International audienceThe uncoordinated nature of IoT networks makes interference management a challenging problem. Motivated by NB-IoT and SCMA protocols, we study the interference statistics of a Poisson spatial field of IoT interferers exploiting OFDM. We show for a sufficiently large number of subcarriers that the interference statistics are well-approximated by a sub-Gaussian α-stable random vector with a non-isotropic underlying Gaussian random vector. This result forms a basis to improve detection and decoding algorithms at the receiver

Clavier, Laurent

Egan, Malcolm

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HAL Id: hal-03018298https://hal.archives-ouvertes.fr/hal-03018298Submitted on 22 Nov 2020HAL 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.Multivariate alpha-Stable Models in OFDM-Based IoTNetworks with Interference From a Poisson Spatial Fieldof InterferersMalcolm Egan, Laurent ClavierTo cite this version:Malcolm Egan, Laurent Clavier. Multivariate alpha-Stable Models in OFDM-Based IoT Networkswith Interference From a Poisson Spatial Field of Interferers. DCCN 2020 - 23rd International Con-ference on Distributed Computer and Communication Networks: Control, Computation, Communi-cations, Sep 2020, Virtual Event, Russia. pp.1-9. ￿hal-03018298￿Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020UDC: 123.456Multivariate α-Stable Models in OFDM-Based IoTNetworks with Interference From a Poisson SpatialField of InterferersMalcolm Egan1 and Laurent Clavier21Univ. Lyon, INSA Lyon, INRIA, CITI, France2IMT Lille Douai, FranceAbstractThe uncoordinated nature of IoT networks makes interference management achallenging problem. Motivated by NB-IoT and SCMA protocols, we study theinterference statistics of a Poisson spatial field of IoT interferers exploiting OFDM.We show for a sufficiently large number of subcarriers that the interferencestatistics are well-approximated by a sub-Gaussian α-stable random vector witha non-isotropic underlying Gaussian random vector. This result forms a basis toimprove detection and decoding algorithms at the receiver.Keywords: α-stable, interference, IoT, NB-IoT, SCMA1. IntroductionA key feature of IoT wireless networks is a lack of coordination between devices.In contrast to mobile cellular networks, with a high level of coordination, onlylimited interference mitigation strategies are available. In particular, transmissionsare typically only constrained by duty cycle guidelines or the time to listen incarrier-sense multiple access (CSMA) protocols [1]. As such, devices must adapt tointerference locally via signal processing tailored to the interference statistics [2].Motivated by the need for appropriate statistical models to perform signalprocessing, there have recently been a number of studies of the interference statisticsin the context of IoT communications. One key observation, obtained from anexperimental campaign in Aalborg, Denmark, is that the interference power oneach subband is heavy tailed[1], with the probability of large interference powersignificantly higher than for Gaussian models. This experimental observation isconsistent with theoretical analysis of interference arising from a Poisson spatial fieldof interferers [3, 4] and variations including the Poisson-Poisson clustered interferersin [5]. In particular, α-stable models have been shown to be a good approximation ofMalcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020the statistics for the interference amplitude when guard zones are sufficiently smalland the network radius is sufficiently large [6, 5].Despite this progress, focusing on the interference on a single subband, there hasbeen limited work investigating interference in the context of OFDM-based systems.In the context of the IoT, OFDM is exploited in narrowband IoT (NB-IoT). In[7, 8, 2], we have studied the scenario where interfering devices randomly select asubset of bands and developed accurate approximations of the interference statisticsexploiting copula models. In [9], we developed a multivariate α-stable model under aGaussian assumption on the product of fading and the baseband emission. However,in both of these papers, the interferers are assumed to have the same subcarrierspacing as the desired link, which is not necessarily the case in NB-IoT.In this paper, we study the interference statistics arising