# A Stochastic Geometry Approach to Full-Duplex MIMO Relay Network

Authors: Mhd Nour Hindia; Moubachir Madani Fadoul; Tharek Abdul Rahman; Iraj Sadegh Amiri
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

Cellular networks are extensively modeled by placing the base stations on a grid, with relays and destinations being placed deterministically. These networks are idealized for not considering the interferences when evaluating the coverage/outage and capacity. Realistic models that can overcome such limitation are desirable. Specifically, in a cellular downlink environment, the full-duplex (FD) relaying and destination are prone to interferences from unintended sources and relays. However, this paper considered two-hop cellular network in which the mobile nodes aid the sources by relaying the signal to the dead zone. Further, we model the locations of the sources, relays, and destination nodes as a point process on the plane and analyze the performance of two different hops in the downlink. Then, we obtain the success probability and the ergodic capacity of the two-hop MIMO relay scheme, accounting for the interference from all other adjacent cells. We deploy stochastic geometry and point process theory to rigorously analyze the two-hop scheme with/without interference cancellation. These attained expressions are amenable to numerical evaluation and are corroborated by simulation results.


---

## Body

## 1. Introduction

Currently, the demand for the high capacity and low latency is dramatically increasing, corresponding to advancements in communication devices such as smartphones. It is expected that the wireless traffic volume of these communication devices will have a 1000-fold increase over the next decade which will be driven by the expected billions of connected devices by 2020 to access and share data anywhere and anytime [1]. The bottom line is that the technology has to move to higher frequency bands of millimeter wave to adapt the 5G future mobile communications, where a very large bandwidth in frequency band is available. With rapid increase in the number of connected devices, some challenges appear such as the capacity shortage, cost, and cochannel, and intercell interference especially at high dense network is rapidly increased [2]. Hence, the fifth-generation (5G) wireless systems should be able to support the ultradense networks to adopt the exponential growth in mobile users and high data demand. To satisfy the quality of service (QoS), it is expected that the 5G network deployment will be much denser compared to that of 4G. However, as the network density increases, the interference will degrade the system’s performance especially at cell edges (dead zone). One of these efficient solutions that allow the 5G network to meet its QoS requirements is to use relay to improve the ultradense network performance (capacity enhancement and coverage extension). Cooperative communication is an alternative way to achieve spatial diversity and multiplexing gain through Multiple-Input Multiple-Output (MIMO). MIMO attracted a lot of attention due to its potential for interference mitigation and capacity increase. FD relaying equipped with MIMO is able to perform spatial self-interference suppression [2–5]. The FD relaying looks to be an alternative solution to satisfy the high capacity demands of these wireless systems. The FD communication has attracted considerable attention of many researchers and it is expected to be integrated into the 5G wireless systems. Most used cooperative protocols are the amplify-and-forward (AF) and the decode-and-forward (DF); the DF cooperative protocols can operate in either a half-duplex (HD) or full-duplex (FD) mode. Multiplexing loss occurred when implementing the HD mode in the DF protocol; this is attributed to the fact that, in the first time/frequency slot, the relay has to wait for the source’s message and then forward the message in the next time slot to the destination. However, FD cooperative protocols can overcome the HD spectral loss via simultaneous transmission from source to relay and from relay to destination; this enables frequency reuse, higher throughput, and lower transmission delay [6]. However, there are two main issues hardening the implementation of FD system in cellular network: (1) the self-interference, that is, the relay’s signal leakage from its transmit and receive antennas, and (2) the fact that the simultaneous transmission creates intercell interference [7].However, cellular networks are usually modeled by placing the sources on a grid (with a regular shape) or arranging them on a line or circle as in the Wyner model with the relay and destination being either deterministically or randomly placed across the network to determine the signal-to-interference-and-noise ratio (SINR). The resulting SINR is complex and depends on multiple random variables. Hence, this fails to account for the randomness in the cellular network distribution [8] and the intercell interference [9]. Such models, however, are highly idealized and not tractable; hence, complex system level simulation is used to evaluate the outage/success probability and ergodic capacity. In order to reduce the dependence on simulations, the closed-form SINR was derived using stochastic geometry [10].Since characterizing the SINR by the grid model and Wyner model is obviously not practicable, recently stochastic geometry has emerged as a powerful tool to model and quantify the capacity, interference, and success probability in cellular networks that are verified to be approximate to the actual networks [11]. The use of the Poisson point processes (PPP) model simplified the analysis and provided insight into the operation of the network in the form of scaling laws. Base station, relay, and destination parameters (e.g., path-loss exponent and transmit power) become the sign of the node in the PPP. Recently, under homogeneity condition, it was shown that the source positions are agnostic to the radio propagation; this makes the received power at the relay from any population of sources as if it generated from PPP, distributed sources [12, 13]. This justifies the modeling assumption of PPP sources and allows computing some metrics performance such as the success probability and Ergodic capacity [14, 15], while [16] derived an upper bound for the success probability. Obtaining full diversity order using distributed space-time codes is detailed in [17, 18], but a distributed space-time code requires precise signaling and very tight coordination among the relays, which increases the complexity and overhead in the system. This paper is motivated by the benefit of MIMO two-hop system, such as performance benefits and reducing implementation complexity.Researchers have dealt with the interference in many different ways. For instance, an active method known as analog cancellation, to cancel the interfering signal at the receiving antenna, utilizes additional radio frequency (RF) chains; and there is another active cancellation method known as digital cancellation, where the RSI is removed in the base-band level after the analog-to-digital converter [19]; another simple passive method is known as antenna separation, where the RSI is attenuated due to the path-loss between the transmitting and receiving antennas on the FD node. Further, reducing the FD interference using directional antennas is analyzed in [20]. References [21, 22] quantify the impact of self-interference of a heterogeneous network consisting of FD and HD nodes. They conclude that the capacity can be maximized by operating all nodes in either HD or FD compared to their mixtures, whereas [23] considers a single-cell setting and [22, 24] consider multicell setting. A cellular system comprising an FD source and HD destination has been illustrated in which the throughput gain is analyzed via extensive simulation [25]. In [26], the numbers of base station antennas and users antennas are increasing with a fixed ratio, while the capacity grows by the number of users and SNR. In this paper, however, we use a stochastic geometry tool to characterize the randomly distributed performance of FD MIMO relay nodes and derive bounds for the probability of successful transmission and ergodic capacity with/without interference cancellation. Finally, numerical results corroborate the theoretical findings with baseline scheme.Next, in Section2, the network system model is described. The SINR characterization is derived in Section 2.3; the success probability and the ergodic capacity are given in Section 2.4 and Section 2.5, respectively. Section 4 depicts the numerical results, and Section 5 offers the conclusion.Throughout this paper, boldface lowercase letters (e.g.,x) represent vectors, and boldface uppercase letters (e.g., X) represent matrices. ·F is the Frobenius norm. ·H stands for conjugate transpose, and E[·] denotes the expectation operator, while • denotes the magnitude and the trace of a matrix is denoted by tr⁡{·}; det(·) is the determinant; diag(x1,…,xn) is the diagonal matrix with diagonal components x1,…,xn; IN×N is the N-by-N identity matrix; XM×N is the M-by-N zero matrix.


