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Alternating Minimization Algorithms for Hybrid Precoding in Millimeter Wave MIMO Systems

Xianghao Yu, Juei-Chin Shen, Jun Zhang, Khaled B. Letaief

arXiv:1601.07340v1cs.IT

TL;DR

Hybrid precoding for mmWave MIMO requires effective designs across fully-connected and partially-connected architectures, whose optimal structures and energy-efficiency trade-offs remain important design questions. The paper develops alternating-minimization algorithms and finds that fully-connected precoders can approach fully digital performance with slightly more RF chains, while partially-connected designs benefit from more RF chains.

  • Problem

    The paper addresses the lack of a settled optimal hybrid-precoder structure and the need to compare spectral- and energy-efficiency trade-offs between fully-connected and partially-connected architectures.

  • Method

    The paper uses alternating minimization to design hybrid precoders for fully-connected and partially-connected structures, including manifold optimization and an orthogonal digital-precoder constraint.

  • Results

    Fully-connected hybrid precoders can approach fully digital performance when the number of RF chains is slightly larger than the number of data streams, while partially-connected designs substantially outperform analog beamforming in spectral efficiency.

  • Takeaways & Limitations

    The results indicate that fully-connected designs need not increase RF chains beyond slightly more than the number of streams, whereas partially-connected designs can benefit spectrally and energetically from relatively many RF chains.

  • Takeaways & Limitations

    The manifold-optimization AltMin algorithm has high computational complexity, and finer convergence and optimality analysis remains for future investigation.

Abstract

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Millimeter wave (mmWave) communications has been regarded as a key enabling technology for 5G networks. In contrast to conventional multiple-input-multiple-output (MIMO) systems, precoding in mmWave MIMO cannot be performed entirely at baseband using digital precoders, as only a limited number of signal mixers and analog-to-digital converters (ADCs) can be supported considering their cost and power consumption. As a cost-effective alternative, a hybrid precoding transceiver architecture, combining a digital precoder and an analog precoder, has recently received considerable attention. However, the optimal design of such hybrid precoders has not been fully understood. In this paper, treating the hybrid precoder design as a matrix factorization problem, effective alternating minimization (AltMin) algorithms will be proposed for two different hybrid precoding structures, i.e., the fully-connected and partially-connected structures. In particular, for the fully-connected structure, an AltMin algorithm based on manifold optimization is proposed to approach the performance of the fully digital precoder, which, however, has a high complexity. Thus, a low-complexity AltMin algorithm is then proposed, by enforcing an orthogonal constraint on the digital precoder. Furthermore, for the partially-connected structure, an AltMin algorithm is also developed with the help of semidefinite relaxation. For practical implementation, the proposed AltMin algorithms are further extended to the broadband setting with orthogonal frequency division multiplexing (OFDM) modulation. Simulation results will demonstrate significant performance gains of the proposed AltMin algorithms over existing hybrid precoding algorithms. Moreover, based on the proposed algorithms, simulation comparisons between the two hybrid precoding structures will provide valuable design insights.

I. INTRODUCTION

mmWave hybrid precoding addresses the RF-chain cost and power constraints that make fully digital precoding infeasible, but its unit-modulus matrix-factorization design remains unresolved. The paper proposes alternating-minimization algorithms for fully-connected and partially-connected structures, with comparisons covering complexity, spectral efficiency, and energy efficiency.

  • mmWave fully digital precoding is infeasible because RF chains require costly, power-hungry mixers and ADCs comparable in number to antenna elements.
  • Hybrid precoding design can be formulated as minimizing the Euclidean distance between hybrid and fully digital precoders under unit modulus constraints.
  • Alternating minimization decouples analog and digital precoder optimization and is applied to both fully-connected and partially-connected architectures.
  • MO-AltMin uses manifold optimization for fully-connected precoding, while PE-AltMin imposes an orthogonal digital-precoder property to reduce complexity.
  • SDR-AltMin directly designs partially-connected hybrid precoders using semidefinite relaxation, offering optimal solutions for both alternating subproblems.
  • The algorithms extend to narrowband and broadband OFDM systems, with simulations reporting near-optimal MO-AltMin solutions and PE-AltMin outperforming OMP.
  • Fully-connected designs approach fully digital performance with RF chains comparable to data streams, whereas partially-connected designs offer energy-efficiency advantages at relatively large RF-chain counts.

