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Coverage Analysis for Millimeter Wave Networks: The Impact of Directional Antenna Arrays
Xianghao Yu, Jun Zhang, Martin Haenggi, Khaled B. Letaief
TL;DR
Mm-wave coverage analysis has been limited by oversimplified antenna patterns despite the importance of directional arrays and angle-dependent interference. The paper develops a general stochastic-geometry framework and accurate, tractable antenna-pattern approximations for ad hoc and cellular networks. Coverage rises as a non-decreasing concave function of array size, while large-scale arrays are needed for satisfactory coverage.
Problem
Previous mm-wave analyses often used oversimplified flat-top antenna patterns, leaving the impact of directional arrays insufficiently investigated with realistic patterns.
Method
The paper presents a coverage framework supporting arbitrary interference distributions and antenna patterns, then applies two accurate, tractable approximations to ad hoc and cellular networks.
Results
Coverage probabilities in both network types increase as non-decreasing concave functions of antenna-array size, and numerical results show that large-scale arrays are needed for satisfactory coverage.
Takeaways & Limitations
Directional-array size is a central determinant of mm-wave coverage, while network density has different effects in cellular and ad hoc deployments.
Abstract
from arXiv · showhide
Millimeter wave (mm-wave) communications is considered a promising technology for 5G networks. Exploiting beamforming gains with large-scale antenna arrays to combat the increased path loss at mm-wave bands is one of its defining features. However, previous works on mm-wave network analysis usually adopted oversimplified antenna patterns for tractability, which can lead to significant deviation from the performance with actual antenna patterns. In this paper, using tools from stochastic geometry, we carry out a comprehensive investigation on the impact of directional antenna arrays in mm-wave networks. We first present a general and tractable framework for coverage analysis with arbitrary distributions for interference power and arbitrary antenna patterns. It is then applied to mm-wave ad hoc and cellular networks, where two sophisticated antenna patterns with desirable accuracy and analytical tractability are proposed to approximate the actual antenna pattern. Compared with previous works, the proposed approximate antenna patterns help to obtain more insights on the role of directional antenna arrays in mm-wave networks. In particular, it is shown that the coverage probabilities of both types of networks increase as a non-decreasing concave function with the antenna array size. The analytical results are verified to be effective and reliable through simulations, and numerical results also show that large-scale antenna arrays are required for satisfactory coverage in mm-wave networks.
I. INTRODUCTION
Mm-wave networks need directional-array-aware coverage analysis because blockage, unusual propagation, and angle-dependent gains make simplified antenna models inadequate. The paper develops tractable analyses for ad hoc and cellular networks using more accurate antenna-pattern approximations and studies how array size and density affect coverage.
- Motivation: Mm-wave signals are sensitive to blockages and have distinct LOS and NLOS path-loss laws, requiring specialized channel models.Limited diffraction and scattering further distinguish mm-wave propagation from sub-6 GHz systems.
- Motivation: Directional arrays provide beamforming gains to offset higher path loss, while making signal and interference powers strongly dependent on AoDs and AoAs.Small angle shifts can cause large array-gain changes, so antenna directionality is central to performance analysis.
- Prior limitations: Flat-top antenna patterns simplify analysis by binary-quantizing array gains but can produce pessimistic coverage and obscure beamwidth, sidelobes, nulls, and beam misalignment.These pattern parameters vary with array size but are represented inaccurately by the flat-top abstraction.
- Contributions: The paper addresses the lack of comprehensive directional-array analysis with a general framework for arbitrary interference distributions and antenna patterns under gamma-distributed signal power.The framework is designed to produce compact, efficiently evaluable coverage expressions.
- Contributions: Two approximate antenna patterns balance accuracy and analytical tractability for coverage analysis in mm-wave ad hoc and cellular networks.The approach also handles gamma-distributed signal powers and interferers in finite regions.
- Findings: Coverage increases monotonically and concavely with antenna-array size in both network types, while density produces a peak in cellular networks and a decline in ad hoc networks.The paper also reports that large-scale arrays are needed for acceptable coverage and that NLOS effects are negligible in its numerical results.
B. Analog Beamforming and Antenna Pattern
The paper models analog beamforming through directional array gains and introduces sinc and cosine antenna-pattern approximations that balance accuracy with analytical tractability.
- Analog beamforming: Analog beamforming uses phase shifters to align each transmitter’s beam with its serving user’s channel direction.This alignment provides the maximum antenna array gain under the paper’s model.
- Actual antenna pattern: The actual antenna pattern is a normalized Fejér kernel whose array gain distribution can be represented using a uniformly distributed angular variable.The substitution preserves the overall distribution of the array gain.
- Approximate antenna patterns: The flat-top pattern quantizes continuously varying gains into main-lobe and side-lobe levels, simplifying analysis but producing substantial coverage discrepancies.Its parameters also make array-size effects and gain rolloff difficult to analyze accurately.
- Approximate antenna patterns: The sinc pattern closely matches the actual pattern and introduces almost no coverage-probability error while remaining more tractable.Its tractability follows from removing the sine function in the denominator.
