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Cooperative Multi-Cell Block Diagonalization with Per-Base-Station Power Constraints

Rui Zhang

arXiv:0910.2304v2cs.IT

TL;DR

The paper addresses weighted sum-rate maximization for BD precoding in fully cooperative multi-cell MIMO systems with per-BS power constraints. It formulates the problem as an auxiliary MIMO BC and applies convex optimization to derive an efficient algorithm and closed-form solution. The optimal vectors are generally non-orthogonal, while the single-antenna case reduces to optimal per-antenna ZF-BF.

  • Problem

    The paper studies optimal BD precoding for weighted sum-rate maximization in cooperative multi-cell systems subject to individual BS power constraints.

  • Method

    It formulates the system as an auxiliary MIMO BC and applies convex optimization to obtain an efficient algorithm and a closed-form optimal BD precoding matrix.

  • Results

    The optimal BD precoding vectors are generally non-orthogonal, and the single-antenna case yields optimal ZF-BF for a MISO BC with per-antenna constraints.

  • Takeaways & Limitations

    Per-BS power constraints require joint BD precoder optimization rather than conventional orthogonal precoder designs; the special case provides optimal ZF-BF.

Abstract

from arXiv · show

Block diagonalization (BD) is a practical linear precoding technique that eliminates the inter-user interference in downlink multiuser multiple-input multiple-output (MIMO) systems. In this paper, we apply BD to the downlink transmission in a cooperative multi-cell MIMO system, where the signals from different base stations (BSs) to all the mobile stations (MSs) are jointly designed with the perfect knowledge of the downlink channels and transmit messages. Specifically, we study the optimal BD precoder design to maximize the weighted sum-rate of all the MSs subject to a set of per-BS power constraints. This design problem is formulated in an auxiliary MIMO broadcast channel (BC) with a set of transmit power constraints corresponding to those for individual BSs in the multi-cell system. By applying convex optimization techniques, this paper develops an efficient algorithm to solve this problem, and derives the closed-form expression for the optimal BD precoding matrix. It is revealed that the optimal BD precoding vectors for each MS in the per-BS power constraint case are in general non-orthogonal, which differs from the conventional orthogonal BD precoder design for the MIMO-BC under one single sum-power constraint. Moreover, for the special case of single-antenna BSs and MSs, the proposed solution reduces to the optimal zero-forcing beamforming (ZF-BF) precoder design for the weighted sum-rate maximization in the multiple-input single-output (MISO) BC with per-antenna power constraints. Suboptimal and low-complexity BD/ZF-BF precoding schemes are also presented, and their achievable rates are compared against those with the optimal schemes.

I. INTRODUCTION

The paper studies BD precoding for fully cooperative multi-cell MIMO systems with per-BS power constraints, rather than conventional single-cell or sum-power settings. It formulates the problem for weighted sum-rate maximization, derives an optimal joint design, and shows that conventional orthogonal designs can be suboptimal.

  • Background: BD precoding makes each MS’s precoding matrix orthogonal to other users’ downlink channels, eliminating inter-user interference.Each MS consequently perceives an interference-free MIMO channel.
  • Cooperative multi-cell transmission: A central processor jointly designs transmissions across cooperative BSs using global channel and transmit-message knowledge.The cooperative setup exploits co-channel interference across cells and provides throughput gains over treating it as noise.
  • Problem formulation: The study formulates BD precoding as an auxiliary MIMO BC with transmit constraints corresponding to individual BS power limits.This differs from prior conventional formulations using one sum-power constraint.
  • Problem formulation: Existing approaches separate conventional BD beamforming from power allocation, but their optimality for weighted sum-rate maximization under per-BS constraints remains unclear.The paper shows that this heuristic is suboptimal and requires joint optimization of the BD precoder.
  • Optimal design: Convex optimization yields an efficient algorithm and a closed-form optimal BD precoding matrix for weighted sum-rate maximization.The analysis also derives bounds on how many BSs transmit at maximum power.
  • Main findings: Under per-BS power constraints, optimal BD vectors are generally non-orthogonal, unlike conventional orthogonal designs, which are generally suboptimal.For single-antenna BSs and MSs, the solution becomes the optimal ZF-BF design for a MISO BC with per-antenna constraints.
  • Low-complexity design: The paper also presents a low-complexity suboptimal BD/ZF-BF scheme, derives power-related bounds, and identifies conditions for sum-rate optimality.Its rates are compared with those of the optimal schemes.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models fully cooperative multi-cell downlink transmission as an auxiliary MIMO broadcast channel and formulates weighted sum-rate maximization with BD and per-BS power constraints.

