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Millimeter-Wave NOMA with User Grouping, Power Allocation and Hybrid Beamforming
Lipeng Zhu, Jun Zhang, Zhenyu Xiao, Xianbin Cao, Dapeng Oliver Wu, Xiang-Gen Xia
TL;DR
The paper addresses how to support multiuser downlink transmission in mmWave systems while balancing hybrid-beamforming constraints and interference. It proposes channel-correlation-based grouping with joint power allocation and hybrid beamforming, and reports higher achievable sum rate and energy efficiency than the cited comparison schemes.
Problem
In mmWave OMA, the number of users served in each resource block is limited by the number of RF chains, motivating mmWave-NOMA with hybrid beamforming.
Method
The paper uses K-means channel-correlation user grouping, solves power allocation under fixed hybrid beamforming, designs digital beamforming by approximate zero forcing, and optimizes constant-modulus analog beamforming with BC-PSO.
Results
The proposed scheme outperforms state-of-the-art schemes and conventional mmWave-OMA in achievable sum rate and is more energy efficient than fully digital MIMO.
Takeaways & Limitations
Joint user grouping, power allocation, and hybrid beamforming provides a supported mmWave-NOMA approach with improved achievable sum rate and energy efficiency.
Abstract
from arXiv · showhide
This paper investigates the application of non-orthogonal multiple access in millimeter-Wave communications (mmWave-NOMA). Particularly, we consider downlink transmission with a hybrid beamforming structure. A user grouping algorithm is first proposed according to the channel correlations of the users. Whereafter, a joint hybrid beamforming and power allocation problem is formulated to maximize the achievable sum rate, subject to a minimum rate constraint for each user. To solve this non-convex problem with high-dimensional variables, we first obtain the solution of power allocation under arbitrary fixed hybrid beamforming, which is divided into intra-group power allocation and inter-group power allocation. Then, given arbitrary fixed analog beamforming, we utilize the approximate zero-forcing method to design the digital beamforming to minimize the inter-group interference. Finally, the analog beamforming problem with the constant-modulus constraint is solved with a proposed boundary-compressed particle swarm optimization algorithm. Simulation results show that the proposed joint approach, including user grouping, hybrid beamforming and power allocation, outperforms the state-of-the-art schemes and the conventional mmWave orthogonal multiple access system in terms of achievable sum rate and energy efficiency.
I. INTRODUCTION
The paper addresses user limitations and efficiency trade-offs in mmWave systems by combining NOMA with hybrid beamforming. It proposes joint user grouping, power allocation, and hybrid beamforming to improve achievable sum rate and energy efficiency.
- Motivation: Hybrid beamforming balances energy and spectrum efficiency by connecting a small number of RF chains to many antennas while enabling beam gain and interference management.Analog beamforming is energy-efficient but generally supports one data stream per RF chain, whereas fully digital beamforming has unaffordable hardware cost and energy consumption for large arrays.
- Motivation: Conventional mmWave OMA serves no more users per resource block than the number of RF chains, limiting total served users.NOMA enables multiple users to share a resource block through power-domain separation, superposition coding, and successive interference cancellation.
- Proposed approach: Users with highly correlated channels are grouped for mmWave-NOMA, while low-correlation users are assigned to different groups to mitigate inter-group interference.The proposed grouping uses K-means with normalized channel correlation as the measure.
- Proposed approach: The paper formulates joint hybrid beamforming and power allocation to maximize achievable sum rate subject to a minimum rate constraint for every user.Power allocation under fixed hybrid beamforming is divided into intra-group and inter-group allocation, with global optimality proved under ideal beam patterns without inter-group interference.
- Proposed approach: Approximate zero-forcing designs digital beamforming to suppress inter-group interference, while boundary-compressed particle swarm optimization solves the constant-modulus analog beamforming problem.Together, these stages realize joint optimization of power allocation and hybrid beamforming.
- Results: The proposed mmWave-NOMA scheme outperforms state-of-the-art schemes and conventional mmWave-OMA in achievable sum rate, while its hybrid-beamforming energy efficiency exceeds that of fully digital MIMO.Its achievable sum rate is close to the ideal no-inter-group-interference case, indicating effective designed hybrid beamforming.
