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Hierarchical Codebook Design for Beamforming Training in Millimeter-Wave Communication

Zhenyu Xiao, Tong He, Pengfei Xia, Xiang-Gen Xia

arXiv:1511.02128v2cs.IT

TL;DR

MmWave beam training needs large-array gain but exhaustive angular search is too costly, making hierarchical codebook design central. This paper proposes coverage criteria and a joint sub-array/deactivation codebook with closed-form generation, reporting superiority over alternatives across tested settings.

  • Problem

    Large mmWave arrays require beam-direction search at both transmitter and receiver, but exhaustive search has O(N^2) complexity and hierarchical performance depends on codebook design.

  • Method

    The paper proposes two hierarchical-codebook criteria and jointly uses sub-array and antenna-deactivation processing to construct a closed-form codebook.

  • Results

    The proposed BMW-SS codebook outperforms deactivation and Sparse alternatives in received power and beam-search success rate under the evaluated power and channel models.

  • Takeaways & Limitations

    BMW-SS provides flatter beams and more active antennas, supporting superior search performance across the evaluated transmission-power models.

Abstract

from arXiv · show

In millimeter-wave communication, large antenna arrays are required to achieve high power gain by steering towards each other with narrow beams, which poses the problem to efficiently search the best beam direction in the angle domain at both Tx and Rx sides. As the exhaustive search is time consuming, hierarchical search has been widely accepted to reduce the complexity, and its performance is highly dependent on the codebook design. In this paper, we propose two basic criteria for the hierarchical codebook design, and devise an efficient hierarchical codebook by jointly exploiting sub-array and deactivation (turning-off) antenna processing techniques, where closed-form expressions are provided to generate the codebook. Performance evaluations are conducted under different system and channel models. Results show superiority of the proposed codebook over the existing alternatives.

I. INTRODUCTION

mmWave systems use large arrays and analog or switched beamforming to overcome high path loss, but exhaustive beam search is costly. The paper therefore focuses on hierarchical codebook design and proposes a joint sub-array/deactivation approach evaluated across channel and power models.

  • Motivation: High mmWave path loss motivates joint Tx/Rx beamforming with large antenna arrays for substantial array gain.The paper notes carrier frequencies around 30–60 GHz and illustrates an array size of 36 antennas.
  • Motivation: Analog beamforming is preferred because mixed-signal components and RF chains are power-hungry and costly, with constant-amplitude weights across antennas.The described structure uses one RF chain shared by the antennas.
  • Search challenge: Exhaustive search tests all sampled transmit and receive beam directions, making search time prohibitive when the number of candidate directions is large.Hierarchical codebooks reduce this burden by organizing coarse and fine beams across resolutions.
  • Search challenge: Search performance depends strongly on hierarchical codebook design, while prior wider-beam methods either lacked design procedures, violated constant-amplitude constraints, or were incomplete.These limitations motivate a new full codebook construction.
  • Contribution: The proposed codebook jointly exploits sub-arrays and antenna deactivation, provides closed-form generation expressions, and is evaluated under LOS/NLOS and total/per-antenna power models.The authors report superiority over existing alternatives, especially under per-antenna transmit-power constraints.
  • System model: The signal model permits activated or deactivated antenna weights, with total and per-antenna power models differing in how transmit power scales with active antennas.The paper states that codebook design is unchanged between the two power models.

B. Channel Model

The channel model represents mmWave propagation as a directionally sparse combination of multipath components. Each path is characterized by a complex gain and transmit/receive angular parameters represented through ULA steering vectors.

  • Channel characteristics: Limited scattering makes mmWave multipath components primarily reflection-generated and directional, with distinct physical AoDs and AoAs.The channel is therefore naturally described in the angle domain.
  • Channel representation: The channel matrix is formed from multipath components whose contributions depend on complex path gains and transmit and receive steering vectors.The number of paths is denoted by L.
  • Channel representation: The model uses normalized angular variables Ωℓ and ψℓ, defined as cosines of the physical AoD and AoA and lying in [−1, 1].The paper subsequently calls these normalized quantities AoDs and AoAs for convenience.
  • Channel representation: For a half-wavelength-spaced ULA, the steering vector is periodic in its angular argument with period 2, and the channel matrix is power normalized.The antenna count N is transmitter- or receiver-specific depending on the array under consideration.

