Analysis of Eavesdropping Region in Hybrid mmWave-Microwave Wireless Systems

. Hybrid communication systems, where millimeter-wave (mmWave) links coexist with microwave links, have been an essential component in the ﬁ fth-generation (5G) wireless networks. Nevertheless, the open feature of the wireless medium makes hybrid systems vulnerable to eavesdropping attacks. Eavesdroppers in hybrid communication systems can enhance their attack performance by opportunistically eavesdropping on mmWave or microwave links. This paper, therefore, aims to answer a natural question: in which region do eavesdroppers prefer the mmWave links? To this end, we ﬁ rst formulate this question as an eavesdropping region characterization problem from the physical layer security perspective, where eavesdroppers select the link to eavesdrop based on the ratio between the security performances of the mmWave and microwave links. To model the security performances of both the mmWave and microwave links, we derive closed-form expressions for the secrecy outage probabilities and lower bounds/exact expressions for the secrecy rates of both links. Finally, we provide numerical results to validate our theoretical analysis and also illustrate the mmWave eavesdropping region under various network parameter settings.


Introduction
In the past decade, the number of wireless devices is increasing exponentially, leading to a critical spectrum scarcity issue in current wireless communication systems. One of the promising solutions is to transmit information over the much wider millimeter-wave (mmWave) frequency band for significantly improved capacity and increased data rate in the fifthgeneration (5G) wireless networks [1,2]. Despite the great capacity and high data rate, mmWave communication suffers from high signal attenuation when mmWave signals encounter obstacles [3]. In this case, users may choose to transmit over conventional microwave links. Therefore, hybrid wireless communication systems, where the mmWave links coexist with microwave links, are expected to be a typical component in the ongoing 5G era [4].
However, due to the open nature of the wireless medium, hybrid communication systems are also vulnerable to eavesdropping attacks like other wireless systems [5][6][7]. Recent research has shown that the emerging physical layer security (PLS) technology can achieve a stronger form of security with less computational cost [8,9]. The key idea of the PLS technology is to exploit physical layer characteristics of wireless channels (e.g., fading and noise) to ensure that almost no information is leaked to eavesdroppers [10]. Moreover, the PLS technology can be combined with existing cryptographic methods to provide a critical security solution that can combat eavesdropping attacks [11,12].
Motivated by the benefits of the PLS technology, extensive research efforts have been devoted to the PLS performance analysis and/or PLS scheme design in wireless communication systems [13][14][15][16][17][18][19][20][21][22][23]. For instance, Zhu et al. [15] explored the potential of PLS in mmWave ad hoc networks. Zhang et al. [16] proposed a sight-based cooperative jamming scheme to improve the PLS performance of mmWave ad hoc networks. Zhang et al. [17] examined the problem of mode selection and spectrum partition in cellular networks with inband device-to-device communication. Some authors analyzed the PLS performance of nonorthogonal multiple access networks [18]. In addition, some researchers discussed the joint resource allocation of artificial noise-assisted multiuser wiretap orthogonal frequency division multiplexing channel [20]. The optimization problem of wireless communication systems with intelligent reflecting surfaces was addressed by Chen et al. [21], Makarfi et al. [22], Shen et al. [23].
Recently, the PLS performance analysis of hybrid wireless communication systems has also attracted considerable attention [24][25][26][27][28][29]. Tokgoz et al. [24] investigated the hybrid freespace optical (FSO) and mmWave wireless system from the perspective of PLS, and different fading channels were considered for FSO and mmWave links, respectively. Vuppala et al. [25,26] analyzed the performance of mmWave-overlaid microwave cellular networks. They developed a mathematical framework to analyze the connection outage probability, secrecy outage probability (SOP), and achievable secrecy rate of the hybrid mmWave network. Umer et al. [27] proposed a tractable method using stochastic geometry to analyze the SOP and secrecy energy efficiency of a hybrid heterogeneous network (HetNet). They also explored the PLS performance of the hybrid HetNet, where mmWave links coexist with sub-6 GHz (microwave) links. Wang et al. [28] first proposed a secure mobile association policy based on an access threshold and then investigated the connection probability and security probability of a randomly located user based on the proposed policy. The results showed that introducing an appropriate access threshold can significantly improve the security throughput performance of heterogeneous cellular networks. Wang et al. [29] studied the PLS of two-tier HetNets with sub-6 GHz massive multi-input multioutput macrocells and mmWave small cells. In contrast to previous studies, the eavesdroppers of this paper sent pilot signals during the channel training phase to improve the quality of the intercepted signals.
The previous studies investigated the security performance of legitimate transmitters but did not consider the possible behavior of eavesdroppers. Eavesdroppers in hybrid systems behave differently than eavesdroppers in systems with only one link type (i.e., mmWave link or microwave link). They can improve their eavesdropping performance by opportunistically selecting the wave (i.e., mmWave or microwave) to eavesdrop on. For example, eavesdroppers may prefer to eavesdrop on mmWave links when they have better connections to mmWave transmitters than microwave transmitters. Motivated by the above finding, this paper aims to answer a natural question in hybrid communication systems: in which region do eavesdroppers prefer the mmWave links? Specifically, this paper considers a hybrid communication system with a mmWave communication pair, a microwave communication pair, and an eavesdropper. We focus on characterizing the region where the eavesdropper prefers to eavesdrop on the mmWave link. We first formulate an eavesdropping region characterization problem, where the eavesdropper selects the link to eavesdrop based on the ratio between the security performance of the mmWave and microwave links. To model the security performance of mmWave and microwave links, we derive a closed-form expression for the SOP and lower bound/exact expressions for the secrecy rate of both links. Finally, we provide numerical results to validate our theoretical analysis and also illustrate the millimeter-wave eavesdropping region under various network parameter settings. A preliminary version of this paper can be found by Qu et al. [30], which only focuses on the SOP performance.
The rest of the paper is structured as follows. Section 2 presents the preliminaries, including the system model and wave selection scheme. In Section 3, we derive the SOPs and secrecy rates of the mmWave link and microwave link and characterize the mmWave eavesdropping regions. We formulate the optimization problem to find the optimal eavesdropping locations in Section 4. Section 5 presents numerical results to validate our theoretical analysis and reveal our findings. Finally, we conclude the paper in Section 6.

