PaperlayerView exact revision on arXiv

Source-linked AI summary

Coordinated Multicast Beamforming in Multicell Networks

Zhengzheng Xiang, Meixia Tao, Xiaodong Wang

arXiv:1210.5813v1cs.IT

TL;DR

The paper addresses coordinated multicast beamforming in multicell networks for power-efficient QoS and fair SINR under intercell interference. It develops decentralized or inversion-based algorithms for these objectives, and simulations report convergence, near-optimality in most cases, and advantages over conventional schemes.

  • Problem

    Multicell multicast beamforming must manage intercell interference while supporting QoS guarantees and max-min fairness with practical per-base-station coordination.

  • Method

    The paper combines decomposition, semidefinite relaxation, and limited-information decentralized coordination for QoS, and uses the inverse relationship between MMS and WPPM for approximate max-min beamforming.

  • Results

    The proposed algorithms converge, achieve the optimal solution in most cases, and significantly outperform conventional multicasting schemes over a large range of individual power constraints.

  • Takeaways & Limitations

    Coordinating beamformers across base stations can provide practical multicast designs for multicell QoS and fairness without requiring full data sharing.

Abstract

from arXiv · show

We study physical layer multicasting in multicell networks where each base station, equipped with multiple antennas, transmits a common message using a single beamformer to multiple users in the same cell. We investigate two coordinated beamforming designs: the quality-of-service (QoS) beamforming and the max-min SINR (signal-to-interference-plus-noise ratio) beamforming. The goal of the QoS beamforming is to minimize the total power consumption while guaranteeing that received SINR at each user is above a predetermined threshold. We present a necessary condition for the optimization problem to be feasible. Then, based on the decomposition theory, we propose a novel decentralized algorithm to implement the coordinated beamforming with limited information sharing among different base stations. The algorithm is guaranteed to converge and in most cases it converges to the optimal solution. The max-min SINR (MMS) beamforming is to maximize the minimum received SINR among all users under per-base station power constraints. We show that the MMS problem and a weighted peak-power minimization (WPPM) problem are inverse problems. Based on this inversion relationship, we then propose an efficient algorithm to solve the MMS problem in an approximate manner. Simulation results demonstrate significant advantages of the proposed multicast beamforming algorithms over conventional multicasting schemes.

I. INTRODUCTION

The paper studies coordinated multicast beamforming in cooperative multicell networks, motivated by intercell interference and the need to support multicell QoS and fairness objectives with limited coordination.

  • I. INTRODUCTION: Physical-layer multicasting uses transmitter-side channel state information to beamform a common message to multiple users, supporting services such as media streaming and mobile TV.The paper places coordinated multicast beamforming within next-generation cellular wireless services.
  • I. INTRODUCTION: Intercell interference limits systems that reuse the total frequency band across cells, motivating cooperative signal processing among base stations.The paper investigates beamformers coordinated across base stations rather than full networked-MIMO data sharing.
  • I. INTRODUCTION: The QoS design minimizes sum power while requiring every user's SINR to exceed a threshold, but multicell interference can make the problem infeasible.The paper derives a necessary feasibility condition and develops a decentralized algorithm based on decomposition, semidefinite relaxation, and iterative interference-parameter updates.
  • I. INTRODUCTION: The MMS design maximizes the minimum user SINR under individual base-station power constraints using coordination only at the beamforming level.This setting avoids the large information-exchange overhead associated with data-sharing networked MIMO.

III. QOS BEAMFORMING

The QoS problem minimizes total transmit power subject to common per-cell SINR targets. Feasibility is constrained by intercell channel correlation and stringent SINR requirements.

  • III. QOS BEAMFORMING: QoS beamforming minimizes total energy consumption while maintaining each user's target SINR through coordinated base-station beamformers.Because each cell transmits common multicast information, users within a cell share a common SINR target.
  • III. QOS BEAMFORMING: The problem is not always feasible in multicell networks, particularly under poor channel conditions or stringent SINR targets.This contrasts with the stated single-cell QoS case, which is always feasible.
  • III. QOS BEAMFORMING: Lemma 1 gives a necessary condition on the common SINR target γ whenever the QoS problem is feasible.The condition is derived after combining corresponding user channel vectors into matrices H_k.
  • III. QOS BEAMFORMING: Intercell-user channel correlation can reduce rank and impose a finite upper bound on the feasible SINR threshold.With independent intercell user channels, H_k is full rank with probability one, whereas correlation may make its rank less than N.

B. Decentralized coordinated beamforming

The paper decentralizes QoS coordinated beamforming by decomposing interference constraints into local subproblems, exchanging subgradients, and iteratively updating interference parameters. The method converges, preserves limited-information implementation, and usually recovers the original problem’s optimum, although SDR randomization may yield near-optimal solutions.