II. SYSTEM MODEL
The system is a single-cell downlink mmWave-NOMA architecture in which a base station with many antennas and M RF chains serves K>M single-antenna users through connected hybrid beamforming. Users are partitioned into groups, with NOMA and SIC within groups and inter-group signals treated as interference.
- A. System model: The system serves K single-antenna users simultaneously, with K>M, and schedules them into M groups associated with independent data streams.Each group contains at least one user, and group user sets are disjoint.
- A. System model: The base station uses a fully connected hybrid beamforming structure with N antennas, M RF chains, and M baseband data streams.Digital precoding operates before each RF chain drives N phase shifters for analog beamforming.
- A. System model: Users within the same group perform NOMA and SIC, whereas signals from different groups are treated as interference.The received signal model includes the user channel vector, Gaussian noise, transmission-signal vector, power allocation matrix, digital beamforming, and analog beamforming.
- A. System model: The hybrid beamforming matrix combines a digital beamforming matrix with an analog beamforming matrix constrained by constant-modulus phase shifters.The formulation separates transmission power from hybrid beamforming and assumes each hybrid-beamforming column has unit norm.
- A. System model: The mmWave channel is modeled as directional and spatially sparse using a Saleh-Valenzuela representation with a uniform linear array and half-wavelength antenna spacing.Multipath components have distinct angles of departure and arrival, and the channel response uses steering vectors.
- A. System model: The base station is assumed to know the channel state information between itself and the users.Channel estimation is outside the paper’s scope.
B. Achievable Rate
The section defines effective channel gains, orders users accordingly for SIC, and formulates achievable rates and sum rate under Gaussian signaling. The downlink model assumes BS-known CSI, while SIC requires additional decoding-order and codebook information at receivers.
- Effective channel gains combine channel gains with beamforming gains, so users must be sorted after the beamforming matrices are fixed.
- NOMA decoding orders users by increasing effective channel gain, allowing each user to successively decode and remove designated higher-index signals.
- Gaussian signaling is assumed, and each user’s achievable rate is derived from its SINR under intra-group and inter-group interference.
- The BS is assumed to know CSI and performs grouping, power allocation, and beamforming, while users need not know other users’ gains.
- SIC requires transmitting prior users’ decoding order and codebook information, creating overhead that grows with users per NOMA group.
- Keeping groups small limits this overhead, and slowly varying channels can reduce it further because decoding order and codebooks change slowly.
III. USER GROUPING AND PROBLEM FORMULATION
The paper groups users by channel correlation and formulates joint hybrid-beamforming and power-allocation optimization. The formulation balances achievable sum rate with per-user minimum-rate guarantees under nonconvex, high-dimensional constraints.
- III. USER GROUPING AND PROBLEM FORMULATION: Because K > M, users are scheduled into M groups before jointly optimizing hybrid beamforming and power allocation.
- A. User Grouping: Highly correlated users are assigned to the same group for multiplexing gain, whereas weakly correlated users are separated to reduce interference.
- A. User Grouping: K-means clustering uses normalized channel correlation, iteratively reassigning users and updating cluster representatives until representatives remain unchanged.
- B. Problem Formulation: The optimization maximizes achievable sum rate while imposing a minimum achievable rate for every user to balance sum-rate performance and fairness.
- B. Problem Formulation: The constraints enforce nonnegative user powers, a total BS power limit P, constant modulus for the analog beamformer, and unit power for the hybrid beamformer.
- B. Problem Formulation: The problem has K + MN + M^2 variables, with entangled variables, nonconvexity, and beamforming-dependent decoding orders preventing direct exhaustive or standard optimization.
- B. Problem Formulation: The proposed two-stage solution first computes low-complexity sub-optimal power allocation for fixed HBF, then designs DBF with AZF and ABF with BC-PSO.
IV. SOLUTION OF POWER ALLOCATION
The power-allocation solution fixes hybrid beamforming, separates intra-group and inter-group allocation, and iteratively handles residual inter-group interference. Its optimality is guaranteed when that interference is small or zero.
- With fixed HBF, the nonconvex power-allocation problem is divided into intra-group and inter-group power-allocation subproblems.
- Introducing group powers preserves the problem’s degrees of freedom while making the allocation formulation more tractable.
- A closed-form sub-optimal intra-group allocation is obtained for fixed inter-group allocation, then used to derive a sub-optimal inter-group solution.
- The proposed allocation is near-optimal when inter-group interference is small, as supported by theoretical analysis and simulation.
