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Modeling and Analyzing Millimeter Wave Cellular Systems

Jeffrey G. Andrews, Tianyang Bai, Mandar Kulkarni, Ahmed Alkhateeb, Abhishek Gupta, Robert W. Heath

arXiv:1605.04283v1cs.IT

TL;DR

mmWave cellular analysis must account for severe blocking and strong directionality from large antenna arrays. The paper compares models for these effects and develops a stochastic-geometry baseline for downlink SINR and rate distributions. Its conclusions include accurate blockage-model comparisons and design implications for initial access, densification, and interference.

  • Problem

    mmWave cellular systems require models that capture blocking and large-array directionality, which conventional cellular analysis does not directly address.

  • Method

    The paper compares blockage models and uses hybrid beamforming and stochastic-geometry analysis to model mmWave cellular systems.

  • Results

    3GPP-like blockage models fit the evaluated regions with root mean squared errors of 2.18% for Austin and 1.45% for LA.

  • Takeaways & Limitations

    The analysis supports studying mmWave SINR and rate coverage while accounting for blocking, directionality, and deployment-specific design trade-offs.

  • Takeaways & Limitations

    The evaluated beamforming setup assumes omnidirectional UEs and base stations with a 10 degree 3 dB beamwidth.

Abstract

from arXiv · show

We provide a comprehensive overview of mathematical models and analytical techniques for millimeter wave (mmWave) cellular systems. The two fundamental physical differences from conventional Sub-6GHz cellular systems are (i) vulnerability to blocking, and (ii) the need for significant directionality at the transmitter and/or receiver, which is achieved through the use of large antenna arrays of small individual elements. We overview and compare models for both of these factors, and present a baseline analytical approach based on stochastic geometry that allows the computation of the statistical distributions of the downlink signal-to-interference-plus-noise ratio (SINR) and also the per link data rate, which depends on the SINR as well as the average load. There are many implications of the models and analysis: (a) mmWave systems are significantly more noise-limited than at Sub-6GHz for most parameter configurations; (b) initial access is much more difficult in mmWave; (c) self-backhauling is more viable than in Sub-6GHz systems which makes ultra-dense deployments more viable, but this leads to increasingly interference-limited behavior; and (d) in sharp contrast to Sub-6GHz systems cellular operators can mutually benefit by sharing their spectrum licenses despite the uncontrolled interference that results from doing so. We conclude by outlining several important extensions of the baseline model, many of which are promising avenues for future research.

I. INTRODUCTION

mmWave cellular systems are attractive for expanding spectrum and broadband capacity but differ fundamentally from Sub-6GHz systems because blocking is severe and large antenna arrays are needed for directionality. The paper develops models and stochastic-geometry analysis to quantify SINR and rate and derive design implications.

  • mmWave cellular attracts interest because traditional Sub-6GHz spectrum is scarce while demand for broadband and wireless data services is increasing.
  • Large antenna arrays of small elements provide the strong directionality that distinguishes mmWave cellular systems and affects their modeling, analysis, and design.
  • mmWave links are especially vulnerable to blocking because high penetration loss, weak diffraction, narrow beams, and low SNR make additional signal loss difficult to tolerate.
  • Analytical approach: The paper presents a baseline model and analytical tools for SINR and per-link rate distributions, including SINR coverage, rate coverage, and load-related analysis.
  • Design implications: The analysis highlights novel initial-access requirements, noise-versus-interference behavior under densification, self-backhauling, spectrum sharing, and extensions beyond the baseline model.

A. Pre-cellular mmWave

Before cellular deployment, mmWave was used mainly for specialized long-range, sensing, vehicular, and short-range cable-replacement applications. Measurements, prototypes, and device innovations later established the feasibility of directional outdoor cellular links.

  • Pre-cellular applications: Early mmWave communication applications included long-range LOS links, satellite and backhaul systems, vehicular communication, and short-range 60 GHz cable replacement.
  • 60 GHz consumer systems: 60 GHz WLAN technologies achieved several Gbps over short indoor ranges and stimulated consumer device, RF, and adaptive-array innovation.
  • Channel measurements: Outdoor measurements found acceptable SNR for directional 38 GHz links up to approximately 200 m using 800 MHz bandwidth.
  • Propagation findings: Measurements show sharp LOS/NLOS differences, stronger NLOS attenuation, high penetration loss, lower delay spread, and sparse angular channels at mmWave.

