Covert Rate Study for Full-Duplex D2D Communications Underlaid Cellular Networks

Device-to-device (D2D) communications underlaid cellular networks have emerged as a promising network architecture to provide extended coverage and high data rate for various Internet of Things (IoT) applications. However, because of the inherent openness and broadcasting nature of wireless communications, such networks face severe risks of data privacy disclosure. This article investigates the covert communications in such networks for providing enhanced privacy protection. Specifically, this article explores the critical covert rate performance in a full-duplex (FD) D2D communication underlaid cellular network consisting of a base station, a cellular user, a D2D pair with a transmitter and an FD receiver, and a warden, where the D2D receiver (DR) can operate over either the FD mode or the half-duplex (HD) mode. We first derive transmission outage probabilities of cellular and D2D links under the FD and HD modes, respectively. Based on these probabilities, we further provide theoretical modeling for the covert rate under each mode and explore the corresponding covert rate maximization by jointly optimizing the transmit powers of the D2D pair and the cellular user. To improve the covert rate performance, we propose a general mode in which the DR can flexibly switch between these two modes. Under the general mode, we also investigate the theoretical modeling and maximization problems of covert rate. Finally, we present extensive numerical results to illustrate the covert rate performances under the FD, HD, and general modes.


I. INTRODUCTION
T HE EXPLOSIVE growth of data traffic from massive Internet of Things (IoT) devices has placed an extremely heavy burden on cellular networks [1].To address this challenge, device-to-device (D2D) communications underlaid cellular networks (DCNTs), which enable direct communications between nearby devices, have been identified as a promising technology to meet the rapidly growing data demand, in terms of increasing spectral efficiency, reducing latency, improving data rate, extending coverage, and enhancing power efficiency [2], [3], [4].Such networks are opening new opportunities for IoT and also vehicle-to-vehicle (V2V) communications.However, due to the inherent features of wireless channels (e.g., broadcast and openness), the networks are facing serious security risks like eavesdropping attacks and privacy violations and thereby hindering their widespread deployment [5].
Covert communications, which intend to hide the existence of wireless communications, have emerged as exciting security technology to preserve users' privacy and also prevent wireless transmission content from being eavesdropped [6].Such a security technology is highly appealing [7], especially, when the DCNTs are deployed to provide some critical services, such as military, healthcare, and location tracking.In these services, if an adversary cannot decide whether wireless communications occur or not, he does not know the transmitter's location achieving a higher level of privacy protection for the transmitter, and also has no opportunity to launch an eavesdropping attack.In DCNTs, the covert rate is a fundamental performance metric to measure the achievable data rate of covert communications.Therefore, it is of paramount importance to explore the covert rate in DCNTs.
By now, the covert rate is extensively studied in single-hop and two-hop/multiple-hop wireless networks (see Section II of Related Work).Notice that these results in the above works without the support of the base station (BS) cannot be applied to DCNTs because of the inherent characteristics (e.g., spectrum sharing and interference) of such networks.Some initial works have been devoted to the study of covert communications in DCNTs [8], [9], [10].By sharing the same spectrum resource between cellular and D2D links, a transmit power control scheme for cellular and D2D transmitters was proposed in [8] with the goal of covert rate maximization of the D2D link.Using the dedicated orthogonal spectrum resource for D2D link, the work in [9] proposed a joint scheme of relay selection and transmit power control of D2D This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/transmitters to maximize the covert rate of cellular links.The work in [10] utilized jamming signals generated from an antenna array of BS to confuse the decision of the warden aiming to achieve covert communications of D2D link.An energy-efficient transmit probability-power control scheme for covert D2D communication was presented in the work [11], which takes into account the requirements with energy consumption and information freshness.Li et al. [12] investigated covert communications in the scenario of multiple wardens in D2D communications underlaid cellular networks.The work in [13] unitized the spectrum reusing technology to implement covert communications for D2D content sharing, based on the trust evaluation mechanism.Shi et al. [14] further investigated the content delivery mode selection and resource management for covert communications.
It is notable that the half-duplex (HD) and full-duplex (FD) are two fundamental communication modes.Under the HD mode, a receiver only receives messages from a transmitter at a specific time, while under the FD mode, the receiver can receive messages and send artificial noise for ensuring covertness simultaneously.In comparison with HD mode, such an FD mode can enhance covertness but may reduce the covert rate of D2D links due to the impact of self-interference.However, all aforementioned works in DCNTs consider the D2D receiver (DR) operates in HD mode.To fully address the advantages of the HD and FD modes, an important issue is how to conduct a joint study on these two modes.As a step toward this direction, this article designs a general mode to flexibly switch between the HD and FD modes for achieving covert rate improvement.Meanwhile, we also model the covert rate performance under the HD, FD, and general modes, respectively.Our study is of great importance to support security-sensitive applications in the sixth generation (6G) wireless networks, where a large number of wireless IoT devices can realize not only the short-range transmission of privacy data (e.g., patient's health information and financial data) but also the remote data transmission with the help of BS.
The contributions of this article are summarized as follows.general model, we also explore the theoretical covert rate and its maximization problem.4) Extensive numerical results are presented to illustrate the covert rate and its maximum value by optimal power control under the FD, HD, and general modes.The remainder of this work is organized as follows.
In Section II, we present the system model.Detection performance at the warden is studied in Section III.In Sections IV and V, we develop theoretical models on covert rate and related maximization problem under the FD and HD modes, respectively.The general mode is proposed in Section VI.Section V provides numerical results.Section VII concludes this article.The abbreviations and notations used in this article are given in Table I.