C. The Problem

The beamforming problem is to identify strong propagation directions under constant-amplitude constraints without estimating a large channel matrix. Hierarchical codebooks address the O(N^2) exhaustive-search cost by organizing beams into progressively finer layers.

  • Problem formulation: Joint Tx/Rx beamforming seeks transmit and receive weights that maximize received SNR.The optimization is constrained by the available transmit and receive beamforming-vector sets.
  • Problem formulation: SVD is impractical because channel entry-wise estimation is too costly for large arrays and the analog beamforming constant-amplitude constraint remains.The paper instead exploits the channel’s dominant angular structure.
  • Angular search: For one-stream beamforming, the search can target the AoD and AoA of a strongest or sufficiently strong multipath component and use steering vectors toward them.The strongest path is especially relevant in LOS, whereas an arbitrary strong path can suffice in NLOS.
  • Hierarchical search: Exhaustive search samples the angle domain and tests all corresponding steering-vector pairs, but its time complexity is O(N^2).This complexity is too high for large arrays, motivating hierarchical search.
  • Hierarchical search: The codebook is a channel-independent relationship among angle-domain codewords, while the beam search selects channel-dependent steering vectors before data transmission.Hierarchical layers use different beam widths to improve search efficiency.
  • Research gap: The paper identifies a lack of general codebook criteria and complete closed-form hierarchical codebooks, then addresses both gaps with a joint sub-array/deactivation design.The contribution is framed as a design methodology rather than instantaneous channel adaptation.

A. Two Criteria

The proposed hierarchical design is governed by coverage and parent–child criteria. Together they ensure angle-domain coverage and define the tree structure needed for staged beam search, with a binary tree used as the main construction.

  • Definitions: Beam coverage is defined from beam gain using a factor ρ, with larger ρ producing smaller coverage; ρ = 1/2 corresponds to 3 dB coverage.Codewords in one codebook may use different ρ values for different beam widths.
  • Criterion 1: Criterion 1 requires the union of all codeword beam coverages within every layer to cover the whole angle domain.This guarantees that hierarchical search does not miss any angle.
  • Criterion 2: Criterion 2 requires each parent codeword’s coverage to be contained in the union of several child-codeword coverages in the next layer.The indexed child set belongs to the immediately finer layer.
  • Hierarchical search: The criteria induce an M-way tree when each parent has M children, enabling tree-search implementation at the receiver and transmitter.The search proceeds through child candidates after selecting a parent at the previous stage.
  • Binary-tree construction: The main construction uses M = 2, yielding a binary tree compatible with antenna counts that are powers of two.The paper states that extension to other branching factors is straightforward.

B. The Deactivation Approach

The DEACT approach builds a binary-tree hierarchical codebook whose layers contain progressively narrower beams, with coverage recursively partitioned across child codewords.

  • Codebook structure: Each parent beam coverage equals the union of its two child beam coverages in the next layer.This recursive relation defines the binary-tree angle partition.
  • Beam quality: The codebook’s beam-gain metric approaches ρ ≈ 0.64 for large N and remains close to this value for small N.For example, ρ = 0.65 when N = 4.
  • Beam coverage: The last layer contains N codewords covering [−1, 1] with narrow beam width 2/N and evenly sampled steering angles.These last-layer codewords are steering vectors distributed across the total angle range.
  • Antenna activation: For DEACT, the k-th layer activates 2^k antennas and turns off the remaining antennas.The approach therefore uses fewer active antennas in lower-index layers.

C. The Joint Sub-Array and Deactivation Approach

The joint sub-array and deactivation approach, BMW-SS, addresses DEACT’s low-layer power limitation while constructing codewords through beam rotation and sub-array processing.

  • Motivation: BMW-SS jointly uses sub-array processing and antenna deactivation to broaden beams while retaining more active antennas than DEACT in low layers.The method is motivated by DEACT activating very few antennas, potentially only one, in the lowest layer.
  • Design requirements: The codebook design requires codewords with layer-dependent widths, including width 2/2^k in the k-th layer and equal widths within each layer.Codewords in the same layer differ by steering angle.
  • Codeword generation: The rotation operation preserves constant-amplitude weights, so the generated codewords remain in the admissible codeword set.The construction uses entry-wise multiplication by a unit-modulus rotation vector.
  • Beam rotation: A beam-rotation theorem generates codewords with shifted beam coverage from a known codeword.The theorem is also applicable to codebook designs beyond BMW-SS.
  • Beam rotation: All codewords in one layer can be obtained from a first codeword by rotating it according to the angle gaps between their steering directions.This follows because same-layer codewords share beam width and coverage shape but have different offsets.

