Coordinated Multi-Point Transmission: A Poisson-Delaunay Triangulation Based Approach
aa r X i v : . [ c s . I T ] J a n IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1
Coordinated Multi-Point Transmission: APoisson-Delaunay Triangulation Based Approach
Yan Li, Minghua Xia,
Member, IEEE , and Sonia A¨ıssa,
Fellow, IEEE
Abstract —Coordinated multi-point (CoMP) transmission is acooperating technique among base stations (BSs) in a cellularnetwork, with outstanding capability at inter-cell interference(ICI) mitigation. ICI is a dominant source of error, and hasdetrimental effects on system performance if not managedproperly. Based on the theory of Poisson-Delaunay triangulation,this paper proposes a novel analytical model for CoMP operationin cellular networks. Unlike the conventional CoMP operationthat is dynamic and needs on-line updating occasionally, theproposed approach enables the cooperating BS set of a userequipment (UE) to be fixed and off-line determined according tothe location information of BSs. By using the theory of stochasticgeometry, the coverage probability and spectral efficiency of atypical UE are analyzed, and simulation results corroborate theeffectiveness of the proposed CoMP scheme and the developedperformance analysis.
Index Terms —Cellular networks, Coordinated multi-point(CoMP) transmission, Poisson-Delaunay triangulation, Poisson-Voronoi tessellation, stochastic geometry.
I. I
NTRODUCTION C OORDINATED multi-point (CoMP) transmission andreception is considered for the 3rd Generation Partner-ship Project (3GPP) long term evolution advanced (LTE-A)as a promising technique to mitigate inter-cell interference,thereby improving the system coverage, the spectral efficiencyand in particular the quality-of-service (QoS) of cell-edge userequipments (UEs) in cellular networks. In the state-of-the-art technical report for physical layer aspects of the studyitem “Coordinated multi-point operation for LTE”, namely,3GPP TR 36.819 [1], two major CoMP strategies are high-lighted: coordinated scheduling/beamforming (CS/CB) andjoint processing. In the CS/CB strategy, data for a UE isonly available at one point of the CoMP cooperation set
Manuscript received April 10, 2019; revised September 16, 2019 and De-cember 24, 2019; accepted January 16, 2020. This work was supported in partby the National Natural Science Foundation of China under Grant 61671488,in part by the Major Science and Technology Special Project of GuangdongProvince under Grant 2018B010114001, in part by the Fundamental ResearchFunds for the Central Universities under Grant 191gjc04, and in part by aDiscovery Grant from the Natural Sciences and Engineering Research Council(NSERC) of Canada. The associate editor coordinating the review of thispaper and approving it for publication was A. Zaidi.Y. Li and M. Xia are with the School of Electronics and InformationTechnology, Sun Yat-sen University, Guangzhou, 510006, China. M. Xia isalso with Southern Marine Science and Engineering Guangdong Laboratory(Zhuhai) (e-mail: [email protected], [email protected]).S. A¨ıssa is with the Institut National de la Recherche Scientifique (INRS-EMT), University of Quebec, Montreal, QC, H5A 1K6, Canada (e-mail:[email protected]).Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.Digital Object Identifier in a time-frequency resource block. In the joint processingstrategy, on the other and, data for a UE is available atmore than one point in the CoMP cooperating set. Clearly,the joint processing strategy outperforms the CS/CB, butat the cost of higher backhaul load. In practice, the jointprocessing strategy has two major implementation schemes:joint transmission, and dynamic point selection/muting [1]. Inthe former, multiple points simultaneously transmit data toa UE in a time-frequency resource block so as to improvedata throughput and/or decrease outage probability. As for thelatter, although data is simultaneously available at multiplepoints, only one point out of the cooperation set transmitsdata to a UE. In this paper, both implementation schemes ofjoint processing will be examined.In the open literature, there are two distinct methodologiesto investigate the performance of CoMP in cellular networks.One is the classic deterministic approach , which is basedon the widely used regular hexagonal cellular model. Thismethod is simple yet highly idealized and, hence, inaccuratein practice. To better reflect the actual deployment of basestations (BSs), the theory of stochastic geometry was in recentyears introduced to model and analyze cellular networks,yielding the novel stochastic approach [2], where a Poissonpoint process (PPP) is used to describe the distribution of BSswhile UEs are uniformly distributed in the coverage area of thenetwork. Each UE is then associated with a target BS by usingthe nearest-neighbor criterion and, accordingly, the polygonalboundaries around BSs form a Poisson-Voronoi tessellation[3]. Using the theory of Poisson-Voronoi tessellation, whenno CoMP transmission is considered among BSs, the coverageprobability of a typical UE inside Poisson-Voronoi cells wasanalyzed in [4], and the performance of the worst-case usersat the vertices of Poisson-Voronoi cells was investigated in[5]. By using CoMP, the performance of the worst-case usersat the vertices of Poisson-Voronoi cells was studied in [6].The performance of a dynamic coordinated beamforming wascharacterized in [7], where each UE is assumed to commu-nicate only with the nearest BS in its CoMP cooperationset. A dynamic interference nulling strategy for small-cellnetworks was proposed in [8], and its average data rate wasanalyzed in [9]. More recently, the stochastic approach wasalso applied to study heterogeneous cellular networks. Forinstance, it was validated that the spatial distribution of macro-and micro-cell BSs can be modeled as the superpositionof two independent PPPs [10]. Further, concerning CoMPamong BSs, the coverage probabilities for a typical UE inheterogeneous downlink networks was studied in [11], [12], (cid:13)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the signal-to-interference ratio (SIR) meta distribution forboth the general UE and the worst-case UE under the Poissonmultiple-tier cellular networks was analyzed in [13]. Mostrecently, stochastic geometry was integrated with optimizationtheory for optimal design and performance analysis of cellularnetworks, see e.g., [14]–[17].When the theory of stochastic geometry is applied tocellular networks, the coverage area of a network is usuallytessellated by Poisson-Voronoi tessellation, where one UE isassociated with its nearest BS. However, since a typical cell ofPoisson-Voronoi tessellation is an irregular polygon with thenumber of edges varying from to , some basic featuresof a typical cell, for instance, the probability density function(PDF) of its area, is still unknown so far. This hinders theanalytical performance evaluation of cellular networks. Asthe dual diagram of Poisson-Voronoi tessellation, in contrast,Poisson-Delaunay triangulation has regular triangular cells,namely, a typical cell of Poisson-Delaunay triangulation isalways triangular. This regularity makes the applications ofPoisson-Delaunay triangulation more mathematically tractable[18]. In our recent work [19], Poisson-Delaunay triangulationwas used to model cellular networks and a novel CoMP trans-mission scheme was proposed. Unlike the conventional user-centric CoMP operations, such as [4]–[9], [12] where on-linesearching and feedback overhead are necessary to determinethe cooperation set of a UE, one of the key features of thesaid CoMP scheme [19] is that the set of cooperative BSspertaining to any UE is fixed and can be off-line determinedonce the geographic locations of BSs are known, which isfeasible in real-world cellular networks. As a companion workto [19], this paper investigates the network performance ofPoisson-Delaunay triangulation based CoMP transmission, interms of the coverage probability and the spectral efficiency.Specifically, this paper studies the performance of CoMPtransmission based on Poisson-Delaunay triangulation. Sincethe UEs at the vertices of conventional Poisson-Voronoi tes-sellation suffer the worst QoS, to characterize this QoS, atypical UE is intentionally chosen to be located at a vertex ofa Poisson-Voronoi cell, which exactly has equal distances fromthe three neighbouring BSs at the vertices of the dual Poisson-Delaunay triangle. On the other hand, two joint processingschemes, namely, joint transmission and dynamic point se-lection/muting, are employed at BSs. By using the theoryof stochastic geometry, the coverage probability and spectralefficiency of a typical UE are