- Neglecting inter-group interference decomposes the intra-group problem into M independent problems, enabling the stated optimality condition.
- The resulting intra-group allocation satisfies the minimum-rate constraints for users ranked second through last in each group.
- The inter-group allocation is formulated after substituting intra-group powers into the sum-rate problem.
B. The Inter-GPA Problem
The inter-group allocation problem remains challenging because interference couples users and allocation variables. The paper proposes an iterative low-complexity scheme whose allocation becomes globally optimal as inter-group interference approaches zero.
- Inter-group interference makes the objective nonconvex even after hybrid beamforming is fixed.
- The proposed inter-GPA method uses an iterative update that repeatedly solves related allocation problems until forming a closed loop.
- Under invariant inter-group interference, the transformed objective is concave with linear constraints and can be solved using convex optimization tools.
- The method provides a low-complexity inter-GPA procedure intended to expose the essential allocation principle.
- As inter-group interference approaches zero, the combined allocation from Algorithm 2 and the intra-GPA solution is globally optimal.
- The optimality of power allocation depends on inter-group interference, which beamforming design should suppress while increasing achievable sum rate.
V. SOLUTION OF HYBRID BEAMFORMING
The paper decomposes hybrid beamforming design into digital beamforming and analog beamforming stages, targeting inter-group-interference suppression and improved achievable sum rate. Approximate zero-forcing handles digital beamforming, while analog beamforming provides additional interference suppression under a constant-modulus constraint.
- Design motivation: The hybrid beamforming design must support multidirectional transmission because multiple users may share each group.Traditional unidirectional analog beamforming cannot support all users in a group.
- Design motivation: The superposition of inter-group and intra-group interference makes the optimal hybrid beamforming solution difficult to obtain.The analog beamforming problem is also high-dimensional and constrained by a non-convex modulus condition.
- Digital beamforming: With fixed analog beamforming, the approximate zero-forcing method designs digital beamforming to reduce inter-group interference.The digital beamforming construction uses an equivalent channel vector selected from the highest-channel-gain user in each group.
- Digital beamforming: Because the digital beamforming matrix has rank no greater than the number of users, digital beamforming cannot completely suppress inter-group interference.The remaining interference can be further suppressed through analog beamforming, which has a higher degree of freedom.
- Digital beamforming: The resulting digital beamforming matrix is obtained from the equivalent channel matrix for any fixed analog beamforming matrix.Each digital beamforming column is normalized to satisfy the unit-power constraint on the hybrid beamforming matrix.
B. ABF Using BC-PSO Alogrithm
The analog beamforming problem is formulated as a high-dimensional, non-convex optimization under constant-modulus constraints. Boundary-compressed particle swarm optimization relaxes the search space during updates and returns particles to feasible boundaries.
- Problem formulation: Analog beamforming is difficult because its achievable-sum-rate expression is complicated and its N × M matrix has a constant-modulus constraint.The matrix dimension makes direct analog beamforming design high-dimensional.
- PSO design: Particle swarm optimization is used to search the analog beamforming space, with each particle representing an analog beamforming position and velocity.Particles retain their locally best positions, while the swarm tracks a globally best position.
- BC-PSO design: BC-PSO updates particle velocities and positions using inertia, cognitive, and social terms before recalculating fitness and best positions.The inertia weight decreases linearly from its maximum to minimum to improve convergence speed.
- BC-PSO design: BC-PSO relaxes the constant-modulus search space and adjusts particles onto outer and inner boundaries during iterations.Boundary updates are applied whenever particle positions fall outside the permitted regions.
- BC-PSO outcome: The particles can traverse the relaxed search space and eventually converge to satisfy the constant-modulus constraint.The paper reports enhanced search capabilities compared with classic PSO.
- Overall procedure: The complete algorithm first groups users, then optimizes analog beamforming while substituting power allocation and digital beamforming as functions of the analog matrix.Each iteration computes digital beamforming, inter-group power allocation, intra-group power allocation, and achievable sum rate.
B. Computational Complexity
The proposed user grouping, hybrid beamforming, and power allocation algorithm has polynomial computational complexity. Under N ≫ K and N ≫ M, the competing method can have lower complexity because it does not jointly optimize hybrid beamforming and power allocation.
