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Coordinated Beamforming for Energy Efficient Transmission in Multicell Multiuser Systems
He Shiwen, Huang Yongming, Jin Shi, Yang Luxi
TL;DR
The paper addresses joint power allocation and beamforming for energy-efficient coordinated multicell multiuser downlinks, where the optimization is non-convex and fractional. It reformulates the problem into a parametric subtractive form and solves it with a two-layer scheme using bisection and an iterative rate–MSE-based algorithm. The method converges to near-optimal performance, simultaneously approaching optimal sum rate and energy efficiency at low transmit power but trading some sum rate for energy efficiency at middle-high power.
Problem
Joint energy-efficient power allocation and beamforming remains open because coupled variables and a fractional objective make the coordinated multicell MU-MISO problem non-convex and difficult to solve directly.
Method
The paper transforms the fractional problem into an equivalent subtractive form, searches its energy-efficiency parameter by bisection, and solves the inner problem with an iterative rate–MSE-based beamforming algorithm.
Results
The algorithm converges within a limited number of iterations and achieves near-optimal performance, including near-optimal energy efficiency and sum rate simultaneously at low transmit power.
Takeaways & Limitations
At low transmit power, near-optimal sum rate and energy efficiency coincide, whereas at middle-high transmit power, higher sum rate can require sacrificing energy efficiency.
Abstract
from arXiv · showhide
In this paper we study energy efficient joint power allocation and beamforming for coordinated multicell multiuser downlink systems. The considered optimization problem is in a non-convex fractional form and hard to tackle. We propose to first transform the original problem into an equivalent optimization problem in a parametric subtractive form, by which we reach its solution through a two-layer optimization scheme. The outer layer only involves one-dimension search for the energy efficiency parameter which can be addressed using the bi-section search, the key issue lies in the inner layer where a non-fractional sub-problem needs to tackle. By exploiting the relationship between the user rate and the mean square error, we then develop an iterative algorithm to solve it. The convergence of this algorithm is proved and the solution is further derived in closed-form. Our analysis also shows that the proposed algorithm can be implemented in parallel with reasonable complexity. Numerical results illustrate that our algorithm has a fast convergence and achieves near-optimal energy efficiency. It is also observed that at the low transmit power region, our solution almost achieves the optimal sum rate and the optimal energy efficiency simultaneously; while at the middle-high transmit power region, a certain sum rate loss is suffered in order to guarantee the energy efficiency.
I. INTRODUCTION
The paper targets joint energy-efficient power allocation and beamforming in coordinated multicell MU-MISO downlinks, where existing work largely focuses on throughput or spectral efficiency. It proposes a tractable algorithmic framework and finds near-optimal performance, with a power-dependent tradeoff between sum rate and energy efficiency.
- Motivation and gap: Existing coordinated beamforming studies primarily optimize throughput, spectral efficiency, power, fairness, or sum rate rather than joint energy-efficient power allocation and beamforming.The paper identifies joint optimization of energy-efficient power allocation and beamforming as an open problem.
- Contribution: The proposed design jointly optimizes transmit powers and beamforming vectors for coordinated multicell MU-MISO downlink systems.The original formulation is non-convex because optimization variables are coupled and the objective is fractional.
- Contribution: The fractional objective is transformed into an equivalent subtractive-form problem and solved through one-dimensional parameter search plus an iterative beamforming algorithm.The inner algorithm uses the relationship between user rate and MSE, with convergence and reasonable complexity established.
- Results: Numerical results validate the algorithm’s effectiveness and show fast convergence with near-optimal energy efficiency.The paper positions energy efficiency as a bit-per-Joule metric for balancing rate and power consumption.
- Results: At low transmit power, the algorithm simultaneously achieves the maximum sum rate and maximum energy efficiency.At middle-high transmit power, high sum rate does not necessarily imply high energy efficiency.
II. SYSTEM MODEL
The system is a coordinated K-cell MU-MISO downlink in which each base station serves multiple single-antenna users using multiuser precoding. Energy efficiency is optimized as weighted sum rate divided by total power consumption under per-base-station power constraints, contrasting with sum-rate-only optimization.
- System configuration: Each base station BS-j has Mj transmit antennas and serves Nj single-antenna users in cell j.The system contains K cells, with users indexed as User-(j, k).
