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Coverage in Multi-Antenna Two-Tier Networks

Vikram Chandrasekhar, Marios Kountouris, Jeffrey G. Andrews

arXiv:0902.3210v5cs.NIcs.IT

TL;DR

Universal frequency reuse makes cross-tier interference a major obstacle to uniform coverage in two-tier cellular systems. This paper derives analytical coverage results with spatial diversity and proposes decentralized, location-assisted femtocell power control; the scheme provides uniform cell-edge cellular and hotspot coverage.

  • Problem

    Cross-tier interference in universal-reuse two-tier cellular systems prevents uniform coverage, including unacceptable interference for macrocell-edge users.

  • Method

    The paper derives analytical coverage zones incorporating spatial diversity and proposes decentralized carrier-sensing power control for femtocell transmissions.

  • Results

    Single-user multiple-antenna transmission provides significantly superior coverage and spatial reuse, while multiuser transmission suffers from residual cross-tier interference.

  • Takeaways & Limitations

    Location-assisted, decentralized femtocell power regulation provides uniform cellular and hotspot coverage at the cell edge.

Abstract

from arXiv · show

In two-tier networks -- comprising a conventional cellular network overlaid with shorter range hotspots (e.g. femtocells, distributed antennas, or wired relays) -- with universal frequency reuse, the near-far effect from cross-tier interference creates dead spots where reliable coverage cannot be guaranteed to users in either tier. Equipping the macrocell and femtocells with multiple antennas enhances robustness against the near-far problem. This work derives the maximum number of simultaneously transmitting multiple antenna femtocells meeting a per-tier outage probability constraint. Coverage dead zones are presented wherein cross-tier interference bottlenecks cellular and hotspot coverage. Two operating regimes are shown namely 1) a cellular-limited regime in which femtocell users experience unacceptable cross-tier interference and 2) a hotspot-limited regime wherein both femtocell users and cellular users are limited by hotspot interference. Our analysis accounts for the per-tier transmit powers, the number of transmit antennas (single antenna transmission being a special case) and terrestrial propagation such as the Rayleigh fading and the path loss exponents. Single-user (SU) multiple antenna transmission at each tier is shown to provide significantly superior coverage and spatial reuse relative to multiuser (MU) transmission. We propose a decentralized carrier-sensing approach to regulate femtocell transmission powers based on their location. Considering a worst-case cell-edge location, simulations using typical path loss scenarios show that our interference management strategy provides reliable cellular coverage with about 60 femtocells per cellsite.

I. INTRODUCTION

The paper studies coverage and spatial reuse in two-tier networks sharing spectrum, focusing on how cross-tier interference and antenna transmission strategies affect per-tier QoS. It also asks how femtocell density and power adaptation can preserve cellular coverage.

  • I. INTRODUCTION: Universal frequency reuse improves spectrum efficiency but creates cross-tier interference because independently deployed femtocells lack scalable coordination.The interference can become capacity-limiting, especially for cellular users near the macrocell edge.
  • I. INTRODUCTION: Multiple antennas are evaluated under single-user and multiuser transmission to determine how spatial diversity changes coverage and spatial reuse.SU transmission benefits from array gain, whereas MU transmission increases simultaneous transmissions while reducing per-user signal strength.
  • I. INTRODUCTION: The paper characterizes near-far effects using the No-Coverage Femtocell Radius and Cellular Coverage Radius.Femtocell users inside Df cannot meet the outage constraint because of cellular interference, while Dc bounds feasible cellular coverage under hotspot interference.
  • I. INTRODUCTION: Greater femtocell spatial reuse trades off against coverage radii, making cellular coverage maximization a central design objective.The macrocell is treated as the primary network for outdoor mobile users.
  • I. INTRODUCTION: The analysis seeks the maximum transmitting femtocell density as a function of user location, power ratio, antenna counts, transmission strategy, QoS targets, and path-loss exponents.It also considers achievable cellular coverage for a given average femtocell count and how femtocells should adapt transmit power.

B. Related Work

The paper extends prior tiered-network and cognitive-radio analyses to opportunistic femtocell deployments, deriving coverage, contention-density, and power-control results for multi-antenna transmission.

