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Codebook Based Hybrid Precoding for Millimeter Wave Multiuser Systems

Shiwen He, Jiaheng Wang, Yongming Huang, Bjorn Ottersten, Wei Hong

arXiv:1706.10181v2cs.IT

TL;DR

The paper tackles downlink hybrid precoding for multiuser mmWave systems with limited RF chains and practical hardware constraints. It develops codebook-based RF precoding with beam-sweep CSIT acquisition, reducing the original optimization to joint codeword selection and precoder design. Numerical results validate the proposed methods, which outperform existing methods and approach fully digital precoding performance.

  • Problem

    The paper addresses joint RF-baseband precoding for multiuser multi-antenna mmWave systems with limited RF chains and practical phase-shifter constraints, targeting sum rate and energy efficiency.

  • Method

    The method uses codebook-based RF precoding and beam sweeping to obtain effective CSIT, transform the system into a virtual multiuser downlink, and formulate joint codeword selection and precoder design problems.

  • Results

    The proposed hybrid precoding methods outperform existing methods and achieve performance close to that of the fully digital precoder.

  • Takeaways & Limitations

    Codebook-based hybrid precoding provides a validated approach for jointly optimizing RF and baseband precoders under sum-rate and energy-efficiency objectives.

Abstract

from arXiv · show

In millimeter wave (mmWave) systems, antenna architecture limitations make it difficult to apply conventional fully digital precoding techniques but call for low cost analog radio-frequency (RF) and digital baseband hybrid precoding methods. This paper investigates joint RF-baseband hybrid precoding for the downlink of multiuser multi-antenna mmWave systems with a limited number of RF chains. Two performance measures, maximizing the spectral efficiency and the energy efficiency of the system, are considered. We propose a codebook based RF precoding design and obtain the channel state information via a beam sweep procedure. Via the codebook based design, the original system is transformed into a virtual multiuser downlink system with the RF chain constraint. Consequently, we are able to simplify the complicated hybrid precoding optimization problems to joint codeword selection and precoder design (JWSPD) problems. Then, we propose efficient methods to address the JWSPD problems and jointly optimize the RF and baseband precoders under the two performance measures. Finally, extensive numerical results are provided to validate the effectiveness of the proposed hybrid precoders.

I. INTRODUCTION

The paper addresses practical mmWave multiuser hybrid-precoding constraints by jointly optimizing RF and baseband precoders for sum rate and energy efficiency. It uses codebook-based RF precoding and beam sweeping to reduce CSIT and optimization complexity, with numerical results showing performance close to fully digital precoding.

  • Motivation: mmWave signals suffer greater path loss, penetration loss, and rain fading than sub-6 GHz signals, while 60 GHz FSPL is 35.6 dB higher than at 1 GHz.These losses must be compensated by the transceiver.
  • Problem: Limited RF chains reduce hardware cost and energy consumption, but pure analog precoding cannot provide multiplexing gains for parallel data streams.Hybrid RF-baseband precoding is therefore studied to combine diversity and multiplexing gains.
  • Problem: Existing multiuser mmWave designs assume perfect CSIT or equal numbers of users and RF chains, motivating more flexible designs for fewer users than RF chains.Perfect CSIT is difficult to obtain with many antennas, and the served-user count per subcarrier may be smaller than the RF-chain count.
  • Contributions: The paper jointly optimizes RF-baseband precoders for system sum rate and energy efficiency under constant-modulus, finite-phase shifters and limited RF chains.The RF design uses codewords to represent RF beamforming vectors and transforms the system into a virtual multiuser downlink MISO system.
  • Contributions: A beam-sweep procedure obtains effective CSIT with less signaling feedback by exploiting the beam-domain sparsity of mmWave channels.The codebook design also simplifies hybrid-precoding optimization into joint codeword selection and precoder design problems.
  • Results: Efficient methods address the joint codeword selection and precoder design problems for sum-rate and energy-efficiency maximization, and numerical results validate the design.The proposed methods outperform existing methods and achieve performance close to that of the fully digital precoder.

II. PROBLEM STATEMENT

The paper formulates downlink transmission for a mmWave multiuser MISO system using hybrid RF-baseband precoding with fewer RF chains than antennas. It specifies the signal model, hardware constraints, and clustered channel assumptions underlying the design problem.

