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Hybrid Analog and Digital Beamforming for mmWave OFDM Large-Scale Antenna Arrays

Foad Sohrabi, Wei Yu

arXiv:1711.08408v1cs.IT

TL;DR

The paper addresses hybrid beamforming for broadband OFDM mmWave systems with frequency-selective channels, where common analog beamformers and subcarrier-specific digital beamformers are desirable. It develops asymptotic and practical designs for SU-MIMO and a heuristic downlink design for MU-MISO, showing performance close to fully-digital beamforming in supported settings.

  • Problem

    Prior hybrid beamforming work largely considers narrowband channels, while broadband mmWave OFDM systems require common analog beamforming across frequency-selective subcarriers.

  • Method

    The paper proposes an asymptotic SU-MIMO design, a unified heuristic algorithm for fully-connected and partially-connected architectures, and a heuristic MU-MISO hybrid precoding design.

  • Results

    The hybrid designs asymptotically realize or approach optimal fully-digital beamforming, while the proposed algorithms outperform existing methods and achieve near-fully-digital performance with fewer RF chains in supported settings.

  • Takeaways & Limitations

    Hybrid beamforming can support broadband frequency-selective mmWave systems while reducing RF-chain requirements, with architecture choice trading beamforming capability against hardware complexity and power consumption.

Abstract

from arXiv · show

Hybrid analog and digital beamforming is a promising candidate for large-scale mmWave MIMO systems because of its ability to significantly reduce the hardware complexity of the conventional fully-digital beamforming schemes while being capable of approaching the performance of fully-digital schemes. Most of the prior work on hybrid beamforming considers narrowband channels. However, broadband systems such as mmWave systems are frequency-selective. In broadband systems, it is desirable to design common analog beamformer for the entire band while employing different digital beamformers in different frequency sub-bands. This paper considers hybrid beamforming design for systems with OFDM modulation. First, for a SU-MIMO system where the hybrid beamforming architecture is employed at both transmitter and receiver, we show that hybrid beamforming with a small number of RF chains can asymptotically approach the performance of fully-digital beamforming for a sufficiently large number of transceiver antennas due to the sparse nature of the mmWave channels. For systems with a practical number of antennas, we then propose a unified heuristic design for two different hybrid beamforming structures, the fully-connected and the partially-connected structures, to maximize the overall spectral efficiency of a mmWave MIMO system. Numerical results are provided to show that the proposed algorithm outperforms the existing hybrid beamforming methods and for the fully-connected architecture the proposed algorithm can achieve spectral efficiency very close to that of the optimal fully-digital beamforming but with much fewer RF chains. Second, for the MU-MISO case, we propose a heuristic hybrid percoding design to maximize the weighted sum rate in the downlink and show numerically that the proposed algorithm with practical number of RF chains can already approach the performance of fully-digital beamforming.

I. INTRODUCTION

The paper addresses broadband frequency-selective mmWave channels, where hybrid beamforming must share analog weights across subcarriers while using subcarrier-specific digital beamformers. It develops asymptotic and practical designs for SU-MIMO and MU-MISO systems.

  • Fully-digital beamforming is impractical for large arrays because it requires one RF chain per antenna, increasing hardware complexity and power consumption.
  • Frequency-selective mmWave channels require a common analog beamformer across subcarriers and digital beamforming weights on a per-subcarrier basis.
  • SU-MIMO: For sufficiently large antenna arrays, channel covariance matrices across subcarriers become approximately identical, enabling shared eigen-beamformers and asymptotic realization of optimal fully-digital beamforming.
  • SU-MIMO: For practical arrays of 32-128 antennas, the proposed unified heuristic designs fully-connected and partially-connected precoders under per-subcarrier power spectral density constraints.
  • SU-MIMO: The analog design uses the average frequency-domain channel covariance, after which closed-form digital beamformers are obtained for each subcarrier.
  • MU-MISO: In MU-MISO, the analog precoder follows the SU-MIMO design, while per-subcarrier WMMSE digital precoding addresses inter-user interference and unequal stream priorities.

