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Base station cooperation on the downlink: Large system analysis

Randa Zakhour, Stephen V. Hanly

arXiv:1006.3360v2cs.ITmath.IT

TL;DR

The paper asks how to optimize minimum supported rates in a two-cell downlink where cells interfere with one another. It analyzes three levels of cooperation using large-system methods and compares their limiting SINRs. The conclusion reports significant rate gains from MCP when its additional coordination and communication costs can be accommodated.

  • Problem

    The paper examines how to maximize the network-wide minimum achievable rate when multiple downlink cells interfere and antenna-enabled BSs can trade off in-cell rates against interference spillover.

    Source: A. Problem scope · p. 1

  • Method

    It analyzes SCP, CBf, and MCP in a two-cell system with large-system asymptotics, comparing their limiting SINRs and using effective interference to assess SINR-target feasibility.

    Source: §VII · p. 11

  • Results

    MCP offers significant rate gains over the other schemes when the additional coordination and communication between BSs can be accommodated.

    Source: §VII · p. 11

  • Takeaways & Limitations

    Effective interference provides a way to establish whether a given SINR target is feasible across the three schemes.

    Source: §VII · p. 11

    The conclusions rely on randomly selected user channels and do not hold when users are selected based on their channels.

    Source: §VII · p. 11

Abstract

from arXiv · show

This paper considers maximizing the network-wide minimum supported rate in the downlink of a two-cell system, where each base station (BS) is endowed with multiple antennas. This is done for different levels of cell cooperation. At one extreme, we consider single cell processing where the BS is oblivious to the interference it is creating at the other cell. At the other extreme, we consider full cooperative macroscopic beamforming. In between, we consider coordinated beamforming, which takes account of inter-cell interference, but does not require full cooperation between the BSs. We combine elements of Lagrangian duality and large system analysis to obtain limiting SINRs and bit-rates, allowing comparison between the considered schemes. The main contributions of the paper are theorems which provide concise formulas for optimal transmit power, beamforming vectors, and achieved signal to interference and noise ratio (SINR) for the considered schemes. The formulas obtained are valid for the limit in which the number of users per cell, K, and the number of antennas per base station, N, tend to infinity, with fixed ratio. These theorems also provide expressions for the effective bandwidths occupied by users, and the effective interference caused in the adjacent cell, which allow direct comparisons between the considered schemes.

I. INTRODUCTION

The paper studies linear precoding in a multicell downlink, where inter-cell interference makes cooperation a system-level design question. It compares three cooperation levels using large-system analysis of a two-cell model.

A. Problem scope

  • The paper compares rate balancing across single cell processing (SCP), coordinated beamforming (CBf), and multicell processing (MCP), which range from independent BS operation to joint transmission.CBf shares channel information so each BS can account for the interference it causes; MCP shares CSI and data so BSs jointly beamform using pooled resources.Source: A. Problem scope · pp. 1–2

B. Related work

  • Large-system analysis extends downlink RZF analysis beyond a single cell and characterizes SCP with RZF and CBf with a new generalized regularized zero-forcing (GRZF) form.The analysis considers the limit where antenna and user counts grow together at fixed cell loading.Source: B. Related work · pp. 2–3, A. Problem scope · p. 2

II. SYSTEM MODEL

  • The two-cell model has N-antenna BSs and K single-antenna users per cell, with independent flat-fading channels and single-user detection that treats interference as noise.Serving-link channels are i.i.d. CN(0, I), while interfering-link channels are i.i.d. CN(0, ϵI), so ϵ controls inter-cell interference.Source: §II · p. 4

III. COOPERATION AND LINEAR BEAMFORMING SCHEMES: PRIMAL PROBLEMS

The section formulates network-wide minimum-rate maximization for three downlink cooperation architectures. They differ in which CSI and user data are available to the base stations and in how transmit power and SINR constraints are handled.

III. COOPERATION AND LINEAR BEAMFORMING SCHEMES: PRIMAL PROBLEMS

  • The three formulations maximize the network-wide minimum achievable rate under different degrees of inter-cell cooperation and coordination.Each architecture is posed through SINR targets for users in both cells.Source: §III · p. 4

A. Single cell processing

  • SCP has each BS serve only its own users with full spectrum reuse, without accounting for the interference it creates in the other cell.For a target SINR, feasibility is tested by minimizing cell transmit power subject to user SINR constraints; the maximum SINR occurs when the power constraint is met with equality.Source: A. Single cell processing · p. 4

B. Coordinated Beamforming

  • CBf shares CSI between BSs, allowing a joint two-cell design that accounts for the interference generated in other cells while each BS serves its own users.The target SINR is feasible when the minimum per-BS power bound is one.Source: B. Coordinated Beamforming · pp. 4–5

C. Multicell processing

  • MCP shares both CSI and data across BSs, which pool their antennas to jointly serve users in both cells.The formulation uses a joint precoding matrix and per-BS power constraints.Source: C. Multicell processing · p. 5

IV. SOLUTION VIA DUALITY THEORY

The paper uses Lagrangian duality to recast difficult downlink beamforming problems as virtual uplink problems, then applies large-system analysis to obtain simpler beamformer structures. The dual formulations retain scheme-specific power and interference constraints.

