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On the Joint Impact of Beamwidth and Orientation Error on Throughput in Directional Wireless Poisson Networks
Jeffrey Wildman, Pedro H J Nardelli, Matti Latva-aho, Steven Weber
TL;DR
Directional wireless networks face a gap in modeling how beam misdirection affects coverage and throughput, because prior work often assumed perfect beam selection or steering. The paper uses stochastic geometry and a bipolar PPP model with orientation error to analyze beamwidth-dependent spatial throughput and transmission capacity. For ideal sectors, concave orientation-error distributions support monotonicity and quasi-concavity results, while numerical studies examine more complex radiation patterns and orientation uncertainty.
Problem
Prior directional-network analyses largely assumed perfect sector selection or beamsteering, while orientation error had not been incorporated into a stochastic-geometry analysis of gain distributions and throughput.
Method
The paper models transmitter-receiver pairs as a marked bipolar PPP and uses stochastic geometry to analyze success probability, spatial throughput, and transmission capacity under orientation error and varied antenna patterns.
Results
For ideal sector antennas without sidelobes, concave orientation-error c.d.f.s are sufficient to prove spatial-throughput monotonicity and transmission-capacity quasi-concavity with respect to beamwidth.
Takeaways & Limitations
Orientation uncertainty generally shifts throughput-maximizing beamwidths toward larger values, while ideal sectors without sidelobes yield zero spatial-throughput-maximizing beamwidth.
Abstract
from arXiv · showhide
We introduce a model for capturing the effects of beam misdirection on coverage and throughput in a directional wireless network using stochastic geometry. In networks employing ideal sector antennas without sidelobes, we find that concavity of the orientation error distribution is sufficient to prove monotonicity and quasi-concavity (both with respect to antenna beamwidth) of spatial throughput and transmission capacity, respectively. Additionally, we identify network conditions that produce opposite extremal choices in beamwidth (absolutely directed versus omni-directional) that maximize the two related throughput metrics. We conclude our paper with a numerical exploration of the relationship between mean orientation error, throughput-maximizing beamwidths, and maximum throughput, across radiation patterns of varied complexity.
I. INTRODUCTION
Directional antennas can improve signal strength and reduce interference, but orientation errors complicate beam alignment and throughput optimization. This paper addresses a gap in stochastic-geometry analyses by modeling orientation error and studying beamwidth-dependent throughput metrics.
- Motivation: Directional antennas increase received signal power while reducing interference, but narrower beams intensify alignment and interference tradeoffs.Narrower beams provide higher main-beam gain and affect fewer interferers, whose individual effects become stronger.
- Approach: The paper models an ad hoc network as a bipolar PPP and uses stochastic geometry to optimize beamwidth for throughput-based metrics under orientation error.The analysis covers spatial throughput and transmission capacity, with comparisons to more complex radiation patterns.
- Related work: The model extends prior directional-network studies that examined capacity, coverage, beamforming, and interference using protocol or stochastic-geometry approaches.Earlier works commonly assumed perfect orientation, sector selection, or beamsteering.
- Research gap: Prior analyses largely assumed perfect sector selection or beamsteering, leaving orientation error insufficiently incorporated into stochastic-geometry models.The paper identifies no prior stochastic-geometry work incorporating orientation error into gain distributions between interferers and a typical receiver.
- Model: Each transmitter-receiver pair uses steerable, symmetric antenna patterns whose boresights are perturbed by additive, zero-centered orientation errors.The gain input angles are defined relative to boresight, and the transmitter and receiver errors determine pairwise alignment.
C. Marks on the Process
The network process marks transmitter locations with paired receiver orientations and independent antenna orientation errors. These marks determine geometry, misalignment, antenna gains, and the interference environment used in success-probability analysis.
- Process marks: The marked homogeneous bipolar PPP encodes transmitter locations, paired receiver locations, and antenna orientation errors.Each transmitter-receiver pair has separation distance d, while marks specify receiver direction and the two beam errors.
