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Modeling the Random Orientation of Mobile Devices: Measurement, Analysis and LiFi Use Case

Mohammad Dehghani Soltani, Ardimas Andi Purwita, Zhihong Zeng, Harald Haas, Majid Safari

arXiv:1805.07999v4cs.IT

TL;DR

Device orientation is inadequately modeled in LiFi despite its effect on optical channel behavior and mobility management. The paper uses measurements from forty participants to derive orientation, channel-gain, SNR, and mobility models. Static polar angles fit Laplace distributions, mobile polar angles fit Gaussian distributions, and random orientation significantly increases handover rates.

  • Problem

    LiFi studies often lack appropriate device-orientation statistics and assume a vertically upward, fixed receiver, limiting orientation-aware system and handover analysis.

  • Method

    The paper combines measurements from forty participants with analytical PDF derivations, truncated-Laplace approximation, channel-gain statistics, correlated orientation processes, and an orientation-based random waypoint model.

  • Results

    Static polar angles fit Laplace distributions, mobile polar angles fit Gaussian distributions, and random orientation significantly increases handover rates across evaluated conditions.

  • Takeaways & Limitations

    Random device orientation should be included in optical wireless channel and handover analyses rather than treating the receiver as vertically upward and fixed.

Abstract

from arXiv · show

Light-fidelity (LiFi) is a networked optical wireless communication (OWC) solution for high-speed indoor connectivity for fixed and mobile optical communications. Unlike conventional radio frequency wireless systems, the OWC channel is not isotropic, meaning that the device orientation affects the channel gain significantly, particularly for mobile users. However, due to the lack of a proper model for device orientation, many studies have assumed that the receiver is vertically upward and fixed. In this paper, a novel model for device orientation based on experimental measurements of forty participants has been proposed. It is shown that the probability density function (PDF) of the polar angle can be modeled either based on a Laplace (for static users) or a Gaussian (for mobile users) distribution. In addition, a closed-form expression is obtained for the PDF of the cosine of the incidence angle based on which line-of-sight (LOS) channel gain is described in OWC channels. An approximation of this PDF based on the truncated Laplace is proposed and the accuracy of this approximation is confirmed by the Kolmogorov-Smirnov distance (KSD). Moreover, the statistics of the LOS channel gain are calculated and the random orientation of a user equipment (UE) is modeled as a random process. The influence of the random orientation on signal-to-noise-ratio (SNR) performance of OWC systems has been evaluated. Finally, an orientation-based random waypoint (ORWP) mobility model is proposed by considering the random orientation of the UE during the user's movement. The performance of ORWP is assessed on the handover rate and it is shown that it is important to take the random orientation into account.

I. Introduction

LiFi offers high-speed indoor optical connectivity, but device orientation remains insufficiently modeled despite its importance for channel analysis and mobility management. The paper develops measurement-based orientation modeling and extends it to channel statistics and mobility.

  • I. Introduction: LiFi uses visible light for downlink communication and illumination, with infrared available for uplink connectivity.The technology is bidirectional, high-speed, and fully networked.
  • I. Introduction: The literature lacks appropriate statistics and rotation models for device orientation in system design and handover management.Prior studies often assumed a vertically upward, fixed receiver because a proper model was unavailable.
  • I. Introduction: The paper bases a new orientation model on practical measurements and derives PDFs and statistics for device orientation, LOS channel gain, and received SNR.The contributions explicitly include measurements, model derivation, and channel-gain and SNR statistics.
  • I. Introduction: The proposed orientation-based random waypoint model incorporates UE orientation during movement and is also applicable to mmWave networks.The model is motivated by the importance of angle of arrival in mmWave systems.
  • I. Introduction: The orientation geometry represents yaw about z, pitch about x, and roll about y, with the rotated device normal obtained by concatenating the three rotations.The polar angle depends only on pitch and roll, which are associated with wrist movements.

III. Mobile Device Orientation Statistics

Measurements from 40 participants produced 222 orientation datasets for sitting and walking smartphone use. Azimuth is approximately uniform, while polar-angle distributions differ by mobility condition: Laplace for sitting and Gaussian for walking.

