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Wideband Beam Tracking in THz Massive MIMO Systems

Jingbo Tan, Linglong Dai

arXiv:2012.04238v2cs.ITmath.IT

TL;DR

Wideband THz beam tracking is challenged by beam split, while mobile users also require frequent retraining. This paper uses delay-phase precoding and beam zooming to control frequency-dependent beam coverage. The proposed scheme reports substantially lower tracking overhead while retaining near-optimal achievable sum-rate.

  • Problem

    Beam split misaligns beams across subcarriers, and mobile users' changing optimal beams make frequent training costly.

    Source: §I · p. 1, A. Channel Model · p. 4

  • Method

    The scheme uses delay-phase precoding, whose time delays create frequency-dependent beams with flexibly adjustable angular coverage.

    Source: B. Delay-Phase Precoding · p. 4, B. Beam Zooming Mechanism · p. 7

  • Results

    At 10 bits/s/Hz, the proposed scheme reduces beam tracking overhead by about 90% compared with the typical scheme.

    Source: §V · p. 14

  • Takeaways & Limitations

    The proposed scheme can track multiple user physical directions using multiple frequency-dependent beams generated by one RF chain.

    Source: §VI · p. 15

Abstract

from arXiv · show

Terahertz (THz) massive multiple-input multipleoutput (MIMO) has been considered as one of the promising technologies for future 6G wireless communications. It is essential to obtain channel information by beam tracking scheme to track mobile users in THz massive MIMO systems. However, the existing beam tracking schemes designed for narrowband systems with the traditional hybrid precoding structure suffer from a severe performance loss caused by the beam split effect, and thus cannot be directly applied to wideband THz massive MIMO systems. To solve this problem, in this paper we propose a beam zooming based beam tracking scheme by considering the recently proposed delay-phase precoding structure for THz massive MIMO. Specifically, we firstly prove the beam zooming mechanism to flexibly control the angular coverage of frequencydependent beams over the whole bandwidth, which can be realized by the elaborate design of time delays in the delay-phase precoding structure. Then, based on this beam zooming mechanism, we propose to track multiple user physical directions simultaneously in each time slot by generating multiple beams. The angular coverage of these beams is flexibly zoomed to adapt to the potential variation range of the user physical direction. After several time slots, the base station is able to obtain the exact user physical direction by finding out the beam with the largest user received power. Unlike traditional schemes where only one frequency-independent beam can be usually generated by one radio-frequency chain, the proposed beam zooming based beam tracking scheme can simultaneously track multiple user physical directions by using multiple frequency-dependent beams generated by one radio-frequency chain. Theoretical analysis shows that the proposed scheme can achieve the near-optimal achievable sum-rate performance with low beam training overhead.

I. INTRODUCTION

Wideband THz massive MIMO offers high bandwidth but faces beam-split effects and costly tracking for mobile users. The paper proposes DPP-based beam zooming to track multiple directions with low overhead and near-optimal achievable sum-rate.

I. INTRODUCTION

  • THz communication offers tens of GHz bandwidth for high data rates, while severe path loss motivates massive MIMO's high-array-gain directional beams.The section introduces a wideband ray-based channel model for a multi-user system using OFDM subcarriers.Source: §I · p. 1, A. Channel Model · p. 3
  • Mobile users require frequent beam training because their narrow optimal beams change quickly, creating high training overhead.The paper motivates beam tracking as an alternative to repeatedly training beams for moving users.Source: §I · p. 1

A. Prior Works

  • Mobility-model tracking depends on potentially inaccurate prior models, while codebook methods search among directional beams using exhaustive, hierarchical, or auxiliary-beam procedures.The auxiliary-beam approach uses two extra RF chains.Source: A. Prior Works · p. 2
  • Existing wideband THz beam tracking remains unaddressed by prior schemes, which are narrowband-suited or constrained by energy use, fixed codebooks, or transmission distance.The paper identifies beam split as the source of severe degradation when narrowband schemes use traditional hybrid precoding.Source: A. Prior Works · p. 2

B. Our Contributions

  • The proposed DPP-based scheme zooms frequency-dependent beams to track multiple user directions per time slot, with theoretical analysis indicating near-optimal sum-rate and low training overhead.Time delays flexibly control angular coverage; the base station selects the optimal beam using received signal power.Source: B. Our Contributions · p. 2

A. Channel Model

  • More than 60% performance loss is reported when bandwidth increases from 1 GHz to 10 GHz, as beam splitting misaligns beams across subcarriers.The beam-split effect is associated with ultra-wide bandwidth and large antenna arrays.Source: A. Channel Model · p. 4

B. Delay-Phase Precoding

The delay-phase precoding structure adds time delays to phase-shifter precoding, producing frequency-dependent beams. Carefully chosen delays can align beams across the bandwidth or adjust their angular coverage.

