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Device-to-Device Millimeter Wave Communications: Interference, Coverage, Rate, and Finite Topologies

Kiran Venugopal, Matthew C. Valenti, Robert W. Heath

arXiv:1506.07158v2cs.IT

TL;DR

Dense, nearby wearable networks can experience interference that threatens Gbps communication. This paper analyzes finite mmWave networks with human blockage and antenna effects, finding that interference and blockage both become large at high crowd density while antenna directivity and gain support performance.

  • Problem

    Dense independent wearable networks in close proximity can create interference problematic for applications requiring Gbps throughput.

  • Method

    The paper uses a finite-region stochastic-geometry model for mmWave networks that incorporates human bodies as both device carriers and blockers of interfering signals.

  • Results

    At high crowd density, both interference and interference-signal blockage are large, causing SINR coverage probability to decrease.

  • Takeaways & Limitations

    Antenna main-lobe directivity and array gain play a crucial role in achieving Gbps performance for wearable networks in a crowd.

  • Takeaways & Limitations

    The analysis relies on an assumed spatial model for self-blockage, while a more refined self-blockage model remains an open direction.

Abstract

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Emerging applications involving device-to-device communication among wearable electronics require Gbps throughput, which can be achieved by utilizing millimeter wave (mmWave) frequency bands. When many such communicating devices are indoors in close proximity, like in a train car or airplane cabin, interference can be a serious impairment. This paper uses stochastic geometry to analyze the performance of mmWave networks with a finite number of interferers in a finite network region. Prior work considered either lower carrier frequencies with different antenna and channel assumptions, or a network with an infinite spatial extent. In this paper, human users not only carry potentially interfering devices, but also act to block interfering signals. Using a sequence of simplifying assumptions, accurate expressions for coverage and rate are developed that capture the effects of key antenna characteristics like directivity and gain, and are a function of the finite area and number of users. The assumptions are validated through a combination of analysis and simulation. The main conclusions are that mmWave frequencies can provide Gbps throughput even with omni-directional transceiver antennas, and larger, more directive antenna arrays give better system performance.

I. INTRODUCTION

Wearable networks may operate in dense environments where interference threatens Gbps applications. This paper analyzes finite mmWave networks with human-body blockage and antenna effects to derive and validate coverage and rate expressions.

  • Motivation: Dense wearable deployments can create interference that threatens Gbps-throughput applications such as virtual and augmented displays.Urban train cars are identified as particularly challenging because many independent wearable networks may be located close together.
  • Motivation: MmWave offers short-range, high-rate connectivity through features including large bandwidth, directional transmission, and dense deployability.The paper notes that Wireless HD and IEEE 802.11ad have already enabled commercial mmWave products.
  • Related work: Prior stochastic-geometry analyses commonly assumed infinitely many devices distributed over an infinite area, unlike the finite wearable settings studied here.Existing mmWave work modeled antenna directionality and blockages, but finite-topology analyses did not account for mmWave-specific channel and antenna characteristics.
  • Modeling perspective: Human users both carry interfering transmitters and block some interfering mmWave signals in crowded environments.Human bodies are described as a significant source of mmWave blockage in train cars and airline cabins.
  • Approach: The paper develops coverage and rate analysis for finite regions with finite interferers, including the effects of antenna beamwidth, gain, and randomly located users.It first derives conditional closed-form coverage expressions, then obtains spatially averaged coverage and rate under simplifying assumptions and validates them through repeated-placement simulations.

II. NETWORK TOPOLOGY AND SIGNAL MODEL

The paper models a finite mmWave wearable network with K interferers in a finite area, incorporating human-body blockage, LOS/NLOS channels, fading, random access, and sectorized antenna arrays. The signal model captures antenna gain and directivity while accounting for finite-region boundaries and selected propagation assumptions.

