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Downlink MIMO HetNets: Modeling, Ordering Results and Performance Analysis

Harpreet S. Dhillon, Marios Kountouris, Jeffrey G. Andrews

arXiv:1301.5034v2cs.ITcs.NI

TL;DR

The paper addresses how to analyze and compare multi-antenna HetNets whose BS tiers differ in deployment and transmission parameters. It develops a general stochastic-order and stochastic-geometry framework for coverage, per-user rate, and ASE, showing that antenna placement and transmission choice produce different performance orderings.

  • Problem

    General multi-antenna HetNets are difficult to compare because heterogeneous tiers and general BS distributions can prevent simple closed-form coverage and rate expressions.

  • Method

    The paper builds a K-tier downlink model and applies stochastic orders for general BS distributions, then uses independent PPP analysis for coverage bounds and ASE comparisons.

  • Results

    The analysis compares SDMA, SU-BF, and SISO, establishing ordering results for coverage and per-user rate and deriving PPP-based coverage bounds and ASE results.

  • Takeaways & Limitations

    For a fixed total antenna count, antennas are preferably spread across many single-antenna BSs; SU-BF exceeds SDMA in coverage and per-user rate, while SDMA can yield higher ASE.

Abstract

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We develop a general downlink model for multi-antenna heterogeneous cellular networks (HetNets), where base stations (BSs) across tiers may differ in terms of transmit power, target signal-to-interference-ratio (SIR), deployment density, number of transmit antennas and the type of multi-antenna transmission. In particular, we consider and compare space division multiple access (SDMA), single user beamforming (SU-BF), and baseline single-input single-output (SISO) transmission. For this general model, the main contributions are: (i) ordering results for both coverage probability and per user rate in closed form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE). The analysis concretely demonstrates, for example, that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Another observation is that SU-BF provides higher coverage and per user data rate than SDMA, but SDMA is in some cases better in terms of ASE.

I. INTRODUCTION

The paper develops a general tractable downlink model for multi-antenna HetNets with heterogeneous tiers and compares transmission techniques using stochastic geometry and stochastic-order tools. It extends analysis beyond PPP-specific models while examining coverage, per-user rate, and ASE.

  • Multi-antenna techniques and HetNets should be analyzed together because future wireless networks are expected to combine them.
  • Ordering results compare coverage probability and per-user rate for SDMA, SU-BF, and SISO under general stationary BS point processes.
  • The ordering analysis avoids requiring explicit coverage and rate expressions, which may not have simple closed forms for general configurations.
  • The ASE analysis uses independent PPP tiers and derives coverage bounds and closed-form ASE results to account for differences in the number of served users.
  • The model allows tiers to differ in transmit power, density, target SIR, antenna count, served users, transmission type, and access restrictions.

III. ORDERING RESULTS FOR COVERAGE AND RATE

This section defines coverage and frames transmission comparison as an SIR-ordering problem. Stochastic-order analysis enables performance comparisons without evaluating complicated coverage and rate expressions, including across general BS spatial models.

  • Coverage means that at least one permitted serving BS provides a downlink SIR above its target.
  • Open-access coverage is always at least as high as closed-access coverage because open access permits connection to more tiers.
  • Coverage probability may lack a simple closed form when BS locations follow a general point process, complicating direct technique comparisons.
  • The proposed method orders random SIR-related quantities using stochastic dominance, rather than comparing only deterministic parameter vectors.
  • Conditioning on BS locations isolates propagation randomness in Gamma-variable ratios, making channel-power ordering central to SIR comparison.

A. Ordering of the Ratios of Gamma Random Variables

The paper establishes stochastic ordering for ratios of Gamma random variables using coupling, with conditions on the shape parameters that ensure dominance. It also shows why comparing only the ratios of shape parameters is insufficient.

  • Coupling constructs two ratio variables from the same randomness so their pointwise ordering establishes stochastic dominance.The coupled variables retain the respective distributions of the original Gamma ratios.
  • Z_k1,m1 stochastically dominates Z_k2,m2 when k1 ≥ k2 and m1 ≤ m2.The result is proved using equivalent sums of exponential random variables and coupling.
  • The condition k1/m1 ≥ k2/m2 alone does not guarantee stochastic dominance because substantially different shape parameters can cause concentration effects.For k1 = 4, m1 = 2 versus k2 = 100, m2 = 100, the first ratio does not dominate despite the mean-ratio condition.
  • A multivariate stochastic-order lemma extends componentwise dominance to expectations of functions that are non-decreasing in every component.This extension supports the later ordering analysis of HetNet performance metrics.

B. Coverage Probability Ordering

The analysis orders coverage across general open and closed access HetNets by comparing Gamma fading shape parameters. It shows consistent coverage advantages for fewer served users and for SU-BF over SDMA and SISO in the stated comparisons.