## 2. System Model

### 2.1. System Setup

In a downlink relay-assisted cellular network, consider a multiple independent FD dual-hop relaying system. Thejth source Sj with message xSj equipped with Ms antennas communicates with Nd destinations Dj receiving antennas through relay set Rj, with messages xRj, equipped with Mr and Nr transmitting and receiving antennas, respectively, as in Figure 1. The relay uses DF protocol; hence, Rj decodes the message from Sj and then reencodes the message before sending it to Dj. The relays are equipped with FD capabilities, where the reception and transmission of relay signal happen simultaneously. The main challenge is that relay’s transmitted signal is coupled with its receiver chain, causing relay self-interference (RSI) via HRj channel and source causes interference to other relay known as source-relay interference (SRI) via HSiRj channel in addition to the interference from the sources to the destinations causing source-destination interference (SDI) via HSiDj. Space division multiplexing is applied so that the two hops are separated.Figure 1
System model depicting cellular network, in which the source (S) is having Ms transmitting antennas, willing to communicate with Nd receiving antennas, at the destination (D), with the help of relay (R), equipped with Mr and Nr transmitting and receiving antennas, respectively. The solid lines show desired signal, and the dashed lines show possible interferences such as RSI, IRI, SDI, and SRI.The locations of sources, relays, and destinations are assumed to be stationary independently marked PPP with intensityλS, λR, and λD on R2×R2×R2, respectively. In order to investigate the performance of a random access wireless network, consider a classical receiver located at the origin. Conditioning on the node’s event lying at the origin does not affect the statistics of the rest of the process as an outcome from Palm probabilities of a Poisson process. Furthermore, the statistics of received signal at the intended receiver are observable by any receiver due to the stationarity of Poisson process [10].


### 2.2. Channel Model

The downlink cellular network is comprised of sourcesSi and Sj, relays Ri andRj, and destinations Di andDj. Previous studies on cellular networks assumed that the sources and relays are positioned regularly as grid model. However, in practice, this is not true and there are some random characteristics. We applied homogeneous PPP to the spatial distribution of the sources and relays to remedy the model as in [11, 15]. Each destination is served by the nearest relay or destination. This means that the cell area of each source or relay constitutes a Voronoi tessellation [27] as in Figure 2, whereas, in grid model in Figure 3, the sources are located on the centers of hexagonal model, while the relays and destinations are distributed in each cell uniformly. This model does not lead to a tractable model; thus, it is evaluated via Monte Carlo simulations. When a destination attempts to access the sources, it chooses to connect directly or through relay in the Voronoi cell. Let the source Sj transmit the message xSj to the jth relay Rj; Rj receives and transmits simultaneously on the same frequency causing RSI through the channel HRj if Rj∈Φ for the same jth relay Rj and receives IRI through the channel HRiRj if RiRj∈Φ from the other ith relay Ri, in addition to interference from other source Si. xSj∈CMs×1 and xRj∈CMr×1 are transmitted signals from Sj and Rj nodes respectively.Figure 2
The cellular network topology, in which each Voronoi cell is the coverage area of a source and relay distributed as PPP; the cell area of each source and relay forms a Voronoi tessellation, where the red circle represents source, blue dot represents relay, and the black circle represents the destination.Figure 3
The grid cellular network topology, in which each Voronoi cell is the coverage area of a source distributed as PPP, where the red circle represents sources, blue dot represents relay, and the black circle represents the destination.The propagation through two-hop wireless channel is subject to path-loss attenuation modeled asR~-α and R^-α∀x∈Φ for distances R~ and R^j, respectively, with path-loss exponent α > 2. The sources and relays transmit with power ρ~ and power ρ^. For such channel, the sources cover large areas, while relays cover small areas with separation distances R~j and R^j(R~i≫R^j) given ρ~HSjRjR~j-α if SjRj∈Φ and ρ^HRjDjR^j-α if RjDj∈Φ. The interfering relays constitute the marked process Φ={(Xi,HRiRj,HRj)}, with Xi denoting the location of the transmitting relay node. Hence, the relay receives interference power from itself, ρ^HRj, and from the ith interfering relay and source as ρ^HRiRjXi-α and ρ~HSiRjXi-α, respectively. The channels HSjRj∈CNr×Ms, HRj∈CNr×Mr, HRiRj∈CNr×Mr, HRjDj∈CNd×Mr, and HSjDi∈CMs×Nd are the channel gain matrices from source-to-relay, relay itself, interrelay, relay-to-destination, and source-to-destination, respectively. The power constraints on transmit signals are E{xSj2}=1, E{xRj2}=1, and E{xRjRj2}=1. yRj∈CNr×1 and yDj∈CNd×1 are received signals at Rj and Dj nodes. All channels experience Rayleigh fading with parameter μ~Exp⁡(μ). The entries of each matrix are independent and identically distributed (i.i.d.) complex Gaussian variables with zero mean and variance σ2. nR and nD are independent circularly symmetric complex Gaussian noise vectors with probability distributions N(0,I) and N(0,ID), respectively, and are uncorrelated to xS and xR.


