Clavier, Laurent

Egan, Malcolm

Gorce, Jean-Marie

Peters, Gareth,

Zheng, Ce,

Zheng, Ce

Hal-Diderot

On the Validity of Isotropic Complex α-Stable Interference Models for Interference in the IoT

Peters, Gareth, W.

INRIA a CCSD electronic archive server

HAL Id: hal-02263787https://hal.archives-ouvertes.fr/hal-02263787Submitted on 5 Aug 2019HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.On the Validity of Isotropic Complex α-StableInterference Models for Interference in the IoTCe Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters, Jean-Marie GorceTo cite this version:Ce Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters, Jean-Marie Gorce. On the Validity ofIsotropic Complex α-Stable Interference Models for Interference in the IoT. GRETSI 2019 - Colloquefrancophone de traitement du signal et des images, Aug 2019, Lille, France. pp.1-4. ￿hal-02263787￿On the Validity of Isotropic Complex α-Stable Interference Models forInterference in the IoTCe ZHENG1, Malcolm EGAN2, Laurent CLAVIER1, Gareth W. PETERS3, Jean-Marie GORCE21IMT Lille Douai, Univ. Lille, IEMN CNRS UMR 852050 Avenue Halley, 59650 Villeneuve-d’Ascq, France2Université de Lyon, INSA Lyon, INRIA, CITI Lab6 Avenue des Arts, 69621 Villeurbanne, France3School of Mathematical and Computer Sciences, Heriot-Watt UniversityEdinburgh, Currie EH14 4AL, UKce.zheng.etu@univ-lille.fr, malcom.egan@insa-lyon.frlaurent.clavier@telecom-lille.fr, g.peters@hw.ac.uk, jean-marie.gorce@insa-lyon.frRésumé – La croissance incessante d’objets communicants contraint les industriels et les chercheurs à développer des solutions permettantaux communications de rester fiables dans des environnements où linterférence n’est pas Gaussienne. Un modèle précis de cette interférenceest avant tout nécessaire et plusieurs propositions ont été faites comme le modèle de Middleton ou les distributions α-stables. Cet article étudiecette dernière proposition qui repose a priori sur des hypothèses peu réalistes, en particulier une puissance reçue non bornée. Nous montrons quece modèle s’avère cependant valide, même si l’on introduit une zone de garde autour du récepteur, limitant la puissance à une valeur finie, enadaptant l’exposant caractéristique de la distribution. Ceci est validé par l’étude des quantiles. La structure de dépendance est également étudiée.Abstract – With the growing numbers of IoT devices, both industries and researchers are facing the challenge of handling interference withnon-Gaussian behavior. To model such interference, theoretical approaches have been proposed such as Middleton Class A and B, α-stable,etc. This paper addresses the validity of the α-stable model. Its theoretical validity is based on idealistic assumptions, especially an unboundedreceived power. We show that the α-stable model remains accurate, if we adapt the characteristic exponent α, when we introduce a guard zonethat limits the received power. To do so, we study three features: the characteristic exponent, the quantiles and the dependence structure.1 IntroductionImpulsiveness is an significant feature of the interference inthe wireless network for Internet of Things (IoT). With largedeployments of transmitting devices, even if they operate atlow power levels, the amplitude of interference is more likelyto be large compared with standard Gaussian models. Such abehavior has been observed both in recent experimental stud-ies [1] and also in theoretical analysis [2]. As a consequence,Gaussian models are often not appropriate and the interferencestatistics lie in a more general class of models.To model the impulsive nature of the interference, the Mid-dleton Class A model has been widely adopted. Such interfer-ence arises when devices are located according to a homoge-neous poisson point process (PPP) with the guard bands, finitenetwork radius, independent fading and baseband emissions.However, the Middleton model has a complicated representa-tion, both in the form of its probability density function (PDF)and its characteristic function (CF). It also only forms an ap-proximation of the interference in more realistic point processmodels, such as the Poisson-Poisson cluster process studiedin [3].A more tractable alternative is the complex α-stable interfer-ence model. This model can be viewed as the limiting case ofthe Middleton model when the guard band and network radiitend to zero and infinity, respectively. Although it is more diffi-cult to work with than Gaussian models, it