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The Impact of Beamwidth on Temporal Channel Variation in Vehicular Channels and its Implications

Vutha Va, Junil Choi, Robert W. Heath

arXiv:1511.02937v1cs.IT

TL;DR

The paper addresses whether mmWave vehicular channels vary too rapidly for practical communication and develops models linking beamwidth, pointing error, and coherence. It finds a non-zero optimal beamwidth, a beam coherence time typically much longer than channel coherence time, and better performance from long-term realignment, while noting limited control over beamwidth in existing systems.

  • Problem

    MmWave vehicular communication faces concerns about severe Doppler variation, while beamwidth and receiver-motion pointing error require explicit treatment.

  • Method

    The paper derives channel temporal correlation and coherence-time expressions incorporating Doppler and pointing error, then defines beam coherence time and evaluates realignment duration.

  • Results

    A non-zero optimal beamwidth maximizes channel coherence time, and beam coherence time is typically an order of magnitude longer than channel coherence time.

  • Takeaways & Limitations

    Beams can be realigned every beam coherence time rather than every channel coherence time, supporting lower-overhead long-term realignment.

  • Takeaways & Limitations

    The channel model does not specify carrier frequency, and numerical examples use parameters from the 60 GHz band; beamwidth control may also be limited in practice.

Abstract

from arXiv · show

Millimeter wave (mmWave) has great potential in realizing high data rate thanks to the large spectral channels. It is considered as a key technology for the fifth generation wireless networks and is already used in wireless LAN (e.g., IEEE 802.11ad). Using mmWave for vehicular communications, however, is often viewed with some skepticism due to a misconception that the Doppler spread would become too large at these high frequencies. This is not true when directional beam is employed for communications. In this paper, closed form expressions relating the channel coherence time and beamwidth are derived. Unlike prior work that assumed perfect beam pointing, the pointing error due to the receiver motion is incorporated to show that there exists a non-zero optimal beamwidth that maximizes the coherence time. To investigate the mobility effect on the beam alignment which is an important feature in mmWave systems, a novel concept of beam coherence time is defined. The beam coherence time, which is an effective measure of beam alignment frequency, is shown to be much larger than the conventional channel coherence time and thus results in reduced beam alignment overhead. Using the derived correlation function, the channel coherence time, and the beam coherence time, an overall performance metric considering both the channel time-variation and the beam alignment overhead is derived. Using this metric, it is shown that beam alignment in every beam coherence time performs better than the beam alignment in every channel coherence time due to the large overhead for the latter case.

I. INTRODUCTION

The paper examines mmWave vehicular communication under concerns about severe Doppler variation, showing how directional beams, pointing error, and beam realignment affect temporal performance.

  • Motivation: 30x decrease in channel coherence time is expected when moving from around 2 GHz to 60 GHz under conventional Doppler reasoning.This reduction would challenge physical-layer design.
  • Directional reception: Directional reception limits incoming angles and Doppler shifts, reducing Doppler spread and increasing channel coherence time.The benefit trades off against beam alignment overhead.
  • Beam coherence time: Beam coherence time captures the slower evolution of propagation-path angles relevant to beam alignment.It can be an order of magnitude longer than channel coherence time, motivating long-term rather than short-term realignment.
  • Beam alignment: Long-term beam realignment performs better than realignment every channel coherence time because short-term alignment overhead exceeds its gain.The paper evaluates alignment duration using both channel variation and alignment overhead.
  • Channel coherence time: Pointing error from receiver motion yields a non-zero optimal beamwidth that maximizes channel coherence time, unlike perfect-pointing models.The paper derives channel temporal correlation while accounting for both Doppler and pointing error.
  • Contributions: The paper derives closed-form correlation functions for line-of-sight and non-line-of-sight cases with pointing error included.These functions support derivations of channel coherence time and beam coherence time.

II. MODEL AND ASSUMPTION

The model combines vehicular channel dynamics with receiver-motion pointing error and represents angular power through a beam-filtered von Mises distribution.

