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Power Scaling of Uplink Massive MIMO Systems with Arbitrary-Rank Channel Means

Qi Zhang, Shi Jin, Kai-Kit Wong, Hongbo Zhu, Michail Matthaiou

arXiv:1405.6952v1cs.IT

TL;DR

The paper asks how massive-MIMO uplink rates and transmit-power scaling behave in Ricean fading when channel means can have arbitrary rank. It analyzes MRC and ZF with perfect and imperfect CSI using tractable large-antenna expressions and finite-antenna approximations. The main result is 1/M power scaling in perfect CSI and in imperfect CSI with nonzero Ricean K-factor, but only 1/√M under imperfect-CSI Rayleigh fading.

  • Problem

    Existing massive-MIMO analyses often assume Rayleigh fading or rank-1 deterministic means, motivating rate and power-scaling analysis for arbitrary-rank Ricean channels.

  • Method

    The paper derives large-antenna achievable-rate expressions and finite-antenna approximations for single-cell uplink MRC and ZF receivers with perfect and imperfect CSI.

  • Results

    1/M transmit-power scaling preserves rates with perfect CSI and with imperfect CSI when K is non-zero, while imperfect-CSI Rayleigh fading permits only 1/√M scaling.

  • Takeaways & Limitations

    With scaling power and the same CSI quality, MRC and ZF uplink rates converge to the same constant value; increasing Ricean K-factor also leads rates toward fixed values.

Abstract

from arXiv · show

This paper investigates the uplink achievable rates of massive multiple-input multiple-output (MIMO) antenna systems in Ricean fading channels, using maximal-ratio combining (MRC) and zero-forcing (ZF) receivers, assuming perfect and imperfect channel state information (CSI). In contrast to previous relevant works, the fast fading MIMO channel matrix is assumed to have an arbitrary-rank deterministic component as well as a Rayleigh-distributed random component. We derive tractable expressions for the achievable uplink rate in the large-antenna limit, along with approximating results that hold for any finite number of antennas. Based on these analytical results, we obtain the scaling law that the users' transmit power should satisfy, while maintaining a desirable quality of service. In particular, it is found that regardless of the Ricean $K$-factor, in the case of perfect CSI, the approximations converge to the same constant value as the exact results, as the number of base station antennas, $M$, grows large, while the transmit power of each user can be scaled down proportionally to $1/M$. If CSI is estimated with uncertainty, the same result holds true but only when the Ricean $K$-factor is non-zero. Otherwise, if the channel experiences Rayleigh fading, we can only cut the transmit power of each user proportionally to $1/\sqrt M$. In addition, we show that with an increasing Ricean $K$-factor, the uplink rates will converge to fixed values for both MRC and ZF receivers.

I. INTRODUCTION

The paper extends massive-MIMO power-scaling analysis to Ricean fading with arbitrary-rank deterministic channel means, considering MRC and ZF receivers under perfect and imperfect CSI. It derives tractable rate expressions and identifies how user transmit power can decrease as the base-station antenna count grows.

  • Motivation: Ricean fading captures specular or line-of-sight components that Rayleigh-only models do not represent.The paper motivates more general fading models for massive-MIMO systems, particularly under millimeter-wave propagation.
  • Contribution: The fast-fading channel combines an arbitrary-rank deterministic component with a Rayleigh-distributed scattered-signal component.Earlier studies commonly assumed a rank-1 deterministic component; this paper relaxes that constraint.
  • System and method: The analysis considers a single-cell uplink with MRC and ZF receivers under perfect and imperfect CSI.The system serves multiple single-antenna users through a base station with many antennas.
  • Power scaling: 1/M is the perfect-CSI transmit-power scaling that maintains a desirable user rate as M grows asymptotically large.This result holds regardless of the Ricean K-factor.
  • Power scaling: 1/M remains achievable with uncertain CSI when the Ricean K-factor is non-zero, whereas the Rayleigh case permits only 1/√M scaling.Under uncertain CSI, uplink rates approach fixed values as M becomes large.

2) Imperfect CSI:

For imperfect CSI, the paper estimates the random channel component from uplink pilots and derives tractable achievable-rate expressions and power-scaling results for finite and asymptotically large antenna arrays.