analytically derived. MonteCarlo simulation results are also provided and corroboratethe effectiveness of the proposed CoMP scheme and thecorresponding performance analysis.The rest of this paper is organized as follows. Section IIdescribes the system model and the principle of constructingthe cooperation set of a UE. Then, Sections III and IV aredevoted to JT and dynamic point selection/muting techniquesat BSs, respectively, where in each case the coverage prob-ability and spectral efficiency of a typical UE are explicitlyderived. Moreover, for comparison purposes, the performanceof transmission without CoMP is investigated. Simulationresults are presented and discussed in Section V. Finally,Section VI concludes the work. Notation : The operator E ( · ) means mathematical expecta- tion and round( · ) takes the nearest integer of a real number.The symbols k x k , k x k , and x H denote the ℓ -norm, ℓ -norm, and Hermitian transpose of vector x , respectively. Thefunction F − ( x ) represents the inverse function of F ( x ) ,and δ ( n ) refers to the Dirac delta function, with δ (0) = 1 and δ ( n ) = 0 if n = 0 . The symbol (cid:0) nm (cid:1) = n ! m ! ( n − m )! refers to binomial coefficient, with n ! being the factorialof a positive integer n . The Gamma, incomplete Gamma,and regularized incomplete Gamma functions are definedas Γ( a ) , R ∞ t a − e − t d t , Γ( a, x ) , R ∞ x t a − e − t d t ,and Q ( a, x ) , Γ( a, x ) / Γ( a ) , for all a > , respec-tively. The generalized hypergeometric function is definedas p F q ( a , · · · , a p ; b , · · · , b q ; x ) , ∞ X n =0 ( a ) n · · · ( a p ) n ( b ) n · · · ( b q ) n x n n ! ,with ( a ) n = a ( a + 1) · · · ( a + n − if n ≥ and ( a ) n = 1 if n = 0 . Notice that these special functions can be readilycomputed by using built-in functions in regular numericalsoftwares, such as Matlab and Mathematica.II. N ETWORK M ODEL
Figure 1 illustrates a cellular network where the BSs andthe UEs are denoted by the ‘ ◦ ’ and ‘ × ’ marks, respectively.The BSs are assumed to be distributed in a two-dimensional(2D) infinite plane as per a homogeneous PPP, denoted Φ ,with intensity λ . If each UE, uniformly distributed in theplane, is associated to its nearest BS in the sense of Euclideandistance, the resulting polygonal boundaries form a Poisson-Voronoi tessellation , as shown by the red dash lines in Fig. 1.On the other hand, the dual
Poisson-Delaunay triangulation is illustrated as the triangles with blue solid boundaries. Eachred polygon associated with a BS is known as a Poisson-Voronoi cell, while each blue triangle associated with threeBSs represents a Poisson-Delaunay cell. Once the locations ofBSs are known, the Poisson-Voronoi tessellation and Poisson-Delaunay triangulation are uniquely determined and they aredual Siamese twins [20].To serve UEs in triangular Poisson-Delaunay cells, each BSis assumed to be equipped with a large number of antennas,which enables multiple narrow directional beams as required.At each UE, a single receive antenna is assumed. The analysisin the rest of this paper confines to the single-antenna UEcase, but it can be extended to the multi-antenna UE case ina straightforward manner, for example, by treating each UEantenna as a separate UE or using maximum ratio combiningat the UE [21, Section 2.2].
A. Principles to Determine the Cooperation Set of a UE
Based on the geometric locations of BSs, the Poisson-Delaunay triangulation dual to the Poisson-Voronoi tessella-tion is uniquely determined and can be efficiently constructedby using, e.g., the radial sweep or divide-and-conquer algo-rithm [22, ch. 4]. Then, for each UE, the CoMP cooperationset can be readily determined. More specifically, as shown inFig. 5a of [19], if a UE is located inside a Poisson-Delaunaytriangular cell, the three BSs at the vertices of the triangleare chosen and form the CoMP cooperating set. On the otherhand, if a UE is exactly located on the edge of a triangle asshown in Fig. 5b of [19], there must be an adjacent triangle IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 3 Fig. 1. An illustrative cellular network modeled by the Poisson-Voronoitessellation (polygons with red dash boundaries) or by the dual Poisson-Delaunay triangulation (triangles with blue solid boundaries), with normalizedcoverage area of one squared kilometers. which shares the same edge and they both form a quadrilateral(the edge effect of the whole cellular network is ignored dueto its large coverage area). Among the four BSs at the verticesof the quadrilateral, the UE on the edge chooses the two BSsat both ends of the edge and a third BS among the remainingtwo opposite BSs which represents the one closer to the UE,so as to form the CoMP cooperating set.
B. Three Types of UEs
According to the Euclidean distances from the three BSsin a CoMP cooperation set determined as per the aboveprinciples, all UEs in the network can be classified into threetypes. Type I UEs are located at the centroids of triangularcells and each of them is equidistant from its three servingBSs. A Type II UE is equidistant from two BSs but has anotherdistance from the third BS. Type III UE has distinct distancesfrom its three serving BSs. For illustration purposes, UE ,UE and UE in Fig. 2 correspond to Type I, Type II andType III, respectively, with the cooperating BS set consistingof BS , BS and BS .Alternatively, by taking a closer look at Fig. 2, it is nothard to recognize that Type I UEs in the proposed Poisson-Delaunay cells are located at the vertices of the dual Poisson-Voronoi cells, while Type II UEs are on the edge of Poisson-Voronoi cells and Type III UEs are inside Poisson-Voronoicells. Clearly, Types I and II UEs are indeed the cell-edge usersin conventional cellular systems without CoMP operation,which suffer the worst QoS [5]. As well-known, by meansof CoMP operation, the performance of all UEs can besignificantly enhanced. To demonstrate the effectiveness ofthe Poisson-Delaunay triangulation based CoMP strategy, therest of this paper focuses on Type I UEs and analyzes itscoverage probability and spectral efficiency. The performanceof Types II and III UEs will be investigated in our future work. C. Received SIR at a Typical UE
By using the Slivnyak-Mecke theorem [3, p. 132], a typicalUE can be assumed to be located at the origin (0 , ∈ R UE UE BS BS BS Fig. 2. An illustration of three types of UEs, where UE has the samedistances from BS , BS and BS , UE has the same distances from BS and BS but another distance from BS , and UE has distinct distances fromthe three serving BSs. of the 2D plane, without loss of generality. The three BSsin the CoMP cooperating set, Φ = { A , B , C } , jointlytransmit signals to a typical UE whereas the BSs in the j th adjacent set, Φ j = { A j , B j , C j } , for all j = 1 , · · · , ∞ , aretreated as external interfering sources, where Φ ∪ Φ j | ∞ j =1 = Φ .As aforementioned, each BS in the network is equipped with M transmit antennas while each UE has a single antenna.Consequently, the received signal at a typical UE can beexpressed as y = X i ∈ Φ P i d − α i h Hi w i x + ∞ X j =1 X k ∈ Φ j P k d − α k, h Hk, w k x j + z, (1)where P i denotes the transmit power of the i th BS; d i isthe Euclidean distance from the i th BS to a typical UE; α > is the path-loss exponent; h i ∈ C M × stands forthe complex channel vector from the i th BS to a typicalUE and w i ∈ C M × is the precoder used at the i th BS; z means the additive white Gaussian noise at a typical UE,with zero mean and variance σ . The parameters P k and w k in the second term on the right-hand side of (1) denote thetransmit power and precoder at the k th interfering BS, for all k ∈ Φ j , while d k, and h k, refer to the distance and channelvector from the k th interfering BS to a typical UE locatedat the origin, respectively. Further, x denotes the desiredsignal that is jointly transmitted by the BSs in the CoMPcooperation set pertaining to a typical UE, while x j refersto the interfering signal transmitted by the BSs belonging tothe adjacent set Φ j , for all j = 1 , · · · , ∞ . x and x j areassumed to be independent and identically distributed (i.i.d.)random variables with zero mean and unit variance. Finally,we note that intra-cell interference is not accounted for in (1)since it can be effectively mitigated by using techniques suchas orthogonal frequency division multiple access (OFDMA). The application of Slyvniak’s theorem is not straightforward due to thelocation correlation between a typical UE and its serving BSs. However,for ease of mathematical tractability, we resort to Slyvniak’s theorem, andits effect on the accuracy of subsequent analyses are explicitly examined inSection V.