  • Model components: The model includes a channel model, receiver-motion pointing error, and a spatial lobe model for angular spread.The spatial lobe model is used to derive beam coherence time.
  • NLOS channel: The NLOS channel follows a narrowband wide-sense stationary uncorrelated-scattering model with a complex channel coefficient.The model assumes stationary scatterers over the considered time scale.
  • Angular model: The effective PAS is defined as P(α|µr) = P′(α)G(α|µr), combining the underlying PAS with the receive antenna pattern.The receive beam points at µr.
  • Angular model: The effective PAS is represented by a von Mises distribution with mean µr and shape parameter kr.For large kr, it is approximated by a Gaussian with variance 1/kr.
  • Beam model: Beamwidth θ is related to the von Mises shape parameter by kr ≃ 1/θ^2.The distribution is chosen for resemblance to real antenna patterns and analytical tractability.
  • LOS channel: The LOS model adds a line-of-sight component to the channel representation used alongside the NLOS model.Angles are defined relative to the receiver’s direction of travel.

K + 1hNLOS(t), (3)

The channel model combines LOS and NLOS components while modeling receiver motion through displacement-induced pointing error. This error changes the effective propagation directions and affects temporal channel correlation.

  • The channel contains both LOS and NLOS components, with the Rician K factor determining their relative power.
  • The motion-induced pointing error is incorporated into the channel model through the change in pointing angle ∆µ(τ).
  • With a fixed receive beam, receiver motion produces a pointing-angle change that causes beam misalignment and alters channel temporal correlation.
  • Receiver motion is modeled as displacement ∆d(τ) = vτ, which changes scatterer distances and the set of observed scatterers.
  • Using f_D = v/λ, the displacement relation becomes ∆d = f_Dλτ, linking receiver motion to Doppler-dependent pointing variation.

C. Channel Spatial Lobe Model

The spatial lobe model statistically describes angular power concentrations produced by multipath clusters. Its lobe width governs beam-alignment difficulty and is central to defining beam coherence time and channel correlation.

  • Channel Spatial Lobe Model: Spatial lobes represent concentrated incoming power from multipath clusters with similar angles of arrival.
  • Channel Spatial Lobe Model: Beam alignment finds the spatial lobe with the highest received power, while narrower lobes make alignment more difficult and motion-induced misalignment easier.
  • Channel Spatial Lobe Model: The lobe width β is modeled with a Gaussian distribution whose mean and standard deviation depend on the propagation environment.
  • Channel Spatial Lobe Model: In urban measurements, the angular-spread parameters are m_AS = 34.8° and σ_AS = 25.7°.
  • Channel Spatial Lobe Model: The study uses amplitude-based channel correlation for coherent detection and derives NLOS correlation functions including receiver-motion pointing error.
  • Channel Spatial Lobe Model: The tractable approximation relies on large k_r and small ∆µ assumptions, and decouples pointing-error effects from Doppler effects.

B. LOS Channel Correlation Function

The LOS correlation is normalized to unity at zero lag and incorporates receiver-motion pointing error. Numerical results examine approximation accuracy and how the Rician K factor and pointing direction shape correlation behavior.

  • LOS Correlation Definition: Receiver-motion pointing error is included in the LOS correlation, and the resulting expression identifies pointing error as the factor affecting LOS correlation.
  • Numerical Verification: The NLOS approximation is evaluated at f_c = 60 GHz, v = 30 m/s, and beamwidth around 8° corresponding to k_r = 50.
  • Numerical Verification: The exact expression, approximate expression, and simulation curves match very well under the stated verification conditions.
  • Effect of K Factor: For µ_r = 10°, channel correlation oscillates, whereas the oscillation is not observed for µ_r = 80°.
  • Effect of K Factor: Channel correlation increases with K regardless of µ_r because the LOS correlation component decreases more slowly than the NLOS component.

