Performance Analysis for UAV-Jammer Aided Covert Communication

This paper investigates the performance analysis for a novel covert communication system, where a friendly UAV-jammer is used to protect the covert transmission from Alice to Bob against the eavesdropping from Dave. In particular, by exploiting the spatial diversity, the UAV can emit artificial noise (AN) to complicate the noise uncertainty at Dave. In this letter, two scenarios are studied, where the channels from Alice to Bob and Dave experience 1) AWGN and 2) Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading. The closed-form expression for privacy rate expression under AWGN is derived, where evaluating the condition for achieving a positive gain of using UAV-jammer. Furthermore, under the Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading scenario, an approximate expression of ergodic privacy rate is obtained by using Taylor expansion. In the end, simulation results show that significant performance gain in terms of privacy rate can be achieved by employing UAV-jammer for covert communication.


I. INTRODUCTION
Security issues are vital important for wireless networks, where a lot confidential information need to be transmitted over wireless medium. Due to the inherent broadcast feature of wireless links, traditional security techniques, including physical layer security (PLS) approaches [1], [2] and encryption methods [3], were developed to protect information transmission from various types of eavesdropping. Nevertheless, physical layer security is not able to guarantee the amount of information being eavesdropped due to the unpredictable characteristics of wireless channels, while encryption techniques also would be in failure when facing the adversaries with intelligent anti-encryption and powerful computing capabilities.
In future wireless networks, some circumstances, such as military applications and national events, usually require a higher level of security by shielding the existence of the wireless transmission instead of protecting the content of transmitted information against unauthorized eavesdroppers. In that case, PLS and encryption methods will become The associate editor coordinating the review of this manuscript and approving it for publication was Bo Zhang . invalid. Hence, more and more research efforts are catering for such security concerns referred to as covert communications [5]- [8]. The authors in [5] found the fundamental limit of the covert rate being O( √ n) over an additive white Gaussian noise (AWGN) channel, where n is the blocklength used with the assumption of n → ∞. Then, Yan et al. [6] extended the work to the case of finite blocklength, i.e. n ≤ N , and proved that the optimal covert rate could be achieved only when n = N . More recently, in [7], [8] the covert rate performance was investigated for a one-way relay network with opportunistic relaying scheme employed. Nevertheless, all the current studies strongly rely on the noise uncertainty of wireless networks, which can not be always guaranteed.
On the other hand, unmanned aerial vehicle (UAV) can exploit the flexibility of deployment, which naturally creates a new degree of noise uncertainty for covert communication.
In [9] the UAV based jammer was used to improve the security communication between a legitimate transmitter-receiver pair against the eavesdropper, where the intercept probability security region (IPSR) was maximized. By contrast, Wang et al. [10] investigated the PLS of the UAV-enabled relaying wireless system, aiming at maximizing the secrecy rate. In addition, the UAV-enabled relaying scheme was extended by the work [11] in the context of the terrestrial point-to-point communications, where the UAV's trajectory is optimized for the secrecy capacity.
In recent, very limited studies [12], [13] have investigated the covert communication in the context of the wireless networks with the existence of UAV. In a little more detail, the authors in [12] have evaluated both the security and covert performance of a multi-hop network under the surveillance of UAV. They maximized the achievable throughput of the network by jointly optimizing the coding rates, transmit power, and required number of hops, and revealed that, there exists a trade-off between the security and efficiency in both secrecy and covert communications. By contrast, the authors in [13] proposed to use UAV link for transmitting covert information, which could exploit the spatial diversity. In particular, this work jointly optimized theUAV's trajectory and transmit power in terms of maximizing the average covert transmission rate, when assuming the locations of both the legitimate receiver Bob and the warden Willie were subject to uncertainties. Unfortunately, so far there is no work considering the UAV as a jamming terminal to protect the covert communication link.
The main goal of the covert communication is to ensure the non detectability of the transmission information [14]- [19]. In that case, by introducing jammer it certainly creates the uncertainty at the eavesdropper side. As done in [20], an uniformed jammer was employed to protect the covert communication between legitimate transmitter and receiver. The study showed that the covert transmitter could remain covert with a transmit power that does not decrease with the blocklength n even when warden employed an optimal detector. Furthermore, the works of [21], [22] have proposed to design a full-duplex receiver, which can generates artificial noise with a varying power causing uncertainty at the adversary. They provided the guidelines for the optimal choice of artificial noise power range, and the optimal transmission probability of covert information to maximize the detection errors at the adversary. In comparison with the above approaches, the use of UAV has a better performance to improve the noise uncertainty, and can certainly assist the covert communication, but it should be considered that the eavesdropper can detect the existence of the covert communication through the relevant information of UAV (the appearance of UAV, the level of interference power, UAV speed). If the frequency and mode of UAV use are random and confusing, this situation can be avoided.
Against the above background, in this paper we are motivated to study the performance analysis of the UAV-jammer aided covert communication system. The main contributions can be summarized as follows.
• First, we develop a novel covert communication system with the aid of introducing a friendly UAV-jammer for creating a new degree of noise uncertainty.
• Second, the closed-form expressions for the privacy rate achieved by our covert system are derived under both the AWGN and Nakagami-m fading scenarios. Specifically, we obtain the condition for achieving a positive gain of privacy rate, which suggests when to and how to use the UAV-jammer.
• Last, we evaluate and analyze the performance of our covert communication system. It is shown that, the achievable privacy rate can be significantly improved by introducing the UAV-jammer, which can be deemed to a promising option for meeting future high-level security requirements.

