A stochastic geometrical approach for full-duplex MIMO relaying model of high-density network

Accepted Manuscript

A Stochastic Geometrical Approach for Full-Duplex MIMO Relaying Model of High-Density Network MHD Nour Hindia , Faizan Qamar , Tharek Abd Rahman , Iraj S Amiri PII: DOI: Reference:

S1570-8705(18)30062-3 10.1016/j.adhoc.2018.03.005 ADHOC 1648

To appear in:

Ad Hoc Networks

Received date: Revised date: Accepted date:

20 July 2017 7 March 2018 19 March 2018

Please cite this article as: MHD Nour Hindia , Faizan Qamar , Tharek Abd Rahman , Iraj S Amiri , A Stochastic Geometrical Approach for Full-Duplex MIMO Relaying Model of High-Density Network, Ad Hoc Networks (2018), doi: 10.1016/j.adhoc.2018.03.005

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ACCEPTED MANUSCRIPT

A Stochastic Geometrical Approach for Full-Duplex MIMO Relaying Model of High-Density Network MHD Nour Hindia1, Faizan Qamar2, Tharek Abd Rahman1, Iraj S Amiri 3,4,* 2 3

Wireless Communication Centre, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310, Johor, Malaysia Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia Computational Optics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

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Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam * Correspondence: [email protected]

Abstract:

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In a high-density wireless communication network, users suffer from low-performance gains due to multiple path loss and scattering issues. Relay nodes, a significant multi-hop communication approach, provide a decent cost-effective solution, which not only provides better spectral efficiency but also enhances the cell coverage area. In this approach, full-duplex topology is the most efficient way in order to provide maximum throughput at the destination, however, it also leads to undesired relay selfinterference. In this paper, we formulated a new Poisson point process approach including a wide variety of interferences by considering a multi-hop high-density cooperative network (source-to-relay and relayto-destination). Performance evaluation is carried out by using stochastic geometric approach for fullduplex MIMO relaying network to model signal-to-interference-plus-noise ratio (SINR) and success probability followed by average capacity and outage probability of the system. The obtained expressions are amenable and provide better performance as compared to conventional multiple antenna ultradensity network approach.

Key Words

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Introduction:

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Stochastic geometry; multi-hop communication; Poisson point process; relay self-interference; fullduplex; success probability

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The latest demands on wireless networks applications are forcing researchers to move ahead in the evolution of 3GPP radio access technology. These demands need cost-effective solutions with high system performance and greater spectral efficiency [1]. The key motivation includes more bandwidth and high capacity to provide higher user data rates. It requires greater throughput to run various advance multimedia services, provide reduced network complexity and high quality of service (QoS) by ensuring low latency [2]. In a wireless high-density network, user suffers from high interference by other nearby devices as these devices are sharing the same frequency spectrum to communicate with the base station (BS). Similarly, scattering becomes an issue in a metropolitan area where a number of buildings are high and no line of sight (LOS) link is possible [3, 4]. It will affect more on millimeter wave (mm-wave) frequency spectrum for the 5G network because scattering mostly occurs when the medium through the wave is traveling contains objects which are much smaller than the wavelength of the electromagnetic Page 1 of 31

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wave [5]. Furthermore, in the source (Base station) to destination (User) communication, the desired signal may not be detected because the received signal strength is too weak to detect the actual transmitted signal. However, deploying a base station (BS) within a cell to fulfill the demands is not an efficient approach, due to the interference handling, coordinated scheduling and high-cost issues [6, 7]. One effective solution to solve this problem is to place an infrastructure-based transceiver by creating a multi-hop communication network such as Relay Nodes (RNs), which provides an efficient solution in a very cost-effective manner [8]. It is one of the major innovations of LTE-A, to meet the growing demand for coverage extension and diversity improvement as shown in Figure 1 [9, 10]. The coverage extension depends on the radial position of RNs in a cell because its location affects the signal to noise ratio (SNR) of the received signal on the evolved-nodeB (eNB) and RN-user equipment (UE) link [11]. On the other hand, diversity improvement can be achieved by deploying multiple RNs, so the received signal coming from both relay and source is maximized, to achieve better modulation and coding (MCS) that gives high user data rates [12, 13].

Type II

UE4

Type I

UE2

UE1

UE3

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Relay Node

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Base Station

Relay Node

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Figure. 1 Relay Application; Type I – Coverage extension, Type II- Diversity improvement

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In the conventional relay communication systems, the relay operates in half-duplex (HD) mode, where the relay utilizes either time-division duplex (TDD) or frequency division duplex (FDD) [14, 15] to transfer information in both directions on the same physical channel but at different timeslots. While, in fullduplex (FD) mode, the BS to UE share a common time-frequency signal-spectrum for transmission and reception simultaneously on the same physical channel [16]. Both modes have different applications [17] like, FD RNs are useful in the urban environment with limited frequency spectrum because it is able to receive and transmit the actual transmitted data with high transparency on the same frequency signal [18]. FD can ideally render up to double spectral efficiency and larger gains when compared to conventional HD operation [19]. It can provide equal requested source–relay and relay–destination data rates to avoid data overflow or congestion at the relay, however, achieved data rates can be unequal due to channel imbalance [16]. Furthermore, FD mode suffers from undesired relay self-interference (RSI), because of the relay’s receiver and transmitter are equipped at the same place on a single access point, and using the same frequency band for simultaneous reception and transmission [20-22]. Sometimes this Page 2 of 31

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is descriptively referred as single-frequency “simultaneous transmit and receive” (STAR) [23]. Three different FD communication scenarios can be possible. The first is the multi-hop relay link, where traffic is symmetric but channels are asymmetric, the second is bidirectional communication link, where traffic is typically asymmetric but channels are symmetric, and the third is simultaneous downlink and uplink, where both the traffic and channels are asymmetric [24, 25].

