# Effect of Node Mobility on MU-MIMO Transmissions in Mobile Ad Hoc Networks

Authors: Wenjian Zhang; Senlin Jiang
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

In this paper, we investigate the expected outage probability and expected throughput of a multi-user multiple-input multiple-output (MU-MIMO) transmission in mobile ad hoc networks (MANET) in the presence of co-channel interference and unpredictable inter-beam interference. In order to achieve multi-user diversity gain, the receiving nodes are required to report measured channel information to the transmitting node. During the time gap between channel measurement and data transmission, the channel may change with the location of moving nodes. The unpredictable behavior may cause a mismatch between the weight of beams and the instantaneous channel and the inter-beam interference in the data transmission phase. In order to obtain the closed form expected outage probability, we categorize the behavior of nodes according to whether the inter-beam interference exists or not and the number of received interference beams. The probability of each category and the closed form outage probability of an instantaneous MU-MIMO transmission of each category are derived. Additionally, the expected throughput of an MU-MIMO transmission which changes with the number of receiving nodes is obtained, and the optimal value of receiving nodes to maximize the expected throughput is discussed. Numeric results show the unpredictable inter-beam interference degrades the outage probability performance. Reducing the duration of the time gap could improve the expected outage probability and expected throughput.


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## Body

## 1. Introduction

The diversity of applications gives rise to the requirement on the flexibility of wireless networks. A conventional cellular network or wireless local area network (WLAN) supplies the inability to satisfy the requirements. Due to the agility network structure, MANET has attracted a lot of attention in different application areas, such as military battlefield [1], postdisaster reconstruction [2], emergency mission, and vehicular communication [3].A MANET is composed of a collection of mobile nodes equipped with a wireless transceiver. The nodes communicate with each other directly or by multihop link with the help of intermediate nodes [4]. Because of the lack of coordination of infrastructure or central node, multiple transmissions may occur on the same channel simultaneously and interfere with each other. Inevitably, the co-channel interference impairs the performance of transmissions. In prior works, the transmission performance in the presence of co-channel interference was investigated [5, 6], and some interference coordination schemes were proposed [7–9].In order to reduce co-channel interference and improve the aggregate throughput of MANET, multiple-input multiple-output (MIMO) has been exploited by many researchers [10–12]. Especially, MU-MIMO, due to its extremely high spectral efficiency, receives a significant attention [13–16]. In the 3rd Generation Partnership Project (3GPP) long term evolution-advanced (LTE-A) [17] and IEEE 802.11 ac [18], the MU-MIMO is adopted. An MU-MIMO transmission is performed in two phases: beam selection phase and data transmission phase. In the beam selection phase, the transmitting node selects a set of orthogonal beams according to the reported channel measurement information from candidate receiving nodes. In the data transmission phase, the transmitting node delivers data packets on the same channel to receiving nodes simultaneously, using the selected orthogonal beams.The orthogonal beams are proven to maximize the received SINR of a receiving node. However, achieving the performance requires the matching between the weight of a beam and instantaneous wireless channel. As shown in Figure1, a time gap for reporting channel measurement information is inevitable in the beam selection phase, whose duration is related to the number of the candidate receiving nodes. Due to the randomness of nodes’ movement, the behavior of nodes is unpredictable during the time gap. If a node moves randomly during the time gap, the wireless channel between the moving node and the transmitting node or a receiving node changes with the location of the moving node. In the data transmission phase, the instantaneous wireless channel mismatches with the weight of beams derived from measured channel information. As a result, the beams for each receiving node may interfere with each other. The unpredicted inter-beam interference will degrade the performance of an MU-MIMO transmission. In previous work [13], the influence of the inter-beam interference on the outage probability performance was exploited, but the randomness of interference was not taken into account.Figure 1
MU-MIMO beamforming model.The expected throughput of an MU-MIMO transmission is the sum of the expected throughput of each receiving node. Intuitively, the increase of receiving nodes could bring more expected throughput of an MU-MIMO transmission. However, with the increase of receiving nodes, the probability of that multiple nodes keep motionless simultaneous during the time gap decreases, but the probability of inter-beam interference increases. Moreover, the increase of receiving nodes reduces the power of the desired signal and potentially increases the power of inter-beam interference. For a receiving node, the instantaneous received SINR and the expected throughput are reduced. Therefore, the expected throughput of an MU-MIMO transmission may not be improved with the increase of receiving nodes. How to balance the expected throughput and the number of receiving nodes is worth discussing.Motivated by the discussion above, in the paper, we investigate the outage probability performance and expected throughput of an MU-MIMO transmission in MANET in the presence of co-channel interference and unpredictable inter-beam interference. Summarily, our main contributions are given as follows:(i)
We categorize the mobility behavior of nodes according to whether the inter-beam interference exists or not and the number of received interference beams. The probability of each category is derived.(ii)
We derive the closed form outage probability of an instantaneous MU-MIMO transmission considering the co-channel interference and unpredictable inter-beam interference.(iii)
We explore the expected throughput of an MU-MIMO transmission and discuss the optimal number of receiving nodes to maximize the expected throughput.The remainder of the paper is structured as follows. Section2 gives the system model. Section 3 derives the instantaneous and expected outage probability of an MU-MIMO transmission in the presence of co-channel interference and unpredictable inter-beam interference. Section 4 explores the expected throughput of an MU-MIMO transmission and discusses the optimal number of receiving nodes to maximize the expected throughput. Section 5 describes the numeric results. Finally, Section 6 gives the conclusions.