II. RELATED WORK
For the single-hop wireless networks, a positive covert rate was proved to be achievable when the warden receives uncertain noise [15], [16], uncertain channel state information [17], [18], [19], and jamming signals [20], [21], [22].Moreover, the work in [23] proposed joint design algorithms based on a penalty decomposition technique in a multiantenna Willie scenario to achieve the improvement of covert rates performance, while a multiantenna Alice Discrete-Time AWGN channel model was applied in [24] to achieve a positive covert rate without any uncertainty or jammer.Besides, the studies of covert communications have been extended to millimetre waves and Terahertz.The work in [25] adopted a beam training approach to establish a covert mmWave directional link and optimized the vital system parameters to maximize the covert rate.The work in [26] designed the covert beam training and covert data transmission in a multiuser mmWave communication system to maximize the covert rate.The work of [27] explored the beamforming scheme based on a random frequency diverse array to enhance the performance of covert mmWave communications.In [28], a novel distance adaptive absorption peak modulation was proposed to enhance the covertness of the Terahertz (THz) covert communications.The work in [29] studied the optimality of Gaussian signaling for covert communications with the covertness constraint of the upper bound of Kullback-Leibler divergence.Regarding a finite number of channel uses, the work in [30] examined the delay-intolerant covert communications with fixed or random transmit power.Recently, regarding the space-airground integrated vehicular networks, the work in [17] applied improper Gaussian signaling techniques to reduce the cochannel interference and improve the covert rate performance of such networks.The work in [31] further explored the covert communications of unmanned aerial vehicles (UAVs) by joint optimization of their flying location and transmit power.Huang et al. [32], [33] and Li et al. [34] studied the covert video surveillance of ground-moving targets based on a UAV or a UAV team, while concealing the existence and movement of the UAVs from the targets' visual system.A 3-D location-based beamforming (LBB) scheme was proposed over the Rician channel to maximize the covert rate in [35].The work in [36] initially focused on the timeless of warden's detection in the binary hypothesis test framework.The work of [37] investigated the optimum strategy of hiding the covert information onto the nonorthogonal multiple access (NOMA) information.
For the two-hop wireless networks, with the help of a relay, the positive covert rates were achievable [6], [38], [39], [40], [41].Remarkably, intelligent reflecting surface (IRS), which is a promising solution to support energy-efficient and cost-effective covert communication, illustrates a huge potential to improve covert rate performance.Deng et al. [42] adopted two-way protocols in IRS-aided relaying networks to improve covert rate performance.An UAV-mounted IRS (UIRS) communication system over THz bands was also proposed to achieve the improvement of covert rate performance [43].Chen et al. [44] and Wang et al. [45] utilized the integration of the IRS and the MIMO techniques to maximize the covert rate.The work [46] investigated the scenario that two-hop signals at the warden be combined to determine the presence of private transmission.Recently, a multihop relaying strategy was designed to improve covet rate performance [47], [48].