2) Beam Broadening:

Beam broadening divides the array into sub-arrays whose widely separated steering directions create wider coverage; combining sub-arrays and deactivation provides additional width control.

  • Sub-array construction: An N-element array divided into M sub-arrays of N_S antennas can broaden a beam by distributing sub-array transmission across widely spaced angles.The construction uses N = M N_S and assigns each sub-array a distinct coverage region.
  • Sub-array construction: The sub-array method broadens beam width by a factor M^2, with one factor from the number of sub-arrays and one from reduced sub-array size.The resulting beam width is 2M^2/N.
  • Scope: The sub-array derivation does not account for mutual effects between different sub-arrays.The analysis relies on their limited interaction at the considered steering directions.
  • Beam flattening: Coefficient phases e^{jθ_m} are adjusted so intersection points between neighboring coverage regions retain high beam gain and beam fluctuation is reduced.The phase choice uses the evenness of N_S and a phase difference Δθ = π.
  • Beam flattening: Setting the sub-array coefficients as specified produces a codeword with beam width 2M/N.The phase-weighted sub-arrays form the broadened codeword.
  • Joint processing: Joint sub-array and deactivation processing can produce codewords with beam width 2N_A/N, where N_A is the number of active sub-arrays.This extends beam-width control beyond sub-array processing alone.

3) Codebook Generation:

The BMW-SS codebook is generated through sub-array decomposition, deactivation, elementwise construction, and normalization, producing hierarchical beams whose coverage nests across layers. Compared with DEACT and Sparse, BMW-SS provides flatter or stronger covered beams while avoiding deep sinks, but its final resolution is limited to 2/N.

  • Codebook generation: The codebook generation procedure computes w(k,n) from w(k,1) by separating it into sub-arrays, setting sub-array factors, combining them elementwise, and normalizing.The number of sub-arrays and active sub-arrays depends on the layer parameter ℓ.
  • Hierarchical beam coverage: For N = 128, w(0,1) covers the union of w(1,1) and w(1,2), while w(1,1) covers the union of w(2,1) and w(2,2).This demonstrates the intended hierarchical coverage relationship across successive layers.
  • Beam-pattern comparison: BMW-SS beams appear flatter than DEACT beams within the covered angle, despite small-scale fluctuations.The comparison is based on the beam patterns shown for the two approaches.
  • Beam-pattern comparison: BMW-SS offers much higher beam gains than DEACT under the per-antenna transmission power model because it exploits more active antennas.BMW-SS codewords use either all antennas or half of them, whereas DEACT can activate few antennas at low layers.
  • Beam-pattern comparison: Sparse exhibits deep sinks within wide-beam coverage when the number of RF chains is small, whereas BMW-SS does not have such deep sinks.A multipath component arriving at a Sparse sink angle may not be detected, and the sink becomes more severe with fewer RF chains.
  • Resolution boundary: The designed hierarchical search converges to steering vectors with final-layer angle resolution 2/N, so higher resolution requires a separate fine codebook.The presented codebook and search are therefore coarse rather than fine.

D. Generalization

The proposed criteria and BMW-SS approach are based on analog beamforming but can also be used within hybrid precoding. The approach can be extended to multi-stream searches and to uniform planar arrays.

  • Generalization: The proposed criteria and BMW-SS approach are designed for analog beamforming and are also feasible for hybrid precoding.Analog beamforming is treated as one branch of the hybrid precoding structure.
  • Generalization: BMW-SS can search the AoD and AoA of each individual multipath component in multi-stream transmission.The extension applies the beamforming-based search separately to each MPC.
  • Generalization: The proposed criteria and BMW-SS approach can be extended from the uniform linear array model to the uniform planar array model.The paper notes that other practical array types include UPA and UCA.

2) For Other Types of Antenna Arrays:

The paper discusses extensions and evaluation boundaries for hierarchical codebooks across array types, transmission-power models, and LOS/NLOS channels. Results characterize when BMW-SS, DEACT, and Sparse succeed or diverge.