## 2. System Model

We assume a MANET consisting of a collection of half-duplex nodes adopts slotted ALOHA protocol [19] and MU-MIMO transmissions. In each slot, a subset of nodes transmits data on the same channel, with the same transmit power Pt. Considering the nodes in MANET are placed arbitrarily, the transmitting nodes in each slot are assumed to be distributed according to a Spatial Poisson Point Process (SPPP) with intensity λ [20]. The model has been used and the validity has been confirmed in prior works. Hence, the set of transmitting nodes is denoted as Φλ=Zi, where Zi is the position of transmitting node i at some time instant.In the beam selection phase of an MU-MIMO transmission, the transmitting node equipped withNt transmit antennas selects K1<K≤Nt orthogonal beams using reported channel information. In the data transmission phase, the transmitting node delivers data packets to K receiving nodes equipped with a single receive antenna simultaneously, using orthogonal beams wkk=1,2,⋯,K, where wk satisfies wkHwk=1 and wkHwl=0k≠l.We assume the duration of the time gap is Tgap, and the wireless channel does not change with time within the time gap. At the start time of the time gap, the nodes in an MU-MIMO system are assumed to be motionless with a probability pstate. After a time interval t, each node changes to the opposite state independently, i.e., the motionless node becomes moving or vice versa. We assume the number of state change in a period of time obeys a Poisson distribution. Hence, the time interval t follows an exponential distribution with mean 1/θ, i.e.,(1)ft=θe−θt,t>0.Further, we assume a node moves in a very short time could result in an obvious variation of location, and the instantaneous wireless channel changes with the location.


## 3. Outage Probability Evaluation

We assume a receiving nodek locates at the origin. In the data transmission phase, the received signal of the receiving node k is denoted as(2)yk=PtKDk−αhkHwksk⏟DesiredSignal+∑j∈K,j≠kPtKDk−αhkHwjsj⏟Inter‐beamInterferenceSignal+∑φ∈Φ/TkPtZφ−αgφ,ksφ⏟Co‐channelInterferenceSignal+nk,where hk∈ℂNt×1 is the channel gain vector between the receiving node k and the transmitting node Tk, whose entries follow complex Gaussian distribution hi,k∼CN0,1. Similarly, gφ,k∈ℂ1×1 represents the channel gain between the receiving node k and another transmitting node φ. sk∈ℂK×1 is the symbol vector transmitted from the transmitting node. Additionally, nk represents the additive white Gaussian noise (AWGN) with zero mean and variance of σ2.For the receiving nodek, whether the inter-beam interference signal exists or not depends on whether the wireless channels change with the location of nodes during the time gap or not. In case all nodes keep motionless during the time gap, the orthogonality of beams could be achieved, and the inter-beam interference is eliminated. In case the transmitting node moves during the time gap, the beams for all receiving nodes interfere with each other. Otherwise, the beam for a motionless receiving node is interfered with the beams for the moving receiving nodes, but the beam for a moving receiving node is interfered with the other beams. With the goal of evaluating the performance of an MU-MIMO transmission, the behavior of nodes is categorized according to whether the inter-beam interference exists or not and the number of received interference beams. In order to simplify the calculations, we define several events to depict the behavior of nodes as follows.When the event E2 or the event E3 with M=K occurs, the beam for a receiving node is interfered with the beams for K−1 receiving nodes. When the event E3 with 0<M<K occurs, the beam for a motionless receiving node is interfered with the beams for M moving receiving nodes, but the beam of a moving receiving node is interfered with the beams for K−1 receiving nodes. With this in mind, we further define the measurement of instantaneous received SINR for motionless and moving receiving nodes as event E3s and E3m, respectively.Definition 1.
An event that the transmitting node and the receiving nodes of an MU-MIMO system both keep motionless during the time gap is denoted as eventE1.Definition 2.
An event that the transmitting node moves during the time gap, independent of the states of the receiving nodes, is denoted as eventE2.Definition 3.
An event that the transmitting node is motionless butM1<M≤K receiving nodes move during the time gap is denoted as event E3.The outage probability of an MU-MIMO transmission is defined as the probability that instantaneous received SINR does not exceed the required SINR threshold. The expected outage probability is the sum of outage probability weighted by the probability of defined events. Assuming the required SINR threshold isγthre, hence, the expected outage probability is expressed as(3)pkexpγthre=∑i=13ℙEiℙγk,Eiins≤γthre=∑i=12ℙEiℙγk,Eiins≤γthre+ℙE3×ℙE3sℙγk,E3sins≤γthre+ℙE3mℙγk,E3mins≤γthre,where ℙEi is the probability of event Ei and γk,Eiins represents the instantaneous received SINR of the receiving node k when the event Ei occurs.