- Orientation variables: Receiver orientations are uniformly distributed, while transmitter and receiver beam errors are zero-mean, bounded, and independently modeled.The absolute error bound satisfies ϵmax ≤ π.
- Communication model: The communication model uses distance-based pathloss, Rayleigh fading, fixed transmit power, and SINR thresholding to define successful transmissions.Success occurs when SINR exceeds β, with interference formed by the aggregate received power from all other transmitters.
- Induced gains: Random transmitter-receiver gains are represented through the absolute orientation-error distribution and become independent of point geometry for interference analysis.These gain distributions and their moments are used to compute the success probability of a typical transmission.
- Error distribution: The analysis assumes a twice-differentiable, concave absolute-error c.d.f. on bounded support and uses truncated distributions parameterized by their pre-truncation mean.Concavity corresponds to a monotonically decreasing second derivative over the error support.
B. Success Probability
The paper derives a general success probability for a typical transmission by combining random antenna gains, pathloss, fading, interference, and noise. Omni-directional antennas emerge as a special case with deterministic unit gains and a closed-form success probability.
- General formulation: Proposition 1 expresses typical-transmission success probability for arbitrary gain patterns and orientation-error distributions in a bipolar PPP.The formulation includes intensity, link distance, pathloss, noise, SINR threshold, orientation error, and Rayleigh fading.
- Gain distributions: The success-probability expression integrates over desired transmitter and receiver gain distributions and uses interference-gain moments.The relevant interference moments are the 2/α-moments of gains between arbitrary interferers and the typical receiver.
- Interference interpretation: Independent gain distributions can be interpreted as thinning probabilities for interferers producing specified combined gains at the typical receiver.Under a dominant-interference approximation, success corresponds to void probabilities across independently thinned PPPs.
- Omni-directional special case: For omni-directional antennas, all four gain variables are deterministic unit gains, yielding a closed-form success probability.The omni-directional pattern is G(θ) = 1, and the resulting expression depends on intensity, pathloss, SINR threshold, link distance, and noise.
C. Ideal Sectors
The ideal-sector model represents antenna gains using beamwidth, mainbeam gain, and sidelobe gain, then derives success probabilities from beam hit and miss events. Orientation error can reverse the ordering of sector configurations: sidelobes may help at low density but hurt at high density.
- Ideal sectors use beamwidth ω, mainbeam gain g1, and sidelobe gain g2, with constant total radiated power across parameters.
- For ideal sectors, p and 1 − p are interferer mainbeam hit and miss rates, while u and 1 − u are typical-pair hit and miss rates caused by orientation error.
- With zero sidelobes, interferer thinning occurs with probability p^2, followed by typical-pair alignment with probability u^2.
- With orientation error, sidelobes benefit low-density networks but harm high-density networks because they support misaligned links while also creating and receiving interference.
- As intensity approaches zero, sidelobe-free sector success is bounded by u^2 = F_|ϵ|(ω/2)^2, which decreases as beamwidth narrows.
IV. MAXIMIZING SPATIAL THROUGHPUT
This section maximizes spatial throughput by balancing active-transmitter intensity against transmission success under ideal sectors and orientation error. For concave orientation-error distributions, narrower beams monotonically increase spatial throughput, while uniform, exponential, and half-normal errors yield throughput decreasing with beamwidth.
- Spatial throughput is the maximum of λps(λ) over the active-transmitter intensity λ, balancing transmission density against success probability.
- At the throughput-optimal intensity, narrowing the beam drives ps(λ*) toward zero while λ* diverges toward infinity.
- For zero-sidelobe sectors, concave F_|ϵ| over [0, π] makes spatial throughput monotone increasing as beamwidth ω approaches zero.
- Uniform, exponential, and half-normal orientation-error distributions produce spatial throughput monotone decreasing in ω over [0, 2π].
- Concavity is sufficient but not necessary for throughput monotonicity: a non-concave distribution with a dimple can still yield monotone spatial throughput.
- Without orientation error, sectorized transmitters with omni-directional receivers achieve a throughput gain of 1/p ∝ 1/ω over omni-directional antennas.