Preliminaries
In this section, we introduce the preliminaries of this paper, including the system model, antenna model, and blockage and propagation models. In addition, we present the metrics used in this paper and the wave selection scheme of the eavesdropper. Figure 1 shows the system model of this paper, and we consider a square network model with the side length D, which consists of one mmWave transmission pair T 1 À! R 1 , one microwave transmission pair T 2 À! R 2 , and one eavesdropper E which can wiretap on mmWave or microwave link, respectively. We assume the distance between T 1 and T 2 is 2l and construct a coordinate system with the origin at the middle point between them. Thus, the coordinate of T 1 and T 2 are −l; ð 0Þ and l; ð 0Þ, respectively. In addition, we define the coordinate of R 1 by x 1 ; ð y 1 Þ, the coordinate of R 2 by x 2 ; ð y 2 Þ, the coordinate of E by x; ð yÞ. As shown in Figure 1, the angle between T 1 R 1 ! and the x-axis

System Model.
Main lobe ð y 1 Þ, one microwave transmission pair T 2 l; ð 0Þ À! R 2 x 2 ; ð y 2 Þ, and one eavesdropper E x; ð yÞ.  [32], where the antenna of T 1 consists of a main lobe and a back lobe. We use ϕ, A m , and a m (A m >a m ) to define the beam width of the main lobe, the main lobe gain, and back lobe gain, respectively. We assume that to obtain the maximum antenna gain, T 1 and T 2 have guided their antennas correctly so that T 1 R 1 ! coincides with the aperture of the antenna. Unlike T 1 , the microwave transmitter T 2 uses an omnidirectional antenna with an antenna gain of A u .
To simplify the analysis, we assume that the eavesdropper E uses an omnidirectional antenna with antenna gain of A E . Note that the effective channel gain G between T 1 and E depends on the location of E. When E is within the main lobe of the antenna of T, G is A m A E . Otherwise, G is a m A E . We need to compare the angle of T 1 R 1 ! with x-axis and the angle between x-axis and T 1 E ! to determine whether E is inside the main lobe of T 1 's antenna.
Based on the locations of T 1 and E as well as the angle θ, ð r 0 sin θÞ, where r 0 denotes the distance between T 1 and R 1 . Thus, the angle between T 1 R 1 ! and T 1 E ! can given by the following: If and only if ϑ x; ð yÞ is smaller than or equal to half of the beamwidth of the main lobe, i.e., ϕ=2, E is inside the main lobe of T's antenna. Formally, we can give the effective channel gain G between T 1 and E by the following: and is NLoS with probability where β represents the blockage density [33]. The blockage effect results in different path losses for LoS and NLoS links, where the exponents are denoted by α L and α N , respectively. In addition, mmWave links are subject to multipath fading, which we characterize with the Nakagami-m fading model. Note that in this paper, we only consider the case where the link T 1 À! R 1 of the mmWave link transmits information only when the link is LoS. Thus the channel gain of the legitimate channel follows a gamma distribution Γ N L ; ð N L Þ with shape N L and rate N L . In contrast, two cases exist for the eavesdropping channel T 1 À! E. When the link is LoS, it follows a gamma distribution Γ N L ; ð N L Þ with shape N L and rate N L , and when the link is NLoS, it follows a gamma distribution Γ N N ; ð N N Þ with shape N N and rate N N . Typically, N L >N N holds. We use h T 1 ;R 1 to denote the channel gain of the T 1 À! R 1 link, and h L T 1 ;E (resp. h N T 1 ;E ) to denote the channel gain of the T 1 À! E link under LoS (resp. NLoS). Thus, the probability density function (PDF) of h T 1 ;R 1 is given by the following: and the PDF of h b T 1 ;E (b ¼ L; N) is given by the following: where Γ ⋅ ð Þ is the gamma function. To describe the fading effect of microwave links, we use a quasi-static Rayleigh fading model. Thus, the legitimate channel gain h T 2 ; R 2 of the link T 2 À! R 2 and the eavesdropping channel gain h T 2 ; E of the link T 2 À! E follow the exponential distribution with unit mean, e.g., h T 2 ; E ∼ Exp 1 ð Þ. In addition, the links of T 2 À! R 2 and T 2 À! E are also impaired by the large-scale path loss. We use α u to denote the path loss of microwave links.

Metrics.
To measure the secrecy performance of the network, we adopt the commonly-used SOP and secrecy rate as the metrics. Note that SOP represents the probability that E succeeds in decoding the transmitted signals. The secrecy rate denotes the difference between the rate of the main communication channel and the rate of the eavesdropping channel.
We use ε m and ε u to denote the minimum required signal-to-noise ratios (SNRs) for decoding the signals from T 1 and T 2 , respectively. Formally, the SOP when E eavesdropps on the mmWave (i.e., transmitter T 1 ) is formulated as follows: and that when E eavesdropps on the microwave (i.e., transmitter T 2 ) is formulated as follows: According to Barros and Rodrigues [34], Bloch et al. [35], Geraci et al. [36], Ozan Koyluoglu et al. [37], the secrecy rate of mmWave link is formulated as follows: Wireless Communications and Mobile Computing and the secrecy rate of the microwave link is formulated as follows:

Eavesdropping Wave Selection.
Based on the SOPs, E chooses between eavesdropping on the mmWave and eavesdropping on the microwave wave. We assume that E uses the ratio between Equations (7) and (8) as the selection criterion and conducts the selection according to the following rule: Similarly, E chooses to eavesdrop on mmWave links or microwave links, depending on the secrecy rate. We propose to utilize the ratio between Equations (9) and (10) as the selection criterion and develop the selection scheme of eavesdroppers according to the following rules: Here, the parameter ρ sop and ρ sr represent the preference of E. If ρ sop resp: ρ sr ð Þ¼1, E treats eavesdropping on the mmWave links and eavesdropping on microwave links as equally important. If ρ sop resp: ρ sr ð Þ<1, E prefers to eavesdrop on the mmWave links rather than the microwave links, and vice versa.

Eavesdropping Region Characterization Modeling
In this section, we characterize the mmWave eavesdropping regions of the eavesdropper, for which we formulate the eavesdropping regions in Section 3.1, derive the SOP p m so and p u so in Section 3.2, and derive the average achievable secrecy rate R s and R u s in Section 3.3, respectively.

