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Joint Transceiver and Large Intelligent Surface Design for Massive MIMO MmWave Systems

Peilan Wang, Jun Fang, Linglong Dai, Hongbin Li

arXiv:2010.05188v1cs.ITeess.SP

TL;DR

The paper addresses spectral-efficiency maximization in LIS-assisted mmWave systems by jointly designing LIS reflection coefficients and hybrid transceiver processing. It develops a manifold-optimization approach that creates a favorable propagation environment, achieving state-of-the-art-comparable performance at lower computational complexity.

  • Problem

    The paper seeks to maximize spectral efficiency in a point-to-point LIS-assisted mmWave system transmitting from a base station to a user equipment.

  • Method

    The paper jointly optimizes LIS reflection coefficients and BS/UE hybrid precoding and combining using a manifold optimization algorithm for the non-convex design problem.

  • Results

    The proposed method creates a favorable propagation environment with a small channel matrix condition number and achieves performance comparable to state-of-the-art algorithms at much lower computational complexity.

  • Takeaways & Limitations

    Exploiting the inherent sparse structure of mmWave channels enables a better tradeoff between spectral-efficiency performance and computational complexity.

Abstract

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Large intelligent surface (LIS) has recently emerged as a potential low-cost solution to reshape the wireless propagation environment for improving the spectral efficiency. In this paper, we consider a downlink millimeter-wave (mmWave) multiple-input-multiple-output (MIMO) system, where an LIS is deployed to assist the downlink data transmission from a base station (BS) to a user equipment (UE). Both the BS and the UE are equipped with a large number of antennas, and a hybrid analog/digital precoding/combining structure is used to reduce the hardware cost and energy consumption. We aim to maximize the spectral efficiency by jointly optimizing the LIS's reflection coefficients and the hybrid precoder (combiner) at the BS (UE). To tackle this non-convex problem, we reformulate the complex optimization problem into a much more friendly optimization problem by exploiting the inherent structure of the effective (cascade) mmWave channel. A manifold optimization (MO)-based algorithm is then developed. Simulation results show that by carefully devising LIS's reflection coefficients, our proposed method can help realize a favorable propagation environment with a small channel matrix condition number. Besides, it can achieve a performance comparable to those of state-of-the-art algorithms, while at a much lower computational complexity.

I. INTRODUCTION

The paper studies joint LIS reflection and hybrid transceiver design for point-to-point mmWave MIMO, targeting higher spectral efficiency under practical hardware constraints. It reformulates the non-convex design using the effective channel structure and develops a manifold-based solution with favorable conditioning and lower complexity.

  • Motivation: mmWave systems face severe path loss and blockage, motivating LIS-assisted links that reshape propagation through programmable reflections.LIS uses many low-cost passive reconfigurable elements whose phase shifts and amplitudes can be controlled.
  • Problem: The central problem is jointly optimizing LIS reflection coefficients and BS active precoding for spectral efficiency in multi-antenna systems.Prior work often focused on SISO or MISO settings, while the paper considers multiple antennas at both BS and UE.
  • Prior limitations: Maximizing the effective channel’s Frobenius norm can produce a large condition number, limiting performance improvement.A large condition number indicates an unfavorable propagation condition because channel power is not uniformly distributed.
  • Prior limitations: Existing alternatives have practical or modeling limitations, including high complexity, single-stream transmission, and reflection design based only on LOS angles.Sequential AO optimization incurs excessively high computational complexity, while other simplifications sacrifice spectral efficiency.
  • Proposed approach: The proposed method decouples LIS and hybrid transceiver design, reformulates reflection optimization through passive beamforming gains, and solves it with manifold optimization.The reformulation exploits the inherent structure of the effective mmWave channel and its composite BS-LIS-UE paths.
  • Results: Simulations show that the method creates a favorable propagation environment with a small condition number and matches state-of-the-art performance at much lower complexity.It significantly improves over the sum-path-gain maximization method and achieves performance similar to the AO-based method.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models a point-to-point LIS-assisted mmWave MIMO downlink with hybrid transceivers and jointly optimizes passive LIS beamforming and transceiver design for spectral efficiency.

