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Spectral Efficiency of Dynamic Coordinated Beamforming: A Stochastic Geometry Approach

Namyoon Lee, David Morales-Jimenez, Angel Lozano, Robert W. Heath Jr

arXiv:1405.1365v1cs.ITmath.IT

TL;DR

The paper addresses how to choose coordination-cluster size when coordination benefits must be weighed against pilot overhead. It analyzes dynamic coordinated beamforming using stochastic geometry and optimizes cluster cardinality using SIR distributions and spectral efficiency. It finds that typical-user optimal clusters are small and scale with fading coherence, with the greatest benefits for users in the inner part of their coordination clusters.

  • Problem

    Determining the optimal cluster cardinality is necessary to assess the true benefits of BS coordination.

    Source: A. Background · p. 1

  • Method

    The paper uses stochastic geometry to derive SIR distributions and ergodic spectral efficiency for dynamic coordinated beamforming, then optimizes cluster cardinality with pilot overhead included.

    Source: C. Contribution · p. 2, §VII · p. 8

  • Results

    The typical user's optimal cluster cardinality is small and scales with fading coherence.

    Source: §VII · p. 8

  • Takeaways & Limitations

    Coordination is most beneficial to users in the outer part of their cells but inner part of their coordination clusters.

    Source: §VII · p. 8

Abstract

from arXiv · show

This paper characterizes the performance of coordinated beamforming with dynamic clustering. A downlink model based on stochastic geometry is put forth to analyze the performance of such base station (BS) coordination strategy. Analytical expressions for the complementary cumulative distribution function (CCDF) of the instantaneous signal-to-interference ratio (SIR) are derived in terms of relevant system parameters, chiefly the number of BSs forming the coordination clusters, the number of antennas per BS, and the pathloss exponent. Utilizing this CCDF, with pilot overheads further incorporated into the analysis, we formulate the optimization of the BS coordination clusters for a given fading coherence. Our results indicate that (i) coordinated beamforming is most beneficial to users that are in the outer part of their cells yet in the inner part of their coordination cluster, and that (ii) the optimal cluster cardinality for the typical user is small and it scales with the fading coherence. Simulation results verify the exactness of the SIR distributions derived for stochastic geometries, which are further compared with the corresponding distributions for deterministic grid networks.

I. INTRODUCTION

The paper studies dynamic BS coordination as a balance between interference reduction and CSI overhead, then develops a stochastic-geometry analysis and optimizes cluster size. It models user-specific clusters through nearest-BS geometry and finds that coordination benefits vary with location.

A. Background

  • The coordination problem balances reduced intercell interference against the CSI acquisition overhead that grows with the coordinated set.The paper identifies optimal cluster cardinality as key to assessing coordination's benefits.Source: A. Background · p. 1

B. Related Work

  • Dynamic clustering groups BSs using user location and channel quality, with prior evidence that such clusters outperform fixed counterparts.Stochastic-geometry analyses had examined fixed cluster structures, motivating their extension to dynamic cooperation.Source: B. Related Work · p. 1

A. Network Model

  • The network model places BSs and users in independent homogeneous Poisson point processes and forms traditional cells from the BS Voronoi tessellation.Users define their own clusters based on location.Source: C. Contribution · p. 1, A. Network Model · p. 2

C. Contribution

  • Dynamic coordinated beamforming serves each user from a location-dependent cluster while nullifying interference from other BSs within that cluster.The paper characterizes this strategy's performance using stochastic geometry.Source: C. Contribution · p. 2
  • The analysis derives SIR and ergodic spectral-efficiency expressions, incorporates pilot overhead, and finds that typical-user cluster cardinality is small and scales with fading coherence.Coordination is most beneficial for users in the outer part of their cells but inner part of their coordination clusters.Source: C. Contribution · p. 2

B. Dynamic Clustering Model

  • Each user coordinates with its K closest BSs, and the corresponding Kth-order Voronoi cell contains locations served by that same BS set.For K = 2, users in V2(d1, d2) connect to BSs at d1 and d2.Source: B. Dynamic Clustering Model · p. 2

C. Signal Model

The signal model uses coordinated beamforming to suppress in-cluster interference, leaving out-of-cluster interference in the user's SIR. The analysis characterizes SIR first for specific relative cluster geometries and then averaged over cluster geometry.

