# Covert Communication for Wireless Networks with Full-Duplex Multiantenna Relay

Authors: Ling Yang; Weiwei Yang; Liang Tang; Liwei Tao; Xingbo Lu; Zhengyun He
Journal: Complexity
 (2022)

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## Abstract

In this work, we investigated a covert communication method in wireless networks, which is realized by multiantenna full-duplex single relay. In the first stage, the source node sends covert messages to the relay, and the relay uses a single antenna to send interference signals to the adversary node to protect the covert information being transmitted. In the second stage, the relay decodes and forwards the covert information received in the first stage; at the same time, the relay uses zero-forcing beamforming to send interference signals to the warden to ensure covert transmission. The detection error rate, transmission outage probability, maximum effective covert rate, and other performance indicators are derived in two stages, and the total performance of the system is derived and analyzed. Then, the performance indicators are verified and analyzed by simulation. Our analysis shows that the maximum effective covert rate of using the characteristics of multiantenna to interfere with Willie in the second stage is taken as the total covert performance of the system, and the transmission interruption probability is significantly less than that of the first stage, so the corresponding maximum effective concealment efficiency will be greater.


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## Body

## 1. Introduction

### 1.1. Background

Networks are ubiquitous in nature and human society. Modern communication has been highly concerned by academic and industrial circles, resulting in many emerging sciences, such as the internet of things [1], complex networks [2–5], cognitive networks [6, 7], virtual reality, and augmented reality technology. Due to the broadcast characteristics of wireless channels, whether users are legal or not, researchers pay more and more attention to the security and privacy of information transmission. Taking full advantage of the uncertainty and unpredictability of wireless channels, physical layer security technology [8, 9] has become a mature technology, which has been applied to achieve secure transmission, focusing on information theory technology [10, 11]. However, in many communication cases, not only the privacy and integrity of information need to be considered, but also the security of communication behavior needs to be protected, such as the existence of hidden communication [12]. In this context, covert communication [13, 14] emerged as a new security technology.In 2013, B. A. Bash proved the reliable communication with low detection probability under additive white Gaussian noise (AWGN) channel theoretically for the first time [15] and proposed the square root rule of covert communication under AWGN channel. Then, the basic research of covert communication has attracted the interest of relevant scholars and has been studied in AWGN channels [16], discrete memoryless channels (DMC) [17, 18], and binary symmetric channel (BSC) [19]. At the same time, the basic limitations of covert communication under different channel conditions have been explored [20]. With the deepening of theoretical research, the concept of covert communication has gradually formed and developed. So far, researchers represent by Dr. B. A. Bash and Dr. Shihao Yan team of Macquarie University in Australia have made contributions to the basic theoretical research and performance analysis of covert communication technology; covert communication technology has attracted more and more researchers’ attention, and its related research has gradually developed and enriched [21–23].


### 1.2. Previous Works and Motivations

Recently, more and more scenes and technologies about covert communication have been studied. In this paper, we focus on full duplex (FD), relay, and multiantenna covert technology.Reference [24] explores the possibility and condition of covert communication in quasi-static wireless fading channel by using an FD receiver to transmit artificial noise (AN) to enhance covert performance. Reference [25] not only studies the influence of channel estimation inaccuracy and randomness on covert wireless transmission performance but also explores the effect of FD relay on covert performance. Furthermore, in ad hoc networks, the internet of things, and other random wireless networks, reference [26] also uses FD technology to effectively enhance the covert of communication.Reference [25] also studies the influence of channel uncertainty of detecting attackers on covert wireless transmission performance in two-hop relay communication system. Reference [27] studies the covert and transmission reliability of covert wireless transmission schemes with the assistance of a single relay and shows that cooperative relay can enhance the performance of covert wireless transmission. In [28], a covert wireless transmission scheme is designed with the assistance of a wireless energy acquisition relay, and the expression of minimum error detection probability is derived. In [29], the problem of the covert wireless transmission assisted by the untrusted relay is studied. References [30, 31] study the covert wireless communication scheme with multihop relay transmission. In particular, reference [31] considers the problem of multihop covert relay transmission in unmanned aerial vehicle communication. Most of these researches focus on reducing transmission power by relay transmission to enhance covertness or improving the communication performance of legal links by using cooperative diversity gain. However, the probability of signal exposure caused by relay transmission increases, and the performance degradation is not clear.From [32], an active eavesdropping scheme assisted by the covert pilot attack is designed for wireless monitoring scenarios, and covert wireless communication is realized by using the uncertainty of the detection channel. The communication process is divided into the channel estimation stage and the data transmission stage. Through the use of a malicious detection node (source node) equipped with multiple antennas to realize hidden pilot attacks on channel status information in the channel estimation stage, the information beam of the communication process is forced to point to the wireless monitoring node in the data transmission stage, and the probability of wireless monitoring success is improved. Reference [33] considers centralized and distributed antenna systems and discusses the coverage and reliability of transmission. In addition, the covert communication model of multiantenna detector is also studied in reference [34], and the influence of the increase of the number of detectors’ antennas on the covert performance is discussed. In recent years, researchers consider using multiantenna jammers to improve the performance of covert communication systems [35]. In [36], the enhancement effect of multiantenna AN nodes on covert performance is studied, and it is pointed out that directional beamforming is the optimal AN strategy. In [14], an FD multiantenna receiver is used to achieve covert communication based on the uncertainty of interference power. The receiver first selects the best antenna to receive the covert information and then randomly selects one of the remaining antennas to generate AN, resulting in the uncertainty of the detector so as to achieve covert communication.In the present work, the FD, multiantenna and cooperative transmission method under covert communication needs to be further studied. On the one hand, the existing research of covert communication often focuses on the covert transmission of its own important information based on the relay forwarding the source’s message and seldom considers that the information sent by the source are covert messages at the beginning, so the established scene has certain limitations. On the other hand, the reliability of relay nodes in covert wireless communications or covert wireless networks in the presence of eavesdropping is rarely involved. How to combine multiantenna, FD, cooperative technology, and covert communication is worthy of further study. Based on this, the research point of covert communication in multiantenna FD relay system is proposed.


### 1.3. Our Approach and Contributions

The main contributions of this paper are summarized as follows:(i)
We prove that the use of multiantenna FD relay is an effective way to achieve covert wireless communication in fading wireless channels. Multiantenna FD relay uses the advantages of multiantenna to design different antenna selection schemes in two stages to transmit signals with different power to cause Willie’s confusion.(ii)
Based on the assumption of the Willie radiometer, we analytically derive Willie’s optimal detection thresholds for the two stages of the system. When we define the optimal decision rules for the minimum detection error rate, the predetermined thresholds of the two stages are consistent, and the optimal detection performance is obtained with the minimum detection error probability.(iii)
For the given covert constraints, we give the design criteria of the optimal interference power of the first stage relay and the optimal transmission power of the second stage relay to forward the covert messages so as to maximize the optimal effective covert rate of the system.(iv)
Our analysis shows that for the same parameters, the transmission outage probabilities of the first stage, the second stage, and the total results have the same trend. Since the total maximum effective covert rate is smaller of the two stages, the total maximum effective covert rate has the same conclusion as that of the first stage.


### 1.4. Organization

The rest of this paper is organized as follows. The system model is introduced in Section2. Sections 3 and 4, respectively, introduce the two stages of the system. The first stage is that the source node sends covert messages to the relay, and the second stage is that the relay forwards covert messages to the destination node. The detection and covert performance of the system are deduced and analyzed in the two stages. In Section 5, the total performance of the system is studied. Section 6 provides the theoretical analyses are verified by numerical results. Finally, Section 7 describes some concluding remarks. A list of the fundamental variables is provided in Table 1.Table 1
List of fundamental variables.
SymbolDescriptionSINRInterference plus noise ratioAWGNThe complex additive Gaussian noisef⋅The cumulative distribution functionPXThe power send byX, X∈Alice,Relay,Bob,WillieyAThe received signal atA, A∈Bob,Relay,Willieha,bThe channel betweena and b that both consist a single antennanAiThe AWGN atA, A∈Bob,Relay,WillieγAThe SINR onA, A∈Bob,WillienThe number of each channel useσA2The variance of AWGN at nodeA, A∈Bob,WillieTWThe average power received at WillieτThe predefined detection thresholdξThe detection error rateτ∗The optimal predefined detection thresholdξ∗The minimum detection error rateRC¯The maximized effective covert rateF⋅The probability density functionεThe predetermined covert communication condition valueRThe transmission rateδThe transmission outage probabilityNAAntenna number ofA, A∈Bob,Relay,Willieha,bThe channel vector betweena and b consisting of one and multiple antennas, respectivelywa,bThe received weighting vector on linka−b†The conjugate transpose operator•FThe Frobenius normT⊥The projection idempotent matrixαFalse alarm rateβMiss detection rate


## 1.1. Background

Networks are ubiquitous in nature and human society. Modern communication has been highly concerned by academic and industrial circles, resulting in many emerging sciences, such as the internet of things [1], complex networks [2–5], cognitive networks [6, 7], virtual reality, and augmented reality technology. Due to the broadcast characteristics of wireless channels, whether users are legal or not, researchers pay more and more attention to the security and privacy of information transmission. Taking full advantage of the uncertainty and unpredictability of wireless channels, physical layer security technology [8, 9] has become a mature technology, which has been applied to achieve secure transmission, focusing on information theory technology [10, 11]. However, in many communication cases, not only the privacy and integrity of information need to be considered, but also the security of communication behavior needs to be protected, such as the existence of hidden communication [12]. In this context, covert communication [13, 14] emerged as a new security technology.In 2013, B. A. Bash proved the reliable communication with low detection probability under additive white Gaussian noise (AWGN) channel theoretically for the first time [15] and proposed the square root rule of covert communication under AWGN channel. Then, the basic research of covert communication has attracted the interest of relevant scholars and has been studied in AWGN channels [16], discrete memoryless channels (DMC) [17, 18], and binary symmetric channel (BSC) [19]. At the same time, the basic limitations of covert communication under different channel conditions have been explored [20]. With the deepening of theoretical research, the concept of covert communication has gradually formed and developed. So far, researchers represent by Dr. B. A. Bash and Dr. Shihao Yan team of Macquarie University in Australia have made contributions to the basic theoretical research and performance analysis of covert communication technology; covert communication technology has attracted more and more researchers’ attention, and its related research has gradually developed and enriched [21–23].


## 1.2. Previous Works and Motivations

Recently, more and more scenes and technologies about covert communication have been studied. In this paper, we focus on full duplex (FD), relay, and multiantenna covert technology.Reference [24] explores the possibility and condition of covert communication in quasi-static wireless fading channel by using an FD receiver to transmit artificial noise (AN) to enhance covert performance. Reference [25] not only studies the influence of channel estimation inaccuracy and randomness on covert wireless transmission performance but also explores the effect of FD relay on covert performance. Furthermore, in ad hoc networks, the internet of things, and other random wireless networks, reference [26] also uses FD technology to effectively enhance the covert of communication.Reference [25] also studies the influence of channel uncertainty of detecting attackers on covert wireless transmission performance in two-hop relay communication system. Reference [27] studies the covert and transmission reliability of covert wireless transmission schemes with the assistance of a single relay and shows that cooperative relay can enhance the performance of covert wireless transmission. In [28], a covert wireless transmission scheme is designed with the assistance of a wireless energy acquisition relay, and the expression of minimum error detection probability is derived. In [29], the problem of the covert wireless transmission assisted by the untrusted relay is studied. References [30, 31] study the covert wireless communication scheme with multihop relay transmission. In particular, reference [31] considers the problem of multihop covert relay transmission in unmanned aerial vehicle communication. Most of these researches focus on reducing transmission power by relay transmission to enhance covertness or improving the communication performance of legal links by using cooperative diversity gain. However, the probability of signal exposure caused by relay transmission increases, and the performance degradation is not clear.From [32], an active eavesdropping scheme assisted by the covert pilot attack is designed for wireless monitoring scenarios, and covert wireless communication is realized by using the uncertainty of the detection channel. The communication process is divided into the channel estimation stage and the data transmission stage. Through the use of a malicious detection node (source node) equipped with multiple antennas to realize hidden pilot attacks on channel status information in the channel estimation stage, the information beam of the communication process is forced to point to the wireless monitoring node in the data transmission stage, and the probability of wireless monitoring success is improved. Reference [33] considers centralized and distributed antenna systems and discusses the coverage and reliability of transmission. In addition, the covert communication model of multiantenna detector is also studied in reference [34], and the influence of the increase of the number of detectors’ antennas on the covert performance is discussed. In recent years, researchers consider using multiantenna jammers to improve the performance of covert communication systems [35]. In [36], the enhancement effect of multiantenna AN nodes on covert performance is studied, and it is pointed out that directional beamforming is the optimal AN strategy. In [14], an FD multiantenna receiver is used to achieve covert communication based on the uncertainty of interference power. The receiver first selects the best antenna to receive the covert information and then randomly selects one of the remaining antennas to generate AN, resulting in the uncertainty of the detector so as to achieve covert communication.In the present work, the FD, multiantenna and cooperative transmission method under covert communication needs to be further studied. On the one hand, the existing research of covert communication often focuses on the covert transmission of its own important information based on the relay forwarding the source’s message and seldom considers that the information sent by the source are covert messages at the beginning, so the established scene has certain limitations. On the other hand, the reliability of relay nodes in covert wireless communications or covert wireless networks in the presence of eavesdropping is rarely involved. How to combine multiantenna, FD, cooperative technology, and covert communication is worthy of further study. Based on this, the research point of covert communication in multiantenna FD relay system is proposed.