### 3.1. Probability of the Events

In terms of Definition1, the event E1 means that all nodes are motionless at the start time of the time gap and keep motionless during a time interval not less than Tgap. Using the probability density function (PDF) in equation (1), the probability of event E1 is denoted as(4)ℙE1=pstate∫Tgap∞θe−θtdtK+1=pstatee−θTgapK+1.When the eventE2 occurs, no matter whether the receiving nodes move or not, the transmitting node is moved during the time gap. Hence, the probability of event E2 is denoted as(5)ℙE2=1K+1⋅1−pstate∫Tgap∞θe−θtdt=1K+1⋅1−pstatee−θTgap.As described in Definition3, the transmitting node and K−M receiving nodes are motionless at the start time of the time gap and keep motionless within a time interval not less than Tgap. But M0<M≤K receiving nodes move during the time gap. Hence, the probability of event E3 is denoted as(6)ℙE3=pstate∫Tgap∞θe−θtdtK−M+11−pstate∫Tgap∞θe−θtdtM=pstatee−θTgapK−M+11−pstatee−θTgapM.Furthermore, the probabilities of eventE3s and E3m are ℙE3s=1−M/K and ℙE3m=M/K, respectively.


### 3.2. Outage Probability of Instantaneous MU-MIMO Transmission

In an MU-MIMO transmission, the instantaneous received SINR of the receiving nodek is denoted as(7)γkins=Pt/KDk−αhkHwk2σ2+ϕ∑j∈UPt/KDk−αhkHwj2+∑φ∈Φ/TkPtZφ−αgφ,k2,where U=M if receiving node k is a motionless node, or U=K−1 if receiving node k is a moving node.For defined events above, the instantaneous received SINRs could be obtained fromγkins by setting ϕ=0 or ϕ=1.(i)
Whenϕ=0, the instantaneous received SINR γkins=γk,E1ins(ii)
Whenϕ=1 and 0<M<K, the instantaneous received SINR γkins=γk,E3sins(iii)
Whenϕ=1 and M=K, the instantaneous received SINR γkins=γk,E2ins=γk,E3minsNeglecting the effect of AWGN in the interference-limit system for analytical simplicity, the outage probability of an instantaneous MU-MIMO transmission is denoted as(8)pkinsγthre=ℙγkins≤γthre=ℙSk≤ϕρIk,IBI+Ik,CCIγthreρ=∫0∞ℙSk≤ωxfϕρIk,IBI+Ik,CCIxdx,where ρ=1/KDkα and ω=γthre/ρ. Sk=hkHwk2 and Ik,IBI=∑j∈UhkHwj2 represent chi-squared random variables with 2Nt−K+1 and 2U degrees of freedom, respectively. Using the PDF of chi-squared random variable Sk with 2Nt−K+1 degrees of freedom, the term ℙSk≤ωx is derived as(9)ℙSk≤ωx=∫0ωxtNt−K2Nt−K+1Nt−K!e−t/2dt=1−1Nt−K!ΓNt−K+1,ωx=1−e−ωx∑m=0Nt−Kωxmm!,where Γq,x=∫x∞tq−1e−tdt=q−1!e−x∑m=0q−1xm/m! is complementary incomplete Gamma function with q>0. Inserting the result into equation (8), the outage probability is expressed as(10)pkinsγthre=1−∫0∞∑m=0Nt−Kωxmm!e−ωxfϕρIk,IBI+Ik,CCIxdx.