  • Coverage ordering follows from system-parameter comparisons through stochastic dominance of the Gamma-ratio fading terms.The result applies to general BS location distributions and extends to closed access with a corresponding tier restriction.
  • Serving fewer users with the same number of antennas provides higher coverage because of higher beamforming gain.The comparison is made tier by tier between two K-tier open access HetNets.
  • SU-BF always has higher coverage than SDMA when corresponding tiers use the same number of antennas.This follows from the different Gamma shape parameters associated with the two transmission techniques.
  • Coverage ranks SU-BF above SISO and SISO above full SDMA in the three-way comparison.The SU-BF and full-SDMA systems need not use the same number of antennas in corresponding tiers.
  • Open access coverage is higher than or equal to closed access coverage because open access permits connections to more tiers.The closed-access ordering result holds under a slightly weaker condition.

C. Ordering Result for Rate per User

The paper extends its stochastic-ordering framework from coverage probability to rate per user under common resource-allocation assumptions. The resulting ordering matches the coverage ordering, while ASE requires separate consideration because transmission schemes serve different numbers of users.

  • Per-user rate depends on SIR and allocated resources, modeled assuming equal user counts per tier and equal time-frequency allocation among users.Exact per-user rate distributions are difficult when BS load and service areas are not explicitly characterized.
  • Open access rate coverage is always higher than closed access rate coverage because the open-access union includes more eligible tiers.The closed-access definition replaces the tier set with B ⊂ K.
  • With the same per-user resource allocation in corresponding tiers, one system has equal or higher rate coverage under the same parameter ordering used for coverage.The theorem applies to both open and closed access networks, with closed access restricted to allowed tiers.
  • The ordering conditions for per-user rate are the same as those for coverage probability.Therefore, the SU-BF, SISO, and SDMA conclusions carry over to rate per user.
  • SU-BF and SISO outperform SDMA in coverage probability and average rate per user, but SDMA may provide a higher sum-data rate by serving more users.This motivates comparing the techniques using area spectral efficiency.

IV. COVERAGE PROBABILITY AND ASE PERFORMANCE

For PPP-modeled HetNets, the paper derives an upper bound on coverage probability to support ASE comparisons. The bound is closed form for full SDMA and otherwise involves derivatives or numerical evaluation.

  • The bound is derived under a PPP model that is less general than the arbitrary point-process model used for ordering results.The PPP assumption is used because ASE comparisons require explicit coverage expressions.
  • The PPP-based analysis uses the Laplace transform of interference and its derivatives to express the coverage bound.Campbell-Mecke and Faà di Bruno methods support the derivation for multi-antenna Gamma fading.
  • Theorem 3 upper bounds typical-user coverage probability in a K-tier open access HetNet.The closed-access version changes the summation from K to the allowed tier set B.
  • The bound is numerically difficult because it involves a derivative of the Laplace transform, although full SDMA has a simple closed form.For Δ_k > 1, numerical computation is reported as fairly easy, especially for small Δ_k.
  • For full SDMA, where Δ_k = 1 for every tier, the coverage bound reduces to a closed-form expression.The full-SDMA case serves M_k users per BS, and the closed form is stated in Corollary 6.

B. Tightness of the Upper Bound

The upper-bound tightness depends on the number of candidate serving BSs and improves as target SIRs increase or BSs serve more users. For full SDMA, the closed-form bound is tight and reveals coverage relationships with SISO.

  • The bound is exact when a typical user has strictly one candidate serving BS; otherwise, its tightness depends on P(X({∆k}, {Ψk}) > 1).
  • Increasing target SIRs tightens the bound because X({∆k}, {Ψk}) decreases element-wise with βk.
  • The bound becomes tighter when BSs serve more users, with full SDMA as the limiting case where each BS serves as many users as transmit antennas.
  • For full SDMA, C(α, M) increases with M and exceeds the SISO-equivalent C(α) for M > 1.
  • With otherwise identical parameters, full SDMA coverage is always lower than SISO coverage.
  • In open-access HetNets with identical target SIRs and antenna counts across tiers, full SDMA coverage is invariant to BS density, tier count, and transmit power.
  • The scale-invariance result does not hold for closed-access HetNets.

C. Area Spectral Efficiency

The ASE analysis accounts for differences in the number of users served by SDMA, SU-BF, and SISO. It compares full SDMA and SISO broadly, and identifies a distinct ordering when target SIRs vanish.

  • ASE measures bits transmitted per unit area, time, and bandwidth, capturing sum-rate differences caused by techniques serving different numbers of users.
  • Under equal per-tier coverage assumptions, the ASE expressions enable analytical comparisons across transmission techniques.
  • For α > 2, the full-SDMA-to-SISO ASE ratio grows with the number of antennas, and full SDMA is reported higher than SISO.
  • When comparing equal user densities, the analysis contrasts λ BSs with M antennas against Mλ single-antenna BSs per unit area.
  • SU-BF comparisons are limited because its coverage upper bound lacks a closed-form reduction and its tightness is questionable.
  • For βk → 0 across all tiers, SU-BF and SISO have equal ASE, while full SDMA has higher ASE than both.