A. Network Model
As shown in Fig. 1, we consider a cellular network 1 consisting of a BS, a cellular user equipment (CE), a D2D pair with a transmitter (DT) and a receiver (DR), and a warden (WD).DT intends to covertly transmit confidential messages to DR, 1 These works mainly consider simple scenarios where a transmitter attempts to covertly transmit the message to its receiver with/without a relay in the presence of a warden/multiple wardens [17], [18], [19], [20], [21], [22], [23], [24].This is because it is very complex to determine the optimal detection error probability, the optimal detection threshold, and the optimal covert rate in the multiple-user scenario.We focus on the uplink transmission scenario in the concerned cellular network, where all transmissions use the uplink spectrum resource.Accordingly, there are six types of links in the network, namely, the link from DT to DR (D2D link), the link from CE to BS (C2B link), the interference link, the selfinference link from DR to itself, the detection link from DT to WD, and the jamming link from DR to WD.We consider that DR is equipped with two omnidirectional antennas for supporting the FD mode while each of the others has a single omnidirectional antenna.DR can adjust its transmit power P r of artificial noise, which is no more than a maximum transmit power P max .CE and DT also employ adjustable transmission powers P c and P t , respectively.The maximum values of P c and P t are denoted as P t max and P c max , respectively.

B. Channel Model
Similar to previous works [18], [49], [50], [51], this article considers that the time is slotted and all nodes are static from on time slot to another.A quasi-static Rayleigh fading channel is used to model these links as shown in Fig. 1, where the channel remains constant in a time slot and independently changes in different time slots.Let the subscripts t, r, c, b, and w denote the DT, DR, CE, BS, and WD, respectively.We use h tr , h tb , h tw , h cr , h cb , h cw , h rb , h rw , and h rr to denote the fading coefficients of these links from DT to DR, DT to BS, DT to WD, CE to DR, CE to BS, CE to WD, DR to BS, DR to WD, and DR to DR, respectively.
We use h ij to denote the channel fading coefficient between nodes i and j and use (1/λ ij ) to denote the mean of |h ij | 2 over different time slots, where i and j ∈ {t, r, c, b, w}.The noise received at j is a complex additive Gaussian random variable with zero mean and variance σ 2 j , i.e., n j ∼ CN (0, σ 2 j ).

C. Full-Duplex Model
We consider DT and CE transmit signal over n available channel uses at each time slot.We use y FD r (i) and y FD b (i) to denote the received signal at DR and BS over the ith time slot.Under the FD mode, they can be given by where x t (i) and x c (i) denote the signal transmitted by DT and CE, respectively, v r (i) denotes the artificial noise generated by DR, and Here, E[ • ] is the expectation operator and i = 1, 2, . . ., n. φ (0 < φ ≤ 1) denotes the self-interference cancellation coefficient corresponding to the cancellation level of the artificial noise signal.The transmit power of DR is treated as AN to cause uncertainty at the warden, which confuses the warden to determine whether the D2D transmitter transmits a message or not.To this end, the transmit power should be random for the warden.Thus, we regard it as a random variable following a continuous uniform distribution in this article.The assumption of the random variable on the AN has also been adopted in previous studies [49], [51], [52].We use f P r (x) to denote the probability density function (PDF) of the random variable, and then Based on its received signal, WD performs a binary hypothesis test to decide whether or not DT did a covert transmission at a time slot.In this test, the null hypothesis (i.e., H 0 ) states that DT did not transmit the covert message, while the alternative hypothesis (i.e., H 1 ) states that DT did a covert transmission.Thus, the received signal at WD under these two hypotheses can be given by and where y FD w (i) denotes the received signal at WD under the FD mode.

D. Half-Duplex Model
Under the HD mode, DR only receives messages without transmission at each time slot.We use y HD r (i) and y HD b (i) to denote the received signal at DR and BS under such a mode, respectively.Then, we have and Under the HD mode, WD also performs a binary hypothesis test to decide whether or not DR did a covert transmission.Then, we have and ) where y HD w (i) denotes the received signal at WD under the HD mode.

