Source-linked AI summary
Cooperative Interference Management with MISO Beamforming
Rui Zhang, Shuguang Cui
TL;DR
The paper addresses Pareto-boundary rate design for cooperative downlink beamforming in a multi-cell MISO interference channel with interference treated as noise. It uses a characterization based on interference-temperature constraints, showing that the target rates can be attained through decentralized per-user capacity optimization and a pairwise implementation algorithm.
Problem
Cooperative beamforming in the MISO-IC requires joint optimization over a generally non-convex achievable rate region, while the Gaussian interference-channel capacity region remains unknown.
Method
The paper characterizes the MISO-IC Pareto boundary through interference-power levels at receivers and relates each user’s constrained optimization to a cognitive-radio MISO channel problem.
Results
Each Pareto-boundary rate-tuple is achievable decentrally, with rank-one optimal transmit covariance matrices and beamforming optimal for the considered boundary points.
Takeaways & Limitations
The characterization yields a decentralized multi-cell cooperative beamforming algorithm that improves rates pairwise until convergence with mutual interference-temperature levels.
Abstract
from arXiv · showhide
This correspondence studies the downlink transmission in a multi-cell system, where multiple base stations (BSs) each with multiple antennas cooperatively design their respective transmit beamforming vectors to optimize the overall system performance. For simplicity, it is assumed that all mobile stations (MSs) are equipped with a single antenna each, and there is one active MS in each cell at one time. Accordingly, the system of interests can be modeled by a multiple-input single-output (MISO) interference channel (IC), termed as MISO-IC, with interference treated as noise. We propose a new method to characterize different rate-tuples for active MSs on the Pareto boundary of the achievable rate region for the MISO-IC, by exploring the relationship between the MISO-IC and the cognitive radio (CR) MISO channel. We show that each Pareto-boundary rate-tuple of the MISO-IC can be achieved in a decentralized manner when each of the BSs attains its own channel capacity subject to a certain set of interference-power constraints (also known as interference-temperature constraints in the CR system) at the other MS receivers. Furthermore, we show that this result leads to a new decentralized algorithm for implementing the multi-cell cooperative downlink beamforming.
I. INTRODUCTION
This correspondence studies decentralized cooperative downlink beamforming in a multi-cell MISO interference channel, targeting Pareto-boundary rate-tuples while treating interference as noise. It characterizes these rates through interference-temperature constraints and derives a decentralized implementation.
- System model: The system models one active single-antenna MS per cell and multiple-antenna BSs as a MISO Gaussian interference channel with interference treated as noise.The simplified model captures cooperative multi-cell downlink transmission under single-user encoding and decoding.
- Problem: The achievable rate region is generally non-convex, making joint beamforming optimization for different Pareto-boundary rate-tuples challenging.Without time-sharing, the rate region is non-convex; time-sharing would produce a convex set.
- Characterization: The work develops a new parametrical characterization of the MISO-IC Pareto boundary using interference-power levels caused by transmitters at other receivers.These interference-power levels are also called interference-temperature levels in cognitive-radio applications.
- Decentralized design: Each Pareto-boundary rate-tuple can be achieved decentrally when every user maximizes its own MISO-channel capacity subject to interference-temperature constraints at other receivers.The resulting optimization has the same solution structure as the corresponding cognitive-radio MISO channel problem.
- Optimality: The optimal transmit covariance matrices for arbitrary Pareto-boundary rate-tuples have new closed-form solutions and are rank-one, so beamforming is optimal.The result establishes beamforming optimality for the considered MISO-IC rate-region boundary.
- Algorithm: A pairwise decentralized algorithm updates mutual interference-temperature constraints and optimizes beamformers until all base-station rates converge with their mutual constraint levels.The algorithm improves base-station rates pairwise while the remaining mobile stations’ interference-temperature constraints stay fixed.
II. SYSTEM MODEL
The system models cooperative multi-cell downlink transmission as a K-user MISO interference channel with single-antenna receivers and interference treated as noise. It defines achievable rates and Pareto-boundary characterization through transmit covariance matrices or beamforming vectors, including rate-profile and convex feasibility methods.