  • 2) For Other Types of Antenna Arrays:: For a UPA with m × n elements, the steering vector is the Kronecker product of steering vectors for m×1 and n×1 ULAs.The search process and codebook design are identified as extensions for future study.
  • 2) For Other Types of Antenna Arrays:: The BMW-SS approach is difficult to extend to UCA models because steering-vector element relations change, although the criteria remain feasible with two-dimensional angle coverage.A new UCA codebook based on the proposed criteria is left as an interesting design direction.
  • 2) For Other Types of Antenna Arrays:: The BMW-SS approach requires a ULA element count N = M^p, limiting direct extension to arbitrary antenna counts.The paper suggests selecting compatible array sizes during system planning or using a reduced sub-array structure before later refinement.
  • A. Total Transmission Power Model: Under total transmission power, antenna deactivation does not change total transmission power, so the compared schemes use equal total power.The simulations consider both total-transmission-power and per-antenna-power models, with the latter reflecting limited amplifier output.
  • A. Total Transmission Power Model: At high SNR, BMW-SS and DEACT have similar received-power search behavior, while both reach the LOS upper bound and approach it under NLOS channels.DEACT is slightly better in the first two steps, BMW-SS slightly better later, and both share the same last-layer codewords; under NLOS, either method may select a nonoptimal MPC.
  • A. Total Transmission Power Model: BMW-SS generally has higher success rates than DEACT and Sparse, but multiple MPCs and spatial fading prevent all schemes from reliably reaching 100%.Under NLOS with L = 1 and sufficiently high γtot, BMW-SS and DEACT reach 100%, whereas Sparse does not because of deep sinks; with L > 1, mutual MPC effects limit success.

B. Per-Antenna Transmission Power Model

Under the per-antenna transmission power model, deactivating antennas reduces total transmission power, giving BMW-SS an advantage in received power and hierarchical-search success rate over DEACT. The advantage is attributed to BMW-SS’s greater number of active antennas and flatter beams.

  • Power model: With equal per-antenna transmission power, fewer active antennas produce lower total transmission power for DEACT.The model compares BMW-SS and DEACT using the same per-antenna transmission power.
  • Received power: BMW-SS’s received-power advantage comes from its wide-beam codewords activating significantly more antennas than DEACT.This gives BMW-SS much higher total transmission power under the same per-antenna power constraint.
  • Search progression: During Steps 1–6, both schemes have the same received-power increase because only Rx array gain is accumulated.Steps 1–6 perform Rx training, while Steps 7–12 perform Tx training.
  • Search progression: During Steps 7–12, DEACT’s received-power increase becomes faster because Tx array gain and total-transmission-power gain accumulate together.BMW-SS’s increase varies because its number of active antennas changes alternately while its Tx beam narrows.
  • Implications: The early-search superiority of BMW-SS can increase beam-search success at the same distance or extend transmission distance at the same success rate.The conclusion also reports that BMW-SS outperforms Sparse because its beam coverage has no deep sinks.
  • Success rate: BMW-SS’s success-rate superiority over DEACT becomes more significant under both LOS and NLOS channels, even at low per-antenna transmission power.The reported success rate for BMW-SS can be close to 100%, outperforming DEACT and Sparse.

APPENDIX A PROOF OF THEOREM 1

Theorem 1 proves how entry-wise multiplication by a steering vector transforms a beamforming vector’s beam gain and coverage. The proof establishes that steering shifts the associated angle set without changing the underlying maximum beam-gain behavior.

  • Theorem setup: The proof considers an arbitrary N-element vector w and two angles while establishing the transformed angle-set relation.The relation is expressed through the vector formed from w and a steering vector.
  • Beam transformation: The vector w ◦ a(N, ψ) is constructed from w and the steering vector a(N, ψ), then analyzed through its beam gain.The proof first evaluates the beam gain of this new vector.
  • Beam-gain proof: The beam-gain derivation follows directly from the beam-gain definition and the definition of entry-wise multiplication.These are the stated bases for steps (a) and (b) of the derivation.
  • Coverage proof: The coverage derivation uses the beam-coverage definition, the established beam-gain relation, and an angle-offset substitution.The proof sets Ω = Ω0 + ψ to express the shifted coverage.
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