  • Experimental setup: 40 participants produced 222 orientation datasets covering portrait and landscape use while sitting and walking.The measurements used instantaneous rotation angles collected with the Physics Toolbox Sensor Suite.
  • Experimental setup: The experiment sampled browsing and video-watching activities in portrait and landscape modes, including walking along a 40 m × 1.5 m corridor.Sitting activities included repeated browsing and streaming-video sessions; walking activities followed a straight corridor path.
  • Azimuth statistics: Azimuth closely follows a uniform distribution, with KSD 0.034 for sitting and 0.019 for walking.Measured sitting and walking skewness values were −0.03 and −0.0045, while kurtosis values were 1.68 and 1.85, respectively.
  • Polar-angle statistics: The polar-angle model uses Laplace and Gaussian distributions, selected by comparing empirical data with fitted distributions.KSD measures the maximum absolute distance between two cumulative distribution functions; smaller values indicate greater similarity.
  • Polar-angle statistics: Sitting polar angles are better matched by Laplace, whereas walking polar angles are better matched by Gaussian according to KSD and kurtosis.Sitting kurtosis is 6.36 and walking kurtosis is 3.77; Laplace and Gaussian kurtoses are 6 and 3, respectively.
  • Polar-angle statistics: Mean polar angles are about 41° for sitting and 30° for walking, with both standard deviations below 9°.The paper mainly uses the Laplace model through Section VI for sitting and the Gaussian model in Section VI for mobile users.
  • Polar-angle statistics: A truncated Laplace distribution is used to fit the polar angle over the experimentally restricted range [0, π/2].The fitted mean and scale parameters are taken from Table I, and the resulting density is approximately normalized over the interval.

IV. Analysis of Random Orientation Effect on Channel Gain

The paper analyzes how random UE orientation changes the LOS channel gain in a downlink optical wireless channel. For a fixed UE position, the gain statistics depend strongly on orientation, while radiance angle remains unaffected by it.

  • Channel-gain dependence: For a fixed UE position, LOS channel-gain statistics depend strongly on the device orientation.The geometry uses UE and AP positions, vertical separation h, distance d, LED half-power semiangle φ1/2, PD field of view Ψc, and incidence angle ψ.
  • Channel-gain dependence: Random orientation affects cos ψ, whose statistics are therefore determined before deriving the LOS channel-gain statistics.The radiance angle φ is not affected by random orientation, so cos φ remains fixed for a given UE–AP geometry.

A. PDF of cos ψ

The incidence-angle variable is expressed through the UE orientation and analyzed conditionally on azimuth. Its PDF is derived in separate monotonic and single-peak cases, with measured orientation approximations supporting the chosen angular representation.

  • Geometric formulation: The PDF of cos ψ is calculated conditioned on azimuth ω.This conditioning isolates the incidence-angle distribution for a specified azimuthal orientation.
  • Geometric formulation: The cosine of the incidence angle ψ is obtained from the rotated UE receiver normal and the UE-to-AP distance vector.The derivation uses spherical-coordinate angles and the azimuth angle ω, which depends on the measured rotation angles.
  • Orientation approximation: Measured angular variability supports approximating roll γ as nearly zero in portrait mode and pitch β as nearly zero in landscape mode.The reported standard deviations are 0.072 and 0.113 radian for γ, and 0.044 and 0.091 radian for β, across sitting and walking conditions.
  • Orientation approximation: Approximating ω by the simplified angular forms has mean absolute errors of about 0.09 radian for sitting and 0.14 radian for mobile users.The paper uses Ω in later equations because it has a better physical interpretation than the approximated or measured azimuth forms.
  • PDF derivation: The analysis assumes b > 0 and constrains the coefficients by −1 < a < 1 and 0 < b ≤ 1.The condition b > 0 follows from the AP being located above the UE.
  • PDF derivation: The coefficients a and b determine whether cos ψ decreases monotonically with θ or forms a concave-down curve with one peak.The two resulting cases are analyzed separately to derive the PDF of cos ψ.
  • PDF derivation: For a < 0, b > 0, and 0 < θ < π/2, cos ψ is a monotonically decreasing function of θ.This monotonic case is used to calculate the corresponding transformed density.

1) For a < 0 (Case 1):

For a < 0, the exact PDF of cos ψ is defined on (a, b), while a first-order Taylor approximation yields a truncated-Laplace model whose accuracy is evaluated against measurements.

