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Common Codebook Millimeter Wave Beam Design: Designing Beams for Both Sounding and Communication with Uniform Planar Arrays

Jiho Song, Junil Choi, David J. Love

arXiv:1606.05634v4cs.ITmath.IT

TL;DR

MmWave links face severe path loss and channel-estimation challenges, while a common codebook must serve conflicting sounding and data-beam needs. The paper designs UPA codebooks for hybrid beamforming by matching each beamformer to an ideal pattern. Its simulations report higher reference gains and data rates than previously reported codebooks in the evaluated comparisons.

  • Problem

    MmWave channel dimensionality can make explicit channel estimation difficult, while a shared codebook must balance wide, low-overhead sounding beams against narrow, high-gain data beams.

    Source: §I · pp. 1–2

  • Method

    The paper defines ideal UPA beam patterns and designs hybrid-feasible codebook beamformers to minimize mean squared error from those patterns.

    Source: §I · p. 2, §III · p. 4

  • Results

    Mean reference gains were 0.553 for the proposed codebook, 0.467 for the guard-band variant, 0.397 for, and 0.440 for; proposed codebooks also scan better and yield higher data rates than,, and.

    Source: A. Beam patterns of codebook examples · p. 9, B. Data rate performance · p. 10, §VI · p. 11

  • Takeaways & Limitations

    The simulations support the proposed codebooks as suitable for sounding mmWave channels and facilitating directional data transmission with UPA structures.

    Source: A. Beam patterns of codebook examples · p. 9, §VI · p. 11

    Electromagnetic interaction between deployed antennas is not modeled, though the approach can be applied using an effective array response that accounts for mutual coupling.

    Source: B. Vector subspace for mmWave channels · p. 3

Abstract

from arXiv · show

Fifth generation (5G) wireless networks are expected to utilize wide bandwidths available at millimeter wave (mmWave) frequencies for enhancing system throughput. However, the unfavorable channel conditions of mmWave links, e.g., higher path loss and attenuation due to atmospheric gases or water vapor, hinder reliable communications. To compensate for these severe losses, it is essential to have a multitude of antennas to generate sharp and strong beams for directional transmission. In this paper, we consider mmWave systems using uniform planar array (UPA) antennas, which effectively place more antennas on a two-dimensional grid. A hybrid beamforming setup is also considered to generate beams by combining a multitude of antennas using only a few radio frequency chains. We focus on designing a set of transmit beamformers generating beams adapted to the directional characteristics of mmWave links assuming a UPA and hybrid beamforming. We first define ideal beam patterns for UPA structures. Each beamformer is constructed to minimize the mean squared error from the corresponding ideal beam pattern. Simulation results verify that the proposed codebooks enhance beamforming reliability and data rate in mmWave systems.

I. INTRODUCTION

The paper addresses UPA beam design for mmWave links where severe channel losses, difficult channel estimation, and conflicting sounding and data-beam requirements complicate reliable transmission. It proposes a hybrid-beamforming codebook designed around ideal beam patterns and reports that its patterns suit codebook-based beam alignment.

I. INTRODUCTION

  • MmWave path loss and atmospheric absorption motivate directional transmission, while UPAs provide a two-dimensional antenna grid and hybrid beamforming uses only a few RF chains.The paper notes that one RF chain per antenna can impose high cost and power consumption.Source: §I · p. 1
  • A shared codebook must balance wide-area, low-overhead sounding against narrow, high-gain data beams.The paper motivates this need because large mmWave channel dimensionality can make explicit channel estimation difficult, while beam alignment can avoid explicitly estimating the channel matrix.Source: §I · pp. 1–2
  • The design derives UPA beam-pattern constraints, defines ideal patterns, and selects hybrid-feasible OMP candidates to minimize pattern MSE.The final beamformer is selected from candidates for each ideal pattern; the paper also reports that other codebook beamformers can be generated from one optimized beamformer, expediting offline construction.Source: §I · p. 2

A. System model

  • The transmitter uses M = MhMv UPA antennas and N≤M RF chains, combining analog phase-shifter beamsteering with baseband weights.The analog phase shifters use quantized phases, and the combined analog and digital beamformer is subject to a power constraint.Source: A. System model · pp. 2–3
  • The sounding codebook contains Q = QhQv beamformers, which are transmitted one by one to sound the channel.The system model chooses the data-transmission beamformer from the sounding codebook.Source: A. System model · p. 3

B. Vector subspace for mmWave channels

The paper models UPA mmWave channels using a beam region of feasible directions, motivated by the strong directivity of channels with a dominant beam. It simplifies the region's geometry and excludes electromagnetic interaction between deployed antennas.