  • B. Related Work: The work extends transmission-capacity analysis from ad hoc networks to a cellular-underlaid ad hoc network.Its setting emphasizes randomly deployed femtocells rather than only operator-planned underlays.
  • B. Related Work: Its distinguishing contribution is decentralized femtocell transmit-power selection that keeps per-tier outage probability below a desired threshold.The scheme regulates power according to femtocell distance from the macrocell BS.
  • B. Related Work: The analysis derives coverage zones where cross-tier interference prevents users in either tier from satisfying QoS requirements.It separately identifies cellular-limited and hotspot-limited operating regimes.
  • B. Related Work: SU femtocell transmission decreases the no-coverage femtocell radius Df relative to MU transmission, indicating superior coverage in either tier.The reported reduction factor is (Tf/ǫ^(1−1/Tf))^(1/αc).
  • B. Related Work: SU macrocell transmission increases the maximum femtocell contention density in the hotspot-limited regime relative to MU transmission.SU transmission is also shown to be preferable for spatial reuse when hotspot interference is considerable, particularly for αfo < 4.
  • B. Related Work: The carrier-sensing strategy provides reliable cellular coverage with up to 60 femtocells per cell site under typical cellular indoor-to-outdoor path loss.The conclusion describes uniform cellular and hotspot coverage at the cell edge.

II. SYSTEM MODEL

The system model represents randomly located multi-antenna femtocells and macrocell users with stochastic geometry, terrestrial path loss, Rayleigh fading, and specified precoding assumptions.

  • II. SYSTEM MODEL: A central macrocell with Tc antennas serves a geographical region containing femtocells modeled as a homogeneous Spatial Poisson Point Process.The average number of active femtocells is Nf = λf|C|, while maximum hotspot intensity varies with macrocell distance D.
  • II. SYSTEM MODEL: Each femtocell has Tf antennas, serves up to Tf indoor users, and uses single-antenna receivers for both cellular and femtocell users.Users are placed on the circumference of a disc of radius Rf centered at their femtocell AP.
  • II. SYSTEM MODEL: Terrestrial propagation is represented by distance-dependent decibel path-loss models for macrocell, femtocell, and cross-tier links.The models distinguish outdoor, indoor, outdoor-to-indoor, and indoor-to-indoor path-loss exponents and wall losses.
  • II. SYSTEM MODEL: The model uses frequency-flat Rayleigh fading with independent complex channel entries distributed as CN (0, 1).Random lognormal shadow fading is omitted and replaced by fixed partition losses.
  • II. SYSTEM MODEL: Perfect CSI to each tier’s own users is assumed, while neighboring macrocell interference is ignored for analytical tractability.User selection and imperfect channel estimation are also deferred or excluded from the model.
  • II. SYSTEM MODEL: The analysis considers transmit beamforming for SU transmission and linear zero-forcing precoding for MU transmission.Zero-forcing is selected for low complexity while retaining the multiplexing gain of higher-complexity schemes.

A. SIR Analysis at a Femtocell User

The femtocell-user analysis models successful reception through SIR under cellular and neighboring-femtocell interference, then derives no-coverage conditions and operating regimes. It shows that SU transmission improves hotspot coverage and that spatial-reuse benefits depend on hotspot isolation and path loss.

  • SIR model: The femtocell-user SIR must meet target Γ with outage probability at most ǫ despite cellular and neighboring-femtocell interference.The analysis uses the constraint P[SIRf(F0, D) ≥ Γ] ≥ 1 − ǫ.
  • Operating regimes: The parameter κ distinguishes interference regimes: increasing cellular-interference strength worsens coverage, whereas κ → 0 leaves neighboring-femtocell interference dominant.Thus, cellular-limited and hotspot-limited behavior arise from different dominant interference sources.
  • Coverage dead zone: Any femtocell within D < Df of the macrocell cannot satisfy its QoS requirement because of excessive cross-tier interference.For D > Df, femtocell users can tolerate both cellular and hotspot interference.
  • Transmission strategy: SU femtocell transmission has a strictly smaller no-coverage radius than MU transmission.For small ǫ, the reduction factor depends on the antenna and outage parameters; the radius also scales as (Pf/Pc)^−1/αc.
  • Spatial reuse: For αfo < 4, SU transmission provides order-wise higher spatial reuse, while MU transmission does so only when αfo > 4.MU multiplexing benefits require relative isolation, or large αfo, between active femtocell access points.

B. SIR Analysis at a Cellular User

The cellular-user analysis derives the maximum femtocell contention density and cellular coverage radius under an outage constraint. It finds that SU macrocell transmission improves cellular coverage and that density and coverage depend on transmit powers and antenna usage.

  • Outage-constrained density: The cellular-user outage constraint requires P[SIRc(B0, D) ≥ Γ] ≥ 1 − ǫ while accounting for interference from transmitting femtocells.The maximum allowable femtocell contention density is derived from this constraint.
  • Power dependence: Increasing the cellular coverage radius by a factor of k requires increasing Pc by 10αc log10 k decibels relative to Pf.This follows from Dc varying as (Pc/Pf)^1/αc.
  • Coverage scaling: Single-user macrocell transmission scales as Dc,SU ∼ Θ(Tc^1/αc), whereas MU transmission scales as Dc,MU ∼ Θ(Tc^−1/αc).The resulting order-wise cellular-coverage improvement is Tc^2/αc for SU relative to MU transmission.