V. MAXIMIZING TRANSMISSION CAPACITY
The paper derives transmission capacity for ideal sector antennas with orientation error and shows that concave error distributions make transmission capacity unimodal in beamwidth. Under suitable endpoint conditions, the transmission-capacity optimum can be omni-directional, contrasting with spatial-throughput optimization.
- Transmission-capacity formulation: Transmission capacity is obtained as the maximum successful-transmission intensity subject to outage constraint pe.For ideal sectors without sidelobes, TCs = λ∗(1 − pe).
- Beamwidth behavior: Orientation error and the outage constraint prevent transmission capacity from increasing monotonically as beamwidth narrows toward zero.Below a minimum beamwidth, misalignment alone makes the required success rate unattainable and transmission capacity effectively zero.
- Beamwidth behavior: Concavity of F_|ϵ| makes TCs unimodal in beamwidth and produces a unique maximizing beamwidth.The proof combines monotonicity outside the relevant domain with quasi-concavity inside it.
- Optimizer conditions: The maximizing beamwidth depends on outage constraint pe and error distribution F_|ϵ|, independently of α, β, η, d, and λ.Corollary 6 characterizes the optimizer using the error-density condition at the support endpoint.
- Opposite beamwidth optima: Under additional endpoint-density conditions, transmission capacity is maximized by the omni-directional limit ω → 2π, whereas spatial throughput is maximized by ω → 0.These opposite extrema arise for concave error distributions with ϵmax = π.
- Transmission-capacity gains: With negligible background noise and no orientation error, sector transmission capacity gains scale as 1/ω^2, equivalent to Θ(M^2) for M sectors.The gain differs from spatial throughput because transmission capacity fixes the success rate at 1 − pe.
VI. RESULTS
Numerical experiments compare ideal sectors, transitional sectors, and 3GPP patterns under orientation error. They show that sidelobes and transition widths alter spatial-throughput monotonicity, while maximum throughput generally decreases and optimal beamwidth increases with orientation uncertainty.
- Experimental setup: The numerical study evaluates success probability, spatial throughput, and transmission capacity across radiation patterns of increasing complexity.It varies beamwidth, transmitter intensity, sidelobe strength, transition width, and mean orientation error.
- Throughput comparisons: At fixed ω = 20 degrees and mean orientation error 3 degrees, directional patterns achieve higher throughput at higher transmitter intensities than omni-directional antennas.Sidelobe strength g2 is more influential than transition width γ in the plotted throughput behavior.
- Spatial throughput: Ideal sectors without sidelobes retain spatial-throughput monotonicity, but sidelobes and transition widths do not preserve it.The distinction becomes more pronounced as beamwidth falls below 20 degrees.
- Transmission capacity: For ideal sectors, reducing sidelobes and transition width increases transmission capacity and shifts its maximizing beamwidth downward.The transmission-capacity optimum appears unimodal for the other displayed radiation patterns as well.
- Transmission-capacity sensitivity: The outage constraint causes a sharp transmission-capacity falloff as beamwidth narrows, while maximum capacity is more sensitive to sidelobe gain than to pattern type.The maximizing beamwidth is more sensitive to whether the pattern is ideal, transitional, or 3GPP.
- Orientation uncertainty: Maximum spatial throughput and transmission capacity decrease as mean orientation error increases, while their maximizing beamwidths generally increase.Across the displayed patterns, both optimal beamwidths are nearly linear in mean orientation error; ideal sidelobe-free sectors retain TP-optimal beamwidth zero.
VII. CONCLUSIONS & FUTURE WORK
The paper models beam misdirection in directional Poisson networks and proves beamwidth-dependent properties for spatial throughput and transmission capacity under ideal sectors without sidelobes. Numerical results extend the analysis to more complex radiation patterns and relate orientation error to throughput-maximizing beamwidths.
- A stochastic-geometry model captures how beam misdirection affects coverage and throughput in directional wireless networks.
