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Joint Power Allocation and User Association Optimization for Massive MIMO Systems

Trinh Van Chien, Emil Björnson, Erik G. Larsson

arXiv:1601.02436v2cs.IT

TL;DR

The paper tackles joint power allocation and BS-user association for multi-cell Massive MIMO downlink transmission while minimizing total power under user QoS requirements. It derives general and Rayleigh-fading spectral-efficiency bounds, converts fixed-QoS optimization into linear programs, and develops a max-min fairness method. Simulations show effective power reduction and load balancing, while optimal solutions usually associate each user with only one BS.

  • Problem

    The problem is to minimize total downlink transmit power while selecting optimized BS subsets for users under QoS constraints.

  • Method

    The method derives a general ergodic-SE lower bound, obtains MRT and ZF closed forms for Rayleigh fading, and formulates association and power jointly.

  • Results

    Fixed-QoS power minimization is a linear program, while weighted max-min fairness is solved through quasi-linear optimization and bisection.

  • Takeaways & Limitations

    The methods demonstrate effective power allocation and association, with max-min optimization providing uniformly strong SE and association supporting load balancing.

Abstract

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This paper investigates the joint power allocation and user association problem in multi-cell Massive MIMO (multiple-input multiple-output) downlink (DL) systems. The target is to minimize the total transmit power consumption when each user is served by an optimized subset of the base stations (BSs), using non-coherent joint transmission. We first derive a lower bound on the ergodic spectral efficiency (SE), which is applicable for any channel distribution and precoding scheme. Closed-form expressions are obtained for Rayleigh fading channels with either maximum ratio transmission (MRT) or zero forcing (ZF) precoding. From these bounds, we further formulate the DL power minimization problems with fixed SE constraints for the users. These problems are proved to be solvable as linear programs, giving the optimal power allocation and BS-user association with low complexity. Furthermore, we formulate a max-min fairness problem which maximizes the worst SE among the users, and we show that it can be solved as a quasi-linear program. Simulations manifest that the proposed methods provide good SE for the users using less transmit power than in small-scale systems and the optimal user association can effectively balance the load between BSs when needed. Even though our framework allows the joint transmission from multiple BSs, there is an overwhelming probability that only one BS is associated with each user at the optimal solution.

I. INTRODUCTION

The paper addresses joint power allocation and BS-user association in multi-cell Massive MIMO downlink systems, where users may be served by multiple BSs. It develops optimization methods intended to reduce power consumption, balance load, and provide fair service.

  • Jointly optimizing power allocation and BS-user association targets lower power consumption in multi-cell Massive MIMO downlink systems.
  • The proposed framework derives an ergodic spectral-efficiency expression for non-coherent joint transmission with multiple BSs, including closed forms for MRT and ZF precoding.
  • Fixed-SE power minimization becomes a linear program, enabling polynomial-time optimization of power allocation and association.
  • The optimal association depends on large-scale fading, estimation quality, SINR, and pilot contamination, and only a subset of BSs serves each user.
  • A weighted max-min formulation optimizes the worst user SE through a quasi-linear problem solved with power minimization and bisection.
  • The system model permits users to associate with multiple BSs, with channel estimation based on uplink pilots in TDD operation.

B. Downlink Data Transmission Model

The downlink model lets each BS send a separate data stream to a user under non-coherent joint transmission. A successive-decoding analysis yields a general lower bound on ergodic spectral efficiency.

  • Non-coherent joint transmission allows each BS to send a different data symbol to the same user without requiring phase synchronization between BSs.
  • A user may be associated with multiple BSs when their allocated transmit powers are nonzero.
  • Users detect desired signals successively using channel statistics rather than current channel realizations, while multi-user interference and noise remain in the received signal.
  • Theorem 1 provides a lower bound on each user's downlink ergodic sum capacity as log2(1 + SINRk) bit/symbol.
  • The effective SINR structure separates desired signal power, beamforming uncertainty, multi-user interference, and additive noise.
  • The lower bound is independent of channel distribution and precoding scheme, making the framework applicable to general channel, precoding, and pilot-allocation settings.

C. Achievable Spectral Efficiency under Rayleigh Fading

For Rayleigh fading, the general ergodic-SE lower bound has closed forms under MRT and ZF precoding. The resulting expressions expose array gain, interference mitigation, pilot contamination, and joint-transmission effects.

  • Closed-form lower bounds on downlink ergodic spectral efficiency are obtained for Rayleigh fading with MRT and ZF precoding.
  • MRT increases signal power proportionally to the number of BS antennas M through array gain.
  • Pilot contamination increases with M under MRT and can saturate the achievable rate as M approaches infinity.
  • Non-coherent joint transmission combines received signal powers from multiple BSs and can provide stronger signal gain than serving a user from one BS.
  • ZF sacrifices some array gain to mitigate multi-user interference, whereas MRT primarily maximizes the SNR.
  • For both MRT and ZF, downlink ergodic SE depends on channel-estimation quality and BS power allocation, with pilot contamination limiting performance.