- Approximate antenna patterns: The cosine pattern approximates main-lobe gains well and achieves a negligible coverage gap relative to the actual pattern.It sacrifices some side-lobe accuracy for an elementary-function representation and better analytical tractability.
- Approximate antenna patterns: Unlike the flat-top pattern, sinc and cosine patterns explicitly depend on array size, enabling analysis of directional-array effects in ad hoc and cellular networks.The paper adopts them for the subsequent network-specific coverage analyses.
III. A GENERAL FRAMEWORK FOR COVERAGE ANALYSIS OF MM-WAVE NETWORKS
The paper develops a general coverage-analysis framework for mm-wave networks with arbitrary antenna patterns and interference distributions, under a gamma-distributed information signal power assumption.
- Framework scope: The framework provides a tractable coverage expression for arbitrary antenna patterns and interference-power distributions.It is later applied to mm-wave ad hoc and cellular networks.
- Signal model: Transmitters align analog-beamforming beams to serving-user angles of departure using full channel-direction information.The model assumes this information is obtained through beam-training protocols.
- Interference model: Interfering channel gains include both small-scale fading and directional antenna-array gains and are modeled as independent and identically distributed non-negative random variables.The gain distributions are specified separately for the ad hoc and cellular settings.
- Signal model: The desired signal power is modeled with a gamma distribution, which introduces additional derivational difficulty compared with exponential Rayleigh-fading power.Gamma-distributed signal power also arises in broader multi-antenna transmission settings.
B. Coverage Analysis Framework
The coverage framework converts gamma-signal coverage probabilities into derivatives of a Laplace transform and a compact lower-triangular Toeplitz-matrix expression, while allowing arbitrary interference and antenna models.
- Coverage formulation: Coverage probability is defined as the probability that received SINR exceeds threshold τ.The framework treats the link distance as random in cellular networks and deterministic in ad hoc networks.
- Coverage formulation: For arbitrary channel-gain distributions, the derivatives of the Laplace transform are expressed recursively and coverage is represented using the induced ℓ1-norm of a lower-triangular Toeplitz matrix.This construction is enabled by the gamma-distributed desired signal and supports compact evaluation.
- Coverage formulation: The analysis derives the Laplace transform of noise and interference and uses its derivatives to represent the coverage probability.The derivative notation is L^(n)(s) = (−1)^n E[I^n e^(−sI)].
- Derivation strategy: The derivation swaps the order of location expectation and channel-gain averaging to preserve analytical tractability for complicated directional-array gains.Fubini’s theorem justifies the swap because the integration function is integrable.
- Generality: The theorem assumes gamma-distributed signal power but permits arbitrary interference distributions and antenna patterns.Its general Laplace-transform exponent specializes to earlier results under particular network and fading settings.
IV. COVERAGE ANALYSIS FOR MM-WAVE AD HOC NETWORKS
Mm-wave ad hoc networks are considered for short-range applications, and the paper derives their coverage expression before examining the role of directional antenna arrays.
- Network setting: Mm-wave ad hoc networks are proposed for short-range applications including military battlefield, high-fidelity video, and device-to-device networks.The section first derives coverage analytically, then investigates directional-array effects.
A. Coverage Analysis
The ad hoc-network analysis uses a tractable sinc antenna-pattern approximation and derives coverage bounds that expose how directional array size affects coverage. The resulting coverage bound is non-decreasing and concave in array size, with asymptotic outage inversely proportional to array size.
- The analysis adopts the sinc antenna pattern as an accurate, analytically tractable approximation of the actual pattern in mm-wave ad hoc networks.
- The proposed framework represents channel gain as the product of gamma-distributed small-scale fading and directional array gain, enabling coverage analysis through matrix coefficients.
- The coverage probability with the sinc pattern is tightly lower bounded by the expression in Proposition 1.
- The infinite series in Proposition 1 converges quickly, so finite terms and offline numerical evaluation of ξ are sufficient in practice.
- The tight coverage lower bound is a non-decreasing concave function of array size, reflecting the combined signal and interference effects of larger arrays.
- For any desired coverage requirement 1 − ϵ, a minimum array size can be numerically determined regardless of other network parameters.
- As array size grows, outage probability is inversely proportional to array size, and the coverage probability approaches one for fixed network parameters.
V. COVERAGE ANALYSIS FOR MM-WAVE CELLULAR NETWORKS
For mm-wave cellular networks, the paper uses a cosine antenna pattern to retain directionality while improving analytical tractability. The resulting expression supports numerical evaluation of coverage, required base-station density, antenna count, and optimal density.
- The cosine antenna pattern provides a more compact and tractable coverage expression while fully reflecting directionality in mm-wave cellular networks.
- The cellular-network coverage probability with the cosine antenna pattern is given by Proposition 2.
- The coefficients in Proposition 2 can be expressed using hypergeometric functions, which modern numerical software can calculate efficiently.
- Proposition 2 enables numerical calculation of the required base-station density and minimum antenna count for a desired coverage probability.
- The same proposition can numerically determine the optimal base-station density that achieves maximum coverage probability.