#### 3.2.1. Outage Probability whenϕ=0

In caseϕ=0, the effect of inter-beam interference is eliminated. The outage probability in equation (10) is rewritten as(11)pkinsγthre=1−∫0∞∑m=0Nt−Kωsmm!e−ωxfIk,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLIk,CCIω,where LIk,CCIω is the Laplace transform of Ik,CCI. The transform of Ltmft=−1mFms is employed and Fms is the mth order differential of the Laplace transform of ft. Note that when K=Nt, the outage probability is simplified as(12)pkinsγthre=1−LIk,CCIω.According to the system model, the transmitting nodes in each slot are distributed according to an SPPP with intensityλ. Consequently, the co-channel interference could be modeled as a general Poisson shot noise process [21]. The Laplace transform of random variable Ik,CCI is given as(13)LIk,CCIω=exp−λ∫ℝ21−EHexp−ωHx−α=exp−2πλ∫0∞x1+xα/ωdx=exp−λQωδ,where δ=2/α and Q=πΓδΓ1−δ. Especially, in case α=4, we have Q=π2. Hence, when K=Nt, the outage probability is expressed as(14)pkinsγthre=1−exp−λQωδ.In order to achieve a high probability of successful reception, it is reasonable that a very strict quality of service (QoS) is required, resulting in that the density of transmitting nodesλ shall be very small. Consequently, the mth order differential of LIk,CCIω could be evaluated using the first-order Taylor series around λQωδ=0. For all m≥1, the mth order differential of LIk,CCIω is given approximately as(15)dmdωmLIk,CCIω≈−λQωδ−m∏l=0m−1δ−me−λQωδ+Ψλ2Q2ω2δ.In caseK<Nt, combining the equation (11) with (15) and discarding the term Ψλ2Q2ω2δ, the outage probability of an instantaneous transmission when the event E1 occurs is derived as(16)pkinsγthre=λQω2/α1+∑m=1Nt−K1m!∏l=0m−1m−δ.


#### 3.2.2. Outage Probability whenϕ=1

In this section, in order to simplify the derivation of outage probability, we first give a definition and a proposition as follows [22].Definition 4.
Given a Beta distributed random variableX with two positive shape parameters a and b, its cumulative distribution function (CDF) is a regularized incomplete beta function, denoted as(17)Fxa,b=ℙX≤x=1−∑i=1aΓb+i−1ΓbΓixi−11−xb=∑i=1bΓa+i−1ΓaΓixa1−xi−1,where Γq=∫0∞tq−1e−tdt is the Gamma function.Proposition 1.
For anyx≥0, and nonnegative integers n, v, and r where n≥r,(18)∑l=0n−rx1+xlv+l−1l=1+xvF1/1+xv,n−r+1.In caseϕ=1, the performance of MU-MIMO transmission is limited by inter-beam interference and co-channel interference simultaneously. The outage probability in equation (10) is rewritten as(19)pkinsγthre=1−∫0∞∑m=0Nt−Kωxmm!e−ωxfρIk,IBI+Ik,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLρIk,IBIωLIk,CCIω.WhenK=Nt, the outage probability is simplified as pkinsγthre=1−LρIk,IBIωLIk,CCIω. Otherwise, the outage probability is further denoted as(20)pkinsγthre=1−∑m=0Nt−K−ωmm!∑l=0mdldωlLρIk,IBIωdm−ldωm−lLIk,CCIω.Note thatLIk,CCIω and its mth order differential were derived in equations (13) and (14). Using the PDF of a chi-squared random variable Ik,IBI with 2U degrees of freedom, the Laplace transform LIk,IBIω and its mth order differential are, respectively, denoted as(21)LIk,IBIω=11+2ρωU,(22)dmdωmLIk,IBIω=−2ρm1+2ρωU+m∏j=0m−1U+j.Hence, whenK=Nt, the outage probability is expressed as(23)pkinsγthre=1−e−λQωδ1+2ρωU.In caseK<Nt, the outage probability could be obtained by combing equation (19) with the results in equations (15) and (22). With the goal of simplifying the expression, we use Definition 4 and Proposition 1 above and perform the first-order Taylor series around λQωδ=0 with discarding the term Ψλ2Q2ω2δ. As a result, the outage probability of an instantaneous transmission is denoted as(24)pkinsγthre=F2γthre/1+2γthreNt−K+1,U+λQω2/αKU−F2γthre/1+2γthreNt−K+1,U,where KU=1+∑m=0Nt−K−1U+m−1m2γthre/1+2γthrem∑i=1Nt−m−11/i!∏j=0i−1j−δ.The outage probability could be obtained withU=M when the event E3s occurs, or with U=K−1 when the event E2 or E3m occurs.