- Signal model: The received signal model includes the intended beamformed signal, intra-cell and inter-cell interference, and complex Gaussian receiver noise.The channel coefficient includes large-scale fading, small-scale fading, and shadow fading.
- Energy-efficiency objective: Energy efficiency is defined as weighted sum rate divided by total power consumption, incorporating amplifier inefficiency, circuit power, and base-station power.User weights represent priorities, while Pc and P0 model non-transmit power consumption.
- Optimization problems: The energy-efficiency problem is subject to per-base-station transmit-power constraints.The traditional comparison problem maximizes weighted sum rate without accounting for power consumption.
- Optimization challenges: Coupled optimization variables make both the energy-efficiency and weighted-sum-rate formulations non-convex, while the fractional objective further prevents direct use of common methods.The paper therefore requires a specialized optimization approach.
III. ENERGY EFFICIENT BEAMFORMING ALGORITHM DESIGN
The paper solves the non-convex energy-efficiency problem with a two-layer scheme. Fractional programming produces an equivalent non-fractional formulation, and the rate–MSE relationship enables an iterative inner algorithm.
- Algorithm framework: The proposed two-layer optimization scheme is designed to solve the non-convex energy-efficiency problem.The scheme separates parameter handling from the non-fractional optimization subproblem.
- Outer reformulation: Fractional programming transforms the original fractional objective into an equivalent non-fractional problem.This reformulation makes the energy-efficiency parameter explicit.
- Inner optimization: The equivalent problem is cast into a tractable form by exploiting the relationship between user rate and the MSE of the optimal receiver.An optimization algorithm is then developed to solve the reformulated problem.
A. Equivalent Optimization Problem
The fractional energy-efficiency problem is reduced to finding the unique zero of a monotone parametric function. Bisection searches for the energy-efficiency factor, with each step solving an associated optimization problem.
- Equivalent Optimization Problem: The energy-efficiency objective is recognized as a classical fractional programming problem.Its fractional form is difficult to solve directly with common optimization approaches.
- Equivalent Optimization Problem: The parametric function F(η) is convex, continuous, and strictly decreasing, and F(η)=0 has a unique solution.These properties establish the solvability of the scalar parameter search.
- Equivalent Optimization Problem: The original fractional problem is equivalent to the parametric problem when the optimal value of the latter is zero.The corresponding optimizer then also solves the original energy-efficiency problem.
- Outer Layer Solution: The scalar parameter η is the energy-efficiency factor and is found sequentially through one-dimensional bisection search.The search updates ηmin or ηmax according to the sign of F(η) until the interval is within a predefined threshold ε.
- Outer Layer Solution: Each outer-layer search step solves the associated parametric optimization problem for a fixed η before updating the search interval.The resulting algorithm terminates when |ηmax − ηmin| ≤ ε.
B. Solution of Sub-Problem
The sub-problem is reformulated using auxiliary variables and the rate–MSE relationship, then solved by alternating updates of receivers, auxiliary variables, and beamformers. The resulting algorithm converges, provides closed-form updates, and supports parallel implementation.
- Reformulation: The non-fractional sub-problem remains non-convex, so the paper reformulates it using the relationship between achievable rate and optimal-receiver MSE.
- Reformulation: Introducing auxiliary variables and receiver filters yields an equivalent objective that is more tractable and convex in each variable block.
- Alternating optimization: Block coordinate descent sequentially updates beamformers, receiver filters, and auxiliary variables while fixing the other two blocks.
- Beamformer update: The beamformer update is a convex quadratic optimization problem whose solution is derived in closed form using a Lagrange multiplier and one-dimensional bisection search.
- Convergence: Algorithm 2 is guaranteed to converge because its block updates monotonically increase a bounded objective function.
- Overall solution: Combining the two algorithms efficiently solves the energy-efficiency problem and yields closed-form beamformers, receiver filters, and auxiliary variables.
- Scope: The formulation maximizes energy efficiency without individual-cell or individual-user rate requirements; adding such constraints is left for future work.
IV.COMPUTATIONAL COMPLEXITY AND IMPLEMENTATION ANALYSIS
The implementation analysis evaluates computational cost using floating-point operations and shows that the proposed algorithm can be executed in a decentralized or parallel manner.
- Computational complexity is analyzed by counting real floating-point operations, and the algorithm supports decentralized or parallel execution.