  • A. System Model: The system uses an LIS to assist downlink transmission from a multi-antenna BS to a multi-antenna UE.The direct BS–UE link is assumed blocked, although the scheme can be extended when it exists.
  • A. System Model: The BS and UE use hybrid analog/digital precoding and combining with far fewer RF chains than antennas.The BS has Nt antennas and Rt RF chains, while the UE has Nr antennas and Rr RF chains, with Rt ≪ Nt and Rr ≪ Nr.
  • A. System Model: Multiple data streams are transmitted simultaneously, with Ns ≤ min{Rt, Rr}.The transmitted symbols satisfy E[ssH] = I, and a transmit-power constraint is imposed.
  • A. System Model: The LIS contains M passive reflecting elements that combine and re-scatter signals using adjustable phase shifts.The BS–LIS and LIS–UE channels are represented by G and R, respectively, while each element applies phase shift φm.
  • BBW H: The received signal depends on the effective cascade channel Heff ≜ RΦG, hybrid precoding and combining, and additive white Gaussian noise.The paper assumes perfect CSI for joint transceiver and LIS design.
  • BBW H: Channel estimation for LIS-assisted systems is identified as an important and challenging issue, but it is not addressed by the perfect-CSI design assumption.Prior works are cited for channel estimation in LIS-assisted mmWave systems.
  • BBW H: The objective is to maximize achievable spectral efficiency under hybrid hardware constraints.The analog precoder and combiner have constant-modulus entries because they are implemented with analog phase shifters.

B. Channel Model

The channel model exploits sparse mmWave multipath structure and reduces transceiver design to passive beamforming design, while seeking balanced singular values for multi-stream transmission.

  • B. Channel Model: MmWave channels have limited diffraction around obstacles and therefore exhibit sparse multipath structure modeled using the Saleh–Valenzuela framework.The model uses ULAs at the BS and UE and a large UPA of passive elements at the LIS.
  • B. Channel Model: The BS–LIS channel G and LIS–UE channel R are expressed through path gains, array response vectors, and path angles.The response vectors are normalized and specified for ULA and UPA geometries.
  • B. Channel Model: The paper jointly designs hybrid precoding, combining, and passive beamforming matrix Φ to maximize spectral efficiency.The resulting optimization is challenging because the objective and per-element unit-modulus constraints are non-convex.
  • B. Channel Model: Hybrid beamforming with few RF chains can asymptotically approach fully digital performance when the transceiver arrays are sufficiently large.This motivates first obtaining a fully digital solution and then approximating it with hybrid matrices.
  • B. Channel Model: For fixed Φ, the optimal fully digital precoder and combiner follow from the ordered SVD of the effective channel Heff.The SVD provides unitary singular-vector matrices and an ordered diagonal matrix of singular values.
  • B. Channel Model: At high effective SNR, equal power allocation is treated as near-optimal, simplifying the fully digital precoder design.The high SNR is attributed to the massive array gain from the LIS and BS.
  • B. Channel Model: The proposed approach targets more balanced singular values because maximizing a spectral-efficiency bound alone can produce a large condition number.Balanced singular values support multi-stream transmission and are reported to substantially improve spectral efficiency empirically.

III. PROPOSED PASSIVE BEAMFORMING DESIGN METHOD

The passive beamforming design exploits the factorized mmWave channel to relate the effective channel’s SVD to pathwise passive gains, replacing a difficult formulation with a diagonalization-oriented problem.

  • III. Proposed Passive Beamforming Design Method: The method exploits mmWave channel structure to gain insight into the SVD of the effective channel.The channel factors are organized according to ordered path gains from the BS–LIS and LIS–UE links.
  • III. Proposed Passive Beamforming Design Method: The effective channel is represented through array-response factors and a matrix D whose entries are path-gain-weighted passive beamforming gains.The gain dij corresponds to a BS–LIS–UE composite path formed by one BS–LIS path and one LIS–UE path.
  • III. Proposed Passive Beamforming Design Method: For sufficiently large transceiver arrays, the relevant array-response matrices can be treated as orthonormal because of asymptotic orthogonality.This property is stated for ULA and extended to UPA geometries.
  • III. Proposed Passive Beamforming Design Method: The phase-shift vector is designed so that off-diagonal entries of D are small relative to its diagonal entries.The diagonal entries satisfy D(i,i) = αiβidii, while |dij| < τ constrains off-diagonal gains.
  • III. Proposed Passive Beamforming Design Method: Under this structure, the effective channel can be approximated by a truncated SVD, converting the original optimization into a more tractable diagonal-focused problem.The approximation uses an effectively non-square diagonal matrix representation.
  • III. Proposed Passive Beamforming Design Method: Although the off-diagonal constraint may restrict the solution space and potentially reduce spectral efficiency, the method omits it in the final formulation.The paper argues that, for sufficiently large M, solving the simplified problem automatically makes off-diagonal entries small relative to diagonal entries.
  • III. Proposed Passive Beamforming Design Method: Solving the simplified optimization problem is expected to provide an effective solution to the original problem.The paper explicitly states this connection after establishing the large-M behavior of D.