## 1.3. Our Approach and Contributions

The main contributions of this paper are summarized as follows:(i)
We prove that the use of multiantenna FD relay is an effective way to achieve covert wireless communication in fading wireless channels. Multiantenna FD relay uses the advantages of multiantenna to design different antenna selection schemes in two stages to transmit signals with different power to cause Willie’s confusion.(ii)
Based on the assumption of the Willie radiometer, we analytically derive Willie’s optimal detection thresholds for the two stages of the system. When we define the optimal decision rules for the minimum detection error rate, the predetermined thresholds of the two stages are consistent, and the optimal detection performance is obtained with the minimum detection error probability.(iii)
For the given covert constraints, we give the design criteria of the optimal interference power of the first stage relay and the optimal transmission power of the second stage relay to forward the covert messages so as to maximize the optimal effective covert rate of the system.(iv)
Our analysis shows that for the same parameters, the transmission outage probabilities of the first stage, the second stage, and the total results have the same trend. Since the total maximum effective covert rate is smaller of the two stages, the total maximum effective covert rate has the same conclusion as that of the first stage.


## 1.4. Organization

The rest of this paper is organized as follows. The system model is introduced in Section2. Sections 3 and 4, respectively, introduce the two stages of the system. The first stage is that the source node sends covert messages to the relay, and the second stage is that the relay forwards covert messages to the destination node. The detection and covert performance of the system are deduced and analyzed in the two stages. In Section 5, the total performance of the system is studied. Section 6 provides the theoretical analyses are verified by numerical results. Finally, Section 7 describes some concluding remarks. A list of the fundamental variables is provided in Table 1.Table 1
List of fundamental variables.
SymbolDescriptionSINRInterference plus noise ratioAWGNThe complex additive Gaussian noisef⋅The cumulative distribution functionPXThe power send byX, X∈Alice,Relay,Bob,WillieyAThe received signal atA, A∈Bob,Relay,Willieha,bThe channel betweena and b that both consist a single antennanAiThe AWGN atA, A∈Bob,Relay,WillieγAThe SINR onA, A∈Bob,WillienThe number of each channel useσA2The variance of AWGN at nodeA, A∈Bob,WillieTWThe average power received at WillieτThe predefined detection thresholdξThe detection error rateτ∗The optimal predefined detection thresholdξ∗The minimum detection error rateRC¯The maximized effective covert rateF⋅The probability density functionεThe predetermined covert communication condition valueRThe transmission rateδThe transmission outage probabilityNAAntenna number ofA, A∈Bob,Relay,Willieha,bThe channel vector betweena and b consisting of one and multiple antennas, respectivelywa,bThe received weighting vector on linka−b†The conjugate transpose operator•FThe Frobenius normT⊥The projection idempotent matrixαFalse alarm rateβMiss detection rate


## 2. System Model

As shown in Figure1, we consider a covert wireless communication model in a single relay multiantenna network, which includes a transmitter (Alice), a relay (Relay), a receiver (Bob), and an adversary (Willie). Among them, Relay is equipped with NR antennas, and other nodes are equipped with a single antenna. Since there is no direct communication link between Alice and Bob, it is necessary to help Alice send covert messages to Bob through Relay, while Willie tries to detect such covert transmissions, and each role knows each other’s existence and location. Information transmission is divided into two stages. In the first stage, Alice sends covert messages to Relay, and Relay sends artificial noise AN signals to Willie while receiving the messages. In order to improve the effective covert rate as much as possible, Relay selects the best antenna between it and Alice for receiving, one of the remaining NR−1 antennas was selected at random to send AN to Willie. In the second stage, Relay selects an optimal antenna to forward Alice’s signal to Bob by decode-and-forward and uses the remaining NR−1 antennas to transmit AN signal to Willie by using zero-forcing beamforming ZFB.Figure 1
System model.Assuming that the transmission of the first stage is completed before the transmission of the second stage, it is necessary to ensure that neither stage can be detected by Willie in order to realize the covert transmission from Alice to Bob. Since Willie needs to detect whether Alice and Relay send covert messages in the first and second stages, respectively, considering the worse case, it is assumed that Willie knowshAW and hRW and that Relay knows hAR, and since Relay knows Willie’s existence, it is assumed that Relay also knows hRW [1]. The wireless channels are subject to independent quasi-static Rayleigh fading; the channels change independently between time slots and remain unchanged within the same communication time slot.


## 3. The First Stage: Alice Transmits Covert Messages to Relay

### 3.1. Relay Receives Covert Messages

The instantaneous signal-to-interference-plus-noise ratioSINR at Relay is given by(1)γRelay=PAmax1≤i≤NRhARi2φPRjhRjRj2+σR2.Relay selects the besti-th antenna to receive the signal according to the channel conditions between Alice and Relay, and its channel coefficient is expressed as hARi, where i=1,2,…,n, PA is the power of Alice sending covert messages, PRj is the power of AN sent by Relay, PRjRj is the channel coefficient of Relay itself, and j is the j-th antenna randomly selected by Relay in NR−1 antennas. Since Relay itself knows AN signal, the residual noise can be reconstructed and eliminated through self-interference elimination. ϕ is used to represent the self-interference elimination coefficient. ϕ=0 is the ideal situation, while 0≤ϕ≤1 refers to different self-interference elimination levels [37]. σR2 is the channel variance of Relay. The maximum power of Relay is PRmax. Assuming that the power is evenly distributed among the antennas of Relay, let PRjmax=PRmax/NR. We assume that PA is fixed [38], and both Relay and Willie know it. The PRj changes from slot to slot, following a continuous uniform distribution over the interval 0,PRjmax, having a probability density function PDF given by(2)fPRjx=1PRjmax,0≤x≤PRjmax,0,otherwise..Reasons for settingPRj to change between slots: the purpose of Relay to send AN power is to make the power received by Willie uncertain. Willie knows hAW in a time slot. If the AN power is constant, Willie can directly detect the covert transmission when Alice sends fixed covert messages, so the PRj is not fixed.In the first stage, when Alice sends covert messages, the signal received by Relay is as follows:(3)YRelayi=PAmaxhARixAi+φPRjhRjRjVRi+nRi,where xAi represents the signals transmitted by Alice, satisfying ExAixA†i=1, i=1,2,…,n represents the symbol index, n is the total number of channels used in each slot, and nRi is the complex additive white Gaussian noise AWGN at Relay with σR2 as its variance, that is, nRi∼CN0,σR2.


### 3.2. Detection Metrics at Willie

In the first stage, the SINR at Willie, in case Alice transmits, is given by(4)γWillie1=PAhAW2PRjhRjW2+σW2,where hAW is the channel coefficient between Alice and Willie, hRjW represents the channel coefficient between the j-th antenna that Relay randomly selects to send interference and Willie, σW2 is the noise variance of Willie. In a communication time slot, Willie has to decide whether Alice has transmitted covert messages to Relay in the first stage. Therefore, Willie is faced with a binary hypothesis testing problem, where the null hypothesis H0 means that Alice has not sent covert messages, and the alternative hypothesis H1 means that Alice has sent covert messages to Relay.The signal received by Willie in the first stage can be expressed as follows:(5)ywilliei=PRjhRjWvRi+nWi,H0,PAhAWxAi+PRjhRjWvRi+nWi,H1,where nWk is the AWGN at Willie with σW2 as its variance, that is, nWk∼CN0,σW2. Willie does not know the value of PRj in the time slot, but the value of PA is fixed and known. In the first stage, Willie attempts to detect whether ywillie is H0 or H1. Through the application of Neyman–Perason criterion [39] and likelihood ratio test, the optimal decision rule for Willie to minimize his detection error is as follows:(6)TW>D1<D0τ,where TW=1/n∑i=1nyWillie1i is the average power received at Willie in the slot and τ is a predefined Willie’s detection threshold. D0 and D1 are binary decisions that infer whether Alice sends covert messages. In this paper, we consider an infinite number of channel uses, that is, n⟶∞. Therefore, we have(7)TW=PRjhRjW2+σW2,H0,PAhAW2+PRjhRjW2+σW2,H1.At the end of the communication slot, Willie has to make a decision. The false alarm rate is defined as the probability of Willie makingD1 decision under condition H0, which is expressed by α=PD1|H0. Similarly, the miss detection rate is defined as the probability of Willie making D0 decision under condition H1, which is expressed by β=PD0|H1. Assuming that the prior probabilities of H0 and H1 are equal, Willie’s detection performance can be judged by the detection error rate, which can be defined as follows:(8)ξ≜α+β.


### 3.3. Detection Performance at Willie

According to Theorems1 and 2 in [37], Willie’s optimal detection threshold, minimum detection error probability, and expected detection error probability can be obtained.The optimal detection threshold is expressed as follows:(9)τ∗=ρ1,ρ2,ρ1<ρ2,ρ2,ρ1,ρ1≥ρ2..The minimum detection error probability is expressed as follows:(10)ξ∗=0,ρ1<ρ2,1−PAhAW2PRjmaxhRjW2,ρ1≥ρ2,where ρ1≜PRjmaxhRjW2+σW2, ρ2≜PAhAW2+σW2, ρ3≜PAhAW2+PRjmaxhRjW2+σW2.The expected detection error probability is expressed as follows:(11)ξ∗¯t=−t2+tlnt+1,where(12)t≜PAhAW2PAhAW2+PRjmaxhRjW2.It can be seen from [2] that the expected detection error rate ξ∗¯ is a monotone increasing function of PRjmax.


### 3.4. Covert Performance

In general, the constraint for covert transmission can be defined asξ∗≥1−ε, where ε is a predetermined value, and there is ε∈0,1. Therefore, the maximized effective covert rate can be expressed as follows:(13)maxPZRC¯,s.t.ξ∗¯≥1−ε,where RC¯≜R1−δ.


#### 3.4.1. The Transmission Outage Probability of Alice to Relay

Assuming that the transmission rateR is known, according to formula (1), hARi2, hRjRj2 and PRj are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the cumulative distribution functionCDF of max1≤i≤NRhARi2 is Fmax1≤i≤NRhARi2x=1−e−xNR; then its PDF is(14)fmax1≤i≤NRhARi2x=∑m=0NRNRm−1me−xm.Theorem 1.
For the first stage, the transmission outage probability from Alice to Relay is derived as follows:(15)δ1=1+PAPRjmaxφ22R−1∑m=1NRNRm−1m1m,exp−m22R−1σR2PA×ln1+mφ22R−1PRjmaxPA.Proof.
see AppendixA.Remark 1.
The comments are as follows:(1.1) In the first stage, the maximum powerPRjmax of transmitting AN of a single antenna randomly selected by Relay can have a direct influence on the transmission outage probability δ1. The larger PRjmax is, the greater δ1 will be.(1.2) In the first stage, the transmission outage probabilityδ1 is a monotonically increasing function of channel noise σR2 and transmission rate R, that is, the larger σR2 and R are, the larger δ1 will be.(1.3) The configuration of antenna numberNR has a direct influence on the transmission outage probability δ1 of the first stage. The larger NR is, the smaller δ1 is.(1.4) In the first stage, the transmission outage probabilityδ1 is a monotonically decreasing function of Alice’s covert message sending power PA, that is, the larger PA is, the smaller δ1 is. And the transmission outage probability δ1 is a monotonically increasing function of the self-interference elimination coefficient ϕ, that is, the larger ϕ is, the larger δ1 is.


#### 3.4.2. The Optimal AN

Theorem 2.
Under any given fixed covert information powerPA sent by Alice, predetermined ε and transmission rate R, the optimal AN power sent by Relay is given by(16)PRjmax∗=1−tεPAhAW2hRjW2tε,where tε is the solution of ξ∗¯t=1−ε.Proof.
see AppendixB.


#### 3.4.3. The Maximized Effective Covert Rate

Theorem 3.
The optimal effective convert rate is derived as follows:(17)RC1∗¯=−RhRjW2tε∑m=1NRNRm−1m×exp−m22R−1σR2PAln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
see AppendixC.Remark 2.
The comments are as follows:(2.1) The larger the predetermined convert constraintε in the first stage, the greater the maximum effective convert rate of the first stage.Due toRC¯≜R1−δ, then there is δ1∗, so there will be RC1∗¯≜R1−δ1∗, and δ1∗ also has the property of δ1, so we can obtain Remarks 2.2–2.4.(2.2) In the first stage, the maximum effective covert rateRC1∗¯ increases with the increase of the antenna number NR and the power PA of covert messages sent by Alice.(2.3) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel noise σR2 of Relay and its self-interference elimination coefficient ϕ, that is, the larger σR2 and ϕ are, the smaller RC1∗¯ is.(2.4) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel coefficient hAW2, that is, the larger hAW2 is, the smaller RC1∗ is. And the maximum effective covert rate RC1∗¯ is the monotonic increasing function of the hRjW2, that is, the larger hRjW2 is, the larger RC1∗¯ is.Corollary 1.
In the first stage, if the powerPA of Alice sending covert messages to Relay increases, the maximum effective covert rate RC1∗¯ tends to a fixed value:(18)RC1∗¯=RhRjW2tε∑m=1NRNRm−1m,ln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
whenPA approaches ∞, then exp−m22R−1σR2/PA approaches 1, so the maximum effective convert rate approaches the result in formula (18) in the first stage.