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RNs are infrastructure-based transceivers that facilitate multi-hop communication by receiving and forwarding the desired signals at single access point [26, 27]. In general, relaying techniques are usually classified into two different relaying protocols, i.e., decode-and-forward (DF) or amplify-and-forward (AF) [28, 29]. DF protocol first decodes the received message signal, then generates a copy of the same signal to be transmitted, eliminating noise in the process due to decoding, quantization and encoding of the signal [30]. However, DF consumes more time in the process and require complex circuitry which is absent in AF protocol [31]. AF protocol would simply amplify the received signal and transmit it to the destination, but is prone to noise which is also amplified in the process and may lead to undesired signal at the destination [32, 33].

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In recent time, the analysis of cellular communication systems has been attractive from the perspective of stochastic geometry, because of the irregular BS location [34]. Poisson Point Process (PPP) is the fundamental of the stochastic geometry modeling tool for communication networks. The main idea of stochastic modeling consists of considering the configuration of the destinations and the transmitters as realizations of stochastic point processes [35]. In many applications, these processes can be taken independent and Poisson. The main advantage of the PPP is its simplicity because the distribution of a Poisson process is completely defined by the intensity measure, representing the mean density of points [36]. On the other hand, some more realistic models are needed, specifically 2-D network of BSs on a square or hexagonal lattice. Signal-to-interference-plus-noise ratio (SINR), user location, relay position and cell corner are some of the few tractable analyses, which can be achieved for a small number of BSs placed in an interference-limited environment. A grid-based model is often referenced to present an ideal scenario where grid size of all the BSs in the network is fixed, BS is always located in the center of the cell with fixed transmit power and therefore, there are no variations in SINR and channel conditions [37]. Though the grid-based model is not practical due to variations in user densities in different cells over a specific region, and therefore ultra-dense network (UDN) model is often used when analyzing the performance of the network. Small cells can be used in UDN to increase the BS to UE throughput as well as save the power consumption of the BS [38]. However, users’ QoS is badly affected due to the random fluctuations in channel conditions, shadowing and fading effects which cause abrupt changes in average user’s data rate and SINR values. There are several parameters which can cause variation in cell radius, such as BS height, transmitting antenna power and user’s density. In this paper, we study multi-hop communication networks by using the stochastic geometrical approach for high-density networks. The FD relaying model has been considered and the location of the source, relays and destination nodes are modeled as point processes on the grid. A new proposed PPP approach considering a wide variety of interferences including Inter-Relay Interference (IRI), Source-Relay Interference (SRI), Relay Self Interference (RSI) and Source-Destination Interference (SDI). As per our research, no one has considered all of these interferences in a single model and this is the first time that Page 3 of 31

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all the mentioned interferences are incorporated in a single model base on PPP. Here, we proposed a new PPP approach which includes: i) analyzing the received SINR by including all the interference faced by the relay node and the destination separately, ii) success probability which defines the minimum SINR threshold required to detect the transmit messages successfully, iii) Outage probability that explained the probability of received signal that falls below a given threshold level and the average capacity, which states the maximum number of users whose probability of success is higher than the set threshold. The remainder of this paper is organized as follows. In Section 1, we first discuss the related work present in the literature. Section 2 evaluates the signal-to-interference-plus-noise ratio calculations for the Sourceto-Relay (SR) and Relay-to-Destination (RD) links separately. Furthermore, Section 3 estimates the success probability of the received signal at both the relay and the destination. Section 4 investigates the outage probability along with the average capacity of the overall network. After this, all numerical results and discussion describe in Section 5. Finally, the conclusions are drawn in Section 6.

1. Related Work:

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In multi-hop communication networks, users suffer from high interference coming from multiple locations, i.e. neighboring sources or relays. A careful action is needed to mitigate this issue, otherwise, the desired results may be ambiguous [39]. Several recent studies were done to understand the different applications of AF and DF mode. Some are mainly focused on FD mode, while some use hybrid FD with HD mode. Hence, it is important to understand the relationship between FD and HD, so that the deployment of the relay based network will get enhanced throughput and extended coverage area along with a cost-effective system [40]. In [41], authors have presented ultra-dense cellular networks to be deployed in 5G scenarios inclusively. Furthermore, network capacity and energy efficiency of the backhaul network is examined to conclude that the performance of the system will start to decrease when density is increased above a certain level of density threshold. However, finding the optimal level of density threshold is still an open issue which can be optimized using different methods. In order to mitigate RSI, [42] proposes a closed form expression for outage probability, which considers both DF and AF protocol for Downlink (DL) and Uplink (UL) path. An approach was proposed in [43] to estimate the self-interference channel and user-to-relay channel by utilizing the maximum-likelihood (ML) criterion for a large scale antenna array. Its results show that the computational complexity is reduced by adopting the expectation-maximization (EM) algorithm.

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Apart from interference cancellation, there are also few schemes presented for interference suppression, as in [44].Different methodologies based on proper beam selection, antenna selection, minimum mean square error (MMSE) filters and null space projection are investigated for different multiple input multiple output (MIMO), even if the imperfect side information is presented. Another approach to control RSI is proposed in [45], which examines the utilization of analog and digital cancellation techniques combined with suppression based on measured suppression values. Recently, several different outage probability techniques were presented such as [46], that investigates the outage probability under Nakagami-m fading scenario for a large scale network. Comparing with the HD mode, it shows that the FD mode is unable to achieve optimal SINR value for high RSI at the relay node. Therefore, suppression of RSI is indispensable to guarantee the best-achieved performance. Another approach in [47] is proposed to calculate the outage probability and diversity-multiplexing tradeoff (DMT). It discusses several distributed Page 4 of 31

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encoding and transmission techniques for the DF protocol, but it deals with a single source to destination path only.

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A combination of the HD mode with the FD mode surely enhances the system performance. It not only efficiently deals with RSI but it also gives a less computationally complex network. Several works like [48] utilize hybrid FD/HD mode, mitigating the RSI by presenting a power control scheme specifically for the asymmetric environment. The basic idea is that the transmit time interval is decomposed into three subintervals, and in each subinterval, the relay operates in either one-way FD or two-way HD mode. Another hybrid approach to enhance the HetNets throughput is proposed in [49]. It takes all the significant parameters of an access point, such as transmission power, user density and self-interference cancellation, either operating in bidirectional FD or DL HD mode.