captures the heavy-tailed nature of the interference observed experimentally. How-ever, an open question is, quantitatively, how well the α-stablemodel approximates more realistic network models.In this paper, we focus on the validity of the α-stable inter-ference assumption. Our study is based on a comparison of theα-stable model with interference simulated in a network of fi-nite radius interferers and the presence guard-zones. We studythree features of the interference models : the estimated charac-teristic exponent α̂ ; the quantiles of the interference distribu-tions, i.e., Q-Q plots ; and the dependence structure (a precisedescription is provided in Section 4). We find for realistic de-vice densities that when the gaurd-zone radii do not exceed 5metres, the α-stable model with theoretical characteristic ex-ponent α = 4η where η is the path-loss exponent, is a goodapproximation. And the estimated stable model with α̂ keepsas a good approximation for different choices of guard zoneradii.2 System ModelConsider a network of devices located according to a homo-geneous (PPP) with intensity λ, denoted by Φ. These devicesform interferers for a receiver located at the origin. While sucha model has been widely studied and will play an importantrole in this paper, practical considerations require that there isa minimum distance between the receiver and the closest inter-ferer, rmin or the expected received power from each interfer-ing device does not exceed a maximum value.To account for the effect of non-zero guard zone radii, theinterference at the origin for a given time slot is given byZ =∑j∈Φa(rj)hjxj (1)where rj is the distance from device j to the origin, η is the pathloss exponent, hj ∼ CN (0, 1) is a Rayleigh fading coefficient,and xj is the baseband emission. To capture the effect of guardzones, the signal attenuation is governed bya(r) ={r−η/2, r ≥ rmin0, r < rmin.(2)A statistical characterization based on the characteristic func-tion of the interference Z have been obtained by Gulati [3] andSousa [4]. However, the resulting models are not analyticallytractable. To this end, we study the isotropic α-stable model,which is known to be the exact distribution ofZ when the guardzone radius rmin → 0 [2, 5, 6]. In the following section, we re-call the definitions and key properties for the class of isotropicα-stable random variables.3 Isotropic α-Stable Random VariablesTheα-stable random variables have heavy-tailed PDF, whichhave been widely used to model impulsive signals [7]. The dis-tribution of an α-stable random variable is parameterized byfour parameters : the characteristic exponent 0 < α ≤ 2 ; thescale parameter γ ∈ R+ ; the skew parameter β ∈ [−1, 1] ; andthe shift parameter δ ∈ R. As such, a common notation for anα-stable random variable X is X ∼ Sα(γ, β, δ).In general, α-stable random variables do not have closed-form PDFs. Instead, they are usually represented by their char-acteristic function, given by [7, Eq. 1.1.6]E[eiθX ]={exp{−γα|θ|α(1−iβ(signθ) tan πα2 )+iδθ}, α 6=1exp{−γ|θ|(1+iβ 2π (signθ) log |θ|)+iδθ}, α=1(3)In the case β = δ = 0, X is said to be a symmetric α-stablerandom variable, denoted as SαS.Isotropic complex α-stable random variables can now be de-fined as follows [7, Definition 2.6.2] :Definition 1. Let Z1, Z2 be two symmetric α-stable randomvariables. The complex α-stable random variable Z = Z1 +iZ2 is isotropic if it satisfies the conditioneiφZ(d)= Z for any φ ∈ [0, 2π). (4)A complex α-stable random variable Z in (4) can be ex-pressed as a vector Z = [Z1, Z2]. A particular class of α-stablerandom vectors is an instance of the sub-Gaussian α-stable ran-dom vectors 1, defined as follows.Definition 2. Any vector distributed as X=√A(G1, . . . , Gd),whereA ∼ Sα/2((cosπα/4)2/α, 1, 0), (5)and G = [G1, . . . , Gd] ∼ N (0, σ2I) is called a sub-GaussianSαS random vector with underlying Gaussian vector G.The following proposition [7, Corollary 2.6.4] highlights thelink between isotropicα-stable random variables and sub-Gaussianα-stable random vectors.Proposition 1. Let 0 < α < 2. A complex α-stable randomvariable Z = Z1 + iZ2 is isotropic if and only if there aretwo independent and identically distributed zero-mean Gaus-sian random variables G1, G2 with variance σ2 and a ran-dom variable A ∼ Sα/2((cos(πα/4))2/α, 1, 0) independentof (G1, G2)T such that (Z1, Z2)T = A1/2(G1, G2)T . That is,(Z1, Z2)T is a sub-Gaussian SαS random vector.4 α-Stable Interference ApproximationThe key