A. Manifold-Based Method for Passive Beamforming Design

The paper solves the unit-modulus passive beamforming problem on a complex circle manifold using Riemannian-gradient updates, retraction, and Armijo line search.

  • A. Manifold-Based Method for Passive Beamforming Design: Manifold optimization is selected to balance computational complexity and performance for the non-convex passive beamforming problem.The unit-modulus search space is modeled as a product of M complex circles, called the complex circle manifold.
  • A. Manifold-Based Method for Passive Beamforming Design: The complex circle manifold is M = SM, where each component has unit magnitude.This directly represents the per-element unit-modulus constraint on the LIS phase-shift vector.
  • A. Manifold-Based Method for Passive Beamforming Design: A line-search procedure updates the phase-shift vector, and retraction maps the tangent-space update back onto the complex circle manifold.The retraction preserves the local gradient behavior at the current point.
  • A. Manifold-Based Method for Passive Beamforming Design: The method computes tangent-space directions and uses the negative Riemannian gradient as the direction of greatest local decrease.The Riemannian gradient is obtained by projecting the Euclidean gradient onto the tangent space.
  • A. Manifold-Based Method for Passive Beamforming Design: The proposed algorithm uses an Armijo step size and stops when the objective-value gap between iterations falls below ε.It then returns the optimized solution v⋆.
  • A. Manifold-Based Method for Passive Beamforming Design: The manifold-based algorithm is guaranteed to converge to a critical point of problem (23).The convergence guarantee is stated for the algorithm summarized in Algorithm 1.

B. Discussions

The discussion shows that the optimized LIS reflection vector produces an approximately diagonal effective structure, with path-gain ordering affecting the solution and high-SNR power allocation becoming nearly uniform.

  • Orthogonality: As M grows, distinct path-related vectors become asymptotically orthogonal under the random AoA and AoD model.The cross-inner products converge to zero when either path index differs.
  • Effective structure: The optimized solution yields exactly zero off-diagonal entries in D under the relaxed phase-shift analysis.This follows from maximizing the reformulated objective while suppressing cross terms.
  • Effective structure: The resulting D is approximately diagonal, with off-diagonal entries small relative to its diagonal entries.The diagonal terms are simultaneously encouraged to approach the aggregate path-gain scale.
  • Power allocation: In the high-SNR regime, equal power allocation across data streams is approximately optimal.The discussion attributes this regime to large effective SNRs enabled by the system dimensions.
  • Path-gain ordering: Descending path-gain order maximizes the relevant rearrangement expression and motivates sorting both gain sets in decreasing order.The paper notes that different orderings can produce different solutions and performance before justifying the descending arrangement.

IV. HYBRID PRECODING/COMBINING DESIGN

The hybrid precoding and combining design approximates the optimal SVD-based transceiver using manifold optimization over the analog components and alternating updates of analog and baseband matrices.

  • Design procedure: The optimal precoder and combiner are obtained from the SVD of the effective channel after passive beamforming.The resulting matrices are then approximated by hybrid analog/baseband designs.
  • Manifold optimization: The analog precoding and combining matrices are optimized on a complex circle manifold.The search space for the vectorized analog precoder is defined as a complex circle manifold.
  • Optimization: Analog and baseband precoding variables are optimized in an alternating manner using a fast manifold-based method.The corresponding hybrid combining design is analogous.
  • Complexity and convergence: The manifold-based method is guaranteed to converge to a critical point with complexity O(NtRtNsL2).L2 denotes the number of iterations required for convergence.

V. SUMMARY AND DISCUSSIONS

The proposed joint design exploits sparse mmWave channel structure to optimize LIS reflections and hybrid transceivers, achieving a favorable performance–complexity tradeoff and guaranteed convergence.