C. Design Interpretations

The design interpretations identify cross-tier cellular interference as the dominant capacity constraint and compare SU and MU transmission across coverage and spatial reuse. SU generally improves coverage and reuse, except when hotspot interference is sufficiently isolated.

  • Coverage improvement: With Tf = 2 antennas, SU transmission reduces the no-coverage femtocell radius Df by nearly 1.5x relative to single-antenna transmission.The comparison is made in the cellular-limited regime.
  • Coverage improvement: SU transmission reduces Df by nearly 1.8x relative to MU transmission.The observation agrees with the predicted improvement and indicates significantly better hotspot coverage.
  • Dominance of cellular interference: Cross-tier cellular interference remains capacity-limiting even in densely populated femtocell networks, while femtocell-to-femtocell interference is negligible under the stated isolation conditions.The explanation invokes proximity of home users to their access points and double-wall partition losses.
  • Spatial reuse: With Tf = 2 antennas, SU achieves NfUf = 1080 versus NfUf = 640 with MU transmission, a nearly 1.7x spatial-reuse gain.This comparison is reported for the hotspot-limited regime with αfo = 3.8.
  • Spatial reuse: MU transmission to Uf = 2 users provides marginally higher spatial reuse than SU only when hotspot interference is significantly diminished.The stated example uses αfo = 4.8 and Tf = 3 antennas; multiplexing benefits require relative isolation between active access points.
  • Design implications: At (Pc/Pf)dB = 0 and D = 0.1, SU tolerates nearly Nf = 62 femtocells/cellsite versus nearly 8 with MU, a 7.75x contention-density improvement.The improvement is approximated by the order-wise factor 8.04x.
  • Design implications: The design recommends single-user macrocell transmission and location-dependent femtocell power adaptation to ensure reliable cellular coverage.These recommendations follow from MU sensitivity to residual hotspot interference and the need to regulate femtocell interference by location.

IV. INTERFERENCE MANAGEMENT USING CARRIER-SENSING AT FEMTOCELLS

The paper proposes decentralized carrier sensing so femtocells regulate transmit power according to nearby cellular activity and their macrocell location. The strategy targets dense deployments while preserving cellular-user QoS.

  • Design motivation: At normalized distance D = 0.4, a femtocell can tolerate hotspot interference from greater than 1000 neighboring femtocells.This supports increasing (Pc/Pf)dB in dense deployments to reduce hotspot interference without violating femtocell QoS.
  • Interference management: The proposed strategy chooses femtocell transmission power when a cellular user is nearby.The approach is presented as a carrier-sensing interference-management strategy.
  • Implementation assumption: The carrier-sensing design assumes femtocells can infer distance to their closest macrocell base station through GPS or calibrated macrocell received-power measurements.This location information supplies the basis for power selection.
  • Carrier sensing: A femtocell performs energy detection on uplink cellular pilot transmissions to detect nearby cellular users.Each cellular user periodically transmits pilot signals for channel information.
  • Power adaptation: When detected cellular-user energy exceeds a threshold, the femtocell selects Pf based on its location D within the macrocell.Without a cellular user, it maintains a constant transmit power Pf.

A. Minimum Required Sensing Range and Per-Tier Transmit Power Ratio Bounds

The section derives a minimum femtocell sensing range and bounds the cellular-to-femtocell transmit-power ratio needed to satisfy per-tier outage constraints. Numerical examples show substantial sensing and power-ratio requirements at the cell edge.

  • A transmitting femtocell within R < Dsense meters of a cellular user violates the maximum outage-probability requirement.
  • The sensing-range derivation lower-bounds outage by considering exactly one hotspot at R = Dsense, the highest-path-loss location.
  • 160 meters of minimum sensing range is required at the cell edge when (Pc/Pf)dB = 20 dB and αc = αfo = 3.8 under SU transmission.
  • The required cellular-to-femtocell power ratio is bounded by (Pc/Pf)lb[D] ≤ Pc/Pf ≤ (Pc/Pf)ub[D].The bounds are derived from cellular-user and femtocell-user outage requirements, respectively.
  • At the cell edge, the required power-ratio bounds are 40 ≤ (Pc/Pf)dB ≤ 55 dB, with a constant decibel gap for all D.

B. Energy Detection based Carrier-Sensing of Cellular Users

The carrier-sensing procedure uses energy detection over pilot slots to infer cellular-user presence and regulate femtocell behavior. Its sensing range depends on transmit power, detection targets, antenna diversity, and the time-bandwidth product.