  • 1) For a < 0 (Case 1):: For a < 0, the PDF of cos ψ has support on the interval (a, b), with supremum b strictly less than 1.
  • 1) For a < 0 (Case 1):: The closed-form expressions for the PDF of cos ψ apply whether θ follows a Laplace or Gaussian distribution.
  • 1) For a < 0 (Case 1):: Under small variance, cos ψ is linearized in the zero-mean Laplace variable θ′, producing the approximation ĝ(θ′).
  • 1) For a < 0 (Case 1):: The resulting approximation models cos ψ with a truncated Laplace distribution over the bounded interval [τ̂min, τ̂max].
  • 1) For a < 0 (Case 1):: 0.055 and 0.026 are the average KSDs for the proposed truncated-Laplace approximation and exact PDF, respectively, across room positions in a 10×10 m2 room.
  • 1) For a < 0 (Case 1):: The first-order Taylor approximation closely matches experimental measurements for stationary sitting users across different UE positions.

C. Behavior of the PDF of cos ψ

The PDF of cos ψ is analyzed through its support, critical points, continuity, and exponential tail behavior, with separate cases determined by the sign of a.

  • C. Behavior of the PDF of cos ψ: The analysis separates the PDF behavior into case 1, a < 0, and case 2, a ≥ 0.
  • C. Behavior of the PDF of cos ψ: For a < 0, τ∗ is the global maximum, and the PDF increases exponentially before τ∗ and decreases exponentially afterward under Proposition 1’s condition.
  • C. Behavior of the PDF of cos ψ: For an indoor L×L m2 room with AP-user height difference h = 2 m and L = 20 m, bθ must be below 0.1414 rad, a condition satisfied by the measurements.
  • C. Behavior of the PDF of cos ψ: For a ≥ 0, the PDF has an exponentially increasing lower tail, but its higher tail is not always exponentially decreasing.
  • C. Behavior of the PDF of cos ψ: When the UE faces the AP along Cw, the approximation becomes a discrete random variable and produces a sudden KSD jump, although Cw has zero area.

V. The Statistics of Channel Gain

This section derives channel-gain statistics under random UE orientation, including conditional and location-dependent PDFs, and models orientation-induced channel variation over time. Analytical PDFs agree with measurement-based simulations, while random orientation produces location- and direction-dependent fading behavior.

  • Channel-gain PDF: The conditional PDF of LOS channel gain is derived for a given UE location and direction, then extended to joint location–orientation statistics.The formulation uses H = H0 cos ψ within the receiver field of view and H = 0 outside it.
  • Channel-gain PDF: Equation (28) provides a simpler approximation for calculating LOS channel gain and related derivations instead of the more complicated exact expressions.The approximation is introduced after justifying the PDF approximation for cos ψ.
  • Validation: The analytical LOS-channel-gain PDF matches measurement-based simulations across different room positions with Ω = π.The Dirac delta magnitude is nearly zero in Fig. 5-(a) and Fig. 5-(c), and equals 0.0336 and 0.006 in Fig. 5-(b) and Fig. 5-(d).
  • Temporal orientation: Random orientation is represented as a stochastic process whose pitch, roll, and polar-angle fluctuations affect normalized LOS channel gain at multiple UE locations.The measured mean pitch is E[β] = −35.81° and the roll angle fluctuates around zero.
  • Temporal orientation: The polar-angle coherence time is a few hundred milliseconds, whereas orientation-induced LOS-gain coherence is a few tens of milliseconds and can be modeled as slow large-scale fading.The channel gain is therefore approximately constant over many transmitted symbols.

VI. Orientation-based Random Waypoint

This section extends random waypoint mobility by incorporating UE orientation during movement. The resulting model represents motion between random waypoints together with time-varying orientation samples.

  • Random waypoint foundation: The conventional RWP model selects random destinations and moves users at constant speed along straight lines between waypoints.Destinations are uniformly randomly distributed in the room area.
  • Motivation: The model is motivated by the fact that orientation changes, alone or combined with mobility, can cause handovers absent under constant UE orientation.This provides a framework for more realistic mobile-wireless-network analysis.
  • Algorithm: The ORWP procedure initializes UE position, speed, polar-angle statistics, coherence time, and simulation runs before generating waypoint transitions.It updates position using the movement direction and coherence-time interval, then generates orientation samples.
  • Orientation-based extension: ORWP combines waypoint movement with the UE’s polar-angle random process during each movement between consecutive waypoints.The orientation process is returned together with consecutive positions, speed, and orientation samples as ORWP specifications.