### 2.3. SINR Characterization

In this section, we formulate the signal-to-interference-plus-noise ratio (SINR), which is needed in the next section to analyze the probability of successful transmission. In doing so, we study the SINR of the SR hop and the SINR of the RD hop.


#### 2.3.1. SR Hop

Consider thejth source node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem [10, Ch. 8.5], we hence express the statistics of relay-received signal l(Rj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth relay node contains the desired signal plus interference and noise ratio, RSI, and SRI, as given as follows:(1)yRj≜ρ~R~j-α/2HSjRjxSj+nRj︸desired  signal  plus  noise+ρ^HRjx^Sj︸RSI+ρ~R~j-α/2HSiRjx^Si︸SRI.The resulting SINR at the relay is obtained as(2)SINRRj≜ρ~R~j-αHSjRj2IRj+σRj2,where IRj is the overall interference (RSI and SRI) at jth relay; that is,(3)IRj≜∑i,j∈Φρ~R~j-αHSiRj2+ρ^HRj2.For simplicity, we assume that this hop is interference-limited and, therefore, IRj≫σRj2.


#### 2.3.2. RD Hop

Consider thejth relay node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem, we express l(Dj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth destination node contains the desired signal plus interference and noise ratio, as shown as follows:(4)yDj≜ρ^R^j-α/2HRjDjx^Sj+nDj︸desired  signal  plus  noise+ρ~R~i-α/2HSiDjx^Dj︸SDIThe resulting SINR at the destination is(5)SINRDj≜ρ^R^j-αHRjDj2IDj+σDj2,where IDj captures the interference from other source to destination for the RD hop:(6)IDj≜∑i,j∈Φρ~R~j-αHSiDj2.For simplicity, we assume that this hop also is interference-limited and therefore IDj≫σDj2. Furthermore, the mean link distance of ISiRj and IRiRj is E[ISiRj]=ρ~R~-α and E[IRiRj]=ρ^R-j-α, respectively, such that EISiRj⋙E[IRiRj].


### 2.4. Success Probability

One of our performance measures is the success probability, which, defined as the success probability of message transmission from the source to destination over the two hops, is given by a joint complementary cumulative distribution function (CCDF) ofSINRRj and SINRDj. We formulate it by Psuc≜P(SINRRj≥θR,SINRDj≥θD), where θ is the given SINR threshold. Due to independent sampling of a point process, there is no correlation between the two hops.(7)Psuc=PsucSRPsucRD,where Psuc(SR)≜P(SINRRj≥θR) and Psuc(RD)≜P(SINRDj≥θD).


#### 2.4.1. SR Hop

Successful SR hop transmission occurs if(8)ρ~R~j-αHSjRj2IRj≥θR;the given threshold θ satisfied a target SINR; the probability of successful transmission for a relay is(9)PSINRRj≥θ=Pρ~R~j-αHSjRj2IRj≥θR=PHSjRj2≥θRρ~-1R~jαIRj=∫0∞PHSjRj2≥sθRρ~-1R~jαfIRJsds=∫0∞FSRcsθρ~-1R~jαfIRJsds,where (9) is obtained by conditioning s and its complementary cumulative distribution function is denoted by Fc(·). Next, we provide the success probability analysis of the SR hop.Theorem 1.
Let the interfering relays transmitters form a Poisson process of intensityλR around relay receivers. The success probability of the SR hop is(10)PsucSR≜∑n=0Nr-1-snn!dndsnLIRjss=θRρ~-1R~jα,where LIRj is the Laplace transform of the interferences IRj.(11)LIRjs≜11+sρ~exp⁡-λSΥs(12)Υs≜∫0∞2π-11+sρ~r-αΨs,rrdrwith(13)Ψs,r≜∫02πdφ1+sρ~R~j2+r2+2R~jrcos⁡φ-α/2.Proof.
Refer to AppendixB.The result in Theorem1 provides a fundamental limit on the SR hop and its performance in an interference-limited scenario.


#### 2.4.2. RD Hop

Successful RD hop transmission occurs if(14)PSINRDj≥θD=Pρ^R^j-αHRjDj2IDj≥θD=PHRjDj2≥θDρ^-1R^jαIDj.Next, we provide the success probability analysis of the RD hop.Theorem 2.
Let the relays transmission form a Poisson process of intensityλD around destinations. The success probability of the RD hop is(15)PsucRD≜∑n=0Nd-1-snn!dndsnLIDjss=θDρ^-1R^jα,where LIDj is the Laplace transform of the interference IDj:(16)LIDjs≜Ψs,R^jexp⁡-λDΥs,where Ψ(s,r) and Υ(s) are already defined in (13) and (12), respectively.Proof.
Refer to AppendixC.Corollary 3.
The Laplace transform in (16) is bounded by LIDj(s)∈[LIDjmin(s),LIDjmax(s)], where(17)LIDjmins≜11+sρ^R~j-R^j-1exp⁡-λΥmaxs,LIDjmaxs≜11+sρ^R~j+R^j-1exp⁡-λΥmins,with Υ(min)(s) and Υ(max)(s) already being given in [21, 28] as follows:(18)Υmaxs≜2ρ~+ρ^π2s2/ααsin⁡2π/α,Υmins≜1+2αρ^+ρ~π2s2/ααsin⁡2π/α.This corollary will help us in proving the interference bound of success probabilityPsucRD of RD hop; this can be done by substituting LIDj(s)∈[LIDjmin(s),LIDjmax] in (15).