- The network has K cells, each with a multi-antenna BS transmitting to one active single-antenna MS over a narrow-band downlink.
- Each BS uses independent circularly symmetric complex Gaussian signaling, while interferences from other transmitters are treated as Gaussian noise.
- The achievable rate region contains simultaneously achievable user-rate tuples under the BS transmit-power constraints, and its upper-right boundary is the Pareto boundary.
- Although prior work often assumes rank-one covariance matrices, the paper shows beamforming has the same Pareto boundary as the general covariance formulation.
- Weighted sum-rate maximization is non-convex and may miss boundary points, whereas rate profiles characterize the complete boundary, including non-convex regions.
- For a fixed rate profile, bisection over the sum-rate target converts the constraints to SINR constraints and then to a convex SOCP solvable efficiently.
III. CHARACTERIZING PARETO BOUNDARY FOR MISO-IC VIA INTERFERENCE TEMPERATURE CONTROL
The paper characterizes the Pareto boundary of the K-user Gaussian MISO-IC through interference-temperature constraints, using its relationship with the CR MISO channel. This yields closed-form optimal covariance solutions and shows that beamforming suffices for every Pareto-boundary rate-tuple.
- Interference-temperature characterization: The method parameterizes the MISO-IC Pareto boundary using K(K −1) nonnegative interference-temperature constraints Γ_kj with practical receiver-protection meaning.Γ_kj limits interference from BS k to MS j.
- Interference-temperature characterization: Each BS’s constrained optimization is equivalent to the capacity problem of a CR MISO channel, with one secondary transmitter and the other BSs as primary transmitters.The other MS receivers are protected by interference-temperature constraints.
- Optimal covariance solutions: The optimal solution is rank-one, so beamforming is optimal for every Pareto-boundary rate-tuple of the MISO-IC.This conclusion combines the rank-one solution result with the Pareto-boundary characterization.
- Optimal covariance solutions: New closed-form solutions are derived for optimal transmit covariance matrices achieving arbitrary Pareto-boundary rate-tuples.The solution uses nonnegative dual variables associated with interference and transmit-power constraints.
- Dimensionality and implementation: The Pareto boundary is parameterized over a lower-dimensional real manifold of interference constraints rather than complex covariance matrices or beamforming vectors.The parameterization uses real Γ values with direct interference-power interpretations.
- Dimensionality and implementation: The practical significance of Γ is that it supports decentralized cooperative beamforming through iterative searches for mutually desirable interference constraints.This distinguishes the proposed parameters from complex coefficients without direct practical meaning.
IV. DECENTRALIZED ALGORITHM FOR MULTI-CELL COOPERATIVE BEAMFORMING
The paper develops a pairwise decentralized algorithm that updates mutual interference-temperature constraints and independently recomputes beamformers. Each update improves the selected pair’s rates without affecting other users, ensuring convergence under bounded achievable rates.
- Algorithm design: The algorithm updates mutual interference-temperature constraints between BS pairs to improve both selected transmit rates while keeping other BS rates unchanged.The constraints affecting MSs outside the updating pair remain fixed.
- Algorithm design: Each BS solves its own constrained optimization problem using local channel knowledge and exchanges only scalar information with its paired BS.The pair exchanges derivative-related quantities before jointly updating their mutual constraints.
- Convergence: The algorithm converges because each iteration improves the updating pair’s rates, leaves other rates non-decreasing, and all achievable rates are bounded by finite Pareto-optimal values.The convergence argument is based on monotonic rate behavior and boundedness.
- Constraint updates: The update direction is chosen so the pair’s rate changes are mutually desirable, while α_ij controls the ratio between their rate increments.When α_ij > 1, the ith BS receives a larger rate increment, provided the step size is sufficiently small.
- Numerical illustration: A larger α_12 produces a larger converged rate for the first MS, matching the prescribed rate-increment trade-off.The comparison is between α_12 = 1 and α_12 = 10.
V. CONCLUDING REMARKS
The paper concludes that interference-temperature control characterizes the complete Pareto boundary and enables decentralized cooperative beamforming with maximal rates and a prescribed fairness guarantee. It also identifies unresolved sufficiency, scalability, and game-theoretic questions.
