Performance Analysis of Two-Way Relaying in mmWave-Based Aerial Links

Two-Way Relaying (TWR) is an efficient relaying technique that doubles the spectral efficiency as compared to the traditional one-way relaying. As such, it has been nominated as a solution to extend the coverage of links operating on the Millimeter Wave (mmWave) band. In this paper, the performance of TWR is investigated for mmWave-based aerial links in which the involved entities are all aerial. Taking into account the orientation fluctuations of the involved aerial entities, the performance is characterized in terms of the outage rate and the bit error rate. Closed form expressions are derived based on the considered performance metrics along with a large set of Monte Carlo simulations results to validate the obtained analytical formulas. Analytical and simulation results address the impact of various factors including the number of antenna elements, the modulation order, the transmission distance, and fluctuation intensity.


I. INTRODUCTION
T HE ADVANCEMENT of wireless networks requires integrating high-frequency spectrum to handle the increasing number of connected devices and data demands.To meet these requirements, millimeter wave (mmWave) bands have emerged as a promising solution, offering ample bandwidth and faster data rates compared to microwave bands [1].However, mmWave faces challenges in nonline-of-sight (NLoS) scenarios due to obstacles and signal loss [2].Another challenge that is usually reported in the mmWave bands is the shorter transmission distances as compared to other lower RF bands.To this end, multi-hop links can be utilized where two nodes exchange data with the help of a relay node, allowing efficient and reliable data transmission even in obstructed environments [3].Two common relaying strategies are popularly used in the literature, namely, one-way relaying (OWR) and two-way relaying (TWR).In OWR, the relay assists in transmitting data from a source node to a destination node.The relay receives the signal from the source, processes it, and then forwards it to the destination.Two more time slots are needed for transmitting in the opposite direction.In TWR, both source nodes communicate simultaneously with the relay, which processes and broadcasts the received signals back to the source nodes.This bidirectional communication in TWR improves spectral efficiency and enables more flexible information exchange [4].Relaying schemes include decode-and-forward (DF), where the relay decodes and reencodes data, and amplify-and-forward (AF), where the relay amplifies data without decoding [5].
Although relays can efficiently boost the link performance and maintain the signal strength acceptable at the destination, their performance over mmWave still subject to the lineof-sight (LoS) requirement which might be unavailable via terrestrial entities.To this end, unmanned aerial vehicles (UAVs) can play a significant role in attaining the LoS for both communication ends due to their mobility, flexibility and adaptability [3].UAVs can efficiently cover large areas, making them valuable for communication in emergencies cases when terrestrial infrastructure is compromised [6].
Moreover, they can operate in hazardous environments, making them suitable for rescue missions.Integrating UAVs in mmWave-based multi-hop networks has gained significant attention due to its potential to revolutionize wireless communication systems.This combination results in a robust and adaptable communication system that enhances the overall performance and reliability of wireless networks ensuring continues connection during critical situations [7].
In spite of the significant research conducted on UAVbased multi-hop networks, a distinct gap remains in the literature regarding the exploration of TWR air-to-air communication utilizing mmWave bands.Literature studies have not sufficiently tackled the effect of the random fluctuations of the UAV's antennas on the performance of the overall link.One of the challenges in aerial links involves the use of hovering UAVs, which are sensitive to continuous vibrations.These vibrations directly impact the performance due to the resulting fluctuations of the antennas.Moreover, the conventional TWR scheme is not applicable in mmWave due to the use of directional antennas, preventing simultaneous service to nodes with the same antenna.To this end, in this work, the performance of TWR in mmWave-based aerial links is investigated, which has never been addressed to the best of our knowledge.The primary contributions of this work can be summarised as follows: • The channel model for aerial links is developed by considering the antenna propagation patterns specified in the 3GPP standards, angular antennas fluctuations, path loss and channel fading.• A comprehensive mathematical framework is followed aiming to obtain closed-form expressions for the bit error rate (BER) and the outage probability (OP) in the TWR mmWave-based aerial links.• Extensive simulation results of the considered system is presented to validate and corroborate the analytical findings.
• The system performance is thoroughly studied by exploring various parameters, including modulation order, the separation distance, UAV's instability parameter, and the number of antenna elements.The rest of this paper is organized as follows.Related works are presented in Section II.Section III outlines the system model and the formulation of the antennas' gain.Further details into the TWR mmWave-based aerial link include transmission and relaying phases are discussed in Section IV.In Section V, the performance analysis of the investigated system covering BER analysis and OP analysis is presented.The validation of the derived analytical formulas through simulation results is illustrated in Section VI.Finally, conclusion presented in Section VII.