III. DOWNLINK TRANSMIT POWER OPTIMIZATION FOR MASSIVE MIMO SYSTEMS

The paper formulates downlink transmit-power minimization under user-specific SE constraints and BS power limits. With the derived MRT and ZF expressions, the problem becomes an optimization over powers whose nonzero entries determine association.

  • Power-amplifier efficiency can differ across BSs and therefore affects both power allocation and user association.
  • The objective is to minimize total power while meeting user-specific SE constraints and respecting BS power-amplifier and peak-output limits.
  • SE targets are transformed into SINR targets, allowing the general formulation to use closed-form MRT and ZF SINRs.
  • For MRT and ZF precoding, the resulting power-minimization formulations are given separately through corresponding lemmas.
  • At the optimum, each user's associated BS subset is identified by the BS-user power variables with nonzero values.
  • Unlike short-term fading-dependent QoS optimization, the approach uses long-term QoS constraints enabled by channel hardening and favorable propagation.

IV. OPTIMAL POWER ALLOCATION AND USER ASSOCIATION BY LINEAR PROGRAMMING

The paper converts total transmit-power minimization for non-coherent joint transmission into a linear program for MRT or ZF precoding. Its solution gives both globally optimal power allocation and BS-user association, while association depends on multiple system factors and usually selects one BS per user.

  • A. Optimal Solution with Linear Programming: The optimal power allocation for MRT or ZF precoding is obtained by solving a linear program.The objective is linear in the users’ power variables, and the constraints are affine.
  • A. Optimal Solution with Linear Programming: The linear program can be solved to global optimality in polynomial time and simultaneously provides the optimal BS-user association.This result applies to multi-cell Massive MIMO with non-coherent joint transmission.
  • B. BS-User Association Principle: The linear-program formulation is specific to non-coherent joint transmission; coherent joint transmission instead yields a second-order cone program.The coherent case is considered for comparison.
  • B. BS-User Association Principle: The association set determines whether a user is served by one BS or multiple BSs at the optimum.A singleton set corresponds to one serving BS, whereas several indices indicate joint service by multiple BSs.
  • B. BS-User Association Principle: Optimal association depends on interference, noise, power allocation, large-scale fading, channel estimation quality, pilot contamination, and QoS constraints.The resulting rule is not a simple max-SNR rule.

V. MAX-MIN QOS OPTIMIZATION

The paper addresses max-min QoS optimization when fixed QoS targets may be infeasible because of propagation conditions and limited pilot power. It solves the problem by repeatedly testing linear-program feasibility over a QoS search range, using bisection to obtain the maximum feasible value.

  • V. MAX-MIN QOS OPTIMIZATION: Fixed QoS targets may be infeasible because of path loss, propagation conditions, and channel-estimation errors caused by limited pilot power.These constraints make target selection difficult for a given network.
  • V. MAX-MIN QOS OPTIMIZATION: Max-min optimization maximizes the lowest QoS value, with optional user-specific weights, to provide uniformly strong service.The weights may reflect propagation, interference, or user priorities.
  • V. MAX-MIN QOS OPTIMIZATION: For a fixed minimum QoS parameter, the max-min problem is a linear program; optimizing that parameter makes the overall problem quasi-linear.The QoS constraints increase with the parameter, enabling feasibility-based search.
  • V. MAX-MIN QOS OPTIMIZATION: Bisection finds the maximum feasible QoS by repeatedly solving the fixed-target linear program and halving the search interval.The procedure terminates when the interval gap is below the line-search accuracy δ.
  • V. MAX-MIN QOS OPTIMIZATION: The algorithm has polynomial complexity, with each iteration dominated by a linear program of complexity O(K^3L^3).The complexity does not depend on the number of BS antennas, and the iteration count grows logarithmically with the initial upper bound and inverse accuracy.

VI. NUMERICAL RESULTS

Simulations evaluate power minimization and max-min QoS in a four-BS Massive MIMO system under varied antenna counts, QoS targets, user loads, and association schemes. Optimal association reduces power or improves fairness, while single-BS association is sufficient in at least 90% of cases.