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Few other techniques to improve overall system performance are presented in [37, 50-54] such as, a method which investigates the feasibility of FD RNs with multiple antennas at BS and relay, presented in [50]. An energy-efficient method in [51] evaluates the analytical model of SINR and achievable rate for both cooperative and non-cooperative users. A physical layer security for RN system is discussed in [52], which uses the geometric programming method under the total power constraint. Its results show that it provides better secrecy rate as compare to conventional HDR operation. Another approach is proposed [53], which enhances the achieved throughput by performing scheduling among all concurrently transmitting nodes, either used it in HD or FD mode. It proves that, if the perfect RSI cancellation techniques are applied, the implantation of ALOHA media access control (MAC) protocol provides more throughput in AF more as compare to DF mode. Another approach has been presented in [54] that deals in DF multi-antenna small cell networks and enhances the probability of success. It characterizes the distribution of the self-interference power under Ricean-fading channel for both arbitrary receive and transmits beamforming strategies. A method in [37] proposes a new relay-assisted grid model that resolves the issue of dense placement of BSs for future networks. Many practical special cases are simplified by closed-form expression or via common integral. The position of BS and relay station is modeled as a homogeneous PPP of density 𝜆 in the Euclidean plane. The users are scattered at random locations and receive signals from the nearest BSs and other interfering BSs. One of the major issue with this approach is that it causes more interference in few cases, however, as some BS can be placed very near to each other because of the independency of PPP. In this paper, boldface lowercase letters (e.g., xs ) represent vectors and boldface uppercase letters (e.g., X ) represent matrices.

F

is the Frobenius norm. The conjugate transpose is denoted by

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h   and Έ  denotes the expectation operator, while denotes the magnitude. The trace of a matrix is

denoted



by

tr   ;

pseudo-inverse

†  ;

det   is

the

determinant;



diag xs 1 ,...., xs n is the diagonal matrix with diagonal components xs 1 ,...., xs n ; I NR  NR is the N R -by- N R

identity matrix; X NT  N R is the NT -by- N R zero matrix. System Model

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Consider the scenario in Figure 2, where a dual-hop relaying cellular network system is designed between the Source (BS) and Destination (UE). The Relays are operating in FD mode where the reception and transmission of signal occur simultaneously on the same frequency band. In our setting, the source S1 is transmitting the signal to the destination D1 by using SR channel hR 1 D 1 and RD channel hR 1 D 1 . The source

S1 and destination D1 are equipped with MIMO transmit antenna NT , S and receiving antenna N R , D respectively, while the relay node consists of two set of antennas for reception N R , R and transmission NT ,S

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NT , R . RNs are utilizing DF protocol, which decodes the message from the signal source xS1  C

and

then encodes the signal xR1  C NT ,R before sending it to the destination. Considering the multi-cell highdensity network, the relay node R1 receives interference IT , R1 from another source S 2 , i.e. source-torelay interference (SRI) via the channel hS 2 R 1 , and from the nearby relay R2 i.e. inter-relay interference (IRI) via the channel hR 2 R 1 and finally relay-self interference (RSI) via the channel hR 1 . Similarly, the

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destination also faces some serious interference such as IT , D1 from nearby source S 2 i.e. source-todestination interference (SDI) via the channel hS 2 D 1 , and from the nearby relay R2 , i.e., relay-todestination interference (RDI) via the channel hR 2 D 1 .

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Relay-Self Interference

Destination 2

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Source 1

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Relay 1

Destination 1

Relay 2

Source 2

Desired Signal

Relay-Self Interference

Desired Signal

Source-to-Relay Interference

Desired Signal

Source-to-Destination Interference Source-to-Relay Interference Source-to-Destination Interference

Desired Signal Inter-Relay Interference Relay-Self Interference

Figure 2. Proposed network scenario Page 6 of 31

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Let the spatial distributions of our source, relay and destination nodes be stationary independently marked

i

PPP

with

 S , R , D  on i

i

I S   DS2 I S , I R   DR2 I R , I D   DD2 I D and

intensity 2

i



2



2

respectively

in

. In a random-access network, the receiver must be located at the

origin in order to examine the network performance. The results will be obtained by using Palm probabilities of Poisson processes as it does not affect the process statistics due to the stationary position of the receiver in PPP [55].

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Channel Model

In a multicell network, we assume that the source and relay nodes are transmitting signals with constant power PS and PR , respectively. The propagation through the multiple-hop wireless channel experiences path loss attenuation which is modeled as PL SR  and PL RD  for distances DSR and DRD  DSR

DRD 

respectively and the path loss exponent is   4 . The channel hS 1 R1  C NR ,R1  NT ,S1 , hR2 R1  C NR ,R1  NT ,R2 and N R ,D1  NT ,R1

are the channel gains matrices from source-to-relay, relay-itself, inter-relay and relay-

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hR 1 D1  C

to-destination respectively with the fading coefficient  . The power constraints on the transmit signals

   1 and Ρ x   1, while

are Ρ xS1

2

2

R1

yR1  C

N R ,R1 1

and yD1  C N R ,D1 1 are the received signal at

R1 and D1 respectively. We assume that all channels except RSI channel experience Rayleigh fading exp    . The entries of each matrix are distributed as independent

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parameter  i.e. hS 1 R1 and hR 1 D1

2

and identically and having complex Gaussian distribution with zero mean and variance μ . The

probability distribution CR

 0, μ

2

, І R  and CD

 0, μ

2

, І D  , which are uncorrelated to message signal

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xS1 and xR1 respectively.

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independent circularly symmetric complex Gaussian noise vectors are zR1  C N R ,R1 and zD1  C N R ,D1 with a

2. SINR characterization

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In this section, we analyze the SINR for the FD nodes which are required to formulate transmission success probability. We study the first and second hop separately i.e. from source-to-relay and relay-todestination respectively, under the point process condition.