II. RELATED WORK
In the literature, there is a set of related works that investigates the performance of mmWave based TWR links.In [8], authors discuss the advantages of using AF-TWR in a mmWave network for bi-directional data exchange between two end users.A relay-selection approach is presented to optimize signal quality, coverage, and spectral efficiency.The work in [9] investigates beamforming design for improving secrecy capacity in mmWave TWR networks, where iterative algorithms are employed to efficiently solve the beamforming problem, and numerical experiments are presented to validate the effectiveness of the proposed designs in enhancing security.
In addition, a study on mmWave TWR systems for advanced cellular networks is presented in [10].The approach studied addresses challenges related to self-interference and sparse mmWave channels, and extensive simulations confirm the algorithm's effectiveness in reducing error rate.
Recently, several works have addressed the performance of the UAVs in various communication systems.A UAVassisted TWR networks for data exchange is investigated in [11] considering two methods; full-duplex and half-duplex.Optimizing bandwidth and power allocation in both cases is shown to be a complex problem, where approximations are presented which reveal that the half-duplex approach is more efficient than the full-duplex.Authors in [12] focus on a UAV-assisted TWR system using a two-slot network coding scheme and an iterative algorithm is introduced to maximize the system's average sum rate along with examination of the impact of traffic patterns on performance.Results demonstrate that a moving relay system offers higher throughput compared to a static relay.Authors in [13] explore a UAV-based multi-hop TWR network with multiple UAVs assisting two ground users in information exchange.The proposed pattern achieves a data rate of (1/2) data packets per time slot, marking a notable enhancement in throughput compared to alternative schemes.The research in [14] concentrates on a TWR system, formulating achievable secrecy energyefficiency by considering various UAV parameters.The problem is solved iteratively through decomposition and optimization techniques, and simulation results show that the proposed methods outperform existing approaches.In [15], an extensive investigation of a cooperative UAV network is demonstrated, employing a two time slots transmission approach, a half-duplex mode, and a DF protocol.Closed-form expressions for both end-to-end Signal-to-Noise Ratio (SNR) and OP are derived.Additionally, Monte Carlo simulations are employed to study the impact of various factors on the OP.In [16], a study is conducted to investigate mmWave-based D2D communication under Nakagami-m fading channels.The OP for D2D users with interference from multiple cellular is analyzed.The study takes into consideration sectorized beamforming gain probabilities and distance from interfering nodes.The results indicate that the D2D OP decreases with higher main lobe gain and increases with higher side lobe gain.
To the best of our knowledge, there are no previous works investigating the TWR performance in mmWavebased aerial links.As mentioned earlier, integrating aerial entities in multi-hop links affects the performance due to the fluctuations of the antennas orientation, which, in turn, results in a performance degradation in both OP and BER.