## 3.1. Probability of the Events

In terms of Definition1, the event E1 means that all nodes are motionless at the start time of the time gap and keep motionless during a time interval not less than Tgap. Using the probability density function (PDF) in equation (1), the probability of event E1 is denoted as(4)ℙE1=pstate∫Tgap∞θe−θtdtK+1=pstatee−θTgapK+1.When the eventE2 occurs, no matter whether the receiving nodes move or not, the transmitting node is moved during the time gap. Hence, the probability of event E2 is denoted as(5)ℙE2=1K+1⋅1−pstate∫Tgap∞θe−θtdt=1K+1⋅1−pstatee−θTgap.As described in Definition3, the transmitting node and K−M receiving nodes are motionless at the start time of the time gap and keep motionless within a time interval not less than Tgap. But M0<M≤K receiving nodes move during the time gap. Hence, the probability of event E3 is denoted as(6)ℙE3=pstate∫Tgap∞θe−θtdtK−M+11−pstate∫Tgap∞θe−θtdtM=pstatee−θTgapK−M+11−pstatee−θTgapM.Furthermore, the probabilities of eventE3s and E3m are ℙE3s=1−M/K and ℙE3m=M/K, respectively.


## 3.2. Outage Probability of Instantaneous MU-MIMO Transmission

In an MU-MIMO transmission, the instantaneous received SINR of the receiving nodek is denoted as(7)γkins=Pt/KDk−αhkHwk2σ2+ϕ∑j∈UPt/KDk−αhkHwj2+∑φ∈Φ/TkPtZφ−αgφ,k2,where U=M if receiving node k is a motionless node, or U=K−1 if receiving node k is a moving node.For defined events above, the instantaneous received SINRs could be obtained fromγkins by setting ϕ=0 or ϕ=1.(i)
Whenϕ=0, the instantaneous received SINR γkins=γk,E1ins(ii)
Whenϕ=1 and 0<M<K, the instantaneous received SINR γkins=γk,E3sins(iii)
Whenϕ=1 and M=K, the instantaneous received SINR γkins=γk,E2ins=γk,E3minsNeglecting the effect of AWGN in the interference-limit system for analytical simplicity, the outage probability of an instantaneous MU-MIMO transmission is denoted as(8)pkinsγthre=ℙγkins≤γthre=ℙSk≤ϕρIk,IBI+Ik,CCIγthreρ=∫0∞ℙSk≤ωxfϕρIk,IBI+Ik,CCIxdx,where ρ=1/KDkα and ω=γthre/ρ. Sk=hkHwk2 and Ik,IBI=∑j∈UhkHwj2 represent chi-squared random variables with 2Nt−K+1 and 2U degrees of freedom, respectively. Using the PDF of chi-squared random variable Sk with 2Nt−K+1 degrees of freedom, the term ℙSk≤ωx is derived as(9)ℙSk≤ωx=∫0ωxtNt−K2Nt−K+1Nt−K!e−t/2dt=1−1Nt−K!ΓNt−K+1,ωx=1−e−ωx∑m=0Nt−Kωxmm!,where Γq,x=∫x∞tq−1e−tdt=q−1!e−x∑m=0q−1xm/m! is complementary incomplete Gamma function with q>0. Inserting the result into equation (8), the outage probability is expressed as(10)pkinsγthre=1−∫0∞∑m=0Nt−Kωxmm!e−ωxfϕρIk,IBI+Ik,CCIxdx.


### 3.2.1. Outage Probability whenϕ=0

In caseϕ=0, the effect of inter-beam interference is eliminated. The outage probability in equation (10) is rewritten as(11)pkinsγthre=1−∫0∞∑m=0Nt−Kωsmm!e−ωxfIk,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLIk,CCIω,where LIk,CCIω is the Laplace transform of Ik,CCI. The transform of Ltmft=−1mFms is employed and Fms is the mth order differential of the Laplace transform of ft. Note that when K=Nt, the outage probability is simplified as(12)pkinsγthre=1−LIk,CCIω.According to the system model, the transmitting nodes in each slot are distributed according to an SPPP with intensityλ. Consequently, the co-channel interference could be modeled as a general Poisson shot noise process [21]. The Laplace transform of random variable Ik,CCI is given as(13)LIk,CCIω=exp−λ∫ℝ21−EHexp−ωHx−α=exp−2πλ∫0∞x1+xα/ωdx=exp−λQωδ,where δ=2/α and Q=πΓδΓ1−δ. Especially, in case α=4, we have Q=π2. Hence, when K=Nt, the outage probability is expressed as(14)pkinsγthre=1−exp−λQωδ.In order to achieve a high probability of successful reception, it is reasonable that a very strict quality of service (QoS) is required, resulting in that the density of transmitting nodesλ shall be very small. Consequently, the mth order differential of LIk,CCIω could be evaluated using the first-order Taylor series around λQωδ=0. For all m≥1, the mth order differential of LIk,CCIω is given approximately as(15)dmdωmLIk,CCIω≈−λQωδ−m∏l=0m−1δ−me−λQωδ+Ψλ2Q2ω2δ.In caseK<Nt, combining the equation (11) with (15) and discarding the term Ψλ2Q2ω2δ, the outage probability of an instantaneous transmission when the event E1 occurs is derived as(16)pkinsγthre=λQω2/α1+∑m=1Nt−K1m!∏l=0m−1m−δ.