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a) SR HOP

Building on Slivnyak’s theorem [56, 57], consider a source S1 located at the origin and, due to the stationarity of  , we express the statistics of the received signal at relay node R1 which contains desired signal plus noise and all interferences it receives.    yR 1   PS PL SR 2 hS 1 R 1 xS1  zR1    

(a)



      PR hR 1 xR1   PS PL SR 2 hS 2 R 1 xS2    PR PL RR 2 hR 2 R 1 xR2  (1)    



(b)

(c)

(d)

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where (a) represents the desired signal plus noise, (b) represents the RSI; relay transmitter and receiver are mounted on a single access point, therefore no path loss in this channel, (c) and (d) indicates the interference coming from neighboring source S 2 i.e. SRI and nearby relay R2 i.e. IRI, respectively. Given the received signal, the resulting SINR can be estimated as [58] 2

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SINRR1 

PS PL SR  hS 1 R 1 IT , R1   2 R1

where IT , R1 is the overall interference at R1 i.e. RSI, SRI and IRI.

  P P

1,2

S



L SR

hS 2 R 1

2

  P P R



L RR

hR 2 R 1

2

  P h  2

R

R1

(3)

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IT , R1 

(2)

In our network, the transmitting powers of relays R1 and R2 are lower as compared to those of S1 and S 2 . Furthermore, RNs are separated by a distance, so signals between the two relays are not interfering each other, therefore IRI can be ignored in this scenario. As our channel is interference-limited, the interference is much larger than the Gaussian noise IT , R1

 2 R , which has a negligible impact on the 1

SINR. The success probability of the SR hop is derived in Section 3.

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b) RD HOP

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Now we consider the next RD hop from the relay R1 to the destination D1 . Again, following the Slivnyak’s theorem [56, 57], we assume that D1 is located at the origin and due to the stationarity of  , we, hence, express the statistics of the received signal at the destination D1 as the combination of the desired signal

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from R1 plus noise and all interferences it receives.

(a)

(b)

(4)

(c)

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         2 2 yD1   PR PL RD hR 1 D 1 xR1  zD1    PS PL SD hS 2 D 1 xS2    PR PL RD 2 hR 2 D 1 xR2       

where (a) represents the desired signal plus noise from R1 , (b) and (c) indicate the interference coming from nearby source S 2 i.e. SDI and nearby relay R2 i.e. RDI, respectively. Given the received signal, the resulting SINR can be estimated as

SINRD1 

PR PL RD  hR 1 D 1 IT , D1   2 D1

2

(5)

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where IT , D1 is the overall interference at D1 i.e. SDI, and RDI. IT , D1 

  P P R

i 1,2



L RD

hR i D 1

2

  P P S



L SD

hS 2 D1

2



(6)

Similar to the first SR hop, our channel here is also interference-limited, and therefore interference is

 2 D , which has a negligible impact on the SINR.

much larger than the Gaussian noise IT , D1

1

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Moreover, the success probability of the RD hop is derived in section 3.

3. Success Probability

The successful transmission from source to destination is given by a joint complementary cumulative distribution function (CCDF) of SINRR1 and SINRD1 . It is which is denoted by  SUC SR





 SINRD1   D , where



 R and  D are the threshold signals at relay and destination

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and  SUC RD



 SINRR1   R

respectively [59]. Due to the independent sampling of the point process, there is no correlation between the two hops and total success probability is the scalar product of  SUC SR and  SUC RD .

SUC T   SUC SR  SUCRD







(7)



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where,  SUC SR and  SUC RD are defined as  SINRR1   R and  SINRD1   D respectively. According

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to Fortuin-Kasteleyn-Ginibre inequality [60], the system performance for the uncorrelated case can be considered as a lower bound. a) SR HOP

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expressed as

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In this section, we analyze the success probability of SR hop. It is defined as the probability that the obtained SINR at the relay R1 , as estimated in eq. (2), exceeds a pre-defined threshold  R and can be

 SUC SR



 SINRR1   R



(8)

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which is also the CCDF of the SINR, here  R is the minimum SINR threshold required to detect the transmit messages successfully. The given threshold satisfies a targeted SINR, and the probability of success at relay node is

 SUC SR

 P P  h S L SR S 1 R1   IT , R1   SUC SR



 hS 1 R 1

2

2

 R   

  R PS 1 PL SR  IT , R1

(9)



(9-a)

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



 SUC SR

0

2



 SUC SR



1    hS 1 R 1   R PS PL SR s  f IT ,R  s  ds

 F  C

SR



(9-b)

1

P 1PL SR  s  f IT ,R  s  ds

R S

(10)

1

0

Next, success probability analysis for the SR hop is calculated as

 SUC SR

NTR 1

 i 0

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where eq. (10) is obtained by conditioning S and its CCDF denoted by F c .

   s i d i  ζ s  i IT ,R1     i ! ds  s  P 1P  R S L SR

Where ζ IT ,R can be estimated as explained in the Theorem 1 of [61].

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1

(11)

Proof: Refer to Appendix I

Theorem 1 of [61] shows how the employment of multiple receive antennas results in an array again: the larger the N R , R more terms added in the summation of  SUC SR as shown in Eq. (11). Note that all terms in the summation are positive since the nth derivative of ζ IT ,R  s  is negative for odd n . 1

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The result in equation (11) provides a fundamental limit on the SR hop and its performance in an interference-limited scenario. It has a looser integral form   s  on the success probability  SUC SR , which

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is not a closed form and that need to be evaluated numerically; however, tight lower and upper bounds can be obtained from the Theorem 1 of [27]

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In order to obtain the success probability of the SR hop  SUC SR ( ) , which is equal to

 SUC SR ( )max  SUC SR ( )min , substitute upper and lower bounds ζ IT ,R  s   ζ min s , ζ max s  in 1  IT ,R1   IT ,R1   

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 SUC SR as shown in Eq. (11).

