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Near-Optimal Hybrid Processing for Massive MIMO Systems via Matrix Decomposition

Weiheng Ni, Xiaodai Dong, Wu-Sheng Lu

arXiv:1504.03777v1cs.IT

TL;DR

Massive MIMO hybrid processing addresses the cost of many RF chains by combining low-dimensional digital processing with phase-shifter-based analog processing. The paper decomposes unconstrained digital precoders and combiners using alternate optimization and an SVD-based initialization, achieving near-optimal performance in simulations across Rayleigh and mmWave channels.

  • Problem

    Traditional massive MIMO processing requires many RF chains, while hybrid processing must satisfy constant-amplitude analog RF constraints that make direct matrix decomposition non-convex.

  • Method

    The MD-HP scheme decomposes unconstrained digital precoders and combiners through alternating optimization, restricted phase updates, and an SVD-based initial point.

  • Results

    The MD-HP scheme achieves near-optimal spectral efficiency with sufficient RF chains and outperforms spatially sparse processing in the evaluated mmWave setting.

  • Takeaways & Limitations

    MD-HP applies to general massive MIMO channels and can sufficiently approach unconstrained digital processing when enough RF chains are provided.

Abstract

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For the practical implementation of massive multiple-input multiple-output (MIMO) systems, the hybrid processing (precoding/combining) structure is promising to reduce the high cost rendered by large number of RF chains of the traditional processing structure. The hybrid processing is performed through low-dimensional digital baseband processing combined with analog RF processing enabled by phase shifters. We propose to design hybrid RF and baseband precoders/combiners for multi-stream transmission in point-to-point massive MIMO systems, by directly decomposing the pre-designed unconstrained digital precoder/combiner of a large dimension. The constant amplitude constraint of analog RF processing results in the matrix decomposition problem non-convex. Based on an alternate optimization technique, the non-convex matrix decomposition problem can be decoupled into a series of convex sub-problems and effectively solved by restricting the phase increment of each entry in the RF precoder/combiner within a small vicinity of its preceding iterate. A singular value decomposition based technique is proposed to secure an initial point sufficiently close to the global solution of the original non-convex problem. Through simulation, the convergence of the alternate optimization for such a matrix decomposition based hybrid processing (MD-HP) scheme is examined, and the performance of the MD-HP scheme is demonstrated to be near-optimal.

I. INTRODUCTION

Massive MIMO offers large-array gains but traditional architectures require many RF chains, motivating hybrid analog/digital processing. The paper decomposes unconstrained precoders and combiners into phase-constrained RF and low-dimensional baseband components.

  • Massive MIMO systems typically require hundreds of antennas, making conventional one-RF-chain-per-antenna architectures costly.
  • Hybrid processing reduces RF-chain dimensionality by combining phase-only analog RF processing with low-dimensional digital baseband processing.
  • The paper designs multi-stream P2P hybrid precoders and combiners by directly decomposing pre-designed unconstrained digital processing matrices.
  • The transceiver supports Ns data streams under Ns ≤ Mt ≤ Nt and Ns ≤ Mr ≤ Nr, with baseband processing before RF precoding and combining.
  • RF precoders and combiners use constant-amplitude entries, whereas baseband matrices can modify amplitudes and phases.

B. Channel Model

The study evaluates hybrid precoders and combiners using large Rayleigh fading and limited-scattering mmWave channel models. The mmWave model uses clustered paths with angular parameters and array-response vectors, simulated primarily with uniform linear arrays.

  • The simulations seek optimal hybrid precoders and combiners based on a general channel matrix H.
  • Two channel models are examined: large Rayleigh fading with i.i.d. CN(0, 1) entries and a limited-scattering mmWave channel.
  • The mmWave model represents clustered propagation through complex path gains, azimuth AoA/AoD angles, and receive/transmit array-response vectors.
  • Simulations use uniform linear arrays, while the proposed precoding scheme can directly apply to arbitrary antenna arrays.
  • The mmWave channel model considers only 2D beamforming, and its array responses are specified using the carrier wavelength and adjacent-element spacing.

III. HYBRID PRECODING/COMBINING DESIGN FOR A GENERAL MASSIVE MIMO CHANNEL

The section formulates hybrid precoder and combiner design as matrix decomposition of optimal unconstrained solutions, then solves the non-convex decomposition through alternating updates and local convexification.