B. Vector subspace for mmWave channels

  • The channel model assumes a single dominant beam, while allowing a LOS path and a few NLOS paths.The paper motivates the dominant-beam subspace by the expected large Ricean K-factor of mmWave channels.Source: B. Vector subspace for mmWave channels · p. 3
  • Electromagnetic interaction between deployed antennas is excluded from the model.The authors state that an effective array response incorporating mutual coupling could be used to generate patterns that account for it.Source: B. Vector subspace for mmWave channels · p. 3
  • The beam region is the feasible set of beam directions across possible AoDs; for d = λ/2, the paper simplifies the direction bounds and ignores the negligible horizontal–vertical pairing areas.The paper calls the two-dimensional geometric model of the vector subspace the beam region.Source: B. Vector subspace for mmWave channels · pp. 3–4

III. FRAMEWORK FOR CODEBOOK DESIGN ALGORITHM

The framework partitions the coverage region into beam-specific target regions and defines ideal patterns with uniform gain inside each target and zero gain outside. These patterns guide practical codebook design, but “ideal” does not mean globally optimal.

III. FRAMEWORK FOR CODEBOOK DESIGN ALGORITHM

  • The common codebook must balance broad geographic coverage for sounding against high, uniform gain within each beam region.The stated design criteria also call for smaller gain outside the target region.Source: §III · p. 4
  • Uniform gain inside a beam region and zero gain outside define the ideal pattern used as a design guide.The paper notes that these patterns are generally unachievable in practice.Source: §III · pp. 4–5
  • “Ideal” describes the prescribed pattern, not a claim that it is globally optimal.Source: §III · p. 4
  • Each beamformer targets a subregion formed by combining one horizontal range with one vertical range.The coverage region is divided into Q=QhQv subspaces under a uniform-user assumption.Source: §III · p. 4
  • With uniformly distributed dominant beam directions, identical nonzero gains in the target region achieve the stated upper bound on expected data rate.The bound is stated for the ideal-pattern analysis.Source: §III · p. 5

A. Problem formulation for beam pattern design

The section formulates practical hybrid beamformer design as minimizing the mismatch between an achievable pattern and a predefined ideal pattern. It quantizes beam directions, reformulates the optimization to obtain feasible candidates, and uses phase-shift structure to derive other codebook beams from one optimized beamformer.

A. Problem formulation for beam pattern design

  • The design objective is to construct a hybrid beamsteering matrix and baseband vector whose beam pattern minimizes MSE from the ideal pattern.The combined beamformer is subject to a unit-power constraint.Source: A. Problem formulation for beam… · pp. 5–6
  • Phase shifting of equal-gain vectors in the optimized (1,1)-th beamsteering matrix generates the beamsteering matrices for the other beamformers.The section states that this structure can avoid optimizing all Q beamformers independently.Source: A. Problem formulation for beam… · p. 6
  • Continuous beam directions are quantized before the actual and ideal patterns are represented as vectors for optimization.The section focuses on optimizing the (1,1)-th beamformer.Source: A. Problem formulation for beam… · p. 6
  • The original minimization has no closed-form solution, so the authors reformulate it to obtain a feasible, usually suboptimal solution.The reformulation compares decomposed actual and ideal pattern vectors for a given pair of equal-gain vectors.Source: A. Problem formulation for beam… · p. 6

B. OMP algorithm constructing beamformer candidates

The OMP procedure builds feasible hybrid beamformer candidates by iteratively selecting equal-gain analog vectors and updating the baseband beamformer to reduce the residual. Equal-gain constraints mean the constructed solution need not be optimal, and phase elements are quantized for digitally controlled RF phase shifters.

B. OMP algorithm constructing beamformer candidates

  • Equal-gain constraints on the analog matrix can prevent the constructed F and v from forming the optimal beamformer.Source: B. OMP algorithm constructing… · p. 7
  • Each added analog vector uses the residual's phase, then quantizes its elements to the available phase set.The phase choice sets ξ=0 for simplicity.Source: B. OMP algorithm constructing… · p. 7
  • OMP selects equal-gain analog vectors one at a time and updates the baseband beamformer to reduce the residual for each candidate.The candidate combines the updated analog matrix and baseband vector.Source: B. OMP algorithm constructing… · p. 7
  • Fig. 6 reports normalized beamforming gains over (θh, θv) for c2,2, with a 16×8 array, 8×4 beam regions, and N=4.Source: Figure 6 · p. 8

C. Final beamformer selection with MSE minimization

The final beamformer is selected to produce a pattern close to the ideal beam pattern. Expanding the candidate sets increases candidate availability but also computational complexity, so their sizes are limited.