## 3.1. Relay Receives Covert Messages

The instantaneous signal-to-interference-plus-noise ratioSINR at Relay is given by(1)γRelay=PAmax1≤i≤NRhARi2φPRjhRjRj2+σR2.Relay selects the besti-th antenna to receive the signal according to the channel conditions between Alice and Relay, and its channel coefficient is expressed as hARi, where i=1,2,…,n, PA is the power of Alice sending covert messages, PRj is the power of AN sent by Relay, PRjRj is the channel coefficient of Relay itself, and j is the j-th antenna randomly selected by Relay in NR−1 antennas. Since Relay itself knows AN signal, the residual noise can be reconstructed and eliminated through self-interference elimination. ϕ is used to represent the self-interference elimination coefficient. ϕ=0 is the ideal situation, while 0≤ϕ≤1 refers to different self-interference elimination levels [37]. σR2 is the channel variance of Relay. The maximum power of Relay is PRmax. Assuming that the power is evenly distributed among the antennas of Relay, let PRjmax=PRmax/NR. We assume that PA is fixed [38], and both Relay and Willie know it. The PRj changes from slot to slot, following a continuous uniform distribution over the interval 0,PRjmax, having a probability density function PDF given by(2)fPRjx=1PRjmax,0≤x≤PRjmax,0,otherwise..Reasons for settingPRj to change between slots: the purpose of Relay to send AN power is to make the power received by Willie uncertain. Willie knows hAW in a time slot. If the AN power is constant, Willie can directly detect the covert transmission when Alice sends fixed covert messages, so the PRj is not fixed.In the first stage, when Alice sends covert messages, the signal received by Relay is as follows:(3)YRelayi=PAmaxhARixAi+φPRjhRjRjVRi+nRi,where xAi represents the signals transmitted by Alice, satisfying ExAixA†i=1, i=1,2,…,n represents the symbol index, n is the total number of channels used in each slot, and nRi is the complex additive white Gaussian noise AWGN at Relay with σR2 as its variance, that is, nRi∼CN0,σR2.


## 3.2. Detection Metrics at Willie

In the first stage, the SINR at Willie, in case Alice transmits, is given by(4)γWillie1=PAhAW2PRjhRjW2+σW2,where hAW is the channel coefficient between Alice and Willie, hRjW represents the channel coefficient between the j-th antenna that Relay randomly selects to send interference and Willie, σW2 is the noise variance of Willie. In a communication time slot, Willie has to decide whether Alice has transmitted covert messages to Relay in the first stage. Therefore, Willie is faced with a binary hypothesis testing problem, where the null hypothesis H0 means that Alice has not sent covert messages, and the alternative hypothesis H1 means that Alice has sent covert messages to Relay.The signal received by Willie in the first stage can be expressed as follows:(5)ywilliei=PRjhRjWvRi+nWi,H0,PAhAWxAi+PRjhRjWvRi+nWi,H1,where nWk is the AWGN at Willie with σW2 as its variance, that is, nWk∼CN0,σW2. Willie does not know the value of PRj in the time slot, but the value of PA is fixed and known. In the first stage, Willie attempts to detect whether ywillie is H0 or H1. Through the application of Neyman–Perason criterion [39] and likelihood ratio test, the optimal decision rule for Willie to minimize his detection error is as follows:(6)TW>D1<D0τ,where TW=1/n∑i=1nyWillie1i is the average power received at Willie in the slot and τ is a predefined Willie’s detection threshold. D0 and D1 are binary decisions that infer whether Alice sends covert messages. In this paper, we consider an infinite number of channel uses, that is, n⟶∞. Therefore, we have(7)TW=PRjhRjW2+σW2,H0,PAhAW2+PRjhRjW2+σW2,H1.At the end of the communication slot, Willie has to make a decision. The false alarm rate is defined as the probability of Willie makingD1 decision under condition H0, which is expressed by α=PD1|H0. Similarly, the miss detection rate is defined as the probability of Willie making D0 decision under condition H1, which is expressed by β=PD0|H1. Assuming that the prior probabilities of H0 and H1 are equal, Willie’s detection performance can be judged by the detection error rate, which can be defined as follows:(8)ξ≜α+β.


## 3.3. Detection Performance at Willie

According to Theorems1 and 2 in [37], Willie’s optimal detection threshold, minimum detection error probability, and expected detection error probability can be obtained.The optimal detection threshold is expressed as follows:(9)τ∗=ρ1,ρ2,ρ1<ρ2,ρ2,ρ1,ρ1≥ρ2..The minimum detection error probability is expressed as follows:(10)ξ∗=0,ρ1<ρ2,1−PAhAW2PRjmaxhRjW2,ρ1≥ρ2,where ρ1≜PRjmaxhRjW2+σW2, ρ2≜PAhAW2+σW2, ρ3≜PAhAW2+PRjmaxhRjW2+σW2.The expected detection error probability is expressed as follows:(11)ξ∗¯t=−t2+tlnt+1,where(12)t≜PAhAW2PAhAW2+PRjmaxhRjW2.It can be seen from [2] that the expected detection error rate ξ∗¯ is a monotone increasing function of PRjmax.


## 3.4. Covert Performance

In general, the constraint for covert transmission can be defined asξ∗≥1−ε, where ε is a predetermined value, and there is ε∈0,1. Therefore, the maximized effective covert rate can be expressed as follows:(13)maxPZRC¯,s.t.ξ∗¯≥1−ε,where RC¯≜R1−δ.


### 3.4.1. The Transmission Outage Probability of Alice to Relay

Assuming that the transmission rateR is known, according to formula (1), hARi2, hRjRj2 and PRj are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the cumulative distribution functionCDF of max1≤i≤NRhARi2 is Fmax1≤i≤NRhARi2x=1−e−xNR; then its PDF is(14)fmax1≤i≤NRhARi2x=∑m=0NRNRm−1me−xm.Theorem 1.
For the first stage, the transmission outage probability from Alice to Relay is derived as follows:(15)δ1=1+PAPRjmaxφ22R−1∑m=1NRNRm−1m1m,exp−m22R−1σR2PA×ln1+mφ22R−1PRjmaxPA.Proof.
see AppendixA.Remark 1.
The comments are as follows:(1.1) In the first stage, the maximum powerPRjmax of transmitting AN of a single antenna randomly selected by Relay can have a direct influence on the transmission outage probability δ1. The larger PRjmax is, the greater δ1 will be.(1.2) In the first stage, the transmission outage probabilityδ1 is a monotonically increasing function of channel noise σR2 and transmission rate R, that is, the larger σR2 and R are, the larger δ1 will be.(1.3) The configuration of antenna numberNR has a direct influence on the transmission outage probability δ1 of the first stage. The larger NR is, the smaller δ1 is.(1.4) In the first stage, the transmission outage probabilityδ1 is a monotonically decreasing function of Alice’s covert message sending power PA, that is, the larger PA is, the smaller δ1 is. And the transmission outage probability δ1 is a monotonically increasing function of the self-interference elimination coefficient ϕ, that is, the larger ϕ is, the larger δ1 is.


### 3.4.2. The Optimal AN

Theorem 2.
Under any given fixed covert information powerPA sent by Alice, predetermined ε and transmission rate R, the optimal AN power sent by Relay is given by(16)PRjmax∗=1−tεPAhAW2hRjW2tε,where tε is the solution of ξ∗¯t=1−ε.Proof.
see AppendixB.


### 3.4.3. The Maximized Effective Covert Rate

Theorem 3.
The optimal effective convert rate is derived as follows:(17)RC1∗¯=−RhRjW2tε∑m=1NRNRm−1m×exp−m22R−1σR2PAln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
see AppendixC.Remark 2.
The comments are as follows:(2.1) The larger the predetermined convert constraintε in the first stage, the greater the maximum effective convert rate of the first stage.Due toRC¯≜R1−δ, then there is δ1∗, so there will be RC1∗¯≜R1−δ1∗, and δ1∗ also has the property of δ1, so we can obtain Remarks 2.2–2.4.(2.2) In the first stage, the maximum effective covert rateRC1∗¯ increases with the increase of the antenna number NR and the power PA of covert messages sent by Alice.(2.3) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel noise σR2 of Relay and its self-interference elimination coefficient ϕ, that is, the larger σR2 and ϕ are, the smaller RC1∗¯ is.(2.4) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel coefficient hAW2, that is, the larger hAW2 is, the smaller RC1∗ is. And the maximum effective covert rate RC1∗¯ is the monotonic increasing function of the hRjW2, that is, the larger hRjW2 is, the larger RC1∗¯ is.Corollary 1.
In the first stage, if the powerPA of Alice sending covert messages to Relay increases, the maximum effective covert rate RC1∗¯ tends to a fixed value:(18)RC1∗¯=RhRjW2tε∑m=1NRNRm−1m,ln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
whenPA approaches ∞, then exp−m22R−1σR2/PA approaches 1, so the maximum effective convert rate approaches the result in formula (18) in the first stage.


## 3.4.1. The Transmission Outage Probability of Alice to Relay

Assuming that the transmission rateR is known, according to formula (1), hARi2, hRjRj2 and PRj are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the cumulative distribution functionCDF of max1≤i≤NRhARi2 is Fmax1≤i≤NRhARi2x=1−e−xNR; then its PDF is(14)fmax1≤i≤NRhARi2x=∑m=0NRNRm−1me−xm.Theorem 1.
For the first stage, the transmission outage probability from Alice to Relay is derived as follows:(15)δ1=1+PAPRjmaxφ22R−1∑m=1NRNRm−1m1m,exp−m22R−1σR2PA×ln1+mφ22R−1PRjmaxPA.Proof.
see AppendixA.Remark 1.
The comments are as follows:(1.1) In the first stage, the maximum powerPRjmax of transmitting AN of a single antenna randomly selected by Relay can have a direct influence on the transmission outage probability δ1. The larger PRjmax is, the greater δ1 will be.(1.2) In the first stage, the transmission outage probabilityδ1 is a monotonically increasing function of channel noise σR2 and transmission rate R, that is, the larger σR2 and R are, the larger δ1 will be.(1.3) The configuration of antenna numberNR has a direct influence on the transmission outage probability δ1 of the first stage. The larger NR is, the smaller δ1 is.(1.4) In the first stage, the transmission outage probabilityδ1 is a monotonically decreasing function of Alice’s covert message sending power PA, that is, the larger PA is, the smaller δ1 is. And the transmission outage probability δ1 is a monotonically increasing function of the self-interference elimination coefficient ϕ, that is, the larger ϕ is, the larger δ1 is.


## 3.4.2. The Optimal AN

Theorem 2.
Under any given fixed covert information powerPA sent by Alice, predetermined ε and transmission rate R, the optimal AN power sent by Relay is given by(16)PRjmax∗=1−tεPAhAW2hRjW2tε,where tε is the solution of ξ∗¯t=1−ε.Proof.
see AppendixB.


## 3.4.3. The Maximized Effective Covert Rate

Theorem 3.
The optimal effective convert rate is derived as follows:(17)RC1∗¯=−RhRjW2tε∑m=1NRNRm−1m×exp−m22R−1σR2PAln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
see AppendixC.Remark 2.
The comments are as follows:(2.1) The larger the predetermined convert constraintε in the first stage, the greater the maximum effective convert rate of the first stage.Due toRC¯≜R1−δ, then there is δ1∗, so there will be RC1∗¯≜R1−δ1∗, and δ1∗ also has the property of δ1, so we can obtain Remarks 2.2–2.4.(2.2) In the first stage, the maximum effective covert rateRC1∗¯ increases with the increase of the antenna number NR and the power PA of covert messages sent by Alice.(2.3) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel noise σR2 of Relay and its self-interference elimination coefficient ϕ, that is, the larger σR2 and ϕ are, the smaller RC1∗¯ is.(2.4) In the first stage, the maximum effective covert rateRC1∗¯ is the monotonic decreasing function of the channel coefficient hAW2, that is, the larger hAW2 is, the smaller RC1∗ is. And the maximum effective covert rate RC1∗¯ is the monotonic increasing function of the hRjW2, that is, the larger hRjW2 is, the larger RC1∗¯ is.Corollary 1.
In the first stage, if the powerPA of Alice sending covert messages to Relay increases, the maximum effective covert rate RC1∗¯ tends to a fixed value:(18)RC1∗¯=RhRjW2tε∑m=1NRNRm−1m,ln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
whenPA approaches ∞, then exp−m22R−1σR2/PA approaches 1, so the maximum effective convert rate approaches the result in formula (18) in the first stage.


## 4. The Second Stage: Relay Forwards Covert Messages to Bob

Relay forwards Alice’s information to Bob and sends zero-forcing AN signal to Willie. Relay selects the best antenna according to the CSI between it and Bob and decodes and forwards the convert messages sent by Alice to Bob, and the remainingNR−1 antennas are all used to send zero-forcing AN to Willie.Relay works in the decoding and forwarding mode. In the second stage, Relay decodes and encodes the received signal and forwards it to Bob, whose transmitted signal isXRi. The PA is fixed, and Relay’s forwarding mode is decoded forwarding, so it is assumed that Alice’s covert messages power is also fixed by Relay forwarding; both Bob and Willie know this.


### 4.1. Reception at Bob

The instantaneous SINR at Bob is given by(19)γBob=PRkmax1≤k≤NRhRkB2σB2,where hRkB is the channel coefficient between the k-th antenna selected by Relay and Bob; here, k=1,2,…,NR, PRk is the fixed power of Relay forwarding Alice covert messages; σB2 is the channel variance of Relay.In the second stage, when Relay sends covert messages, the signal received by Bob can be expressed as follows:(20)yBobi=PRkmax1≤k≤NRhRkBxRi+nBi,where xRi is the covert signal forwarded by Relay, satisfying ExRixR†i=1; i=1,2,…,n represents the symbol index, n is the total number of channels used in each slot, and nBi is the AWGN at Bob with σB2 as its variance, i.e., nBi∼CN0,σB2.