Problem Formulation.
In this section, we will formulate the eavesdropping regions characterized by SOPs and secrecy rates, respectively.

Eavesdropping Region Characterized by SOP.
Note that both p m so and p u so vary with the location of E. Thus, the eavesdropping wave of E, i.e., the wave on which E eavesdrops, varies with its location. Therefore, this paper aims to characterize the eavesdropping region, i.e., the mmWave eavesdropping region where E eavesdrops on the mmWave based on the proposed wave selection scheme in Section 2.5. We use R m to denote the mmWave eavesdropping regions characterized by SOPs, which can be given by the following: We can see that, to determine the mmWave eavesdropping region R m , we need to derive the SOPs p m so x; ð yÞ and p u so x; ð yÞ when E is located in an arbitrary location x; ð yÞ.
3.1.2. Eavesdropping Region Characterized by Secrecy Rate. Similar to SOPs, both R s and R u s vary with the location of E. Therefore, the wave that E chooses to eavesdrop on varies with its location. Thus, we also use the secrecy rates to characterize the mmWave eavesdropping region. We use M m to denote the mmWave eavesdropping regions characterized by secrecy rates, which can be given by the following: It is easy to see that we need to derive the secrecy rate R s and R u s to determine the mmWave eavesdropping region M m .

SOP Analysis.
In this section, we derive the expression of the SOPs p m so x; ð yÞ and p u so x; ð yÞ to determine the regions R m . With the help of the SOPs, we show the mmWave eavesdropping regions characterized by SOPs in Section 5.
3.2.1. SOP of mmWave Link. According to Equation (7), to derive the SOP, we first need to determine the SNR T 1 ;E . Note that SNR T 1 ;E varies depending on whether the link T 1 À! E is LoS or NLoS. We use SNR L T 1 ;E (resp. SNR N T 1 ;E ) to denote the SNR when the link is LoS (resp. NLoS). SNR L T 1 ;E and SNR N T 1 ;E can be given by the followings: and where P m represents the transmit power of T 1 , d T 1 ;E denotes the distance between T 1 and E, and σ 2 is the noise power.
ð yÞ when E eavesdrops on the mmWave link is as follows: and γ ⋅; ð ⋅Þ is the lower incomplete gamma function.
Proof. Applying the law of total probability, we have the following: Next, we derive Q L and Q N . Applying the PDF of gamma random variables, we have the following: Similarly, we have the following: Substituting (17) and (18) into Equation (16) completes the proof. □ 3.2.2. SOP of Microwave Link. Likewise, to derive the SOP in this case, we need to determine the SNR SNR T 2 ;E , which is given by the following: where d T 2 ; E is the distance between T 2 and E. Based on SNR T 2 ;E , we derive the SOP in the following theorem.
Theorem 2. The SOP p u so x; ð yÞ when E eavesdrops on the microwave link is as follows: Proof. Following the definition of SOP, we have the following: Substituting in Equation (21) completes the proof. □ Using the SOPs in Theorems 1 and 2, we can determine the mmWave eavesdropping region R m based on the definition in Equation (11). 3.3. Secrecy Rate Analysis. In this section, we show the derivation steps of the secrecy rates of the mmWave link and microwave link, respectively. With the help of these two rates, we determine the mmWave eavesdropping region characterized by secrecy rates.