RFHFRFFBBs + WH

The mmWave system uses a clustered channel model and hybrid transceiver architecture with analog phase-shifter constraints. Fully-connected mapping maximizes beamforming gain, while partially-connected mapping reduces implementation complexity by connecting each RF chain to fewer antennas.

  • Perfect channel state information is assumed at both transmitter and receiver, and achievable spectral efficiency is evaluated for Gaussian-distributed transmitted symbols.
  • Phase shifters adjust only signal phases, so nonzero analog precoder and decoder entries satisfy unit modulus constraints.
  • Fully-connected mapping sends each RF-chain output to all antennas, whereas partially-connected mapping connects each RF chain to fewer antennas and lowers hardware complexity at some beamforming-gain cost.
  • The Saleh–Valenzuela model represents mmWave propagation with clusters and rays whose gains are modeled as independent complex Gaussian variables.
  • The considered channel uses a uniform square planar array, with array responses determined by antenna spacing, wavelength, and azimuth and elevation angles.
  • The channel model is used in simulations, while the precoder design is stated to apply to more general channel models.

C. Problem Formulation

The paper formulates hybrid precoding as constrained matrix factorization and solves it by alternating digital and analog updates. For the fully-connected structure, manifold optimization handles unit-modulus variables directly, while conjugate-gradient iterations converge to a critical point.

  • C. Problem Formulation: Precoding and decoding have similar mathematical formulations, but precoding additionally includes a power constraint; the paper focuses on precoder design.
  • C. Problem Formulation: The objective approximates spectral-efficiency maximization by minimizing the distance between the fully digital precoder and analog–digital factors under structure-specific unit-modulus feasibility sets.
  • C. Problem Formulation: Alternating minimization decouples the analog and digital variables because jointly optimizing their product with element-wise unit-modulus constraints is highly complicated.
  • III. MANIFOLD OPTIMIZATION BASED HYBRID PRECODING FOR THE FULLY-CONNECTED STRUCTURE: For the fully-connected structure, every analog-precoder entry has unit modulus, so the feasible set is non-convex and is treated as a Riemannian manifold.
  • III. MANIFOLD OPTIMIZATION BASED HYBRID PRECODING FOR THE FULLY-CONNECTED STRUCTURE: With at least twice as many RF chains as data streams, the Frobenius-norm objective can be zero; this paper instead focuses on Ns ≤ Nt_RF < 2Ns.
  • A. Digital Baseband Precoder Design: With a fixed analog precoder, the digital update has a least-squares solution, while the temporary removal of the power constraint is addressed later.
  • B. Analog RF Precoder Design via Manifold Optimization: The analog search space is a product of complex circles, whose tangent spaces, Riemannian gradients, and retractions support manifold-based optimization.
  • B. Analog RF Precoder Design via Manifold Optimization: A conjugate-gradient method with line search, transport, and retraction updates the analog precoder and is guaranteed to converge to a critical point.

C. Hybrid Precoder Design

The fully-connected hybrid precoder is designed with alternating minimization, while manifold optimization handles unit-modulus constraints and normalization preserves proximity to the optimal digital precoder.

  • MO-AltMin iteratively solves the fully-connected hybrid precoder design using manifold optimization and alternating minimization.The algorithm updates the analog and digital precoders alternately and normalizes the digital precoder to satisfy the power constraint.
  • The normalization step maintains a small distance to the optimal digital precoder when the unnormalized hybrid precoder is already sufficiently close.Lemma 1 bounds the post-normalization distance using the pre-normalization distance.
  • Each alternating-minimization iteration is non-increasing in the objective and therefore converges to a feasible solution.The objective is minimized in the alternating updates and is non-negative, although global optimality for general non-convex problems remains open.
  • MO-AltMin has high complexity because its analog-precoder update uses nested line searches and large Kronecker-product matrices.The resulting matrix dimensions scale with antenna size, producing an exponential increase in computational complexity.

IV. LOW-COMPLEXITY HYBRID PRECODING FOR THE FULLY-CONNECTED STRUCTURE

The low-complexity fully-connected design imposes orthogonality on the digital precoder, enabling closed-form alternating updates for the analog and digital components while reducing complexity relative to manifold optimization.