B. Delay-Phase Precoding

  • The delay-phase precoding structure uses time-delay networks between RF chains and phase shifters to make analog beamforming frequency-dependent and mitigate beam-split loss.The paper describes each RF chain as connected to time-delayers, which connect to antenna elements through phase shifters.Source: B. Delay-Phase Precoding · p. 4
  • The analog beamformer is formed from frequency-independent phase-shifter settings and time delays, with the number of time-delayers set as a design parameter.The time delays introduce frequency-dependent phase shifts into the analog beamformer.Source: B. Delay-Phase Precoding · pp. 4–5
  • Elaborately designed time delays rotate beams across subcarrier frequencies so that they align with the user's physical direction throughout the bandwidth.The frequency-dependent beamforming vector combines the phase-shifter beamformer with time-delay phase shifts.Source: B. Delay-Phase Precoding · p. 5

C. Frame Structure

The frame structure separates channel changes caused by LoS-path events from faster user-direction changes. Beam tracking updates directions at frame starts, before equivalent-channel estimation and data transmission.

C. Frame Structure

  • A block spans successive LoS-path appearances or disappearances, while a frame captures the faster changes in the LoS path's physical direction.Full-channel estimation occurs at the beginning of each block; beam tracking occurs at the beginning of each frame.Source: C. Frame Structure · p. 5
  • After beam tracking, the base station forms the analog beamformer, estimates the low-dimensional equivalent channel, selects a digital precoder, and transmits data.Zero-forcing and MMSE are given as example digital precoding methods.Source: C. Frame Structure · p. 5

III. THE PROPOSED FLEXIBLE BEAM ZOOMING BASED BEAM TRACKING METHOD

  • Narrowband tracking schemes with traditional hybrid precoding cannot handle beam split, motivating the proposed DPP-based wideband beam-zooming method.The method section first considers a typical tracking scheme as a benchmark, then presents the beam-zooming mechanism and tracking scheme.Source: C. Frame Structure · p. 5, §III · p. 5

A. Typical Beam Tracking Scheme Using DPP

The DPP-adapted typical scheme searches candidate physical directions across time slots and selects the direction with the largest received power. Although it can track users simultaneously, its exhaustive search requires high training overhead.

A. Typical Beam Tracking Scheme Using DPP

  • The adapted scheme selects the optimal beamforming vector from a codebook by training with different codewords over successive time slots.Each codeword corresponds to a potential beam aligned with a unique physical direction.Source: A. Typical Beam Tracking Scheme Using… · pp. 5–6
  • The base station selects a candidate beam for each time slot, then estimates the next-frame direction from the direction with the largest received power.The candidate directions span the user’s assumed potential variation range, which the base station knows in advance.Source: A. Typical Beam Tracking Scheme Using… · p. 6
  • Because the DPP structure can generate K beams simultaneously, the typical tracking procedure can run for all K users at once.Each time slot still tracks only one physical direction per user.Source: A. Typical Beam Tracking Scheme Using… · p. 6
  • With N = 256 and αk = 0.1, the typical scheme requires T > 52 time slots, making its exhaustive search overhead unacceptable in practice.The source identifies this as a consequence of searching the tracking range exhaustively, particularly with narrow THz beams.Source: A. Typical Beam Tracking Scheme Using… · p. 6

B. Beam Zooming Mechanism

The DPP structure uses time delays to adjust frequency-dependent beam directions and flexibly set their angular coverage. Lemma 2 shows that the resulting beams can cover the user’s tracking range simultaneously.