E. Detection at Warden
Based on its received signals over the n channel uses, WD makes a binary decision on whether DT transmits covert messages or not under each of the FD and HD modes.The corresponding detection error probability ξ can be expressed as follows: where D 1 and D 0 denote the binary decisions that WD conducts a covert transmission or not, respectively.P(D 1 |H 0 ) denotes the false alarm probability that H 0 is true but WD makes a decision to approve D 1 .P(D 0 |H 1 ) denotes the miss detection probability that H 1 is true but WD approves D 0 .
According to Pinsker's inequality [53], we have a lower bound of ξ given by where Here, D(P 0 P 1 ) is the relative entropy between two probability distributions P 0 to P 1 , and P 0 and P 1 are the probability distributions of WD's channel observations of n channel uses when H 0 and H 1 are true, respectively.γ w is the signal-tointerference-plus-noise ratio (SINR) at WD.Note that when the value of D(P 0 P 1 ) is small, the distance between these two distributions P 0 and P 1 is also small such that WD has a high detection error probability ξ .
In covert communications, we require a covert constraint ξ ≥ 1 − ε, where ε is an arbitrarily small constant.We adopt D(P 0 P 1 ) as covertness requirement, which is determined as follows:

F. Performance Metrics
Covert rate is defined as the achievable rate from DT to DR, with which the transmission from DT to DR and that from CE to BS do not occur outage such that DR and BS can decode their received messages, while maintaining a high detection error probability at WD.

IV. COVERT RATE UNDER FULL-DUPLEX MODE
This section first provides theoretical modeling for the covert rate under FD mode, and then further explores the covert rate maximization.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

A. Covert Rate Modeling
Based on the definition of covert rate, we have where η FD denotes the covert rate under FD mode, R FD is a desired rate, δ FD r and δ FD b denote the transmission outage probabilities that DR and BS cannot successfully decode their received message, respectively.Specially, the transmission outage from CE to BS occurs when C FD cb < ζ cb , and that from DT to DR occurs when C FD tr < ζ tr .Here, ζ cb and ζ tr are two preset thresholds, C FD cb and C FD tr denote the channel capacity from CE to BS and that from DT to DR, respectively.
We need to determine the unknown δ FD b and δ FD r .To obtain them, we first give the SINRs at BS and DR, which are denoted by γ FD b and γ FD r , respectively.They are given by and Then, we obtain δ FD r and δ FD b in the following lemma.Lemma 1: Under FD mode, the transmission outage probabilities δ FD r and δ FD b are determined as follows: where i and j ∈ {t, r, c, b, w}.Then, we can determine the transmission outage probability δ FD b as follows: where We further determine the transmission outage probability δ FD r as follows: where β = λ tr (2 ζ tr − 1)/P t .We finish the proof of Lemma 1.

B. Covert Rate Maximization
Our goal is to maximize the covert rate under the FD mode by jointly optimizing the transmit powers of DT and CE and the AN power of DR.To this end, covert rate maximization can be formulated as the following optimization problem: where (22b) represents the covert requirement, and (22c) and (22d) represent the ranges of transmit powers of DT and CE, respectively.Constraint (22e) represents that the range of the maximum transmit power of DR is between 0 and P r max .Here where the SINR at WD under the FD mode is given by and the terms P c |h cw | 2 and P r |h rw | 2 are the interference caused by CE and DR, respectively.The solution to the optimization problem (22) is given in the following theorem.
Theorem 1: We use η * FD to denote the maximum covert rate, and use P * t and P * c to denote the optimal transmit powers of DT and CE, respectively.Then, we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where w .Based on these optimal transmit powers, the maximum covert rate can be determined as follows:

V. COVERT RATE UNDER HALF-DUPLEX MODE
In this section, we first provide the theoretical modeling for covert rate and then explore the covert rate maximization by optimizing the transmit powers of DT and CE under the HD mode.

A. Covert Rate Modeling
Under the HD mode, the covert rate η HD can be expressed as follows: where R HD is a desired rate, δ HD and We can obtain the transmission outage probabilities δ HD b and δ HD r in the following lemma.Lemma 2: Under the HD mode, the transmission outage probabilities at BS and DR can be determined as follows: and where α = λ cb (2 ζ cb − 1)/P c and β = λ tr (2 ζ tr − 1)/P t .

B. Covert Rate Maximization
We aim to maximize the covert rate under the HD mode by jointly optimizing the transmit powers of DT and CE and thus, formulate the covert rate maximization as the following optimization problem: where (35b) represents the covert requirement, and (35c) and (35d) represent the ranges of transmit powers of DT and CE, respectively.In (35b), we have where is the SINR at WD under HD mode.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.We can obtain the solution of the optimization problem (35) in the following theorem.
Theorem 2: Under the HD mode, the optimal transmit powers of DT and CE can be determined as follows: and where Based on these optimal transmit powers, we further determine the maximum covert rate η * HD as where α = λ cb (2 ζ cb − 1)/P c * and β = λ tr (2 ζ tr − 1)/P t * .