  • Decentralized implementation: Local channel knowledge and exchanged real-valued subgradients enable decentralized coordinated beamforming with limited backhaul information.Each base station solves a local subproblem and broadcasts its subgradient vector to the other cells.
  • Problem decomposition: Introducing an interference-temperature vector decouples the coupled constraints into N parallel base-station subproblems solved using semidefinite relaxation.The resulting subproblems require only local channel state information, while a master problem updates the interference parameters.
  • Iterative optimization: The distributed subgradient algorithm is guaranteed to converge exactly to the optimal solution of the convex decomposed problem.The interference-temperature vector is updated iteratively using a projected subgradient method with a diminishing step size.
  • Beamformer recovery: The original nonconvex problem lacks a general optimality guarantee after relaxation, so rank-one solutions use eigenvalue extraction while higher-rank solutions use randomization and scaling.These procedures are implemented locally and preserve the distributed nature of the algorithm.
  • Performance and scope: In simulations, the algorithm converges in several iterations, achieves most beamforming gains quickly, and reaches the original optimum in most cases, especially for small networks.The scheme also applies to the conventional interference channel as the K = 1 special case.
  • Communication overhead: Each iteration exchanges subgradient vectors with only N × K nonzero entries per base station, giving a signaling overhead that scales with the network size.The total signaling also depends on the number of iterations Nb.

IV. MAX-MIN SINR BEAMFORMING

The MMS beamforming problem maximizes the worst-user SINR in a multicell multicast network while respecting individual base-station power constraints.

  • IV. MAX-MIN SINR BEAMFORMING: MMS beamforming maximizes the minimum SINR among all users under individual power constraints at the base stations.The formulation uses a common SINR target γ and a power constraint vector p for the network’s base stations.

A. Connection with power optimization

The paper connects max-min SINR optimization with weighted peak-power minimization through an inverse relationship. This connection supports solving the MMS problem by solving a related power optimization problem under multiple base-station constraints.

  • A. Connection with power optimization: The MMS SINR optimization and weighted peak-power minimization problems are inverse problems linked through the base-station power vector.The relationship extends inverse optimization to multicell multicasting with individual, rather than only sum-power, constraints.
  • A. Connection with power optimization: The optimal objective values of the two problems are numerically observed to be monotonically non-decreasing in the power constraints and SINR target.The paper explains that more available power permits larger achievable SINR, and conversely.

B. Inversion-property-based algorithm

The MMS beamforming algorithm exploits an inversion relationship between S1(p) and Q1(γ, p), solving the latter repeatedly with bisection before recovering beamformers. Semidefinite relaxation enables efficient subproblem solution, while EVD or randomization and scaling handles the final beamformer construction.

  • B. Inversion-property-based algorithm: The algorithm solves S1(p) by iteratively solving its inverse SDP Q1(γ, p) and using one-dimensional bisection over γ.Q1(γ, p) has strong duality and can be solved efficiently with an interior method; monotonicity supports the bisection search.
  • B. Inversion-property-based algorithm: The inversion property links the optimal value of S1(p) to a Q1(γ, p) value of 1 at the corresponding optimal γ.This equality provides the criterion used to locate the MMS optimum.
  • B. Inversion-property-based algorithm: After obtaining the relaxed matrices, the method uses EVD for rank-1 solutions and randomization with scaling otherwise to produce final base-station beamformers.The central controller performs randomization and scaling when the rank-1 condition is not satisfied.
  • B. Inversion-property-based algorithm: The algorithm initializes an interval for γ, tests its midpoint through Q1(γ, p), and updates the interval according to whether the optimal value exceeds 1.Iterations continue until the stopping condition is met.
  • B. Inversion-property-based algorithm: Each iteration solves a weighted peak power minimization problem, and the resulting algorithm is approximate but reaches the optimal solution in most simulated cases.This weighted peak power step is identified as the main difference from the algorithm in [2].

V. SIMULATION RESULTS

The simulations evaluate the proposed multicell multicast beamforming designs under normalized Rayleigh fading and specified intercell-channel assumptions. Results are averaged over 200 channel realizations, with 100 Gaussian randomizations when required.

  • V. SIMULATION RESULTS: The simulations use normalized Rayleigh fading channels, with independent circularly symmetric zero-mean complex Gaussian channel elements of unit variance within each cell.The passage specifies the within-cell channel model used for the numerical examples.
  • V. SIMULATION RESULTS: Each experiment uses 200 channel realizations and generates 100 Gaussian randomizations whenever the randomization method is needed.System configurations are denoted by N−K−Nt for cells, users per cell, and antennas per base station.

