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Hybrid Beamforming for Millimeter Wave Systems Using the MMSE Criterion

Tian Lin, Jiaqi Cong, Yu Zhu, Jun Zhang, Khaled B. Letaief

arXiv:1902.08343v1cs.IT

TL;DR

The paper tackles joint HBF design for mmWave systems under MMSE objectives, addressing costly hardware, non-convex constant-modulus constraints, and broadband analog-beamformer sharing. It uses alternating optimization with manifold, eigen-decomposition, and matching-pursuit methods, and reports performance approaching full-digital beamforming while reducing complexity through low-complexity alternatives.

  • Problem

    mmWave systems need large arrays for beamforming gains, while full-digital architectures are costly and existing HBF work mainly optimizes spectral efficiency rather than MSE-based transmission reliability.

  • Method

    The paper directly minimizes MMSE using alternating transmit-receive optimization, a virtual full-digital initialization, manifold optimization, GEVD, EVD, and OMP-based algorithms for narrowband and broadband systems.

  • Results

    The proposed algorithms significantly improve over existing HBF methods and approach full-digital performance, while low-complexity methods balance performance and computational complexity.

  • Takeaways & Limitations

    MMSE-based HBF provides a direct design framework for transmission reliability that also extends through WMMSE to spectral-efficiency maximization.

Abstract

from arXiv · show

Hybrid analog and digital beamforming (HBF) has recently emerged as an attractive technique for millimeter-wave (mmWave) communication systems. It well balances the demand for sufficient beamforming gains to overcome the propagation loss and the desire to reduce the hardware cost and power consumption. In this paper, the mean square error (MSE) is chosen as the performance metric to characterize the transmission reliability. Using the minimum sum-MSE criterion, we investigate the HBF design for broadband mmWave transmissions. To overcome the difficulty of solving the multi-variable design problem, the alternating minimization method is adopted to optimize the hybrid transmit and receive beamformers alternatively. Specifically, a manifold optimization based HBF algorithm is firstly proposed, which directly handles the constant modulus constraint of the analog component. Its convergence is then proved. To reduce the computational complexity, we then propose a low-complexity general eigenvalue decomposition based HBF algorithm in the narrowband scenario and three algorithms via the eigenvalue decomposition and orthogonal matching pursuit methods in the broadband scenario. A particular innovation in our proposed alternating minimization algorithms is a carefully designed initialization method, which leads to faster convergence. {Furthermore, we extend the sum-MSE based design to that with weighted sum-MSE, which is then connected to the spectral efficiency based design.} Simulation results show that the proposed HBF algorithms achieve significant performance improvement over existing ones, and perform close to full-digital beamforming.

I. INTRODUCTION

mmWave systems need large antenna arrays for beamforming gains, but full-digital beamforming is costly, motivating HBF designs optimized directly for MMSE across narrowband and broadband settings.

  • Large mmWave antenna arrays compensate severe path and penetration loss, but full-digital beamforming requires one RF chain per antenna and becomes costly and power-hungry.
  • HBF reduces hardware demands by separating beamforming into low-dimensional digital and high-dimensional analog components implemented with phase shifters.
  • The paper addresses joint transmit-receive HBF under non-convex analog constraints and broadband frequency selectivity, where digital beamformers vary by subcarrier but analog beamformers remain invariant.
  • A. Related Works and Motivations: Existing HBF work predominantly maximizes spectral efficiency, while MSE directly characterizes transmission reliability and relates to SINR and error-rate metrics.
  • B. Contributions and Paper Organization: The proposed framework directly optimizes MMSE for narrowband and broadband systems, using alternating minimization, a new initialization method, MO, GEVD, EVD, and OMP-based algorithms.

RFHVRFVBs + WH

The design formulates modified MSE minimization for hybrid transmit and receive beamformers under constant-modulus and power constraints, then alternates between precoder and combiner subproblems.