VI. COVERT RATE UNDER GENERAL MODE
In this section, we propose a general mode.Under the general mode, the DR can flexibly switch between the FD and HD modes.Such a general mode can well overcome the passive impact of strong self-interference on covert performance under the FD mode.The system structure diagram of the general mode is shown in Fig. 2. and the covert rate optimization algorithm under the general mode is described in Algorithm 1. Regarding the time complexity of the algorithm, we have obtained the closed-form solutions for the maximum covert rate under the FD and HD modes (see Theorems 1 and 2).Thus, the complexity of the algorithm is a constant O(1).

B. Covert Rate Maximization
The objective is to maximize the covert rate under the general mode, while being subject to the covertness requirement and transmit power constraints.The covert rate maximization can be formulated as the following optimization problem: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where (44b) represents the covert requirement, and (44c) and (44d) represent the ranges of transmit powers of DT and CE, respectively.While, (44e) represents the range of the maximum AN power.
For the optimization problem in (45), the maximum covert rate can be expressed as follows: where η * FD and η * HD are given in Theorems 1 and 2, respectively.
Regarding the tradeoff among the data rate, energy efficiency, and covert rate, we formulate covert rate maximization as the following optimization problem with the constraints of the data rate of cellular link and total power of transmitters under the general model max where (46f) represents the range of data rate of cellular link, and (46g) represents the range of the total power of transmitters DT, and CE.Here, R c max represents the maximum value of the date rate, and P all represents the maximum value of the total power of P t and P c .A 3-D search over (P c , P t , P max ) can be used to find the optimal solution for the optimization problem (46).

C. Covert Rate Analysis
This section explores the impact of some critical parameters on the covert rate performance and corresponding performance optimization under the FD, HD, and general modes.Unless stated otherwise, the parameter values are provided in Table II.
We first explore the impact of the transmit power of DT P t on covert rates under the three modes.Fig. 3 summarizes how covert rates vary with P t for the setting of the transmit power of CE P c = {1.0,1.5} W. It can be observed from Fig. 3 that as Fig. 3. Covert rates versus the transmit power of DT P t under the FD, HD, and general modes.P t increases, the covert rate η FD under the FD increases, while the covert rate η HD under the HD mode first increases and then remains at zero.This is because the effect of increasing P t is twofold.On the one hand, it can increase the covert rates under the two modes.On the other hand, it can also increase the probability that the warden WD detects the transmission of DT.When WD can detect its transmission, η HD remains at zero.However, the receiver DR works in the FD mode, and thus, the interference emitted by DR can confuse the detection of WD, which leads to the increase of η FD as P t continues to increase.We know that the covert rate under the general mode corresponds to the maximum value of η FD and η HD as shown in Fig. 3.We further observe from Fig. 3 that for each fixed P t , η HD is higher than η FD due to the self-interference under the FD mode.We can also observe from Fig. 3 that for each fixed P t , the maximum covert rate with the setting of P c = 1.5 W is smaller than that with the setting of P c = 1.0 W for each mode.The reason behind this is that a large P c can cause more interference to the DR, which leads to a small maximum covert rate.
We further investigate the impact of the covert requirement ε on covert rates under these three modes.We summarize in Fig. 4 how covert rates vary with ε for the setting of the transmit power of DT P t = {1.0,1.5} W. We can see from Fig. 4 that as ε increases, η FD increases, while η HD first keeps at zero and then increases.The phenomenon can be explained as follows.We know that an increase of ε leads to the increases of η FD and η HD .However, as ε is relatively small, the covert constraint under the HD mode cannot be satisfied, and thus, η HD is zero.As for the FD mode, the covert constraint is satisfied with the help of interference from DR.We can also see from Fig. 4 that the covert rate under the general mode is the maximum one of η FD and η HD for each fixed ε.We can also observe from Fig. 4 that for each ε, the maximum covert rate with the setting of P t = 1.5 W is larger than that with the setting of P t = 1.0 W for each mode.The reason behind this phenomenon is that a larger P t results in a stronger received signal at the DR, which leads to the improvement of covet performance.We now explore the impact of the transmit power of CE P c on covert rates for the setting of the transmit power of DT P t = {1.0,1.5} W under these three modes, as shown in Fig. 5.It can be seen from Fig. 5 that the covert rates under these three modes decrease with the increase of P c .This is because increasing P c can increase the interference to the receiver DR, which leads to the decrease of the covert rates under these three modes.It can also seen from Fig. 5 that for each fixed P c , the maximum covert rate with the setting of P t = 1.5 W is larger than that with the setting of P t = 1.0 W for each mode.The reason is the same as that in Fig. 4.
Finally, we examine the impact of the transmit maximum power of DR P max on the covert rates under these three modes.Fig. 6 illustrates how the covert rates vary with P max for the setting of the transmit power of DT P t = {1.0,1.5} W. One observation from Fig. 6 indicates that as P max increases, η FD first remains at zero, then achieves a maximum value and finally decreases.Meanwhile, η HD always remains at zero.This is due to the following reason.When P max is relatively small, the covert constraints under the FD and HD modes Fig. 6.Covert rates versus the transmit power of DR P max under the FD, HD, and general modes.Fig. 7. Maximum covert rates versus the maximum transmit power of DT P t max under the FD, HD, and general modes.cannot be satisfied, and thus, both η FD and η HD remain at zero.When P max increases up to a threshold, the covert constraint under the FD mode can be satisfied with the help of the artificial noise, which leads to a maximum covert rate.On the other hand, increasing P max can also increase the selfinterference at DR, which leads to the decrease of covert rate under the FD mode.Note that there is no effect of the artificial noise/self-interference under the HD mode, and thus, η HD still remains at zero.It can also seen from Fig. 6 that for each fixed P max , the maximum covert rate with the setting of P t = 1.5 W is larger than that with the setting of P t = 1.0 W for each mode.The reason has been explained in Figs. 4 and 5.