  • Network topology: The network contains a reference receiver and K potentially interfering transmitters positioned in a finite area A, with interferer density λ = K/|A|.Transmitter locations are represented relative to a receiver at the origin, and interferers are assumed coplanar for analytical simplicity.
  • Blockage model: Human users are modeled as diameter-W circular blockages that can obstruct direct transmitter-receiver paths or self-block a transmitter.The model distinguishes LOS and NLOS paths and assigns them different path-loss and fading parameters.
  • Blockage model: Interferers are blocked by other users under a no-self-blocking assumption, using sorted blockages and blocking cones to determine which signals remain unblocked.The procedure identifies initially unblocked transmitters, orders blockages by distance, constructs cones, and tests transmitters against those cones.
  • Antenna model: The sectorized 3D antenna model uses azimuth and elevation half-power beamwidths plus main-lobe and side-lobe gains to approximate practical UPA patterns.An N-element UPA provides the reference case for antenna directivity; one element gives an omni-directional model with unit main- and side-lobe gains.
  • Signal model: The channel model combines Nakagami fading, LOS/NLOS-dependent path-loss exponents, transmit powers, activity probability p_t, and noise normalized without antenna gains.Excluding antenna gains from the noise normalization makes array-related SNR enhancement explicit in the resulting analysis.
  • Assumptions and scope: The reference link is assumed LOS, while finite-area boundaries exclude external interference and reflections are represented only coarsely through LOS/NLOS parameters.The paper notes that a refined model should capture self-blockage of the reference link and that omitted reflections are most plausible for poorly reflecting boundaries.

III. INTERFERENCE MODEL

The interference model conditions on transmitter and blockage locations, then derives SINR coverage by combining random activity, antenna orientation, blockage state, fading, and LOS/NLOS channel parameters. A general formulation permits per-interferer channel parameters, while fixed LOS/NLOS parameters enable tractable spatial averaging.

  • SINR formulation: Conditioned on transmitter and blockage locations, the analysis derives the SINR complementary distribution function and coverage probability.The formulation adapts prior stochastic-geometry analysis to the finite wearable-network setting.
  • Channel parameterization: The general analysis allows individual interferers to have separate path-loss exponents α_i and Nakagami factors m_i, without requiring fixed LOS or NLOS parameter pairs.Fixing channel parameters by LOS/NLOS class yields tractable expressions for spatially averaged SINR coverage.
  • Interferer power: Each interferer contributes a discrete relative power gain that is zero when inactive and otherwise depends on whether the receiver lies within its random antenna main lobe.Activity is governed by p_t, while antenna orientation is randomized over azimuth and elevation.
  • Interferer orientation and access: Random antenna orientations are justified by user motion and by multiple wearable devices having different orientations and random activity.The model assumes one device per user transmits at a time, while devices belonging to different users may collide.
  • Reference link: The reference transmitter is assumed within the reference receiver’s main beam, and short reference-link distance makes moderate pointing errors relatively unimportant under the sectorized model.The paper states that a beam offset of half the beamwidth can leave performance unchanged in this model.

A. Coverage Probability

The section derives conditional SINR coverage probability by modeling the desired signal and interference contributions under a fixed interferer geometry. An exact expression requires integer Nakagami parameter m0; otherwise, only an upper bound is available.

  • Conditional coverage probability is defined as the probability that SINR exceeds threshold β for a given geometry Ω.
  • Assuming positive integer m0, the desired-signal variable S is gamma distributed, enabling the exact SINR coverage derivation.
  • The derivation expands interference terms using the binomial theorem, multinomial sequences, and independence of the interference variables.
  • When m0 is not an integer, exact evaluation is unavailable to the authors’ knowledge, and only an upper bound can be obtained.

B. Ergodic Spectral Efficiency

The section connects SINR coverage to rate coverage through the logarithmic spectral-efficiency mapping and derives ergodic spectral efficiency by integrating coverage over SINR thresholds. Practical rate limits bound the integration range.

  • Spectral efficiency is η = log2(1 + γ) bits per channel use when SINR is γ.
  • Rate coverage at spectral-efficiency threshold η is equivalent to SINR coverage at threshold 2^η − 1.
  • Ergodic spectral efficiency is obtained from the coverage probability using a change of variables between η and β.
  • Practical modulation, RF-front-end, and receiver-sensitivity limits impose minimum and maximum SINR thresholds βmin and βmax.
  • The ergodic-efficiency integral can be computed numerically by evaluating coverage on a fine β grid and applying the trapezoidal rule.

IV. NUMERICAL RESULTS FOR FIXED GEOMETRY

For a fixed finite geometry, the numerical evaluation studies coverage and spectral efficiency across transmission probabilities and antenna configurations. Larger, more directive arrays improve rate, while higher interferer activity lowers coverage.