C. Paper Organization and Notations

The paper models OFDM hybrid beamforming with subcarrier-specific digital processing and shared analog processing, then distinguishes fully-connected and partially-connected analog architectures.

  • Organization: The paper is organized around SU-MIMO modeling, asymptotic analysis, practical hybrid design, MU-MISO design, simulations, and conclusions.
  • Signal Model: In the transmitter, digital precoding processes each subcarrier before IFFTs, while the post-IFFT analog precoder remains identical across all subcarriers.
  • Signal Model: The receiver applies a common analog combiner before FFT processing and then uses a low-dimensional digital combiner separately on each subcarrier.
  • Analog Architectures: The fully-connected architecture connects every RF chain to every antenna through phase shifters, whereas the partially-connected architecture assigns each RF chain to one sub-array.
  • Analog Architectures: The partially-connected structure uses Nt phase shifters, reducing RF-beamformer hardware complexity by a factor of NRF relative to the stated fully-connected count.
  • Analog Architectures: The paper explicitly frames these architectures as a performance-complexity trade-off: full connectivity offers full phase control, while partial connectivity lowers implementation complexity and power consumption.

C. Problem Formulation

The SU-MIMO formulation designs hybrid analog and digital beamformers for fully and partially connected architectures to maximize OFDM spectral efficiency under per-subcarrier power constraints. It uses a sparse geometric mmWave channel model with common analog processing across subcarriers.

  • C. Problem Formulation: The optimization jointly designs transmit and receive hybrid beamformers to maximize overall spectral efficiency under a power spectral density constraint for each subcarrier.
  • C. Problem Formulation: The analog precoder and combiner may use fully connected or partially connected structures, represented by their respective nonzero-element sets.
  • C. Problem Formulation: For Gaussian signaling, each subcarrier’s achievable rate depends on the overall hybrid precoder, combiner, and combiner-induced matrix C[k].
  • D. Channel Model: The mmWave channel is modeled geometrically with Nc scattering clusters and Nsc scatterers per cluster, reflecting a limited number of propagation paths.
  • D. Channel Model: Each path is characterized by a scaled complex gain and arrival and departure angles, while antenna response vectors depend on array structure.
  • D. Channel Model: Antenna response-vector elements typically satisfy a constant-modulus constraint, supporting implementation with analog phase-shifting hardware.
  • D. Channel Model: The model assumes the operating frequency greatly exceeds total OFDM bandwidth, so all subcarriers use approximately the same signal wavelength.
  • D. Channel Model: The channel matrix is also expressed in a compact form for subsequent beamforming analysis.

III. ASYMPTOTIC BEAMFORMING DESIGN FOR SU-MIMO

For frequency-selective mmWave SU-MIMO channels, sparsity makes subcarrier covariance matrices share approximately common eigenvectors as antenna counts grow. Consequently, only Ns RF chains can asymptotically realize optimal fully-digital beamforming.

  • With fixed data streams and scatterers, Ns RF chains can asymptotically realize the optimal fully-digital beamformer as Nt and Nr approach infinity.
  • Channel covariance matrices across subcarriers are approximately similar because of mmWave channel sparsity, so they share approximately common eigenvectors.
  • As antenna counts grow, the receive array-response Gram matrix approaches the identity because its off-diagonal elements become small with high probability.
  • The transmit array-response columns therefore become approximate eigenvectors of each subcarrier’s channel covariance matrix.
  • Selecting the Ns transmit responses with largest |αcℓ|2 yields the dominant fully-digital precoder directions.
  • The optimal fully-digital precoders can be realized with VRF equal to the selected responses and VD[k] equal to the diagonal power-allocation matrix Γ[k].
  • At the receiver, the asymptotically optimal analog combiner selects array-response columns associated with the Ns largest complex gains.
  • When AHr Ar approximately equals I, hybrid beamforming with Ns RF chains realizes optimal fully-digital beamforming.