V. NUMERICAL RESULTS

The numerical procedure estimates coverage using simulated tier-specific PPP deployments and independent channel marks. The reported setup includes a two-tier full-SDMA coverage evaluation against its analytical upper bound.

  • Coverage is estimated by repeatedly simulating BS locations as independent PPPs, assigning channel marks, and evaluating a typical user at the origin.
  • Figure 4 evaluates two-tier HetNets with both tiers using full SDMA, equal antenna counts and target SIRs, and λ2 = 2λ1.
  • The simulations validate the location model and assess the tightness of the SDMA upper bound.
  • In the interference-limited regime, absolute transmit powers and deployment densities are irrelevant; results depend only on their ratios.

A. Model validation and tightness of the upper bound on Pc

The model is tested against hexagonal-grid, real-deployment, and PPP macrocell layouts. The PPP model is then used for numerical evaluation, where the full-SDMA upper bound remains tight to very low target SIRs.

  • Model validation: The validation compares hexagonal-grid macrocells, an actual 4G deployment, and independently generated PPP macrocells in a two-tier HetNet.
  • Coverage comparisons: Figure 5 compares coverage across two-tier combinations of SISO, full SDMA, and SU-BF under the stated common parameters.
  • Model validation: The PPP model is considered about as accurate as the grid model, which provides an upper bound, and is used thereafter.
  • Upper-bound validation: The full-SDMA upper bound is tight down to very low target SIRs and remains tight to about −4 dB for M = 2.
  • Upper-bound validation: A negligible moderate-to-high-SIR gap is attributed to finite-window simulation border effects absent from the infinite-plane analysis.

B. Effect of adding additional tier on coverage probability

Adding a tier changes coverage according to its access mode and transmission technique. In the closed-access case, SISO and SU-BF interference links can yield identical typical-user coverage, while SU-BF remains highest and full SDMA lowest in the open-access comparison.

  • Open access: SU-BF yields the highest coverage when both tiers use it, while full SDMA yields the lowest coverage in the open-access comparison.The reported explanation is that SU-BF combines beamforming gain with the proximity gain also enjoyed by SISO.
  • Comparison scope: The model compares coverage across SISO, full SDMA, and SU-BF tiers, including closed-access configurations.
  • Closed access: The closed-access comparison uses a two-tier HetNet with P = [1, .01], λ2 = 2λ1, β1 = β2, α = 3.8, and M = 4 for multi-antenna tiers.
  • Closed access: Coverage probability is the same whether the new closed-access tier uses SISO or SU-BF.The interfering-link channel power distribution is Γ(1, 1), or exp(1), in both cases.
  • Closed access: The closed-access result preserves the transmission-technique ordering reported for the open-access case.
  • Partial access: Under PPP modeling, independently splitting a tier into open- and closed-access BSs creates two independent PPP tiers with appropriate densities.This enables coverage analysis when a fraction 1−θ of second-tier BSs is closed access.

APPENDIX A SIGNALING PRELIMINARIES

The signaling preliminaries specify a linear-precoding downlink with independent Rayleigh channel vectors and characterize desired and interfering channel powers for multiuser and single-user transmission.

  • Signal model: The received signal from a kth-tier BS is modeled using per-user transmit power Pk and a normalized transmit signal vector zk.
  • Signal model: Each BS uses linear precoding, multiplying user data symbols by columns of a precoding matrix.
  • Zero-forcing transmission: For zero-forcing beamforming with perfect CSI, the precoding matrix has Ψk columns corresponding to users served in tier k.
  • Channel powers: The desired channel power under zero-forcing follows Γ(∆k, 1), with ∆k = Mk − Ψk + 1.
  • Interference model: Interfering channel power is Γ(Ψj, 1) because it sums Ψj independent exponential contributions when spatial correlation is neglected.
  • Transmission techniques: Full SDMA is defined by ∆k = 1 and Ψj = Mj, whereas SU-BF serves one user with Ψk = 1 and uses the channel-direction beamformer.
  • Transmission techniques: For SU-BF, desired channel power is Γ(Mk, 1), while each interfering mark is Γ(1, 1).

APPENDIX B PROOF OF LEMMA 4

The proof evaluates a PPP-based expression by applying independence, the PPP probability generating functional, gamma-mark transforms, the binomial theorem, coordinate conversion, and an Euler beta-function substitution.

  • Proof steps: The derivation uses tier independence and independence between channel powers and BS locations before applying the PPP probability generating functional.
  • Proof steps: The interference-mark transform uses gjy ∼ Γ(Ψj, 1), followed by the binomial theorem.
  • Integral evaluation: Converting Cartesian to polar coordinates and substituting (1 + r−α)−1 → t reduces the integral to Euler’s beta function B(x, y).
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