### 4.2. Detection Metrics at Willie

In order to cause the uncertainty of Willie’s detection power, Relay uses the remainingNR−1 antennas for zero-forcing beamforming, which interferes with Willie’s transmission without affecting Bob’s reception. The optimal weighted vector WZF is the solution of the following optimization problem:(21)maxWZFhRW†WZF,s.t.hRB†WZF=0WZFF=1,where † is the conjugate transpose operator, •F denotes the Frobenius norm, and hRW or hRB represents the n-dimensional channel vectors between Relay and Willie or Bob, respectively. According to the theorems in [40, 41], the solution of the optimization problem in formula (21), i.e., the precoding vector WZF, can be described as follows:(22)WZF=T⊥hRWT⊥hRW,where T⊥=I−hRBhRB†hRB−1hRB† is the projection idempotent matrix with rank NR−2.Let us defineZ1≜PZhRW†WZF2/σW2 and Z≜PZhRW†WZF2. According to equations (11) and (12) in [40], we have(23)fZ1Z=ZNR−3exp−σw2Z/PZNR−3!PZ/σw2NR−2,NR≥3,Z≥0,(24)FZ1Z=1−exp−ZσW2PZ∑l=0NR−31l!ZσW2PZl,Z≥0.The SINR at Willie is given by(25)γwillie2=PRkhRkW2PZhRW†WZF+σW2,where hRkW is the channel coefficient between the k-th antenna that Relay chooses to forward the covert messages and Bob and PZ is the AN power of ZFB sent by Relay. In a communication slot of the second stage, Willie has to decide whether Relay has forwarded covert messages to Bob [42]. Therefore, Willie is faced with a binary hypothesis testing problem again, where the zero hypothesis M0 means Relay has not forwarded covert messages, and the alternative hypothesis M1 means Relay has forwarded covert messages to Bob. Based on these assumptions, Willie receives signals is given by(26)ywillie2i=PZhRW†WZFVR2i+nWi,M0,PRkhRkWxRi+PZhRW†WZFVR2i+nWi,M1.Willie does not know the value ofPZ in the time slot, but the value of PRK is known. In the second stage, Willie attempts to detect whether ywillie2 is M0 or M1. Through the application of Neyman–Perason criterion and likelihood ratio test, the optimal decision rule for Willie to minimize his detection error is as follows:(27)TW2>U0<U1τ,where TW2=1/n∑i=1nyWillie2i is the average power received at Willie in the slot, τ is a predefined Willie’s detection threshold, and U0 and U1 are binary decisions that infer whether Relay forwards covert messages, respectively. In this paper, we consider an infinite number of channel uses, that is, n⟶∞. Therefore, we have(28)TW2=PZhRW†WZF2+σW2,U0,PRkhRkW2+PZhRW†WZF2+σW2,U1..At the end of the communication slot, Willie has to make a decision. The false alarm rate is defined as the probability of Willie makingU1 decision under condition M0, which is expressed by α2=PU1|M0. Similarly, the miss detection rate is defined as the probability of Willie making U0 decision under condition M1, which is expressed by β2=PU0|M1. Assuming that the prior probabilities of M0 and M1 are equal, Willie’s detection performance can be judged by detecting error rate, which can be defined as follows:(29)ξ2≜α2+β2.


### 4.3. Detection Performance at Willie

#### 4.3.1. False Alarm Rate and Miss Detection Rate

Theorem 4.
In the second stage, Willie’s false alarm rate is derived as follows:(30)α2=1,τ≤σW2,xEi−x+exp−x,τ>σW2NR=3,exp−x,τ>σW2NR=4,1+∑l=2NR−31ll−1∑k=0l−2xk+1k!exp−x,τ>σW2NR≥5,and the miss detection rate is formula (31) where x=τ−σw2/NR−1PRjmax, y=τ−σw2−PRkhRkW2/NR−1PRjmax.ThePRjmax is the maximum AN power transmitted by Relay’s single antenna, which has been introduced in Section 3.1.(31)β2=0,τ≤σW2+PRkhRkW2,1−yEi−y+exp−y,τ>σW2+PRkhRkW2NR=3,1−exp−y,τ>σW2+PRkhRkW2NR=4,1−1+∑l=2NR−31ll−1∑k=0l−2yk+1k!exp−y,τ>σW2+PRkhRkW2NR≥5.Proof.
see AppendixD.


#### 4.3.2. Optimal Detection Threshold and Minimum Detection Error Rate

Theorem 5.
According to Willie’s hypothesis, the optimal detection threshold can be expressed as follows:(32)τ∗=σW2+PRKWhRKW2.
The corresponding minimum detection error rate is derived as follows:(33)ξ2∗=x∗Ei−x∗+exp−x∗,NR=3,exp−x∗,NR=4,1+∑l=2NR−31ll−1∑k=0l−2x∗k+1k!exp−x∗,NR≥5,where(34)x∗=PRkhRkW2NR−1PRjmaxλRW.Proof.
see AppendixE.Remark 3.
The comments are as follows:(3.1) In the second stage, the minimum detection error rateξ2∗ is a monotonic increasing function of the maximum AN power PRjmax of an antenna in Relay using ZFB method, that is, the larger PRjmax is, the larger ξ2∗ is. The minimum detection error rate ξ2∗ is also monotonically increasing with respect to the number of antennas NR, that is, the larger NR is, the larger ξ2∗ is.(3.2) In the second stage, the minimum detection error rateξ2∗ is a monotonic decreasing function of Relay forwarding covert message power PRk, that is, the larger PRk is, the smaller ξ2∗ is.


### 4.4. Covert Performance

In the second stage, the maximized effective covert rate can be expressed as follows:(35)argmaxPRkRC¯,s.t.ξ2∗≥1−ε.


#### 4.4.1. The Transmission Outage Probability of Relay to Bob

Assuming that the transmission rateR is known, according to formula (19), hRkB2 and PRK are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the (CDF) of max1≤i≤NRhRkB2 is Fmax1≤k≤NRhRkB2x=1−e−xNR; then its PDF is(36)fmax1≤k≤NRhRkB2x=∑m=0NRNRm−1me−xm.Theorem 6.
For the second stage, the transmission outage probability from Relay to Bob is derived as follows:(37)δ=1−exp−22R−1σB2PRkNR.Proof.
see AppendixF.Remark 4.
The comments are as follows:(4.1) In the second stage, the larger the powerPRk of Relay forwarding Alice covert messages is, the smaller the transmission outage probability δ2 is(4.2) In the second stage, the transmission outage probabilityδ2 is a monotonically increasing function of σB2 and R, that is, the larger σB2 and R are, the greater δ2 is(4.3) In the second stage, the transmission outage probabilityδ2 decreases with the increase of the number of antennas NR, that is, the larger NR is, the smaller δ2 is


#### 4.4.2. Optimal Transmission Power of Relay Forwarding Covert Messages

Theorem 7.
Under any givenε and transmission rate R, the optimal transmission power of Relay forwarding Alice’s covert message is expressed as follows:(38)PRk∗=NR−1PRjmaxtε1hRkW2,NR=3,NR−1PRjmaxhRkW2ln11−ε,NR=4,NR−1PRjmaxtε2hRkW2,NR≥5,where tε1 is the solution of xEi−x+exp−x=1−ε and tε2 is the solution of 1+∑l=2NR−31/ll−1∑k=0l−2xk+1/k!exp−x=1−ε.Proof:.
see AppendixG.


#### 4.4.3. Optimal Effective Convert Rate

Theorem 8.
In the second stage, the optimal effective convert rate is derived as follows:(39)RC2¯∗=R−R1−exp−22R−1σB2PRk∗NR.Proof.
see AppendixH.Remark 5.
The comments are as follows:(5.1) In the second stage, the optimal effective convert rateRC2∗¯ is a monotonic increasing function of the AN power PRjmax, that is, the larger PRjmax is, the larger RC2∗¯ is(5.2) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of antenna number NR and ε, that is, the larger NR and ε, the greater RC2∗¯(5.3) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of the convert message power PRk∗ forwarded by Relay, and it decreases with the increase of channel noise σB2 and channel coefficient hRkW2Corollary 2.
In the second stage, if the optimal power of Relay forwarding Alice’s covert messages is increased, the maximum effective covert rate approaches a fixed valueR.(40)limPRk∗⟶∞RC2∗¯=R.Proof.
according to formula (39), when PRk∗ tends to ∞, the exp−22R−1σB2/PRk∗ tends to 1; then 1−exp−22R−1σB2/PRk∗NR⟶0, so RC2∗¯⟶R.


## 4.1. Reception at Bob

The instantaneous SINR at Bob is given by(19)γBob=PRkmax1≤k≤NRhRkB2σB2,where hRkB is the channel coefficient between the k-th antenna selected by Relay and Bob; here, k=1,2,…,NR, PRk is the fixed power of Relay forwarding Alice covert messages; σB2 is the channel variance of Relay.In the second stage, when Relay sends covert messages, the signal received by Bob can be expressed as follows:(20)yBobi=PRkmax1≤k≤NRhRkBxRi+nBi,where xRi is the covert signal forwarded by Relay, satisfying ExRixR†i=1; i=1,2,…,n represents the symbol index, n is the total number of channels used in each slot, and nBi is the AWGN at Bob with σB2 as its variance, i.e., nBi∼CN0,σB2.


## 4.2. Detection Metrics at Willie

In order to cause the uncertainty of Willie’s detection power, Relay uses the remainingNR−1 antennas for zero-forcing beamforming, which interferes with Willie’s transmission without affecting Bob’s reception. The optimal weighted vector WZF is the solution of the following optimization problem:(21)maxWZFhRW†WZF,s.t.hRB†WZF=0WZFF=1,where † is the conjugate transpose operator, •F denotes the Frobenius norm, and hRW or hRB represents the n-dimensional channel vectors between Relay and Willie or Bob, respectively. According to the theorems in [40, 41], the solution of the optimization problem in formula (21), i.e., the precoding vector WZF, can be described as follows:(22)WZF=T⊥hRWT⊥hRW,where T⊥=I−hRBhRB†hRB−1hRB† is the projection idempotent matrix with rank NR−2.Let us defineZ1≜PZhRW†WZF2/σW2 and Z≜PZhRW†WZF2. According to equations (11) and (12) in [40], we have(23)fZ1Z=ZNR−3exp−σw2Z/PZNR−3!PZ/σw2NR−2,NR≥3,Z≥0,(24)FZ1Z=1−exp−ZσW2PZ∑l=0NR−31l!ZσW2PZl,Z≥0.The SINR at Willie is given by(25)γwillie2=PRkhRkW2PZhRW†WZF+σW2,where hRkW is the channel coefficient between the k-th antenna that Relay chooses to forward the covert messages and Bob and PZ is the AN power of ZFB sent by Relay. In a communication slot of the second stage, Willie has to decide whether Relay has forwarded covert messages to Bob [42]. Therefore, Willie is faced with a binary hypothesis testing problem again, where the zero hypothesis M0 means Relay has not forwarded covert messages, and the alternative hypothesis M1 means Relay has forwarded covert messages to Bob. Based on these assumptions, Willie receives signals is given by(26)ywillie2i=PZhRW†WZFVR2i+nWi,M0,PRkhRkWxRi+PZhRW†WZFVR2i+nWi,M1.Willie does not know the value ofPZ in the time slot, but the value of PRK is known. In the second stage, Willie attempts to detect whether ywillie2 is M0 or M1. Through the application of Neyman–Perason criterion and likelihood ratio test, the optimal decision rule for Willie to minimize his detection error is as follows:(27)TW2>U0<U1τ,where TW2=1/n∑i=1nyWillie2i is the average power received at Willie in the slot, τ is a predefined Willie’s detection threshold, and U0 and U1 are binary decisions that infer whether Relay forwards covert messages, respectively. In this paper, we consider an infinite number of channel uses, that is, n⟶∞. Therefore, we have(28)TW2=PZhRW†WZF2+σW2,U0,PRkhRkW2+PZhRW†WZF2+σW2,U1..At the end of the communication slot, Willie has to make a decision. The false alarm rate is defined as the probability of Willie makingU1 decision under condition M0, which is expressed by α2=PU1|M0. Similarly, the miss detection rate is defined as the probability of Willie making U0 decision under condition M1, which is expressed by β2=PU0|M1. Assuming that the prior probabilities of M0 and M1 are equal, Willie’s detection performance can be judged by detecting error rate, which can be defined as follows:(29)ξ2≜α2+β2.


## 4.3. Detection Performance at Willie

### 4.3.1. False Alarm Rate and Miss Detection Rate

Theorem 4.
In the second stage, Willie’s false alarm rate is derived as follows:(30)α2=1,τ≤σW2,xEi−x+exp−x,τ>σW2NR=3,exp−x,τ>σW2NR=4,1+∑l=2NR−31ll−1∑k=0l−2xk+1k!exp−x,τ>σW2NR≥5,and the miss detection rate is formula (31) where x=τ−σw2/NR−1PRjmax, y=τ−σw2−PRkhRkW2/NR−1PRjmax.ThePRjmax is the maximum AN power transmitted by Relay’s single antenna, which has been introduced in Section 3.1.(31)β2=0,τ≤σW2+PRkhRkW2,1−yEi−y+exp−y,τ>σW2+PRkhRkW2NR=3,1−exp−y,τ>σW2+PRkhRkW2NR=4,1−1+∑l=2NR−31ll−1∑k=0l−2yk+1k!exp−y,τ>σW2+PRkhRkW2NR≥5.Proof.
see AppendixD.