A. Complexity Analysis
The paper counts the flop cost of each alternating-optimization step and compares the proposed method with WMMSE in per-iteration complexity and convergence behavior.
- Complexity accounting: Algorithm 2 complexity is decomposed into receiver-filter, auxiliary-variable, and beamformer-update costs measured in flops.
- Complexity accounting: The beamformer update is the major computational component and uses matrix construction, decomposition, and inversion-related operations.
- Overall complexity: The total cost depends on the iteration counts of the outer and inner algorithms, denoted by 1 and 2, respectively.
- Baseline comparison: WMMSE has the same order of per-iteration computational complexity as Algorithm 2.
- Baseline comparison: The proposed method may need more iterations because it updates the energy-efficiency factor, but its search generally takes only a few iterations and does not significantly increase complexity.
B. Parallel Implementation
The proposed beamforming updates can run across parallel processors associated with base stations, coordinated by a controller that distributes shared information and the energy-efficiency parameter.
- Parallel structure: Beamforming vectors can be optimized simultaneously by K parallel processors, enabling a parallel implementation of the algorithm.
- Coordination: Initialization requires a central controller to collect and share all CSI, then distribute an initial energy-efficiency factor η to the processors.
- Iteration flow: Each processor optimizes its beamforming vector and associated user variables for the fixed η(n), after which the controller updates η(n).
- System assumptions: The K base stations can serve as parallel processors when connected by backhaul with sufficient capacity.
- System assumptions: Without a central controller, one base station must be assigned as controller, adding a coordination requirement.
- Communication overhead: Each iteration exchanges updated parameters and transmit power among processors and the controller, creating signalling overhead.
V. SIMULATION RESULTS
Simulations evaluate the proposed energy-efficiency algorithm against optimal, sum-rate, WMMSE, and fixed-beamformer baselines across transmit-power, channel, antenna, user, and circuit-power settings.
- Optimality and convergence: Over 96% of the optimal energy efficiency is achieved in most tested channel realizations.The optimum is approximated by solving the inner problem across η with 10000 random beamforming initializations.
- Optimality and convergence: Algorithm 2 always converges to a stable point in a limited number of iterations and achieves over 99% of the optimal sum rate.Different initializations can slightly affect performance and convergence speed.
- Transmit-power comparisons: At 26–34 dBm, the proposed algorithm and WMMSE achieve almost the same energy efficiency, whereas the proposed algorithm outperforms WMMSE at high transmit power.The low-power result suggests that transmitting at maximum available power is most energy efficient in that region; WMMSE’s capacity gain cannot offset maximum-power consumption at high power.
- System-size effects: Energy efficiency increases with served users and transmit antennas, but user-related gains shrink and the proposed algorithm’s antenna advantage over WMMSE narrows as antenna count grows.At Pj = 46 dBm, the proposed algorithm has an especially clear advantage when the number of transmit antennas is not large.
- Beamforming comparisons: The proposed algorithm and WMMSE outperform fixed-beamformer power allocation in energy efficiency, indicating that transmit beamforming optimization is important.Joint optimization of transmit beamforming vectors and power allocation is used for energy-efficient transmission design.
- Circuit-power effects: Reducing circuit power per antenna from 40 dBm to 30 dBm improves the proposed algorithm’s energy efficiency by over 100%.The upper transmit-power boundary where the proposed algorithm and WMMSE match moves from 42 dBm to 38 dBm.
VI. CONCLUSIONS
The paper addresses non-convex fractional energy-efficient coordinated beamforming and power allocation through an equivalent subtractive formulation and iterative optimization. The algorithm converges to near-optimal performance, especially achieving near-optimal energy efficiency and sum rate in the low-transmit-power region.
- The multicell multiuser downlink optimization problem is non-convex and fractional, making energy-efficient beamforming and power allocation difficult.
- Fractional programming transforms the original problem into an equivalent subtractive-form problem.
- The proposed iterative algorithm has proved convergence and admits a closed-form solution.
- The algorithm converges to a stable point within a limited number of iterations and achieves near-optimal performance.
- At low transmit power, the method obtains near-optimal energy efficiency and near-optimal sum rate simultaneously.
- Future work includes studying imperfect CSI and finding the global optimum for each fixed energy-efficiency parameter.