IV. HYBRID BEAMFORMING DESIGN FOR SU-MIMO

For practical antenna arrays, the paper decouples transmitter and receiver design and develops a heuristic procedure for fully and partially connected hybrid architectures. Digital precoders are optimized per subcarrier, while analog constraints are handled through large-array approximations.

  • Joint transmit–receive optimization is computationally complex, so the design first optimizes the transmitter with an ideal receiver and then designs the receiver.
  • The hybrid-beamforming optimization is nonconvex, including in single-carrier flat-fading systems.
  • For a fixed analog precoder, each subcarrier’s digital precoder has an optimal closed-form solution and can be optimized independently.
  • The method simplifies digital-precoder structure for large arrays and uses Jensen’s inequality to derive an upper bound on spectral efficiency.
  • The digital precoder uses dominant right singular vectors of the effective channel and diagonal per-symbol power allocation obtained by water-filling.
  • In fully connected arrays, FH_RFF_RF is approximately N I with high probability as N grows, whereas partially connected arrays satisfy FH_RFF_RF = N/NRF I exactly.
  • Under moderate and high SNR, equal stream power gives VD[k] approximately equal to γUe[k] without significant performance degradation.

2) Analog Precoding Design:

The analog design replaces frequency-selective covariance information with its average across subcarriers, then applies an iterative coordinate-descent procedure under analog connectivity constraints.

  • The analog precoder is designed by maximizing an upper bound on the overall OFDM spectral efficiency.
  • The upper bound uses the average of frequency-domain channel covariance matrices, linking the broadband problem to a flat-fading analog-design formulation.
  • The resulting optimization replaces HHH with the covariance matrix averaged over all subcarriers and reuses the single-carrier design algorithm.
  • Because the analog constraints decouple, the method performs coordinate descent over individual analog-precoder elements.
  • Each analog element’s contribution is isolated using the precoder with its corresponding column removed and associated coefficients ηij.
  • The algorithm initializes a feasible analog precoder and sequentially updates elements, with convergence to a local optimum guaranteed because every update increases the objective.

B. Receiver Design

The receiver design fixes the digital combiner as the MMSE solution for each subcarrier, then designs the analog combiner through a transformed optimization problem. The overall procedure includes separate analog and digital combiner updates and has practical phase-shifter and coordination considerations.

  • For a fixed analog combiner, the optimal digital combiner on each subcarrier is the MMSE solution.
  • The analog combiner design exploits the approximately white effective noise produced when the analog combiner satisfies W_RF^H W_RF ∝ I.
  • Jensen’s inequality is used to replace the analog combiner objective with an upper-bound maximization problem.
  • Algorithm 1 first designs V_RF and V_D[k], then designs W_RF and computes W_D[k] for each subcarrier.
  • The receiver design depends on the already designed transmitter, requiring either transmit-beamformer feedforward or additional receiver-side computation.
  • O(KN^3) is the stated computational complexity of the overall algorithm when the antenna counts at both ends have the same order.
  • The design initially assumes infinite-resolution phase shifters, while low-resolution implementations restrict analog entries to a finite phase codebook G.

V. HYBRID PRECODING DESIGN FOR MU-MISO

For OFDM MU-MISO, the paper designs a common analog precoder before applying per-subcarrier digital beamforming to the effective channels. The heuristic handles user interference and priority weights through iterative WMMSE while retaining practical per-subcarrier power constraints.

  • The design serves the same users across all subcarriers because this is more spectrally efficient than multiplexing users across frequencies.
  • The weighted-sum-rate problem includes inter-user interference and unequal priority weights, unlike the SU-MIMO spectral-efficiency objective.
  • The heuristic first designs the analog precoder for cooperative users with equal priority weights, emphasizing their direct channels while neglecting interference and priority differences.
  • Digital precoders are then designed for the effective channel using iterative WMMSE, which accounts for inter-user interference and different priority weights.
  • O(KN^2_t) is the stated overall computational complexity of the proposed hybrid beamforming algorithm under min{K, N_t} ≫ N_RF.
  • Compared with prior WMMSE-based design, the proposed method supports per-subcarrier power constraints and avoids repeatedly updating the analog precoder inside WMMSE.
  • When the analog precoder is fixed, zero-forcing or maximum-ratio transmission can replace WMMSE digital precoders to reduce design complexity.