II. SYSTEM MODEL & MAIN ASSUMPTIONS
As shown in Fig. 1, we conceive a covert communication system, which includes, a legitimate transmitter-receiver pair (Alice and Bob), the UAV-enabled jammer, and the eavesdropper Dave. Suppose that, Alice transmits information to Bob with a pre-allocated spectrum, while Dave tries to detect the existence of Alice's transmission. Further, the UAV-enabled jammer is employed to transmit artificial noise (AN) from air to ground, which creates the noise uncertainty at Dave side for protecting the legitimate transmission. Note that, all the nodes in our system are equipped with single antenna.
Let us assume that all the communications happen during the time period t 0 , consisting of J consecutive time slots, whose indexes are collected in J = {1, 2, . . . , J }. Suppose that, Alice, Bob and Dave stay still, while the UAV keeps a uniform motion with speed V during t 0 . The positions of all the nodes are reflected by the coordinate, such as c i = (x i , y i , z i ). In particular, the coordinates of the ground nodes are denoted by c A for Alice, c B for Bob, and c D for Dave. Hence, the distance for Alice→Bob is d AB = c A − c B 2 , and that for Alice→Dave is d AD = c A − c D 2 . Without loss of generality, we assume that, the trajectory of UAV is random, which is not the main focus of this work. At the jth where δ t = t 0 /J . Note that, similar constraints can be found for d UB [j]. Let us assume, Alice is (or is not) transmitting information during the time slots collected in the setĴ (orJ ), where J =Ĵ ∪J . For instance, during time slot j (j ∈Ĵ ), Alice transmits N complex valued symbols to Bob, denoted by is the constant transmit power of each symbol and n = 1, 2, . . . , N . On the other hand, Dave is always passively collecting the symbols, and assume the number of collected channel usesÑ is relatively large during each time slot. In that case, the received signal power of Bob or Dave is proportion to the distance to Alice, which can be characterized by the free space path loss model, such that P r, AD when Alice is transmitting information. On the other hand, let us denote the AWGN at Bob and Dave as w B and w D , where w B ∼ CN (0, B ) is perfectly known by Bob. Whereas, we assume that Dave observes the noise variance uncertainty in the range of where D is the true noise variance at Dave, and ρ is the uncertainty parameter [16]. In addition, the UAV-jammer can emit AN with power P U [j], which further complicates the uncertainty of noise variance at Dave. Hence, the uncertainty of Dave's measurements on noise becomeŝ where P U [j] > 0 for UAV emitting, otherwise P U [j] = 0.

A. HYPOTHESIS TESTING AT DAVE
When all the channels are AWGN, to detect Alice's transmission, Dave is always trying to distinguish between the following two signal hypotheses, expressed as Above, Y [j, n] is the received signal at Dave for the nth sampling during time slot j, and ) α is the interference power imposed by the UAV. Further, H 1 (or H 0 ) denotes the hypothesis that Alice is (or not) transmitting. Let us define that, P F = Pr(D 1 | H 0 ) is false alarm probability and P M = Pr(D 0 | H 1 ) is misdetection probability, where D 0 and D 1 are the decisions at Dave for noise or signal plus noise. The ultimate goal of Dave is to minimize the error rate of detection, i.e. to minimize the probability of ξ = P F +P M ≥ 1 − for an arbitrary small .
When radiometer, i.e. energy detector, is employed by Dave, the test statistics for time slot j can be given by where ϒ is the detection threshold used by Dave. As a pioneering work, we are motivated to evaluate the covert rate of the system in the presence of friendly UAV-jammer under both AWGN and Nakagami-m fading scenarios.

