Source: A. MRC Receivers · p. 10
  • More severe channel aging lowers the MRC achievable rate, which increases with temporal correlation α.The finite-M lower bound is described as tight, and its large-M limit matches the asymptotic result.Source: A. MRC Receivers · p. 10

B. ZF Receivers

For ZF with aged CSI, the analysis finds the same power-scaling law as for MRC. In the asymptotic regime, ZF and MRC achieve the same uplink rate.

B. ZF Receivers

  • ZF with aged CSI attains the same power-scaling law as MRC, with no rate degradation under that scaling.The section states that each user's transmit power can be reduced at most by 1/√M while preserving rate.Source: B. ZF Receivers · p. 11
  • With aged CSI, ZF achieves the same asymptotic uplink rate as MRC.Source: B. ZF Receivers · p. 11
  • The section derives an achievable-rate lower bound for ZF in the finite-M regime.The lower bound is described as asymptotically tight.Source: B. ZF Receivers · pp. 11–12

IV. ACHIEVABLE UPLINK RATE WITH CHANNEL PREDICTION

Channel aging reduces achievable rate, motivating channel prediction. The section introduces a linear Wiener predictor and derives closed-form achievable-rate lower bounds for predicted CSI.

IV. ACHIEVABLE UPLINK RATE WITH CHANNEL PREDICTION

  • The section derives closed-form lower bounds on achievable uplink rates with predicted CSI for MRC and ZF receivers.The predictor order is also considered in characterizing the power scaling law.Source: §IV · p. 12
  • The paper uses a p-th order linear Wiener predictor to estimate the next channel from past observations.The predicted channel is separated from the true channel by a prediction-error vector independent of the prediction.Source: §IV · pp. 12–14
  • The predictor’s covariance structure is a scaled identity, supporting the subsequent analysis of predicted-channel and prediction-error variances.The paper states that Θ_k(p, α) is a scaled M × M identity matrix.Source: §IV · p. 14
  • For predicted CSI, the achievable uplink rate is analyzed through its SINR, with the expectation taken over small-scale fading.The section substitutes the predicted channel estimate into the rate expression.Source: §IV · pp. 14–15

A. MRC Receivers

For MRC with predicted CSI, channel prediction improves achievable rate without changing the asymptotic power scaling law. Higher predictor order can yield larger gains, with increased processing complexity.

A. MRC Receivers

  • With MRC and predicted CSI, users can scale transmit power down at most by 1/√M without changing the power scaling law.The result holds for fixed E_u and matches the scaling law without channel prediction.Source: A. MRC Receivers · pp. 9, 10, 16
  • Channel prediction increases achievable rate by enhancing effective SINR, and higher prediction order yields a larger rate gain.The reported gain is most pronounced for large α and becomes negligible for small α.Source: A. MRC Receivers · p. 17
  • When predictor order p becomes large, processing complexity increases substantially, requiring a system-design trade-off.The paper advises balancing this complexity against prediction order.Source: A. MRC Receivers · p. 17
  • The section also gives a tight finite-M lower bound on achievable uplink rate for MRC with predicted CSI.The bound is stated for the k-th user.Source: A. MRC Receivers · pp. 9, 16, 17

B. ZF Receivers

For ZF with predicted CSI, the uplink retains the MRC power-scaling law. The single-cell downlink extension likewise finds that base-station power can scale down with antenna count while preserving a nonzero rate limit.

B. ZF Receivers

  • With ZF and predicted CSI, users can scale transmit power down at most by 1/√M, matching the MRC asymptotic power-scaling law.The ZF analysis considers power scaling p_u = E_u/M^γ for fixed E_u.Source: B. ZF Receivers · pp. 17–18

V. EXTENSION TO SINGLE-CELL DOWNLINK

  • For the single-cell downlink with MRC receivers, the achievable rate is bounded for aged CSI and the BS power can be scaled with antenna count.At most, BS transmit power scales down proportionally to 1/√M while the rate converges to a nonzero limit, the same scaling as in the uplink.Source: §V · pp. 20–21

VI. EXTENSION TO MULTI-CELL SYSTEMS

In the multi-cell setting, the asymptotic rate depends on power scaling, channel aging, and inter-cell interference. Channel prediction yields a strictly higher achievable rate than aged CSI in the stated comparison.