B. Beam Zooming Mechanism

  • The DPP structure’s time delays let the base station control beam split and align frequency-dependent beams with multiple physical directions across the tracking range.Unlike a traditional hybrid structure that generates one frequency-independent beam per RF chain, DPP can generate multiple frequency-dependent beams from one RF chain.Source: B. Beam Zooming Mechanism · p. 6
  • TD-provided frequency-dependent phase shifts adjust each beam’s physical direction, but their value is constrained to [−1, 1] for sufficient array gain.The direction can be adjusted from the beam-split-determined direction; outside the stated phase-shift range, a directional beam with sufficient array gain cannot be formed.Source: B. Beam Zooming Mechanism · pp. 6–7
  • Lemma 2 establishes that carefully designed phase shifts and time delays can flexibly zoom the angular coverage of frequency-dependent beams.The M beam directions can cover the whole angular tracking range, allowing multiple potential user directions to be tracked simultaneously.Source: B. Beam Zooming Mechanism · p. 7
  • With θ(0)k = −0.025, αk = 0.025, M = 32, N = 256, fc = 100 GHz, and B = 10 GHz, the generated beams cover [−0.05, 0].The paper presents this beam pattern as an illustration of simultaneous tracking of different physical directions.Source: B. Beam Zooming Mechanism · p. 7, Figure 6 · p. 7

C. Beam Zooming Based Beam Tracking Scheme

The proposed scheme zooms DPP beam coverage to track multiple candidate user directions in each time slot. Across T slots it searches the full variation range, then identifies each user’s direction from received pilot power.

C. Beam Zooming Based Beam Tracking Scheme

  • Each time slot zooms frequency-dependent beam coverage onto a fraction of the tracking range, and the process repeats across T slots until the full range is searched.The scheme chooses the next-frame direction corresponding to the largest received signal power.Source: C. Beam Zooming Based Beam Tracking… · pp. 7–8
  • The scheme builds a target set of directions for the time slots, then designs PS phase shifts and TD time delays so generated beams align with those directions.It transmits training pilots and uses the received signals to identify the user’s direction.Source: C. Beam Zooming Based Beam Tracking… · p. 8
  • Different users’ tracking procedures run simultaneously because their DPP beamforming vectors are generated independently using different TDs and PSs.The scheme supposes orthogonal pilot sequences for different users to mitigate interference when their tracking ranges overlap.Source: C. Beam Zooming Based Beam Tracking… · p. 8
  • Setting θ(0)k = 0 and αk = 1 enables whole-angle beam tracking, which the paper proposes for quickly rediscovering a user after blockage.Multiple directions are tracked simultaneously during this accelerated beam training.Source: C. Beam Zooming Based Beam Tracking… · p. 9
  • At equal training overhead T, the proposed scheme tracks M times as many user physical directions as the typical scheme, improving tracking accuracy while reducing overhead relative to its one-direction-per-slot search.The paper attributes the advantage to actively controlling the angular coverage of DPP beams.Source: C. Beam Zooming Based Beam Tracking… · p. 9

D. Extension to Multi-Antenna Users

For multi-antenna users, the scheme tracks the base-station and user-side directions in sequence, holding the other side fixed. The user-side omnidirectional beam may lose array gain, though the paper says this may have little effect in most scenarios.

D. Extension to Multi-Antenna Users

  • The multi-antenna-user setup assumes each user has Nr antennas, one RF chain, and a DPP structure to mitigate beam split.The tracking target includes physical directions at both the base station and user sides.Source: D. Extension to Multi-Antenna Users · p. 9
  • The extension tracks the base-station direction first with a fixed user-side omnidirectional beam, then tracks the user-side direction while the base station holds its new direction.After both directions are obtained, the base station and user beamform toward them for data transmission.Source: D. Extension to Multi-Antenna Users · p. 9
  • Using an omnidirectional user beam may cause array-gain loss, but the paper expects little effect in most scenarios because the base station has more antennas and the scheme needs only subcarrier-label feedback.The authors suggest a wide beam covering the user-side tracking range as a way to overcome the issue.Source: D. Extension to Multi-Antenna Users · p. 9

IV. PERFORMANCE ANALYSIS

The performance analysis derives a lower bound on beam-training overhead and examines achievable sum-rate, concluding that the scheme can approach near-optimal sum-rate with low overhead. Figure 8 also depicts its three-stage tracking and transmission procedure for multi-antenna users.