  • The system assumes perfect CSI and synchronization, with additive Gaussian noise at the receiver.
  • The modified MSE jointly optimizes hybrid beamformers and a scaling factor β, while analog entries satisfy constant-modulus constraints.
  • Introducing β incorporates noise effects into precoder optimization and helps obtain closed-form digital-precoder and MSE expressions.
  • Because the joint problem contains five variables and non-convex constraints, the design separates hybrid precoding and combining and solves them independently.
  • The analog precoder subproblem is reduced to optimizing VRF after deriving optimal digital precoding and β for fixed receive combining.
  • Two analog-precoder algorithms are proposed: manifold optimization and generalized eigenvalue decomposition.

1) Analog Precoder Design Based on the MO Method:

The MO-HBF method handles the analog precoder’s constant-modulus constraint by optimizing on a Riemannian manifold using a derived Euclidean conjugate gradient.

  • The method defines a Riemannian manifold for VRF and iteratively updates it along the projected Euclidean gradient.
  • A complex-valued Euclidean conjugate gradient is derived through differential and trace identities to enable computation of the Riemannian gradient.
  • Algorithm 1 initializes VRF randomly, repeatedly computes the gradient, performs manifold optimization, and stops when its condition is satisfied.
  • After the analog iterations, β and the unnormalized digital precoder VU are computed according to the derived closed-form updates.
  • The MO-HBF procedure is a gradient-based algorithm whose complexity depends on iteration count, objective form, and stopping condition.

2) Analog Precoder Design Based on the GEVD Method:

The GEVD-HBF method lowers complexity by approximating analog-beamformer orthogonality, optimizing columns sequentially through generalized eigenvectors and phase extraction.

  • The method approximates VRF^HVRF as proportional to the identity because optimized analog beamforming vectors for different streams are likely orthogonal.
  • Under this approximation, the analog precoder can be optimized column-by-column after rewriting the MSE using rank-one matrix identities.
  • Algorithm 2 initializes VRF randomly, computes Am, Um, and Wm, updates each column, and then recalculates β and VU.
  • For each column, the optimal unconstrained vector is the eigenvector associated with the maximum generalized eigenvalue of Um and Wm.
  • The constant-modulus constraint is enforced by extracting the phase of every element in the generalized eigenvector.

B. Hybrid Receive Combiner Design

The broadband receive-combiner problem is formulated alongside hybrid precoding and solved within an alternating minimization framework. Different subcarriers use separate digital combiners, while the analog combiner remains common across the band.

  • Hybrid Receive Combiner Design: Fixing the hybrid precoder and scaling factor reduces the optimization to a hybrid receive-combiner problem.The formulation uses H2 = HVRF VU and W = WRF WB.
  • Hybrid Receive Combiner Design: The receive-combiner subproblem has the same form as the corresponding precoder subproblem, enabling direct reuse of the MO-HBF and GEVD-HBF algorithms.This structural equivalence applies the previously introduced optimization procedures to the hybrid combiner.
  • Alternating Optimization: Transmit and receive beamformers are optimized alternately until a stopping condition is satisfied.Each outer iteration updates the hybrid precoder and then the hybrid combiner using the same algorithmic procedure.
  • Beamforming Initialization: The VFD initialization uses an optimal full-digital combiner virtually at one side, significantly improving convergence speed over random initialization.At least one outer iteration is needed to obtain implementable hybrid beamformers on both sides, while the added complexity is much lower than the main algorithms.
  • Broadband Extension: The broadband design uses OFDM, with digital beamformers optimized per subcarrier and analog beamformers invariant across the frequency band.The sum-MSE objective aggregates modified MSE across all subcarriers and streams, subject to constant-modulus analog components.

B. Broadband Hybrid Transmitter Design

In the broadband transmitter design, the digital precoder and scaling factor are optimized separately for each subcarrier while the analog precoder is shared across subcarriers. The resulting reduced problem optimizes the analog precoder after eliminating the digital variables.

