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Beamforming Design for Large-Scale Antenna Arrays Using Deep Learning

Tian Lin, Yu Zhu

arXiv:1904.03657v2cs.IT

TL;DR

Large-scale mmWave beamforming with limited RF chains and phase-shifter hardware is difficult, especially with imperfect CSI. The paper develops a BFNN that directly optimizes the beamformer using an SE-related objective and two-stage training, achieving improved performance and stronger robustness to imperfect CSI than conventional HBF algorithms.

  • Problem

    Phase-shifter constraints, imperfect CSI, and difficult optimization make large-scale mmWave beamforming challenging.

  • Method

    The paper develops a BFNN that directly outputs the analog beamformer from estimated CSI, using an SE-related loss and two-stage training.

  • Results

    The BFNN significantly outperforms traditional HBF algorithms under imperfect CSI, with larger gains as channel estimates become less accurate.

  • Takeaways & Limitations

    The BFNN provides a DL-based beamforming approach that handles hardware limitation and imperfect CSI, with extensions discussed for broadband, multiple-RF-chain, joint, and multi-user designs.

Abstract

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Beamforming (BF) design for large-scale antenna arrays with limited radio frequency chains and the phase-shifter-based analog BF architecture, has been recognized as a key issue in millimeter wave communication systems. It becomes more challenging with imperfect channel state information (CSI). In this letter, we propose a deep learning based BF design approach and develop a BF neural network (BFNN) which can be trained to learn how to optimize the beamformer for maximizing the spectral efficiency with hardware limitation and imperfect CSI. Simulation results show that the proposed BFNN achieves significant performance improvement and strong robustness to imperfect CSI over the conventional BF algorithms.

I. INTRODUCTION

HBF design for large-scale mmWave arrays offers high beamforming gains at affordable hardware cost, but phase-shifter constraints, non-convex optimization, imperfect CSI, and costly iterative algorithms make the problem difficult. The paper motivates deep learning as a potential approach to address these challenges.

  • HBF provides high beamforming gains for severe mmWave path loss with affordable hardware cost and power consumption.
  • The phase-shifter-based analog beamformer imposes a constant modulus constraint, making HBF optimization especially difficult.
  • Existing model-based algorithms either approximate the original objective or rely on many serial, time-consuming iterations.
  • Deep learning is considered because it can address intractable problems and learn complicated wireless-channel characteristics from many training samples.
  • After offline training, neural networks use limited matrix operations and parallel computation, potentially enabling faster online BF deployment.

B. Novelty and Contributions

The paper introduces a BFNN that directly designs the analog beamformer under phase-shifter hardware constraints, uses a spectral-efficiency-related loss, and trains for robustness to imperfect CSI. The approach is presented for a one-RF-chain setting and is described as extensible to more complex beamforming problems.

  • The BFNN directly outputs an optimized analog beamformer from estimated CSI instead of replacing the phase-shifter beamformer with a digital neural network.
  • The system model is illustrated as an MISO mmWave system with one RF chain.
  • The proposed loss function is closely related to spectral efficiency rather than the conventional mean square error between transmitted and recovered symbols.
  • A two-stage design trains the BFNN offline to approach ideal spectral efficiency from practical channel estimates, then deploys it online under channel-estimation errors.
  • The initial study focuses on analog beamforming for a large-scale antenna array with only one radio-frequency chain.

II. SYSTEM MODEL

The system studies narrowband analog beamforming for a one-RF-chain MISO mmWave link, using spectral efficiency as the optimization objective under hardware and transmit-power constraints. The channel includes one LoS and multiple NLoS paths, while the estimated SNR is assumed accurate and CSI may be imperfect.

  • System configuration: A base station with one RF chain and N_t antennas transmits one data stream to a single-antenna user through scalar digital and phase-shifter-based analog precoders.The transmitted signal is x = v_RFv_Ds.
  • Signal and channel model: The received signal is r = h^Hv_RFv_Ds + n, with circularly symmetric complex Gaussian noise of covariance σ^2.The channel vector h^H links the base station and user.
  • Signal and channel model: The adopted Saleh–Valenzuela channel model contains one line-of-sight path and L−1 non-line-of-sight paths.Each path has a complex gain and an associated base-station antenna array response; the l = 1 term is the LoS component.
  • Scope: Although the letter focuses on narrowband analog beamforming, the design approach can be generalized to broadband MIMO mmWave systems with hybrid beamforming.The broader generalization is stated as a scope extension discussed later in the paper.
  • Optimization objective: Spectral efficiency is selected as the optimization objective for the studied system.The objective is evaluated for the analog beamformer under the system constraints.
  • Optimization constraints: The analog beamformer has constant-modulus entries, while the combined precoder obeys the maximum transmit-power constraint ∥v_RFv_D∥^2 ≤ P.The digital precoder has an optimal closed-form solution under these constraints, after which the analog beamformer is optimized.
  • Imperfect CSI setting: The estimated SNR is assumed equal to the actual SNR, so the study focuses on handling imperfect CSI rather than SNR estimation error.The paper argues that SNR is generally estimated more accurately than CSI.

III. DL MODEL AND DESIGN OF BFNN

This section frames the deep-learning beamforming design problem around imperfect CSI, the BFNN architecture, and deployment complexity. It introduces the organization of the model-design discussion rather than specifying the architecture itself.

  • Section overview: The section introduces challenges in applying deep learning to the beamforming optimization problem with imperfect CSI, then describes the BFNN architecture and analyzes its complexity.The discussion is organized around these three topics.