C. Final beamformer selection with MSE minimization

D. Overlapped beam regions

The design widens each beam region with a guard band to reduce sharp dips between consecutive beams. The ideal pattern is redefined for the widened region, and beamformers are recomputed using the design algorithm.

D. Overlapped beam regions

V. SIMULATION RESULTS

The simulations compare the proposed codebook with extended codebooks from prior work and a 2D KP codebook. In the beam-pattern examples, the proposed codebook has the highest mean reference gain, while its guard-band variant reduces sharp dips between consecutive beams.

V. SIMULATION RESULTS

  • The simulations use four RF chains, a six-bit phase control register, and 20 directions in each beam-width to compute MSE.The codebook uses Q = Q_hQ_v beamformers; the equal-gain-set parameters are L_h = 8, L_v = 8, and I = 3.Source: §V · p. 8

A. Beam patterns of codebook examples

  • 0.553, 0.467, 0.397, and 0.440 are the mean reference gains for the proposed codebook, its guard-band variant,, and, respectively.Fig. 7 compares codebook examples with (M_h, M_v) = (12, 6), (Q_h, Q_v) = (8, 8), and N = 4.Source: Figure 7 · p. 9, A. Beam patterns of codebook examples · p. 9
  • The proposed codebook in Section IV-D alleviates sharp dips between consecutive beams, though its reference gains may be lower than those without a guard band.The guard-band variant's reference gains are reported as higher than those of and.Source: A. Beam patterns of codebook examples · p. 9
  • The proposed codebook suppresses MSE better than, whose design uses a single set of all-ones vectors and is one candidate of the proposed algorithm.Source: A. Beam patterns of codebook examples · p. 9

B. Data rate performance

Simulations compare codebook data rates for UPA channels with a LOS path and three NLOS paths, and for channels with three NLOS paths only. The proposed codebooks outperform the previously reported codebooks and 2D KP codebook in both scenarios.

B. Data rate performance

  • Data-rate simulations use 10,000 independent channel realizations and two ray-like street-geometry scenarios, with channel-vector norm squared set to M for a fair comparison.The first scenario has one LOS and three NLOS paths, with a Ricean K-factor of 13.5 dB; the second has three NLOS paths without LOS.Source: B. Data rate performance · pp. 9–10
  • The proposed codebook scans LOS-plus-NLOS channels better and provides higher data rates than codebooks in,, and the 2D KP codebook in.Fig. 8 compares data rates for UPA sizes (12, 6), (16, 8), (20, 10), and (24, 12).Source: Figure 8 · p. 10, B. Data rate performance · p. 10
  • For channels with three NLOS paths and no LOS path, the proposed codebooks select a dominant NLOS path better than codebooks in,, and the 2D KP codebook in.Fig. 9 compares the codebooks for the same four UPA sizes; the authors note that selecting and combining multiple beamformers could fully support channels with multiple NLOS paths.Source: B. Data rate performance · pp. 10–11, Figure 9 · p. 11

VI. CONCLUSIONS

  • The conclusion reports that the proposed codebooks outperform previously reported codebooks in mmWave channels using UPA structures.The design minimizes MSE between each actual beam pattern and its corresponding ideal pattern, using a simplified solution and OMP for hybrid beamforming.Source: §VI · p. 11

APPENDIX A PROOF OF LEMMA 3

The appendix shows that shifting the ideal beam pattern and phase-shifting variables reduces the optimization for each beamformer to the (1, 1)-th case. It also derives the reference-gain relation used in that argument.

APPENDIX A PROOF OF LEMMA 3

  • Any ideal beam pattern is obtained by shifting the (1, 1)-th ideal pattern, and the corresponding MSE problem has the same solution as the (1, 1)-th beamformer.The appendix also states that the actual beam pattern shifts in direction, as illustrated in Fig. 10.Source: Appendix A · pp. 11–12, Figure 10 · p. 12

APPENDIX B PROOF OF LEMMA 4

  • The proof of Lemma 4 rewrites the reference gain using phase-shifting functions and array-vector properties, with a unit-gain condition for the phase-shifted vector.The proof also uses decomposition of 1_M,N and the associative property of the Hadamard product.Source: Appendix A · p. 12, Appendix B · p. 13

APPENDIX A PROOF OF LEMMA 3

  • The proof rewrites the (q, p)-th beamformer’s minimization using shifted variables and a phase-shifted matrix, relying on periodicity over φ_h and φ_v.This converts its MSE problem to the form for the (1, 1)-th beamformer.Source: Appendix A · p. 12
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