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b) RD HOP

In this section, we analyze the success probability of the RD hop. It is defined as the probability that the obtained SINR at the destination D1 , as estimated in eq. (5), exceeds a pre-defined threshold  D and can be expressed as

SUC RD



 SINRD1   D



(12)

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which is also the CCDF of the SINR, here  D is the minimum SINR threshold required to detect the transmit messages successfully i.e., the given threshold should satisfy a targeted SINR. The probability of success at the destination is

 SUC RD

 SUC RD



 hR 1 D 1

 h

2



 SUC RD

 D   

  D PR 1PL RD  IT , D1 2

R1 D1

2



 F  C

RD

0



1

P 1PL RD  s  f IT ,D  s  ds

D R

1

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



  D PR 1PL RD  s  f IT ,D  s  ds

0

 SUC RD

(13)

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 P P  h R L RD R1 D1   IT , D1 

(14-a)

(14-b)

(15)

where Eq. (10) is obtained by conditioning S and the CCDF is denoted by F c . Next, success probability analysis for the RD hop is calculated as

 i 0

   s i d i  ζ s     i IT ,D1  i ! ds  s 

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 SUC RD

NTD 1

D PR

1

PL RD

(16)



where ζ I,D can be estimated as explained in the Theorem 2 of [61]. Proof: Refer to Appendix II

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Theorem 2 of [61] explains the employment of multiple receive antennas resulting an array again: the

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larger the number of antennas, N R , R , the more terms added in the summation of  SUC RD as shown in Eq. (16). Note that all terms in the summation are positive since the nth derivative of ζ IT ,D  s  is negative

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for odd n .

The result in Eq. (16) provides a fundamental limit on the RD hop and its performance in an interference-

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limited scenario. It has a looser integral form   s  on the success probability  SUC RD , which is not a closed form and that need to be evaluated numerically; however, tight lower and upper bounds can be obtained from the Theorem 1 of [27]. In order to obtain the success probability of RD hop  SUC RD ( ) ,

denoted by  SUC RD ( )max and

 SUC RD ( )min , can be found by substituting upper and lower bounds ζ IT ,D  s   ζ max s , ζ min s  in 1  IT ,D1   IT ,D1  

 SUC RD as shown in Eq. (16). Page 11 of 31

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Proof: Refer to Appendix III

4. Outage Probability and Average Capacity

a) Outage Probability For the first SR hop, that has been served by

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In this section, we calculate the outage probability of the received signal at relay and destination nodes simultaneously. Furthermore, we also estimate the average capacity value for both hops, which is defined as the number of active users that can be accommodated with respect to threshold SINR values.

S1 , the following equations give the outage probability at

R1 . The probability that the SINRR falls below a given threshold level  R , is known as outage 1

probability, which is the CDF of the SINRR1 of the SR hop denoted by



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

OP, R1  R    SINRR1   R given by

OP, R1  R   1  e

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Similarly, for the RD hop, where signals are received at

(17)

2 I S 

(18)

D1 . The probability that the SINRD falls below a 1

set threshold level  D , which is the CDF of the SINRD1 and can be expressed as



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OP , D1  D    SINRD1   D

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given by



OP, D1  D   1  e 2 I R 

(19)

(20)

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After finding the individual outage probability of each hop, the overall total outage probability that is

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denoted as OP ,T 



is given as the scalar product of the individual probabilities as

OP,T 

  OP,R  R  1

OP , D1  D 

(21)

b) Average Capacity Capacity defines the maximum number of users whose probability of success is higher than the set threshold. Here, we calculate the total average capacity achieved at R1 and D1 . First, the average capacity for SR hop can be expressed by using  SUC with  R as SR

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N R ,R

CS1R1    SUC log 1   R  SR

i 1

(22)

and similarly, the average capacity for the RD hop capacity can be expressed by using  SUC with  D as RD

N R ,D

CR1D1    SUC log 1   D  RD

(23)

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i 1

After calculating the individual average capacity or each hop, the overall total average capacity, denoted as CT , is the scalar product of the individual capacities CT  CS1R1

(24)

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Proof: Refer to Appendix IV

CR1D1

5. Results and Discussion

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In this section, we present numerical results to prove our theoretical findings that the proposed PPP model is robust to interference. Furthermore, we compare the performance of proposed PPP model with the ideal grid model and conventional multiple antenna Ultra-dense network (UDN) approaches with respect to both SINR threshold and user’s density [37]. Monte Carlo simulations are presented to validate the proposed scheme and averages are performed over 10000 independent channel realizations. The relay and the BS are assumed to have transmitted power of 46 dBm and 20 dBm, respectively [62]. Here, we assumed fixed relays nodes, therefore, the IRI channel may not be deeply faded, and due to the limited distance between transmitter and receiver compared to IRI case, the RSI is stronger than the IRI. Practically, the relay station is designed to mitigate the RSI, thus we assume the IRI and RSI have the same effects. Table 1 shows the list of simulation parameters with corresponding values.

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Table 1. Simulation parameters Parameters Values Number of Users 200 Number of Base Stations 25 Network Scenario Urban (Random user deployment) Cell size 400km2 Number of Transmit Antenna 2 Number of Receive Antenna 2 Path loss model flat earth model (  => 4) Simulation Iteration SNR Threshold BS transmission power RN transmission power UE height

100 30 dB 46 dBm 20 dBm 1m Page 13 of 31

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BS height Fading Model Number of RNs in a cell Radius of Number of Interference Sources

25 m Rayleigh fading 1 800 m 16

The cellular network model consists of BSs arranged according to some homogeneous PPP i of intensity

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I S in the Euclidean plane. We assume each mobile user is associated with the closest BS in the Voronoi cell of a BSs, resulting in coverage areas that comprise a Voronoi tessellation on the plane [63]. Voronoi cells are constructed based on the distance from mobile users to base station. The regions are adjusted to include those mobile users whose distance from the base station in the selected cell is less than any other base station of the neighbor cells. Here, we take two independent Poisson processes Ð and Ş as an example which represents the users and the BSs respectively. The parameters of this model come to be the intensity measures Ð and Ş of two processes. As the density of subscribers is bigger than that of

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BSs, we can assume   2 . Figure 3 depicts the basic cell model, where the subscribers are connected

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with the BS of their corresponding Voronoi cell. Furthermore, a traditional square lattice grid model for cellular BSs with one tier of eight interfering BSs is shown in Figure 4 and an Ultra Dense Network BS deployment is shown in Figure 5.