D. Covert Rate Optimization
This section explores the impact of system parameters on the maximum covert rates under the FD, HD, and general modes.We summarize in Fig. 7 how the maximum covert rates vary with P t max under these three modes with the setting of the transmit power of CE P c = {1.0,1.5} W. We can see from Fig. 7 that for a given P c , as P t max increases, the maximum covert rates increase under the three modes, while as P t max increases up to some threshold, the maximum covert rates under the HD and general modes keep unchanged.This is due to the following reason.As P t max is relatively small, P t has not reached the optimal value corresponding to the maximum value of the maximum covert rates.Thus, the maximum covert rates increase with the increase of P t max under the three modes.As P t max becomes larger, P t reaches the optimal value and keeps unchanged under the HD and general modes, at which the maximum covert rates remain at the largest constant.However, due to effect of self-interference under the FD mode, P t has not reached the optimal value.Thus, the maximum covert rate under the FD mode increases with the increase of P t .We can also observe from Fig. 7 that for each fixed P t max , the maximum covert rates with the setting of P c = 1.5 W are smaller than these with the setting of P c = 1.0 W under these three modes.This is due to the fact that a large P c can cause more interference to the DR, which leads to small maximum covert rates under these three modes.
Fig. 8 illustrates how the maximum covert rates vary with the covertness requirement ε under these three modes with the setting of the transmit power of CE P c = {1.0,1.5} W. It can be observed from Fig. 8 that as ε increases, the maximum covert rates under the FD and general modes increase, while the maximum covert rate under the HD mode first remains at zero and then increases.The reason behind this phenomenon is the same as that of Fig. 4. Similar to Fig. 7, we can see from Fig. 8 that for each fixed ε, the maximum covert rates with the setting of P c = 1.5 W are smaller than these with the setting of P c = 1.0 W under these three modes.
We proceed to explore the impact of P c max on the maximum covert rates under these three modes with the setting of the transmit power of DT P t = {1.0,1.5} W. It can be seen from Fig. 9 that as P c max increases, the maximum covert rates under these three modes first increase, then achieve maximum  values and keep unchanged.This can be explained as follows.According to the definition of covert rate, we know that the maximum covert rate under each mode is inversely proportional to the transmission outage probability of the cellular link from CE to BS. Increasing P c max can reduce the transmission outage probability, which leads to the increase of the maximum covert rate under each mode.On the other hand, it can also incur more interference to the D2D link from DT to DR, which leads to the decrease of the maximum covert rate.As P c max is relatively small, the former dominates the latter, and thus, the maximum covert rate under each mode increases with the increase of P c max .As P c max further increases, the maximum covert rate achieves a maximum value and thus keeps unchanged under each mode.We can also observe from Fig. 9 that the maximum covert rates with the setting of P t = 1.5 W are higher than these with the setting of P t = 1.0 W under these three modes.This is because a large P t leads to a high SINR at DR, which leads to a large maximum covert rate under each mode when the constraint of covert requirement can be satisfied.
To explore the tradeoff among the data rate, energy efficiency, and covert rate, we summarize in Fig. 10 how P max affects the maximum covert rate with/without the constraints of data rate and total power for these three modes.We can see from Fig. 10 that as P max increases, the maximum covert rates under all three modes without the constraint are always greater than that with the constraint.This is because the constraint of data rate and total power leads to the decrease of maximum covert rate, which achieves the tradeoff among the data rate, energy efficiency, and covert rate.