  • Problem formulation: Hybrid precoding and combining maximize spectral efficiency under constant-amplitude RF constraints and digital baseband constraints.The joint optimization is generally intractable because the RF feasible regions are non-convex.
  • Problem formulation: The unconstrained precoder and combiner formed from dominant singular vectors provide an upper bound on achievable hybrid spectral efficiency.The first Ns right and left singular vectors of the channel construct the unconstrained solutions.
  • Matrix decomposition: The design approximates the unconstrained precoder and combiner by products FRFB and WRWB rather than requiring exact equality.A residual between the unconstrained and hybrid products is unavoidable under the hybrid structure.
  • Matrix decomposition: Mutual-information maximization is approximately equivalent to minimizing the Frobenius decomposition error ||F⋆−FRFB||F.After precoder optimization, the hybrid combiner is designed to further increase spectral efficiency.
  • Alternate optimization: Alternate minimization fixes one parameter set while optimizing the other, alternating between digital baseband and analog RF variables.The digital update is convex, while the RF update is locally convexified by restricting phase increments around the preceding iterate.
  • Alternate optimization: Small phase-increment restrictions transform the RF update into a convex quadratic program with a unique global solution within the restricted neighborhood.The phase update uses a local approximation of the complex exponential and imposes linear bounds on each increment.

1) An Error Measure:

The algorithm controls convergence through an error-based stopping rule and adaptive phase-increment thresholds, while recognizing that neighborhood restrictions can slow convergence.

  • 1) An Error Measure:: Alternate iterations stop when the change in the normalized decomposition error falls below a prescribed convergence tolerance.The last iterate is then taken as the solution of the relaxed decomposition problem.
  • 1) An Error Measure:: Restricting RF phase increments to a small neighborhood can affect the algorithm’s convergence rate.Each phase remains within the effective range [0, 2π), and the threshold controls the permitted local movement.
  • 1) An Error Measure:: The phase-increment threshold increases when error is decreasing far from tolerance and decreases when error rises or approaches tolerance.This adaptation permits larger steps during productive progress and smaller steps when precision or correction is needed.
  • 1) An Error Measure:: The threshold strategy compensates for invalid local exponential approximations caused by overly large phase increments.Reducing the threshold also supports higher precision near the required error tolerance.

3) Re-formulation of Problem (19):

The RF-update formulation is decomposed into independent row-wise convex quadratic programs so standard interior-point solvers can handle it explicitly.

  • 3) Re-formulation of Problem (19):: The reformulated objective separates into Nt sub-problems, one for each row index p.Each sub-problem is an explicitly formulated convex quadratic program.
  • 3) Re-formulation of Problem (19):: Interior-point algorithms can efficiently solve each row-wise convex quadratic program.The decomposition preserves convexity while making the problem compatible with standard solvers.
  • 3) Re-formulation of Problem (19):: A suitable initial point is important because the original non-convex problem may contain multiple local minimizers.The likelihood of reaching a global or good local minimizer depends strongly on initialization.

4) Choosing an Initial Point:

The initialization uses an SVD-based decomposition of the unconstrained precoder, then modifies it to satisfy the RF constant-amplitude constraint while retaining its phases.

  • 4) Choosing an Initial Point:: Ignoring the RF amplitude constraint, the unconstrained precoder admits an exact SVD-based matrix decomposition.This decomposition motivates constructing an initialization from the SVD of F⋆.
  • 4) Choosing an Initial Point:: The auxiliary RF columns are generated with amplitude 1/Nt and uniformly distributed phases over [0, 2π).These columns supply the dimensions needed for the hybrid RF and baseband factors.
  • 4) Choosing an Initial Point:: The unconstrained SVD construction is globally optimal for the decomposition objective before imposing constant-amplitude RF constraints.The resulting factorization is infeasible for the constrained problem but provides a target for initialization.
  • 4) Choosing an Initial Point:: The feasible initial RF matrix retains the SVD factor phases while forcing the corresponding amplitudes to 1/Nt.This modification produces a feasible point close to the unconstrained SVD-based factor.
  • 4) Choosing an Initial Point:: The baseband normalization step preserves consistent transmission power after precoding.The normalized factors are used in the step-by-step hybrid precoder design.

B. Hybrid Combiners Design

The paper extends matrix-decomposition hybrid processing to receiver combiners by decomposing the unconstrained combiner into RF and baseband components. It uses linear combining rather than minimum-distance decoding and applies the same alternating optimization procedure.

  • Receiver processing: Linear combining is employed at the receiver because Nr-dimensional minimum-distance decoding is difficult to implement due to its high complexity.If the hybrid precoder matched the unconstrained optimum, the corresponding unconstrained combiner would be U1.
  • Alternating optimization: Algorithm 1 iterates until the change in its error measure satisfies the prescribed stopping condition, then returns FR and FB.The supplied algorithm excerpts show the stopping loop and returned hybrid precoders.
  • Hybrid combiners design: The hybrid precoder design is adapted to decompose the unconstrained combiner W⋆ into hybrid combiners WR and WB.The decomposition follows the same alternate optimization method used for the precoder.
  • Evaluation: The paper states that the proposed hybrid processing scheme is evaluated through simulations after extending the decomposition procedure to combiners.The evaluation targets the performance of the overall hybrid processing scheme.