from a Poisson spatialfield of interferers in the scenario where the desired transmitter exploits OFDM witha different subcarrier spacing to the interfering devices. As a consequence, the signalon each subband—as observed by the receiver—is affected by each OFDM symbol ofthe interfering devices. This introduces non-trivial statistical dependence betweenthe signal observed on each subband at the receiver, even if each interferer transmitson all subcarriers.Under the assumption of interferers adopting PSK modulation, we show that theresulting interference is well approximated by a multivariate α-stable model wheneach interferers transmits on all of their subcarriers. In particular, the interferencemodel is sub-Gaussian α-stable with an underlying non-isotropic Gaussian randomvector. In contrast, when the interferers use the same subcarrier spacing as the desiredtransmitter, the interference is sub-Gaussian α-stable with an isotropic Gaussianrandom vector [8].2. System ModelConsider a desired system with receiver at the origin exploiting OFDM with asymbol period T ; that is, the subcarriers have a bandwidth of ∆f satisfying 1∆f = T .The desired system coexists with a network of interfering devices with locationsforming a homogeneous Poisson point process Φ with intensity λ. Devices in theinterfering network also transmit using OFDM with a total of L subcarriers andsubcarrier spacing ∆f ′, which in general does not satisfy the OFDM orthogonalityconditions with the symbol period utilized by the desired receiver.We assume that the symbol period of the interfering system, T ′ = 1∆f ′ , satisfiesT ′ > T . As such, at most two interfering symbols can be present in each time periodof duration T . We also assume that each interferer utilizes M -PSK with averagepower P , with equally likely symbols.Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020In continuous time, the interference observed by the desired receiver at the origindue to the l-th subcarrier of the interfering system is then given byYl(t) =∑k∈Φr−η/2k hk,l[√P2√2Tcos(2π(fc + l∆f′)t+ θk,l + φk,l)u(Dk − t)√P2√2Tcos(2π(fc + l∆f′)t+ θ′k,l + φk,l)u(t−Dk)], 0 ≤ t ≤ T. (1)where rk is the distance from interferer k ∈ Φ to the origin at the receiver, η is thepathloss coefficient, hk,leiφk,l is the baseband fading coefficient on the l-th subcarrierfrom interferer k, eiθk,l is the baseband data transmitted in the first symbol, eiθk,ll isthe baseband data transmitter in the second symbol, and Dk ∼ Unif[0, T ] is the timethat transmission of the second symbol begins, assumed to be uniformly distributedon [0, T ].Using the linear correlation demodulator [4], the desired receiver is subject tointerference on subcarrier i corresponding to a frequency fc + i∆f . In particular, thein-phase component of the interference contributed by the l-th interfering subcarrieris given byY1,i,l =∫ T0Yl(t)√2Tcos(2π(fc + i∆f)t)dt, (2)which yields in-phase interference given byY1,i =bL2c−1∑l=b−L2cY1,i,l. (3)Similarly, the quadrature component of the interference contributed by the l-thinterfering subcarrier is given byY2,i,l = −∫ T0Y (t)√2Tsin(2π(fc + i∆f)t)dt, (4)yielding quadrature interference given byY2,i =bL2c−1∑l=b−L2cY2,i,l. (5)Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020In this paper, we are concerned with characterizing the interference randomvectorY = [Y1,1, Y2,1, . . . , Y1,K , Y2,K ], (6)which consists of the inphase and quadrature interference components on eachsubcarrier of the desired link. In the following, we derive a tight approximation ofthe statistics of Y as L→∞.3. Multivariate α-Stable Interference ModelIn this section, we derive an approximation for the interference statistics forsufficiently large L. As the receiver exploits the correlation demodulator, the in-phasecomponent of the signal observed on subcarrier i of the receiver due to subcarrier lof the interfering system is given byY1,i,l =∫ T0Yl(t)√2Tcos(2π(fc + i∆ft)dt=∑k∈Φr−η/2k hk,lPT∫ T0[cos(2π(fc + l∆f′)t+ θk,l + φk,l)u(Dk − t)+ cos(2π(fc + l∆f′)t+ θ′k,l + φk,l)u(t−Dk)]cos(2π(fc + i∆f)t)dt≈ PT∑k∈Φr−η/2k hk,l·[12π(l∆f ′ − i∆f)(sin(2π(l∆f ′ − i∆f)Dk + θk,l + φk,l)− sin(θk,l + φk,l))+12π(l∆f ′ − i∆f)sin(2π(l∆f ′ − i∆f)T + θ′k,l + φk,l)− 12π(l∆f ′ − i∆f)sin(2π(l∆f ′ − i∆f)Dk + θ′k,l + φk,l)]=∑k∈ΦPr−η/2kTX1,k,l, (7)where the approximation is tight when fcT  1. Using a similar