### 2.5. Ergodic Capacity

The other performance measure is the ergodic capacity; the goal of FD relay is to increase the network capacity; for the relay served by source, the following theorems give the outage probability (CDF of SINR) and the SR hop capacity.Theorem 4.
The probability thatSINRRj falls below a given target level θR is known as outage probability, which is the CDF of SINRRj of the SR hop denoted by Zj(θR)=P{SINRRj≤θR} and given by(19)ZRjθR=1-exp⁡-2πλS,and similarly for the RD hop(20)ZRjθD=1-exp⁡-2πλR.Hence, the SR hop capacity can be shown as(21)CSjRj≜∑j=1Nrlog⁡1+θRPsucSRand the RD hop capacity is(22)CRjDj≜∑j=1Ndlog⁡1+θDPsucRD.The total channel capacity is denoted by(23)C=min⁡CSjRj,CRjDj.Proof.
Refer to AppendixA.For the single-antenna case, the following corollary shows that the FD relay outperforms the HD relay in terms of capacity.Corollary 5.
Consider the SR hop where relay and destination nodes are having only single antennaNr=Nd=1. Letting s=θRρ~-1R~α, the FD relay capacity is lower-bounded by(24)CSjRj≜21+sρ~exp⁡-λSΥmaxs∑j=1Nrlog⁡1+θR;on the other hand, the HD relay capacity is upper-bounded by(25)CSjRjHD≜exp⁡-λSρ~2π2s2/ααsin⁡2π/α∑j=1Nrlog⁡1+θR;therefore, CSjRj≥CSjRjHD as long as the following condition holds:(26)λS≤αsin⁡2π/α2π2s2/αlog2⁡21+s.Proof.
The proof is obtained by combining Corollary3 and Theorem 2.It is evident that increasing the densityλS beyond the threshold does not compensate for the additional RSI, even if we use twice the FD capacity; this is due to FD simultaneous transmission and reception. Hence, the HD relay is optimal in this case.


### 2.6. Minimum Throughput Gain

We demonstrate the feasibility of FD relay over its counterpart, the HD relay, for the SR hop capacity only, but similar approach can be applied to the RD hop as well since the total capacity follows (23). We define a performance metric called the minimum throughput gain, which shows the worst case FD relay performance over HD relay in terms of capacity as follows:(27)TGmin≜CSjRjCSjRjHD.when TG>1, the FD relay outperforms the HD relay.


## 2.1. System Setup

In a downlink relay-assisted cellular network, consider a multiple independent FD dual-hop relaying system. Thejth source Sj with message xSj equipped with Ms antennas communicates with Nd destinations Dj receiving antennas through relay set Rj, with messages xRj, equipped with Mr and Nr transmitting and receiving antennas, respectively, as in Figure 1. The relay uses DF protocol; hence, Rj decodes the message from Sj and then reencodes the message before sending it to Dj. The relays are equipped with FD capabilities, where the reception and transmission of relay signal happen simultaneously. The main challenge is that relay’s transmitted signal is coupled with its receiver chain, causing relay self-interference (RSI) via HRj channel and source causes interference to other relay known as source-relay interference (SRI) via HSiRj channel in addition to the interference from the sources to the destinations causing source-destination interference (SDI) via HSiDj. Space division multiplexing is applied so that the two hops are separated.Figure 1
System model depicting cellular network, in which the source (S) is having Ms transmitting antennas, willing to communicate with Nd receiving antennas, at the destination (D), with the help of relay (R), equipped with Mr and Nr transmitting and receiving antennas, respectively. The solid lines show desired signal, and the dashed lines show possible interferences such as RSI, IRI, SDI, and SRI.The locations of sources, relays, and destinations are assumed to be stationary independently marked PPP with intensityλS, λR, and λD on R2×R2×R2, respectively. In order to investigate the performance of a random access wireless network, consider a classical receiver located at the origin. Conditioning on the node’s event lying at the origin does not affect the statistics of the rest of the process as an outcome from Palm probabilities of a Poisson process. Furthermore, the statistics of received signal at the intended receiver are observable by any receiver due to the stationarity of Poisson process [10].