  • The system has A cooperative cells, each with one BS, and jointly uses all M=M_BA transmitting antennas for downlink transmission.
  • Each kth MS receives y_k=H_kx_k+z_k, where H_k is the channel from all cooperative BS antennas and z_k is unit-covariance Gaussian noise.
  • The transmitted signal is parameterized as x_k=T_ks_k, with T_k specifying beamforming vectors and power allocation across D_k data streams.
  • The optimization maximizes weighted sum-rate over transmit covariance matrices subject to Tr(B_aS_k)≤P for every BS a.Using covariance matrices makes the problem convex because the objective is concave and the constraints define a convex set.
  • BD imposes H_jT_k=0 for every j≠k, eliminating inter-user interference, while requiring NK≤M for the assumed D_k=N streams per MS.
  • The paper uses Lagrange duality rather than only standard convex solvers to expose the optimal BD precoder structure and derive a closed-form expression.

III. PROPOSED SOLUTION

Section III presents an algorithm for solving the weighted sum-rate problem and examining the resulting optimal BD structure in general and single-antenna settings.

  • The proposed algorithm targets the optimal BD precoding matrix for arbitrary numbers of BS and MS antennas.
  • The section also studies the single-antenna BS and MS case and compares the developed solution with existing schemes.

A. General Case

The general-case solution removes the zero-forcing constraints through a null-space parameterization, then solves the resulting convex problem by Lagrange duality and obtains closed-form optimal covariance and precoding matrices.

  • Null-space reduction: Each covariance matrix is represented within the null space of the other users’ channels, which makes all BD zero-forcing constraints automatically satisfied.
  • Closed-form solution: The optimal Q_k is obtained from a reduced SVD and a water-filling solution, while the dual variables are updated with a subgradient-based method such as the ellipsoid method.
  • Dual formulation: The reduced problem is convex, and its dual uses non-negative variables μ_a associated with the per-BS power constraints.
  • Dual formulation: For fixed dual variables, the maximization separates into K independent subproblems, one for each user covariance matrix Q_k.
  • Closed-form solution: Theorem 3.1 gives the optimal covariance and BD precoding matrices using the optimal dual solutions of the reduced dual problem.
  • Structural properties: A unitary receiver decoder diagonalizes each BD-precoded MIMO channel into N scalar sub-channels while preserving single-user MIMO capacity.
  • Structural properties: Under per-BS constraints, the optimal BD precoding vectors are generally non-orthogonal, whereas the corresponding sum-power solution has orthogonal precoding columns.

B. Special Case: MISO BC with Per-Antenna Power Constraints

In the single-antenna BS and MS setting, the general solution becomes optimal ZF beamforming for a MISO broadcast channel with per-antenna power constraints.

  • Special-case reduction: When M_B=N=1, the cooperative MIMO broadcast channel reduces to an equivalent MISO broadcast channel with per-antenna power constraints.
  • Comparison with prior designs: With a single sum-power constraint, the solution reduces to conventional ZF beamforming based on the channel pseudo-inverse.
  • Comparison with prior designs: The channel-pseudo-inverse ZF-BF design is generally suboptimal under per-antenna or per-BS power constraints.
  • Special-case reduction: When N=1 and M=K, the proposed and conventional beamformers can both be written as scaled null-space vectors, regardless of M_B.
  • Proposed MISO solution: The proposed method obtains optimal ZF-BF precoders from a closed-form expression and searches numerically only over the dual variables using the ellipsoid method.
  • Comparison with prior designs: The alternative beamforming formulation is non-convex because its objective is not necessarily concave in the beamforming vectors.
  • Proposed MISO solution: Unlike the alternative covariance approach, which is not guaranteed to return rank-one solutions, the proposed closed-form solution is guaranteed to be rank-one.

IV. SUBOPTIMAL SOLUTION

The paper proposes a lower-complexity suboptimal BD solution by optimizing power allocation over a fixed precoding structure. This structure is generally suboptimal under per-BS constraints, except in identified special cases.

  • Suboptimal solution: The suboptimal solution reduces the problem to convex power allocation with dual variables associated with per-BS power constraints.The resulting power allocation is obtained through Lagrange duality and an iterative search over the dual variables.
  • Algorithm: The proposed algorithm computes singular-value decompositions, evaluates power allocation, updates the dual variables, and repeats until convergence.Its iterations avoid recomputing the full precoding matrix.
  • Complexity: A2 has lower complexity than A1 because each loop performs only power allocation rather than precoding-matrix computation.This reduction lowers the per-loop computational burden, although the resulting covariance structure is generally suboptimal.
  • Optimality conditions: A2 is generally suboptimal for P1, becoming optimal when N = 1 and M = K; it is also optimal under a single sum-power constraint.For N = 1 and M = K, A2 can serve as an alternative to A1 for obtaining the optimal solution.
  • MISO special case: In the MISO BC special case, the suboptimal precoder has the form T = H†Θ, with Θ diagonal and optimized through per-antenna power allocation.When M > K, this construction remains suboptimal for P1.
  • Power constraints: The optimal solution for P6 activates at most NK per-BS power constraints, limiting the number of BSs that can transmit at full power when A/(NK) ≫ 1.Under that condition, most BSs cannot transmit with their full power levels using A2.