### 4.3.2. Optimal Detection Threshold and Minimum Detection Error Rate

Theorem 5.
According to Willie’s hypothesis, the optimal detection threshold can be expressed as follows:(32)τ∗=σW2+PRKWhRKW2.
The corresponding minimum detection error rate is derived as follows:(33)ξ2∗=x∗Ei−x∗+exp−x∗,NR=3,exp−x∗,NR=4,1+∑l=2NR−31ll−1∑k=0l−2x∗k+1k!exp−x∗,NR≥5,where(34)x∗=PRkhRkW2NR−1PRjmaxλRW.Proof.
see AppendixE.Remark 3.
The comments are as follows:(3.1) In the second stage, the minimum detection error rateξ2∗ is a monotonic increasing function of the maximum AN power PRjmax of an antenna in Relay using ZFB method, that is, the larger PRjmax is, the larger ξ2∗ is. The minimum detection error rate ξ2∗ is also monotonically increasing with respect to the number of antennas NR, that is, the larger NR is, the larger ξ2∗ is.(3.2) In the second stage, the minimum detection error rateξ2∗ is a monotonic decreasing function of Relay forwarding covert message power PRk, that is, the larger PRk is, the smaller ξ2∗ is.


## 4.3.1. False Alarm Rate and Miss Detection Rate

Theorem 4.
In the second stage, Willie’s false alarm rate is derived as follows:(30)α2=1,τ≤σW2,xEi−x+exp−x,τ>σW2NR=3,exp−x,τ>σW2NR=4,1+∑l=2NR−31ll−1∑k=0l−2xk+1k!exp−x,τ>σW2NR≥5,and the miss detection rate is formula (31) where x=τ−σw2/NR−1PRjmax, y=τ−σw2−PRkhRkW2/NR−1PRjmax.ThePRjmax is the maximum AN power transmitted by Relay’s single antenna, which has been introduced in Section 3.1.(31)β2=0,τ≤σW2+PRkhRkW2,1−yEi−y+exp−y,τ>σW2+PRkhRkW2NR=3,1−exp−y,τ>σW2+PRkhRkW2NR=4,1−1+∑l=2NR−31ll−1∑k=0l−2yk+1k!exp−y,τ>σW2+PRkhRkW2NR≥5.Proof.
see AppendixD.


## 4.3.2. Optimal Detection Threshold and Minimum Detection Error Rate

Theorem 5.
According to Willie’s hypothesis, the optimal detection threshold can be expressed as follows:(32)τ∗=σW2+PRKWhRKW2.
The corresponding minimum detection error rate is derived as follows:(33)ξ2∗=x∗Ei−x∗+exp−x∗,NR=3,exp−x∗,NR=4,1+∑l=2NR−31ll−1∑k=0l−2x∗k+1k!exp−x∗,NR≥5,where(34)x∗=PRkhRkW2NR−1PRjmaxλRW.Proof.
see AppendixE.Remark 3.
The comments are as follows:(3.1) In the second stage, the minimum detection error rateξ2∗ is a monotonic increasing function of the maximum AN power PRjmax of an antenna in Relay using ZFB method, that is, the larger PRjmax is, the larger ξ2∗ is. The minimum detection error rate ξ2∗ is also monotonically increasing with respect to the number of antennas NR, that is, the larger NR is, the larger ξ2∗ is.(3.2) In the second stage, the minimum detection error rateξ2∗ is a monotonic decreasing function of Relay forwarding covert message power PRk, that is, the larger PRk is, the smaller ξ2∗ is.


## 4.4. Covert Performance

In the second stage, the maximized effective covert rate can be expressed as follows:(35)argmaxPRkRC¯,s.t.ξ2∗≥1−ε.


### 4.4.1. The Transmission Outage Probability of Relay to Bob

Assuming that the transmission rateR is known, according to formula (19), hRkB2 and PRK are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the (CDF) of max1≤i≤NRhRkB2 is Fmax1≤k≤NRhRkB2x=1−e−xNR; then its PDF is(36)fmax1≤k≤NRhRkB2x=∑m=0NRNRm−1me−xm.Theorem 6.
For the second stage, the transmission outage probability from Relay to Bob is derived as follows:(37)δ=1−exp−22R−1σB2PRkNR.Proof.
see AppendixF.Remark 4.
The comments are as follows:(4.1) In the second stage, the larger the powerPRk of Relay forwarding Alice covert messages is, the smaller the transmission outage probability δ2 is(4.2) In the second stage, the transmission outage probabilityδ2 is a monotonically increasing function of σB2 and R, that is, the larger σB2 and R are, the greater δ2 is(4.3) In the second stage, the transmission outage probabilityδ2 decreases with the increase of the number of antennas NR, that is, the larger NR is, the smaller δ2 is


### 4.4.2. Optimal Transmission Power of Relay Forwarding Covert Messages

Theorem 7.
Under any givenε and transmission rate R, the optimal transmission power of Relay forwarding Alice’s covert message is expressed as follows:(38)PRk∗=NR−1PRjmaxtε1hRkW2,NR=3,NR−1PRjmaxhRkW2ln11−ε,NR=4,NR−1PRjmaxtε2hRkW2,NR≥5,where tε1 is the solution of xEi−x+exp−x=1−ε and tε2 is the solution of 1+∑l=2NR−31/ll−1∑k=0l−2xk+1/k!exp−x=1−ε.Proof:.
see AppendixG.


### 4.4.3. Optimal Effective Convert Rate

Theorem 8.
In the second stage, the optimal effective convert rate is derived as follows:(39)RC2¯∗=R−R1−exp−22R−1σB2PRk∗NR.Proof.
see AppendixH.Remark 5.
The comments are as follows:(5.1) In the second stage, the optimal effective convert rateRC2∗¯ is a monotonic increasing function of the AN power PRjmax, that is, the larger PRjmax is, the larger RC2∗¯ is(5.2) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of antenna number NR and ε, that is, the larger NR and ε, the greater RC2∗¯(5.3) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of the convert message power PRk∗ forwarded by Relay, and it decreases with the increase of channel noise σB2 and channel coefficient hRkW2Corollary 2.
In the second stage, if the optimal power of Relay forwarding Alice’s covert messages is increased, the maximum effective covert rate approaches a fixed valueR.(40)limPRk∗⟶∞RC2∗¯=R.Proof.
according to formula (39), when PRk∗ tends to ∞, the exp−22R−1σB2/PRk∗ tends to 1; then 1−exp−22R−1σB2/PRk∗NR⟶0, so RC2∗¯⟶R.


## 4.4.1. The Transmission Outage Probability of Relay to Bob

Assuming that the transmission rateR is known, according to formula (19), hRkB2 and PRK are random variables, which can still cause transmission interruption.Since the wireless channel is subject to independent quasi-static Rayleigh fading and independent identically distributed, the (CDF) of max1≤i≤NRhRkB2 is Fmax1≤k≤NRhRkB2x=1−e−xNR; then its PDF is(36)fmax1≤k≤NRhRkB2x=∑m=0NRNRm−1me−xm.Theorem 6.
For the second stage, the transmission outage probability from Relay to Bob is derived as follows:(37)δ=1−exp−22R−1σB2PRkNR.Proof.
see AppendixF.Remark 4.
The comments are as follows:(4.1) In the second stage, the larger the powerPRk of Relay forwarding Alice covert messages is, the smaller the transmission outage probability δ2 is(4.2) In the second stage, the transmission outage probabilityδ2 is a monotonically increasing function of σB2 and R, that is, the larger σB2 and R are, the greater δ2 is(4.3) In the second stage, the transmission outage probabilityδ2 decreases with the increase of the number of antennas NR, that is, the larger NR is, the smaller δ2 is


## 4.4.2. Optimal Transmission Power of Relay Forwarding Covert Messages

Theorem 7.
Under any givenε and transmission rate R, the optimal transmission power of Relay forwarding Alice’s covert message is expressed as follows:(38)PRk∗=NR−1PRjmaxtε1hRkW2,NR=3,NR−1PRjmaxhRkW2ln11−ε,NR=4,NR−1PRjmaxtε2hRkW2,NR≥5,where tε1 is the solution of xEi−x+exp−x=1−ε and tε2 is the solution of 1+∑l=2NR−31/ll−1∑k=0l−2xk+1/k!exp−x=1−ε.Proof:.
see AppendixG.


## 4.4.3. Optimal Effective Convert Rate

Theorem 8.
In the second stage, the optimal effective convert rate is derived as follows:(39)RC2¯∗=R−R1−exp−22R−1σB2PRk∗NR.Proof.
see AppendixH.Remark 5.
The comments are as follows:(5.1) In the second stage, the optimal effective convert rateRC2∗¯ is a monotonic increasing function of the AN power PRjmax, that is, the larger PRjmax is, the larger RC2∗¯ is(5.2) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of antenna number NR and ε, that is, the larger NR and ε, the greater RC2∗¯(5.3) In the second stage, the optimal effective convert rateRC2∗¯ increases with the increase of the convert message power PRk∗ forwarded by Relay, and it decreases with the increase of channel noise σB2 and channel coefficient hRkW2Corollary 2.
In the second stage, if the optimal power of Relay forwarding Alice’s covert messages is increased, the maximum effective covert rate approaches a fixed valueR.(40)limPRk∗⟶∞RC2∗¯=R.Proof.
according to formula (39), when PRk∗ tends to ∞, the exp−22R−1σB2/PRk∗ tends to 1; then 1−exp−22R−1σB2/PRk∗NR⟶0, so RC2∗¯⟶R.


## 5. Total Performance

### 5.1. Transmission Outage Probability

Theorem 9.
The total transmission outage probability is formula (41).(41)δ=1+PAPRjmaxφ22R−1∑m=1NRNRm−1m1mexp−m22R−1σR2PAln1+mφ22R−1PRjmaxPA×1−1−exp−22R−1σB2PRkNR.Proof.
see AppendixI.


### 5.2. Maximum Effective Covert Rate

Theorem 10.
TheσR2=σB2,PA=PRk∗ is assumed; then the overall optimal effective covert rate is given by(42)RC∗¯=RC1∗¯=−RhRjW2tε∑m=1NRNRm−1m,exp−m22R−1σR2PAln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
see AppendixJ.


## 5.1. Transmission Outage Probability

Theorem 9.
The total transmission outage probability is formula (41).(41)δ=1+PAPRjmaxφ22R−1∑m=1NRNRm−1m1mexp−m22R−1σR2PAln1+mφ22R−1PRjmaxPA×1−1−exp−22R−1σB2PRkNR.Proof.
see AppendixI.


## 5.2. Maximum Effective Covert Rate

Theorem 10.
TheσR2=σB2,PA=PRk∗ is assumed; then the overall optimal effective covert rate is given by(42)RC∗¯=RC1∗¯=−RhRjW2tε∑m=1NRNRm−1m,exp−m22R−1σR2PAln1+mφ22R−11−tεhAW2/tεhRjW2mφ22R−11−tεhAW2.Proof.
see AppendixJ.


## 6. Numerical Result

### 6.1. The First Stage

In this section, we present some detailed numerical results to illustrate the influence of system parameters on the detection and covert performance of the system. By observing the results in these figures, it can be found that the analysis results are completely consistent with the derivation results, which verifies the correctness of the theoretical analysis. If there are no special cases, we will set some general parameter values. Due to the channel does not change in a time slot, that is, it remains a constant, the value of the channel coefficient can be set to 0 dB. At the same time, the value of channel noise is set to 0 dB. The transmitting power of a single antenna (including nodes with a single antenna) is set to 10 dB. We define the predetermined covert communication condition value as 0.05 and determine the transmission rate as 1 or 1.5. Specific simulation data can be referred to in the instructions below each figure.


#### 6.1.1. Transmission Outage Probability

Figures2–4 illustrate the relationship between the transmission outage probability δ1 and the maximum power PRjmax of a single antenna randomly selected by Relay based on different parameter values. It can be observed that the larger PRjmax is, the greater δ1 is. This is because Relay will also generate self-interference while sending interference to Willie. Such self-interference will affect the communication between Alice and the i-th antenna of Relay, and the transmission is more likely to be interrupted. This is consistent with Remark 1.1. In Figure 2, it can be observed that the transmission outage probability δ1 is a monotonic increasing function with respect to σR2. This is because the larger the channel noise of Relay is, the smaller the SINR of the channel is, and the easier the transmission is interrupted. It can also be seen from Figure 2 that δ1 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 1.2. In Figure 3, with the increase of the number of antennas, the transmission outage probability δ1 is smaller. This is because the more the number of antennas is configured, the more security gain can be obtained, so that the more secure the transmission is, the less likely the transmission is interrupted. This confirms the correctness of Remark 1.3. In Figure 4, we can see that δ1 is a monotonic increasing function of ϕ, that is, the larger the interference cancellation coefficient is, the easier the transmission is interrupted. This is because the self-interference coefficient determines the degree of self-interference. The greater the self-interference is, the less secure the transmission from Alice to Relay is. At the same time, it can be observed from Figure 4 that δ1 is a monotonic decreasing function of PA. The higher the power of hidden message transmission, the more favorable it is for Relay’s reception, and the lower the transmission outage probability. This is consistent with Remark 1.4. In addition, since the expected detection error rate is cited in [2], it can be seen from Figure 3 of [2]: (1) the expected detection error rate ξ∗¯ is a monotone increasing function of PRjmax and (2) the expected detection error rate ξ∗¯ is a monotonic decreasing function of PA.Figure 2
Transmission outage probability versusPRjmax with different value of σR2 and R, where PA=10dB, NR=5, and ϕ=0.5.Figure 3
Transmission outage probability versusPRjmax with different value of NR, where PA=10dB, σR2=0dB, R=1.5, and ϕ=0.5.Figure 4
Transmission outage probability versusPRjmax with different value of PA and ϕ, where NR=5, σR2=0dB, and R=1.5.