D. Performance analysis

Prior studies motivated tractable mathematical analysis because simulation results were difficult to relate transparently to their many parameters. Stochastic-geometry models incorporate blockage and directionality to analyze SINR, rate, density, and deployment trade-offs.

  • Thermal noise can dominate out-of-cell interference in mmWave coverage, making the systems substantially noise-limited under relevant configurations.
  • Densification: Increasing base-station density improved mean throughput from 3 to 5.8 Gbps and cell-edge rates from 25 to 1400 Mbps in a 0.72 km^2 region.
  • Antenna arrays: Increasing antenna configuration from (4,2) to (32,8) raised mean rates from about 500 Mbps to more than 4 Gbps and cell-edge rates from about 50 to 200 Mbps.
  • Large-array mmWave systems were predicted to provide comparable SINR coverage and much higher rates than conventional cellular networks.
  • Blockage modeling: Shadowing alone cannot accurately capture dense-network blockage because blockage can dramatically change the effective path-loss exponent.

A. 3GPP model for incorporating blockages

The blockage models classify links by LOS probability and apply different path-loss laws, using either standardized distance functions or stochastic building geometry. Random-shape models can derive LOS parameters from building statistics, while frequency-independent mmWave models are supported above 15 GHz.

  • The 3GPP approach maps link distance d to a non-increasing LOS probability P_LOS(d), with environment-specific urban, suburban, and rural forms.
  • LOS and NLOS links must be classified separately because they use different path-loss laws.
  • Above 15 GHz, the narrow Fresnel zone supports using a frequency-independent building-blockage model across mmWave bands, unlike Sub-6GHz.
  • Random shape theory model: Random-shape Boolean models reproduce the negative-exponential suburban LOS form and allow parameter C to be computed from building statistics.
  • Parameter estimation: Equations based on average building area, perimeter, density, and covered-area fraction provide quick LOS-parameter approximations without extensive simulations or measurements.

C. LOS ball model

The paper compares simplified and geometry-based blockage models for mmWave systems, emphasizing tractability, LOS modeling, and agreement with SINR coverage. Simple LOS-ball variants can closely approximate more detailed models, but link-correlation assumptions and environment-specific fitting remain important.

  • C. LOS ball model: The LOS ball model represents the LOS area around a typical user as a ball with maximum LOS length R_B.Its radius can be fitted to match the average LOS area obtained from other blockage models or geographic data.
  • F. Comparison and conclusions on blockage models: At high base-station density, the fitted LOS ball model produces only a minor SINR-distribution gap relative to random shape theory.This supports its use as a simple approximation in dense deployments.
  • C. LOS ball model: A generalized LOS ball model uses a LOS fraction p_l within radius R_B, reducing to the standard model when p_l = 1.It was validated using real building locations in Manhattan and Chicago.
  • C. LOS ball model: The Poisson line model captures correlations in LOS probabilities between links, which prior analyses and simulations ignored, and can change SINR-distribution tail behavior.It models streets as Poisson-line grids and declares outdoor links LOS when their endpoints lie on the same line.
  • F. Comparison and conclusions on blockage models: Matching LOS-probability curves does not guarantee accurate SINR or rate-coverage estimates because several models neglect correlations from shared obstructions.The Poisson line model handles correlations but is difficult to validate because of its specific street and user geometry.
  • F. Comparison and conclusions on blockage models: For real-building comparisons, the generalized LOS ball model accurately fits SINR coverage in Austin and downtown LA, whereas random shape theory can underestimate LA coverage.The generalized model used p_l = 0.3027 for Austin and p_l = 0.2419 for LA; useful ball radii were typically 150–300 m in the studied dense deployments.

IV. NOVEL MODELING ASPECTS: LARGE ANTENNA ARRAYS

MmWave systems rely on large antenna arrays, but hardware constraints make fully digital processing difficult. Analog and hybrid architectures provide practical alternatives, with hybrid precoding retaining multiple-stream capability and near-optimal performance.