3.3.1. Secrecy Rate of mmWave Link. We analyze the average achievable secrecy rate of mmWave communication networks. Based on Equation (9) and [38][39][40], the average secrecy rate can be lower bounded by the following: where R ¼ E log 2 1 þ SNR T 1 ;R 1 À Á Â Ã is the average achievable secrecy rate of the channel between the transmitter T 1 and its receiver R 1 , and R e ¼ E log 2 1 þ SNR T 1 ;E À Á Â Ã is the average achievable rate of the channel between transmitter T 1 and eavesdropper E.
According to Equation (9), to derive the secrecy rate, we first need to determine the SNR T 1 ;R 1 and SNR T 1 ;E . As mentioned above, the link T 1 À! R 1 transmits information only when the link is LoS. Thus, the SNR T 1 ;R 1 is as follows: where x 1 ; ð y 1 Þ denotes the coordinate of R 1 and d T 1 ;R 1 denotes the distance between T 1 and R 1 .
The SNR T 1 ;E varies depending on whether the link T 1 À! E is LoS or NLoS. We use SNR L T 1 ;E (resp. SNR N T 1 ;E ) to denote the SNR when the link is LoS (resp. NLoS). Thus, the SNR b T 1 ;E b ¼ L; ð NÞ is given by the following: Wireless Communications and Mobile Computing 5 Next, we first derive R, then show the derivation of R e , and finally derive R s based on Equation (22). Lemma 1. The average rate of the channel between the transmitter T 1 and its receiver R 1 is as follows: where Proof. To simplify the calculation, we ignore the 1 in The reason is that we focus on the high SNR regime, i.e., SNR ≫ 1. Then, we have the following: Applying the PDF of gamma random variables, we have the following: where (a) follows the series expansions of the incomplete gamma function by Olver et al. [41]. Let Then, R can be rewritten as follows: Next, we derive the limit lim uÀ!1 D u ð Þ, which is given by the following: Based on the series expansions of lower incomplete gamma function by Olver et al. [41], we have the following: Letting ln u ¼ ln N L u N L ¼ ln N L u − ln u, we have the following: where step (b) follows from Equation (33) and step (c) follows the Equation (8.8.13) by Olver et al. [41], Then, we derive the lim uÀ!0 D u ð Þ, which is given by the following: where step (d) follows from the series expansion of the lower incomplete gamma function by Olver et al. [41] and step (e) due to the L'Hospital's rule. When u À! 0, Equation (34) is equal to 0. Substituting Equations (32) and (34) into Equation (29) completes the proof.
□ Lemma 2. The average rate of the channel between the transmitter T 1 and the eavesdropper E is as follows: Proof. To simplify the calculation, we ignore the 1 in log 2 1 þ SNR T 1 ;E À Á . Following the definition of SNR T 1 ;E , Equations (3) and (4), and the law of total probability, we have the following: Applying the PDF of gamma random variables, we have the following: Next, we derive I L and I N , Then, we let Thus, the I L can be rewritten as follows: where step (f ) follows after substituting Equations (32) and (34) into R e . Similarly, we have the following: Wireless Communications and Mobile Computing Substituting Equations (40) and (41) to Equation (37), we complete the proof. □ Theorem 3. Based on Equation (18), Lemma 1 and 2, the average secrecy rate of mmWave link can be lower bounded by the following: Proof. Substituting Equations (3) and (4) to Equation (9) and Equation (4) to Equation (4), we complete the proof.