### 3.2.2. Outage Probability whenϕ=1

In this section, in order to simplify the derivation of outage probability, we first give a definition and a proposition as follows [22].Definition 4.
Given a Beta distributed random variableX with two positive shape parameters a and b, its cumulative distribution function (CDF) is a regularized incomplete beta function, denoted as(17)Fxa,b=ℙX≤x=1−∑i=1aΓb+i−1ΓbΓixi−11−xb=∑i=1bΓa+i−1ΓaΓixa1−xi−1,where Γq=∫0∞tq−1e−tdt is the Gamma function.Proposition 1.
For anyx≥0, and nonnegative integers n, v, and r where n≥r,(18)∑l=0n−rx1+xlv+l−1l=1+xvF1/1+xv,n−r+1.In caseϕ=1, the performance of MU-MIMO transmission is limited by inter-beam interference and co-channel interference simultaneously. The outage probability in equation (10) is rewritten as(19)pkinsγthre=1−∫0∞∑m=0Nt−Kωxmm!e−ωxfρIk,IBI+Ik,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLρIk,IBIωLIk,CCIω.WhenK=Nt, the outage probability is simplified as pkinsγthre=1−LρIk,IBIωLIk,CCIω. Otherwise, the outage probability is further denoted as(20)pkinsγthre=1−∑m=0Nt−K−ωmm!∑l=0mdldωlLρIk,IBIωdm−ldωm−lLIk,CCIω.Note thatLIk,CCIω and its mth order differential were derived in equations (13) and (14). Using the PDF of a chi-squared random variable Ik,IBI with 2U degrees of freedom, the Laplace transform LIk,IBIω and its mth order differential are, respectively, denoted as(21)LIk,IBIω=11+2ρωU,(22)dmdωmLIk,IBIω=−2ρm1+2ρωU+m∏j=0m−1U+j.Hence, whenK=Nt, the outage probability is expressed as(23)pkinsγthre=1−e−λQωδ1+2ρωU.In caseK<Nt, the outage probability could be obtained by combing equation (19) with the results in equations (15) and (22). With the goal of simplifying the expression, we use Definition 4 and Proposition 1 above and perform the first-order Taylor series around λQωδ=0 with discarding the term Ψλ2Q2ω2δ. As a result, the outage probability of an instantaneous transmission is denoted as(24)pkinsγthre=F2γthre/1+2γthreNt−K+1,U+λQω2/αKU−F2γthre/1+2γthreNt−K+1,U,where KU=1+∑m=0Nt−K−1U+m−1m2γthre/1+2γthrem∑i=1Nt−m−11/i!∏j=0i−1j−δ.The outage probability could be obtained withU=M when the event E3s occurs, or with U=K−1 when the event E2 or E3m occurs.


## 3.2.1. Outage Probability whenϕ=0

In caseϕ=0, the effect of inter-beam interference is eliminated. The outage probability in equation (10) is rewritten as(11)pkinsγthre=1−∫0∞∑m=0Nt−Kωsmm!e−ωxfIk,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLIk,CCIω,where LIk,CCIω is the Laplace transform of Ik,CCI. The transform of Ltmft=−1mFms is employed and Fms is the mth order differential of the Laplace transform of ft. Note that when K=Nt, the outage probability is simplified as(12)pkinsγthre=1−LIk,CCIω.According to the system model, the transmitting nodes in each slot are distributed according to an SPPP with intensityλ. Consequently, the co-channel interference could be modeled as a general Poisson shot noise process [21]. The Laplace transform of random variable Ik,CCI is given as(13)LIk,CCIω=exp−λ∫ℝ21−EHexp−ωHx−α=exp−2πλ∫0∞x1+xα/ωdx=exp−λQωδ,where δ=2/α and Q=πΓδΓ1−δ. Especially, in case α=4, we have Q=π2. Hence, when K=Nt, the outage probability is expressed as(14)pkinsγthre=1−exp−λQωδ.In order to achieve a high probability of successful reception, it is reasonable that a very strict quality of service (QoS) is required, resulting in that the density of transmitting nodesλ shall be very small. Consequently, the mth order differential of LIk,CCIω could be evaluated using the first-order Taylor series around λQωδ=0. For all m≥1, the mth order differential of LIk,CCIω is given approximately as(15)dmdωmLIk,CCIω≈−λQωδ−m∏l=0m−1δ−me−λQωδ+Ψλ2Q2ω2δ.In caseK<Nt, combining the equation (11) with (15) and discarding the term Ψλ2Q2ω2δ, the outage probability of an instantaneous transmission when the event E1 occurs is derived as(16)pkinsγthre=λQω2/α1+∑m=1Nt−K1m!∏l=0m−1m−δ.