  • Hardware constraints and architectures: Large antenna arrays are central to mmWave systems, but their use differs from lower frequencies because of transceiver hardware limitations.Array sizes of 32–256 antennas at base stations and 4–16 at mobile users are suggested in initial research and prototypes.
  • Hardware constraints and architectures: Fully digital processing is difficult at mmWave because one RF chain per antenna requires costly, power-hungry mixed-signal components and many full-resolution ADCs.The text identifies 32–256 full-resolution ADCs at mmWave receivers as presently infeasible.
  • Analog beamforming: Analog beamforming uses phase-shifter networks to steer beams, but phase-shifter constraints limit it to single-stream transmission and complicate multi-user MIMO.Analog beamforming is nevertheless described as commercially viable in the near term.
  • Hybrid precoding: Hybrid architectures divide precoding and combining between analog and digital domains, using NRF ≪ Ntx RF chains while supporting several independent data streams.This enables spatial multiplexing gains with substantially fewer RF chains than antennas.
  • Hybrid precoding: Hybrid architectures were shown to achieve near-optimal performance compared with fully digital transceivers despite using far fewer RF chains.Extending the system analysis to all hybrid-precoding facets remains future work.

B. Spatial channel modeling

MmWave spatial channel models represent propagation through a small number of dominant scattering clusters and directional array responses. Virtual-channel and sectored-beam approximations make these channels tractable for network-level analysis and precoder design.

  • Spatial channel representation: Outdoor mmWave measurements typically show a small number of dominant scattering clusters, motivating geometric sparse-channel models.These models are used for system capacity analysis and precoder design.
  • Spatial channel representation: The channel model combines path fading, distance-dependent path loss, path count, and array response vectors determined by spatial AoAs and AoDs.For a ULA, spatial angle depends on physical angle, antenna spacing, and carrier wavelength.
  • Virtual channel model: The virtual channel representation samples AoAs and AoDs on uniform spatial grids, making sparse mmWave channels suitable for network-level analysis.Most entries are zero or negligible, reducing the representation to distinct propagation paths.
  • Antenna modeling: Analytical models commonly approximate directional array patterns with step functions having constant main-lobe and side-lobe gains.The approximation supports tractable coverage and rate analysis.
  • Beamforming models: Hybrid precoding provides more beamforming freedom than purely analog beamforming, while uniform planar arrays support azimuthal and vertical steering.Existing cellular-network analyses have largely focused on two-dimensional deployments.
  • Precoding design: MmWave hardware constraints make approximate SVD precoding less suitable and motivate low-complexity hybrid precoding algorithms.Prior work uses matching-pursuit and related optimization methods to design RF and baseband precoders.

E. MU-MIMO

The paper models multi-user mmWave transmission with hybrid precoding and directional combining, then analyzes downlink SINR and rate using a tractable stochastic-geometry framework. Directionality, blockage, fading, user geometry, and noise jointly shape performance, with mmWave often operating in a noise-limited regime.

  • MU-MIMO: A two-stage hybrid precoding strategy assigns analog beams to users and applies baseband processing to cancel inter-user interference.It was shown to achieve results very close to unconstrained digital solutions with low training and feedback overhead.
  • MU-MIMO: The multi-user hybrid model serves U users through a hybrid base station and analog-only user combiners, with a lower bound derived for single-path channels.The resulting SINR distribution is analytically tractable in the stochastic-geometry framework.
  • Performance metrics: The baseline analysis focuses on SINR coverage and per-user rate, with rate depending on log(1 + SINR) and the user’s allocated resource fraction.Rate coverage is presented as the more direct performance metric for user experience.
  • Performance metrics: MmWave systems are often noise-limited because path loss and blocking combine with large bandwidths that increase thermal noise power.In this regime, the SNR distribution can approximate the SINR distribution.
  • Baseline model: The baseline model uses a typical user in a Poisson base-station deployment, associates by smallest path loss, and assumes continuously transmitting base stations.The always-active assumption is pessimistic for SINR, while user-load extensions are treated separately for rate distributions.
  • Baseline model: Blocking is represented by independent LOS and NLOS base-station processes whose distance-dependent intensities follow the LOS probability function.Analog beamforming uses perfect desired-link steering, randomly oriented interfering beams, and a sectored antenna approximation.

C. SINR downlink coverage probability

The baseline mmWave analysis computes SINR coverage by modeling blockage, LOS/NLOS base stations, Nakagami fading, and directional interference within a stochastic-geometry framework.