C. Signal Model

  • Beamforming vectors are constructed from CSI to nullify intra-cluster interference while maximizing desired signal strength, and the resulting SIR includes aggregate out-of-cluster interference.The beamforming solution exists when Nt ≥ K.Source: C. Signal Model · p. 3
  • The SIR randomness includes user location relative to the serving BS and interfering BSs, desired-link fading, and interference fading.The serving-location randomness can equivalently be represented by the distance to the furthest cluster BS and δ1.Source: C. Signal Model · p. 3

D. Performance Metrics

1) Specific Relative Cluster Geometry:

  • The SIR distribution is first conditioned on δ1, with absolute cluster distances and out-of-cluster interference marginalized over.This conditional CCDF represents average behavior among cluster geometries sharing δ1, not the experience of an actual user.Source: 1) Specific Relative Cluster Geometry · p. 3

2) Average Cluster Geometry:

  • Marginalizing further over δ1 yields the SIR distribution across geometries, which supports typical-user performance analysis but cannot distinguish favorable from adverse geometries.This fully marginalized distribution is used to characterize the typical user's performance.Source: 2) Average Cluster Geometry · p. 3

III. SPECIFIC RELATIVE CLUSTER GEOMETRY

The section characterizes the conditional SIR CCDF for specific relative cluster geometries in terms of K, Nt, and β. It gives general bounds and examines special cases, including how coordination’s benefits depend on a user’s relative location within the cluster.

A. General Characterization

  • The exact general characterization is unwieldy, so the section also gives closed-form upper and lower bounds; they coincide when Nt = K, yielding an exact CCDF for that case.The bounds are presented for the full generality of Theorem 1.Source: A. General Characterization · p. 4

1) Noncoordinated Network:

  • Without coordination, setting K = 1 and Nt = 1 recovers the SIR CCDF in, which Theorem 2 generalizes; for β = 4, the upper bound closely matches the exact CCDF across the SIR range shown for different Nt.Figure 2 presents the CCDF for β = 4 and different transmit-antenna counts in the absence of coordinated beamforming.Source: 1) Noncoordinated Network · p. 4, Figure 2 · p. 4

B. Special Cases

  • With coordinated beamforming and Nt = K, Corollary 1 gives a simple CCDF expression, and for β = 4 the function D reduces to a simpler form.The text identifies β = 4 as a standard value in terrestrial outdoor wireless systems.Source: 2) Coordinated Network with Nt = K · p. 4

1) Noncoordinated Network:

  • For K = Nt = 2 and β = 4, a user with δ1 = 0.5 has a better SIR distribution than one with δ1 = 0.8; versus no coordination, δ1 = 0.5 is better up to γ = 16 dB.For δ1 = 0.8, the SIR distribution is worse than without coordination across the range of γ; the text relates this to protection area and the choice between interference cancellation and desired-signal maximization.Source: 2) Coordinated Network with Nt = K · pp. 4–5
  • For Nt = K = 2 and β = 4, the largest ergodic spectral-efficiency gains from coordination occur at small δ1, where users face less out-of-cluster interference.The comparison across relative locations is reported in Table I.Source: 2) Coordinated Network with Nt = K · p. 5

IV. AVERAGE CLUSTER GEOMETRY

The SIR CCDF is characterized through bounds in K, Nt, and β; for β = 4 and Nt = K, the bounds coincide, and simulations agree with the approximations across the tested SIR range.

IV. AVERAGE CLUSTER GEOMETRY

  • The section derives tight lower and upper CCDF bounds as functions of K, Nt, and β.Source: §IV · p. 5

B. Bound Approximations

C. Low-SIR Analysis

The low-SIR analysis relates coordination to outage behavior, while the cluster-size analysis incorporates pilot overhead and user geometry. For a large initial protection area, adding BSs does not outweigh overhead, so the optimal cluster remains small.

C. Low-SIR Analysis

  • The low-SIR expansion indicates that, for K = Nt, outage probability in a quasi-static setting decreases linearly with K.Source: C. Low-SIR Analysis · p. 6

V. OPTIMAL CLUSTER CARDINALITY

  • Pilot overhead is modeled as Lp = ηKNt, with η representing SINR-dependent pilots per transmit antenna and fading-coherence interval.Source: §V · p. 6

A. Optimal Cluster Cardinality for a Specific Relative Cluster Geometry

B. Optimal Cluster Cardinality for an Average Cluster Geometry

The section optimizes cluster size using effective spectral efficiency and compares stochastic-geometry SIR distributions with a deterministic grid model. Its results show that optimal cluster size depends on coherence, and that the stochastic and grid SIR curves have analogous shapes.