#### 6.1.2. Maximum Effective Convert Rate

Figures5–7 shows the probability simulation curves of the maximum effective convert rate RC1∗¯ under different power PA that Alice sends covert messages. It can be observed that RC1∗¯ increases with the increase of PA. This is because the larger the transmission power of the covert message, the larger the SINR of the antenna used to receive the covert message at Relay, and the smaller the transmission outage probability in the first stage, so that RC1∗¯ will be larger. It can be observed from Figure 5 that the larger ε is, the larger the maximum effective covert rate is. This is because the larger ε is and the smaller 1−ε is, the lower Willie’s minimum detection error rate is required, and the constraint condition of the maximum effective covert rate is more relaxed. Therefore, the maximum effective covert rate RC1∗¯ is a monotonic increasing function of ε, which is consistent with the content of Remark 2.1. It can also be observed from Figure 5 that with the increase of the antenna’s number NR, the maximum effective covert rate increases. The reason is that the more the number of antennas, the smaller the transmission outage probability, but the larger the RC1∗¯. This is consistent with Remark 2.2. It can be seen from Figure 6 that the maximum effective covert rate RC1∗¯ is a monotonic decreasing function of σR2 and ϕ. The larger the channel noise of Relay, the smaller the SINR of Relay, the easier the transmission is interrupted, and the smaller the maximum effective covert rate will be. The larger the self-interference cancellation coefficient ϕ, the greater the self-interference to Relay, and the greater the influence on the covert transmission from Alice to Relay, so that the maximum effective covert rate RC1∗¯ is smaller. This proves the correctness of inference Remark 2.3. Figure 7 illustrates the relationship between hAW2 and hRjW2 and the maximum effective covert rate RC1∗¯. When Alice sends covert messages, the larger the channel coefficient hAW2 between Alice and Willie is, the smaller the maximum effective covert rate RC1∗¯ is. Because the larger the channel gain is, the more favorable Willie’s detection is, the smaller the maximum effective covert rate is. And it can be observed that the larger the channel coefficient hRjW2 between Relay randomly selected single antenna and Willie, the greater the maximum effective covert rate. The reason is that the more the channel gain of transmit interference is, the more difficult Willie is to detect, and the easier Willie is to perform covertness. This is consistent with Remark 2.4. In addition, a common point can be seen from Figures 5–7, that is, when increased to a certain value, the maximum effective covert rate of the first stage tends to be a fixed value, which confirms the correctness of Corollary 1.Figure 5
Maximum effective covert rate versusPA with different value of ε and NR, where NR=5, σR2=0dB, R=1, hAW2=1, hRjW2=1, and ϕ=0.01.Figure 6
Maximum effective covert rate versusPA with different value of σR2 and ϕ, where R=1, hAW2=1, hRjW2=1, ε=0.05, and NR=3.Figure 7
Maximum effective covert rate versusPA with different value of hAW2 and hRjW2, where R=1, σR2=0dB, ϕ=0.01, ε=0.05, and NR=3.


### 6.2. The Second Stage

#### 6.2.1. Detection Error Probability

Figure8 shows the relationship between the false alarm rate α2, missed detection rate β2, detection error probability ξ2, and the threshold τ. It can be observed that the smaller the false alarm rate, the larger the missed detection rate, the smaller the missed detection rate, and the larger the false alarm rate. It is difficult for Willie to detect the covert transmission with probability 1, which is consistent with our expectation, and there is a minimum detection error probability; Figure 9 draws the detection error rate curve according to different antenna number n and further illustrates the feasibility of minimum detection error rate under different parameters. This verifies the correctness of Theorem 5.Figure 8
Minimum detection error rate, false alarm rate, and missed detection rate versusτ, where σB2=0dB, NR=3, PRjmax=PRk=10dB, and hRkW2=1.Figure 9
Minimum detection error rate versusτ with different value of NR, where σB2=0dB, PRjmax=PRk=10dB, and hRkW2=1.In Figures10 and 11, the relationship between the minimum detection error probability ξ2 and the maximum interference transmission power PRjmax of a single antenna formed by Relay using zero-forcing beamforming can be observed at the same time. The larger the PRjmax is, the greater the ξ2 is. And Figure 10 also describes the relationship between the minimum detection error probability and the number of antennas NR. It is easy to understand that the minimum detection error rate will increase with the increase in the number of antennas. This is because the larger the interference power and the number of antennas of a single antenna, the greater the total interference to Willie, and the greater the interference will hinder Willie’s detection. Willie will be more prone to make errors, resulting in a higher error rate of Willie. This is consistent with the conclusion of Remark 3.1. In addition to describing the relationship between ξ2 and PRjmax, Figure 11 also plots the relationship between the minimum detection error probability ξ2 and the power PRK of Relay forwarding Alice covert messages. The larger the PRK is, the smaller the ξ2 is. This is because the higher the power of Relay forwarding covert messages is, the easier Willie is to detect the transmission of covert messages, and the smaller the probability of error is. This verifies the correctness of Remark 3.2.Figure 10
Minimum detection error rate versusPRjmax with different value of NR, where PRk=10dB and hRkW2=1.Figure 11
Minimum detection error rate versusPRjmax with different value of PRk=10dB, where NR=3 and hRkW2=1.


#### 6.2.2. Transmission Outage Probability

It can be seen from Figures12 and 13 that the transmission outage probability δ2 is a monotonic decreasing function of Relay covert message transmission power PRK, that is, the larger the covert message transmission power PRK is, the smaller the transmission interruption probability δ2 of the second stage is. This is because increasing PRK will increase the SINR at Bob, so the more secure the transmission is, the less likely it is to be interrupted. This verifies the correctness of Remark 4.1. The relationship between the transmission outage probability δ2 and Bob’s channel noise σB2 is also described in Figure 12. δ2 increases with the increase of σB2. This is because the increase of channel noise e will reduce the signal-to–interference-noise ratio at Bob, which makes the transmission easier to be interrupted. It can also be observed from Figure 12 that δ2 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 4.2. It can be observed from Figure 13 that the transmission outage probability δ2 is a monotonic decreasing function of the number of Relay antennas NR, that is, the larger NR is, the smaller the transmission outage probability δ2 of the second stage is. This is because the increase in the number of antennas will increase the gain of the channel, thus making the transmission more secure. This verifies the correctness of Remark 4.3.Figure 12
Transmission outage probability versusPRk with different value of σB2 and R, where NR=3.Figure 13
Transmission outage probability versusPRk with different value of NR, where R=1.5 and σB2=0dB.


#### 6.2.3. Maximum Effective Covert Rate

It can be seen from Figures14 and 15 that the optimal effective covert rate RC2∗¯ in the second stage is a monotonic increasing function of the interference transmission power PRjmax of a single Relay’s antenna, that is, the greater the interference transmission power PRjmax of a single Relay’s antenna, the greater the optimal effective covert rate RC2∗¯ in the second stage. This is because the greater the interference, the more likely Willie’s detection is to make mistakes, and the higher the effective covert rate is. This is consistent with Remark 5.1. Similar to the first stage, the optimal effective covert rate RC2∗¯ in the second stage increases with the increase of the number of antennas NR and ε. The reason has been described in the first stage, and it is not described here. This is consistent with the content of Remark 5.2.Figure 14
Maximum effective covert rate versusPRjmax with different value of NR and ε, where R=1.5, hRjW2=1, and σB2=0dB.Figure 15
Maximum effective covert rate versusPRk with different value of σB2, where R=1.5, hRjW2=1, NR=5, and ε=0.05.Figure16 shows the relationship between the optimal effective covert rate RC2∗¯ and the power PRK of Relay forwarding covert messages and the noise σB2 of Bob channel. The higher the power PRK of Relay forwarding covert messages, the higher the RC2∗¯. The larger the noise σB2 of Bob channel is, the smaller the RC2∗¯ is. This is because the larger the PRK is, the larger Bob’s SINR is, which is the more conducive to the transmission of covert messages, the larger the σB2 is, the smaller the SINR of Bob will be, which is not conducive to covert message transmission. Figure 15 also describes the relationship between the optimal effective covert rate RC2∗¯ and the channel coefficient hRkW2. The larger the channel coefficient hRkW2 is, the smaller the RC2∗¯ is. Because it is the channel gain between Relay and Willie when Relay forwards Alice’s covert messages, it is obvious that the larger the coefficient is, the worse the transmission of covert messages will be. This is not only similar to the first stage conclusion but also verifies the correctness of Remark 5.3. In addition, Figures 14–16 shows that the effective covert rate of the second stage tends to a fixed value, which verifies the correctness of Corollary 2.Figure 16
Maximum effective covert rate versusPRjmax with different value of hRjW2, where R=1.5, NR=5, ε=0.05, and σB2 = 0 dB.


### 6.3. Total Performance

#### 6.3.1. Transmission Outage Probability

As in the first and second stages, we also graphically plot the relevant parameters of the total transmission outage probability, as shown in Figures17–20. By comprehensively comparing the transmission outage probability in the first and second stages, we can get the contents in Table 2. Among them, the upward front represents the increase of the value, and the downward arrow represents the decrease of the value.Figure 17
Total transmission outage probability versusPRjmax with different value of PRk and PA, where σR2=0dB, σB2=0dB, R=1.5, NR=3, and ϕ=0.05.Figure 18
Total transmission outage probability versusPRjmax with different value of σR2 and σB2, where PRk=10dB, PA=10dB, R=1.5, NR=3, and ϕ=0.05.Figure 19
Total transmission outage probability versusPRjmax with different value of NR, where PRk=10dB, PA=10dB, R=1.5, ϕ=0.05, PRk=10dB, and PA=10dB.Figure 20
Total transmission outage probability versusPRjmax with different value of ϕ and R, where PRk=10dB, PA=10dB, NR=3, σR2=0dB, and σB2=0dB.Table 2
Transmission outage probability results.
ParameterParameter variationδ1δ2δInterference power↑↑↑↑Channel noise from the receiver↑↑↑↑Transmission rate↑↑↑↑Antennas number↑↓↓↓Covert message sending power↑↓↓↓Self-interference coefficient↑↑—↑It can be observed from Table2 that for the same parameters, the first stage, the second stage, and the total transmission outage probability have the same conclusion, which is consistent with our expectation. The causes of these graph phenomena have been described in detail in the previous stages and will not be repeated here.


#### 6.3.2. Maximum Effective Covert Rate

Because the total maximum effective covert rate is smaller in the first and second stages, the maximum effective covert rates of the first and second stages are drawn synthetically to facilitate the comparison. The details are shown in Figure21. Obviously, the maximum effective covert rate of the first stage is less than that of the second stage.Figure 21
Comparison of stage 1 and stage 2 maximum effective covert rates, whereσR2=0dB, σB2=0dB, PRk=10dB, R=1.5, ϕ=0.01, NR=3, ε=0.05, hRjW2=1, and hAW=1.By comprehensively comparing the maximum effective covert rate in the first and second stages, we can get the contents in Table3.Table 3
Maximum effective covert rate results.
ParameterParameter variationRC1∗¯RC2∗¯RC∗¯Covert message sending power↑↑↑↑Covert constraints,ε↑↑↑↑Antennas number↑↑↑↑Channel noise from the receiver↑↓↓↓Self-interference coefficient↑↓—↓The channel coefficient between covert message sender and Willie↑↓↓↓The channel coefficient between the AN sender and Willie↑↑—↑Interference power↑—↑—It can be seen from Table3 that for the same parameters, the total maximum effective concealment rate has the same conclusion as the maximum effective convert rate in one stage. This is because zero-forcing beamforming is used in the second stage to interfere with Willie without any impact on Bob, the transmission interruption probability will be smaller, and the corresponding maximum effective convert rate will be larger, so the maximum effective convert rate in the second stage is significantly higher than that in the first stage. This also verifies the correctness of Theorem 10.


## 6.1. The First Stage

In this section, we present some detailed numerical results to illustrate the influence of system parameters on the detection and covert performance of the system. By observing the results in these figures, it can be found that the analysis results are completely consistent with the derivation results, which verifies the correctness of the theoretical analysis. If there are no special cases, we will set some general parameter values. Due to the channel does not change in a time slot, that is, it remains a constant, the value of the channel coefficient can be set to 0 dB. At the same time, the value of channel noise is set to 0 dB. The transmitting power of a single antenna (including nodes with a single antenna) is set to 10 dB. We define the predetermined covert communication condition value as 0.05 and determine the transmission rate as 1 or 1.5. Specific simulation data can be referred to in the instructions below each figure.


### 6.1.1. Transmission Outage Probability

Figures2–4 illustrate the relationship between the transmission outage probability δ1 and the maximum power PRjmax of a single antenna randomly selected by Relay based on different parameter values. It can be observed that the larger PRjmax is, the greater δ1 is. This is because Relay will also generate self-interference while sending interference to Willie. Such self-interference will affect the communication between Alice and the i-th antenna of Relay, and the transmission is more likely to be interrupted. This is consistent with Remark 1.1. In Figure 2, it can be observed that the transmission outage probability δ1 is a monotonic increasing function with respect to σR2. This is because the larger the channel noise of Relay is, the smaller the SINR of the channel is, and the easier the transmission is interrupted. It can also be seen from Figure 2 that δ1 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 1.2. In Figure 3, with the increase of the number of antennas, the transmission outage probability δ1 is smaller. This is because the more the number of antennas is configured, the more security gain can be obtained, so that the more secure the transmission is, the less likely the transmission is interrupted. This confirms the correctness of Remark 1.3. In Figure 4, we can see that δ1 is a monotonic increasing function of ϕ, that is, the larger the interference cancellation coefficient is, the easier the transmission is interrupted. This is because the self-interference coefficient determines the degree of self-interference. The greater the self-interference is, the less secure the transmission from Alice to Relay is. At the same time, it can be observed from Figure 4 that δ1 is a monotonic decreasing function of PA. The higher the power of hidden message transmission, the more favorable it is for Relay’s reception, and the lower the transmission outage probability. This is consistent with Remark 1.4. In addition, since the expected detection error rate is cited in [2], it can be seen from Figure 3 of [2]: (1) the expected detection error rate ξ∗¯ is a monotone increasing function of PRjmax and (2) the expected detection error rate ξ∗¯ is a monotonic decreasing function of PA.Figure 2
Transmission outage probability versusPRjmax with different value of σR2 and R, where PA=10dB, NR=5, and ϕ=0.5.Figure 3
Transmission outage probability versusPRjmax with different value of NR, where PA=10dB, σR2=0dB, R=1.5, and ϕ=0.5.Figure 4
Transmission outage probability versusPRjmax with different value of PA and ϕ, where NR=5, σR2=0dB, and R=1.5.