Figure 3. 20 × 20 km view of a current base station deployment for Poisson point process

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Figure 5. 20 × 20 km view of Ultra-Dense Network

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Figure 4. Grid multicell topology

The SINR threshold indicates the minimum signal quality required at the relay node R1 and the destination node D1 . It better quantifies the relationship between radio received signals and interference plus noise of hS 1 R 1 and hR 1 D 1 . Figure 6(a) shows the relationship between the set SINR values and the total success probability (  SUC T ) as we defined in Eq. (2), Eq. (5) and Eq. (7) respectively. We consider the

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urban asymmetric traffic requirements scenario for our network, which indicates the real random user's distribution environment. It can be noticed from Figure 6(a) that, the effect of  SUC T for each approach is

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very small and all achieved probabilities are almost higher than 90 % when the set SINR values are very low. However, the behavior starts changing significantly with the increase of set SINR values. At 10 dB SINR value, the proposed PPP scheme gives 11 and 6 % better probability rate for multi-antenna UDN and grid approach, respectively. Moreover, when it approaches higher SINR values up to 30 dB, it can clearly be seen that the  SUC T achieved for the proposed model outperforms both the multi-antenna UDN

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approach by 15 % and grid approach by 7 %. This happens because, for each subinterval, all interferences including RSI, SRI, and IRI at relay station as defined in Eq. (1) and SDI, and RDI at destination node as shown in Eq. (4), are optimized for calculating SINR, which makes the proposed technique robust to interference and helps to maximize the success rate for both SR Hop and RD Hop.

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As our focus of the study is on the higher density networks as discussed in Figure 3, the effect of different success probability due to various users’ density investigated here. The success probability is highly affected by the number of users presents in BS; as the number of users increases the SINR of each user affected which tends to reduce the success probability of the cell. Here in Figure 6(b) we plot the success probability  SUC T in over different values of the density 𝜆. As discussed, it can clearly be observed the success probability is decreasing with the increase in user’s density for all cases. The achieved probability for multi-antenna UDN and grid approach are much correlated which giving 74 and 77 % for the minimum for 50 users and then decreasing and reaches up to 45 and 44 % for the maximum case of around 200 Page 15 of 31

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users per cell, respectively. However, the proposed PPP approach providing the upper bound throughout the different user’s density and delivering up to maximum of 80 % to minimum of 56 % for higher to lower user’s density.

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Figure 6(a). Success Probability vs SINR Threshold

Figure 6(b). Success Probability vs User’s Density

Another result in Figure 7(a) illustrates the total achieved capacity CT with respect to set SINR values as defined in Eq. (24). All three Grid, multi-antenna UDN, and PPP proposed approaches give significant increment in the capacity values according to Shannon–Hartley theorem stated

C  Bandwidth * log(1  SINR) [64, 65]. It is obvious that grid approach can deliver the higher average capacity value of 2.25 bits/sec, due to the ideal square cell model as shown in Figure 4. Besides, Figure 7(a) shows that the effects of the proposed PPP and multi-antenna UDN approaches are almost similar Page 16 of 31

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for lower SINR values up to 5 dB, due to low signal quality to the Ri and Di . However, when the SINR

Figure 7(a). Average Capacity vs SINR Threshold

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reaches higher values i.e. greater than 10 dB, the proposed technique starts giving the slightly better average capacity value and reaches up to 2.10 bits/sec, as compared to the multi-antenna UDN approach. This proves the effectiveness of applying all interference including RSI, which makes the proposed approach more precise to handle all channel interference that leads to achieving greater SINR and improved average capacity at the relay and destination. The total capacity is also affected because of the number of users in a cell. Here in Figure 7(b), the average capacity has been estimated for various user’s density. Similarly, in Figure 7(a), here also grid approach is delivering higher value of 0.94 to 1.38 for minimum 50 users to maximum 200 users, respectively. Moreover, the proposed PPP approach gives slightly better capacity results of 3.7 % as compared to conventional multi-antenna UDN approach. This relation can be understood by the fact that when the number of users increases in the cell, more users facing interferences effect which might affect the average capacity of the cell. However, due to the proposed PPP approach that can able to mitigate all the interferences which include SRI, IRI, SDI, RDI and RSI, it helps to perform better overall average capacity performance.

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Figure 7(b). Average Capacity vs User’s Density

Figure 8(a) compares the outage probability of the proposed model with multi-antenna UDN and grid model. It has been calculated by OP,T      SINRT    , which is defined in Eq. (21). We can

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observe that when the SINR values are less than 2dB, the outage probabilities for all models are similar in values. However, when the SINR reaches 15 dB, the proposed scheme outage probability is 20 and 8 % lower than the multi-antenna UDN and grid approach, respectively. Furthermore, at the higher SINR value

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of 30 dB, the outage performance OP ,T achieved for the proposed model is 15 % lower than the multi-

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antenna UDN and 7 % lower than the grid model. It indicates that the proposed scheme efficiently deals with the multi-hop channel interference such as SRI, IRI, SDI, RDI including RSI. As similar to Figure 6(b), here we estimate the outage probability with respect to different values of the user’s density 𝜆. The number of users are varying from minimum 50 to maximum of 200. As it showing in Figure 8(b), here the response of outage is very much similar to Figure 8(a). The maximum outage results are showing my multi-antenna UDN approach, which varies from 24.5 to 56 % for minimum to maximum density. Similarly, the grid approach also delivers higher outage which is correlated to the multi-Antenna UDN approach. On the other side, the proposed PPP approach gives the minimum outage probability of 10 an 12 % as compared to grid and multi-antenna UDN approaches, respectively. This result may be explained by the fact that the even if the number of users increasing in the cell, the proposed PPP approach delivering bettering outage performance by mitigating all the channel interferences at relay and destination nodes.