B. Optimal Cluster Cardinality for an Average Cluster Geometry

VI. COMPARISONS

  • The deterministic-grid SIR CCDF is more favorable than the stochastic-geometry CCDF, but their analogous shapes help gauge coordination gains in actual deployments.Under the PPP model, the nearest interferer can lie arbitrarily close to the in-cluster BSs.Source: §VI · p. 8, Figure 8 · p. 8

VII. CONCLUSION

  • The analysis finds coordinated beamforming most beneficial for users inside their coordination clusters, and the typical user's optimal cluster is small and scales with fading coherence.The benefit is smaller near cluster edges, where out-of-cluster interference is strong.Source: §VII · p. 8

APPENDIX A PROOF OF THEOREM 1

The proof combines desired- and interfering-link fading distributions with the PPP serving-distance distribution to derive the SIR CCDF.

APPENDIX A PROOF OF THEOREM 1

  • The desired-link fading is chi-squared with 2(Nt −K + 1) degrees of freedom, while out-of-cluster interference fading is independent exponential with unit mean.These distributions apply to the desired link and all out-of-cluster links, respectively.Source: Appendix A · p. 8
  • For a planar PPP of intensity λ, the distance from a typical user to its Kth serving BS follows a generalized Gamma distribution.This distance distribution supplies the serving-location averaging used in the proof.Source: Appendix A · p. 8
  • Conditioning on the Kth BS location, the proof evaluates the interference Laplace transform and averages over the relevant distributions to obtain the SIR CCDF.The desired-link chi-squared fading and the out-of-cluster interference enter this calculation.Source: Appendix A · p. 9

APPENDIX B PROOF OF THEOREM 2

The proof bounds the conditional SIR CCDF using Alzer’s inequality, then removes conditioning on the Kth serving-BS distance through PPP-based interference calculations.

APPENDIX B PROOF OF THEOREM 2

  • Alzer’s inequality bounds the CDF of chi-squared desired-link fading, with strict equality when its shape parameter M = 1.The proof uses the inequality to form an upper bound on the conditional SIR CCDF.Source: Appendix B · p. 9
  • Conditioning on the Kth BS distance, the proof applies the fading bound and binomial expansion to upper-bound the SIR CCDF, where M = Nt −K + 1.It then unconditions with respect to the Kth BS location.Source: Appendix B · p. 9
  • The interference Laplace transform is evaluated using independent distances and unit-mean exponential fading, then marginalized over the Kth-BS distance distribution.The probability generating functional of the PPP is used in the conditional transform.Source: Appendix B · pp. 9–10

APPENDIX C PROOF OF LEMMA 1

The proof derives the joint distance density for the nearest and Kth-nearest BSs from independent PPP events in nonoverlapping regions, then uses it to obtain the distribution of δ1.

APPENDIX C PROOF OF LEMMA 1

  • The joint PDF of the nearest- and Kth-nearest-BS distances is constructed by partitioning the plane into four nonoverlapping regions.The proof considers the two distances falling in thin ring areas.Source: Appendix C · p. 10
  • The probability of the two distances falling in their respective ring areas is the product of four independent PPP events.Source: Appendix C · p. 10
  • The joint distance density is then used to derive the CDF of δ1 = ∥d1∥/∥dK∥.Source: Appendix C · p. 10

APPENDIX D PROOF OF EQ. (25)

The proof approximates an integral by a constant capturing randomness induced by δ1, then substitutes the result into Eq. (25) and marginalizes over δ1.

  • The integral is approximated by a constant that captures randomness induced by δ1.Source: Appendix D · p. 10
  • The result is substituted into the left side of Eq. (25) and marginalized with respect to δ1.Source: Appendix D · p. 11

APPENDIX E PROOF OF PROPOSITION 1

The proof uses the exact upper bound for K = Nt to expand outage probability around γ = 0. It applies a Gauss hypergeometric series expansion, drops the second-order error term, and marginalizes over δ1 to obtain average outage probability.

  • For K = Nt, the upper bound in Theorem 2 is exact, and the outage probability is expanded at γ = 0.Source: Appendix E · p. 11
  • The expansion invokes the Gauss hypergeometric function's series at γ = 0.Source: Appendix E · p. 11
  • Dropping the second-order error term and marginalizing with respect to δ1 yields the average outage probability.Source: Appendix E · p. 11
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