C. Approach To Waterfilling Spectral Efficiency

Waterfilling is incorporated by applying power allocation to the unconstrained precoder before matrix decomposition. Streams receiving zero power are represented through baseband precoding rather than RF phase shifts.

  • Waterfilling processing: Waterfilling updates the unconstrained precoder to F⋆ = V1Γ while leaving the unconstrained combiner as W⋆ = U1.Γ is a diagonal matrix that performs waterfilling power allocation.
  • Zero-power streams: When waterfilling assigns zero power to lower-singular-value streams, the precoder can be written as F∗ = [F′, 0].F′ contains the non-zero columns after power allocation.
  • Zero-power streams: The non-zero portion F′ is decomposed first, and the complete precoder decomposition then appends the zero-power columns.This yields F∗ = [F′, 0] = FRFB.
  • Zero-power streams: Zero-power allocation is realized through baseband precoding rather than phase shifting in the RF domain.The baseband component represents the zero columns in the complete decomposition.

D. Quantized RF Phase Control

The paper addresses practical RF phase control by quantizing phase-shifter settings and examines convergence of the alternating optimization under adaptive and constant phase-increment thresholds. The adaptive threshold enables larger early updates and progressively finer adjustments near convergence.

  • Quantized RF phase control: Practical RF precoders and combiners use quantized phase values when arbitrary phase assignment is unavailable.Each phase is selected from L-bit candidates using the nearest neighbor under Euclidean distance.
  • Convergence setup: The convergence study uses a 256 × 64 MIMO system with Ns = 4 and Mt = 6, initializing the phase increment threshold at 0.1.The initial RF precoder is selected using the SVD-based initialization technique.
  • Adaptive thresholding: The adaptive threshold increases phase increments when error changes remain large and reduces them when the algorithm approaches convergence.The threshold is adjusted according to differences between successive error indicators.
  • Adaptive thresholding: The phase increment threshold is clamped to the interval [0.1, 0.5] to avoid excessively small or large feasible regions.The paper notes that the listed parameters can be revised for other specific cases.
  • Convergence results: In the trace example, the iterate begins with relatively large updates and gradually approaches e^-j1.0026 ≈ 0.5381−j0.8429.The two displayed traces update simultaneously and remain close to each other.
  • Convergence results: The error measure converges to about 0.2, while adaptive phase increments converge more quickly than constant increments.Adaptive updates use larger steps initially and then move closer to the solution as iterations proceed.

B. Spectral Efficiency Evaluation

The spectral-efficiency evaluation compares MD-HP with optimal unconstrained SVD processing under large i.i.d. Rayleigh and mmWave channel settings. With 12 RF chains, MD-HP is near-optimal, while using 8 RF chains reduces spectral efficiency by about 3 bps/Hz.

  • Evaluation setup: The simulations evaluate MD-HP under large i.i.d. Rayleigh and mmWave channels across SNR values from −40 dB to 0 dB.The comparison focuses on spectral efficiency.
  • Rayleigh-channel comparison: For Ns = 8 data streams in a 256 × 64 MIMO system, MD-HP is compared with optimal unconstrained SVD-based processing.The comparison includes configurations with 8 and 12 RF chains and quantized versions.
  • Rayleigh-channel comparison: With 12 RF chains at both transmitter and receiver, MD-HP achieves near-optimal performance relative to unconstrained SVD-based processing.This result is reported for the evaluated spectral-efficiency comparison.
  • Rayleigh-channel comparison: Using 8 RF chains, equal to the number of data streams, decreases spectral efficiency by around 3 bps/Hz.The paper reports this decrease relative to the higher-RF-chain MD-HP configuration.

1) Large i.i.d Rayleigh Fading Channels:

In large i.i.d. Rayleigh channels, MD-HP remains close to optimal spectral efficiency with RF chains equal to the data streams and approaches SVD-based processing as RF chains increase. The scheme also outperforms spatially sparse processing in the reported mmWave comparison.

  • MD-HP consistently remains close to optimal spectral efficiency for Ns = 2, 4, and 8 when Mt = Mr = Ns.
  • Increasing RF chains reduces the gap between MD-HP and SVD-based processing; with 12 RF chains, MD-HP is sufficiently close to the SVD benchmark.
  • 2.5 dB loss occurs for the quantized MD-HP versions with L = 2 phase quantization.
  • The simulations examine spectral efficiency for 256 × 64 systems with 8 data streams and either 8 or 12 RF chains in both i.i.d. Rayleigh and mmWave channels.
  • MD-HP outperforms spatially sparse processing with the same number of RF chains in the reported 256 × 64 mmWave system.With 8 RF chains, MD-HP can exceed spatially sparse processing using 12 RF chains.
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