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Fig. 3. An illustrative application of CoMP transmission based on Poisson-Delaunay triangulation in two-tier heterogeneous networks, where ‘ ⊲ ’ and ‘ ◦ ’denote macro- and micro-cell BSs, respectively, and ‘ × ’ indicate the UEs. Since we consider the downlink transmission without powercontrol in a single-tier cellular network, the transmit powersof all BSs are assumed identical and normalized to unity.Also, full downlink channel state information (CSI) is assumedavailable at BSs interconnected via high-speed optical links.Accordingly, the channel-inverse precoder w i used by the i th BS is given by w i = h i k h i k . (2)Also, as the network performance under study is typicallyinterference-limited, the noise term in (1), i.e., z , is negli-gible. Thus, by substituting (2) into (1), we can express theinstantaneous received signal-to-interference ratio (SIR) at atypical UE as Γ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P i ∈ Φ d − α i k h i k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ P j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k ∈ Φ j d − α k, h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3)In the next section, the coverage probability and spectralefficiency of a typical UE in the case of joint trasnmisisonamong BSs in the cooperation set is investigated, followed bythe case of dynamic point selection/muting among BSs. Remark 1 (Extension from single-tier to multi-tier networks) . Although a single-tier cellular network is considered in thispaper, the idea of CoMP transmission based on Poisson-Delaunay triangulation can be readily applied to multi-tiernetworks. For instance, Fig. 3 shows a two-tier heterogenouscellular network where the macro- and micro-cell BSs aremodeled as homogeneous PPPs, denoted Φ and Φ ofintensity λ and λ , respectively. It is well-known that all BSsconsisting of macro- and micro-cell BSs can be modeled asthe superposition of Φ and Φ [10]. With the resulting PPPof intensity λ + λ , a Poisson-Delaunay triangulation can beobtained, as shown in Fig. 3. Clearly, the BSs at the verticesof a triangle may be either macro- or micro-cell BSs, withwhich each UE can be associated. III. J
OINT T RANSMISSION
In this section, joint transmission (JT) is applied at the threeBSs in the cooperation set pertaining to a typical UE, andthe coverage probability and spectral efficiency are used tocharacterize its performance. Mathematically, given an outagethreshold on the received SIR at a typical UE, say γ , thecoverage probability is defined as [4] P , − Pr { Γ ≤ γ } . (4)To calculate (4), next we address the distribution characteris-tics of Γ shown in (3). A. Received SIR at a Typical UE
By recalling Fig. 3, a UE at the vertex of a triangular cell,e.g., UE , is chosen as a typical point and is set to be theorigin in space, which implies that the Euclidean distancesbetween a typical UE and its serving BSs are identical, i.e., d i = d , for all i ∈ Φ . In such a case, (3) reduces to Γ = d − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P i ∈ Φ k h i k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ P j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k ∈ Φ j d − α k, h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d − α UI , (5)where U , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈ Φ k h i k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (6) I , ∞ X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Φ j d − α k, h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7)In the following, the distribution functions of d and U andthe Laplace transform of I are discussed in sequence.
1) The Distribution of the Distance d : Based on the theoryof Palm distribution, the PDF of distance d involved in (5) isderived in [23], and given by f d ( x ) = 2( λπ ) x exp (cid:0) − λπx (cid:1) . (8)
2) The Distribution of the Desired Signal U : Since eachelement of the channel vector h i ∈ C M × is a complexGaussian variable with zero mean and unit variance, it isevident that k h i k is of Nakagami distribution with PDF givenby f k h i k ( x ) = 2 x M − Γ( M ) exp (cid:0) − x (cid:1) . (9)Let an intermediate variable T , P i ∈ Φ k h i k = P i =1 k h i k , then T is clearly the sum of three independentNakagami random variables. In theory, an exact PDF of T isobtainable by using an approach similar to [24], yielding f T ( x ) = 8 √ π Γ(2 M )Γ ( M ) 2 M − exp (cid:0) − x (cid:1) × ∞ X n =0 Γ(2 M + n ) Γ(4 M + 2 n ) x M + n ) − Γ(2 M + n + ) Γ(6 M + 2 n ) Γ( n + 1) 2 n IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 5 × F (cid:18) M, M + 2 n ; 3 M + n + 12 , M + n ; 12 x (cid:19) . (10)Albeit accurate, (10) is too complex to be further processed.For ease of further proceeding, an approximate and accuratePDF of T is used in this paper. Specifically, by using a similarapproach to [25], an approximate PDF of T can be derivedand given by f T ( x ) ≈ m m x m − Γ( m )Ω m exp (cid:18) − mx Ω (cid:19) , (11)where Ω = E [ T ] , (12) m , round (cid:18) Ω E [ T ] − Ω (cid:19) . (13)To calculate the moments E [ T ] and E [ T ] required in (12)-(13), by recalling the formula of multinomial expansion, theexact n th -order moment of T can be written in terms of themoments of its three components, such that E [ T n ] = n X n =0 n X n =0 (cid:18) nn (cid:19)(cid:18) n n (cid:19) E (cid:2) k h k n − n (cid:3) × E (cid:2) k h k n − n (cid:3) E [ k h k n ] , (14)where E [ k h i k n ] , Z ∞ x n f k h i k ( x ) d x = Γ (cid:0) M + n (cid:1) Γ ( M ) . (15)Next, since U = | T | = T by noting that T is a non-negative real number, in light of (11) the PDF of U can beexpressed as f U ( x ) = (cid:0) m Ω (cid:1) m Γ( m ) x m − exp (cid:16) − m Ω x (cid:17) . (16)Meanwhile, the complementary cumulative density function(CCDF) of U is readily given by F U ( x ) = 1Γ( m ) Γ (cid:16) m, m Ω x (cid:17) = m − X k =0 k ! (cid:16) m Ω x (cid:17) k exp (cid:16) − m Ω x (cid:17) . (17) Remark 2 (The accuracy of the approximation given byEq. (11)) . For ease of analytical tractability, the value ofparameter m is rounded in (13) to its nearest integer. Forinstance, if M = 2 , after some tedious yet straightforwardcalculation, we get m = 5 . . The value m = 6 is taken inpractice for subsequent numerical calculations such that thefinite series expansion shown in (17) holds. Since the value of m is large enough, this approximation yields little deviationfrom the exact PDF given by (10) , as illustrated in Fig. 4.3) The Laplace Transform of the Interference I : Accord-ing to (7), the aggregate interference received at a typical UE T P D F The sum of three Nakagami-M random variables
SimulationEq. (10)Eq. (11)
M=2 M=4
Fig. 4. The accuracy of PDFs of T given by Eqs. (10) and (11), comparedwith simulation results. Interference P D F = 0.02, M = 2 Eq. (18)Eq. (19) 0 0.005 0.01 0.015 0.02
Interference P D F = 0.02, M = 4 Eq. (18)Eq. (19)
Fig. 5. PDFs of the interference calculated as per (18) and (19), with λ =0 . and M = 2 (left) or M = 4 (right). is given by I , ∞ X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Φ j d − α k, h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (18) ≈ ∞ X j =1 d − αj, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Φ j h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (19)where, for ease of mathematical tractability, the distancesfrom the three BSs in the interfering set Φ j to a typical UEare assumed identical and given by d j, in (19). Intuitivelyspeaking, this assumption is feasible in practice since all theBSs are supposed to be distributed in the infinite 2D planeand, as such, the three BSs in a CoMP cooperation set arerelatively close to each other. For illustration purposes, thePDFs of the interference calculated as per (18) and (19) areplotted in Fig. 5, where λ = 0 . and M = 2 (left-panel)or M = 4 (right-panel). As observed, the PDFs of (18) and(19) coincide with each other. Also, Fig. 5 shows that thePDF of the interference is surprisingly independent of M , i.e.,the number of transmit antennas at BSs, as mathematicallyjustified next. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS
Let ˆ h k , P k ∈ Φ j h Hk, h k / k h k k , since h k, is the channelvector from the k th BS to a typical UE whereas h k isthe precoder for a local UE, h k, and h k / k h k k are notmatched. By noting that h k / k h k k is isotropic, it is clear that h Hk, h k / k h k k is of normalized complex Gaussian distributionwith unit mean and, hence, | ˆ h k | is of exponential distributionwith mean µ = 3 , independent of the number of transmitantennas at BSs. As a result, substituting | ˆ h k | into (19), theLaplace transform of the interference I is given by L I ( s ) ≈ E ˆΦ , ˆ h exp − s X k ∈ ˆΦ d − αk, | ˆ h k | (20) = E ˆΦ Y k ∈ ˆΦ E ˆ h h exp (cid:16) − sd − αk, | ˆ h k | (cid:17)i (21) = E ˆΦ Y j ∈ ˆΦ
11 + sµd − αj, (22) = exp (cid:18) − λ ′ π Z ∞ d sµx − α +1 sµx − α d x (cid:19) (23) = exp µλ ′ πsd − α − α F " − α − α ; − µd − α s , (24)where ˆΦ is a thinning process of Φ \ Φ with intensity λ ′ = λ/ , and where (21) follows from the fact that ˆ h k are i.i.d.for all k ∈ ˆΦ , (22) is due to the fact that | ˆ h k | ∼ exp( µ ) , and(23) is based on the probability generating functional of theunderlying PPP [26].With the obtained PDFs of d and U shown in (8) and (16),respectively, and the Laplace transform of I given by (24),the coverage probability of a typical UE can be analyzed. B. Coverage Probability
Now, we are in a position to formalize the coverage prob-ability of a typical UE in the following theorem.