A. Performance comparison with existing schemes

The proposed coordinated algorithms are evaluated for convergence, relaxation tightness, feasibility, energy efficiency, and minimum-SINR performance against bounds and conventional schemes. Results show near-optimal QoS performance in most cases and strong MMS performance over competing methods.

  • QoS beamforming: The QoS algorithm converges exactly to the optimal value of its convex distributed subproblem, with rapid progress during the first few iterations.The convergence guarantee depends on the convexity of P1(γ, Γ), while the step size affects convergence speed and accuracy.
  • QoS beamforming: The proposed QoS coordinated beamforming algorithm achieves the relaxed lower-bound performance in most cases despite semidefinite relaxation.The lower bound is obtained by relaxing the rank-one constraint in problem P1(γ).
  • QoS beamforming: The proposed and M-BD beamformers remain feasible across the considered SINR-target range, whereas open-loop STBC and L-SLNR nearly fail at large targets.Feasibility is measured over 200 channel realizations; L-SLNR cannot guarantee receiver-side SINR maximization.
  • MMS beamforming: The proposed MMS algorithm achieves a minimum SINR 6dB higher than L-SLNR, 8dB higher than M-BD, and 9dB higher than open-loop STBC at 10dB per-base-station power.These results are reported for the (3−2−5) system and the proposed method significantly outperforms the alternatives over a large range of individual power constraints.

B. Effects of channel correlation

The paper models correlated user channels and finds that intercell correlation reduces QoS feasibility, whereas intracell correlation reduces consumed power. These effects are illustrated for the QoS scheme, with similar intracell benefits observed for MMS.

  • Channel-correlation model: The simulations use a Kronecker channel model with exponentially parameterized correlation matrices.The correlation ratio r satisfies 0 ≤ r ≤ 1.
  • Intercell-user channel correlation: Intercell-user channel correlation decreases the feasibility of the QoS problem.The comparison uses correlation ratios r = 0.5, 0.7, and 0.9, with r = 0 representing independent channels.
  • Intracell-user channel correlation: Intracell-user channel correlation reduces the total consumed power in QoS beamforming.The reported correlation ratios are r = 0.5, 0.7, and 0.9.
  • Intracell-user channel correlation: The study also observes that intracell-user channel correlation is useful for the MMS scheme.Further MMS results are omitted because of page limitations.
  • Scope: The paper investigates both intercell-user and intracell-user correlation effects within its coordinated multicell beamforming study.The QoS and MMS schemes are among the coordinated designs considered.

APPENDIX A PROOF OF LEMMA 1

The appendix proves a necessary feasibility condition for the QoS beamforming problem by bounding user SINR through matrix rank and singular-value arguments. The proof concludes by relating the SINR bound to the target threshold.

  • Beamformer representation: The beamformer matrix W is defined as a full-rank matrix with diagonal beamforming entries.The displayed matrix contains diagonal elements w_1 through w_N.
  • SVD-based bound: The proof bounds SINR_i,k by bounding an associated parameter through the singular value decomposition of H_kW.The decomposition writes H_kW = U_kΣ_kV_k^H and uses the rank r_k = rank(H_kW).
  • SVD-based bound: The proof applies the monotonicity of f(x) = 1/(1/x − 1) for x < 1 to transfer bounds between the auxiliary parameter and SINR.This monotonicity connects the parameter bound to the SINR bound.
  • Feasibility condition: If problem (3) is feasible, its minimum SINR must exceed the threshold γ.The appendix obtains this conclusion by substituting the preceding bound into the relevant expression.
  • Conclusion: The appendix completes the lemma after deriving the final inequality labeled (55).The result follows the preceding SINR and threshold arguments.

APPENDIX B PROOF OF THEOREM 1

The appendix derives subgradients for the decomposed QoS subproblems and the global master problem. It uses the subproblem Lagrangian and strong duality to complete the theorem’s proof.

  • Subproblem subgradients: The proof computes each subproblem’s subgradient from its Lagrangian.The Lagrangian is formed for the ith subproblem.
  • Duality: Convexity of the relevant function yields strong duality for the subproblem.The appendix explicitly links convexity to strong duality.
  • Duality: The optimal dual Lagrange multiplier is used to obtain the subproblem’s equivalent characterization.The appendix denotes this multiplier by λ⋆.
  • Master problem: The global subgradient g is computed for P^mas(γ, Γ).This extends the subproblem calculations to the master problem.
  • Conclusion: The appendix concludes the derivation by completing the proof of Theorem 1.
Loading 1210.5813v1…