III. SYSTEM MODEL
The system under consideration involves three hovering UAVs, particularly quadrotor.Quadrotor UAVs are capable of remaining stationary and flying at zero-speed [17], [18], [19], [20], [21], [22].The considered UAVs utilizing mmWave frequencies to establish communication between each other.Among these UAVs, one serves as a relay (R) positioned between the other two UAV nodes A and B, as depicted in Fig. 1.Both source nodes communicate simultaneously with the relay.The relay operates in a DF mode, where it decodes the received data before forwarding it to the destination node.Note that the direct communication link between nodes A and B is not available due to obstacles, or high path loss.The modulators of the nodes are considered to use the same modulation order, denoted by M. The signals transmitted from nodes A, B, and R are affected by the channel fading, and path loss.Nodes A and B are equipped with single uniform square array antennas, serving for both transmission and reception tasks.In contrast, relay node R is equipped with a dual-antenna configuration, each antenna is dedicated to separately serve nodes A and B which are assumed to be sufficiently spaced.
As illustrated in Fig. 2, operating within a hovering UAVs system introduces the challenge of vibrations stemming from internal or external factors.These vibrations lead to dynamic shifts in the positions and orientations of the antennas on nodes A, B and R.These fluctuations exert a significant influence on the directional mmWave links, thereby highlighting their pivotal role in shaping the link's overall performance.These fluctuations manifest as angular deviations in both x and y directions, denoted by θ k = (θ kx , θ ky ).These angles follow a Gaussian distribution, with θ kx ∼ N {θ kx , σ 2 θx } and θ ky ∼ N {θ ky , σ 2 θy }, where k ∈ {A, B, R}, and σ 2 θ is the UAV's fluctuation intensity.Each antenna consists of N × N elements spaced by λ 2 in both x, and y directions, with an antenna gain denoted as G k .Note that λ is the antenna wavelength and it is equal to λ = c f c , where c is the speed of light, and f c is the carrier frequency.Node k's array radiation can be formulated in the direction of θ k , and φ k as follows [17] These angles, represented in (1), play an important role in defining the antenna gain at node k, i.e., G k , as follows where G e corresponds to the radiation pattern of an individual antenna element, and G a represents the antenna array factor.G o can be determined through the following expression [17] G o = 1 The exact form of G e can be inferred from the 3GPP report as follows [23].
where G e,3dB can be defined as follows [23] G e,rdB,2 = − min 12 ) where the maximum directional gain of the antenna element, identified as G max , which is fixed to 10 dBi.Additionally, F m represents the front-to-back ratio, set to 30 [17].The side-lobe level limit, denoted as G SL , is also set at 30 [17].
The array factor G a of the square array antenna at node k with dimensions N × N, can be formulated as follows [24] It is worth emphasizing that the array factor G a remains consistent across all antennas, implying that it exhibits an identical pattern for each antenna.However, its magnitude can vary based on the direction.Hence, the instantaneous SNR γ kl can be expressed as follows where kl ∈ {AR, RB, BR, RA} represents different links pairs, G kl = G k G l is the overall gain, P denotes the transmit power, η stands for the small-scale fading, σ 2 signifies the noise power, and ρ kl represents the path loss which can be defined based on the 3GPP report as follows [25] ρ kl = 20 log 10 (40π (d kl )(f c /3)) + min(0.03h The parameter d kl signifies the link distance between k and l, and h represents the average height of the buildings. To account for the channel fading, Nakagami-m model is utilized, where η is Nakagami random variable (RV).As a result, the probability density function (pdf) of the RV ζ = |η| 2 , denoted as f ζ (ζ ), can be formulated as outlined in [26].
where m is the Nakagami fading parameter, and (•) represents the Gamma function.

IV. TWR MMWAVE-BASED AERIAL LINKS
The considered TWR mmWave-based aerial communication system involves two phases carried out across two consecutive time slots, as illustrated in Fig. 3.The first phase is the transmission phase, followed by the relaying phase.

A. TRANSMISSION PHASE
During the transmission phase, each of the nodes A and B transmits a symbol to the relay within the first time slot.Specifically, node A transmits a bit vector denoted as a, containing log 2 M bits.Similarly, node B has its own bit vector b of the same length.These bit vectors, a and b, modulated through an M-ary modulator at their respective nodes, generating modulated symbols x AR at node A and x BR at node B. Note that all nodes are sufficiently spaced, and antennas are directed toward the respective nodes to prevent mutual interference between the transmitter and receiver.The received signal at the relay node R from node k upon accomplishing the first phase is y kR , where kR ∈ {AR, BR}, and it can be expressed as follows where w R the additive white Gaussian noise at node R.
The considered system employs the Maximum Likelihood (ML) detection method, assuming that the channel state information is available at the receiver end.It is noteworthy that ML is considered the optimal detection method [27].In this phase, the relay node employs ML detection to detect symbols from nodes A and B separately, as follows xkR = arg min