Theorem 1 ( ℓ -Toeplitz matrix form of the coverage prob-ability) . Given that joint transmission is applied to BSs inthe cooperation set of a typical UE, with a prescribed outagethreshold γ , the coverage probability of a typical UE can becalculated as P ( γ, λ, α ) = Z x> f d ( x ) k exp( Q ( d )) k d x, (25) where f d ( x ) is shown in (8) , and Q ( d ) is an m × m lowertriangular Toeplitz matrix, expressed as Q ( d ) = q q q q q q ... ... . . . q m − · · · q q q , (26) with the entry q n given by q n = λ ′ πd δ ( n ) − λ ′ πd − nα (cid:18) mµ (cid:19) n γ n × F " n + 1 n − α n + 1 − α ; − mµγ . (27) Proof:
See Appendix A.As an application of Theorem 1, we consider the specialcase of single transmit antenna at each BS, i.e., M = 1 . Insuch a case, we get a simple expression as summarized in thefollowing corollary. Corollary 1.
In the case of M = 1 , the coverage probabilitygiven by (25) reduces to P ( γ, α ) = 1 (cid:16) V ( γ )3 (cid:17) − mγV ( γ )3Ω (cid:16) V ( γ )3 (cid:17) + ( mγ ) V ( γ ) (cid:16) V ( γ )3 (cid:17) + ( mγ ) V ( γ )3Ω (cid:16) V ( γ )3 (cid:17) , (28) where V ( γ ) , µmγ ( α − F " − α − α ; − µmγ Ω , (29) V ( γ ) , µ − α F " − α − α ; − µmγ Ω + µ mγ ( α − F " − α − α ; − µmγ Ω , (30) V ( γ ) , µ α − F " − α − α ; − µmγ Ω − µ mγ (3 α − F " − α − α ; − µmγ Ω . (31) Proof:
See Appendix B.Notice that (28) demonstrates that the coverage probabilityin the case of M = 1 is independent of the intensity of the BSs( λ ). By using a similar approach as described above, it is nothard to show that this conclusion holds as well even if M > .To sum up, this means that increasing the number of BSs willnot benefit the coverage probability. An intuitive interpretationof this conclusion is that an increase in desired signal poweris exactly counter-balanced by that in unwanted interferencepower. This conclusion agrees with empirical observations ininterference-limited urban networks [4]. C. Spectral Efficiency
By using a similar method to [9, Eq. (12)], the spectralefficiency of the JT scheme can be approximated as τ ( α ) ≈ Z x> E ln d − α E [ U ] P k ∈ ˆΦ d − αk, E (cid:20)(cid:12)(cid:12)(cid:12) ˆ h k (cid:12)(cid:12)(cid:12) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) d f d ( x )d x (32) = Z x> E ln Ω µ P k ∈ ˆΦ d − αk, d α (cid:12)(cid:12)(cid:12)(cid:12) d f d ( x )d x IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 7 = Z s> Z x> s (cid:20) − exp (cid:18) − s Ω µ (cid:19)(cid:21) exp (cid:8) − λ ′ πd × (cid:20) exp( − s ) + s α γ (cid:18) − α , s (cid:19)(cid:21) − (cid:27) d s f d ( x ) d x (33) = 9 Z s> s (cid:18) − exp (cid:18) − s Ω µ (cid:19)(cid:19) × (cid:20) exp( − s ) + s α γ (cid:18) − α , s (cid:19) + 2 (cid:21) − d s, (34)where (33) is derived by using the lemma in [27], and (34)is obtained by substituting (8) into (33) as well as performingsome basic calculus.For comparison purposes, the exact expression for thespectral efficiency is derived as τ ( α ) , E (ln (1 + Γ ))= Z x> Z t> P (cid:20) ln (cid:18) d − α UI (cid:19) > t (cid:21) d t f d ( x )d x = Z x> Z t> m − X k =0 k ! (cid:16) − m Ω d α (exp( t ) − (cid:17) k × ∂ k L I ( s ) ∂s k (cid:12)(cid:12)(cid:12)(cid:12) s = m Ω d α ( e t − d t f d ( x )d x (35) = Z x> Z t> k exp( Q ( d )) k (cid:12)(cid:12)(cid:12)(cid:12) γ = e t − d t f d ( x )d x. (36)In the special case of M = 1 , we have Ω = 3 accordingto (12). Then, assuming the path-loss exponent α = 4 ,the spectral efficiency of the proposed JT scheme can benumerically calculated as per (36), yielding τ ( α = 4) = 2 . nats/sec/Hz . (37) Remark 3 (Performance analysis of Types II and III UEs) . Notice that the performance analysis developed in this sectionfor Type I UEs exploits the fact that the typical UE isequidistant from three serving BSs, as shown in Eq. (5) . Asfar as Types II and III UEs are concerned, the typical UEhas different distances to three serving BSs and, thus, thereceived SIR given by Eq. (3) cannot be reduced to Eq. (5) ,and the subsequent analyses of coverage probability andspectral efficiency cannot be repeated in a similar way. As analternative, some advanced technique needs to be developed toattain the distribution function of the desired signal expressedby the numerator of Eq. (3) . This is beyond the scope of thispaper and will be tackled in our future work.
IV. D
YNAMIC P OINT S ELECTION /M UTING
Although the JT scheme described above benefits loweroutage probability, it requires all BSs in the cooperation set tosimultaneously serve a target UE, leading to higher hardwareand coordination costs. To get a tradeoff between the highercosts and lower outage probability, the technique of dynamicpoint selection/muting can be applied [13], [28]. Specifically,not all BSs in the cooperation set but only the one with thebest channel quality (i.e., the product of the large-scale pathloss and small-scale fading) is chosen to serve the target UE while the remaining BSs keep silent. This scheme is called optimal point selection (OPS) in the sequel.