question that we address in this paper is under whichconditions is the isotropic α-stable model a good approxima-tion of the interference arising in the system model detailedin Section 2. To address this question, we proceed numeri-cally by studying the quantiles of the marginals and the de-pendence between the real and imaginary parts via copula. Inparticular, we generate a simulated data set based on the systemmodel in Section 2 and compare it with the theoretical and fittedisotropic α-stable model. In the numerical results, the intensityλ = 0.001 m−2 and the pathloss η = 5, which corresponds toan average of one device in a disc of radius of approximately18 m and a non-line of sight pathloss environment (e.g., indooror urban).4.1 Properties of the Marginal DistributionsWe first examine the distributions of real and imaginary partsof the interference in Section 2. In the isotropic α-stable model,the real and imaginary parts are SαS random variables.It has been shown that the marginal distribution is α-stablewith α = 4η when rmin → 0 [6]. Hence, to obtain further in-sights into the impact of the guard zone radius rmin on the be-havior of the distributions, we first study Q-Q plots of the realor imaginary parts of the interference Z. In particular, we plotthe quantiles for the interference estimated from the simulateddata set with rmin > 0, denoted byQI , against the quantiles forthe theoretical model with rmin = 0, denoted by Q0. Note that1. There exist also sub-Gaussian stable vectors allowing for more generaldependence structure [7], but are not necessary for the purposes of this paper.when the two distributions are the same, the Q-Q plot exhibitsa straight line.Fig. 1 shows the Q-Q plots for varying rmin. When rmin ≤5 m, the theoretical model is in good agreement with the sim-ulated data. In particular, the figures exhibit straight lines. Forrmin > 5 m, the quantiles begin to significantly differ, imply-ing that α = 4η is not a good approximation. Hence, we fit thedata set to the stable model to get the estimated characteristicexponent α̂.FIGURE 1 – Q-Q plots, the system model (rmin > 0) againstthe theoretical model (rmin = 0)Fig. 2 shows the effect of increasing the guard zone radiusrmin. Observe that for small rmin, α̂ is approximately 0.8, asexpected from the theoretical model. On the other hand, as rmingoes beyond 5 m, α̂ increases from 0.9 to approximately 2 forrmin > 30 m. This implies that for very large choices of rmin, aGaussian model is the best fit. Nevertheless, for rmin < 30 m,the Gaussian model is not a good choice.FIGURE 2 – Estimated α̂ under different guard zone radiiFurthermore, we keep QI and replace Q0 with the quantilesfor the fitted stable model with α̂ in Fig. 3. In Fig. 3, whenrmin ≤ 5 m, the estimated model is in good agreement withthe simulated data, which is consistent with Fig. 1 due to α̂ ≈0.8 as shown in Fig. 2. For rmin > 5 m, the quantiles start toslightly differ, but still remains approximately as a straight line.However, when rmin > 30 m, the Q-Q plot is approximately astraight line again for the reason that α̂ is close to 2 as shownin Fig. 2, indicating a more Gaussian behavior. This validatesthat the estimated stable model keeps a good approximation fordifferent rmin.4.2 Dependence StructureIt is first important to mention that the isotropic α-stable ran-dom vector contains some dependency between the differentdimensions as long as α < 2, while the isotropic Gaussian vec-tor results in uncorrelated dimensions. As such, it is useful tohave tractable representations of the joint probability densityfunction.A popular method in statistics for tractably modeling non-Gaussian multivariate distributions is based on copulas. In thecopula modeling approach, the joint distribution function of arandom vector in Rn, say X = [X1, . . . , Xn]T , is given asF (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)), (6)where C : [0, 1]n → [0, 1] is called a copula function, andFi, i = 1, . . . , n are the marginal distribution functions.The copula of a multivariate distribution can be viewed ascapturing the dependence structure of the distribution. More-over, this dependence structure can be investigated by trans-forming a random vector X in Rn to the copula space via thetransformation (x1, . . . , xn) 7→ (F1(x1), . . . , Fn(xn)). Thisperspective is useful as it reveals non-linear dependencies indata, which are often more important than linear dependen-cies in the case of heavy-tailed