III. PRIVACY RATE FOR NON-FADING SCENARIO
Let us first consider the non-fading scenario that all the links are AWGN. In [16], the SNR wall was given: Dave cannot detect Alice even if he gathers an infinite number of samples N → ∞ when Alice transmits with an SNR below ρ − 1/ρ. Similarly, Dave is assumed to employ the detection metric of the max-min approach on the error rate of ξ over the interval of its noise uncertainty. In this case, the average privacy rate of Alice's transmission can be given bỹ and defining In this case, upon applying the Central Limit Theorem (CLT) on the chi square distribution of the test statistics for H 0 in (5) and H 1 in (6), the false alarm probability and the detection probability at Dave can be given by whereˆ D [j] ∈ D [j]. In order to maximize the privacy rate in (9), it needs to find the highest transmit power allowed for Alice while guaranteeing the error rate of detection at Dave is approaching 1, i.e. ξ → 1, which corresponds to the condition that satisfying P F [j] → 1 or P D [j] → 0. When considering N → ∞, we can derive the following results: For the AWGN scenario, when Alice is not transmitting signals, the following theorem can be derived.  (14) for the case of Dave having the threshold ϒ < ρ d . Proof 3.1: When the UAV-jammer is not emitting AN, the noise uncertainty at Dave falls in the range of (4) for P U [j] = 0. To achieve the result in (10), it simply needs to choose the upper bound of D forˆ D . On the other hand, when the UAV-jammer is emitting AN, the noise uncertainty at Dave becomes that in (4) for P U [j] > 0. In that case, by subtracting the received power from the UAV it readily derives the result in the second case of (14) so that P F [j] → 1 is guaranteed.
By contrast, the following theorem can be given for deriving the achievable privacy rate when Alice is transmitting signals.
Theorem 2: The privacy rate for Alice's transmission over time period t 0 is given bỹ for the case of Dave having ϒ > ρ D , where P max A [j] is the maximal transmit power allowed for Alice, defined as

Proof 3.2:
The proof is given in Appendix A.
Known from Theorem 2, by introducing UAV's emission it can complicate the noise uncertainty at Dave, which in turn enhances the covert rate achieved by the legitimate pair. Nevertheless, there is always a cost that the legitimate receiver, i.e. Bob, also suffers from the UAV's interference. Hence, the following lemma is derived to investigate what is the best condition for employing UAV.
Lemma 1: In order to achieve a positive gain on privacy rate of the legitimate transmitter-receiver pair, the UAV-jammer should emit AN if the following condition is satisfied: Proof 3.3: The proof is given in Appendix B.

IV. PRIVACY RATE FOR NAKAGAMI-m FADING SCENARIO
Under Nakagami-m fading scenario, all the system setups are identical to those for AWGN scenario, except that Alice→Bob and Alice→Dave links experience Nakagami-m fading, which are respectively denoted by h AB [j] and h AD [j]. Without loss of generality, let us assume that, the channel gains are static during each time slot, and the channel state information (CSI) of the fading channels is known by Alice only. In that case, Alice and Dave maintain the same objectives as the AWGN scenario.
When Alice is not transmitting information, the conclusions derived in Theorem 1 keep the same for Nakagami-m fading scenario. Hence, we are now focus on the case that Alice is transmitting information. In that case, the privacy rate during each time slot, such that in (9), becomes R pr,naka [j] = log 2 (18) where j ∈Ĵ . In particular, we have In order to achieve P D [j] → 0, it needs to find the maximum allowable transmit power. By applying the same approach used in Theorem 2, the expression for P max A [j] is derived as follows ∼ (m 0 , θ 0 ), |h AD [j]| 2 ∼ (m 1 , θ 1 ). Specifically, the ergodic privacy rate can be found as follows.
Theorem 3: The ergodic privacy rate for Alice's transmission over time period t 0 is given bỹ for the case of Dave having the detection threshold ϒ > ρ D . Proof 4.1: The proof is given in Appendix C.
In Table 1, it evaluates the accuracy of the approximated expression for the ergodic privacy rate, where the degree of difference with the simulation ones are compared. Let us assume that m 0 = m 1 = m and θ 0 = θ 1 = 1. From the results, we readily obtain the remark below.
Remark 1: When Taylor expansion of order 2 is used, the difference is very small and is acceptable in practice. Furthermore, the higher degree of Taylor polynomials it uses, the worse approximations it derives. This is due to the logarithmic function in (22), which is similar to Runge's phenomenon.