V. NUMERICAL EXAMPLES

The numerical examples evaluate cooperative multi-cell downlink transmission under independent Rayleigh-like channel matrices and equal user weights. The experiments focus on sum-rate maximization.

  • Simulation setup: The simulations assume channel matrices independent across users, with entries distributed as zero-mean unit-variance CSCG random variables.These assumptions define the channel realizations used in the numerical examples.
  • Simulation setup: The numerical study considers sum-rate maximization by setting all user weights equal to one in P1.The reported results are presented for cooperative multi-cell downlink transmission.

A. Convergence Behavior

Both algorithms converge through searches over the per-BS dual variables. In the reported two-BS example, Algorithm A1 reaches fixed rate and power values after about 30 iterations.

  • Convergence behavior: After around 30 iterations, Algorithm A1’s achievable sum-rate and consumed powers converge to fixed values.The example uses A = 2, M_B = 4, K = 4, N = 2, and per-BS power P = 10.
  • Convergence behavior: The converged transmit powers of both BSs equal their constraint value of 10.Thus, both BSs use their full prescribed power in this example.
  • Complexity: The convergence speed of A1 and A2 depends critically on A, the number of per-BS constraints and searched dual variables.With the ellipsoid method, the search complexity is O(A^2) for large A, so iteration counts grow asymptotically quadratically with the number of BSs.

B. MISO BC with Per-Antenna Power Constraints

For single-antenna BSs and MSs, the cooperative system becomes a MISO BC with per-antenna constraints. The optimal ZF-BF design increasingly outperforms the suboptimal design as M grows beyond K.

  • Rate comparison: When M = K = 2, the optimal and suboptimal ZF-BF precoders achieve identical rates.The comparison uses per-antenna power P = 10, K = 2, and M ranging from 2 to 10.
  • Rate comparison: When M > K, the sum-rate gain of the optimal precoder over the suboptimal one increases with M.This trend is reported in the comparison of the two precoding schemes.
  • Power utilization: The optimal design can use full transmit powers from at least M − K + 1 antennas, whereas the suboptimal design can use at most K.The active-constraint observations support the widening rate gap as M increases with fixed K.
  • Rate comparison: The rate gap between optimal and suboptimal ZF-BF designs enlarges as M increases with fixed K.The paper connects this trend to their differing abilities to exploit full antenna powers.

C. MIMO BC with Per-Antenna Power Constraints

The section examines multi-antenna receiver settings for BD under per-BS constraints, comparing optimal and low-complexity precoders. The low-complexity method can approach optimal performance in a specified antenna regime, but may remain suboptimal when receivers have multiple antennas.

  • For A = 4, MB = 1, K = 2, and N = 2, (A2) generally gives a suboptimal solution for (P1) because N > 1.
  • In Fig. 4, the optimal precoder performs better than the suboptimal precoder under per-BS and per-antenna transmit-power constraints.
  • The rate gap between the optimal and suboptimal precoders is smaller than in Fig. 2 when M > NK and P = 10.The section attributes this comparison to the M = NK setting and the resulting antenna-power limitation described around Lemma 4.1.
  • When MB = 1 and A is not substantially larger than NK, (A2) achieves sum-rate performance close to optimal (A1).This is presented as a practical rule for applying the low-complexity suboptimal BD precoder.

VI. CONCLUSION

The conclusion presents an optimal BD design for cooperative multi-cell downlink transmission with individual BS power constraints. It derives the solution using convex optimization, identifies non-orthogonal optimal vectors, and connects special cases and extensions to established precoding settings.

  • The paper applies BD to fully cooperative multi-cell downlink transmission with individual BS power constraints.
  • Convex optimization yields a closed-form optimal BD precoding matrix that maximizes the users’ weighted sum-rate.
  • The optimal BD precoding vectors are generally non-orthogonal, unlike conventional BD under a single sum-power constraint.
  • A suboptimal heuristic combines conventional orthogonal BD precoding with optimized power allocation to satisfy per-BS constraints.
  • For MISO BC with per-antenna constraints, the proposed optimal BD solution becomes the optimal ZF-BF solution.
  • The results extend to general linear transmit-power constraints, including per-antenna and per-BS constraints as special cases.

PROOF OF LEMMA 3.2

The proof constructs a contradiction from the assumed structure of positive power-constraint multipliers and uses rank and feasibility arguments. It concludes that the assumed multiplier-count bound must hold, while the supplied passages also include a separate active-constraint argument.

  • The proof assumes a set of strictly positive μ_a values and partitions the precoder basis rows according to their associated diagonal multipliers.
  • If the active multiplier count is too small, a unit vector q_k can be chosen so that B_μ Ṽ_k q_k = 0 while Ṽ_k q_k ≠ 0.
  • The resulting nonzero direction makes the objective unbounded under the assumed solution structure.
  • This contradiction establishes the stated lower bound on the number of positive multipliers, using independent channel realizations where required.
  • A separate contradiction argument shows that more than NK active per-BS constraints would produce more independent equations than the NK unknowns, making the system infeasible for P > 0.
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