## 2.2. Channel Model

The downlink cellular network is comprised of sourcesSi and Sj, relays Ri andRj, and destinations Di andDj. Previous studies on cellular networks assumed that the sources and relays are positioned regularly as grid model. However, in practice, this is not true and there are some random characteristics. We applied homogeneous PPP to the spatial distribution of the sources and relays to remedy the model as in [11, 15]. Each destination is served by the nearest relay or destination. This means that the cell area of each source or relay constitutes a Voronoi tessellation [27] as in Figure 2, whereas, in grid model in Figure 3, the sources are located on the centers of hexagonal model, while the relays and destinations are distributed in each cell uniformly. This model does not lead to a tractable model; thus, it is evaluated via Monte Carlo simulations. When a destination attempts to access the sources, it chooses to connect directly or through relay in the Voronoi cell. Let the source Sj transmit the message xSj to the jth relay Rj; Rj receives and transmits simultaneously on the same frequency causing RSI through the channel HRj if Rj∈Φ for the same jth relay Rj and receives IRI through the channel HRiRj if RiRj∈Φ from the other ith relay Ri, in addition to interference from other source Si. xSj∈CMs×1 and xRj∈CMr×1 are transmitted signals from Sj and Rj nodes respectively.Figure 2
The cellular network topology, in which each Voronoi cell is the coverage area of a source and relay distributed as PPP; the cell area of each source and relay forms a Voronoi tessellation, where the red circle represents source, blue dot represents relay, and the black circle represents the destination.Figure 3
The grid cellular network topology, in which each Voronoi cell is the coverage area of a source distributed as PPP, where the red circle represents sources, blue dot represents relay, and the black circle represents the destination.The propagation through two-hop wireless channel is subject to path-loss attenuation modeled asR~-α and R^-α∀x∈Φ for distances R~ and R^j, respectively, with path-loss exponent α > 2. The sources and relays transmit with power ρ~ and power ρ^. For such channel, the sources cover large areas, while relays cover small areas with separation distances R~j and R^j(R~i≫R^j) given ρ~HSjRjR~j-α if SjRj∈Φ and ρ^HRjDjR^j-α if RjDj∈Φ. The interfering relays constitute the marked process Φ={(Xi,HRiRj,HRj)}, with Xi denoting the location of the transmitting relay node. Hence, the relay receives interference power from itself, ρ^HRj, and from the ith interfering relay and source as ρ^HRiRjXi-α and ρ~HSiRjXi-α, respectively. The channels HSjRj∈CNr×Ms, HRj∈CNr×Mr, HRiRj∈CNr×Mr, HRjDj∈CNd×Mr, and HSjDi∈CMs×Nd are the channel gain matrices from source-to-relay, relay itself, interrelay, relay-to-destination, and source-to-destination, respectively. The power constraints on transmit signals are E{xSj2}=1, E{xRj2}=1, and E{xRjRj2}=1. yRj∈CNr×1 and yDj∈CNd×1 are received signals at Rj and Dj nodes. All channels experience Rayleigh fading with parameter μ~Exp⁡(μ). The entries of each matrix are independent and identically distributed (i.i.d.) complex Gaussian variables with zero mean and variance σ2. nR and nD are independent circularly symmetric complex Gaussian noise vectors with probability distributions N(0,I) and N(0,ID), respectively, and are uncorrelated to xS and xR.


## 2.3. SINR Characterization

In this section, we formulate the signal-to-interference-plus-noise ratio (SINR), which is needed in the next section to analyze the probability of successful transmission. In doing so, we study the SINR of the SR hop and the SINR of the RD hop.


### 2.3.1. SR Hop

Consider thejth source node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem [10, Ch. 8.5], we hence express the statistics of relay-received signal l(Rj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth relay node contains the desired signal plus interference and noise ratio, RSI, and SRI, as given as follows:(1)yRj≜ρ~R~j-α/2HSjRjxSj+nRj︸desired  signal  plus  noise+ρ^HRjx^Sj︸RSI+ρ~R~j-α/2HSiRjx^Si︸SRI.The resulting SINR at the relay is obtained as(2)SINRRj≜ρ~R~j-αHSjRj2IRj+σRj2,where IRj is the overall interference (RSI and SRI) at jth relay; that is,(3)IRj≜∑i,j∈Φρ~R~j-αHSiRj2+ρ^HRj2.For simplicity, we assume that this hop is interference-limited and, therefore, IRj≫σRj2.


### 2.3.2. RD Hop

Consider thejth relay node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem, we express l(Dj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth destination node contains the desired signal plus interference and noise ratio, as shown as follows:(4)yDj≜ρ^R^j-α/2HRjDjx^Sj+nDj︸desired  signal  plus  noise+ρ~R~i-α/2HSiDjx^Dj︸SDIThe resulting SINR at the destination is(5)SINRDj≜ρ^R^j-αHRjDj2IDj+σDj2,where IDj captures the interference from other source to destination for the RD hop:(6)IDj≜∑i,j∈Φρ~R~j-αHSiDj2.For simplicity, we assume that this hop also is interference-limited and therefore IDj≫σDj2. Furthermore, the mean link distance of ISiRj and IRiRj is E[ISiRj]=ρ~R~-α and E[IRiRj]=ρ^R-j-α, respectively, such that EISiRj⋙E[IRiRj].


## 2.3.1. SR Hop

Consider thejth source node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem [10, Ch. 8.5], we hence express the statistics of relay-received signal l(Rj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth relay node contains the desired signal plus interference and noise ratio, RSI, and SRI, as given as follows:(1)yRj≜ρ~R~j-α/2HSjRjxSj+nRj︸desired  signal  plus  noise+ρ^HRjx^Sj︸RSI+ρ~R~j-α/2HSiRjx^Si︸SRI.The resulting SINR at the relay is obtained as(2)SINRRj≜ρ~R~j-αHSjRj2IRj+σRj2,where IRj is the overall interference (RSI and SRI) at jth relay; that is,(3)IRj≜∑i,j∈Φρ~R~j-αHSiRj2+ρ^HRj2.For simplicity, we assume that this hop is interference-limited and, therefore, IRj≫σRj2.


## 2.3.2. RD Hop

Consider thejth relay node Sj located at the origin and, according to the stationarity of Φ and building on Slivnyak’s theorem, we express l(Dj,x)=Xx-α, with Xx being the distance of x from the origin. Therefore the received signal at the jth destination node contains the desired signal plus interference and noise ratio, as shown as follows:(4)yDj≜ρ^R^j-α/2HRjDjx^Sj+nDj︸desired  signal  plus  noise+ρ~R~i-α/2HSiDjx^Dj︸SDIThe resulting SINR at the destination is(5)SINRDj≜ρ^R^j-αHRjDj2IDj+σDj2,where IDj captures the interference from other source to destination for the RD hop:(6)IDj≜∑i,j∈Φρ~R~j-αHSiDj2.For simplicity, we assume that this hop also is interference-limited and therefore IDj≫σDj2. Furthermore, the mean link distance of ISiRj and IRiRj is E[ISiRj]=ρ~R~-α and E[IRiRj]=ρ^R-j-α, respectively, such that EISiRj⋙E[IRiRj].