IV. CHANNEL COHERENCE TIME

The section defines channel coherence time through a correlation threshold and derives tractable LOS/NLOS expressions under narrow-beam approximations. For small pointing angles, coherence time decreases inversely with beamwidth squared.

  • Channel coherence time Tc is defined as the delay at which channel correlation magnitude falls to a preset threshold R, typically 0.3–0.9.A larger required R produces a smaller Tc for a given channel.
  • Exact coherence-time solutions are intractable because LOS and NLOS correlation components are complex, with the NLOS term involving a Bessel function.The derivation therefore treats LOS and NLOS cases separately to obtain insight into coherence-time bounds.
  • For a general beam direction and beamwidth, the Bessel-function expression remains intractable, so the analysis assumes narrow beams and separates small and non-small pointing angles.The approximation is generally valid up to around 20° beamwidth, matching the narrow beams expected in mmWave systems.
  • 1) When |µr| is small:: When receiver pointing changes are ignored, the resulting expression removes the receiver-motion angular change from the coherence-time calculation.This corresponds to the limiting case Dr,λ → ∞.
  • 1) When |µr| is small:: For small |µr|, the coherence time is inversely proportional to θ^2.This result follows from the small-angle approximation applied to the channel correlation function.

2) When |µr| is not small:

For non-small pointing angles, the paper derives an approximate coherence-time expression by expanding the correlation function and neglecting higher-order terms. The approximation has a parameter-dependent validity range and yields worst-case scaling at 90° pointing.

  • 2) When |µr| is not small:: The non-small-|µr| derivation expands the Bessel-function correlation and neglects higher-order terms to obtain approximate expression (46).The procedure includes logarithmic rearrangement, squaring, and a quadratic approximation in τ.
  • 2) When |µr| is not small:: For fixed µr, expression (46) can become invalid when its denominator is negative.Its valid-solution range increases with µr.
  • 2) When |µr| is not small:: The approximation is valid up to approximately half the pointing angle, µr/2, and therefore covers most beamwidths of practical interest.This practical range is supported by the narrow beams required to compensate for mmWave path loss.
  • 2) When |µr| is not small:: At µr = 90°, the fastest-fading case, coherence time increases on the order of 1/θ for small θ.This is the stated worst-case channel-coherence scaling at perpendicular beam pointing.

B. LOS Channel

The LOS analysis defines beam coherence time from receive-power loss caused by motion-induced pointing error and derives its relation to beamwidth. Numerical comparisons assess channel-coherence approximations across pointing angles.

  • For small pointing angles, the approximation performs poorly at very small θ but still captures the effect of receiver motion.The error becomes more severe as |µr| increases.
  • Under v = 30 m/s, fc = 60 GHz, Dr,λ = 100, and R = 0.5, the small-|µr| approximation is compared numerically with the exact correlation-function solution.The comparison uses µr = 1° and µr = 5° and also includes the no-angular-difference expression.
  • For non-small pointing angles, the approximation diverges at a denominator singularity, while its valid θ range increases with µr and reaches about µr/2.The authors state that this covers most beamwidths of practical interest because mmWave systems use narrow beams.
  • Beam coherence time is the average duration for which the receive beam remains aligned before power falls below ζ of its peak value.The analysis focuses on the receive beam and attributes power loss to motion-induced pointing error.
  • The LOS beam-coherence expression is obtained from the receive beam pattern using the motion-induced pointing change, with kr = 1/θ^2.The resulting expression is closely related to the LOS channel-coherence expression.

B. NLOS Case

The NLOS analysis models angular spread statistically and derives beam coherence time from the combined spatial-lobe and receive-beam patterns. Numerical results show strong dependence on mean arrival angle and near-linear beamwidth scaling in the LOS case.