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Figure 8(a). Outage Probability vs SINR Threshold

Figure 8(b). Outage Probability vs User’s Density

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Figure 9 shows the impact of success probability for the proposed PPP model at different MIMO conditions by changing NT i.e. NT  2 , NT  4 , NT  6 and NT  8 with similar N R  2 . The path loss coefficient is set as   4 for relatively lossy environments. It can clearly be observed that, at lower SINR values, all antennas probabilities are similar with greater than 90%, but when SINR values increase, major differences in probability rate can be observed with respect to all higher SINR values. The maximum  SUC we attained is at NT  8, N R  2 , while the minimum  SUC is achieved at

NT  2, N R  2 . It shows that the achieved  SUC T becomes higher as the number of transmitting Page 19 of 31

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antennas increases across the whole SINR range. It can be stated that deploying multiple antennas at the source and relay nodes, results in increasing the spectral efficiency of the channel, while reducing the effect of shadowing, hence obtaining considerably better success rate.

Figure 9. Success Probability vs SINR (MIMO)

6. Conclusion

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In high-density multi-hop communication networks, users suffer from several serious types of interference such as source-to-relay interference, inter-relay interference, relay-to-destination interference and relay-self interference. This paper analyzes the performance of wireless networks with the full-duplex relaying mode, which offers large potential, which is exploitable after solving the main technical problem, i.e., mitigation of relay-self interference. In this paper, we proposed Poisson point process approach based on stochastic geometrical approach, to calculate the success probability, average system capacity and outage probability. Its results also verified that adding more antennas at the source provides extra degrees of freedom which provides a better probability of success for the desired received signal. Finally, our simulations illustrated that the proposed Poisson point process scheme offers significant interference mitigation and gives better overall system performance as compared to the conventional multiple antenna ultra-density network approach. Overall, this paper investigated how dense networks should be deployed while taking into account a model that more accurately describes the channel characteristic compared to the more conventional multiple antenna ultra-density networks, which is the current standard reference model when studying high-density networks. For future work, we will analyze more robust stochastic PPP approach by including additional interferences coming from neighboring cell and evaluate our results with some other QoS parameters such as time delay and coverage probability as well. Acknowledgment The authors would like to thank Wireless Communication Centre (WCC) Universiti Teknologi Malaysia (UTM) for their support. Page 20 of 31

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Author Contributions: MHD Nour Hindia and Faizan Qamar conceived and designed the experiments; MHD Nour Hindia and Faizan Qamar performed the experiments; MHD Nour Hindia, Faizan Qamar, Tharek Abd Rahman, Iraj S Amiri analyzed the data; MHD Nour Hindia, Tharek Abd Rahman and Iraj S Amiri contributed reagents/materials/analysis tools; Faizan Qamar and MHD Nour Hindia wrote the paper.

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Conflicts of Interest: The authors declare no conflict of interest.

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Appendix I The success probability of the first hop can be calculated by

 SUC SR



2

 R   

2

 hS 1 R 1   R PS 1PL SR  IT , R1

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(25)

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 P P  h S L SR S 1 R1   IT , R1 



(26-a)

where IT , R1 is defined in eq. (3) as the overall interference received at R1 i.e.

  P P S

1,2



hS 2 R 1

L SR





2

  P P R



L RR

hR 2 R 1

2

  P h  2

R

AN US

IT , R1 

R1

(26-b)

Let us define f IT ,R  t   d  IT , R1  t as the PDF of IT , R1 . The integration of f IT ,R (t ) using the CCDF 1

FSR  t  , transforms it as C

 s



 F  st   f t  dt C

M

ED

Note that the power of desired signal is distributed as H S 1 R 1



PT

 SINRR1   R

(26-c)

IT ,R1

SR

SR

0

given is by

1





2

 hS 1 R 1

X12 NR ,R , and the probability of success 2

  R PS 1PL SR  IT , R1



(26-d)



 F  st   f t  dt C

SR

IT ,R1

(26-e)

CE

0

 s

s  R PS 1PL SR 

(26-f)

AC

Finally, using the CCDF FSR  st   et and Laplace transform, the success probability can be expressed C

using the transformation of f IT ,R  t  as 1

 s



 FSR  st   f IT ,R t  dt C

1

(27)

0





ζ f IT ,R  t   s   ζ IT ,R  s  1

(27-a)

1

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Finally, using the CCDF F  t    e nt  eant the transformation of f IT ,R  t  is given by SR 1 i

C

n

i



 s

 F  st   f t  dt C

(27-b)

IT ,R1

SR

0



i  ani  st    nst e e i   f IT ,R1  t  dt 0   n 





ani s i ζ t i f IT ,R  t   ns 

n

s

i

n



i



1



(27-d)

(27-e)



  di i ζ ns  ani   s     I i T ,R1 d  ns   

(27-f)



  s i d i  ζ ns    ani    i IT ,R1    n  d  ns  

(27-g)

 s

n

 s



n

AN US

 s

   nt i  ani st   e t f IT ,R  t  dt  1 0  i

(27-c)

CR IP T

s

i

i

M

dn ζ  f  t   s . Using ds n  the moment-generating function of X 12 N R ,R and Gamma distributions, and applying it to [53] (Thr. 1) we n

ED

where eq. (27-g) is obtained by using Laplace transform property t n f  t    1

PT

can get eq. (11)

Appendix II

AC

CE

The success probability of the RD hop is calculated as

 SUC RD



 SINRD1   D



(28)

 SUC RD

 P P  h R L RD R1 D1   IT , D1 

 SUC RD

 hR 1 D 1   D PR 1PL RD  IT , D1



2

2

 D   

(28-a)



(28-b)

where IT , D1 is defined in eq. (6) as,

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IT , D1 

  P P



hR i D 1

R L RD

i 1,2

2

The desired signal is distributed as H S 1 R 1

2

  P P S



L SD

hS 2 D1

2



(28-c)

X12 NR ,R . On the other hand, the moment generating

function of X 12 N R ,R distribution can provide the Laplace transform of IT , D1 [56] (P. 125) and can be computed as

Appendix III

2

AN US

Given the definition of   s, r 



1

  s, DRD   

1  sPR  DSR  DRD 



        

CR IP T

   P ζ I,D  s  Ē   Ē exp   s R PL RD  hR 1 D 1 1  1,2   N R , D  

,

1

1  sPR  DSR  DRD 

3

DRD  , then the maximum

M

where we can recall that distances between S to R and R to D i.e.,  DSR



  

(28-d)

and minimum ζ IT ,D can be formulated as stated in (28) and (29).