  • Simulation setup: The simulations use four BSs at square corners, randomly located users, LTE-like large-scale fading, orthogonal pilots, and both MRT and ZF precoding.Users are uniformly distributed over joint coverage, with a 100 m minimum distance from BSs; shadow fading has 7 dB standard deviation.
  • Power minimization: Massive MIMO substantially reduces transmit power for a fixed QoS, with larger antenna arrays requiring less power than systems with fewer antennas.The comparison considers total transmit power for 20 users and QoS of 1 bit/symbol.
  • Feasibility and fairness: The optimal association is more robust than max-SNR association, while infeasibility can reach about 80% with few antennas or high QoS demands.This motivates max-min QoS optimization, which provides feasible solutions for all user locations and channel realizations in the reported experiments.
  • Feasibility and fairness: The optimal association improves max-min QoS by up to 22% at the 95%-likely level and up to 11% on average compared with max-SNR association.With 300 antennas per BS, every user can achieve more than 2 bit/symbol with high probability, and QoS can reach 4 bit/symbol.
  • User load and association: With 200 antennas per BS and non-coherent joint transmission, the system serves 20 and 40 users at QoS requirements of 2.28 and 1.87 bit/symbol, respectively.The max-min QoS decreases with 40 users because of increased interference; MRT can outperform ZF when antennas are not much more numerous than users.
  • User load and association: At least 90% of users are served by a single BS, while multiple-BS service is mainly needed under severe shadow fading or high user loads.Users near a BS are more likely to associate with it, whereas users near coverage boundaries are more likely to use multiple BSs.

VII. CONCLUSION

The paper jointly optimizes power allocation and user association in multi-cell Massive MIMO, minimizing transmit power under QoS constraints and maximizing weighted worst-user QoS. Experiments show effective methods, while max-SNR association performs well in many scenarios but is not optimal.

  • The proposed method jointly optimizes power allocation and user association for downlink non-coherent joint transmission.
  • The fixed-QoS transmit-power minimization problem with MRT or ZF precoding is formulated as a linear program.
  • Weighted max-min optimization maximizes the worst QoS value among users with user-specific weights.
  • Experimental results show that max-SNR association works well in many Massive MIMO scenarios but is not optimal.

APPENDIX

The appendix derives ergodic-rate bounds for successive detection under non-coherent joint transmission and specializes the required signal and interference terms to MRT and ZF precoding. These derivations produce closed-form SINR expressions for the optimization framework.

  • Successive detection processes signals from multiple BSs by subtracting previously detected signals over known average channels.
  • A lower bound on each transmission’s ergodic capacity is obtained by treating uncorrelated noise as Gaussian noise.
  • The appendix computes desired-signal and uncorrelated-noise powers before forming the downlink ergodic rate between each BS and user.
  • The resulting user spectral-efficiency bound is obtained by summing the rates from the successively detected BS signals.
  • MRT: For Rayleigh fading, channel moments and pilot-reuse relationships yield closed-form MRT SINR expressions.

C. Proof of Corollary 2

The proof transforms the optimization into a quadratic Lagrangian and derives dual conditions identifying which BSs can receive positive power for each user. The resulting association rule sets other BS-user powers to zero while QoS constraints require service from at least one BS.

  • A variable transformation and diagonal matrix representation convert the Lagrangian into a quadratic function.
  • The dual function is bounded below when the transformed matrix is positive semidefinite.
  • The first-order conditions provide necessary and sufficient conditions for the optimal transformed variables.
  • If a BS associates with a user, its association condition follows from the corresponding dual optimality relation.
  • BSs that fail the condition receive zero transmit power, while QoS constraints ensure every user is served by at least one BS.

E. Proof of Corollary 3

The proof bounds the max-min QoS objective through an upper bound on the achievable SINR. It uses an optimistic orthogonal-pilot case and then derives separate MRT and ZF bounds.

  • Because pilot reuse reduces spectral efficiency, the upper-bound analysis considers mutually orthogonal pilot sequences as an optimistic special case.
  • The upper bound is expressed through log2(1 + SINR_k).
  • The proof first computes the maximal SINR value to solve the upper-bound problem.
  • For MRT and ZF precoding, separate inequalities and precoding-specific expressions produce SINR upper bounds.

F. Joint Power Allocation and User Association for Massive MIMO Systems with Coherent Joint Transmission

This section derives an ergodic spectral-efficiency lower bound and formulates power minimization for coherent joint transmission in Rayleigh-fading Massive MIMO systems. The resulting optimization uses precoding-specific parameters and is solved as a convex program with second-order-cone constraints.

  • Coherent joint transmission has all base stations precode and send the same signal to each user.
  • A lower bound on user k’s ergodic spectral efficiency is obtained using an added-and-subtract technique and Gaussian noise as the worst-case uncorrelated-noise distribution.
  • For Rayleigh fading with MRT or ZF precoding, the total transmit-power minimization problem is expressed using precoding-dependent parameters.MRT gives g_i,k = Mθ_i,k and z_i,k = β_i,k, while ZF gives g_i,k = (M − K)θ_i,k and z_i,k = β_i,k − θ_i,k.
  • The reformulated optimization has a convex quadratic objective and second-order-cone constraints, enabling interior-point solution with CVX.The passage states that optimal power allocation and user association can be obtained this way, alongside max-min QoS levels from the referenced formulation.
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