  • Modeling assumptions: Nakagami fading and random directional gains introduce the principal differences from conventional Sub-6GHz analysis.The sectored antenna approximation models interference-channel gain as a discrete random variable.
  • Analytical technique: Alzer’s Lemma converts the gamma-fading calculation into a weighted sum involving exponential random variables, with a generally tight approximation in numerical simulations.This enables tractable evaluation of the SINR coverage expression.
  • Modeling assumptions: The model treats LOS and NLOS base stations as two independent inhomogeneous PPP tiers with distance-dependent densities.This approximation ignores correlations between nearby LOS probabilities.
  • Coverage result: Theorem 1 computes SINR coverage as a mixture of LOS- and NLOS-associated conditional coverage probabilities.The interference Laplace functional is evaluated before deconditioning over link state and serving-link distance.
  • Coverage result: In noise-limited networks, the SINR distribution can be replaced by the simpler SNR distribution; setting σ2 = 0 instead yields the SIR distribution.These are limiting forms of the baseline SINR analysis.

D. Rate coverage probability

The rate-coverage analysis extends SINR coverage by accounting for per-user resource sharing and average load, using tractable approximations for irregular mmWave association cells.

  • Rate model: Per-user rate depends on perceived SINR and the amount of time-frequency resources allocated by the serving base station.Under round-robin scheduling, the load is represented by the number of connected users Ψ.
  • Rate model: Rate coverage is defined as R(τ) = P(R > τ), where τ is the rate threshold in bps.The metric builds on SINR coverage and incorporates the user’s available resources.
  • Load approximation: Joint association-area and SINR distributions are needed because larger association cells imply greater load and longer links, but the required Poisson-Voronoi distribution remains open.Blockage makes mmWave association cells more irregular than conventional Voronoi cells.
  • Load approximation: The tractable model approximates mmWave association areas using typical Poisson-Voronoi cells and assumes independent homogeneous PPP user locations and load-related quantities.The mean number of users in a typical cell is λu/λ, while the serving-cell mean is 1 + 1.28λu/λ.
  • Rate result: Theorem 2 derives rate coverage by combining the load distribution with the SINR coverage from Theorem 1.The infinite summation can be truncated at nmax terms with little accuracy loss.

VI. IMPLICATIONS OF MODELS AND ANALYSIS

The baseline analysis indicates that mmWave systems are often noise-limited, although densification, bandwidth, antenna configuration, loading, and blockage can shift interference behavior.

  • Noise versus interference: mmWave systems are more likely to be noise-limited because beamforming improves SNR while blocking and beam misalignment reduce interference.This differs from the longstanding interference-management emphasis in Sub-6GHz systems.
  • Densification: SINR coverage varies non-monotonically with base-station density, transitioning toward interference-limited behavior once enough interfering base stations are present.No crisp transition threshold exists because many system parameters jointly determine whether SIR or SNR is the better description.
  • Noise versus interference: More building blockages make systems more noise-limited by blocking stronger interfering signals as well as desired signals.The impact of interference is smaller in Los Angeles than Austin because of denser building blockages.
  • Noise versus interference: Interference increases with smaller bandwidth, wider beams, smaller antenna arrays, and more users in MU-MIMO.For 28 GHz in Austin, the simulated INR rises under each of these conditions.
  • Noise versus interference: A factor-of-2 smaller ISD, or four times the base-station density, is required at 73 GHz to produce interference relative to noise comparable to 28 GHz.This comparison holds for both Austin and Los Angeles under the stated criterion.
  • Deployment implications: Initial deployments should prioritize sufficient SNR, while interference becomes significant with extremely dense deployments and heavily loaded cells.Interference may matter earlier when dense deployment is needed to fill coverage holes because of small arrays or high blockage.

B. Initial access in mmWave systems is challenging

Initial access is difficult in mmWave because communication requires beam alignment at both ends despite low pre-alignment SNR, creating substantial search and overhead tradeoffs.

  • Access procedure: Beam alignment is harder than in LTE because beams must be searched at both the base station and UE sides.The challenge is driven by the need for large antenna gains and low pre-alignment SNR.
  • Access procedure: Initial access must establish downlink cell search and uplink random access before communication is possible.The base station and UE must each acquire the other’s signal and suitable beam direction.
  • Beam training: Analog beamforming requires initial access across the entire band, increasing overhead and forcing difficult beam-search tradeoffs.Testing more combinations improves alignment and directionality but consumes slots that could carry data.
  • Beam training: More frequent beam training improves robustness for new or moving users but reduces time available for data transmission.The tradeoff is between connectivity robustness and data-plane resources.
  • Analytical findings: Unless beams are very wide or coherence blocks are very short, exhaustive search with full pilot reuse is nearly as good as perfect beam alignment.The result comes from an effective reliable-rate analysis of beam sweeping and downlink control-pilot reuse.