A. Correlated Gaussian Random Process

This section generates a correlated Gaussian polar-angle process for walking users and embeds it in the orientation-based random waypoint model. The process is calibrated to measured moments and coherence time rather than an exact autocorrelation function.

  • Model choice: Walking measurements show that the polar angle follows a Gaussian distribution and that adjacent orientation samples are correlated.A correlated Gaussian random process is therefore required to match the experimental orientation behavior.
  • AR process: The AR model uses a bias term, autoregressive coefficients, and white-noise variance to generate successive polar-angle samples.The general AR(p) formulation contains p + 2 unknown parameters, including c0, c1, …, cp, and σ_w^2.
  • AR process: A first-order AR(1) model is sufficient because the generated process is matched to measured moments and coherence time rather than its exact ACF.The AR order is set to p = 1.
  • ORWP integration: The generated process is inserted into ORWP after its parameters are determined from the measured mean, variance, and coherence time.The resulting samples specify the UE orientation during waypoint movement.
  • Handover evaluation: Random orientation significantly increases handover rate across the evaluated room dimensions and UE speeds.Handover rate generally decreases with room length and increases with speed, with the orientation effect more pronounced in smaller networks.

Appendix

The appendix derives the exact PDF of cos ψ by transforming the polar-angle distribution, treating separately the cases a < 0 and a ≥ 0 under b > 0.

  • Case classification: The derivation assumes b > 0 and classifies the transformation according to whether a is negative or nonnegative.For a < 0 the transformation is monotonically decreasing, while for a ≥ 0 it is concave downward.
  • Case 1: For a < 0, the transformed distribution uses the inverse of the angle-to-cos ψ mapping and its Jacobian over the valid interval.Combining this Jacobian with the polar-angle distribution yields equation (19).
  • Case 2: For a ≥ 0, the domain is divided into two intervals where the transformation is one-to-one, and inverse functions are derived for each interval.Both inverse functions share the same Jacobian, leading to equation (22).

B. Proof of Proposition 1

The proof establishes that fcos ψ is well defined and continuous at its peak, increasing exponentially before a global maximum and decreasing exponentially afterward under stated inequalities.

  • Critical point: The critical point τ∗ lies within the support and produces a peak for fcos ψ when 0 < µθ < π.This peak is analogous to those of Laplace or Gaussian distributions.
  • Support and continuity: fcos ψ is well defined on a < τ < b and continuous at τ = τ∗.
  • Monotonicity analysis: The positivity and monotonicity of the exponential terms reduce the monotonicity analysis to the coefficients C1(τ) and C2(τ).
  • Monotonicity result: Under the stated inequalities, fcos ψ increases exponentially for a < τ < τ∗ and decreases exponentially for τ∗ < τ < b.
  • Monotonicity result: Consequently, τ∗ is the global maximum of fcos ψ over (a, b).

C. Proof of Proposition 2

The proof characterizes a second support case in which the peak may lie outside the support, with monotonic behavior determined by coefficient signs and possible divergence at an endpoint.

  • Higher-tail analysis: The region Cw is introduced so the CDF of cos ψ is nearly one near its maximum value for the higher-tail analysis.
  • Case 2 support: When τ∗ is outside the support, fcos ψ may have no interior peak and can increase toward infinity at the support boundary.For (xu, yu) ∈ Cw, only a monotonic increasing function remains.
  • Case 2 support: For min{a, b} < τ < τ∗, C1(τ) is always positive, so fcos ψ is monotonically increasing.
  • Coefficient analysis: In case 2, the behavior of C2(τ) determines whether fcos ψ remains monotonic, and τd separates coefficient-sign regimes.
  • LOS channel gain: The PDF of the LOS channel gain is derived from the CDF, with support hmin ≤ ℏ ≤ hmax and normalization chosen to ensure unit integral.The derivation defines hn ≜ H0/dm+2 and accounts for a Dirac delta caused by a CDF discontinuity at ℏ = 0.
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