VI. SIMULATIONS

The simulations use half-wavelength uniform linear arrays and a clustered mmWave propagation environment, evaluating SU-MIMO spectral efficiency across antenna counts or SNR.

  • The simulations use uniform linear arrays with half-wavelength antenna spacing and an environment containing 5 clusters with 10 scatterers per cluster unless otherwise stated.
  • SU-MIMO performance is measured by average spectral efficiency versus antenna elements or per-subcarrier SNR over 100 channel realizations.

A. Asymptotic Hybrid Beamforming Design Analysis for SU-MIMO systems

The asymptotic hybrid design is evaluated against optimal fully-digital beamforming as antenna counts increase in two propagation environments. Its achievable rate converges to the fully-digital benchmark, with faster convergence for single-scatterer clusters.

  • The experiment uses an N × N MIMO system with 4 RF chains, 4 data streams, K = 32 subcarriers, and SNR = 20dB.
  • The achievable rate of asymptotic hybrid design converges to optimal fully-digital beamforming for sufficiently large antenna counts in both propagation scenarios.
  • Convergence is faster with one scatterer per cluster because independent arrival and departure angles yield asymptotic array-response orthogonality at smaller N.

B. Hybrid Beamforming Analysis in SU-MIMO systems

The proposed hybrid beamforming designs improve OFDM SU-MIMO spectral efficiency and, in the fully-connected case, approach fully-digital performance with few RF chains. Performance gains are also observed under practical antenna sizes and low-resolution phase shifters.

  • The evaluation compares the proposed designs with existing hybrid beamforming methods across two OFDM-based SU-MIMO settings.One setting uses hybrid beamforming at both transceiver sides; another uses it only at the transmitter.
  • With 4 transceiver RF chains, the proposed fully-connected design approaches optimal fully-digital beamforming using 64 and 32 RF chains.This result is reported for a 64 × 32 system with Ns = 2 and K = 64.
  • The proposed fully-connected and partially-connected designs achieve higher spectral efficiency than the method in for the 64 × 32 system.The method in minimizes distance to optimal fully-digital beamformers rather than directly maximizing spectral efficiency.
  • About 2dB gain is achieved over the asymptotic design when the number of antennas is not extremely large.The comparison is reported for the 64 × 32 OFDM-based SU-MIMO setting.
  • For the fully-connected transmitter-only case, the proposed algorithm has performance very similar to the design in [20].The passage attributes this similarity to [20] using the average covariance matrix of the frequency-domain channels.

C. Hybrid Beamforming Analysis in MU-MISO systems

The MU-MISO evaluation studies weighted sum rate and user-rate distributions under practical large-array OFDM settings. The proposed hybrid design benefits from large antenna arrays and can approach fully-digital performance with a practical RF-chain count, while user scheduling remains unoptimized.

  • Users are randomly and uniformly scheduled, although same-cluster scheduling may increase intended signal power while also increasing inter-user interference.The paper leaves scheduling optimization for future work.
  • With 64 antennas and 8 RF chains, the proposed hybrid algorithm achieves much higher spectral efficiency than fully-digital WMMSE using 8 antennas and 8 RF chains.The first experiment sets each user’s priority weight proportional to the inverse of its expected rate at −55 dBm/Hz.
  • With 16 RF chains, the proposed hybrid design approaches fully-digital WMMSE performance with 64 antennas.This comparison is reported for the MU-MISO weighted-sum-rate experiment.
  • The paper reports that the proposed algorithm outperforms existing methods for both hybrid architectures and can approach optimal fully-digital performance with fewer RF chains than antennas.This conclusion covers the SU-MIMO and MU-MISO designs considered in the paper.
  • The MU-MISO experiments evaluate weighted sum rate and empirical user-rate distributions for 4 users, 64 antennas, and 32 subcarriers.Figures 7 and 8 use these OFDM-based MU-MISO parameters.
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