  • The analysis assumes co-located blockage and interferer locations, neglects self-blocking, and assigns equal transmit power to all interferers.
  • Array windowing can improve side-lobe isolation but adds design complexity, and its suitability for wearable devices remains uncertain because of power and heating concerns.
  • The fixed geometry uses a 7 × 7 lattice restricted to an annulus; with 0.6 m spacing and n = 7, it contains K = 36 interferers at density λ = 2.25 passengers/m2.
  • Higher transmission probability pt produces lower coverage probability for a given SINR threshold, including with omni-directional antennas.
  • Larger antenna arrays significantly improve rate by increasing desired-link gain and reducing the likelihood that interfering main-lobes point toward the reference receiver.
  • Larger transmitter arrays Nt are more advantageous than larger receiver arrays Nr for the fixed geometry considered.
  • The asymmetric Nt–Nr behavior is attributed to the lower probability that interferers point their main-lobes toward the reference receiver when Nt is large.

V. SPATIAL AVERAGING FOR RANDOM GEOMETRIES

For random user locations in a finite region, the paper spatially averages SINR coverage and rate over network realizations. It derives closed-form expressions by removing conditioning on interferer locations and blockages, then validates them against simulation.

  • In the fixed-geometry results, spectral efficiency improves significantly with more antennas, and contour plots examine ergodic efficiency over Nt and Nr.
  • The random-geometry analysis considers a finite mmWave device-to-device network with users placed at random locations.
  • Users are independently placed, allowing overlaps for mathematical tractability even though real users are generally spaced apart.
  • Spatially averaged SINR coverage is obtained by taking the expectation of conditional coverage over network geometries.
  • Simulation repeatedly samples K interferers from a binomial point process and computes coverage and rate for each realization.
  • Simulation approaches exact spatial averages with infinitely many trials but is computationally expensive.
  • The analytical approach develops a closed-form spatially averaged SINR CCDF by unconditioning results over interferer locations and blockages, then validates it against simulation.

A. Assumptions

The paper adopts sequential assumptions to make finite-network blockage and interference analysis tractable while retaining distance-dependent blocking. Simulations closely match the analytical blockage approximations, supporting the modeling sequence.

  • Motivation: Users can both generate interference and block interfering signals, motivating a sequence of simplifying assumptions for tractable analysis.The model explicitly treats users as potential interference sources and blockage sources.
  • Assumption 1: Orbital model: The orbital model places each transmitter randomly on a circle around its associated blockage, making blockage depend on relative geometry.A transmitter is positioned randomly on a circle of radius d centered at its blockage, with d > W/2.
  • Assumption 2: Independent point processes: Independent blockage and interferer point processes provide a tractable stepping stone, allowing both sets of K locations to be generated independently.The corresponding simulation lays out K interferers and K blockers independently.
  • Assumption 3: Independent blockage states: Independent blockage states simplify the coupling between interferers and blockages, although nearby transmitters may be correlated in reality.The paper explicitly notes that a transmitter close to a blocked transmitter is likely also to be blocked.
  • Assumption 4: LOS ball: The LOS-ball approximation treats interferers inside RB as LOS and those beyond RB as NLOS, with RB obtained by matching mean non-blocked interferers or average rate.The irregular random LOS boundary is replaced by an equivalent ball.
  • Validation: Simulation closely matches the analytical blockage approximations, while larger K or W increases blockage probability at a given distance r.The comparison uses an annulus with rin = 1 and rout = 7.

C. Analysis of Coverage Probability

Under the stated assumptions, the paper derives the distribution of each interferer's effective received-power factor and uses it to obtain spatially averaged SINR coverage and ergodic spectral efficiency.

  • Effective interferer distribution: The analysis derives the distribution of each effective interferer factor Ωi under assumptions 1–4.The resulting variables depend on interferer locations and incorporate the relevant LOS/NLOS cases.
  • Coverage probability: A closed-form spatially averaged SINR CCDF is computed by evaluating the expectation of the per-interferer terms over Ωi.The expectation EΩi[Gti(Ωi)] is evaluated using the derived formulation.
  • Ergodic spectral efficiency: Numerical integration of the SINR expression yields the spatially averaged ergodic spectral efficiency.The SINR CCDF appears in the integrand used for the spectral-efficiency calculation.