VII. CONCLUSION
In this article, we investigated the covert rate performance in an FD DCNT.We theoretically model the covert rate under the FD and HD modes, respectively.Then, we optimize the transmit powers of DT, DR, and CE for maximizing the corresponding covert rate.Specially, we designed a general mode to improve such performance.Remarkably, the extensive numerical simulations illustrate that our proposed general mode can achieve significant improvement in covert rate performance compared to pure HD mode or FD mode.An interesting future work is to explore the impact of node mobility on covert rate performance in DCNTs.Another promising work is to focus on the studies of covert rate performance in the multiuser scenario, where one D2D pair may be detected by more than one warden and one warden may be confused by more than one D2D receiver.Further, it is deserved to explore the covert communications of the complex multilevel structure with obstacles in our future study.

APPENDIX A PROOF OF THEOREM 1
Proof: Based on a step-by-step manner [49], we solve the optimization problem of (22).We first determine the monotonicity of η FD with respect to P t , and the monotonicity of D(P FD 0 P FD 1 ) with respect to P t and P c , respectively.Then, we derive the optimal transmit powers of DT and CE.Finally, we derive the optimal maximum transmit power of DR, i.e., P max .

A. Monotonicity of η FD
To determine the nonotonicity of η FD with respect to P t , we first calculate the first derivative of γ FD r with respect to P t in (16) as Thus, γ FD r monotonically increases with P t .Then, we determine the monotonicity of δ FD r with respect to γ FD r .According to [54], we know that where and We now derive the first derivative of δ FD r with respect to γ FD r , and then To determine (51), let where We determine the first derivative of h(x) as follows: We can see that the sign of h (x) is the same as We further derive the first derivative of v(x) with respect to x, and then Thus, v(x) decreases with x, where x > 1.Since v(x) < v(1) = 0, we have h (x) < 0. Thus, h(x) is a decreasing function of x.We use h max (x) to denote the maximum value of h(x).Then Thus, regarding (51), we have and then We obtain (δ FD r ) | γ FD r < 0. Thus, δ FD r monotonically decreases with γ FD r .Similarly, δ FD b monotonically decreases with γ FD b .We further calculate the first derivative of η FD with respect to δ FD r as follows: Thus, η FD deceases with δ FD r .
Finally, we obtain that η FD monotonically increases with P t .

B. Monotonicity of D(P FD
We calculate the first derivative of D(P FD 0 P FD 1 ) with respect to γ FD w as We continue to calculate the first derivative of γ FD w with respect to P t , P r , and P c , respectively.Then, we have and Thus, D(P FD 0 P FD 1 ) monotonically increases with P t , while monotonically decreases with P r and P c , respectively.

C. Optimal Transmit Power of DT
Since both the objective function η FD and D(P FD 0 P FD 1 ) in the constraint of (22b) monotonically increase with P t , we can determine the optimal value P * t of P t for maximizing η FD and ensuring D(P FD 0 P FD 1 ) = 2ε 2 with any given P r an P c .Substituting (12) into D(P FD 0 P FD 1 ) = 2ε 2 , we have To guarantee a low detection probability at WD, γ FD w is normally very small.Thus, ln(1 + x) ≈ x.Then, (64) can be rewritten as follows: By solving (65), we have Substituting ( 24) into (66), we have and thus, we obtain the optimal P * t given in (25).