  • Imperfect CSI: Uplink pilots are used to estimate the random channel component after removing the known line-of-sight component.The pilot sequences are orthogonal and occupy τ symbols within the channel coherence time T.
  • Imperfect CSI: The MMSE estimator produces an estimate of the random channel component from the received noisy pilot matrix.The training stage uses simultaneous orthogonal pilots and pilot power p_p = τp_u.
  • Imperfect CSI: Channel estimation error contributes a distinct impairment term alongside intracell interference and noise in the achievable-rate expression.The imperfect-CSI receiver depends on the estimated channel matrix.
  • Rate metric: The achievable uplink sum rate is defined over users while accounting for pilot duration within the channel coherence time.The coherence interval contains T symbols, of which τ are used for channel estimation.
  • Analysis: The paper derives closed-form large-antenna rate expressions and finite-antenna approximations that apply to arbitrary-rank Ricean mean matrices.The approximations become particularly accurate in massive-MIMO systems because their accuracy improves with the number of summed terms.

A. Perfect CSI

For perfect CSI, the analysis uses large-antenna limits and inner-product expectations to characterize achievable rates with linear receivers, beginning with MRC.

  • Perfect CSI: The law of large numbers supplies asymptotic channel limits as the number of base-station antennas grows.The convergence is stated almost surely.
  • Perfect CSI: The analysis evaluates expectations of same-column inner products and squared inner products between different channel columns.These quantities support the rate analysis for multiuser reception.
  • MRC: MRC uses the channel matrix itself as the linear receiver, so the nth combining vector equals the nth channel column.Substituting this receiver into the achievable-rate expression gives the MRC user rate.

1) MRC Receivers:

For MRC with perfect CSI, the paper derives exact limits and finite-antenna approximations for Ricean fading, showing that transmit power can scale as 1/M while maintaining rate. Rates also approach fixed values as the Ricean K-factor increases.

  • Power scaling: 1/M is the maximum transmit-power reduction for MRC with perfect CSI without degrading the nth user’s rate.This scaling maintains a non-zero limiting rate as M grows.
  • Large-antenna limit: The exact large-M uplink-rate limit is independent of the underlying fading model because the Ricean K-factor does not appear in the limiting expression.The same 1/M scaling conclusion therefore aligns with the Rayleigh-fading result.
  • Rate approximation: Finite-antenna approximations for the achievable MRC uplink rate expose the impact of the Ricean K-factor more effectively than the exact asymptotic limit.The approximation is intended to hold for any finite number of antennas.
  • Ricean K-factor: As the Ricean K-factor grows, the MRC uplink rate approaches a constant value.This fixed-value limit is obtained from the rate approximation when the relevant K-factors tend to infinity.
  • Approximation accuracy: For power scaling pu = Eu/M, the finite-antenna MRC approximation converges to the exact rate as M becomes large.The convergence follows because random quantities become deterministic asymptotically.

2) ZF Receivers:

For ZF with perfect CSI, the paper establishes the power-scaling law and develops closed-form finite-antenna rate approximations. The transmit power can scale as 1/M, while the approximation converges to the exact large-system limit and rates approach fixed values for large Ricean K-factors.

  • Power scaling: 1/M is the maximum transmit-power reduction for ZF with perfect CSI without degrading the nth user’s rate.The result applies to the achievable uplink rate in the large-antenna regime.
  • Rate approximation: The ZF uplink-rate approximation is derived using non-central Wishart analysis and applies to finite antenna numbers.The derivation is more involved than for MRC because the channel Gram matrix is non-central Wishart.
  • Ricean K-factor: As the Ricean K-factor tends to infinity, the ZF rate approximation converges to a fixed limit.Corollary 4 gives this limit under the stated full-rank condition on the mean channel matrix.
  • Approximation accuracy: As M grows, the ZF rate approximation with pu = Eu/M converges to the exact limiting rate.This parallels the corresponding MRC convergence result.
  • Scope condition: The full-rank condition required for the large-K ZF result is likely in practice when users have different angles of arrival.The paper connects this condition to randomly distributed users.

B. Imperfect CSI

For imperfect CSI, the paper establishes large-antenna inner-product results for the estimated channel matrix and uses them to analyze MRC and ZF uplink rates.