Secrecy
Rate of Microwave Link. The average achievable secrecy rate of the microwave link is given by the following: where R u ¼ log 2 1 þ SNR T 2 ;R 2 À Á is the secrecy rate of the channel between the transmitter T 2 and its receiver R 2 , and R u e ¼ log 2 1 þ SNR T 2 ;E À Á is the secrecy rate of the channel between transmitter T 2 and eavesdropper E.
According to Equation (10), to derive the secrecy rate of the microwave link, we first need to determine the SNR T 2 ;R 2 and SNR T 2 ;E . Therefore, the SNR T 2 ;R 2 is as follows: and the SNR T 2 ;E is as follows: where d T 2 ;R 2 denotes the distance between T 2 and R 2 , and d T 2 ;E denotes the distance between T 2 and E. We then derive the average achievable secrecy rate of the microwave link, which is given by the following theorem.
Theorem 4. The average secrecy rate of the microwave link is as follows: Proof. Using Equations (10), (44), and (45), we have the following: where step (i) follows after letting C ¼ C 3 C 4 . Then, applying the PDF of exponential distribution, we have the following: We derive A first, where x dx denotes the Exponential Integral Function [42]. Hence, Applying the rule of integral by parts yields, To calculate the above limit, we can use the following expansion of the exponential integral function [42], 8 Wireless Communications and Mobile Computing where γ * is the Euler constant. Therefore, Equation (51) can be expressed as follows: Substituting Equations (44) and (45) into Equation (53) completes the proof. Using the secrecy rates in Theorems 3 and 4, we can determine the mmWave eavesdropping region M m characterized by secrecy rates based on the definition in Equation (12).