IV. SOLUTION VIA DUALITY THEORY

  • Downlink beamforming formulations are nonconvex, so the paper uses Lagrangian duality to connect them to easier virtual uplink optimization problems.The approach aims to derive simpler, asymptotically optimal beamformers from the dual problems.Source: §IV · p. 5

A. Dual UL SCP problem

  • For SCP, the dual uplink powers are Lagrange multipliers, and the resulting beamforming directions are equivalent to uplink MMSE filters.The dual-power equations form an interference function, so iterative power-control approaches converge to the optimal dual variables.Source: A. Dual UL SCP problem · pp. 5–6
  • The paper’s UL-DL duality concerns optimization without dirty paper coding, not the capacity-region duality for the MIMO broadcast channel.The reviewed optimization approach can be extended to include dirty paper coding, but this paper does not use it.Source: A. Dual UL SCP problem · p. 6

B. Dual UL CBf problem

  • For CBf, the dual uplink includes uncertain noise, with the multipliers for the two BS power constraints interpreted as virtual-uplink noise levels.Once these levels are set, the optimal dual powers are given implicitly by a fixed-point equation.Source: B. Dual UL CBf problem · pp. 6–7

C. Dual UL MCP problem

The MCP dual formulation expresses the optimization through dual variables and a fixed-point equation for the virtual-uplink powers. The downlink and dual beamforming vectors are related at the optimum.

C. Dual UL MCP problem

  • The dual of the MCP problem is written using the per-BS power multipliers and is equivalent to a virtual-uplink formulation.The excerpt introduces normalized channel notation and a diagonal matrix whose entries are the dual powers.Source: C. Dual UL MCP problem · p. 7
  • The optimal dual powers satisfy a fixed-point equation obtained by imposing the target SINR with the optimal dual variables.The duality gap is zero for a feasible primal problem, and the original and dual beamforming vectors are related.Source: C. Dual UL MCP problem · p. 7
  • The downlink transmit powers are obtained by substituting the beamformers into the downlink SINR constraints.The passage identifies this step as providing the solution for the powers.Source: C. Dual UL MCP problem · p. 7

V. LARGE SYSTEM RESULTS

For fixed cell loading as antennas and users grow, the section characterizes beamforming and SINR for SCP, coordinated beamforming, and MCP. It also explains how dependent optimal dual powers are analyzed and how scheme-specific loading conditions constrain feasibility.

V. LARGE SYSTEM RESULTS

  • The analysis bounds SINR using deterministic dual uplink powers, then uses monotonicity and a sandwich argument to show optimal powers converge to deterministic values.Direct random-matrix analysis is complicated because the optimal dual powers depend on the channel matrix and on one another.Source: §V · p. 8

A. Asymptotically Optimal Beamformers

  • The theorems characterize asymptotically optimal beamforming for SCP, CBf, and MCP.The analysis takes the large-system limit with the number of users and antennas growing while K/N remains fixed.Source: §V · p. 7, A. Asymptotically Optimal Beamformers · pp. 8–9
  • SCP's optimal beamformer has a regularized zero-forcing structure, and its target-SINR condition is both necessary and sufficient.The optimal beamformer exists even when a zero-forcing beamformer does not.Source: A. Asymptotically Optimal Beamformers · p. 8
  • CBf achieves target SINR γ when β < 1, while the target is asymptotically unachievable when β > 1.Its generalized RZF beamformer zero-forces interference to users in both its own and the other cell; its effective bandwidth for other-cell users is reduced versus SCP when ϵ is non-negligible.Source: A. Asymptotically Optimal Beamformers · pp. 8–9
  • For MCP, target SINR γ is achievable when βγ/(1+γ) < 1 and unachievable when βγ/(1+γ) > 1.MCP has the same effective bandwidth per user as SCP in a single isolated cell, while its power consumption is reduced by the factor (1+ϵ).Source: A. Asymptotically Optimal Beamformers · p. 9

B. Effective interference

The section expresses target-SINR feasibility through effective interference, distinguishing the criterion for SCP and CBf from that for MCP. This characterization also connects to large-system analysis of linear uplink receivers through uplink-downlink duality.