## 3.2.2. Outage Probability whenϕ=1

In this section, in order to simplify the derivation of outage probability, we first give a definition and a proposition as follows [22].Definition 4.
Given a Beta distributed random variableX with two positive shape parameters a and b, its cumulative distribution function (CDF) is a regularized incomplete beta function, denoted as(17)Fxa,b=ℙX≤x=1−∑i=1aΓb+i−1ΓbΓixi−11−xb=∑i=1bΓa+i−1ΓaΓixa1−xi−1,where Γq=∫0∞tq−1e−tdt is the Gamma function.Proposition 1.
For anyx≥0, and nonnegative integers n, v, and r where n≥r,(18)∑l=0n−rx1+xlv+l−1l=1+xvF1/1+xv,n−r+1.In caseϕ=1, the performance of MU-MIMO transmission is limited by inter-beam interference and co-channel interference simultaneously. The outage probability in equation (10) is rewritten as(19)pkinsγthre=1−∫0∞∑m=0Nt−Kωxmm!e−ωxfρIk,IBI+Ik,CCIxdx=1−∑m=0Nt−K−ωmm!dmdωmLρIk,IBIωLIk,CCIω.WhenK=Nt, the outage probability is simplified as pkinsγthre=1−LρIk,IBIωLIk,CCIω. Otherwise, the outage probability is further denoted as(20)pkinsγthre=1−∑m=0Nt−K−ωmm!∑l=0mdldωlLρIk,IBIωdm−ldωm−lLIk,CCIω.Note thatLIk,CCIω and its mth order differential were derived in equations (13) and (14). Using the PDF of a chi-squared random variable Ik,IBI with 2U degrees of freedom, the Laplace transform LIk,IBIω and its mth order differential are, respectively, denoted as(21)LIk,IBIω=11+2ρωU,(22)dmdωmLIk,IBIω=−2ρm1+2ρωU+m∏j=0m−1U+j.Hence, whenK=Nt, the outage probability is expressed as(23)pkinsγthre=1−e−λQωδ1+2ρωU.In caseK<Nt, the outage probability could be obtained by combing equation (19) with the results in equations (15) and (22). With the goal of simplifying the expression, we use Definition 4 and Proposition 1 above and perform the first-order Taylor series around λQωδ=0 with discarding the term Ψλ2Q2ω2δ. As a result, the outage probability of an instantaneous transmission is denoted as(24)pkinsγthre=F2γthre/1+2γthreNt−K+1,U+λQω2/αKU−F2γthre/1+2γthreNt−K+1,U,where KU=1+∑m=0Nt−K−1U+m−1m2γthre/1+2γthrem∑i=1Nt−m−11/i!∏j=0i−1j−δ.The outage probability could be obtained withU=M when the event E3s occurs, or with U=K−1 when the event E2 or E3m occurs.


## 4. Throughput of MU-MIMO Transmission

Assume adaptiveN-QAM modulation with L discrete rates is applied in an MU-MIMO transmission. When an event Ei occurs, if the instantaneous received SINR lies in Γl,Γl+11≤l≤L, the instantaneous rate is given as(25)bl=log21+ΓlGbps/Hz,γk,Eiins∈Γl,Γl+1,where G is Shannon Gap with variable rate N-QAM transmission [23]. Hence, the long-term expected throughput of receiving node k when the event Ei occurs is denoted as(26)Ck,Eiexp=∑l=1L‐1l⋅ℙΓl≤γk,Eiins<Γl+1+L⋅ℙγk,Eiins≥ΓL=∑l=1L‐1l⋅pk,EiinsΓl+1−pk,EiinsΓl+L⋅1−pk,EiinsΓL.The expected total throughput of an MU-MIMO transmission is given as(27)Ctotexp=∑k=1KℙE1Ck,E1exp+ℙE2Ck,E2exp+ℙE3ℙE3sCk,E3sexp+ℙE3mCk,E3mexp.Intuitively, the increase of receiving nodes could bring more expected total throughput. However, with the increase of the number of receiving nodes, the probability that multiple nodes keep motionless during the time gap decreases, causing the probability that the eventE1 or E3 occurs decreases. In addition, the increase of receiving nodes reduces the power of the desired signal and potentially increases the power of inter-beam interference. For a receiving node, the instantaneous received SINR and the long-term expected throughput are reduced. Hence, there is an optimal number of receiving nodes K∗ which could maximize the expected total throughput of an MU-MIMO transmission, i.e.,(28)K∗=argmax1<K≤NtCtotexp.