argument, thequadrature component of the signal observed on subcarrier i of the receiver due toMalcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020subcarrier l of the interfering system is given byY2,i,l ≈PT∑k∈Φr−η/2k hk,l·[12π(l∆f ′ − i∆f)(− cos(2π(l∆f ′)Dk + θk,l + φk,l) + cos(θk,l + φk,l))− 12π(l∆f ′ − i∆f)sin(2π(l∆f ′ − i∆f)T + θ′k,l + φk,l)+12π(l∆f ′ − i∆f)sin(2π(l∆f ′ − i∆f)Dk + θ′k,l + φk,l)]=∑k∈ΦPr−η/2kTX2,k,l. (8)We now considerY1,i =bL2c−1∑l=b−L2cY1,i,l=bL2c−1∑l=b−L2c∑k∈Φr−η/2kTX1,k,l=∑k∈Φ√Lr−η/2kT1√LbL2c−1∑l=b−L2cX1,k,l. (9)Conditioned on Dk, the terms {X1,k,l}l are independent and identically distributedzero mean Gaussian random variables, with variance independent of Dk. This is dueto the fact that Θk,l = θk,l + φk,l has distributionPr(θk,l + φk,l ≤ x) =M∑j=11M((x− θj)2π1{0≤x−θj<2π} + 1{x−θj>2π}), (10)where θj is the j-th symbol in the M -PSK constellation. We then have, for anyz ∈ R,E[sin(z + Θ)] =M∑j=11ME[sin(z + θj + φ)] = 0. (11)Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020Similarly,E[sin2(z + Θ)] =M∑j=11ME[sin2(z + θj + φ)] =12. (12)Let σ2Y = Var(X1,k,l) = Var(X2,k,l). As L→∞, by the central limit theorem, wethen obtainY1,i ≈ Ỹ1,i =∑k∈ΦP√Lr−η/2kTG1,k, (13)where G1,k ∼ N (0, σY ). Similarly,Y2,i ≈ Ỹ2,i∑k∈ΦP√Lr−η/2kTG2,k, (14)where G2,k ∼ N (0, σY ).WritingỸ1,i =P√LT∑k∈Φ(r−η/4k )2G1,k (15)and applying the mapping theorem for Poisson point processes [3], it follows thatỸ1,i is a symmetric 4/η-stable random variable by the LePage series expansion. Formore details, see [3, 4]. By the same argument, Ỹ2,i is also a symmetric 4/η-stablerandom variable.We now turn to the random vector Y. So far, we have shown that each marginalof Y, i.e., Y1,1, Y2,1, . . ., is well approximated by an identically distributed symmetricα-stable random variable. If each term G1,1, G2,1, . . . is independent, then by [8, 2],it follows that Y is a sub-Gaussian α-stable random vector with underlying Gaussianrandom vector G ∼ N (0, σI).However, in the present case, the terms G1,1, G2,1, . . . are not in general inde-pendent due to the presence of {θk,l, φk,l, hk,l} in the expressions for Y1,i and Y2,i.As such, there is a non-zero correlation between each pair Yj,i, Yj′,i′ for j 6= j′ andi, i′ ∈ {1, . . . , I}. An approximation for the statistics of Y is given in Theorem 1.Theorem 1. Ỹ is a sub-Gaussian 4/η-stable random vector with a non-isotropicunderlying Gaussian random vector.Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020Proof. We first writeỸ =√LT∑k∈Φr−η/2k Gk=√LTC∑k∈Φr−η/2k G̃k, (16)where G̃k ∼ N (0, σY I) and C is the Cholesky decomposition of the covariance matrixof Gk. It then follows that∑k∈Φ r−η/2k G̃k is a sub-Gaussian α-stable random vectorwith isotropic underlying Gaussian random vector Z by [8, 2]. Using the fact that∑k∈Φ r−η/2k G̃k is sub-Gaussian α-stable,Ỹ =√LTC√AZ, (17)where A is a skewed stable random variable (see Definition 1), it then follows that Ỹis a sub-Gaussian α-stable random vector with non-isotropic underlying Gaussianrandom vector. 4. ConclusionMultivariate interference models naturally arise in OFDM-based communicationsystems, such as NB-IoT. In this paper, we considered the impact of a mismatchbetween the subcarrier spacing of the desired link and the interfering signals. In thecase of interferer locations forming a Poisson spatial field, we have shown that theinterference is well approximated by a sub-Gaussian α-stable model with non-isotropicunderlying Gaussian random vector when the number of interferer subcarriers L islarge, bearing similarities with the analysis of passband additive symmetric α-stablenoise in [10]. The result forms a basis for improved receiver design in IoT systems.A. α-Stable PreliminariesIn this appendix, we overview key properties of α-stable random vectors. Theprobability density function of an α-stable random variable is parameterized byfour parameters: the exponent 0 ≤ α ≤ 2; the scale parameter γ ∈ R+; the skewparameter β ∈ [−1, 1]; and the shift parameter δ ∈ R. As such, a common notationfor an α-stable random variable X is X ∼ Sα(γ, β, δ). In the case β = δ = 0, X issaid to be a symmetric α-stable random variable.In general, α-stable random variables do not have closed-form probability densityfunctions. Instead, they are more compactly represented by their characteristicMalcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020function, given