Problem Formulation.
To better observe the eavesdropping behavior of eavesdroppers, we propose the optimal eavesdropping location problem to investigate the optimal eavesdropping location in the given network. We formulate the optimization problem as follows: Due to the complexity of the optimization problem, it is difficult to obtain closed-form solutions. Thus, we use the Arg max function in Mathematica to calculate the numerical results [43]. Lastly, we present the numerical results of the optimization problem in the next section.

Numerical Results
In this section, we first provide simulation results to validate the expressions of SOPs and secrecy rates for the mmWave link and microwave link, respectively. We then provide the mmWave eavesdropping regions results characterized by different metrics, i.e., SOPs and secrecy rates, and demonstrate the selection behavior of the eavesdropper in the considered hybrid communication scenario under different parameter settings. Finally, we show the numerical results of the optimal eavesdropping location. Table 1 summarizes the parameter settings used in this section.

Parameter
Value Beamwidth φ of main lobe of T 1 π=6 Main (back) lobe gain A m (a m ) of T 1 10 (0.1) Antenna gain A u of T 2 1 Antenna gain A r of T 1 10 Antenna gain A E of E 5 Path loss exponent α L (α N , α u ) 2 (4, 3) Nakagami-m fading parameter N L (N N ) 3 (2) Transmit power P m (P u ) of T 1 (T 2 ) 1 (1) (w) Noise power σ 2 10 −5 (w) Distance 2ℓ between T 1 and T 2 80 (m) The coordinates of mmWave receiver The coordinates of microwave receiver Minimum required SNR ε m for decoding the signals from T 1 0.03 Minimum required SNR ε u for decoding the signals from We next show in Figure 3 the simulation results vs. theoretical ones for the SOP of the microwave transmission. We also consider three different locations of E, which are −90; ð 0Þ, −60; ð 40Þ, and 80; ð 80Þ. The results in Figure 3 show that the theoretical results match nicely with the simulation ones, demonstrating the correctness of the SOP expression of the microwave transmission. Similar to the SOP of the mmWave transmission, the results show that the SOP of the microwave also decreases as the decoding threshold ε u increases.  Figure 5. We set the antenna gain of E as A E ¼ 5, the coordinates of mmWave receiver R 1 x 1 ; ð y 1 Þ and that of microwave receiver as 40; ð 20 ffiffi ffi 3 p Þ and 60; ð −40Þ, respectively. The results in Figure 4 show that the theoretical results match the simulation results very well, indicating that the lower bound on the secrecy rate of the mmWave link is tight enough to be used as an approximation. Figure 4 also indicates that the secrecy rate of the mmWave link increases as the blockage density β increases. Figure 5 demonstrates the validation of secrecy rate R u s for various target antenna gain A u of microwave transmitter T 1 . Notice that the simulation results match nicely with theoretical values, indicating the correctness of the expression of the secrecy rate of the microwave link. The results in Figure 5 also show that the secrecy rate of the microwave transmission increases as the microwave transmitter T 1 's antenna gain A u increases.

Performance Evaluation.
We investigate the impacts of several important parameters on the mmWave eavesdropping region R m characterized by the SOPs and that on the mmWave eavesdropping region M m characterized by the secrecy rates.

Eavesdropping Region R m .
To understand the impact of the selection parameter ρ sop on the mmWave eavesdropping region R m , we summarize in Figure 6 the mmWave eavesdropping region R m under three different values of ρ sop . Figure 6 shows that R m enlarges as the selection parameter ρ sop decreases. Recall that ρ sop represents the selection preference of E. The smaller ρ sop is, the more E prefers the mmWave over the microwave. Thus, for a fixed location, the SOP of the microwave remains unchanged, and as ρ sop decreases, this location is more likely to be included in the mmWave  eavesdropping region R m . As a result, the size of the mmWave eavesdropping region R m increases. Figure 7 shows the impact of the blockage density β on the mmWave eavesdropping region R m . It can be observed that the size of R m decreases as β increases. The major reason for this phenomenon is that the larger β is, the more blockage exists in the network. As a result, the link from T 1 to E is more likely to be NLoS, leading to a smaller SOP. Thus, the possibility of a fixed location being included in R m is reduced, resulting in a smaller R m .
Finally, we explore the impact of the angle θ between T 1 R 1 ! and the x-axis (i.e., the boresight of the mmWave transmitter's antenna) on the mmWave eavesdropping region R m . As Figure 8 shows, the size of R m changes as the angle θ changes. In general, the region size is minimized when the mmWave transmitter's antenna points toward the microwave transmitter (i.e., the case of θ ¼ 0 in Figure 8), while it is maximized when the mmWave transmitter's antenna points towards the opposite direction of the microwave transmitter (i.e., the case of θ ¼ π in Figure 8). This is intuitive since the closer E is to the microwave transmitter, the larger the SOP under the microwave and, thus, the less likely E prefers the mmWave.