B. RELAYING PHASE
Upon completing the transmission phase, the relay proceeds to modulate the decoded symbols for subsequent transmission to nodes A and B. The received signal at node l from the relay can be mathematically formulated as follows where l ∈ {A, B}.Nodes A and B will utilize ML detection on the received signal y Rl to decode the received data as follows After completing the second phase, each of the nodes has received a symbol from the other node, and the detected bit vectors at nodes A and B are denoted as â and b, respectively.

V. PERFORMANCE ANALYSIS
In this section, we will comprehensively evaluate the performance of the investigated scheme using two important measures, namely, BER analysis and OP.

A. BIT ERROR RATE ANALYSIS
Considering the studied system described earlier, it is crucial to note that the average BER at both nodes A and B is identical.This uniformity arises because both nodes share identical average channel characteristics, path loss models, antenna gains, and separation distances.Hence, the BER analysis can be limited to only one of the nodes, such as node B. When node B receives bit sequence, denoted as â, from node A through the relay R, the average BER at node B in the data originating from node A, referred to as BER AB , can be mathematically expressed as follows where e AR and e RB represent the error vectors corresponding to the data transmitted from node A to the relay R, and from the relay R to node B, respectively.Each bit in these error vectors can take a value of either 1, indicating an incorrect reception, or 0, indicating a successful reception.Eq. ( 17) is derived by incorporating different expressions for â and â, both of which involve these error vectors.The last line in (17) highlights that the situation where e AR ⊕ e RB = 0 arises exclusively when the bits within the error vectors differ.This condition can be expanded as follows Let us consider α AR = Pr.(eAR = 1), and α RB = Pr.(eRB = 1).These terms can now be utilized to reformulate (19) as follows where α AR and α RB represent the probabilities of incorrectly decoding the received bit vector at R from A and the received bit vector at B from R, respectively.For a coherent PSK modulation scheme, the average BER over a fading channel for the link A − R can be expressed as follows [28] where erf(•) is the error function, γ AR is the instantaneous SNR of the A − R link.δ and can be determined respectively as follows and The pdf f γ AR (γ ) in ( 21) is the pdf of γ AR and it can be computed as follows [17] where D is considered as the number of antenna sectors assumed.While R ij can be expressed as follows where μ AR can be expressed as follows and R i,j in ( 25) can be calculated as follows where G (N) o can be expressed as follows Accordingly, (21) can be rewritten as follows The integral appearing in ( 27) can be evaluated using the formula provided in [29, eq. (4.3.8)].Consequently, α AR can be expressed in a closed-form expression as demonstrated in (28), shown at the bottom of the page.Here, 2 F 1 (.; .; .)refers to the hypergeometric Gauss function.
Employing a similar approach as in ( 21)-( 28), a closedform expression for α RB can also be derived, as shown in (29), shown at the bottom of the page.With these results in hand, the average BER AB can be computed by substituting the expressions for α AR and α RB from ( 28) and ( 29) into (20).

B. OUTAGE PROBABILITY ANALYSIS
The OP is a measure of how often the overall SNR in a system falls below a specific threshold due to channel effects.It can be represented mathematically as follows In the considered TWR mmWave-based aerial link system, the average OP at node A or node B is identical.Hence, the OP analysis can be computed on only one node, say node B, and it can be expressed as follows where γ AR , γ RB are the SNR for the A − R, and R − B links, respectively.Assuming that γ min = min{γ AR , γ RB }, the OP can be expressed in terms of CDF as follows where F γ min (γ ) is the CDF of γ min , and it can be obtained as follows [30] The OP for the kl link can be derived by calculating the CDF of the RV γ kl using (24), and [31, eq. ( 8.350.1)] as follows where V(., .) is an incomplete Gamma function.Hence, the OP at node B can be expressed in (35) at the bottom of the page by substituting (34) into (33).