  • System architecture: The downlink mmWave system serves K single-antenna users with M transmit antennas and S RF chains, where K ≤ S ≤ M.The system uses hybrid precoding because installing one RF chain per antenna is expensive and impractical.
  • System architecture: Hybrid precoding jointly processes the transmitted signals in the digital baseband and analog RF domains.The baseband precoder maps the user signals to the RF chains, while the RF precoder uses analog circuitry such as phase shifters.
  • Signal model: The received-signal model includes user data, channel vectors, and independent zero-mean AWGN, with each user's SINR determined from the hybrid precoders.The channel matrix is formed from the user channel vectors, and the baseband precoder columns correspond to users.
  • Precoding constraints: The RF precoder is constrained by implementation hardware, with elements typically required to have constant modulus and restricted phase variation.These constraints distinguish the RF stage from the baseband precoder.
  • Channel model: Each user channel follows a narrowband clustered Saleh-Valenzuela model with scattering clusters, multiple propagation rays, random complex gains, and Laplacian angle distributions.The model supports both uniform linear and uniform planar antenna arrays through their array response vectors and angular parameters.

C. Problem Formulation

The paper formulates spectral-rate and energy-efficiency hybrid-precoding problems under constant-modulus RF, power, QoS, and RF-chain constraints. A codebook and beam-sweep design converts the system into a virtual multiuser MISO problem and then into JWSPD optimization.

  • C. Problem Formulation: The design targets system sum-rate and energy-efficiency maximization for downlink multiuser mmWave hybrid precoding.The formulation includes SRmax and EEmax objectives.
  • C. Problem Formulation: The feasible RF precoder has constant-modulus entries, while transmit power, user target rates, and RF-chain hardware constraints remain explicit.Power-amplifier inefficiency and dynamic circuit consumption are included in the energy-efficiency formulation.
  • C. Problem Formulation: The original problems are difficult because of nonconvex user-rate constraints, fractional energy-efficiency objectives, constant-modulus structure, and costly CSIT acquisition.The paper assumes the set of user target rates is feasible.
  • III. CODEBOOK BASED MMWAVE PRECODING DESIGN WITH BEAM SWEEPING: A constant-modulus RF codebook restricts analog precoding to predefined phase-shifted codewords, including quantized and beamsteering designs.Each codebook column represents a phase rotation across antenna elements and generates a beam.
  • III. CODEBOOK BASED MMWAVE PRECODING DESIGN WITH BEAM SWEEPING: Beam sweeping sends training packets over codebook directions, allowing users to report received strengths and effective channels instead of exact CSIT.Directional sparsity means only a few nonzero effective channel coefficients may need feedback.
  • III. CODEBOOK BASED MMWAVE PRECODING DESIGN WITH BEAM SWEEPING: Beam sweeping represents the original system as a virtual multiuser MISO downlink whose virtual antennas are RF codewords.The effective channel is sparse in the beam domain, reducing the feedback burden.
  • III. CODEBOOK BASED MMWAVE PRECODING DESIGN WITH BEAM SWEEPING: Expanding the baseband precoder makes codeword selection equivalent to row sparsity, transforming hybrid precoding into JWSPD problems for SRmax and EEmax.A codeword is selected exactly when the corresponding expanded-precoder row is nonzero, with at most S selected codewords.

IV. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR SRMAX PROBLEM

The JWSPD SRmax problem is NP-hard because it combines a nonconvex sum-rate objective with an ℓ0 constraint. The paper therefore seeks an efficient, likely suboptimal, solution rather than a globally optimal one.

  • IV. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR SRMAX PROBLEM: The JWSPD SRmax problem is NP-hard because of its nonconvex sum-rate objective and ℓ0-(quasi)norm constraint.These features make globally optimal solution search prohibitively complex.
  • IV. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR SRMAX PROBLEM: An efficient probably suboptimal solution is preferred in practice because obtaining the global optimum has prohibitive complexity.The paper introduces such an efficient solution in the following development.

A. Joint Codeword Selection and Precoder Design for SRmax problem

The SRmax JWSPD method replaces hard sparsity and rank constraints with tractable approximations, then solves successive convex problems. Its objective sequence is nondecreasing, convergence is guaranteed, and the method reaches a KKT solution of the relaxed problem.