IV. PERFORMANCE ANALYSIS

  • The analysis concludes that the scheme can approach near-optimal achievable sum-rate with very low beam-training overhead.It derives a lower bound on required training overhead and analyzes achievable sum-rate.Source: §IV · p. 9
  • In the multi-antenna user procedure, the BS tracks while the user beams omnidirectionally, then the user tracks while the BS beams toward the new direction, followed by data transmission.Figure 8 presents these three stages.Source: Figure 8 · p. 10

A. The Beam Training Overhead

The analysis bounds the training time needed to cover the angular tracking range using frequency-dependent beams. The bound increases with tracking range or bandwidth, while simultaneous tracking can keep overhead below that of a typical scheme in the stated example.

A. The Beam Training Overhead

  • Each time slot's frequency-dependent beams cover a fraction of the angular tracking range, but the DPP's bounded TD phase shifts prevent arbitrary coverage at any overhead.The TD-provided phase shifts satisfy βk,m ∈ [−1, 1], limiting physical-direction adjustment.Source: A. The Beam Training Overhead · p. 10
  • The overhead lower bound is derived by imposing constraints on the TD-delayed periods and selecting the smallest training time that satisfies them.The derivation relates the phase-shift limit to inequalities that must hold across time slots, subcarriers, and directions.Source: A. The Beam Training Overhead · p. 10
  • The lower bound on required training overhead increases as the angular tracking range or system bandwidth grows.The text states that larger range or bandwidth requires longer tracking time.Source: A. The Beam Training Overhead · pp. 10–11
  • For N = 256, Kd = 16, M = 128, αk = 0.1, fc = 100 GHz, and B = 10 GHz, the proposed scheme needs T_min = 2 slots versus T = 52 for the typical scheme.The comparison is reported as an example of the lower bound on required training overhead.Source: A. The Beam Training Overhead · p. 11

B. The Achievable Sum-Rate Performance

The analysis relates achievable sum-rate to beam-tracking accuracy: smaller direction-quantization error supports higher rates. Under the stated assumptions, the proposed scheme's many simultaneous direction tracks can make that error small enough for near-optimal sum-rate, including with adjacent-subcarrier detection errors.

B. The Achievable Sum-Rate Performance

  • Beam-tracking accuracy matters because inaccurate tracking prevents beams from aligning with users, while achievable rate is mainly determined by array gain on the user's physical direction.The analysis derives a lower bound on achievable rate and then on achievable sum-rate.Source: B. The Achievable Sum-Rate Performance · pp. 11–12
  • The achievable rate for each user and subcarrier is defined through its SINR, and the data-period sum-rate is formed from those rates.The analysis uses zero-forcing digital precoding and a diagonal power-normalization matrix under the transmission power constraint.Source: B. The Achievable Sum-Rate Performance · p. 11
  • The rate analysis assumes a single LoS path, high SNR, zero-forcing digital precoding, and users separable in the angle domain.Under angular separability, beams for one user have near-zero array gain on other users' directions.Source: B. The Achievable Sum-Rate Performance · p. 11
  • The scheme tracks M times as many user physical directions simultaneously, yielding a smaller maximum quantization error than the typical scheme and supporting better achievable sum-rate.The lower rate bound is governed mainly by maximum quantization error; increasing tracking accuracy improves the rate bound.Source: B. The Achievable Sum-Rate Performance · p. 12
  • The maximum quantization error is 4.3 × 10^-4; because it is small, the proposed scheme is expected to achieve near-optimal achievable sum-rate performance.Source: B. The Achievable Sum-Rate Performance · p. 12
  • Even with subcarrier-detection errors, the analysis expects near-optimal sum-rate because errors are mainly between adjacent subcarriers and direction error remains bounded with high probability.The bound is expressed using a small ζ̄; ζ̄ = 4 is given for the simulation results in Fig. 10.Source: B. The Achievable Sum-Rate Performance · p. 12

V. SIMULATION RESULTS

The simulations show that beam zooming tracks user directions accurately and achieves near-optimal sum-rate with less training overhead than typical beam tracking. The gains persist across user variation ranges, carrier settings, SNRs, and multi-antenna users.