## 2.4. Success Probability

One of our performance measures is the success probability, which, defined as the success probability of message transmission from the source to destination over the two hops, is given by a joint complementary cumulative distribution function (CCDF) ofSINRRj and SINRDj. We formulate it by Psuc≜P(SINRRj≥θR,SINRDj≥θD), where θ is the given SINR threshold. Due to independent sampling of a point process, there is no correlation between the two hops.(7)Psuc=PsucSRPsucRD,where Psuc(SR)≜P(SINRRj≥θR) and Psuc(RD)≜P(SINRDj≥θD).


### 2.4.1. SR Hop

Successful SR hop transmission occurs if(8)ρ~R~j-αHSjRj2IRj≥θR;the given threshold θ satisfied a target SINR; the probability of successful transmission for a relay is(9)PSINRRj≥θ=Pρ~R~j-αHSjRj2IRj≥θR=PHSjRj2≥θRρ~-1R~jαIRj=∫0∞PHSjRj2≥sθRρ~-1R~jαfIRJsds=∫0∞FSRcsθρ~-1R~jαfIRJsds,where (9) is obtained by conditioning s and its complementary cumulative distribution function is denoted by Fc(·). Next, we provide the success probability analysis of the SR hop.Theorem 1.
Let the interfering relays transmitters form a Poisson process of intensityλR around relay receivers. The success probability of the SR hop is(10)PsucSR≜∑n=0Nr-1-snn!dndsnLIRjss=θRρ~-1R~jα,where LIRj is the Laplace transform of the interferences IRj.(11)LIRjs≜11+sρ~exp⁡-λSΥs(12)Υs≜∫0∞2π-11+sρ~r-αΨs,rrdrwith(13)Ψs,r≜∫02πdφ1+sρ~R~j2+r2+2R~jrcos⁡φ-α/2.Proof.
Refer to AppendixB.The result in Theorem1 provides a fundamental limit on the SR hop and its performance in an interference-limited scenario.


### 2.4.2. RD Hop

Successful RD hop transmission occurs if(14)PSINRDj≥θD=Pρ^R^j-αHRjDj2IDj≥θD=PHRjDj2≥θDρ^-1R^jαIDj.Next, we provide the success probability analysis of the RD hop.Theorem 2.
Let the relays transmission form a Poisson process of intensityλD around destinations. The success probability of the RD hop is(15)PsucRD≜∑n=0Nd-1-snn!dndsnLIDjss=θDρ^-1R^jα,where LIDj is the Laplace transform of the interference IDj:(16)LIDjs≜Ψs,R^jexp⁡-λDΥs,where Ψ(s,r) and Υ(s) are already defined in (13) and (12), respectively.Proof.
Refer to AppendixC.Corollary 3.
The Laplace transform in (16) is bounded by LIDj(s)∈[LIDjmin(s),LIDjmax(s)], where(17)LIDjmins≜11+sρ^R~j-R^j-1exp⁡-λΥmaxs,LIDjmaxs≜11+sρ^R~j+R^j-1exp⁡-λΥmins,with Υ(min)(s) and Υ(max)(s) already being given in [21, 28] as follows:(18)Υmaxs≜2ρ~+ρ^π2s2/ααsin⁡2π/α,Υmins≜1+2αρ^+ρ~π2s2/ααsin⁡2π/α.This corollary will help us in proving the interference bound of success probabilityPsucRD of RD hop; this can be done by substituting LIDj(s)∈[LIDjmin(s),LIDjmax] in (15).


## 2.4.1. SR Hop

Successful SR hop transmission occurs if(8)ρ~R~j-αHSjRj2IRj≥θR;the given threshold θ satisfied a target SINR; the probability of successful transmission for a relay is(9)PSINRRj≥θ=Pρ~R~j-αHSjRj2IRj≥θR=PHSjRj2≥θRρ~-1R~jαIRj=∫0∞PHSjRj2≥sθRρ~-1R~jαfIRJsds=∫0∞FSRcsθρ~-1R~jαfIRJsds,where (9) is obtained by conditioning s and its complementary cumulative distribution function is denoted by Fc(·). Next, we provide the success probability analysis of the SR hop.Theorem 1.
Let the interfering relays transmitters form a Poisson process of intensityλR around relay receivers. The success probability of the SR hop is(10)PsucSR≜∑n=0Nr-1-snn!dndsnLIRjss=θRρ~-1R~jα,where LIRj is the Laplace transform of the interferences IRj.(11)LIRjs≜11+sρ~exp⁡-λSΥs(12)Υs≜∫0∞2π-11+sρ~r-αΨs,rrdrwith(13)Ψs,r≜∫02πdφ1+sρ~R~j2+r2+2R~jrcos⁡φ-α/2.Proof.
Refer to AppendixB.The result in Theorem1 provides a fundamental limit on the SR hop and its performance in an interference-limited scenario.