  • Estimated-channel behavior: As M grows asymptotically, the inner product of any two columns in the estimated channel matrix can be characterized using the law of large numbers.These results provide the large-system channel quantities used in the imperfect-CSI rate analysis.
  • Estimated-channel behavior: The paper also derives expectations for same-column inner products and squared cross-column inner products in the estimated channel matrix.These expectations support subsequent imperfect-CSI rate expressions.
  • MRC specialization: With MRC under imperfect CSI, the combining vector equals the estimated channel column, allowing the uplink rate to be written directly from the general rate expression.This specializes the receiver model to ˆan = ˆgn.

1) MRC Receivers:

With imperfect CSI, MRC’s power-scaling law depends on the Ricean K-factor: Rayleigh fading permits only 1/sqrt(M), whereas non-zero K permits 1/M. The paper derives rate approximations and their large-K and large-M limits.

  • Approximation accuracy: For the appropriate power scaling, the imperfect-CSI MRC approximation converges to the exact rate limit as M tends to infinity.This convergence is stated after the large-antenna analysis of Theorem 6.
  • Power scaling: 1/sqrt(M) is the maximum power reduction for MRC with imperfect CSI when the nth user has zero Ricean K-factor.Other scaling exponents make the relevant rate term vanish or grow without bound.
  • Power scaling: 1/M is the maximum power reduction for MRC with imperfect CSI when the nth user has a non-zero Ricean K-factor.The paper attributes the stronger scaling to reduced fading fluctuations from the LOS component.
  • Rate approximation: The imperfect-CSI MRC approximation is derived from the estimated-channel inner-product results and is obtained using MMSE channel estimation.Theorem 6 gives a closed-form approximation for the achievable uplink rate.
  • Ricean K-factor: As the Ricean K-factor grows without bound, the imperfect-CSI MRC rate approximation approaches the same constant as the perfect-CSI case.The paper states that purely deterministic channels yield the same limiting constant regardless of CSI quality.

2) ZF Receivers:

For ZF receivers, the paper derives imperfect-CSI approximations and shows that power scaling depends on whether the Ricean K-factor is zero. In the large-antenna limit, ZF and MRC share the same exact uplink-rate limit.

  • Power-scaling analysis: Theorem 7 analyzes ZF with imperfect CSI when user power scales as pu = Eu/M^α for α > 0 and fixed Eu.The result provides the basis for determining which scaling exponents preserve a non-zero asymptotic rate.
  • Power-scaling analysis: When Kn = 0, imperfect-CSI ZF can scale each user’s transmit power down by at most 1/M without degrading the nth user’s rate.The corresponding achievable rate becomes the same as the MRC rate in the cited asymptotic expression.
  • Power-scaling analysis: When Kn is non-zero, each user’s power can be scaled to pu = Eu/M while retaining a non-zero asymptotic rate.The limiting rate depends on the Ricean factor in the imperfect-CSI case.
  • Asymptotic receiver behavior: For both perfect and imperfect CSI, the exact asymptotic ZF uplink-rate limit equals the MRC limit.The equality holds regardless of receiver type and is consistent with the paper’s perfect-CSI conclusion.
  • Asymptotic receiver behavior: As the Ricean K-factor tends to infinity, ZF uplink-rate approximations converge to the same fixed value as MRC, regardless of CSI quality.This is stated for the case where all users share the same increasing K-factor.

IV. NUMERICAL RESULTS

Numerical experiments compare analytical approximations with simulations for MRC and ZF under perfect and imperfect CSI. The results support the predicted power-scaling laws and K-factor limits, while revealing receiver- and CSI-dependent behavior.

  • Baseline comparisons: For N = 10 users and pu = 10dB, simulations closely agree with analytical approximations across the tested Ricean K-factors.The tested K-factors are 0, 3dB, 6dB, and 10dB, under both perfect and imperfect CSI.
  • Baseline comparisons: Without power normalization, both MRC and ZF sum rates grow without bound as the number of BS antennas increases.This establishes the unscaled-power baseline for the numerical comparisons.
  • 1/M power scaling: With pu = Eu/M and Eu = 20dB, analytical approximations converge to exact results as M grows large.Under perfect CSI, MRC and ZF approach the same value independently of K; under imperfect CSI, the common limit depends on K.
  • Stronger power scaling: With the stronger tested power reduction, imperfect-CSI rates converge to constants for K = 0 but increase with M when K ≠ 0.The observations are consistent with the predicted distinction between Rayleigh fading and nonzero Ricean factors.
  • Ricean K-factor behavior: As K increases, sum rates approach fixed values for all receiver and CSI cases, except that perfect-CSI ZF can decline because the channel becomes ill-conditioned.The perfect-CSI ZF limitation is attributed to the large singular-value spread and high condition number of the mean channel matrix.