data (e.g., in α-stable models(α < 2), covariances are infinite) [8, 9].Fig. 4 shows the scatter plots for the simulated data set cor-responding to different rmin. Observe that in Fig. 4e, the scat-ter plot is approximately uniform, which implies that the realand imaginary parts of the interference are independent. Thisis expected because the interference Z is isotropic and in theregime rmin > 30 m approximately Gaussian. In particular, ifa complex Gaussian random variable is isotropic, then it is wellknown that its real and imaginary parts are independent.On the other hand, when rmin < 5 m the scatter points areconcentrated in a non-uniform manner. In this regime, we havealready seen that the distributions for the real and imaginaryparts of the interference are approximately symmetric α-stable.For α < 2, isotropic complex α-stable random variables do nothave independent real and imaginary parts. This is particularlyevident by the concentration of points in the corners of the scat-ter plot. The consequence is that when the magnitude of the realpart of the interference is large, there is a greater probability ofthe imaginary part of the interference to have a large magnitudethan in the Gaussian model.Fig. 4c and Fig. 4d show the impact of further increasingrmin. In these cases, the scatter plot is again non-uniform, whichmeans that the Gaussian model is not an appropriate choice.(a) rmin = 2.5m (b) rmin = 5m (c) rmin = 10m (d) rmin = 30mFIGURE 3 – Q-Q plots, the system model (rmin > 0) against the fitted model (α̂)(a) rmin = 0 (b) rmin = 2.5m (c) rmin = 5m (d) rmin = 10m (e) rmin = 30mFIGURE 4 – Samples in Copula space under different rmin5 ConclusionsInterference modeling for the IoT is challenging due to thefact that the interference is often impulsive, ruling out Gaus-sian models. While there has been significant work investigat-ing good approximations for the interference distribution, anextensive investigation into the impact of practical network pa-rameters has not been carried out. In this work, we have studiedwhen an isotropic α-stable model forms a good approximationfor the interference distribution. This study investigated boththe impact of non-zero guard zone radii on the marginal distri-butions and the joint distribution. For realistic network param-eters, isotropic α-stable models are a significantly better choicethan Gaussian models.AcknowledgementsThis work has been (partly) funded by the French NationalAgency for Research (ANR) under grant ANR-16-CE25-0001- ARBURST.Fruitful discussions in the framework of the COST ACTIONCA15104, IRACON and in IRCICA, USR CNRS 3380 havealso contributed to this work.Références[1] B. Vejlgaard, M. Lauridsen, H. Nguyen, I. Kovács, P. Mo-gensen, and M. Sørensen, “Interference impact on cover-age and capacity for low power wide area IoT networks,”in IEEE Wireless Communications and Networking Con-ference (WCNC), 2017.[2] M. Egan, L. Clavier, M. de Freitas, L. Dorville, J.-M.Gorce, and A. Savard, “Wireless communciation in dy-namic interference,” in IEEE Global CommunciationsConference (GLOBECOM), 2017.[3] K. Gulati, B. Evans, J. Andrews, and K. Tinsley, “Statis-tics of co-channel interference in a field of Poisson-Poisson clustered interferers,” IEEE Transactions on Sig-nal Processing, vol. 58, no. 12, pp. 6207–6222, 2010.[4] E. S. Sousa, “Performance of a spread spectrum packetradio network link in a poisson field of interferers,”IEEE Transactions on Information Theory, vol. 38, no. 6,pp. 1743–1754, 1992.[5] M. Egan, M. de Freitas, L. Clavier, A. Goupil, G. Peters,and N. Azzaoui, “Achievable rates for additive isotropicalpha-stable noise channels,” in Proc. of the IEEE Inter-national Symposium on Information Theory, 2016.[6] M. Egan, L. Clavier, C. Zheng, M. de Freitas, and J.-M.Gorce, “Dynamic interference for uplink SCMA in large-scale wireless networks without coordination,” EURASIPJournal on Wireless Communications and Networking,vol. 1, 2018.[7] G. Samorodnitsky and M. Taqqu, Stable Non-GaussianRandom Processes. New York, NY : CRC Press, 1994.[8] X. Yan, L. Clavier, G. Peters, N. Azzaoui, F. Septier, andI. Nevat, “Skew-t copula for dependence modelling ofimpulsive (alpha-stable) interference,” in IEEE Interna-tional Conference on Communications (ICC), 2015.[9] C. Zheng, M. Egan, L. Clavier, G. Peters, and J.-M. Gorce, “Copula-based interference models for IoTwireless networks, in Proc. IEEE International Confer-ence on Communications (ICC), 2019.