## 2.4.2. RD Hop

Successful RD hop transmission occurs if(14)PSINRDj≥θD=Pρ^R^j-αHRjDj2IDj≥θD=PHRjDj2≥θDρ^-1R^jαIDj.Next, we provide the success probability analysis of the RD hop.Theorem 2.
Let the relays transmission form a Poisson process of intensityλD around destinations. The success probability of the RD hop is(15)PsucRD≜∑n=0Nd-1-snn!dndsnLIDjss=θDρ^-1R^jα,where LIDj is the Laplace transform of the interference IDj:(16)LIDjs≜Ψs,R^jexp⁡-λDΥs,where Ψ(s,r) and Υ(s) are already defined in (13) and (12), respectively.Proof.
Refer to AppendixC.Corollary 3.
The Laplace transform in (16) is bounded by LIDj(s)∈[LIDjmin(s),LIDjmax(s)], where(17)LIDjmins≜11+sρ^R~j-R^j-1exp⁡-λΥmaxs,LIDjmaxs≜11+sρ^R~j+R^j-1exp⁡-λΥmins,with Υ(min)(s) and Υ(max)(s) already being given in [21, 28] as follows:(18)Υmaxs≜2ρ~+ρ^π2s2/ααsin⁡2π/α,Υmins≜1+2αρ^+ρ~π2s2/ααsin⁡2π/α.This corollary will help us in proving the interference bound of success probabilityPsucRD of RD hop; this can be done by substituting LIDj(s)∈[LIDjmin(s),LIDjmax] in (15).


## 2.5. Ergodic Capacity

The other performance measure is the ergodic capacity; the goal of FD relay is to increase the network capacity; for the relay served by source, the following theorems give the outage probability (CDF of SINR) and the SR hop capacity.Theorem 4.
The probability thatSINRRj falls below a given target level θR is known as outage probability, which is the CDF of SINRRj of the SR hop denoted by Zj(θR)=P{SINRRj≤θR} and given by(19)ZRjθR=1-exp⁡-2πλS,and similarly for the RD hop(20)ZRjθD=1-exp⁡-2πλR.Hence, the SR hop capacity can be shown as(21)CSjRj≜∑j=1Nrlog⁡1+θRPsucSRand the RD hop capacity is(22)CRjDj≜∑j=1Ndlog⁡1+θDPsucRD.The total channel capacity is denoted by(23)C=min⁡CSjRj,CRjDj.Proof.
Refer to AppendixA.For the single-antenna case, the following corollary shows that the FD relay outperforms the HD relay in terms of capacity.Corollary 5.
Consider the SR hop where relay and destination nodes are having only single antennaNr=Nd=1. Letting s=θRρ~-1R~α, the FD relay capacity is lower-bounded by(24)CSjRj≜21+sρ~exp⁡-λSΥmaxs∑j=1Nrlog⁡1+θR;on the other hand, the HD relay capacity is upper-bounded by(25)CSjRjHD≜exp⁡-λSρ~2π2s2/ααsin⁡2π/α∑j=1Nrlog⁡1+θR;therefore, CSjRj≥CSjRjHD as long as the following condition holds:(26)λS≤αsin⁡2π/α2π2s2/αlog2⁡21+s.Proof.
The proof is obtained by combining Corollary3 and Theorem 2.It is evident that increasing the densityλS beyond the threshold does not compensate for the additional RSI, even if we use twice the FD capacity; this is due to FD simultaneous transmission and reception. Hence, the HD relay is optimal in this case.


## 2.6. Minimum Throughput Gain

We demonstrate the feasibility of FD relay over its counterpart, the HD relay, for the SR hop capacity only, but similar approach can be applied to the RD hop as well since the total capacity follows (23). We define a performance metric called the minimum throughput gain, which shows the worst case FD relay performance over HD relay in terms of capacity as follows:(27)TGmin≜CSjRjCSjRjHD.when TG>1, the FD relay outperforms the HD relay.


## 3. Interference Cancellation

This section considers the interference cancellation at the MIMO receiving nodes, that is, for SR hop at the relay and for RD hop at the destination. We present a low complexity spatial interference cancellation scheme known as partial zero forcing (PZF) [29, 30]. During the SR hop and RD hop, PZF can be applied at the relay and destination to cancel K≤Nr-1 and L≤Nd-1 interferences, respectively, while using the remaining degree of freedom to boost the intended received signal.


### 3.1. SR Hop: Canceling RSI and SRI

It is beneficial to increase the capacity by canceling the interference (RSI and SRI) from nearby nodes in an interference-limited scenario. Let us denote the points of{i,j}∈Φ, in an increasing order from the source, that is, R~i≤R~i+1i=1∞; the relay applies PZF to cancel its K interference. From (3), the overall interference can be rewritten as(28)IRjK≜∑j∈Φj>Kρ~R~j-αHSiDj2+ρ^HRj2.For the SR hop, with PZF at the relay, the success probability is given next.Theorem 6.
The success probability of the SR hop with PZF at the relay is(29)PsucSR≜∑n=0Nr-K-1-snn!dndsnLIRjkss=θRρ~-1R~jα,where LIRj(K)(s) is the Laplace transform of the interferences:(30)LIRjKs≜11+sρ~EΦ∏j∈Φj≤K11+sρ~r-α.Proof.
See AppendixA and follow similar steps to those in [30].


### 3.2. RD Hop: Canceling SDI

It is beneficial to increase the capacity by canceling the interference (SDI) from nearby nodes in an interference limited scenario. Let us denote the points of{i,j}∈Φ, in an increasing order from the source, that is, R^i≤R^i+1i=1∞; the destination applies PZF to cancel its L interference. From (6), the overall interference can be rewritten as(31)IDjL≜∑j∈Φj>Lρ^R^j-αHSiDj2and its corresponding success probability is given below.Theorem 7.
The success probability of the RD hop with PZF at the destination is(32)PsucRD≜∑n=0Nd-L-1-snn!dndsnLIDjLss=θDρ^-1R^jα,where LIDj(L)(s) is the Laplace transform of the interferences:(33)LIDjLs≜11+sρ^EΦ∏j∈Φj≤L11+sρ^r-α.Proof.
See AppendixA and follow similar steps as those in [30].