- User grouping: Algorithm 1 has maximal complexity O(K^2N) when the number of antennas is much larger than the number of RF chains.Channel-correlation and channel-vector calculations dominate the grouping procedure.
- Power allocation: Algorithm 2 computes effective channel gains in O(MKN) and updates inter-group power allocation through bounded iterative subcycles.The inter-group allocation update requires at most M iterations per subcycle.
- Complexity result: The proposed algorithm has total computational complexity O(TmaxIM^2KN^2 + TmaxFmaxIM^2K^2N), which is polynomial.This expression accounts for user pairing, hybrid beamforming, and power allocation procedures.
- Overall complexity: The proposed method invokes user grouping once and Algorithm 2 TmaxIMN times within Algorithm 3.The overall complexity follows from these invocation counts and the component costs.
- Comparison: Under N ≫ K and N ≫ M, the algorithm in [13] has lower complexity than the proposed method because it does not jointly optimize hybrid beamforming and power allocation.The compared complexity is O(MK^2 + MN + TK^4.5 log2(1/ε)).
P + NRFPRF + NPSPPS
Simulations compare achievable sum rate and energy efficiency across channel models, power and rate constraints, RF-chain counts, and multiple baselines. The proposed mmWave-NOMA approach generally outperforms the compared hybrid and orthogonal schemes, while retaining an energy-efficiency advantage over fully digital MIMO.
- Minimum-rate constraints: The proposed mmWave-NOMA approach outperforms mmWave-OMA, TDMA, and the mmWave-NOMA scheme in under varying minimum-rate constraints.For r from 1 to 2 bps/Hz, its ASR is nearly 10 bps/Hz larger than the scheme in.
- Minimum-rate constraints: For sufficiently large minimum-rate constraints, both compared schemes can have zero ASR when channel realizations fail to satisfy all minimum-rate constraints.The proposed scheme nevertheless has larger average ASR and higher feasibility than the scheme in.
- Inter-group interference: The proposed approach remains close to the ideal case, with an ASR gap no more than 0.5 bps/Hz attributed to inter-group interference.This supports the paper’s approximation of neglecting inter-group interference during intra-group power allocation.
- Energy efficiency: The proposed HBF scheme achieves nearly fourfold EE compared with fully digital MIMO when the minimum-rate constraint is no more than 1.5 bps/Hz.Fully digital MIMO has higher ASR but lower EE than the hybrid structures.
- RF-chain count: The proposed approach is more robust against channel model changes because its ASR is almost unchanged between LOS and NLOS conditions.By contrast, the ASR of the scheme in is lower under NLOS than LOS conditions.
VIII. CONCLUSION
The paper develops mmWave-NOMA downlink transmission with hybrid beamforming by combining channel-correlation-based user grouping, joint beamforming and power allocation, and dedicated interference-suppression methods. Simulations report improved achievable sum rate and energy efficiency relative to benchmark schemes, alongside higher computational complexity.
- The study considers downlink mmWave-NOMA with a hybrid beamforming structure.
- A K-means user-grouping algorithm uses channel correlations to organize multiple users.
- The joint hybrid beamforming and power-allocation problem maximizes achievable sum rate subject to each user’s minimum-rate constraint.
- Power allocation is solved under fixed hybrid beamforming through separate intra-group and inter-group subproblems, while approximate zero-forcing digital beamforming reduces inter-group interference.
- Boundary-compressed particle swarm optimization solves the analog-beamforming problem under the constant-modulus constraint.
- The proposed scheme outperforms mmWave-OMA in achievable sum rate and is more energy efficient than fully digital MIMO, but requires higher computational complexity.
APPENDIX A PROOF OF LEMMA 1
Appendix A establishes the optimality conditions for a power-allocation subproblem using the Lagrange multiplier method and a contradiction argument. The proof shows that the optimal allocation satisfies the stated total-power condition.
- Because the objective increases with P_m, the total power constraint is active at the optimum.
- The constraint-free subproblem is solved using the Lagrange Multiplier Method and its KKT equation set.
- Solving the KKT equations yields the optimal solution of Problem (21).
- The proof uses contradiction by perturbing one user’s allocated power by a sufficiently small nonnegative ε.
- The derivative analysis shows the relevant objective difference is monotone in ε, supporting the contradiction argument.
- The contradiction establishes that every user in the considered set satisfies the required optimality condition.