C. The promise of self-backhauling

Self-backhauling is attractive for mmWave because directional, weakly interfering links can provide in-band wireless backhaul in dense deployments. However, densification eventually increases LOS interference and can reduce SINR and rate performance.

  • The promise of self-backhauling: Self-backhauling lets wired-backhaul base stations provide in-band wireless backhaul to remaining base stations.Directional backhaul links usually do not interfere strongly with access links.
  • The promise of self-backhauling: Increasing the fraction of wired-backhaul base stations improves peak rates, but per-user rate saturates when base-station density grows while wired-backhaul density remains fixed.The saturation density is proportional to the density of wired-backhaul base stations.
  • The promise of self-backhauling: Operators can benefit from uncoordinated spectrum sharing because mmWave directionality keeps inter-operator interference low.Static coordination can further improve overall network performance while differentiating spectrum access.
  • The promise of self-backhauling: Densification initially improves mmWave SINR and rate coverage by shortening serving links and increasing LOS probability, but eventually introduces enough LOS interferers to degrade SINR.If α_L < 2, SINR coverage eventually reaches zero as BS density tends to infinity; if α_L > 2, it converges to a nonzero value.
  • The promise of self-backhauling: The critical density is proportional to average LOS range and decreases with narrower beams because narrower beams reduce interference.The reported average LOS ranges are 43 meters for Los Angeles and 85 meters for Austin.

VII. EXTENSIONS TO THE BASELINE MODEL

The baseline downlink analysis can be extended to uplink, heterogeneous deployments, dual connectivity, and indoor scenarios. These extensions expose practical constraints involving noise limitation, coordination, indoor propagation, and incomplete deployment models.

  • Uplink extensions: Uplink analysis must account for scheduled users whose spatial distribution differs from the downlink base-station process because of Voronoi cell structure.Non-homogeneous PPP models have been proposed for LOS and NLOS uplink interferers.
  • Uplink extensions: The uplink tends to be even more noise-limited than the downlink because mmWave mobile transmit power is expected to be smaller than in Sub-6GHz systems.A simple approximation uses SINR ≈ SNR and substitutes mobile transmit power into downlink coverage expressions.
  • Multi-band coexistence: Sub-6GHz systems can provide control signaling, fallback coverage, and dual connectivity when mmWave access or link quality is inadequate.Users may revert to an LTE macrocell when mmWave link quality falls below a threshold.
  • Multi-band coexistence: Beamforming-aware association increases connections to high-bandwidth mmWave base stations, but excessive bias toward mmWave eventually reduces rate because of weak mmWave-link SNR.An optimum bias exists, after which rate decreases.
  • Multi-band coexistence: Independent PPP assumptions for mmWave and Sub-6GHz base-station locations omit possible co-location and traffic-driven correlation.Future analyses need to consider correlations in both base-station locations and load.
  • Indoor extensions: The outdoor baseline does not model indoor first-order reflections or partition losses, limiting its applicability to indoor coverage.Indoor users away from windows are unlikely to be served by outdoor mmWave base stations and generally require indoor base stations or lower carrier frequencies.

D. MIMO techniques beyond analog beamforming

Beyond analog beamforming, mmWave analyses must address hybrid and multi-user MIMO, channel rank, imperfect channel information, and hardware constraints. These extensions preserve the importance of density and beam alignment while exposing major modeling gaps.

  • MIMO architectures: Multi-user scheduling and beamforming affect cell-edge rates: round-robin scheduling suffers, while denser single-user beamforming can outperform less-dense MU-MIMO at equal power density.These results highlight the importance of user selection when employing MU-MIMO.
  • MIMO architectures: Network-level studies can compare analog, hybrid, low-resolution-ADC, and CAP-MIMO architectures under density, antenna, RF-chain, and power constraints.The virtual-channel representation offers a way to incorporate different precoders and combiners.
  • Channel information: Imperfect channel state information and channel aging are important extensions because incorrect departure or arrival angles can significantly reduce post-processing SINR.Existing stochastic-geometry analyses commonly assume perfect transmitter channel information.
  • Open extensions: The baseline model leaves outdoor-to-indoor coverage, load balancing, offloading, and mobility insufficiently understood.These gaps are especially important because directionality and blocking complicate association and handoff behavior.
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