- Contributions: The proposed method characterizes the complete Pareto boundary of the achievable rate region for the K-user Gaussian MISO-IC with interference treated as noise.The result is stated for the modeled cellular downlink setting.
- Contributions: The method leads to a decentralized downlink beamforming algorithm that achieves maximal rates with a prescribed fairness guarantee.The algorithm iteratively updates mutual interference-temperature constraints between BS pairs.
- Limitations: The algorithm was verified across many random channels and system parameters to converge to Pareto-optimal rate-pairs in the two-user MISO-IC.This empirical verification does not replace a general proof.
- Future directions: Future work includes extending the design to multiple active MSs, multiple-antenna MSs, and a game-theoretic analysis of the decentralized algorithm.These directions broaden the cellular setting beyond one single-antenna active MS per cell.
- Limitations: It remains unproved whether the necessary interference-temperature conditions are sufficient for Pareto optimality, even in the two-user case.This proof is identified as essential for establishing global convergence to Pareto-optimal rates.
PROOF OF PROPOSITION 3.1
The proof solves the convex beamforming problem through Lagrange duality and derives a rank-one optimal transmit covariance structure, establishing Proposition 3.1.
- Dual formulation: Problem (9) is convex, so zero duality gap permits solving it through its Lagrange dual problem.The dual function is minimized over component-wise nonnegative multipliers, using standard methods such as the ellipsoid algorithm.
- Dual formulation: Boundedness of the dual objective requires B_k(λ_k) to be full rank.A null direction would allow the primal variable to grow without bound with probability one, contradicting bounded optimality.
- Dual formulation: B_k(λ_k) is full rank when the transmit-power multiplier is positive or at least M_k interference-temperature multipliers are positive.The second case requires M_k ≤ K−1 and corresponds to at least M_k tight interference constraints.
- Rank-one structure: After introducing a transformed covariance and its eigenvalue decomposition, the optimal transformed covariance has rank at most one.The eigenvalue optimization shows that only one eigenmode needs positive weight at optimum.
- Rank-one structure: The resulting covariance solution has the form stated in Proposition 3.1, and the converged dual solution yields the optimizer of Problem (9).Combining the rank-one solution with the covariance transformation establishes the proposition.
PROOF OF PROPOSITION 3.2
The proof shows that each covariance matrix from a Pareto-optimal MISO-IC rate tuple must solve its corresponding constrained single-user problem, proving Proposition 3.2.
- Feasibility and optimality: Each covariance matrix S_k is feasible for Problem (9) because its power and interference constraints match the problem’s constraints.Its objective function also coincides with the objective of Problem (9) for user k.
- Feasibility and optimality: If S_k were not optimal for Problem (9), replacing it with an optimizer would strictly improve user k’s rate without reducing other users’ rates.This would produce a rate tuple that contradicts Pareto optimality.
- Feasibility and optimality: The contradiction establishes S_k = S_k⋆ for every k.Thus, each Pareto-optimal rate R_k equals the optimal value C_k(Γ_k) of the corresponding Problem (9).
PROOF OF PROPOSITION 4.1
The proof establishes Proposition 4.1 by perturbing two interference constraints and showing that any nonzero coupled slack would contradict Pareto optimality.
- Contradiction argument: The proof assumes a pair (i,j) has nonzero D_ij and constructs Γ′ by perturbing only Γ_ij and Γ_ji.The perturbation uses a positive step size and a vector d_ij satisfying D_ij d_ij > 0 component-wise.
- Contradiction argument: Under Γ′, optimizers for users other than i and j remain unchanged, while the optimizers for i and j change.The resulting rates are then evaluated in the original MISO-IC setting.
- Contradiction argument: The perturbation produces rates r_i > R_i, r_j > R_j, and r_k ≥ R_k for all other users.This strictly improves two users while leaving the remaining rates no worse.
- Contradiction argument: That rate tuple contradicts Pareto optimality, so no pair can have |D_ij| ≠ 0.Therefore, the assumed nonzero coupled quantity cannot occur at a Pareto-optimal rate tuple.