VI. EXTENSION TO MULTI-CELL SYSTEMS

  • For γ > 1/2, the multi-cell achievable rate converges to zero; for γ = 1/2, it converges to a nonzero limit.The same 1/√M power-scaling law is reported to hold in the multi-cell setting.Source: §VI · p. 23
  • In the multi-cell scenario, the 1/√M power-scaling law still holds, while the nonzero asymptotic rate is affected by channel aging and pilot-contamination interference.The rate also includes contributions from intra-cell and inter-cell interference.Source: §VI · p. 23
  • For 0 < γ < 1/2, the multi-cell rate converges to a nonzero limit rather than growing unbounded, and channel aging vanishes from that limit.The paper attributes this behavior to an interference-limited regime in which terminals scale received power by the same factor.Source: §VI · p. 23
  • With predicted CSI, the achievable multi-cell rate is strictly higher than with aged CSI in the stated comparison.The analysis considers MRC for the channel-prediction case.Source: §VI · pp. 22–24

VII. NUMERICAL RESULTS

In the single-cell uplink results, channel prediction’s rate benefit depends on aging severity: it helps at small normalized Doppler shifts but can underperform current CSI at large shifts. With user power scaled by 1/sqrt(M), the achievable sum-rate approaches a nonzero limit as the antenna count grows.

VII. NUMERICAL RESULTS

A. Single-cell uplink scenario

B. Single-cell downlink scenario

The paper extends its channel-aging analysis to the single-cell downlink and multi-cell uplink, then concludes that aging and prediction affect achievable sum-rate without changing the power-scaling law. In the multi-cell case, pilot contamination is associated with a non-zero limiting rate when power is not reduced at the maximum rate.

B. Single-cell downlink scenario

C. Multi-cell scenario

  • For multi-cell uplink with aged CSI, achievable sum-rate tends to zero as M grows when γ > 1/2, but converges to a non-zero limit when γ ≤ 1/2.The passage describes this as the multi-cell power-scaling law.Source: C. Multi-cell scenario · pp. 28–29, Figure 6 · p. 28
  • With channel prediction, the multi-cell achievable sum-rate has a non-zero limit for γ ≤ 1/2 and that limit grows with predictor order.The passage also states that the rate reduces to zero for γ > 1/2.Source: C. Multi-cell scenario · pp. 28–29, Figure 7 · p. 29

VIII. CONCLUSION

  • The paper derives tractable sum-rate lower bounds for MRC and ZF, with and without channel prediction, valid for arbitrary antenna and user counts.It characterizes how channel aging and prediction affect power scaling as well as achievable sum-rate.Source: §VIII · p. 29
  • Aged CSI degrades achievable sum-rate, with more severe aging causing greater reduction; channel prediction enhances sum-rate, and higher predictor order improves performance.The paper also states that prediction benefits are more pronounced in a particular scenario, but the supplied passage does not identify it.Source: §VIII · pp. 29–30
  • The single-cell downlink achieves the same power-scaling law as the single-cell uplink, while the multi-cell rate has a non-zero limit for 0 < γ < 1/2 due to pilot contamination.The multi-cell result applies when user power scales as 1/M^γ.Source: §VIII · p. 30

APPENDIX A

Appendix A derives the result through substitutions and expectation bounds, including Jensen’s inequality and Gaussian-distribution properties. Appendix B uses the ZF detector identity to complete its proof.

APPENDIX A

  • The derivation substitutes ˆak[n + 1] = ¯gk[n + 1] = αˆgk[n] into equation (15).The appendix then plugs (91) and (92) into (90) to obtain the desired result.Source: Appendix A · pp. 30–31
  • A tight lower bound is obtained by applying Jensen’s inequality to the convex function log2.The passage describes log2 as convex with respect to x.Source: Appendix A · p. 30
  • Conditioned on ˆgk[n], ˜gi[n] is Gaussian with zero mean and a variance that does not depend on ˆgk[n].The passage states that ˜gi[n] is therefore Gaussian distributed.Source: Appendix A · p. 30
  • The remaining expectation is evaluated using [35, Lemma 2.10].The passage identifies this as the remaining task in the derivation.Source: Appendix A · p. 31

APPENDIX B

  • The Appendix B proof uses the ZF detector relation ˆA†[n + 1] ¯G[n + 1] = I_K.The passage also gives ˆA†[n + 1] = ( ¯G†[n + 1] ¯G[n + 1])−1 ¯G†[n + 1].Source: Appendix B · p. 31
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