by [11, 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(18)It is possible to extend the notion of an α-stable random variable to the multi-variate setting. In particular, a random vector X is a symmetric α-stable randomvector if for all a, b > 0 there exists c > 0 such thataX(1) + bX(2)d= cX, (19)where X(1),X(2) are independent copies of X.A sufficient condition for a random vector X to be a symmetric α-stable randomvector is that all linear combinations of the marginal distributions for the elements ofX are symmetric α-stable [11]. In general, d-dimensional symmetric α-stable randomvectors are also represented via their characteristic function, given by [11]E[eiθ·X] = exp(−∫Sd−1∣∣∣∣∣d∑k=1θksk∣∣∣∣∣αΓ(ds)), (20)where Γ is the unique symmetric measure on the surface of the d-dimensional unitsphere.In the case that a d-dimensional symmetric α-stable random vector X is trulyd-dimensional, there exists a joint probability density function pX(x) on Rd. Notethat a simple necessary and sufficient condition for X to be truly d-dimensional isfor the support of the spectral measure Γ to span Rd [?]. This condition meansthat degenerate symmetric α-stable random vectors (e.g., when Xi = Xj for somei 6= j, i, j ∈ {1, . . . , d}) are not considered.An important class of α-stable random vectors is the sub-Gaussian α-stablefamily.Definition 1. Any vector X distributed as X = (A1/2G1, . . . , A1/2Gd), whereA ∼ Sα/2((cosπα/4)2/α, 1, 0), (21)and G = [G1, . . . , Gd]T ∼ N (0, σ2I) is called a sub-Gaussian α-stable random vectorin Rd with underlying Gaussian vector G.Malcolm Egan and Laurent ClavierMultivariate α-Stable InterferenceDCCN 202014-18 September 2020REFERENCES1. M. Lauridsen, B. Vejlgaard, I. Kovács, H. Nguyen, P. Mogensen, Interferencemeasurements in the European 868 MHz ISM band with focus on LoRa andSigFox, in: IEEE Wireless Communications and Networking Conference (WCNC),2017.2. C. Zheng, M. Egan, L. Clavier, G. Peters, J.-M. Gorce, Impulsive multivariateinterference models for iot networks, https://hal.archives-ouvertes.fr/hal-02533821/file/Multivariate_Interference_IoT.pdf (2020).3. J. Ilow, D. Hatzinakos, Analytic alpha-stable noise modeling in a poisson fieldof interfers or scatterers, IEEE Transactions on Signal Processing 46 (6) (1998)1601–1611.4. P. Pinto, M. Win, Communication in a Poisson field of interferers-part I: in-terference distribution and error probability, IEEE Transactions on WirelessCommunications 9 (7) (2010) 2176–2186.5. K. Gulati, B. Evans, J. Andrews, K. Tinsley, Statistics of co-channel interferencein a field of Poisson-Poisson clustered interferers, IEEE Transactions on SignalProcessing 58 (12) (2010) 6207–6222.6. E. Sousa, Performance of a spread spectrum packet radio network link in aPoisson field of interferers, IEEE Transactions on Information Theory 38 (6)(1992) 1743–1754.7. C. Zheng, M. Egan, L. Clavier, G. Peters, J.-M. Gorce, Copula-based interfer-ence models for iot wireless networks, in: ICC 2019-53rd IEEE InternationalConference on Communications, 2019, pp. 1–6.8. M. Egan, L. Clavier, C. Zheng, M. de Freitas, J.-M. Gorce, Dynamic interferencefor uplink SCMA in large-scale wireless networks without coordination, EURASIPJournal on Wireless Communications and Networking 1.9. C. Zheng, M. Egan, L. Clavier, T. Pedersen, J.-M. Gorce, Linear combining independent α-stable interference, in: IEEE International Conference on Commu-nications (ICC), 2020.10. A. Mahmood, M. Chitre, M. Armand, PSK communication with passbandadditive symmetric α-stable noise, IEEE Transactions on Communications 60 (10)(2012) 2990–3000.11. G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes, CRCPress, New York, NY, 1994.

Multivariate $alpha$-Stable Models in OFDM-Based IoT Networks with Interference From a Poisson Spatial Field of Interferers

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Multivariate $alpha$ -Stable Models in OFDM-Based IoT Networks with Interference From a Poisson Spatial Field of Interferers

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Multivariate alphaalphaalpha-Stable Models in OFDM-Based IoT Networks with Interference From a Poisson Spatial Field of Interferers

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Multivariate $alpha$ -Stable Models in OFDM-Based IoT Networks with Interference From a Poisson Spatial Field of Interferers