VI. RESULTS FOR RANDOM GEOMETRY

Random-geometry simulations and analytic results validate the proposed expressions and reveal how antenna configuration, blockage, interferer density, and noise shape performance. Larger transmit arrays and larger blockage diameters improve outcomes, while high density initially reduces SINR coverage.

  • Setup: Simulation and numerical results evaluate coverage probability and spectral efficiency for users distributed in a finite annular region.The region uses inner radius rin = 0.3 m and outer radius rout = 2.1 m.
  • Antenna configurations: Larger transmit arrays outperform larger receive arrays, and performance is not symmetric with respect to Nt and Nr.This behavior appears for configurations with the same Nt × Nr product and is summarized across antenna arrays.
  • Modeling limitation: The assumptions treat blockage and user locations as independent even though the orbital model makes them dependent in reality.This is presented as a modeling limitation while the corresponding approximations are evaluated through simulation.
  • Validation: The analytic expressions match the SINR CCDF simulation exactly under the LOS-ball setting, supporting the approximation used for coverage analysis.The comparison uses analytic expressions against simulations under assumptions 1 and 4.
  • Blockage diameter: Larger blockage diameter improves throughput because interfering signals are more effectively blocked.The throughput is defined as pt times ergodic spectral efficiency; the illustrated case uses Nt = Nr = 4.
  • Interferer density: As interferer density λ increases, SINR coverage decreases rapidly at first, then declines more slowly because blockage probability also rises.This behavior supports modeling users carrying interferers as the source of blockages in the indoor wearables environment.
  • Noise and random access: At low noise, changing random-access probability significantly affects performance, whereas at higher noise the system becomes noise limited and the effect becomes small.The latter regime produces little change in the SINR distribution and ergodic spectral efficiency.

VII. CONCLUSION

The paper analyzes finite, crowded mmWave wearable networks, accounting for interference, human-body blockage, antenna parameters, and finite user populations. Closed-form coverage results agree with simulations, while antenna directivity and array gain support high-rate operation.

  • VII. CONCLUSION: The model characterizes mmWave wearable-network performance with finite interferer numbers and a finite spatial region.It targets SINR, coverage, and rate in crowded environments.
  • VII. CONCLUSION: Human bodies both contribute interference through carried devices and block some interfering signals.The analysis uses different path-loss and small-scale-fading parameters for blocked and unblocked links.
  • VII. CONCLUSION: High crowd density produces large interference and blockage probabilities, causing SINR coverage probability to decrease at a much lower rate.The conclusion considers both fixed and random interferer positions.
  • VII. CONCLUSION: Antenna array gain and beamwidth affect coverage and ergodic spectral efficiency, with main-lobe directivity and array gain crucial for gigabits-per-second performance.The study evaluates these antenna parameters directly.
  • VII. CONCLUSION: Closed-form expressions are obtained for spatially averaged coverage probability when a user is centered in a dense crowd with finitely many users.The analytic results and simplifying assumptions are confirmed against simulations.
  • VII. CONCLUSION: The model is a first step toward finite-region mmWave ad-hoc SINR analysis and can avoid simulations for performance prediction.Future refinements include 3D device locations, boundary reflections, and self-blockage of the reference link.

APPENDIX A

Appendix A derives the blocking-region geometry for an interferer in a finite annular network. It converts uniformly placed blockages into blockage probabilities using the blocking-region area, including two boundary cases.

  • APPENDIX A: An interferer is blocked when a blockage lies within its blocking region between the transmitter and reference receiver.The blocking-region geometry depends on the interferer distance and the annulus boundaries.
  • APPENDIX A: Uniformly random blockages yield a blockage probability equal to the blocking-region area divided by the annular network area.This converts the geometric construction into the distance-dependent probability pb(r).
  • APPENDIX A: The interior-case area is decomposed into a sector, two right triangles, and a semicircular disk.Fig. 17 evaluates this area over rin ≤ r ≤ rout − W.
  • APPENDIX A: The boundary-case area uses the corresponding region B1 and accounts for independent blockages through the probability that no blockage lies in the blocking region.The resulting expression is stated in Lemma 1.
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