### 6.1.2. Maximum Effective Convert Rate

Figures5–7 shows the probability simulation curves of the maximum effective convert rate RC1∗¯ under different power PA that Alice sends covert messages. It can be observed that RC1∗¯ increases with the increase of PA. This is because the larger the transmission power of the covert message, the larger the SINR of the antenna used to receive the covert message at Relay, and the smaller the transmission outage probability in the first stage, so that RC1∗¯ will be larger. It can be observed from Figure 5 that the larger ε is, the larger the maximum effective covert rate is. This is because the larger ε is and the smaller 1−ε is, the lower Willie’s minimum detection error rate is required, and the constraint condition of the maximum effective covert rate is more relaxed. Therefore, the maximum effective covert rate RC1∗¯ is a monotonic increasing function of ε, which is consistent with the content of Remark 2.1. It can also be observed from Figure 5 that with the increase of the antenna’s number NR, the maximum effective covert rate increases. The reason is that the more the number of antennas, the smaller the transmission outage probability, but the larger the RC1∗¯. This is consistent with Remark 2.2. It can be seen from Figure 6 that the maximum effective covert rate RC1∗¯ is a monotonic decreasing function of σR2 and ϕ. The larger the channel noise of Relay, the smaller the SINR of Relay, the easier the transmission is interrupted, and the smaller the maximum effective covert rate will be. The larger the self-interference cancellation coefficient ϕ, the greater the self-interference to Relay, and the greater the influence on the covert transmission from Alice to Relay, so that the maximum effective covert rate RC1∗¯ is smaller. This proves the correctness of inference Remark 2.3. Figure 7 illustrates the relationship between hAW2 and hRjW2 and the maximum effective covert rate RC1∗¯. When Alice sends covert messages, the larger the channel coefficient hAW2 between Alice and Willie is, the smaller the maximum effective covert rate RC1∗¯ is. Because the larger the channel gain is, the more favorable Willie’s detection is, the smaller the maximum effective covert rate is. And it can be observed that the larger the channel coefficient hRjW2 between Relay randomly selected single antenna and Willie, the greater the maximum effective covert rate. The reason is that the more the channel gain of transmit interference is, the more difficult Willie is to detect, and the easier Willie is to perform covertness. This is consistent with Remark 2.4. In addition, a common point can be seen from Figures 5–7, that is, when increased to a certain value, the maximum effective covert rate of the first stage tends to be a fixed value, which confirms the correctness of Corollary 1.Figure 5
Maximum effective covert rate versusPA with different value of ε and NR, where NR=5, σR2=0dB, R=1, hAW2=1, hRjW2=1, and ϕ=0.01.Figure 6
Maximum effective covert rate versusPA with different value of σR2 and ϕ, where R=1, hAW2=1, hRjW2=1, ε=0.05, and NR=3.Figure 7
Maximum effective covert rate versusPA with different value of hAW2 and hRjW2, where R=1, σR2=0dB, ϕ=0.01, ε=0.05, and NR=3.


## 6.1.1. Transmission Outage Probability

Figures2–4 illustrate the relationship between the transmission outage probability δ1 and the maximum power PRjmax of a single antenna randomly selected by Relay based on different parameter values. It can be observed that the larger PRjmax is, the greater δ1 is. This is because Relay will also generate self-interference while sending interference to Willie. Such self-interference will affect the communication between Alice and the i-th antenna of Relay, and the transmission is more likely to be interrupted. This is consistent with Remark 1.1. In Figure 2, it can be observed that the transmission outage probability δ1 is a monotonic increasing function with respect to σR2. This is because the larger the channel noise of Relay is, the smaller the SINR of the channel is, and the easier the transmission is interrupted. It can also be seen from Figure 2 that δ1 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 1.2. In Figure 3, with the increase of the number of antennas, the transmission outage probability δ1 is smaller. This is because the more the number of antennas is configured, the more security gain can be obtained, so that the more secure the transmission is, the less likely the transmission is interrupted. This confirms the correctness of Remark 1.3. In Figure 4, we can see that δ1 is a monotonic increasing function of ϕ, that is, the larger the interference cancellation coefficient is, the easier the transmission is interrupted. This is because the self-interference coefficient determines the degree of self-interference. The greater the self-interference is, the less secure the transmission from Alice to Relay is. At the same time, it can be observed from Figure 4 that δ1 is a monotonic decreasing function of PA. The higher the power of hidden message transmission, the more favorable it is for Relay’s reception, and the lower the transmission outage probability. This is consistent with Remark 1.4. In addition, since the expected detection error rate is cited in [2], it can be seen from Figure 3 of [2]: (1) the expected detection error rate ξ∗¯ is a monotone increasing function of PRjmax and (2) the expected detection error rate ξ∗¯ is a monotonic decreasing function of PA.Figure 2
Transmission outage probability versusPRjmax with different value of σR2 and R, where PA=10dB, NR=5, and ϕ=0.5.Figure 3
Transmission outage probability versusPRjmax with different value of NR, where PA=10dB, σR2=0dB, R=1.5, and ϕ=0.5.Figure 4
Transmission outage probability versusPRjmax with different value of PA and ϕ, where NR=5, σR2=0dB, and R=1.5.


## 6.1.2. Maximum Effective Convert Rate

Figures5–7 shows the probability simulation curves of the maximum effective convert rate RC1∗¯ under different power PA that Alice sends covert messages. It can be observed that RC1∗¯ increases with the increase of PA. This is because the larger the transmission power of the covert message, the larger the SINR of the antenna used to receive the covert message at Relay, and the smaller the transmission outage probability in the first stage, so that RC1∗¯ will be larger. It can be observed from Figure 5 that the larger ε is, the larger the maximum effective covert rate is. This is because the larger ε is and the smaller 1−ε is, the lower Willie’s minimum detection error rate is required, and the constraint condition of the maximum effective covert rate is more relaxed. Therefore, the maximum effective covert rate RC1∗¯ is a monotonic increasing function of ε, which is consistent with the content of Remark 2.1. It can also be observed from Figure 5 that with the increase of the antenna’s number NR, the maximum effective covert rate increases. The reason is that the more the number of antennas, the smaller the transmission outage probability, but the larger the RC1∗¯. This is consistent with Remark 2.2. It can be seen from Figure 6 that the maximum effective covert rate RC1∗¯ is a monotonic decreasing function of σR2 and ϕ. The larger the channel noise of Relay, the smaller the SINR of Relay, the easier the transmission is interrupted, and the smaller the maximum effective covert rate will be. The larger the self-interference cancellation coefficient ϕ, the greater the self-interference to Relay, and the greater the influence on the covert transmission from Alice to Relay, so that the maximum effective covert rate RC1∗¯ is smaller. This proves the correctness of inference Remark 2.3. Figure 7 illustrates the relationship between hAW2 and hRjW2 and the maximum effective covert rate RC1∗¯. When Alice sends covert messages, the larger the channel coefficient hAW2 between Alice and Willie is, the smaller the maximum effective covert rate RC1∗¯ is. Because the larger the channel gain is, the more favorable Willie’s detection is, the smaller the maximum effective covert rate is. And it can be observed that the larger the channel coefficient hRjW2 between Relay randomly selected single antenna and Willie, the greater the maximum effective covert rate. The reason is that the more the channel gain of transmit interference is, the more difficult Willie is to detect, and the easier Willie is to perform covertness. This is consistent with Remark 2.4. In addition, a common point can be seen from Figures 5–7, that is, when increased to a certain value, the maximum effective covert rate of the first stage tends to be a fixed value, which confirms the correctness of Corollary 1.Figure 5
Maximum effective covert rate versusPA with different value of ε and NR, where NR=5, σR2=0dB, R=1, hAW2=1, hRjW2=1, and ϕ=0.01.Figure 6
Maximum effective covert rate versusPA with different value of σR2 and ϕ, where R=1, hAW2=1, hRjW2=1, ε=0.05, and NR=3.Figure 7
Maximum effective covert rate versusPA with different value of hAW2 and hRjW2, where R=1, σR2=0dB, ϕ=0.01, ε=0.05, and NR=3.


## 6.2. The Second Stage

### 6.2.1. Detection Error Probability

Figure8 shows the relationship between the false alarm rate α2, missed detection rate β2, detection error probability ξ2, and the threshold τ. It can be observed that the smaller the false alarm rate, the larger the missed detection rate, the smaller the missed detection rate, and the larger the false alarm rate. It is difficult for Willie to detect the covert transmission with probability 1, which is consistent with our expectation, and there is a minimum detection error probability; Figure 9 draws the detection error rate curve according to different antenna number n and further illustrates the feasibility of minimum detection error rate under different parameters. This verifies the correctness of Theorem 5.Figure 8
Minimum detection error rate, false alarm rate, and missed detection rate versusτ, where σB2=0dB, NR=3, PRjmax=PRk=10dB, and hRkW2=1.Figure 9
Minimum detection error rate versusτ with different value of NR, where σB2=0dB, PRjmax=PRk=10dB, and hRkW2=1.In Figures10 and 11, the relationship between the minimum detection error probability ξ2 and the maximum interference transmission power PRjmax of a single antenna formed by Relay using zero-forcing beamforming can be observed at the same time. The larger the PRjmax is, the greater the ξ2 is. And Figure 10 also describes the relationship between the minimum detection error probability and the number of antennas NR. It is easy to understand that the minimum detection error rate will increase with the increase in the number of antennas. This is because the larger the interference power and the number of antennas of a single antenna, the greater the total interference to Willie, and the greater the interference will hinder Willie’s detection. Willie will be more prone to make errors, resulting in a higher error rate of Willie. This is consistent with the conclusion of Remark 3.1. In addition to describing the relationship between ξ2 and PRjmax, Figure 11 also plots the relationship between the minimum detection error probability ξ2 and the power PRK of Relay forwarding Alice covert messages. The larger the PRK is, the smaller the ξ2 is. This is because the higher the power of Relay forwarding covert messages is, the easier Willie is to detect the transmission of covert messages, and the smaller the probability of error is. This verifies the correctness of Remark 3.2.Figure 10
Minimum detection error rate versusPRjmax with different value of NR, where PRk=10dB and hRkW2=1.Figure 11
Minimum detection error rate versusPRjmax with different value of PRk=10dB, where NR=3 and hRkW2=1.


### 6.2.2. Transmission Outage Probability

It can be seen from Figures12 and 13 that the transmission outage probability δ2 is a monotonic decreasing function of Relay covert message transmission power PRK, that is, the larger the covert message transmission power PRK is, the smaller the transmission interruption probability δ2 of the second stage is. This is because increasing PRK will increase the SINR at Bob, so the more secure the transmission is, the less likely it is to be interrupted. This verifies the correctness of Remark 4.1. The relationship between the transmission outage probability δ2 and Bob’s channel noise σB2 is also described in Figure 12. δ2 increases with the increase of σB2. This is because the increase of channel noise e will reduce the signal-to–interference-noise ratio at Bob, which makes the transmission easier to be interrupted. It can also be observed from Figure 12 that δ2 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 4.2. It can be observed from Figure 13 that the transmission outage probability δ2 is a monotonic decreasing function of the number of Relay antennas NR, that is, the larger NR is, the smaller the transmission outage probability δ2 of the second stage is. This is because the increase in the number of antennas will increase the gain of the channel, thus making the transmission more secure. This verifies the correctness of Remark 4.3.Figure 12
Transmission outage probability versusPRk with different value of σB2 and R, where NR=3.Figure 13
Transmission outage probability versusPRk with different value of NR, where R=1.5 and σB2=0dB.


### 6.2.3. Maximum Effective Covert Rate

It can be seen from Figures14 and 15 that the optimal effective covert rate RC2∗¯ in the second stage is a monotonic increasing function of the interference transmission power PRjmax of a single Relay’s antenna, that is, the greater the interference transmission power PRjmax of a single Relay’s antenna, the greater the optimal effective covert rate RC2∗¯ in the second stage. This is because the greater the interference, the more likely Willie’s detection is to make mistakes, and the higher the effective covert rate is. This is consistent with Remark 5.1. Similar to the first stage, the optimal effective covert rate RC2∗¯ in the second stage increases with the increase of the number of antennas NR and ε. The reason has been described in the first stage, and it is not described here. This is consistent with the content of Remark 5.2.Figure 14
Maximum effective covert rate versusPRjmax with different value of NR and ε, where R=1.5, hRjW2=1, and σB2=0dB.Figure 15
Maximum effective covert rate versusPRk with different value of σB2, where R=1.5, hRjW2=1, NR=5, and ε=0.05.Figure16 shows the relationship between the optimal effective covert rate RC2∗¯ and the power PRK of Relay forwarding covert messages and the noise σB2 of Bob channel. The higher the power PRK of Relay forwarding covert messages, the higher the RC2∗¯. The larger the noise σB2 of Bob channel is, the smaller the RC2∗¯ is. This is because the larger the PRK is, the larger Bob’s SINR is, which is the more conducive to the transmission of covert messages, the larger the σB2 is, the smaller the SINR of Bob will be, which is not conducive to covert message transmission. Figure 15 also describes the relationship between the optimal effective covert rate RC2∗¯ and the channel coefficient hRkW2. The larger the channel coefficient hRkW2 is, the smaller the RC2∗¯ is. Because it is the channel gain between Relay and Willie when Relay forwards Alice’s covert messages, it is obvious that the larger the coefficient is, the worse the transmission of covert messages will be. This is not only similar to the first stage conclusion but also verifies the correctness of Remark 5.3. In addition, Figures 14–16 shows that the effective covert rate of the second stage tends to a fixed value, which verifies the correctness of Corollary 2.Figure 16
Maximum effective covert rate versusPRjmax with different value of hRjW2, where R=1.5, NR=5, ε=0.05, and σB2 = 0 dB.