ZHENG, Ce,

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-02263787
https://hal.archives-ouvertes.fr/hal-02263787
Submitted on 5 Aug 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
On the Validity of Isotropic Complex -Stable
Interference Models for Interference in the IoT
Ce Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters, Jean-Marie Gorce
To cite this version:
Ce Zheng, Malcolm Egan, Laurent Clavier, Gareth Peters, Jean-Marie Gorce. On the Validity of
Isotropic Complex -Stable Interference Models for Interference in the IoT. GRETSI 2019 - Colloque
francophone de traitement du signal et des images, Aug 2019, Lille, France. pp.1-4. ￿hal-02263787￿
On the Validity of Isotropic Complex α-Stable Interference Models for
Interference in the IoT
Ce ZHENG1, Malcolm EGAN2, Laurent CLAVIER1, Gareth W. PETERS3, Jean-Marie GORCE2
1IMT Lille Douai, Univ. Lille, IEMN CNRS UMR 8520
50 Avenue Halley, 59650 Villeneuve-d’Ascq, France
2Universite´ de Lyon, INSA Lyon, INRIA, CITI Lab
6 Avenue des Arts, 69621 Villeurbanne, France
3School of Mathematical and Computer Sciences, Heriot-Watt University
Edinburgh, Currie EH14 4AL, UK
ce.zheng.etu@univ-lille.fr, malcom.egan@insa-lyon.fr
laurent.clavier@telecom-lille.fr, g.peters@hw.ac.uk, jean-marie.gorce@insa-lyon.fr
Re´sume´ – La croissance incessante d’objets communicants contraint les industriels et les chercheurs a` de´velopper des solutions permettant
aux communications de rester fiables dans des environnements ou` linterfe´rence n’est pas Gaussienne. Un mode`le pre´cis de cette interfe´rence
est avant tout ne´cessaire et plusieurs propositions ont e´te´ faites comme le mode`le de Middleton ou les distributions α-stables. Cet article e´tudie
cette dernie`re proposition qui repose a priori sur des hypothe`ses peu re´alistes, en particulier une puissance rec¸ue non borne´e. Nous montrons que
ce mode`le s’ave`re cependant valide, meˆme si l’on introduit une zone de garde autour du re´cepteur, limitant la puissance a` une valeur finie, en
adaptant l’exposant caracte´ristique de la distribution. Ceci est valide´ par l’e´tude des quantiles. La structure de de´pendance est e´galement e´tudie´e.
Abstract – With the growing numbers of IoT devices, both industries and researchers are facing the challenge of handling interference with
non-Gaussian behavior. To model such interference, theoretical approaches have been proposed such as Middleton Class A and B, α-stable,
etc. This paper addresses the validity of the α-stable model. Its theoretical validity is based on idealistic assumptions, especially an unbounded
received power. We show that the α-stable model remains accurate, if we adapt the characteristic exponent α, when we introduce a guard zone
that limits the received power. To do so, we study three features: the characteristic exponent, the quantiles and the dependence structure.
1 Introduction
Impulsiveness is an significant feature of the interference in
the wireless network for Internet of Things (IoT). With large
deployments of transmitting devices, even if they operate at
low power levels, the amplitude of interference is more likely
to be large compared with standard Gaussian models. Such a
behavior has been observed both in recent experimental stud-
ies [1] and also in theoretical analysis [2]. As a consequence,
Gaussian models are often not appropriate and the interference
statistics lie in a more general class of models.
To model the impulsive nature of the interference, the Mid-
dleton Class A model has been widely adopted. Such interfer-
ence arises when devices are located according to a homoge-
neous poisson point process (PPP) with the guard bands, finite
network radius, independent fading and baseband emissions.
However, the Middleton model has a complicated representa-
tion, both in the form of its probability density function (PDF)
and its characteristic function (CF). It also only forms an ap-
proximation of the interference in more realistic point process
models, such as the Poisson-Poisson cluster process studied
in [3].
A more tractable alternative is the complex α-stable interfer-
ence model. This model can be viewed as the limiting case of
the Middleton model when the guard band and network radii
tend to zero and infinity, respectively. Although it is more diffi-
cult to work with than Gaussian models, it captures the heavy-
tailed nature of the interference observed experimentally. How-
ever, an open question is, quantitatively, how well the α-stable
model approximates more realistic network models.
In this paper, we focus on the validity of the α-stable inter-