  • The orthogonal digital-precoder constraint removes the analog–digital product form and substantially simplifies analog-precoder design.The constraint is motivated by the mutual orthogonality of the unconstrained optimal precoder’s columns.
  • PE-AltMin replaces direct objective minimization with an upper-bound objective and uses closed-form analog and digital precoder updates.The analog update extracts phases from an equivalent precoder, while the digital update is obtained through an SVD-related solution.
  • The PE-AltMin analog update uses phase extraction, avoiding the nested conjugate-gradient search required by MO-AltMin.MO-AltMin repeatedly searches the complex-circle manifold and computes large Kronecker-product matrices, whereas PE-AltMin uses a phase-extraction operation.
  • PE-AltMin has much lower complexity than MO-AltMin, as also observed numerically in simulations.The digital updates have comparable closed-form complexity; the main difference comes from the higher-dimensional analog-precoder updates.
  • The approximation’s effectiveness depends on the tightness of the upper bound used instead of the original objective.The paper notes that increasing antenna-related dimensions can enlarge the bound gap and cause performance loss.

V. HYBRID PRECODING FOR THE PARTIALLY-CONNECTED STRUCTURE

For the partially-connected structure, each RF chain connects to a subarray, reducing RF hardware complexity while imposing a block-structured, unit-modulus analog precoder constraint.

  • The partially-connected architecture connects each RF chain only to Nt/N_RF antennas, reducing hardware complexity in the RF domain.It is also called the array-of-subarrays structure and is advocated for energy-efficient mmWave MIMO systems.
  • Its analog precoder consists of block matrices whose nonzero entries are unit-modulus phase-shifter coefficients.Each block has dimension Nt/N_RF and the phase of each phase shifter determines its entries.
  • An alternating-minimization algorithm is developed for this structure, with optimal solutions available for both analog and digital subproblems.

A. Analog RF Precoder Design

The partially-connected analog-precoder subproblem is simplified by its block structure and solved through phase rotation, yielding a closed-form expression for the nonzero RF entries.

  • The analog-precoder design is formulated under the partially-connected feasible set of block matrices.The reformulated problem exploits the special structure of the analog precoder.
  • The reformulated problem becomes a vector approximation using phase rotation, which has a closed-form solution for nonzero analog-precoder entries.
  • The partially-connected structure makes the unit-modulus constraint tractable by simplifying the analog-precoder design.

B. Digital Baseband Precoder Design

The digital baseband precoder design is formulated through alternating minimization, with a semidefinite-relaxation route for solving the partially connected structure's non-convex subproblem.

  • The transmit-side precoder design is rewritten as an optimization problem based on the preceding formulation.
  • The resulting problem is a non-convex quadratic constraint quadratic programming problem that can be reformulated as a homogeneous QCQP.
  • The homogeneous formulation introduces an auxiliary variable and lifts the variables into a complex Hermitian matrix Y = yyH.
  • Semidefinite relaxation drops the non-convex rank constraint on Y to produce a relaxed problem.
  • With fewer than three constraints, the relaxation is tight for the stated complex homogeneous QCQP, yielding an SDP solvable by standard convex optimization algorithms.
  • The SDR-AltMin procedure summarizes the resulting semidefinite-relaxation-based hybrid precoding algorithm for the partially-connected structure.

C. Comparison Between Two Hybrid Precoding Structures

The paper contrasts fully-connected and partially-connected architectures by their phase-shifter counts, spectral-efficiency degrees of freedom, and power-related trade-offs, then extends the AltMin framework to OFDM.

  • Comparison Between Two Hybrid Precoding Structures: The two structures differ primarily in the number of phase shifters used for fixed data streams, RF chains, and antennas.
  • Comparison Between Two Hybrid Precoding Structures: The fully-connected structure has more RF-domain design degrees of freedom and is expected to provide higher spectral efficiency than the partially-connected structure.
  • Comparison Between Two Hybrid Precoding Structures: Energy efficiency is defined as the ratio of spectral efficiency to total power consumption.
  • Hybrid Precoding in mmWave MIMO-OFDM Systems: The OFDM extension uses frequency-domain digital precoding per subcarrier while sharing one common analog precoder across subcarriers after the IFFT.
  • Hybrid Precoding in mmWave MIMO-OFDM Systems: For OFDM, the hybrid-precoder objective approximates the spectral-efficiency design through distances to the optimal digital precoders for each subcarrier.
  • Hybrid Precoding in mmWave MIMO-OFDM Systems: Alternating minimization updates all digital subcarrier precoders in parallel and adapts the analog-precoder step for the common OFDM analog precoder.
  • Hybrid Precoding in mmWave MIMO-OFDM Systems: The proposed AltMin algorithms can be directly extended to mmWave MIMO-OFDM systems for subsequent performance evaluation.