  • Broadband Hybrid Transmitter Design: The broadband digital precoder is separated as VB,k = βkVU,k, with a subcarrier-specific scaling factor and unnormalized precoder.This separates amplitude scaling from the unnormalized digital precoding matrix at each subcarrier.
  • Broadband Hybrid Transmitter Design: The optimal subcarrier scaling factor βk is derived before optimizing the unnormalized precoder.The paper states that the optimal scaling follows from the broadband formulation.
  • Broadband Hybrid Transmitter Design: With the hybrid receive combiner fixed, KKT conditions yield the optimal VU,k as a function of the analog precoder VRF.The original broadband problem is thereby reduced to an optimization over VRF.

1) Analog Precoding Based on the MO Method:

The broadband analog precoder can be optimized by manifold optimization or by eigenvalue-based lower- and upper-bound approximations. These alternatives address the constant-modulus constraint with different complexity–approximation trade-offs.

  • Analog Precoding Based on the MO Method: Manifold optimization projects the Euclidean gradient onto the Riemannian tangent space to preserve feasible analog precoders.With an appropriate step size, the method converges to a feasible local optimum.
  • Analog Precoding Based on the EVD Method: The broadband objective cannot be rewritten in the narrowband GEVD form because channel matrices and digital beamformers vary across subcarriers.The sum-MSE across subcarriers creates this structural limitation.
  • Analog Precoding Based on the EVD Method: The EVD-LB-HBF algorithm minimizes a lower bound and extracts the phase of the resulting eigenvector-based solution to satisfy constant modulus.The lower-bound construction uses eigenvalues of positive-definite matrices and Jensen’s inequality.
  • Analog Precoding Based on the EVD Method: The EVD-UB-HBF algorithm instead minimizes an upper bound, obtaining its solution through eigenvalue decomposition and phase extraction.This provides a second low-complexity eigenvalue-based design for the broadband analog precoder.
  • Analog Precoding Based on the EVD Method: The upper-bound derivation relies on an eigenvalue inequality for positive-definite Hermitian matrices and para-unitary projections.The inequality relates eigenvalues of compressed matrices to those involving the inverse matrix.

3) Analog Precoding Based on the OMP Method:

The OMP-based broadband design restricts the analog precoder to a dictionary of array-response vectors and greedily selects vectors correlated with the residual error. The framework also extends the MMSE design toward spectral-efficiency optimization through WMMSE.

  • Analog Precoding Based on the OMP Method: The OMP-MMSE-HBF algorithm restricts the analog precoder to a finite set of array-response basis vectors.This converts the analog design into a sparse selection problem with NRF nonzero rows in the expanded digital representation.
  • Analog Precoding Based on the OMP Method: OMP selects dictionary columns most strongly correlated with the residual error to construct the analog precoder.The digital precoder is obtained in closed form during the iterative selection process.
  • Extension to Spectral Efficiency: The WMMSE extension connects weighted sum-MSE minimization with spectral-efficiency maximization in both narrowband and broadband settings.The paper reports better or comparable spectral efficiency to conventional HBF methods in simulations.
  • Extension to Spectral Efficiency: The spectral-efficiency formulation uses hybrid precoders and combiners while enforcing constant-modulus analog entries and a transmit-power constraint.The achievable rate is expressed through a log-determinant involving the hybrid precoder and combiner.
  • Extension to Spectral Efficiency: WMMSE optimization proceeds by alternately optimizing the combiner, weighting matrix, and precoder-related variables.The first step optimizes W while fixing Λ and V, and its digital combiner has the same form as in the MMSE design.

2 WRF

The broadband WMMSE formulation introduces subcarrier-specific weighting and MSE matrices, while adapting the proposed HBF procedures to the weighted objective under constant-modulus constraints.