  • Method: The method is designed specifically for mmWave systems and exploits sparse scattering to approximate the effective channel’s truncated SVD.This distinguishes it from methods not tailored to mmWave hybrid precoding and combining.
  • Algorithm: The algorithm first optimizes LIS reflection coefficients and then solves the hybrid beamforming problem.The first step uses a manifold-based passive beamforming method with per-iteration gradient complexity O(M).
  • Complexity: The overall proposed-algorithm complexity is O(ML1 + NrNt min(Nr, Nt) + NrRrNsL2 + NtRtNsL2).L1 and L2 denote iteration counts for the two optimization stages.
  • Convergence: The two algorithmic steps are executed once rather than through an alternating process, and convergence is guaranteed.Each step is stated to be convergent independently.
  • Comparison: The proposed method has much lower complexity than the AO-based method under the stated practical scaling assumptions.The comparison identifies the proposed method as dominated by O(ML1 + NrNt), versus O(2NrNtMI) for AO.
  • Comparison: Both the WMMSE- and SPGM-based methods scale cubically with the number of reflecting elements M.This comparison is made under the stated assumption M ≥ Nt > Nr > Ns.

COMPUTATIONAL COMPLEXITY COMPARISON

The simulations evaluate the proposed joint transceiver and LIS design in a three-dimensional mmWave setup using a 16×16 LIS and 64-antenna BS and UE arrays.

  • Convergence: The proposed manifold-based algorithm converges within only a few, approximately ten, iterations.This iteration count is reported for the simulation results.
  • Simulation setup: The simulations use a three-dimensional BS–LIS–UE geometry with specified coordinates and 148 m and 9.8 m link distances.The setup places the LIS between the BS and UE in the modeled environment.
  • Simulation setup: The default configuration uses M = 16×16 LIS elements, Nt = Nr = 64 antennas, Rt = Rr = 6, Ns = 4, and L = P = 7.The carrier frequency is 28 GHz with a 251.1886 MHz bandwidth.
  • Baselines: The proposed method is compared with SPGM, AO-based, and WMMSE-based algorithms.SPGM maximizes the Frobenius norm of the effective channel, while WMMSE is adapted from multicell-multiuser settings.

A. Perfect Channel State Information

Under perfect CSI, the proposed T-SVD-BF method provides competitive spectral efficiency while substantially reducing computational cost, and its LIS design produces a better-conditioned effective channel than SPGM.

  • Hybrid transceiver: Hybrid precoding and combining achieve performance very close to fully digital processing with a small number of RF chains.The result supports approaching fully digital beamforming and combining as the number of antennas becomes sufficiently large.
  • Spectral efficiency: T-SVD-BF achieves performance close to the AO-based method while outperforming SPGM and WMMSE in spectral efficiency.The performance advantage over SPGM is attributed to a smaller effective-channel condition number.
  • Convergence: The proposed algorithm reaches the maximum objective-function value within only ten iterations.Its convergence behavior is evaluated for the passive beamforming design problem at ρ = 30dBm.
  • LIS scaling: As M increases, T-SVD-BF spectral efficiency increases and its gap over SPGM and WMMSE becomes more pronounced.The proposed method also outperforms WMMSE in both spectral efficiency and computational complexity.
  • Complexity trade-off: 4% lower spectral efficiency than AO is reported for T-SVD-BF, while its computational efficiency is substantially higher.At M = 192, T-SVD-BF requires 0.1988 second versus about 26.7357 seconds for AO.
  • Complexity trade-off: 0.1988 second versus about 26.7357 seconds: T-SVD-BF requires far less runtime than AO when M = 192.The comparison is presented as evidence of a substantial complexity reduction for LIS-assisted mmWave communications.
  • Channel conditioning: T-SVD-BF maintains a small, nearly unchanged truncated condition number as M increases, unlike SPGM.This indicates a more favorable wireless channel and explains SPGM’s limited gains from additional passive elements.

B. Imperfect Channel State Information

Under channel-estimation errors, the proposed method experiences performance loss similar to AO but remains advantageous over SPGM and WMMSE.

  • Robustness to estimation errors: The proposed method does not exhibit higher sensitivity to inaccurate CSI than the other evaluated methods.The comparison is conducted under imperfect channel knowledge because perfect CSI is usually unavailable in practice.
  • Robustness to estimation errors: The proposed method and AO suffer nearly the same performance loss as AoA/AoD estimation error increases.The evaluation models estimation errors as independent and identically distributed uniform random variables.
  • Robustness to estimation errors: The proposed method retains a clear performance advantage over SPGM and WMMSE in the presence of channel-estimation errors.The simulation uses Nt = Nr = 64, M = 256, L = P = 7, and ρ = 50dBm.
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