VII. SIMULATION RESULTS

Simulations evaluate the proposed algorithms in clustered mmWave channels and compare spectral-efficiency performance across RF-chain counts and hybrid-precoding structures.

  • Simulation Setup: The simulations use a 144-antenna transmitter, a 36-antenna receiver, five clusters, ten rays per cluster, and 1000 channel realizations.
  • Spectral Efficiency Evaluation: When RF chains equal data streams, the fully-connected MO-AltMin algorithm achieves near-optimal spectral efficiency across the considered SNR range.
  • Spectral Efficiency Evaluation: In the same limited-RF-chain setting, MO-AltMin outperforms the existing OMP algorithm, while SDR-AltMin outperforms the SIC-Based benchmark for the partially-connected structure.
  • Spectral Efficiency Evaluation: When NRF ≥ 2Ns, the proposed algorithm can achieve optimal spectral efficiency, whereas OMP cannot achieve it.
  • Spectral Efficiency Evaluation: The partially-connected structure uses fewer phase shifters but incurs non-negligible spectral-efficiency loss relative to the fully-connected structure.

B. Energy Efficiency Evaluation

The evaluation compares energy efficiency across fully-connected and partially-connected hybrid precoding structures, alongside spectral-efficiency performance of proposed algorithms. The results show a crossover between structures and strong performance from AltMin methods, including in OFDM systems.

  • Energy efficiency comparison: When N_RF increases, fully-connected energy efficiency decreases because power consumption grows faster than spectral efficiency.The spectral efficiency approaches the optimal digital precoder and then stops increasing, while power consumption continues to rise.
  • Energy efficiency comparison: When N_RF increases, partially-connected energy efficiency rises because spectral efficiency improves while dominant power consumption remains almost unchanged.The number of phase shifters is independent of N_RF in this structure.
  • Energy efficiency comparison: At N_RF = 5, the energy-efficiency curves intersect: fully-connected is better with few RF chains, whereas partially-connected is better with many.This crossover provides a design reference for choosing the hybrid-precoding structure and RF-chain count.
  • Low-complexity fully-connected design: PE-AltMin nearly matches MO-AltMin spectral efficiency when N_RF = N_s, while OMP performs poorly under the same setting.The result indicates that the orthogonal-column constraint has negligible spectral-efficiency impact in this case.
  • Low-complexity fully-connected design: For N_s = 6, PE-AltMin has a small gap from MO-AltMin but substantially outperforms OMP, especially with few RF chains.The gap arises because PE-AltMin minimizes an upper bound that becomes looser as N_RF increases.
  • Hybrid precoding in mmWave MIMO-OFDM systems: In mmWave MIMO-OFDM systems, MO-AltMin achieves the highest spectral efficiency, while PE-AltMin nearly matches it when N_RF = N_s and outperforms OMP-based precoding.The OFDM evaluation uses K = 128 subcarriers.
  • Conclusions: Fully-connected precoders approach fully digital performance when the RF-chain count is slightly larger than the data-stream count, limiting the value of further RF chains.Partially-connected precoders benefit from relatively many RF chains by improving spectral and energy efficiency.

APPENDIX A FORMULATION OF THE HOMOGENEOUS QCQP PROBLEM

The appendix reformulates the original non-homogeneous QCQP into a homogeneous form suitable for semidefinite relaxation. Vectorization and an auxiliary variable are used to express the objective and constraints in matrix form.

  • The partially-connected hybrid-precoder design is formulated as a non-homogeneous QCQP.
  • Vectorization rewrites the objective using f = vec(F_opt), b = vec(F_BB), and E = I_Ns ⊗ F_RF.
  • An auxiliary variable t homogenizes the original QCQP before semidefinite-relaxation reformulation.
  • The reformulated problem includes a positive-semidefinite matrix constraint and a rank(Y) = 1 constraint.
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