## 6.2.1. Detection Error Probability

Figure8 shows the relationship between the false alarm rate α2, missed detection rate β2, detection error probability ξ2, and the threshold τ. It can be observed that the smaller the false alarm rate, the larger the missed detection rate, the smaller the missed detection rate, and the larger the false alarm rate. It is difficult for Willie to detect the covert transmission with probability 1, which is consistent with our expectation, and there is a minimum detection error probability; Figure 9 draws the detection error rate curve according to different antenna number n and further illustrates the feasibility of minimum detection error rate under different parameters. This verifies the correctness of Theorem 5.Figure 8
Minimum detection error rate, false alarm rate, and missed detection rate versusτ, where σB2=0dB, NR=3, PRjmax=PRk=10dB, and hRkW2=1.Figure 9
Minimum detection error rate versusτ with different value of NR, where σB2=0dB, PRjmax=PRk=10dB, and hRkW2=1.In Figures10 and 11, the relationship between the minimum detection error probability ξ2 and the maximum interference transmission power PRjmax of a single antenna formed by Relay using zero-forcing beamforming can be observed at the same time. The larger the PRjmax is, the greater the ξ2 is. And Figure 10 also describes the relationship between the minimum detection error probability and the number of antennas NR. It is easy to understand that the minimum detection error rate will increase with the increase in the number of antennas. This is because the larger the interference power and the number of antennas of a single antenna, the greater the total interference to Willie, and the greater the interference will hinder Willie’s detection. Willie will be more prone to make errors, resulting in a higher error rate of Willie. This is consistent with the conclusion of Remark 3.1. In addition to describing the relationship between ξ2 and PRjmax, Figure 11 also plots the relationship between the minimum detection error probability ξ2 and the power PRK of Relay forwarding Alice covert messages. The larger the PRK is, the smaller the ξ2 is. This is because the higher the power of Relay forwarding covert messages is, the easier Willie is to detect the transmission of covert messages, and the smaller the probability of error is. This verifies the correctness of Remark 3.2.Figure 10
Minimum detection error rate versusPRjmax with different value of NR, where PRk=10dB and hRkW2=1.Figure 11
Minimum detection error rate versusPRjmax with different value of PRk=10dB, where NR=3 and hRkW2=1.


## 6.2.2. Transmission Outage Probability

It can be seen from Figures12 and 13 that the transmission outage probability δ2 is a monotonic decreasing function of Relay covert message transmission power PRK, that is, the larger the covert message transmission power PRK is, the smaller the transmission interruption probability δ2 of the second stage is. This is because increasing PRK will increase the SINR at Bob, so the more secure the transmission is, the less likely it is to be interrupted. This verifies the correctness of Remark 4.1. The relationship between the transmission outage probability δ2 and Bob’s channel noise σB2 is also described in Figure 12. δ2 increases with the increase of σB2. This is because the increase of channel noise e will reduce the signal-to–interference-noise ratio at Bob, which makes the transmission easier to be interrupted. It can also be observed from Figure 12 that δ2 is a monotonic increasing function of R, that is, the higher the transmission rate is, the easier the transmission is interrupted. These observations are consistent with Remark 4.2. It can be observed from Figure 13 that the transmission outage probability δ2 is a monotonic decreasing function of the number of Relay antennas NR, that is, the larger NR is, the smaller the transmission outage probability δ2 of the second stage is. This is because the increase in the number of antennas will increase the gain of the channel, thus making the transmission more secure. This verifies the correctness of Remark 4.3.Figure 12
Transmission outage probability versusPRk with different value of σB2 and R, where NR=3.Figure 13
Transmission outage probability versusPRk with different value of NR, where R=1.5 and σB2=0dB.


## 6.2.3. Maximum Effective Covert Rate

It can be seen from Figures14 and 15 that the optimal effective covert rate RC2∗¯ in the second stage is a monotonic increasing function of the interference transmission power PRjmax of a single Relay’s antenna, that is, the greater the interference transmission power PRjmax of a single Relay’s antenna, the greater the optimal effective covert rate RC2∗¯ in the second stage. This is because the greater the interference, the more likely Willie’s detection is to make mistakes, and the higher the effective covert rate is. This is consistent with Remark 5.1. Similar to the first stage, the optimal effective covert rate RC2∗¯ in the second stage increases with the increase of the number of antennas NR and ε. The reason has been described in the first stage, and it is not described here. This is consistent with the content of Remark 5.2.Figure 14
Maximum effective covert rate versusPRjmax with different value of NR and ε, where R=1.5, hRjW2=1, and σB2=0dB.Figure 15
Maximum effective covert rate versusPRk with different value of σB2, where R=1.5, hRjW2=1, NR=5, and ε=0.05.Figure16 shows the relationship between the optimal effective covert rate RC2∗¯ and the power PRK of Relay forwarding covert messages and the noise σB2 of Bob channel. The higher the power PRK of Relay forwarding covert messages, the higher the RC2∗¯. The larger the noise σB2 of Bob channel is, the smaller the RC2∗¯ is. This is because the larger the PRK is, the larger Bob’s SINR is, which is the more conducive to the transmission of covert messages, the larger the σB2 is, the smaller the SINR of Bob will be, which is not conducive to covert message transmission. Figure 15 also describes the relationship between the optimal effective covert rate RC2∗¯ and the channel coefficient hRkW2. The larger the channel coefficient hRkW2 is, the smaller the RC2∗¯ is. Because it is the channel gain between Relay and Willie when Relay forwards Alice’s covert messages, it is obvious that the larger the coefficient is, the worse the transmission of covert messages will be. This is not only similar to the first stage conclusion but also verifies the correctness of Remark 5.3. In addition, Figures 14–16 shows that the effective covert rate of the second stage tends to a fixed value, which verifies the correctness of Corollary 2.Figure 16
Maximum effective covert rate versusPRjmax with different value of hRjW2, where R=1.5, NR=5, ε=0.05, and σB2 = 0 dB.


## 6.3. Total Performance

### 6.3.1. Transmission Outage Probability

As in the first and second stages, we also graphically plot the relevant parameters of the total transmission outage probability, as shown in Figures17–20. By comprehensively comparing the transmission outage probability in the first and second stages, we can get the contents in Table 2. Among them, the upward front represents the increase of the value, and the downward arrow represents the decrease of the value.Figure 17
Total transmission outage probability versusPRjmax with different value of PRk and PA, where σR2=0dB, σB2=0dB, R=1.5, NR=3, and ϕ=0.05.Figure 18
Total transmission outage probability versusPRjmax with different value of σR2 and σB2, where PRk=10dB, PA=10dB, R=1.5, NR=3, and ϕ=0.05.Figure 19
Total transmission outage probability versusPRjmax with different value of NR, where PRk=10dB, PA=10dB, R=1.5, ϕ=0.05, PRk=10dB, and PA=10dB.Figure 20
Total transmission outage probability versusPRjmax with different value of ϕ and R, where PRk=10dB, PA=10dB, NR=3, σR2=0dB, and σB2=0dB.Table 2
Transmission outage probability results.
ParameterParameter variationδ1δ2δInterference power↑↑↑↑Channel noise from the receiver↑↑↑↑Transmission rate↑↑↑↑Antennas number↑↓↓↓Covert message sending power↑↓↓↓Self-interference coefficient↑↑—↑It can be observed from Table2 that for the same parameters, the first stage, the second stage, and the total transmission outage probability have the same conclusion, which is consistent with our expectation. The causes of these graph phenomena have been described in detail in the previous stages and will not be repeated here.


### 6.3.2. Maximum Effective Covert Rate

Because the total maximum effective covert rate is smaller in the first and second stages, the maximum effective covert rates of the first and second stages are drawn synthetically to facilitate the comparison. The details are shown in Figure21. Obviously, the maximum effective covert rate of the first stage is less than that of the second stage.Figure 21
Comparison of stage 1 and stage 2 maximum effective covert rates, whereσR2=0dB, σB2=0dB, PRk=10dB, R=1.5, ϕ=0.01, NR=3, ε=0.05, hRjW2=1, and hAW=1.By comprehensively comparing the maximum effective covert rate in the first and second stages, we can get the contents in Table3.Table 3
Maximum effective covert rate results.
ParameterParameter variationRC1∗¯RC2∗¯RC∗¯Covert message sending power↑↑↑↑Covert constraints,ε↑↑↑↑Antennas number↑↑↑↑Channel noise from the receiver↑↓↓↓Self-interference coefficient↑↓—↓The channel coefficient between covert message sender and Willie↑↓↓↓The channel coefficient between the AN sender and Willie↑↑—↑Interference power↑—↑—It can be seen from Table3 that for the same parameters, the total maximum effective concealment rate has the same conclusion as the maximum effective convert rate in one stage. This is because zero-forcing beamforming is used in the second stage to interfere with Willie without any impact on Bob, the transmission interruption probability will be smaller, and the corresponding maximum effective convert rate will be larger, so the maximum effective convert rate in the second stage is significantly higher than that in the first stage. This also verifies the correctness of Theorem 10.


## 6.3.1. Transmission Outage Probability

As in the first and second stages, we also graphically plot the relevant parameters of the total transmission outage probability, as shown in Figures17–20. By comprehensively comparing the transmission outage probability in the first and second stages, we can get the contents in Table 2. Among them, the upward front represents the increase of the value, and the downward arrow represents the decrease of the value.Figure 17
Total transmission outage probability versusPRjmax with different value of PRk and PA, where σR2=0dB, σB2=0dB, R=1.5, NR=3, and ϕ=0.05.Figure 18
Total transmission outage probability versusPRjmax with different value of σR2 and σB2, where PRk=10dB, PA=10dB, R=1.5, NR=3, and ϕ=0.05.Figure 19
Total transmission outage probability versusPRjmax with different value of NR, where PRk=10dB, PA=10dB, R=1.5, ϕ=0.05, PRk=10dB, and PA=10dB.Figure 20
Total transmission outage probability versusPRjmax with different value of ϕ and R, where PRk=10dB, PA=10dB, NR=3, σR2=0dB, and σB2=0dB.Table 2
Transmission outage probability results.
ParameterParameter variationδ1δ2δInterference power↑↑↑↑Channel noise from the receiver↑↑↑↑Transmission rate↑↑↑↑Antennas number↑↓↓↓Covert message sending power↑↓↓↓Self-interference coefficient↑↑—↑It can be observed from Table2 that for the same parameters, the first stage, the second stage, and the total transmission outage probability have the same conclusion, which is consistent with our expectation. The causes of these graph phenomena have been described in detail in the previous stages and will not be repeated here.


## 6.3.2. Maximum Effective Covert Rate

Because the total maximum effective covert rate is smaller in the first and second stages, the maximum effective covert rates of the first and second stages are drawn synthetically to facilitate the comparison. The details are shown in Figure21. Obviously, the maximum effective covert rate of the first stage is less than that of the second stage.Figure 21
Comparison of stage 1 and stage 2 maximum effective covert rates, whereσR2=0dB, σB2=0dB, PRk=10dB, R=1.5, ϕ=0.01, NR=3, ε=0.05, hRjW2=1, and hAW=1.By comprehensively comparing the maximum effective covert rate in the first and second stages, we can get the contents in Table3.Table 3
Maximum effective covert rate results.
ParameterParameter variationRC1∗¯RC2∗¯RC∗¯Covert message sending power↑↑↑↑Covert constraints,ε↑↑↑↑Antennas number↑↑↑↑Channel noise from the receiver↑↓↓↓Self-interference coefficient↑↓—↓The channel coefficient between covert message sender and Willie↑↓↓↓The channel coefficient between the AN sender and Willie↑↑—↑Interference power↑—↑—It can be seen from Table3 that for the same parameters, the total maximum effective concealment rate has the same conclusion as the maximum effective convert rate in one stage. This is because zero-forcing beamforming is used in the second stage to interfere with Willie without any impact on Bob, the transmission interruption probability will be smaller, and the corresponding maximum effective convert rate will be larger, so the maximum effective convert rate in the second stage is significantly higher than that in the first stage. This also verifies the correctness of Theorem 10.


## 7. Conclusion

In this work, we consider a wireless network covert communication system achieved by multiantenna full-duplex relay. The system is divided into two stages; each stage needs to complete the transmission of covert messages. In the first stage, Relay receives covert messages and sends interference to Willie. In the second stage, Relay decodes and forwards covert messages while interfering with Willie’s detection. The system must ensure that neither stage can be detected. The two stages and the total system performance are derived and analyzed. Under the constraint of covert transmission, the maximum effective covert rate of multiantenna single relay decode and forward relay network is studied, and the simulation results are analyzed in detail, the system can achieve excellent covert performance. In the future, multiantenna and multirelay wireless network covert communication systems can be further considered.



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*Source: 1456570-2022-01-07.xml*