ference assumption. Our study is based on a comparison of the
α-stable model with interference simulated in a network of fi-
nite radius interferers and the presence guard-zones. We study
three features of the interference models : the estimated charac-
teristic exponent αˆ ; the quantiles of the interference distribu-
tions, i.e., Q-Q plots ; and the dependence structure (a precise
description is provided in Section 4). We find for realistic de-
vice densities that when the gaurd-zone radii do not exceed 5
metres, the α-stable model with theoretical characteristic ex-
ponent α = 4η where η is the path-loss exponent, is a good
approximation. And the estimated stable model with αˆ keeps
as a good approximation for different choices of guard zone
radii.
2 System Model
Consider a network of devices located according to a homo-
geneous (PPP) with intensity λ, denoted by Φ. These devices
form interferers for a receiver located at the origin. While such
a model has been widely studied and will play an important
role in this paper, practical considerations require that there is
a minimum distance between the receiver and the closest inter-
ferer, rmin or the expected received power from each interfer-
ing device does not exceed a maximum value.
To account for the effect of non-zero guard zone radii, the
interference at the origin for a given time slot is given by
Z =
∑
j∈Φ
a(rj)hjxj (1)
where rj is the distance from device j to the origin, η is the path
loss exponent, hj ∼ CN (0, 1) is a Rayleigh fading coefficient,
and xj is the baseband emission. To capture the effect of guard
zones, the signal attenuation is governed by
a(r) =
{
r−η/2, r ≥ rmin
0, r < rmin.
(2)
A statistical characterization based on the characteristic func-
tion of the interference Z have been obtained by Gulati [3] and
Sousa [4]. However, the resulting models are not analytically
tractable. To this end, we study the isotropic α-stable model,
which is known to be the exact distribution ofZ when the guard
zone radius rmin → 0 [2, 5, 6]. In the following section, we re-
call the definitions and key properties for the class of isotropic
α-stable random variables.
3 Isotropic α-Stable Random Variables
Theα-stable random variables have heavy-tailed PDF, which
have been widely used to model impulsive signals [7]. The dis-
tribution of an α-stable random variable is parameterized by
four parameters : the characteristic exponent 0 < α ≤ 2 ; the
scale parameter γ ∈ R+ ; the skew parameter β ∈ [−1, 1] ; and
the shift parameter δ ∈ R. As such, a common notation for an
α-stable random variable X is X ∼ Sα(γ, β, δ).
In general, α-stable random variables do not have closed-
form PDFs. Instead, they are usually represented by their char-
acteristic function, given by [7, Eq. 1.1.6]
E[eiθX ]=
{
exp
{−γα|θ|α(1−iβ(signθ) tan piα2 )+iδθ} , α 6=1
exp
{−γ|θ|(1+iβ 2pi (signθ) log |θ|)+iδθ} , α=1
(3)
In the case β = δ = 0, X is said to be a symmetric α-stable
random variable, denoted as SαS.
Isotropic complex α-stable random variables can now be de-
fined as follows [7, Definition 2.6.2] :
Definition 1. Let Z1, Z2 be two symmetric α-stable random
variables. The complex α-stable random variable Z = Z1 +
iZ2 is isotropic if it satisfies the condition
eiφZ
(d)
= Z for any φ ∈ [0, 2pi). (4)
A complex α-stable random variable Z in (4) can be ex-
pressed as a vector Z = [Z1, Z2]. A particular class of α-stable
random vectors is an instance of the sub-Gaussian α-stable ran-
dom vectors 1, defined as follows.
Definition 2. Any vector distributed as X=
√
A(G1, . . . , Gd),
where
A ∼ Sα/2((cospiα/4)2/α, 1, 0), (5)
and G = [G1, . . . , Gd] ∼ N (0, σ2I) is called a sub-Gaussian
SαS random vector with underlying Gaussian vector G.
The following proposition [7, Corollary 2.6.4] highlights the
link between isotropicα-stable random variables and sub-Gaussian
α-stable random vectors.
Proposition 1. Let 0 < α < 2. A complex α-stable random
variable Z = Z1 + iZ2 is isotropic if and only if there are
two independent and identically distributed zero-mean Gaus-
sian random variables G1, G2 with variance σ2 and a ran-
dom variable A ∼ Sα/2((cos(piα/4))2/α, 1, 0) independent
of (G1, G2)T such that (Z1, Z2)T = A1/2(G1, G2)T . That is,
(Z1, Z2)
T is a sub-Gaussian SαS random vector.
4 α-Stable Interference Approximation
The key question that we address in this paper is under which
conditions is the isotropic α-stable model a good approxima-
tion of the interference arising in the system model detailed
in Section 2. To address this question, we proceed numeri-
cally by studying the quantiles of the marginals and the de-