  • Femtocells test H0 for cellular-user absence and H1 for an active cellular user using received pilot-slot signals and complex Gaussian noise.
  • The energy detector compares Y = 2/N0 ∫|x(t)|2dt against a threshold λ to infer cellular-user presence.
  • The sensing model uses m = TW as an integer time-bandwidth product and calibrates noise power against a cell-edge downlink SNR.
  • Selection combining across Tf available diversity branches chooses the maximum-SNR branch before computing detection and false-alarm probabilities.
  • 230 meters of sensing range requires a minimum time-bandwidth product m = 500 with PUT,pilot = 20 dBm, Pdetect = 0.9, and Pfalse = 0.1.

V. NUMERICAL RESULTS

Simulations evaluate decentralized femtocell power control under randomized hotspot deployments and show reliable cellular and hotspot coverage with about 60 femtocells per cell-site.

  • Simulation setup: 1000 random hotspot drops with 1000 trials per drop evaluate fixed and location-based femtocell power-ratio selection.The simulations use Rayleigh fading and compare fixed Pc/Pf against sensing-triggered power adjustment.
  • Simulation setup: 230 meters is the cellular-user sensing radius determined by computer simulations.This exceeds the minimum required sensing range.
  • Cellular coverage: The proposed scheme ensures uniform cell-edge cellular coverage with large numbers of femtocells.The result is reported for the worst-case cellular-user locations considered in the simulations.
  • Hotspot coverage: 2.15, 3.63, 3.56, 3.32 and 3.22 b/s/Hz are the hotspot 10 percentile outage capacities across the five tested femtocell locations.These values exceed the minimum desired spectral-efficiency target of 2.06 b/s/Hz.
  • Transmission strategy: Single-user transmission provides significantly superior coverage and spatial reuse compared with multiuser transmission.The simulations adopt single-user transmission in both tiers.
  • Interference management: Location-assisted femtocell power control regulates transmit powers through a fully decentralized scheme providing uniform cellular and hotspot cell-edge coverage.The scheme contrasts with randomized hotspot transmissions without carrier-sensing.
  • Implications: The results motivate closed-access tiered cellular architectures requiring minimal network overhead.This conclusion follows the reported carrier-sensing and power-control results.

APPENDIX I

The appendix derives reception-probability expressions and QoS conditions using chi-squared and F-distributed random variables, Laplace transforms, and a low-outage approximation.

  • Signal distributions: The desired signal power is modeled with chi-squared distributions whose degrees of freedom depend on Tf − Uf + 1.The derivation covers both single-user beamforming and multiuser transmission cases.
  • Ratio distribution: The interference-to-signal ratio Z is represented through a canonical F-distribution with parameters determined by Tf − Uf + 1 and Uc.The numerator and denominator have 2(Tf − Uf + 1) and 2Uc degrees of freedom, respectively.
  • QoS condition: A necessary indoor-user QoS condition is κ+1(Tf − Uf + 1, Uc) ≤ ǫ, giving κ* = I−1(ǫ; Tf − Uf + 1, Uc).Substituting κ into the system relation yields the no-coverage femtocell radius Df.
  • Reception probability: The femtocell reception probability is derived by conditioning on aggregate interference and applying Laplace transforms.Independent cellular and femtocell interference terms allow the transform of their sum to decouple into a product.
  • Interference model: The analysis models neighboring femtocell interference as a Poisson shot-noise process with independently marked interferers.Its Laplace transform is used in the success-probability calculation.
  • Density bound: A first-order Taylor expansion around λfCfθ^δf = 0 and omission of Θ(λf^2) terms produce an upper bound on femtocell density.The bound determines the maximum number of simultaneous femtocell transmissions satisfying the outage requirement.

APPENDIX III

The appendix develops algebraic identities involving binomial coefficients and the incomplete Beta function, alongside figures characterizing coverage, transmission capacity, sensing, and data-rate behavior.

  • APPENDIX III: The proof applies a combinatorial identity to transform a polynomial coefficient expression.The derivation multiplies by (1 + x)^(n−r) and evaluates the coefficient of x^q.
  • APPENDIX III: The derivation then uses the incomplete Beta function and an index substitution to obtain the stated result.The passage explicitly invokes It(a, b) = 1 − I1−t(b, a) before combining equations (44) and (45).
  • APPENDIX III: The figures examine no-coverage radii, simultaneous femtocell transmissions, outage-constrained transmission counts, sensing ranges, and cellular or femtocell data rates.The listed figures vary antenna counts, path-loss parameters, user distance, sensing conditions, and transmission settings.
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