B. Effective interference

  • The effective-interference characterization directly compares the limiting SINRs of SCP, CBf, and MCP and tests target feasibility.SNR denotes P/σ².Source: B. Effective interference · p. 9
  • Target γ is feasible for SCP or CBf when SNR I_eff(SNR,ϵ,γ) > γ, while MCP requires (1+ϵ)SNR I_eff(SNR,ϵ,γ) > γ.The criterion uses the effective interference at target SINR γ.Source: B. Effective interference · pp. 9–10
  • Uplink-downlink duality accounts for the close parallel with effective interference in large-system linear uplink receivers.Source: B. Effective interference · p. 10

C. Asymptotically optimal cell loading

The section characterizes how optimal cell loading depends on the scheme and compares cooperation levels across interference settings. It reports finite-system checks and concludes that multicell processing can yield substantial rate gains, subject to its coordination and data-sharing costs.

C. Asymptotically optimal cell loading

  • The optimal-loading propositions characterize when each scheme’s normalized rate keeps increasing with β and when it instead has a finite maximizing β∗.For SCP, the finite optimum may be found by line search; the supplied proposition fragments do not preserve all scheme-specific conditions.Source: C. Asymptotically optimal cell loading · p. 10
  • In the noise-limited region, loading can increase indefinitely, but rates per user decrease and more user channels must be learned.Source: C. Asymptotically optimal cell loading · p. 10

A. When is half re-use SCP better than CBf?

B. Numerical results

  • With optimized loading and ϵ < 1, CBf outperforms SCP, SCP-ZF, and GZF; MCP gains over CBf and other schemes are significant and grow with ϵ.At small ϵ, CBf gains little over SCP; at ϵ = .8, its gains over SCP are significant but gains over half-reuse or optimized GZF are small.Source: B. Numerical results · pp. 10–11

VI. PERFORMANCE RESULTS

  • MCP offers the most gain when inter-cell interference is large, but requires significant BS cooperation and full data sharing whose backhaul cost is not accounted for.The conclusion reports significant MCP rate gains when additional coordination and communication between BSs can be accommodated.Source: B. Numerical results · p. 11, §VII · p. 11

B. Numerical results

  • For K = 3 and N = 4, the large-system analysis approximates average rates from finite-system optimization quite well; the authors also test asymptotically optimal beamformers in the finite system.The optimization results use SNR = 10 dB and ϵ values .01, .1, .5, .8, and 1; the beamformers are slightly modified for the MCP power constraint.Source: B. Numerical results · p. 11

VII. CONCLUSIONS

  • The analysis compares SCP, CBf, and MCP in a two-cell network and characterizes their limiting SINRs; Monte Carlo simulations verify the results.Source: §VII · p. 11
  • The conclusions assume randomly selected user channels; they do not apply to channel-based user selection, where scheduled ZF can be almost optimal.Source: §VII · p. 11

APPENDIX A USEFUL THEOREMS

The appendices supply random-matrix results used to derive the paper’s large-system expressions. They apply these results to SCP and establish almost-sure convergence under the stated model conditions.

APPENDIX B USEFUL ASYMPTOTIC EXPRESSIONS

  • The appendix applies these theorems to derive asymptotic expressions for SCP, CBf, and MCP in the symmetric two-cell channel model.For this model, the diagonal matrix in the theorem reduces to one or two variables because many entries are equal.Source: Appendix B · p. 13

A. SCP

  • For SCP, with user power multipliers fixed and bounded, the quantity of interest converges almost surely as N and K grow with their ratio fixed.The appendix states boundedness is required for the cited results to apply; otherwise the dual objective is unbounded and the primal problem infeasible.Source: A. SCP · p. 13

B. CBf

The appendix derives large-system expressions for coordinated beamforming and multicell processing. Its MCP derivation also treats symmetric parameter settings and relates quantities across the two cells.

B. CBf

  • For CBf, the appendix states that the quantity of interest for a user converges almost surely as N and K grow with their ratio fixed.The derivation holds the per-cell user power multipliers and the μ parameters fixed.Source: B. CBf · p. 13

C. MCP

  • For MCP, the derivation includes a symmetric case with equal μ and λ parameters across cells and uses cross-cell identities for its trace quantities.The stated trace identities are V1(D1,2) = V2(D1,1) and V2(D1,2) = V1(D1,1).Source: C. MCP · pp. 14–15

APPENDIX C A SIMPLE MONOTONICITY RESULT

The appendix characterizes feasible power control through a standard interference function. When feasible powers exist, iteration converges to a unique positive fixed point that meets all users’ SINR targets.