- For ideal sector antennas without sidelobes, concave orientation-error distributions suffice to prove monotonicity of spatial throughput and quasi-concavity of transmission capacity versus beamwidth.
- Numerical results show spatial-throughput monotonicity is not preserved by more complex antenna models, whereas unimodality appears across studied radiation patterns for both metrics.
- Sidelobe strength can strongly affect transmission capacity maximized over beamwidth, while the corresponding maximizing beamwidths tend to be approximated well by the sidelobe-free sector.
- An apparent linear relationship connects mean orientation error with throughput-maximizing beamwidths across both metrics and the explored sector patterns.
APPENDIX
The appendix derives the success-probability and throughput expressions using independence, point-process transformations, Laplace methods, and derivative tests. It establishes global maximizers for successful-transmission intensity under the stated model.
- The proof begins from the SINR success condition and evaluates interference through its Laplace transform.
- Figure 9 plots beamwidth-optimized spatial throughput and transmission capacity together with their corresponding maximizing beamwidths.
- Independence among locations, fading, and antenna gains allows expectations to separate before applying a probability generating functional for the Poisson process.
- The derivation exchanges integration order, substitutes variables, integrates by parts, evaluates the resulting integral, and then averages over antenna gains.
- The appendix states the resulting success-probability expression in terms of transmit power, antenna gains, and their gain distributions.
- A single stationary point with negative second derivative establishes the global maximizer of successful-transmission intensity.
C. Proof of Prop. 3 (Concave F|ϵ| Implies Monotonicity of TPs in Beamwidth)
The proof rewrites spatial throughput using half-beamwidth and shows it decreases monotonically under a concave orientation-error c.d.f. The argument uses derivative bounds implied by concavity.
- Spatial throughput is rewritten using x = ω/2 and analyzed over x ∈ [0, π].
- To prove monotonic decrease, the proof reduces the task to showing that the derivative of spatial throughput is nonpositive.
- Concavity of F_|ϵ| bounds its values by the first-order Taylor approximation over [0, π].
- After evaluating the bound at y = 0 and adding the positive term Bx^2, the proof concludes the required derivative inequality.
- The derivative-based maximization framework uses positive constants and identifies λ* = 1/A as a global maximizer in the associated intensity expression.
E. Proof of Prop. 4 (TC with Sectors without Sidelobes)
The proof analyzes transmission capacity over regions determined by the orientation-error c.d.f. It establishes increasing, unimodal, and decreasing behavior across those regions, yielding a unique maximizer.
- Transmission capacity is rewritten with x = ω/2 and partitioned into increasing, quasi-concave, and decreasing regions.
- Monotonicity follows on the lower interval because 2 log(F(x)) + B increases with x.
- A sufficient quasi-concavity condition is applied so that stationary points correspond to local maxima and a global maximum.
- Transmission capacity is unimodal on the remaining interval, and its unique maximizer lies between the lower and upper boundaries.
- Beyond the upper endpoint, F(x) = 1 and f(x) = 0, simplifying the expression used to establish decreasing behavior.
G. Proof of Cor. 6 (Conditions on the Maximizing ω∗for TCs)
The proof uses unimodality of TCs(x), with x = ω/2, to locate its unique maximizer relative to ϵmax. Derivative signs at ϵmax determine whether the maximizer lies below or exactly at that boundary.
- TCs(x) is unimodal and has a unique maximizer x∗ within the stated interval, using Prop. 5 and its appendix proof.
- A sharp turn may occur at x = ϵmax, so the proof evaluates the left derivative of TCs there using (52).
- When the derivative at ϵmax is negative, the maximizing x∗ lies strictly below ϵmax.
- When f(ϵmax) > Bϵmax, the derivative at ϵmax is positive and unimodality forces x∗ to equal ϵmax.
- When the derivative at ϵmax is zero, the proof uses the left second derivative and the stationary-point relation in (59) to establish the boundary maximizer.
- The argument reuses the proof of Proposition 4 with A = πκd^2β^2/α, B = βdαη/Pt, and C = 1.