Simulation setup

  • The simulations use a DPP THz massive MIMO system with N = 256, M = 128, K = 4, Kd = 16, fc = 100 GHz, and B = 10 GHz.They assume one quasi-optical path and a training pilot sequence length Q = 10.Source: §V · p. 12

Tracking accuracy

  • The proposed scheme correctly tracks the user trajectory, with very small, negligible deviation between actual physical directions and tracked physical directions, even for user 4, who changes direction in motion.For SNR larger than 10 dB, the probability that the physical-direction estimation error satisfies the stated bound is larger than 99%.Source: §V · p. 13

Sum-rate performance

  • Nearly 98% of optimal fully-digital achievable sum-rate is reached for T > 2 with the proposed tracking and beam-selection precoding.The measured sum-rate is tightly lower-bounded by the theoretical bound.Source: Figure 11 · p. 13, §V · pp. 13–14

Beam training overhead

  • About 90% less training overhead achieves 10 bits/s/Hz than typical beam tracking, while the proposed scheme has negligible rate gap from perfect-direction beam selection for T > 2 at αk,max = 0.1 or T > 4 at αk,max = 0.2.At equal overhead, it also achieves higher sum-rate than the typical scheme.Source: Figure 12 · p. 14, Figure 13 · p. 14, §V · p. 14

Higher carrier frequency

  • With fc = 300 GHz, B = 5 GHz, and αk,max = 0.2, the scheme still achieves near-optimal sum-rate with less overhead than typical tracking.This case tests performance when beam split is relatively less serious.Source: Figure 14 · p. 14, §V · p. 14

SNR performance

  • For SNR comparisons at T = 5 and αk,max = 0.1, the proposed scheme outperforms typical tracking and approaches optimal fully-digital MMSE precoding.Perfect-direction beam selection under hybrid precoding loses achievable rate from beam-split-related array-gain loss.Source: Figure 15 · p. 15, §V · p. 15

Multi-antenna users

  • With multi-antenna users, the proposed scheme retains near-optimal sum-rate and outperforms typical tracking; the two benchmark schemes cross at SNR = 5 dB.At low SNR, typical tracking error is dominant; at higher SNR, typical tracking more often finds correct directions and DPP can overcome beam-split loss.Source: Figure 16 · p. 15, §V · p. 15

VI. CONCLUSIONS

The paper proposes beam zooming for wideband THz beam tracking with delay-phase precoding, using controlled frequency-dependent beams to track multiple possible user directions per time slot. Theoretical analysis and simulations report accurate tracking with about 90% less training overhead than typical tracking and nearly 98% of optimal sum-rate; future work includes additional THz channel effects and RIS-aided systems.

VI. CONCLUSIONS

  • The delay-phase precoding structure enables beam zooming by controlling angular coverage, allowing one RF chain to track multiple physical directions simultaneously with frequency-dependent beams.The scheme selects the optimal beam after covering the user-direction variation range.Source: §VI · p. 15
  • The scheme tracks user motion accurately with about 90% less beam training overhead than typical tracking and achieves nearly 98% of optimal sum-rate.These results are based on theoretical analysis and simulations.Source: §VI · p. 15
  • Future work will consider angle spread, distance-aware effects, and beam tracking in reconfigurable intelligent surface aided THz massive MIMO systems.Source: §VI · p. 15

APPENDIX A. PROOF OF LEMMA 2

The proof relates delay-induced frequency-dependent phase shifts to beam directions and shows that the resulting beams cover the angular tracking range. The following lemma derives a constraint for DPP beam-split elimination at arbitrary physical directions.

APPENDIX A. PROOF OF LEMMA 2

  • The proof maps the delay-induced phase shift to the physical direction of each frequency-dependent beam, which increases with subcarrier index and spans the angular tracking range.The beam direction is obtained by applying Lemma 1 to the beamforming vector.Source: Appendix A · p. 16

APPENDIX B. LEMMA 3

  • For arbitrary physical directions, eliminating beam split requires the frequency-dependent phase shifts to satisfy the DPP bound βk,m ∈ [−1, 1].The proof converts this bound into a condition involving the ratio of antennas to time delays.Source: Appendix B · p. 16
  • The target direction set uses θ(0)k,i − αk as its minimum element.Source: Appendix B · p. 16

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