APPENDIX C A SIMPLE MONOTONICITY RESULT

  • A user’s standard interference function is written as I_k(λ) = γ_k F_k(λ), where γ_k is its SINR target and F is a standard interference function.Source: Appendix C · p. 15
  • A power vector is feasible when each λ_k is at least γ_k F_k(λ).Source: Appendix C · p. 15
  • If a feasible solution exists, the interference function has a unique positive fixed point, and iteration from any power vector converges to powers meeting every SINR target.Source: Appendix C · p. 15

APPENDIX D PROOF OF THEOREM 1

For SCP, the large-system analysis establishes asymptotically optimal beamforming directions and equal dual powers across users. Deterministic downlink powers attain the SINR constraints with vanishing duality gap, subject to the stated feasibility condition.

APPENDIX D PROOF OF THEOREM 1

  • The optimal dual powers converge to a common constant, yielding asymptotically optimal downlink beamforming directions through UL-DL duality.The downlink directions in one cell do not depend on the other cell’s directions, although optimal power levels do depend on the other cell’s power levels.Source: Appendix D · p. 16
  • Deterministic downlink powers meet the SINR constraints asymptotically and are optimal, as the primal-dual gap tends to zero.The primal objective converges to the same value as the dual objective.Source: Appendix D · p. 18
  • The coupled primal problem is feasible only when its effective-bandwidth condition holds; otherwise required transmit power diverges and no feasible solution remains asymptotically.The passage states the condition as an effective-bandwidth condition less than 1.Source: Appendix D · p. 18

APPENDIX E PROOF OF THEOREM 2

For coordinated beamforming, the asymptotically optimal dual variables are symmetric across cells, and the optimal downlink directions follow from them. The resulting deterministic power allocation meets SINR targets with zero asymptotic duality gap when the feasibility condition holds.

APPENDIX E PROOF OF THEOREM 2

  • The asymptotically optimal dual variables are μ1 = μ2 = 1, and all user powers converge to a common constant that determines the optimal downlink directions.The proof identifies the equal dual-variable choice as optimal for the stated optimization problem.Source: Appendix E · pp. 19–20
  • Equal deterministic downlink powers meet the SINR constraints asymptotically and are optimal because the duality gap tends to zero.The primal and dual objective values converge to the same value.Source: Appendix E · p. 21
  • The primal problem has an asymptotic solution only when its stated condition is satisfied; otherwise no feasible solution exists in the large-system limit.If the condition fails, required transmit power diverges as the denominator approaches zero.Source: Appendix E · p. 21

APPENDIX F PROOF OF THEOREM 3

For full cooperative processing, the asymptotically optimal dual variables are equal across cells, and the resulting downlink beamforming directions are fixed by the dual analysis. Equal per-user powers meet the SINR constraints and attain asymptotic optimality when the feasibility condition holds.

APPENDIX F PROOF OF THEOREM 3

  • Under the stated condition, the SINRs of all users converge to the common target γ.The deterministic dual power assignment converges to the unique fixed point (γ, γ).Source: Appendix F · p. 22
  • The optimal dual variables are μ1 = μ2 = 1, and all user powers converge to a common constant that determines the asymptotically optimal downlink directions.The directions are given by the vectors identified in the theorem’s analysis.Source: Appendix F · p. 22
  • The two base stations transmit with equal power asymptotically under this scheme.Both total powers converge to the same value.Source: Appendix F · pp. 23–24
  • Equal per-user downlink powers meet the SINR constraints asymptotically and are optimal because the duality gap tends to zero.The primal and dual objective values converge to the same value.Source: Appendix F · p. 24
  • The primal problem has no asymptotic feasible solution when its stated condition is not satisfied.Source: Appendix F · p. 24

APPENDIX G PROOF OF THEOREM 1

The appendix characterizes the loading that maximizes normalized rate per cell through the rate’s behavior as a function of the target SINR. Depending on parameter conditions, the optimum corresponds either to infinite loading or a finite loading.

APPENDIX G PROOF OF THEOREM 1

  • When η < 1, r(γ*) is initially increasing and then decreasing.Source: Appendix G · p. 24
  • For the stated ranges of a and ε, r decreases and its optimum occurs at γ* = 0, corresponding to β* = ∞.These ranges are a ≥ 1 with 0 ≤ ε ≤ ε0,2, or 7/8 ≤ a ≤ 1 with ε0,1 ≤ ε ≤ ε0,2.Source: Appendix G · p. 25
  • In another case, r initially increases and eventually decreases, so it reaches a global maximum at a finite γ* > 0.Source: Appendix G · p. 25
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