A. Challenges

The BFNN design must address the analog phase-shifter architecture, choose an appropriate input, enforce the analog beamformer constraint, and define a useful training label. These issues prevent directly applying conventional neural-network communication designs.

  • Architectural constraint: The analog phase-shifter architecture prevents replacing the analog beamformer with a conventional multilayer neural network trained across the communication link.The proposed alternative is to design a BFNN that directly outputs v_RF.
  • Input design: Received baseband signals are unsuitable BFNN inputs because they depend on the analog precoder being optimized and have fewer dimensions than the N_t-dimensional precoder.The input must therefore be chosen differently from many existing deep-learning beamforming works.
  • Output constraint: Most deep-learning frameworks poorly support complex outputs, making it difficult to impose the constant-modulus constraint directly on the BFNN output v_RF.This is an output-design challenge specific to the analog beamformer.
  • Training label: Using a conventionally optimized analog beamformer as the training label would prevent the BFNN from outperforming that traditional algorithm.The beamforming problem therefore requires a label-selection strategy beyond simply copying an existing optimized output.
  • Two-stage design: Figure 2 depicts the two-stage deep-learning hybrid-beamforming design approach, comprising offline training and online deployment.The figure provides the high-level process context for addressing the design challenges.

B. BFNN Architecture

The section introduces the considerations used to address the BFNN design challenges and transitions to the two-stage deep-learning beamforming approach. The supplied passage does not specify the network layers or activation functions.

  • Architecture overview: The subsection first considers the three BFNN design challenges and then describes the two-stage design approach.The supplied text provides the subsection’s organization but not architectural implementation details.

1) Three Specific Considerations: •

The BFNN takes estimated channel and SNR information, directly produces a constrained analog beamformer, and trains without labels using a spectral-efficiency-related loss.

  • The BFNN generates an optimized analog beamforming vector from the channel estimate hest and SNR estimate γest.
  • A Lambda layer converts real-valued phase inputs into a complex-valued vRF that satisfies the constant modulus constraint.Each phase input corresponds to one analog beamforming coefficient phase.
  • The phase-output design directly generates coefficient phases, requiring fewer neurons than generating and normalizing real and imaginary components.
  • The BFNN uses a label-free loss function directly related to spectral efficiency rather than the conventional MSE between transmitted and recovered symbols.
  • Reducing the loss corresponds exactly to increasing the average spectral efficiency.

2) Two-stage Design Approach:

The two-stage design trains the BFNN offline with simulated channel and estimation data, then deploys the fixed network online using imperfect CSI to output the analog beamformer.

  • Two-stage Design Approach: During offline training, simulated channel, pilot, and noise samples are processed by a practical channel estimator to obtain partial CSI.
  • Two-stage Design Approach: During online deployment, the channel estimate from the same estimator is input to the trained BFNN, which outputs the optimized beamformer.
  • Two-stage Design Approach: Perfect CSI is required only to calculate the offline training loss, not during online inference.
  • Two-stage Design Approach: For Nt = 64, the real and imaginary parts of hest are concatenated with γest to form a (2Nt + 1) × 1 real-valued input.
  • Two-stage Design Approach: The network applies dense layers with 256, 128, and 64 neurons, with layer sizes selected through empirical trials.

C. Complexity Analysis

The BFNN has competitive online complexity with conventional HBF methods and is more compatible with parallel acceleration, while simulations show improved spectral efficiency and robustness to CSI errors.

  • 0.15 million FLOPs are required by the BFNN online deployment for Nt = 64.Offline training complexity is excluded from the count.
  • 0.26 million complex multiplications characterize the conventional model-based HBF complexity for Nt = 64, under the stated coefficient assumption.
  • The BFNN has competitive computational complexity with traditional model-based HBF algorithms and uses matrix operations suitable for GPU acceleration.
  • At 8 bits/s/Hz and PNR = 20dB, the BFNN provides around a 1.5dB SNR gain over traditional algorithms, with larger gains at smaller PNRs.
  • The BFNN outperforms traditional HBF algorithms more strongly as the estimated number of channel paths becomes less accurate.
  • The BFNN exhibits stronger robustness to imperfect CSI, with a larger performance gap as channel estimates become less accurate.The paper attributes this to training on mmWave propagation characteristics and the relationship between imperfect CSI and ideal spectral efficiency.

V. DISCUSSION OF THE GENERALITY OF BFNN

The BFNN is presented as a generalizable design framework that can be adapted from the paper’s simple scenario to broadband, multi-RF-chain, joint transmit-receive, and multi-user beamforming problems.

  • The BFNN can extend to broadband mmWave channels by concatenating multi-tap channel vectors and redefining the loss for broadband spectral efficiency.
  • For multiple RF chains, the BFNN can output an Nt × NRF analog beamforming matrix by increasing its output dimensions to NRFNt and using a new loss function.
  • The same extension idea can support joint transmit-receive beamforming and multi-user beamforming.

VI. CONCLUSION AND FUTURE WORK

The paper proposes a deep-learning BF design approach for large-scale mmWave antenna arrays and reports competitive BFNN performance under hardware limitations and imperfect CSI. Future work includes extending BFNN to more complex problems and clarifying the physical meaning of its layers.

  • The proposed BFNN uses specialized Lambda-layer and loss-function designs to handle hardware limitations and imperfect CSI.
  • Simulation results show competitive BFNN performance and provide insights for future beamforming designs.
  • Future work includes extending BFNN to more complex beamforming problems and investigating the physical meaning of each layer.
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