V. CONCLUSION

The paper develops tractable large-antenna uplink-rate expressions for Ricean massive MIMO and uses them to characterize power scaling and K-factor behavior. Its conclusions cover MRC and ZF with perfect and imperfect CSI.

  • Contributions: The analysis treats Ricean fading channels with arbitrary-rank means using both MRC and ZF receivers under perfect and imperfect CSI.The paper derives large-antenna expressions and finite-antenna approximations for achievable uplink rates.
  • Power-scaling conclusions: With perfect CSI, user transmit power can scale proportionally to 1/M while maintaining a desirable uplink rate.This result is stated for Ricean fading regardless of the Ricean K-factor.
  • Receiver comparison: Under power scaling and matched CSI quality, MRC and ZF uplink rates tend to the same constant value.The conclusion concerns the asymptotic behavior of both receiver types.
  • K-factor behavior: As the Ricean K-factor increases, perfect- and imperfect-CSI uplink rates converge to the same fixed value for a given receiver.The conclusion applies to the rate behavior studied for both MRC and ZF.
  • Approximation accuracy: The paper’s approximation becomes increasingly accurate as the numbers of random variables increase and their variances become small.This accuracy observation follows from the law-of-large-numbers argument for the approximation offset.

APPENDIX B PROOF OF LEMMA 2

The appendix proves a lemma by expressing channel quantities through independent real and imaginary Gaussian components and evaluating their large-M limits. Cross-user terms vanish asymptotically, yielding the desired result.

  • Asymptotic setup: The proof begins by applying the law of large numbers to channel-derived random quantities as M grows.This establishes convergence toward their mean values in the asymptotic analysis.
  • Channel decomposition: The fast-fading entries are decomposed into independent real and imaginary parts, each with zero mean and variance 1/2.The deterministic phase factor is also separated before evaluating the limit.
  • Cross-user terms: For distinct users, the last three terms vanish asymptotically, leaving only the remaining contribution.This is the key simplification used to obtain the cross-user part of the lemma.
  • Same-user terms: For the same user, the proof extracts real and imaginary parts before computing norm squares and removing zero-expectation terms.The resulting expectations provide the final expression needed by the lemma.
  • Conclusion: Substitution of the evaluated expectations produces the final result and completes the proof.The appendix explicitly concludes after obtaining all expectations used in the lemma.

APPENDIX D PROOF OF LEMMA 4

The proof derives the imperfect-CSI channel-matrix terms and simplifies their expectations using MMSE estimation, the law of large numbers, and zero-mean independence. It then evaluates remaining norm-square expectations separately for equal and unequal user indices.

  • Imperfect-CSI channel model: The imperfect-CSI channel model supplies the channel-matrix entries used in the proof.The model is referenced from Section II-B2, specifically equation (12).
  • Imperfect-CSI channel model: MMSE estimation is applied to the estimated random component, while the non-estimated term matches the perfect-CSI case.The estimation details are stated to have been introduced earlier in Section II-B2.
  • Large-antenna simplification: As M →∞, the proof evaluates expectations, removes zero-expectation terms, and simplifies the resulting expressions.The simplification uses the law of large numbers and expectation calculations across equations (131)–(133).
  • Large-antenna simplification: Because the entries of H and W are i.i.d. zero-mean with unit variance, only the channel-mean element remains in one expanded expectation.The same reasoning is used to simplify equations (137) and (139).
  • Norm-square expectations: The remaining norm-square expectations are handled separately for i = n and i ≠ n using expansions, zero-expectation removal, and algebraic manipulation.For i ≠ n, the inner product has non-zero real and imaginary parts before the final algebraic simplification.
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