  • The WMMSE optimization iteratively updates the weighting matrix and beamformers, using closed-form weighting updates and the existing MO-HBF or GEVD-HBF procedures.The weighting matrix is updated as Λ = T−1, and the beamformer subproblem retains the same form with Λ replacing the identity matrix.
  • The broadband WMMSE problem uses a weighting matrix Λk and MSE matrix Tk for each subcarrier.
  • The MO-HBF and EVD-LB-HBF algorithms extend directly to WMMSE with slight modification because Λk is the only difference.
  • The EVD-UB-HBF algorithm cannot be generalized to WMMSE because Lemma 3 does not hold with weighting matrices.

VI. SYSTEM EVALUATION

The proposed algorithms differ in convergence guarantees and computational cost. MO-HBF has a strict convergence proof, whereas the other methods rely partly on approximations or empirical convergence, with complexity evaluated by multiplication counts.

  • VI. SYSTEM EVALUATION: Digital beamformer updates have closed-form KKT solutions and always decrease the objective for fixed analog beamformers.
  • A. Convergence: MO-HBF strictly converges because manifold optimization drives the objective downward until the gradient reaches zero.
  • A. Convergence: GEVD-HBF lacks a strict convergence proof because its derivation uses orthogonal approximations and phase extraction.
  • A. Convergence: EVD-UB-HBF and EVD-LB-HBF do not ensure monotonic objective decrease because of approximation, bounding, and phase-extraction operations.
  • A. Convergence: WMMSE preserves strict convergence for MO-HBF, while convergence of the other WMMSE algorithms remains unproved despite simulation evidence.
  • B. Computational Complexity: WMMSE complexity is effectively the same as MMSE complexity because the additional Ns × Ns weighting-matrix computation is negligible.
  • B. Computational Complexity: MO-HBF complexity scales with NoutNin, while GEVD-HBF can reduce generalized-eigenvector computation to O(NpNant^2) using the power method.

2) Broadband algorithms:

Broadband evaluation considers frequency-selective complexity and simulation settings for comparing proposed HBF algorithms with existing and full-digital methods.

  • 2) Broadband algorithms:: Broadband MO-HBF complexity is approximately N times its narrowband complexity.
  • 2) Broadband algorithms:: EVD-LB-HBF and EVD-UB-HBF complexity includes subcarrier-dependent matrix computation and eigenvalue decomposition, reducible through the power method.
  • 2) Broadband algorithms:: OMP-MMSE-HBF complexity is dominated by computing subcarrier-dependent terms and scales with the number of outer iterations.
  • 2) Broadband algorithms:: Complexity results are summarized in Table I, alongside average inner and outer iteration counts from 1000 independent channel realizations.
  • VII. SIMULATION RESULTS: Simulations use clustered mmWave channels with NC = 5 clusters, NR = 10 rays per cluster, Nt = Nr = 64, perfect channel estimation, and uncoded QPSK.
  • VII. SIMULATION RESULTS: Figure 3 evaluates average MSE against outer iterations at SNR = −16dB in the narrowband scenario.
  • VII. SIMULATION RESULTS: Figure 4 compares BER against SNR for HBF algorithms with NRF = Ns = 2 in the narrowband scenario.

A. Performance in the Narrowband Scenario

In the narrowband evaluation, the proposed HBF algorithms converge quickly, substantially improve BER over OMP, approach full-digital performance, and slightly improve spectral efficiency under WMMSE.

  • Convergence: The VFD initialization lets both proposed algorithms converge within a few outer iterations, with some MSE improvement.It initializes the hybrid receive combiner using a virtual full-digital combiner.
  • BER performance: The proposed MO-HBF and GEVD-HBF algorithms significantly outperform OMP in BER and approach full-digital performance within 1dB.GEVD-HBF performs closely to MO-HBF while offering a low-complexity alternative.
  • Spectral efficiency: Under the WMMSE criterion, the proposed narrowband algorithms perform slightly better in spectral efficiency than the conventional HBF algorithms.The paper attributes this to WMMSE's connection to maximum spectral efficiency and alternating optimization between transmit and receive sub-problems.
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