B. Impact of Directional Antenna Arrays
A lower bound removes the matrix-exponential norm that complicates direct analysis of array-size effects in cellular networks. This bound is non-decreasing and concave in array size, with behavior similar to the ad hoc case despite user association terms.
- The cellular coverage expression is difficult to use for array-size analysis because it contains an integral involving the induced ℓ1-norm of a matrix exponential.
- A lower bound eliminates the induced ℓ1-norm of the matrix exponential from the integrand, enabling explicit analysis of antenna-array effects.
- The cellular coverage lower bound is a non-decreasing concave function of array size.
- For increasing antenna count, the asymptotic cellular outage probability is characterized in the limit t → 0, where t = 1/N_t.
- The cellular lower bound has array-size dependence similar to the ad hoc result, with additional terms introduced by user association.
- The similarity between ad hoc and cellular results indicates that directional-array effects do not depend strongly on the user-association strategy.
VI. NUMERICAL RESULTS
Numerical results validate the analytical coverage expressions and show how NLOS components, noise, interference, and antenna-array size affect mm-wave ad hoc and cellular networks.
- Simulation Settings: The simulations use 1 GHz bandwidth, 1-Watt BS transmit power, quarter-wavelength antenna spacing, and 5 × 10^5 realizations.The path-loss exponent is set near 2 with intercept β = −61.4 dB.
- A. The Role of NLOS Signals and Interference: NLOS signals and interference are negligible when the serving link is LOS in both cellular and ad hoc networks.The cellular simulation curves nearly coincide with and without NLOS components, while NLOS interference is likewise negligible for LOS ad hoc links.
- A. The Role of NLOS Signals and Interference: Cellular SINR coverage peaks as BS density increases, whereas ad hoc SINR coverage decreases with densification.Cellular coverage reflects changing LOS/NLOS fading and eventually interference limitation; ad hoc networks face fixed dipole distances and arbitrarily close interferers.
- B. Coverage Analysis: Analytical coverage results closely match simulations for ad hoc and cellular networks, validating Proposition 1 and approximating Proposition 2 and Corollary 2.Only five summation terms are needed for the ad hoc evaluation, while Corollary 2 provides a tractable approximation and lower bound for Proposition 2.
- C. Impact of Directional Antenna Arrays: Coverage probability increases as a non-decreasing concave function of antenna-array size in both network types.For ad hoc networks, beam narrowing reduces the probability that interferers point main lobes toward the receiver, while increasing array gain alone offers little benefit.
APPENDIX A
Appendix A derives the ad hoc-network coverage expression through Laplace-transform expansions, recursive coefficients, and a matrix-exponential representation.
- Coverage Derivation: The coverage probability is expressed through derivatives of the Laplace transform and recursively defined coefficients.The derivation introduces x_n recursively from L(s) = exp{η(s)} and uses Leibniz’s formula.
- Coefficient Representation: Power-series manipulations convert the coefficient recursion into a differential equation and a matrix-exponential form.The first M coefficients of e^C(z) form the first column of exp{C_M}.
- Proof Structure: Positivity of the coefficients c_k for k ≥ 1 completes the proof of the theorem used in the coverage derivation.The proof relies on monotonicity of z^(2−α_k)E_(1+δ−k)(z).
APPENDIX C
Appendix C establishes antenna-array monotonicity and concavity by analyzing the Laplace-transform coefficients and their matrix representation.
- Monotonicity: The Laplace-transform term x_0 is non-decreasing with antenna-array size N_t because its defining factor is positive.The positivity follows from the monotonic decrease of the generalized exponential integral.
- Monotonicity: The recursive relationship implies that every coefficient x_n is non-decreasing with N_t, proving the monotonicity in Corollary 1.The result is combined with p_c(τ) expressed through the coefficients.
- Matrix Structure: The matrix C_M decomposes into a scalar matrix and a nilpotent strictly lower-triangular Toeplitz matrix.The nilpotent component satisfies (C_M − c_0I_M)^n = 0 for n ≥ M.
- Concavity: Positive entries in the matrix powers support the concavity proof for the coverage lower bound.The appendix uses the finite matrix-exponential expansion and a Taylor expansion near t → 0.
- Cellular Extension: For the cellular case, the cosine antenna pattern yields the Laplace-transform exponent and the entries of C_M used in Proposition 2.The derivation uses generalized hypergeometric-function expansions and derivatives.
- Cellular Extension: The resulting matrix representation supplies the analytical ingredients for Proposition 2.The entries in C_M are obtained by combining the derived expression with Theorem 1.
APPENDIX E
Appendix E derives a cellular-network lower bound by applying a power-series representation and Jensen’s inequality to the antenna-pattern expression.
- Lower-Bound Construction: The derivation defines a power series and uses X(z) = exp{C(z)} to formulate a lower bound.Because C(z) depends on the serving distance r_0 in cellular networks, the notation is adjusted accordingly.
- Lower-Bound Construction: Jensen’s inequality is applied to the exponential function to obtain the coverage lower bound.The serving-distance distribution enters through the stated density-dependent expression.
- Final Expression: The resulting expression is further written in the form of Corollary 3.