pendence between the real and imaginary parts via copula. In
particular, we generate a simulated data set based on the system
model in Section 2 and compare it with the theoretical and fitted
isotropic α-stable model. In the numerical results, the intensity
λ = 0.001 m−2 and the pathloss η = 5, which corresponds to
an average of one device in a disc of radius of approximately
18 m and a non-line of sight pathloss environment (e.g., indoor
or urban).
4.1 Properties of the Marginal Distributions
We first examine the distributions of real and imaginary parts
of the interference in Section 2. In the isotropic α-stable model,
the real and imaginary parts are SαS random variables.
It has been shown that the marginal distribution is α-stable
with α = 4η when rmin → 0 [6]. Hence, to obtain further in-
sights into the impact of the guard zone radius rmin on the be-
havior of the distributions, we first study Q-Q plots of the real
or imaginary parts of the interference Z. In particular, we plot
the quantiles for the interference estimated from the simulated
data set with rmin > 0, denoted byQI , against the quantiles for
the theoretical model with rmin = 0, denoted by Q0. Note that
1. There exist also sub-Gaussian stable vectors allowing for more general
dependence structure [7], but are not necessary for the purposes of this paper.
when the two distributions are the same, the Q-Q plot exhibits
a straight line.
Fig. 1 shows the Q-Q plots for varying rmin. When rmin ≤
5 m, the theoretical model is in good agreement with the sim-
ulated data. In particular, the figures exhibit straight lines. For
rmin > 5 m, the quantiles begin to significantly differ, imply-
ing that α = 4η is not a good approximation. Hence, we fit the
data set to the stable model to get the estimated characteristic
exponent αˆ.
FIGURE 1 – Q-Q plots, the system model (rmin > 0) against
the theoretical model (rmin = 0)
Fig. 2 shows the effect of increasing the guard zone radius
rmin. Observe that for small rmin, αˆ is approximately 0.8, as
expected from the theoretical model. On the other hand, as rmin
goes beyond 5 m, αˆ increases from 0.9 to approximately 2 for
rmin > 30 m. This implies that for very large choices of rmin, a
Gaussian model is the best fit. Nevertheless, for rmin < 30 m,
the Gaussian model is not a good choice.
FIGURE 2 – Estimated αˆ under different guard zone radii
Furthermore, we keep QI and replace Q0 with the quantiles
for the fitted stable model with αˆ in Fig. 3. In Fig. 3, when
rmin ≤ 5 m, the estimated model is in good agreement with
the simulated data, which is consistent with Fig. 1 due to αˆ ≈
0.8 as shown in Fig. 2. For rmin > 5 m, the quantiles start to
slightly differ, but still remains approximately as a straight line.
However, when rmin > 30 m, the Q-Q plot is approximately a
straight line again for the reason that αˆ is close to 2 as shown
in Fig. 2, indicating a more Gaussian behavior. This validates
that the estimated stable model keeps a good approximation for
different rmin.
4.2 Dependence Structure
It is first important to mention that the isotropic α-stable ran-
dom vector contains some dependency between the different
dimensions as long as α < 2, while the isotropic Gaussian vec-
tor results in uncorrelated dimensions. As such, it is useful to
have tractable representations of the joint probability density
function.
A popular method in statistics for tractably modeling non-
Gaussian multivariate distributions is based on copulas. In the
copula modeling approach, the joint distribution function of a
random vector in Rn, say X = [X1, . . . , Xn]T , is given as
F (x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)), (6)
where C : [0, 1]n → [0, 1] is called a copula function, and
Fi, i = 1, . . . , n are the marginal distribution functions.
The copula of a multivariate distribution can be viewed as
capturing the dependence structure of the distribution. More-
over, this dependence structure can be investigated by trans-
forming a random vector X in Rn to the copula space via the
transformation (x1, . . . , xn) 7→ (F1(x1), . . . , Fn(xn)). This
perspective is useful as it reveals non-linear dependencies in
data, which are often more important than linear dependen-
cies in the case of heavy-tailed data (e.g., in α-stable models
(α < 2), covariances are infinite) [8, 9].
Fig. 4 shows the scatter plots for the simulated data set cor-
responding to different rmin. Observe that in Fig. 4e, the scat-
ter plot is approximately uniform, which implies that the real
and imaginary parts of the interference are independent. This