A. Optimal Point Selection
This subsection derives the coverage probability and spec-tral efficiency of the OPS scheme in sequence. To start with,the aggregate interference at a typical UE, denoted I , comesfrom all BSs in the complement set Φ \ Φ , and is given by I = X k ∈ Φ \ Φ d − αk, (cid:12)(cid:12)(cid:12)(cid:12) h Hk, h k k h k k (cid:12)(cid:12)(cid:12)(cid:12) = X k ∈ Φ \ Φ d − αk, g k , (38)where g k , (cid:12)(cid:12)(cid:12) h Hk, h k k h k k (cid:12)(cid:12)(cid:12) is of exponential distribution withunit mean. Then, the Laplace transform of I can be derivedas L I ( s ) = E Φ ,g j exp − s X j ∈ Φ \ Φ g j d − αj, (39) = exp λπsd − α − α F " − α − α ; − d − α s . (40)
1) Coverage Probability:
By jointly applying the theoriesof order statistics and stochastic geometry, the coverage prob-ability of a typical UE in case the OPS is applied, can beformalized as follows.
Theorem 2.
Given that the optimal point selection techniqueis applied to BSs in the cooperation set of a typical UE, witha prescribed outage threshold γ , the coverage probability ofa typical UE can be calculated as P ( γ, λ, α ) = Z x> f d ( x ) { k exp( Q ′ ( d )) k − E I (cid:2) Q ( M, γd α I | d ) − Q ( M, γd α I | d ) (cid:3)(cid:9) d x, (41) where Q ′ ( d ) is an M × M lower triangular Toeplitz matrix,expressed as Q ′ ( d ) = q ′ q ′ q ′ q ′ q ′ q ′ ... ... . . . q ′ M − · · · q ′ q ′ q ′ , (42) with the entry q ′ n given by q ′ n = λπd δ ( n ) − λπd − nα γ n F " n + 1 n − α n + 1 − α ; − γ , (43) and where E I [ Q n ( M, γd α I | d )]= M − X k =0 k ! ( − γd α ) k ∂ k L I ( s ) ∂s k ! n (cid:12)(cid:12)(cid:12)(cid:12) s = γd α , n = 2 , . (44) Proof:
See Appendix C.As an application of Theorem 2, we consider the specialcase with M = 1 , i.e., there is only a single transmit antenna IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS at each BS. Recalling that Q (1 , x ) = exp( − x ) , (94) inAppendix C reduces to Pr [
G > γd α I | d ] = 3 L I ( γd α ) − L I (2 γd α )+ L I (3 γd α ) . (45)On the other hand, by virtue of (40), performing some basiccalculus yields the coverage probability: P ( γ, λ, α ) = 3 (1 + V ( γ )) − − V (2 γ )) − + (1 + V (3 γ )) − , (46)where V ( γ ) , γα − F " − α − α ; − γ . (47)Like (28), (46) also demonstrates that the coverage probabil-ity in the case of dynamic point selection/muting with M = 1 is independent of the intensity of BSs (i.e., λ ). By using asimilar approach as above, this conclusion can be shown tohold as well even if M > .
2) Spectral Efficiency:
By using a similar approach to(32)-(34), the spectral efficiency of the OPS scheme can beapproximated as τ ( α ) ≈ Z s> s − exp ( − sN ( M )) h exp( − s ) + s α γ (cid:0) − α , s (cid:1)i d s, (48)where N ( M ) , R u x ( u )d u , with x ( u ) = F − g ′ i ( u ) and F g ′ i ( x ) given by (91).By definition, the exact spectral efficiency of the OPSscheme can be derived and given by τ ( α ) = Z t> Z x> f d ( x ) { k exp( Q ′ ( d )) k − E I (cid:2) Q ( M, γd α I | d ) − Q ( M, γd α I | d ) (cid:3)(cid:9) d x d t. (49)where γ = e t − . In the case of M = 1 and α = 4 , thespectral efficiency of the OPS scheme, numerically computedas per (49), is τ ( α = 4) = 1 . nats/sec/Hz . (50)For comparison purposes, next a random point selection(RPS) scheme is discussed, where one BS in the cooperationset is randomly chosen to serve the target UE while the otherBSs serve other UEs at the same time. Compared with the OPSscheme descried above, RPS has higher resource utilization,yet with lower spectral efficiency, as shown below. B. Random Point Selection
In this case, a typical UE is randomly served by only oneBS without CoMP. Without loss of generality, the serving BSis denoted A . Unlike the preceding JT and OPS cases, theaggregate interference at a typical UE, denoted I , comesfrom all BSs in the complement set of Φ , i.e., Φ \ { A } .Mathematically speaking, we have I = X k ∈ Φ \{ A } d − αk, g k , (51) where g k is as defined right after (38). The Laplace transformof I can be readily computed as L I ( s ) = 1 (cid:0) sd α (cid:1) E Φ Y k ∈ Φ \{ A } E g k h exp (cid:16) − sg k d − αk, (cid:17)i = (cid:16) sd α (cid:17) − exp λπsd − α − α F " − α − α ; − sd α . (52)
1) Coverage Probability:
By using the Alzer’s lemma in[29], the coverage probability of a typical UE in the case ofRPS can be derived, as formalized below.
Theorem 3.
The coverage probability of a typical UE in thecase of random point selection is upper bounded by P ( γ, λ, α ) = Z x> f d ( x ) M − X k =0 ( − γd α ) k k ! ∂ k L I ( s ) ∂s k (cid:12)(cid:12)(cid:12)(cid:12) s = γd α d x (53) ≤ M X k =1 ( − k +1 (cid:18) Mk (cid:19) Z x> f d ( x ) L I ( kβγd α ) d x, (54) where β = ( M !) − /M . In particular, if single transmit antenna is deployed at eachBS ( M = 1 ) and Rayleigh fading is assumed, (53) reduces to P ( γ, λ, α ) = Z x> f d ( x ) L I ( γd α ) d x = [(1 + γ )(1 + V ( γ ))] − , (55)where V is previously defined in (47). Like (28), (55) alsodemonstrates that the coverage probability of a typical UE isindependent of the intensity of BSs, i.e., λ .
2) Spectral Efficiency:
Using a similar approach as inSection III-C, the spectral efficiency of the RPS scheme canbe approximated as follows τ ( α ) ≈ Z s> s (1 − exp ( − sM )) exp( − s ) h exp( − s ) + s α γ (cid:0) − α , s (cid:1)i d s. (56)On the other hand, by definition, the exact spectral efficiencyof the RPS scheme can be computed as τ ( α ) = Z t> Z x> f d ( x ) M − X k =0 ( − (exp( t ) − d α ) k k ! × ∂ k L I ( s ) ∂s k (cid:12)(cid:12)(cid:12)(cid:12) s =( e t − d α d x d t. (57)In the case of M = 1 and α = 4 , the spectral efficiency ofthe RPS scheme, numerically computed according to (57), is τ ( α = 4) = 0 . nats/sec/Hz . (58)To sum up, with the obtained (37), (50), and (58), cor-responding to the spectral efficiencies of the JT, OPS, andRPS schemes, respectively, it is obvious that the JT schemeattains the highest spectral efficiency whereas the RPS schemeyields the lowest. This result is not surprising since increasingcoordination costs benefit higher spectral efficiency. IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 9 -10 -5 0 5 10 15 SIR threshold (in dB) C ov e r a g e p r ob a b ilit y JT: Sim.JT: Eq. (28)OPS: Sim.OPS: Eq. (46)RPS: Sim.RPS: Eq. (55)-10 -5 0 5 10 15
SIR threshold (in dB) C ov e r a g e p r ob a b ilit y JT: Sim.JT: Eq. (25)OPS: Sim.OPS: Eq. (41)RPS: Sim.RPS: Eq. (54)
Fig. 6. Coverage probabilities versus SIR threshold, with three differenttransmission schemes (JT: joint transmission, OPS: optimal point selection,and RPS: random point selection).