D. Optimal Transmit Powers of CE
We know that δ FD r and δ FD b monotonically decreases with γ FD r and γ FD b , respectively.We can also determine the first derivative of η FD with respect to δ FD b as Thus, based on (59), we obtain that η FD decreases with δ FD r and δ FD b , respectively.Then, η FD monotonically increases with γ FD r and γ FD b , respectively.
Based on (67), we have and By substituting (69) into ( 16), we have We now calculate the first derivative of γ FD r with respect to P c as follows: Then, we need to find an optimal P max to maximize the covert rate η FD , This corresponds to the case that determining an optimal P max maximizes ν(P max ), which can be easily solved by a 1-D search method.We complete the proof of Theorem 1.

APPENDIX B PROOF OF THEOREM 2
Proof: Using the step-by-step manner, we solve this optimization problem.We first determine the monotonicity of D(P HD 0 P HD 1 ) with respect to P t and P c .To this end, we calculate the first derivative of D(P HD Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. We now continue to calculate the monotonicity of γ HD w with respect to P t and P c by solving its first derivative.Then, we have Thus, we obtain that γ HD w monotonically increases with P t , while decreases with P c .Accordingly, D(P HD 0 P HD 1 ) monotonically increases with P t and decreases with P c .
Based on the fact that both D(P HD 0 P HD 1 ) and η HD are increasing functions of P t , we know that regarding an optimal value P * t of P t , the constraint D(P HD 0 P HD 1 ) ≤ 2ε 2 is always guaranteed for any fixed P c and P r .We now determine the optimal P * t .Substituting (36)  Thus, the optimal P * t can be obtained in (38).We proceed to derive the optimal P * c .Since δ HD r monotonically decreases with γ HD r , the objective function η HD monotonically increases with γ HD r .We only need to determine the monotonicity of γ HD Case 2: When |h cw | 2 σ 2 r < |h cr | 2 σ 2 w , we have (γ HD r ) | P c < 0. Thus, γ HD r monotonically decreases with P c .To achieve the maximization of η HD , P c needs to set as small as possible.So, we have P * c = 0.Then, (38) and ( 39) follow.We complete the proof of Theorem 2.

r
and δ HD b denote the transmission outage probabilities at DR and BS cannot successfully decode their received message, respectively.If C HD cb < ζ cb , the transmission outage from CE to BS occurs.If C FD tr < ζ tr , the transmission outage from DT to DR occurs.C HD cb and C HD tr denote the channel capacity from CE to BS and that from DT to DR, respectively.We need to determine the unknown δ HD b and δ HD r .First, we use γ HD b and γ HD r to denote the SINRs at BS and DR, respectively.The SINRs are given by

Fig. 2 .
Fig. 2. System structure diagram of the general mode.

) Algorithm 1
Covert Rate Optimization Algorithm Under the General Mode Input: P r , P t , P c : Transmit powers of DR, DT and CE, respectively.h xy , x, y ∈ {t, r, b, w, c}: The channel fading coefficients.σ 2 w , σ 2 b , σ 2 r : The noise variances at WD, DR and BS, respectively.Output: η * : The optimal covert rate under the general mode./ * Calculate the maximum covert rate η * FD under the FD mode * / 1: Calculate the SINRs at BS and DR, i.e. γ FD b and γ FD r under FD mode, respectively; 2: According to γ FD b and γ FD r , calculate the transmission outage probabilities δ FD b and δ FD r , respectively; 3: Calculate the covert rate η FD = R FD (1 − δ FD r )(1 − δ FD b ) under the FD mode; 4: Calculate η * FD according to (46); / * Calculate the maximum covert rate η * HD under the HD mode * / 5: Calculate the SINRs at BS and DR, i.e. γ HD b and γ HD r under HD mode, respectively; 6: According to γ HD b and γ HD r , calculate the transmission outage probabilities δ HD b and δ HD r , respectively; 7: Calculate the covert rate η HD = R HD (1−δ HD r )(1−δ HD b ) under the HD mode; 8: Calculate η * HD according to (46); / * Calculate the maximum covert rate η * under the general mode * / 9: if η * FD ≥ η * HD then

Fig. 5 .
Fig. 5. Covert rates versus the transmit power of CE P c under the FD, HD, and general modes.

Fig. 8 .
Fig. 8. Maximum covert rates versus the covertness requirement ε under the FD, HD, and general modes.

Fig. 9 .
Fig. 9. Maximum covert rates versus the maximum transmit power of CE P c max under the FD, HD, and general modes.

Fig. 10 .
Fig.10.Impact of P max on maximum covert rate with/without the constraints of data rate and total power.

0 P HD 1 ) 1 )
with respect to γ HD w as follows: monotonically increases with γ HD w .