ED

1

Appendix IV

A source is located at the origin O , and relay is situated at R2 . Since the relay is communicating with the

PT

nearest source, the CDF of R2 can be calculated by,



CE

 DSR  R1





1   no source closer than R1 1  e I S R1



(30)

2

(30-a)

Then the pdf of DSR is f  DSR   e I S DSR 2 I S D SR , which yields its SINR. The CDF of SINR regarding the

AC

2

IT , R1 is given by, R1

   1  Ē   SINRR   R 

(30-b)

1



R    1   2 I S D 1

0

 I S D SR SR e

2

 P P  h S L SR S 1 R1   IT , R   2 R 1 1 

2

   R d D SR  

(30-c)

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 S

D SR

2

SR

0

2

Using the fact that H S 1 R 1

NT ,S

  hS 1 R 1

2

with h

exp 1 simplifies to   PL SR  R  IT ,R1   2R1     Ē e PS d D SR     

   1   2 I S D

SR

e I S DSR

0



R    1   2 I S D 1

SR e

2

  R  2 R1  2  PL   I S D SR   SR   PS  

0

 PL   R ζ IT ,R  SR 1   PS

NT ,R

2

i 1

(30-f)

(31)

    s PR gi      N ζ IT ,R  s   Ē  Ē gi  e T ,R    1    i   

(31-a)

    1   ζ IT ,R  s   Ē   1  i   1   s PS    N    T ,S

(31-a)

ED

PT CE

 d D SR 

(30-e)

  hS 1 R 1 with h exp 1 simplifies to

M

2

AN US

   s PR  H S 1R1 2   N  ζ IT ,R  s   Ē e T ,R  i  1   Using the fact that H S 1 R 1

(30-d)

i 1



R1

  PL   R 2   hS 1 R 1  SR IT , R1   2 R1 d D SR   PS  

CR IP T

R1

   1   2 I S D e I

          

The probability generating function of PPP  with interference I R , which states for functions v  x  is

AC

denoted by

    Ē  v  x    exp   I  1  v  x   dx     x   R2 

(32)

Apply the probability generating function to the ζ IT ,R yields, 1

   1    exp  2 I S  1  xdx   1  sx   D    SR 

(32-a)

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ζ IT , D

      1   exp  2 I R  1   PL   R DRD   1   RD  PR     

1

1

      xdx     x     

(33)

(34)

AC

CE

PT

ED

M

AN US

The distance between DSR and DRD is constant.

      xdx     x    

CR IP T

ζ IT , R

       1  exp  2 I S  1   PL   R DSR   1   SR  PS     

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CR IP T

Author Biography

CE

PT

ED

M

AN US

MHD Nour Hindia Obtained his Ph.D In University of Malaya, Faculty of Engineering in Telecommunication in 2015. He is working in the field of Wireless Communications Especially in Channel Sounding, Network Planning, Converge Estimation, Handover, Scheduling and Quality of Service Enhancement for 5G Networks. Besides That, He is working with Research Group in Modulation and Coding Scheme for Internet of Thing for Future Network. He Has Authored and Co-Authored a Number of Science Citation Index (SCI) Journals and Conference Papers. He has also participated as a Reviewer and a Committee Member of a Number of ISI Journals and Conferences.

AC

Faizan Qamar received the M.E. Degree in Telecommunication in 2013 from NED University, Karachi, Pakistan and B.E. Degree in Electronics in 2010 from Hamdard University, Karachi, Pakistan. He has more than 5 years of research and teaching experience. He is currently pursuing towards his Doctorate Degree in Electrical Engineering (major in wireless communication) from Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia. He has authored and co-authored a number of ISI Journals and IEEE Conference Papers. His Research interest includes Wireless networks, Interference management and Millimeter wave communication for 5G.

Page 30 of 31

CR IP T

ACCEPTED MANUSCRIPT

PT

ED

M

AN US

Prof. Dr. Tharek A. Rahman currently is a professor in wireless communication at faculty of electrical engineering, Universiti Teknologi Malaysia. He obtained his BSc (Hons) (Electrical Engineering) from University of Strathclyde, UK, MSc in Communication Engineering from UMIST, Manchester, UK and PhD in Mobile Communication from University of Bristol, UK. He is the Director of Wireless Communication Centre (WCC), Faculty of Electrical Engineering, Universiti Teknologi Malaysia and currently conducting research related to mobile communications, antenna and propagation. He has also conducted various short courses related to mobile and satellite communication to the telecommunication industry and government agencies since 1988. Prof. Tharek has published more than 300 scientific papers in journals and conferences and obtained many national and international awards. He is also a consultant for many communication companies and an active member in several research academic entities.

AC

CE

Iraj Sadegh Amiri received his BSc (Applied Physics) from Public University of Oroumiyeh, Iran in 2001 and a gold medalist MSc from Universiti Teknologi Malaysia (UTM), in 2009. He was awarded a Ph.D. degree in photonics in 2014. He has published over 100 ISI journal papers and 250 research papers including Scopus papers, conference papers, books/chapters and international journal papers in Optical Soliton Communications, Laser Physics, Photonics, Fiber Optics, Nonlinear Optics, Quantum cryptography and Nanotechnology Engineering.

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