## 3.1. SR Hop: Canceling RSI and SRI

It is beneficial to increase the capacity by canceling the interference (RSI and SRI) from nearby nodes in an interference-limited scenario. Let us denote the points of{i,j}∈Φ, in an increasing order from the source, that is, R~i≤R~i+1i=1∞; the relay applies PZF to cancel its K interference. From (3), the overall interference can be rewritten as(28)IRjK≜∑j∈Φj>Kρ~R~j-αHSiDj2+ρ^HRj2.For the SR hop, with PZF at the relay, the success probability is given next.Theorem 6.
The success probability of the SR hop with PZF at the relay is(29)PsucSR≜∑n=0Nr-K-1-snn!dndsnLIRjkss=θRρ~-1R~jα,where LIRj(K)(s) is the Laplace transform of the interferences:(30)LIRjKs≜11+sρ~EΦ∏j∈Φj≤K11+sρ~r-α.Proof.
See AppendixA and follow similar steps to those in [30].


## 3.2. RD Hop: Canceling SDI

It is beneficial to increase the capacity by canceling the interference (SDI) from nearby nodes in an interference limited scenario. Let us denote the points of{i,j}∈Φ, in an increasing order from the source, that is, R^i≤R^i+1i=1∞; the destination applies PZF to cancel its L interference. From (6), the overall interference can be rewritten as(31)IDjL≜∑j∈Φj>Lρ^R^j-αHSiDj2and its corresponding success probability is given below.Theorem 7.
The success probability of the RD hop with PZF at the destination is(32)PsucRD≜∑n=0Nd-L-1-snn!dndsnLIDjLss=θDρ^-1R^jα,where LIDj(L)(s) is the Laplace transform of the interferences:(33)LIDjLs≜11+sρ^EΦ∏j∈Φj≤L11+sρ^r-α.Proof.
See AppendixA and follow similar steps as those in [30].


## 4. Numerical Results

Several Monte Carlo simulations are presented to validate the proposed scheme and averaged over 10000 number of channel realizations. Rayleigh fading assumption results in a convenient coverage probability form in terms of the Laplace transform of the interference, which is easier to characterize than the probability density function of the interference [10]. The sources and relays are transmitting with ρ~=1 W and ρ^=0.5 W with their corresponding distances R~=5 m and R^=0.5 m, respectively. The SINR threshold is θ=0 dB and the path-loss exponent is α=2.3, with 100 destination nodes, 100 source nodes, and 100 relay nodes, with N=8 interfering source and relay nodes. We assume that, in this open-access case, destinations (users) are able to connect to any source (base station). Open access requirements often are not satisfied, particularly deploying Wi-Fi in the multi-RAT (radio access technology) femtocells network [31].Figure4 shows the ergodic capacity in (23) of Ms=Nr=2, Mr=Nd=2 and Ms=Nr=2, Mr=Nd=4 and Ms=Nr=2, Mr=Nd=8, antenna configurations. From (21) and (22) it is obvious that increasing the number of antennas employed at the relay and destination adds a positive term to the ergodic capacity; however, the total capacity is limited by the weakest hop (23). It can be observed that the ergodic capacity scales linearly with the number of antennas.Figure 4
Ergodic capacity versus SINR threshold for different relay and destination transmitted and received antennas, respectively.Figure5 shows that the success probability in (7) increases with increasing the number of transmitting antennas at the source across the whole SINR. Deploying multiple antennas at the relay allows reducing the effect of interference, hence obtaining a substantial SINR gain. Figure 6 indicates that the success probability decreases with an increase in the SINR threshold. It compares the traditional grid model constituted by a Voronoi tessellation (see Figure 3) to the random PPP source model. Considering N=8 and N=24, it can be observed that the case of N=8 is almost overlapped with N=24, for α=4. As expected, the grid model provides high coverage area across the whole SINR. A small gap has been observed when considering the SNR = 10 and SNR → ∞, due to the noise effect in dense cellular networks, which are known to be interference-limited. This validates the assumption that the noise can be ignored in interference-limited scenario.Figure 5
Success probability versus SINR threshold.Figure 6
Success probability comparison between our proposed PPP source model and grid model withN=8 and N=24 and α=4. No noise assumption is quite correct, and it can be observed that the case with N=24 is slightly lower than N=8.Figure7 presents a crucial issue in FD relay communications by evaluating the performance of multiple receive antennas in canceling the self-interference. The capacity gain is plotted against the self-interference attenuation with θ=0 dB and λ=10-3. We observed that TGmin>1, even for moderate self-interference attenuation values ≤4, ≤8, and ≤10 dBs for Nr=Mr=1, Nr=Mr=2, and Nr=Mr=4, respectively.Figure 7
Comparison between FD and HD relay in the SR hop: minimum throughput gain for different values of self-interference attenuation withλ=10-3 and θ=0 dB.For the SR hop, the relays are equipped with two receive antennasNr=2 and deploy PZF to cancel either the RSI or the SRI. Figure 8 shows the success probability of the SR hop against the density λ and compares it with the multiantenna UDN model [30] and the grid model. Note that the grid model is best suited for higher densities, whereas our proposed model performs slightly better than the multiantenna UDN model, due to the use of both antennas at source and relay for array gain to exploit the MIMO capability in boosting the intended received signal.Figure 8
Success probability of the SR hop with interference cancellation versus the densityλ with Ms=Nr=Mr=2 and SINR threshold θ=0 dB for different schemes.


## 5. Conclusion

In a MIMO relay cellular network, RSI, SRI, and SDI will likely be the main obstacle reducing the capacity. This paper, however, developed a tractable model to analyze the success probability and ergodic capacity expressions for the random nodes, using tools from stochastic geometry. It is more realistic than the grid-based models and requires an extensive comparison with real base station deployments to validate its accuracy. We showed the effect of PZF in mitigating the interferences including the self-interference to demonstrate the feasibility of FD technology even for moderate values of self-interference attenuation. Future work probably uses this model extensively due to the interference modeling from neighboring nodes.



---

*Source: 8342156-2018-01-03.xml*