## 5. Numeric Results

This section reports the results of computer simulations. In the simulation, the MANET nodes are assumed to be located randomly in a circle region with 1 km radius. The simulation configurations are listed in Table1.Table 1
Simulation configurations.
ParameterValueTotal power of transmitting nodePt23 dBmTransmit antennaNt8Receive antennaNr1Receiving nodesK4,8Mobile receiving nodesM1,K/2, KPath loss exponentα4SINR thresholdγthre-30 dBDistance between Tx and RxDk100 mDistribution density of nodesλ10~100/km2Probability of motionless statepstate0.5Average time intervalθ1 sNumber of discrete ratesL4Figure2 shows the probabilities of the defined events. With the increase of the duration of the time gap, the probability that all the nodes keep motionless decreases, i.e., the probability of the event E1 decreases. But the probability that the transmitting node is moving increases, causing the increase of the probability of the event E2. When the event E3 occurs, the probability is dependent on the number of moving nodes. If the moving nodes are less than the receiving nodes, the probability that the motionless nodes keep the same state decreases; hence, the probability of the event E3 decreases. Otherwise, the probability that all receiving nodes are moving increases with the duration of the time gap.Figure 2
Probability of events vs. duration of time gap.The outage probabilities of instantaneous MU-MIMO transmissions are given in Figure3. On the one hand, with the increase of the intensity of the MANET nodes, a receiving node experiences more severe co-channel interference signals. On the other hand, the increasing receiving nodes reduce the power of the desired signal but enhance the power of inter-beam interference. As a result, the outage probabilities of instantaneous MU-MIMO transmissions increase with the intensity of the MANET nodes and the number of receiving nodes. Further, because of the moving receiving nodes in the event E2 and E3, i.e., M>0, the receiving nodes will receive the inter-beam interference signal. Hence, the outage probabilities of the event E2 and E3 (M>0) are higher than that of the event E1 (M=0). And the outage probability increase with the number of moving receiving nodes.Figure 3
Outage probability of instantaneous MU-MIMO transmissions vs. average number of ad hoc nodes.Figure4 describes the relationship between the expected outage probability and the duration of the time gap. In terms of the definition of the events, the probability of the event E1 decreases but the probabilities of the event E2 and E3 increase with the duration of the time gap. As a result, the expected outage probability increases with the probabilities of the event E2 and E3. However, the increase of the expected outage probability is related to the number of receiving nodes. For instance, the increase of K=8 is faster than that of K=4. On the other hand, with the increase of the number of moving receiving nodes, the probability of event E3 and further the expected outage probability increase.Figure 4
Expected outage probability of MU-MIMO transmissions vs. duration of time gap.Figure5 reflects the impact of the number of receiving nodes on the expected throughput of an MU-MIMO transmission. When the number of the receiving nodes is less than a threshold, the increasing receiving nodes reduce the expected throughput of each receiving node and the total expected throughput. While the number of receiving nodes excesses the threshold, the total expected throughput begins to grow with the increase of the receiving nodes. As a result, an optimal value of K equal to 2 or the number of the transmit antennas could maximize the total expected throughput. On the other hand, it is obvious that the increase of the duration of the time gap reduces the total expected throughput. In case of a fixed duration of the time gap, the total expected throughput increases with the number of moving receiving nodes.Figure 5
Expected throughput of MU-MIMO transmissions vs. number of receiving nodes.


## 6. Conclusions

This paper evaluates the expected outage probability and expected throughput of an MU-MIMO transmission in MANET in the presence of co-channel interference and unpredictable inter-beam interference. To derive the closed form results, the unpredictable behavior of nodes is categorized. Based on the categories, the closed form outage probability of instantaneous MU-MIMO transmissions and further expected outage probability are obtained. The analytical and numeric results indicate the unpredictable inter-beam interference degrades the outage probability performance. Reducing the duration of the time gap could improve the expected outage probability and expected throughput.As a future work, we will explore the performance optimization by interference management or resource assignment based on the derived results. At the same time, we believe that the application of integrated cognition-control-communication technology to make network intelligent is an important direction.



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*Source: 9954940-2021-05-04.xml*