V. S
IMULATION R ESULTS AND D ISCUSSIONS
In this section, numerical results computed as per thepreviously obtained analytical expressions are presented anddiscussed, in comparison with extensive Monte-Carlo simula-tion results. In the simulation experiments, a cellular networkwith a coverage area of × squared meters is considered,where the path-loss exponent and BS intensity are set to α = 4 and λ = 0 . , respectively. The channel fading from eachtransmit antenna at BSs to a typical UE is subject to Rayleighfading. Moreover, for the case with single transmit antenna ateach BS, i.e., M = 1 , according to (12)-(15) and after somealgebraic calculations, we have Ω = 7 . and m = 3 . Similarly, Ω = 16 . and m = 6 in the case of M = 2 . A. Coverage Probability
Figure 6 shows the coverage probability of a typical UEversus the outage threshold γ , where the top panel correspondsto the case of M = 1 while the bottom panel to the case of M = 2 . For comparison purposes, the coverage probabilitiesof three different transmission schemes, namely, JT, OPS, andRPS, are plotted. For a particular outage threshold value, it isseen that the JT has the highest coverage probability whereasthe RPS gets the lowest, as expected. On the other hand,taking the JT for instance, it is observed that the numericalresults computed as per (28) are slightly smaller than thecorresponding simulation results. Similar observations can bemade in the OPS and RPS cases, as shown in Fig. 6. In otherwords, the coverage probability of a typical UE is slightlyunderestimated in the preceding analyses. The reason behindthis interesting observation is far more complex than whatseems at first sight. Specifically, although (11) is an approx-imate PDF, it is accurate and has little effect on the derivedcoverage probability as previously discussed in Remark 2 atthe end of Section III-A2. The approximation given by (19)is also accurate, as previously illustrated in Fig. 5.In fact, this underestimation is introduced by the assumptionof independence between BSs and a typical UE. More concreteevidence is provided below. By recalling the Slivnyak-Mecke S defined in Eq. (59) P D F Sim: at a typical UESim: at the originAnalysis: at the origin
Fig. 7. PDFs of the aggregate signal defined in Eq. (59) at a typical UE andat the origin. theorem in stochastic geometry [3, p. 132], a typical UE canbe assumed to be located at the origin (0 , ∈ R , withoutloss of generality. This implies that the location of a typicalUE is independent of the locations of BSs. However, as far asthe worst-case UEs under study in this paper is concerned, atypical UE has the same distance from its three nearest BSs.Clearly, the location of a typical UE in our work is dependentupon the locations of BSs. For better clarity, let us take a closelook at the aggregate signal power at the origin, defined as S , X k ∈ Φ d − αk, g k , (59)where g k is defined immediately after (38). It is well-knownthat S is subject to a skewed stable distribution [30], whoseprobability densities exist but, with a few exceptions, they arenot known in closed form. In particular, if α = 4 , the PDF of S is of the L´evy type, explicitly expressed as [31] f S ( x ) = λ (cid:16) πx (cid:17) exp (cid:18) − λ π x (cid:19) . (60)Figure 7 shows the simulation results of the aggregate signalpower at the origin and at a typical UE which has the samedistance to its nearest three BSs, compared with the numericalresults computed with (60), given the BS intensity λ = 0 . . Itis seen that the former simulation results accord fully with thenumerical results whereas the latter simulation results deviatefrom the numerical ones significantly. In particular, the PDF ofthe aggregate signal power at a typical UE has higher kurtosisthan that of the power at the origin. This means that the formerhas infrequent extreme deviations or, equivalently, this reflectsthe dependence of different signals transmitted from BSs to atypical UE.As far as the JT scheme under study is concerned, theinterference can be expressed as S , X k ∈ Φ \ Φ d − αk, g k . (61)Figure 8 shows the simulated PDFs of the powers of S atthe origin and at a typical UE which has the same distancefrom its three nearest BSs, respectively (exact PDF of S S defined in Eq. (61) P D F At the originAt a typical UE
Fig. 8. PDFs of the interference defined in Eq. (61) at the origin and at atypical UE. is not mathematically tractable, to the best of the authors’knowledge). Clearly, the difference between these two PDFcurves is much smaller than that in Fig. 7. As a consequence,we may conclude that the difference between the powers of S received at the origin and at a typical UE under study in thispaper is mainly caused by the signals from the nearest threeBSs. This is indeed the reason why the obtained analyticalresults has led to a slight underestimation of the coverageprobability, as shown in Fig. 6. B. Spectral Efficiency
Figure 9 compares the spectral efficiencies of the JT, OPSand RPS schemes. As expected, the spectral efficiency of allschemes increases with larger number of transmit antennasat BS (i.e., M ). For a fixed M , the JT scheme has thehighest spectral efficiency while the RPS gets the lowest, sincethe former requires higher cooperation and hardware costs.Importantly, it is observed from Fig. 9 that the numericalresults pertaining to the JT and RPS schemes agree very wellwith the simulation results, whereas those of the OPS schemeunderestimate the simulation ones. This observation impliesthat the dependence discussed in the previous subsection haslittle effect on the accurate analysis of the spectral efficienciesof the JT and RPS schemes. Also, it is seen that the approxi-mated numerical results match well with the exact analyticalresults, which illustrates the effectiveness of the analysis. C. Poisson-Delaunay Triangulation vs. Poisson-Voronoi Tes-sellation1) Comparison with Poisson-Voronoi tessellation withoutCoMP:
To illustrate the effectiveness of the Poisson-Delaunaytriangulation based JT scheme, besides the performance analy-sis of Type I UEs, this subsection shows the simulation resultspertaining to Types II and III UEs defined in Section II-B, incomparison with their counterparts in the conventional cellularsystems based on Poisson-Voronoi tessellation.Figure 10 illustrates the coverage probabilities of threetypes of UEs, compared with their counterparts in the dualPoisson-Voronoi tessellation. It is seen that, in the conven-tional Poisson-Voronoi scenario, the performance of Type I S p ec t r a l e ff i c i e n c y ( n a t s / s ec / H z ) Sim.AnalysisApprox.
RPS OPS JT
Fig. 9. Spectral efficiency versus the number of transmit antennas at eachBS (i.e., M ) (JT: (36) vs. (34); OPS: (49) vs. (48); RPS: (57) vs. (56)). -10 -5 0 5 10 15 20 SIR threshold (in dB) C ov e r a g e p r ob a b ilit y Delaunay: Type III UEsDelaunay: Type II UEsDelaunay: Type I UEsVoronoi: Type III UEsVoronoi: Type II UEsVoronoi: Type I UEs
Fig. 10. Coverage probabilities of different types of UEs versus the SIRthreshold, with M = 1 . UEs is the worst and Type II UEs perform worse thanType III UEs. The reason is that they have shorter and shorterdistances from the serving BS by noting that Type I UEsare at the vertices of each Poisson-Voronoi cell, Type IIUEs are on the edge of each cell, and Type III UEs areinside each cell (cf. Fig. 2). In the scenario of Poisson-Delaunay triangulation, similar observations can be made,namely, Type I UEs perform the worst since the strengthof desired signals received at Type I UEs is the lowestwhereas the strengths of interfering signals on them are almostidentical. More specifically, if we consider only the path-losseffect, by recalling the inequality of arithmetic and geometricmeans, we have d − α + d − α + d − α ≥ p ( d d d ) − α , wherethe equality holds if and only if d = d = d (correspondingto Type I UEs). Finally, Fig. 10 shows that all UEs inthe scenario of Poisson-Delaunay triangulation significantlyoutperform their counterparts in the conventional Poisson-Voronoi scenario, indicating the effectiveness of the proposedtriangulation scheme. Clearly, this performance gain comesfrom the cooperation of BSs. Next, we will demonstrate theinterference mitigation capability of our triangulation scheme.