  • A. Joint Codeword Selection and Precoder Design for SRmax problem: The method approximates the nonconvex ℓ0 sparsity constraint with a convex ℓ1,∞-norm squared surrogate.A group-sparsity parameter controls the number of selected codewords.
  • A. Joint Codeword Selection and Precoder Design for SRmax problem: Auxiliary variables and semidefinite lifting reformulate the rate and SINR constraints, while rank-one constraints are subsequently relaxed.The relaxed formulation remains nonconvex before successive convex approximation.
  • A. Joint Codeword Selection and Precoder Design for SRmax problem: Dropping rank-one constraints and applying successive convex approximation produces tractable convex subproblems solvable with modern convex optimization tools.The difficult quadratic constraints are approximated iteratively.
  • A. Joint Codeword Selection and Precoder Design for SRmax problem: Algorithm 1 iteratively initializes variables, solves the approximate problem, updates the objective, and stops when the improvement is below ζ.The procedure uses a fixed sparsity parameter λ.
  • A. Joint Codeword Selection and Precoder Design for SRmax problem: The algorithm generates a nondecreasing objective sequence because each iterate remains feasible for the next approximation, and convergence follows from bounded transmit power.The method converges to a KKT solution of the relaxed problem.
  • A. Joint Codeword Selection and Precoder Design for SRmax problem: The nonzero diagonal entries of the lifted selection matrix identify selected virtual antennas, which determine the selected RF codewords and reduced effective channels.The number of selected entries is denoted Lλ.

B. Sparse Parameter for SRmax problem

The sparse parameter λ balances sum-rate maximization against the number of selected RF codewords. A one-dimensional search chooses λ large enough to satisfy the RF-chain limit while retaining rate performance.

  • B. Sparse Parameter for SRmax problem: Increasing λ promotes sparsity and reduces the number of selected RF chains, but excessive sparsity can conflict with user target-rate requirements.The parameter must therefore balance system sum rate and RF-chain usage.
  • B. Sparse Parameter for SRmax problem: Because system sum rate increases with the number of RF chains, the algorithm seeks the minimum λ that satisfies the RF-chain constraint.This search can use classical one-dimensional methods such as bisection.
  • B. Sparse Parameter for SRmax problem: Algorithm 2 uses iterative SRmax solutions and temporary objectives to adjust λ until the selected-codeword count is at most S.The procedure updates lower and upper search bounds according to whether Lλ exceeds S.
  • B. Sparse Parameter for SRmax problem: The search terminates when objective change is within ζ and the RF-chain constraint is satisfied.The output is the corresponding sparse JWSPD solution.

C. Refined Solution for SRmax Problem

The SRmax refinement reduces the problem size by omitting antennas associated with zero entries, then iteratively solves convex approximations and refines the resulting beamformers.

  • Convex approximation: The resulting nonconvex constraints are approximated through a sequence of convex lower-bound problems.The iterations use the mixed ℓ1,∞-norm approximation and update the constraints at iteration I.
  • Beamformer reconstruction: The final beamforming vectors are reconstructed from the obtained uplink-related quantities and refined to fit the original problem.The procedure derives the power vector and combines it with the beamforming solution to obtain an explicit transmit beamformer.
  • Problem reduction: The refined SRmax procedure omits antennas corresponding to zero diagonal entries in the approximated sparse solution Z.This yields a size-reduced SRmax problem before subsequent optimization.
  • Initialization: The initialization uses a virtual uplink formulation, where dual uplink powers and combiners support transmit beamforming initialization.The virtual uplink problem is solved with an iterative power-allocation and beamformer procedure.

V. JOINT CODEWORD SELECTION AND PRECODER OPTIMIZATION FOR EEMAX PROBLEM

The EEmax problem is harder because its fractional objective and RF-chain sparsity create an NP-hard optimization over sparse patterns. The proposed solution combines sparsity regularization, structural simplification, and successive convex approximation.

  • Problem formulation: EEmax is more difficult than SRmax because the ℓ0-(quasi)norm appears in both the constraint and objective denominator.Finding the global solution requires searching possible sparse patterns, and each pattern yields an NP-hard problem.
  • Sparsity modeling: The method uses a mixed ℓ1,∞-norm squared to approximate the nonconvex ℓ0-(quasi)norm and control RF-chain sparsity.A tunable sparse parameter λ regulates the sparsity of the solution.
  • Relaxation: Dropping the nonconvex rank constraints produces a relaxed formulation with dynamic power consumption represented through the sparse variables.The resulting formulation introduces auxiliary variables before convex approximation.
  • Structural simplification: Theorem 2 shows that setting Xi,j = 0 for i ≠ j preserves optimality while simplifying the EEmax formulation.The diagonal solution remains optimal after the off-diagonal variables are set to zero.
  • Convex solution: Successive convex approximation replaces the nonconvex inequalities with convex constraints, yielding a convex program solvable by a procedure similar to Algorithm 1.The approximation is applied at each iteration I.

B. Sparse Parameter for EEmax problem

The sparse parameter λ controls the number of selected RF chains, but energy efficiency is non-monotonic and piecewise in λ. The paper therefore searches λ using dynamic interval compression.