V. SIMULATION RESULTS
In this section, we provide the numerical results to evaluate the performance of the UAV-jammer aided covert communication system. Unless otherwise stated, we have the following system setups: Fig. 2 evaluates the privacy rate of our covert system under AWGN channel, when considering different noise uncertainties and different UAV speeds. First, it observes that, regardless of UAV speed, the privacy rate achieved can be enhanced as the noise uncertainty gets higher. Second, it finds   that, by introducing the UAV-jammer, the privacy rate can be significantly improved, compared to the case without UAV (i.e. the curve of V = 0m/s). Furthermore, the faster the UAV speed uses, the higher privacy rate it achieves. At last, when Dave obtains the noise information exactly, that is, when ρ = 1, Alice will not be able to transmit covert information without the help of UAV-jammer. Fig. 3 shows the influence of UAV speed on covert communication rate under different noise uncertainty in AWGN channel. First, it observes that, regardless of noise uncertainty, the privacy rate achieved can be enhanced as the UAV speed gets higher. Secondly, it can be found that the influence of low-speed jamming UAV on the covert communication rate is very small. At this time, the covert communication rate P U is mainly affected by the noise uncertainty ρ. In addition, when the speed of UAV increases, the rate of covert communication is significantly increased, and the noise uncertainty can no longer effectively affect the rate of covert communication.
In Fig. 4, it shows the privacy rate versus interference power P U when varying d UB [j − 1] and d UD [j − 1]. It is apparent that, a positive gain in terms of privacy rate can be obtained under the case of V = 20m/s as P U increases, and is much higher than that without UAV. By contrast, for the case of V = 5m/s, we set the distances d UB [j − 1] = 25m, d UD [j − 1] = 50m, thereby resulting in a negative gain of privacy rate compared to the case without UAV. Therefore, the above two observations can verify the conditions deduced in lemma1, which provides the guidelines when and how to employ UAV-jammer for covert communication. Fig. 5 provides the performance of ergodic privacy rate under the Nakagami-m fading channel. In the simulation, we set m 0 = m 1 = m and θ 1 =θ 1 =1, and the argument b defined in Appendix C is the effective SINR of legitimate receiver. As shown, the analytical results almost perfectly match the simulation under different values of m, which validates the accuracy of the approximation approach introduced by Theorem3. Furthermore, as the fading effect becomes less severe, the achievable privacy rate decreases. This reveals that the condition of eavesdropping channel determines how much covert information can be transmitted.

VI. CONCLUSION
This letter has studied the performance analysis for the UAV-jammer aided covert communication system. In order to create a new degree of noise uncertainty at Dave, the UAV can emit artificial noise to protect the covert transmission from Alice to Bob against Dave. Two scenarios have been investigated: the channels from Alice to Bob and Dave experience AWGN and Nakagami-m fading, respectively. We have derived the privacy rate expressions for both scenarios. Simulation results have shown that, the privacy rate can be significantly improved by introducing the UAVjammer, where the positive gain condition under the AWGN scenario is analyzed. Furthermore, for the Nakagami-m fading scenario, the approximation of ergodic privacy rate has also been validated with very high accuracy. Therefore, the proposed UAV-jammer aided covert communication can be deemed to a promising option for meeting future high-level security requirements.

APPENDIX A PROOF OF THEOREM 1
When Dave has the threshold ϒ > ρ D , it is impossible to achieve the target of P F [j] → 1, since the condition in (12) can never be met regardless of UAV emits AN or not. Instead, it should aim to find achieve the target of P D [j] → 0. On the other hand, due to the independency of every time slot, one should motivate to maximize the privacy rate during each time slot, formulated as which is subject to (13). The expression of P pr,awag [j](P A [j]) can be found in (9), which is a monotonic function of P A [j].
In that case, solving problem (23) is equivalent to finding the maximal transmit power for Alice. Hence, the optimal solution can be derived by setting upˆ D = 1 ρ D − P U [j]θ [j] and finding the upper bound of the condition in (13). As a result, we can obtain the following equation: From (24), the expression for P max A [j] is derived. Then, by substituting it into (9) the proof of Theorem 1 is completed.

APPENDIX B PROOF OF LEMMA 1
To ensure a positive gain on the privacy rate, the following condition should be met: where R pr,0 is the privacy rate without UAV's emission. From Theorem 1, we readily know that R pr,0 = log 2 1 When substituting (26) into (25), the following inequality holds Applying some manipulations on (27), the condition in Lemma 1 can be obtained, and the proof is completed.

APPENDIX C PROOF OF THEOREM 3
In order to obtain an approximate solution of (21), a Taylor expansion is performed thereon. In order to minimize the approximation error of Taylor expansion, (21)  . Then, a second-order Taylor expansion is performed at (m 0 θ 0 , m 1 θ 1 ) on the first term in (28).Therefore, Q 1 can be rewritten as follows (29) VOLUME 8, 2020 Note that, m 0 θ 0 and m 0 θ 2 0 are the mean and variance of |h AB [j]| 2 , and the same conclusion applies to ||h AD [j]| 2 .The second term in (28) can be calculated as where ψ is the digamma function. From (29) and (30), the expression for (22) is derived, then the proof of Theorem 3 is completed.