2) Comparison with Poisson-Voronoi tessellation with dy-namic cooperation set:
For fairness, we compare the triangu-lation scheme with the conventional Poisson-Voronoi scheme IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 11 with three dynamic cooperating BSs, in terms of spectralefficiency. Since the three serving BSs of a Type I UE inthe triangulation scheme are exactly the nearest ones, thedesired signal powers under these two schemes are identical,say d . Then, by recalling (16), the MGF of the desired signalconditioned on d can be computed as M S = (cid:18) m d − α z (cid:19) − m . (62)Now, we compare their interference powers I and I givenby (7) and (38), respectively. By inserting µ = 3 into (22),for a given d , the MGF of I can be readily shown as M I = E ˆΦ Y k ∈ ˆΦ (cid:16) zd − αk, (cid:17) − . (63)Likewise, the MGF of I given by (38) can be derived as M I = E exp − z X k ∈ Φ \ Φ d − αk, g k ≈ E exp − z ∞ X j =1 d − αj, X k ∈ Φ j g k (64) = E ˆΦ Y k ∈ ˆΦ E h exp (cid:16) − zd − αk, ˆ g k (cid:17)i (65) = E ˆΦ Y k ∈ ˆΦ (cid:16) zd − αk, (cid:17) − , (66)where ˆΦ is a thinning process of Φ \ Φ with intensity λ ′ = λ/ , and ˆ g k , P k ∈ Φ j g k , which follows a Gammadistribution with shape parameter and unit scale factor.To derive the spectral efficiency, we exploit the lemmareported in [27], which reads ln (cid:18) XY (cid:19) = Z z> z (1 − exp ( − zX )) exp( − zY )d z. (67)Then, the spectral efficiency can be readily computed as R = Z x> E (cid:20) ln (cid:18) SI (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) d (cid:21) f d ( x ) d x (68) = Z x> f d ( x ) Z z> z (1 − M S ) M I d z d x, (69)where M S and M I denote the MGFs of the desired signalpower S and interference power I , respectively. Comparing(63) with (66), since (cid:16) zd − αk, (cid:17) − > (cid:16) zd − αk, (cid:17) − forall z > and d k, > , we know that M I > M I .Therefore, the spectral efficiency of the proposed scheme,computed by substituting (62) and (63) into (69), is larger thanthat of the Poisson-Voronoi scheme, computed by substituting(62) and (66) into (69). Simulation results shown in Fig. 11corroborates this analysis. For completeness of presentation,Fig. 11 also illustrates the spectral efficiency of the other twotypes of UEs. It is observed that the spectral efficiency ofthe Poisson-Voronoi scheme is slightly higher than that of thePoisson-Delaunay scheme, for either Type II UEs or Type III S p ec t r a l e ff i c i e n c y ( n a t s / s ec / H z ) DelaunayVoronoi
Type III UEs Type I UEsType II UEs
Fig. 11. Spectral efficiency for different types of UEs simulated underPoisson-Delaunay triangulation based JT scheme and Poisson-Voronoi tes-sellation with three dynamic cooperating BSs.
UEs. This is because the former can always choose the threenearest BSs through exhaustive searching.VI. C
ONCLUDING R EMARKS
This paper analyzed the performance of a novel coordinatedmulti-point (CoMP) transmission scheme based on Poisson-Delaunay triangulation. Using the theory of stochastic geom-etry, the coverage probabilities and the spectral efficiencies ofthe worst-case UEs pertaining to three different transmissionschemes, namely, joint transmission, optimal point selection,and random point selection, were analytically derived andcompared. Numerical results demonstrated the effectiveness ofthe performance analyses and the superiority of the proposedapproach. Thanks to the simplicity of cooperation strategy andsuperiority of network performance, the proposed transmissionscheme is promising in the emerging small-cell networksand/or heterogeneous networks, where CoMP transmission isindispensable for higher resource utilization.A
PPENDIX AP ROOF OF T HEOREM d , the coverage probability can beexplicitly computed as P ( γ, λ, α ) = E [Pr[Γ > γ | d ]]= Z x> Pr (cid:20) d − α UI > γ | d (cid:21) f d ( x ) d x = Z x> E I [Pr [ U > γd α I | d, I ]] f d ( x ) d x = Z x> f d ( x ) m − X k =0 k ! (cid:18) mγd α (cid:19) k × E I (cid:20) I k exp (cid:18) − mγd α I (cid:19)(cid:21) d x = Z x> f d ( x ) m − X k =0 k ! (cid:18) − mγd α Ω (cid:19) k × ∂ k L I ( s ) ∂s k (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α d x, (70) where (70) follows the relationship between the moments andthe Laplace transform of a RV.Then, by virtue of (21), the L I ( s ) used in (70) can beexpressed as L I ( s ) ≈ E ˆΦ Y k ∈ ˆΦ E ˆ h h exp (cid:16) − sd − αk, | ˆ h k | (cid:17)i = exp (cid:26) − λ ′ π Z ∞ d (cid:16) − E ˆ h h exp (cid:16) − s ˆ hv − α (cid:17)i v d v (cid:17)(cid:27) = exp [ η ( s )] , (71)where η ( s ) , − λ ′ π Z ∞ d (cid:16) − E ˆ h h exp (cid:16) − s ˆ hv − α (cid:17)i v d v (cid:17) = λ ′ πd + 2 α λ ′ πs α E ˆ h (cid:20) ˆ h α γ (cid:18) − α , sd − α ˆ h (cid:19)(cid:21) (72) = λ ′ πd − λ ′ πd E ˆ h " F " − α − α ; − sd − α ˆ h (73) = λ ′ πd − λ ′ πd F " − α − α ; − sµd − α , (74)where [32, Eq. (8.351)] is exploited to reach (73), and (74)follows the fact ˆ h ∼ exp( µ ) .By using a similar method to that in [33], [34], the recursiverelations between the derivatives of L I ( s ) can be attained,based on which a compact Toeplitz matrix expression for thecoverage probability is finally derived. Specifically, let q n , ( − s ) n n ! L nI ( s ) . Then, it is clear that q , L I ( s ) | s = mγd α = exp ( λ ′ πd − λ ′ πd F " − α − α ; − mµγ = exp( t ) , (75)where t , λ ′ πd − λ ′ πd F h − α − α ; − mµγ i . Next, com-bining (71) with (75) yields L (1) I ( s ) = η (1) ( s ) L I ( s ) . After-wards, by recursion, for all n ≥ , we have L ( n ) I ( s ) = d n − d s n − L (1) I ( s ) = n − X i =0 (cid:18) n − i (cid:19) η n − i ( s ) L ( i ) I ( s ) , (76)followed by ( − s ) n n ! L ( n ) I ( s ) = n − X i =0 n − in ( − s ) n − i ( n − i )! η ( n − i ) ( s ) ( − s ) i i ! L ( i ) I ( s ) . (77)Let q n = ( − s ) n n ! L ( n ) I ( s ) . Then, for all n ≥ , (77) implies that q n = n − X i =0 n − in t n − i q i , (78)wher t n = ( − s ) n n ! η ( n ) ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α = ( − s ) n n ! (cid:18) λ ′ πd − λ ′ πd F " − α − α ; − sµd − α ( n ) (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α = − λ ′ πd − nα ) (cid:18) mµ (cid:19) n γ n × F " n + 1 n − α n + 1 − α ; − mµγ . (79)Combining (75) with (79) yields the intended (27).Next, to explicitly express q n , we define two power seriesas follows: T ( z ) , ∞ X n =0 t n z n , Q ( z ) , ∞ X n =0 q n z n . (80)By taking the first-order derivative of T ( z ) and Q ( z ) , we have T (1) ( z ) = ∞ X n =0 ( n + 1) t n +1 z n , Q (1) ( z ) = ∞ X n =0 nq n z n − . (81)Combining (78), (80) and (81) yields T (1) ( z ) Q ( z ) = ∞ X n =0 n − X i =0 ( n − i ) t n − i z n − q i z i − = ∞ X n =0 nq n z z − = Q (1) ( z ) , (82)which implies that Q ( z ) = a exp( T ( z )) . (83)By recalling (75), q = exp ( t ) leads to a = 1 and,consequently, the coverage probability given by (25) can beexplicitly computed as P ( γ, λ, α ) = E d " m − X n =0 q n = E d " m − X n =0 n ! Q ( n ) ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z =0 = E d " m − X n =0 n ! d n d z n exp( T ( z )) (cid:12)(cid:12)(cid:12)(cid:12) z =0 . (84)Finally, by using a similar technique to that in [34], thefirst m − coefficients of the power series exp( Q ( z )) canbe derived and expressed as the first column of the matrixexponential exp ( Q ) , which completes the proof.A PPENDIX BP ROOF OF C OROLLARY M = 1 , it is clear that k h i k is Rayleigh distributed and, as per (13), we get m = 3 .Then, the coverage probability given by (70) reduces to P ( γ, λ, α ) = Z x> f d ( x ) (cid:20) L I ( s ) − mγd α Ω ∂L I ( s ) ∂s + ( mγd α ) ∂ L I ( s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α d x. (85)Next, we calculate the three terms in the square brackets onthe right-hand side of (85). To start with, according to (24), itis straightforward that L I (cid:18) mγd α (cid:19) IA et al. : COORDINATED MULTI-POINT TRANSMISSION: A POISSON-DELAUNAY TRIANGULATION BASED APPROACH 13 = exp µλ ′ πmγd (2 − α )Ω F " − α − α ; − µmγ Ω = exp (cid:0) − λ ′ πd V (cid:1) , (86)where V is shown in (29). Then, by recalling the first-order derivative of Gaussian hypergeometric function [35, Eq.(7.2.1.10)], we have ∂L I ( s ) ∂s = L I ( s ) ( µλ ′ πd − α − α F " − α − α ; − µd − α s + µ λ ′ πd − α sα − F " − α − α ; − µd − α s . (87)Substituting s = mγd α into (87) yields ∂L I ( s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α = L I (cid:18) mγd α (cid:19) " µλ ′ πd − α − α F " − α − α ; − µmγ Ω + µ λ ′ π d − α mγ ( α − F " − α − α ; − µmγ Ω = λ ′ πd − α V L I (cid:18) mγd α (cid:19) , (88)where V is shown in (30). By using a similar approach asabove, we have ∂ L I ( s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s = mγd α = (cid:0) ( λ ′ π ) d − α V + λ ′ πd − α V (cid:1) L I (cid:18) mγd α (cid:19) , (89)where V is expressed as (31). Finally, substituting Eqs. (8),(86), (88), and (89) into (85), and performing some calculus,yields the intended (28).A PPENDIX CP ROOF OF T HEOREM G , max { g ′ , g ′ , g ′ } . Since g ′ i , k h i k , i = 1 , , ,are i.i.d. Gamma random variables, with parent PDF and CDFgiven by f g ′ i ( x ) = 1Γ( M ) x M − exp( − x ) , (90)and F g ′ i ( x ) = 1 − Γ( M, x )Γ( M ) = 1 − Q ( M, x ) , (91)respectively, then by recalling the theory of order statistics,the CCDF of G is expressed as Pr(
G > x ) = 1 − (cid:0) F g ′ i ( x ) (cid:1) . (92)Consequently, the coverage probability can be computed as Pr( γ, λ, α ) = Z x> Pr (cid:20) r − α GI > γ | d (cid:21) f d ( x ) d x = Z x> Pr [
G > γd α I | d ] f d ( x ) d x. (93) By virtue of (91) and (92), the conditional CCDF of G neededin (93) can be expressed as Pr [
G > γd α I | d ] = E I [3 Q ( M, γd α I | d ) − Q ( M, γd α I | d ) + Q ( M, γd α I | d ) (cid:3) , (94)where E I [ Q ( M, γd α I | d )] = M − X k =0 k ! ( − γd α ) k ∂ k L I ( s ) ∂s k (cid:12)(cid:12)(cid:12)(cid:12) s = γd α . (95)By using a similar approach to the proof of Theorem 1, theconditional expectation of Q ( M, γd α I ) with respect to I given d can be explicitly expressed as E I [ Q ( M, γd α I | d )] = k exp( Q ′ ( d )) k , (96)where Q ′ is an M × M lower triangular Toeplitz matrix withnone-zero entries shown in (43). Finally, substituting (94) into(93) yields the desired (41).R IEEE Commun. Surveys & Tuts , vol. 19, no. 1, pp. 167–203, First quarter2017.[3] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke,
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Yan Li received the B.S. degree in Electronic Infor-mation Engineering from Hunan Normal University,Changsha, China, in 2013, and the M.S. degree inElectronics and Communication Engineering fromSun Yat-sen University, Guangzhou, China, in 2016.She is currently working towards the Ph.D. degree inInformation and Communication Engineering at SunYat-sen University. Her research interests includemodeling and analysis of cellular networks based onstochastic geometry theory, as well as cooperativecommunications.
Minghua Xia (M’12) received his Ph.D. degree inTelecommunications and Information Systems fromSun Yat-sen University, Guangzhou, China, in 2007.Since 2015, he has been a Professor with Sun Yat-sen University.From 2007 to 2009, he was with the Electronicsand Telecommunications Research Institute (ETRI)of South Korea, Beijing R&D Center, Beijing,China, where he worked as a member and then asa senior member of engineering staff. From 2010to 2014, he was in sequence with The Universityof Hong Kong, Hong Kong, China; King Abdullah University of Scienceand Technology, Jeddah, Saudi Arabia; and the Institut National de laRecherche Scientifique (INRS), University of Quebec, Montreal, Canada, as aPostdoctoral Fellow. His research interests are in the general areas of wirelesscommunications and signal processing.Dr. Xia received the Professional Award at the IEEE TENCON, heldin Macau, in 2015. He was recognized as Exemplary Reviewer by IEEET
RANSACTIONS ON C OMMUNICATIONS in 2014, IEEE C
OMMUNICATIONS L ETTERS in 2014, and IEEE W
IRELESS C OMMUNICATIONS L ETTERS in2014 and 2015. Dr. Xia serverd as TPC Symposium Chair of IEEE ICC’2019and now serves as Associate Editor for the IEEE T
RANSACTIONS ON C OGNITIVE C OMMUNICATIONS AND N ETWORKING and the IET S
MART C ITIES . Sonia A¨ıssa (S’93-M’00-SM’03-F’19) received herPh.D. degree in Electrical and Computer Engineer-ing from McGill University, Montreal, QC, Canada,in 1998. Since then, she has been with the InstitutNational de la Recherche Scientifique-
Energy, Mate-rials and Telecommunications
Center (INRS-EMT),University of Quebec, Montreal, QC, Canada, whereshe is a Full Professor.From 1996 to 1997, she was a Researcher withthe Department of Electronics and Communicationsof Kyoto University, and with the Wireless SystemsLaboratories of NTT, Japan. From 1998 to 2000, she was a Research Associateat INRS-EMT. In 2000-2002, while she was an Assistant Professor, shewas a Principal Investigator in the major program of personal and mobilecommunications of the Canadian Institute for Telecommunications Research,leading research in radio resource management for wireless networks. From2004 to 2007, she was an Adjunct Professor with Concordia University,Canada. She was Visiting Invited Professor at Kyoto University, Japan, in2006, and at Universiti Sains Malaysia, in 2015. Her research interests includethe modeling, design and performance analysis of wireless communicationsystems and networks.Dr. A¨ıssa is the Founding Chair of the IEEE Women in Engineering AffinityGroup in Montreal, 2004-2007; acted as TPC Symposium Chair or Cochairat IEEE ICC ’06 ’09 ’11 ’12; Program Cochair at IEEE WCNC 2007; TPCCochair of IEEE VTC-spring 2013; TPC Symposia Chair of IEEE Globecom2014; TPC Vice-Chair of IEEE Globecom 2018; and serves as the TPCChair of IEEE ICC 2021. Her main editorial activities include: Editor, IEEET
RANSACTIONS ON W IRELESS C OMMUNICATIONS , 2004-2012; AssociateEditor and Technical Editor, IEEE C
OMMUNICATIONS M AGAZINE , 2004-2015; Technical Editor, IEEE W
IRELESS C OMMUNICATIONS M AGAZINE ,2006-2010; and Associate Editor,
Wiley Security and Communication Net-works Journal , 2007-2012. She currently serves as Area Editor for the IEEET