  • The NLOS derivation combines motion-induced pointing change with Gaussian approximations to the von Mises distributions, yielding a beam-coherence expression in the same form as LOS.The convolution of Gaussian approximations has variance β^2 + θ^2.
  • In the NLOS case, beam coherence time depends on the spatial-lobe angular-spread parameter β, modeled statistically through the angle-of-arrival distribution.The incoming power is modeled as reflection from scatterers represented by von Mises spatial lobes.
  • For the LOS case, beam coherence time is almost linear in beamwidth because the relevant cosine expression admits a first-order approximation.At equal traveled distance, smaller µr causes less pointing-angle change and produces larger TB.
  • For mean arrival angle 80°, increasing beamwidth does not effectively increase TB as it does for mean arrival angle 10°.The NLOS example uses σAS = 25.7° and Dr,λ = 1000.
  • Beam realignment can be performed every beam coherence time rather than every channel coherence time to maximize performance.The stated motivation is the overhead associated with more frequent realignment at channel-coherence intervals.

A. Lower Bound on Mutual Information

The paper derives a mutual-information lower bound that incorporates channel estimation error and temporal channel variation. The bound shows that pilot spacing has an optimum depending on channel variation and arrival angle.

  • The signal model represents each received sample as the transmitted signal multiplied by the channel plus additive noise.
  • A Kalman filter estimates the time-varying channel from equally spaced pilot symbols under a Gauss-Markov model.Pilot spacing is ν samples, and the steady-state estimation-error variance converges to ψ for long sequences.
  • Channel time variation further increases estimation error when pilot-based channel estimates are used to decode data.The additional error is determined by the channel correlation function.
  • At high SNR or with narrow beams, channel time-variation loss acts like interference and cannot be mitigated by increasing transmit power.
  • For θ = 10° and SNRs = SNRv = 0 dB, an optimal pilot spacing ν exists; it increases with coherence bandwidth and decreases in channel time variation measured in samples.The optimal ν also increases as the arrival angle µr decreases.

B. How Often Should the Beams Be Realigned?

The paper compares beam realignment every channel coherence time Tc with realignment every beam coherence time TB, accounting for received-power selection, beam-sweeping overhead, and channel variation. Long-term realignment performs better in the considered NLOS comparisons because its lower overhead outweighs the limited benefit of more frequent sweeping.

  • The two candidate realignment intervals are the channel coherence time Tc and beam coherence time TB, with TB ≫ Tc for NLOS channels.
  • Short-term realignment selects the highest-power path at each Tc, whereas long-term realignment keeps the initially selected beam until the next TB.The long-term scheme can therefore experience suboptimal received power after channel fading changes.
  • The overall performance metric combines channel time-variation loss, temporal efficiency, and the mutual-information lower bound.The temporal efficiencies and metric depend on beamwidth θ.
  • For beamwidths up to 30°, pilot spacing ν = 64 is optimal or incurs negligible loss near the optimum for both realignment schemes.Sensitivity to ν is small near the optimum.
  • Long-term realignment has higher spectral efficiency for path loss ratios Δ = 3 dB and Δ = 10 dB, with a larger gap at Δ = 10 dB.A larger Δ makes beam sweeping less likely to select a suboptimal path, so short-term realignment's added overhead provides insufficient benefit.
  • The short-term scheme is penalized more as Δ increases because its larger beam-sweeping overhead outweighs the smaller benefit from more frequent realignment.

VII. CONCLUSION

The paper accounts for both Doppler variation and receiver-motion pointing error, identifies an optimal beamwidth, and introduces beam coherence time for beam-alignment decisions. It concludes that realignment every beam coherence time offers the best performance in the studied setting, while measured-channel validation remains future work.

  • A non-zero optimal beamwidth maximizes channel coherence time because narrow beams are limited by pointing error while wide beams are limited by Doppler spread.
  • Beam coherence time is a beam-alignment-specific quantity that is typically an order of magnitude longer than conventional channel coherence time.
  • The paper recommends realigning beams every beam coherence time to reduce the overhead associated with realigning every channel coherence time.
  • A natural next step is computing beam coherence times from measured channel data and categorizing them across different environments.Existing vehicular mmWave measurements have mostly characterized path loss because the apparatus had limited control over beamwidth.
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