is expected because the interference Z is isotropic and in the
regime rmin > 30 m approximately Gaussian. In particular, if
a complex Gaussian random variable is isotropic, then it is well
known that its real and imaginary parts are independent.
On the other hand, when rmin < 5 m the scatter points are
concentrated in a non-uniform manner. In this regime, we have
already seen that the distributions for the real and imaginary
parts of the interference are approximately symmetric α-stable.
For α < 2, isotropic complex α-stable random variables do not
have independent real and imaginary parts. This is particularly
evident by the concentration of points in the corners of the scat-
ter plot. The consequence is that when the magnitude of the real
part of the interference is large, there is a greater probability of
the imaginary part of the interference to have a large magnitude
than in the Gaussian model.
Fig. 4c and Fig. 4d show the impact of further increasing
rmin. In these cases, the scatter plot is again non-uniform, which
means that the Gaussian model is not an appropriate choice.
(a) rmin = 2.5m (b) rmin = 5m (c) rmin = 10m (d) rmin = 30m
FIGURE 3 – Q-Q plots, the system model (rmin > 0) against the fitted model (αˆ)
(a) rmin = 0 (b) rmin = 2.5m (c) rmin = 5m (d) rmin = 10m (e) rmin = 30m
FIGURE 4 – Samples in Copula space under different rmin
5 Conclusions
Interference modeling for the IoT is challenging due to the
fact that the interference is often impulsive, ruling out Gaus-
sian models. While there has been significant work investigat-
ing good approximations for the interference distribution, an
extensive investigation into the impact of practical network pa-
rameters has not been carried out. In this work, we have studied
when an isotropic α-stable model forms a good approximation
for the interference distribution. This study investigated both
the impact of non-zero guard zone radii on the marginal distri-
butions and the joint distribution. For realistic network param-
eters, isotropic α-stable models are a significantly better choice
than Gaussian models.
Acknowledgements
This work has been (partly) funded by the French National
Agency for Research (ANR) under grant ANR-16-CE25-0001
- ARBURST.
Fruitful discussions in the framework of the COST ACTION
CA15104, IRACON and in IRCICA, USR CNRS 3380 have
also contributed to this work.
Re´fe´rences
[1] B. Vejlgaard, M. Lauridsen, H. Nguyen, I. Kova´cs, P. Mo-
gensen, and M. Sørensen, “Interference impact on cover-
age and capacity for low power wide area IoT networks,”
in IEEE Wireless Communications and Networking Con-
ference (WCNC), 2017.
[2] M. Egan, L. Clavier, M. de Freitas, L. Dorville, J.-M.
Gorce, and A. Savard, “Wireless communciation in dy-
namic interference,” in IEEE Global Communciations
Conference (GLOBECOM), 2017.
[3] K. Gulati, B. Evans, J. Andrews, and K. Tinsley, “Statis-
tics of co-channel interference in a field of Poisson-
Poisson clustered interferers,” IEEE Transactions on Sig-
nal Processing, vol. 58, no. 12, pp. 6207–6222, 2010.
[4] E. S. Sousa, “Performance of a spread spectrum packet
radio network link in a poisson field of interferers,”
IEEE Transactions on Information Theory, vol. 38, no. 6,
pp. 1743–1754, 1992.
[5] M. Egan, M. de Freitas, L. Clavier, A. Goupil, G. Peters,
and N. Azzaoui, “Achievable rates for additive isotropic
alpha-stable noise channels,” in Proc. of the IEEE Inter-
national Symposium on Information Theory, 2016.
[6] M. Egan, L. Clavier, C. Zheng, M. de Freitas, and J.-M.
Gorce, “Dynamic interference for uplink SCMA in large-
scale wireless networks without coordination,” EURASIP
Journal on Wireless Communications and Networking,
vol. 1, 2018.
[7] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian
Random Processes. New York, NY : CRC Press, 1994.
[8] X. Yan, L. Clavier, G. Peters, N. Azzaoui, F. Septier, and
I. Nevat, “Skew-t copula for dependence modelling of
impulsive (alpha-stable) interference,” in IEEE Interna-
tional Conference on Communications (ICC), 2015.
[9] C. Zheng, M. Egan, L. Clavier, G. Peters, and J.-
M. Gorce, “Copula-based interference models for IoT
wireless networks, in Proc. IEEE International Confer-
ence on Communications (ICC), 2019.


English

https://hal.archives-ouvertes.fr/hal-02263787/document

On the Validity of Isotropic Complex α-Stable Interference Models for Interference in the IoT

Abstract

Similar works

Full text

Available Versions

Hal-Diderot

INRIA a CCSD electronic archive server

Archive Ouverte en Sciences de l'Information et de la Communication