Eavesdropping
Region M m . Figure 9 illustrates the impact of the blocking density β on the mmWave eavesdropping region M m . We can observe that the size of M m decreases as β increases. A larger β means that more blockages exist in the network. Consequently, the link from T 1 À! E is more likely to be NLoS, which leads to a larger secrecy rate of the mmWave link. Therefore, the likelihood of any location being included in the mmWave eavesdropping region decreases, which leads to a smaller mmWave eavesdropping region M m .
We then demonstrate the impact of the selection parameter ρ sr on the mmWave eavesdropping region M m in Figure 10. It shows that as the selection parameter ρ sr    increases, the size of M m decreases. Note that we use ρ sr to represent the selection preference of E in this paper, and a smaller ρ sr means that E prefers mmWave. Since the secrecy rate of the microwave link remains constant, as ρ sr increases, it is more difficult for any fixed location to be included in the mmWave eavesdropping region. Therefore, the mmWave eavesdropping region M m becomes smaller.
An interesting phenomenon can be observed from Figure 10, when ρ sr ¼ 0:5 or 1, the change of the mmWave eavesdropping region is extremely small. This is because, in this paper, we assume that the eavesdropper E treats eavesdropping on the mmWave link and the microwave link as equally important when ρ sr ¼ 1. Also, the transmit power of the mmWave link is much smaller than that of the microwave link. Therefore, when ρ sr is large (i.e., 0:5<ρ sr <1), although the eavesdropper E is in the vicinity of the mmWave transmitter T 1 , it still prefers to eavesdrop on the microwave link with stronger transmit power in order to improve their eavesdropping performance.

Optimal Eavesdropping Location.
We demonstrate the numerical result of the optimization eavesdropping locations in Section 4. We set the beam width ϕ of the main lobe of T 1 as ϕ ¼ π=6, the blockage density as β ¼ 0:1 and the antenna gain A E of E as 5. Moreover, we set the minimum required SNRs ε m and ε u as 0:03 and 0:1, respectively.
We illustrate the optimal eavesdropping locations by considering three different angles of θ (the angle between T 1 R 1 ! and x-axis), i.e., π=6, π=8, and π=10. Figure 11 shows the optimal eavesdropping locations of the eavesdropper according to the objective function that we proposed when the angle θ is at three different angles. Note that the sectors represent the main lobe of the mmWave transmitter T 1 −40; ð 0Þ, T 2 40; ð 0Þ represent the microwave transmitter, the three different colors represent three different angles θ, and the continuous dots on the borders denote the optimal eavesdropping locations.
According to Figure 11, we can observe that the optimal eavesdropping location changes as the angle θ changes. Moreover, it is easily seen that the optimal eavesdropping location is always on the lower border of the main lobe of the mmWave transmitter's antenna. As W m increases, the optimal location is close to mmWave transmitter T 1 .

Conclusion
This paper investigates the millimeter-wave (mmWave) eavesdropping region characterization problem in hybrid wireless communication systems where mmWave links and microwave links coexist. We first derived the secrecy outage probabilities and secrecy rates of both the mmWave link and microwave link, respectively, based on which we identify the eavesdropping region, where eavesdroppers prefer the mmWave links. We then demonstrate the numerical results of optimization eavesdropping locations. The results in this paper showed that the mmWave eavesdropping region decreases as the selection parameter ρ sop and ρ sr increases. In addition, the eavesdropping region decreases when there are more blockages in the network (i.e., when blockage density β becomes larger).

Data Availability
No underlying data were collected or produced in this study.

Disclosure
This paper was presented in part at the International Conference on Networking and Network Applications (NaNA), Haikou, China, December 2020.

Conflicts of Interest
The authors declare that they have no conflicts of interest.