  • Sparse-parameter behavior: A larger λ produces a sparser approximated EEmax solution, while excessively large λ can drive the solution toward zero.The parameter must therefore be selected rather than increased without bound.
  • Search challenge: Energy efficiency is not monotonic in either the number of RF chains or λ, so classical bisection cannot optimize λ.The system EE is described as a piecewise function of λ.
  • RF-chain transitions: The search spans RF-chain counts from LMin to LMax and defines key λ values at transitions between adjacent nonzero-diagonal counts.These key values support the interval-based search over the continuous sparse parameter.
  • Dynamic interval compression: The paper introduces dynamic interval compression to search for a suitable λ and obtain the corresponding codewords.The method exploits intervals whose endpoint solutions have the same number of nonzero diagonal entries.
  • Algorithm 5: Algorithm 5 initializes lower and upper λ values, solves the associated problems, and iteratively generates selected codeword-index sets.It merges overlapping intervals and records candidate RF-chain sets during the search.

C. Refined Solution for EEmax problem

After selecting codewords, the EEmax problem is reduced to the corresponding channel dimension and solved through a convex iterative approximation, while retaining the selected RF-chain structure.

  • RF-chain selection: The sparse RF-chain selection relies on the mixed ℓ1,∞-norm squared introduced for RF-chain selection.The selected structure is carried into the reduced optimization rather than directly extracting the energy-efficient beamforming vector.
  • Reduced-size formulation: The selected codewords determine a reduced-size channel for the subsequent EEmax optimization.The reduction is constructed from the codewords generated by Algorithm 5.
  • Reduced-size formulation: The reduced-size EEmax problem retains the dynamic power term associated with the selected RF-chain configuration.The formulation includes Pdyn as part of the energy-efficiency objective.
  • Algorithm 5: Algorithm 5's generated codeword sets provide the inputs for the final reduced-size EEmax optimization.The algorithm's later stages enumerate λ values and solve the associated problems for candidate configurations.
  • Iterative optimization: The convex approximated reduced-size problem is solved by a procedure similar to Algorithm 3.The formulation is iterated using iteration index I.

VI. NUMERICAL RESULTS

Numerical experiments compare hybrid precoding designs with fully digital precoding for spectral-rate and energy-efficiency objectives under varied antenna, RF-chain, and user configurations. The proposed design achieves the highest hybrid sum rate, while energy efficiency depends strongly on codebook and hardware configuration.

  • Simulation setup: Simulations average results over 1000 random channel realizations and compare hybrid designs with a fully digital precoder under a common total-power constraint.The setup uses a half-wavelength uniform linear array and specified circuit-power parameters.
  • Spectral efficiency: The proposed hybrid precoding design achieves the highest sum rate among the evaluated hybrid RF-baseband precoders.The authors attribute this to greater flexibility in achieving beam diversity gain.
  • Spectral efficiency: Fully analog beamforming has the worst sum-rate performance because inter-user interference cannot be effectively suppressed.
  • RF-chain selection: As λ increases, selected RF chains decrease at breaking points; between them, the selected set stays fixed while χ decreases.The same RF-chain set is selected over an interval of λ, consistent with Theorem 3.
  • Energy efficiency: The DFT codebook provides better energy efficiency than the 802.15.3c codebook, while both produce similar sum-rate performance.
  • Trade-offs: Hybrid precoders incur some sum-rate loss relative to fully digital precoding because the architecture may not fully exploit multipath diversity gain.Circuit power also increases with the number of phase shifters and mixers, linking energy efficiency to antenna and RF-chain counts.
  • Energy efficiency: For M = N = 32, S = 4, and K = 2, hybrid precoders achieve better energy efficiency than the fully digital precoder.The fully digital design uses M = 32 RF chains versus S = 8 RF chains for the hybrid architecture in the cited comparison.

VII. CONCLUSIONS

The paper develops codebook-based hybrid RF-baseband precoding for multiuser mmWave downlink systems, targeting both sum rate and energy efficiency. Beam sweeping supplies channel information, the design reduces optimization to JWSPD problems, and numerical simulations validate the proposed approach.

  • The study designs hybrid RF-baseband precoding for multiuser multi-antenna downlink systems to maximize system sum rate and energy efficiency.
  • A codebook-based RF precoding method obtains channel state information through a beam-sweep procedure.
  • The codebook design transforms the hybrid-precoder optimization into joint codeword selection and precoder design problems.The paper denotes these problems as JWSPD problems.
  • Efficient methods address the JWSPD problems for maximizing both system sum rate and energy efficiency.
  • Extensive numerical simulations validate the effectiveness of the proposed hybrid precoding design.
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