VI. SIMULATION RESULTS
This section presents a comprehensive evaluation of the TWR mmWave-based aerial communication system.Through a combination of simulations and analytical formulas, we assess the system performance and reliability.The simulations provide practical evidence, while the analytical approach offers valuable theoretical insights.The simulations parameters that are fixed are listed in Table 1.
In Fig. 4, the average BER performance of the considered dual-hop TWR mmWave-based aerial links is explored considering different operational parameters.In Fig. 4-(a), the influence of varying the modulation order within the studied system is depicted.Notably, increasing the modulation order leads to a negative impact on the BER performance within mmWave-based aerial communication scenarios.This outcome signifies that employing higher modulation order results in higher probabilities of errors during data transmission.Moreover, the matching between the analytical and simulation results is distinctly evident, underscoring the precision of the derived closed-form expressions.
In the investigated dual-hop setup, the separation between the UAVs plays a crucial role in determining the BER performance.As indicated in Fig. 4-(b), where the variable d denotes the distance between A − R or R − B. As the UAVs are positioned closer together, the signal strength becomes stronger, resulting in a lower BER and enhanced communication effectiveness.However, as the distance between UAVs expands, the signal strength diminishes, leading to an increased BER and decreased data reliability.By examining how the BER changes across different distances, we gain insights into designing a more effective model for successful communication within mmWave-based UAV networks.
The number of antenna elements in UAVs has a significant impact on the BER in TWR mmWave-based aerial communication.As visualized in Fig. 4-(c), as the number of antenna elements increases, the system's beamforming capabilities improves, allowing for more precise and focused signal transmission and reception.This increased beamforming ability enhances the directivity of the antennas, concentrating the transmitted power towards the intended direction and increasing the received signal power at the destination UAV's node.Hence, the BER decreases as the number of antenna elements increases.
The instability factor of UAVs σ 2 θ , which refers to the random fluctuations in the antennas during communication, can significantly impact the BER performance in TWR mmWave-based aerial communication system.When UAVs experience higher instability, their positions may vary more rapidly, leading to changes in the link quality.As a result, the received signal strength will fluctuate, affecting the overall communication performance.As visualized in Fig. 4-(d), as the UAV's instability factor increases from σ 2 θ = 0.5 • to σ 2 θ = 1.25 • , the BER tends to rise due to the increases in the probability of communication errors caused by signal fluctuations and misalignment between antennas beams.Therefore, minimizing UAV instability becomes crucial to ensure reliable and stable communication.
The OP of the investigated system is studied in Fig. 5 considering different operational parameters.The relationship between the SNR threshold and the OP in the studied system is shown in Fig. 5-(a).A notable trend is observed: as the SNR threshold increases, there is a simultaneous increase in the OP.To elaborate, when a higher signal quality is required for successful communication by increasing the SNR threshold, the OP also increases.This indicates that, a stricter SNR threshold might reduce the overall reliability of the system, resulting in a higher likelihood of communication outage.
The influence of the distance between nodes on the OP within the investigated system is presented in Fig. 5-(b).The graph demonstrates a clear trend: as the distance between nodes increases, the OP tends to increase as well.This observation highlights the crucial role of node spacing in determining the system's performance.Specifically, maintaining a sufficient distance between nodes contributes to reducing the chances of signal degradation due to path loss and other propagation effects.
Fig. 5-(c) vividly demonstrates the relationship between the number of antenna elements and the OP of the TWR mmWave-based aerial communication system.This visual aid illustrates that increasing the number of antenna elements corresponds to a lower OP.This emphasizes the significance of how the antenna design plays in enhancing the performance, and minimizing outages in the investigated communication systems.Furthermore, Fig. 5-(d) sheds light on the impact of the UAV's instability parameter on the OP within the studied system.This illustration offers a clear understanding of how varying the UAV's fluctuation parameter can influence the likelihood of communication outages.As the UAV's fluctuation parameter increases, the OP tends to rise.

VII. CONCLUSION
The performance of TWR in mmWave-based aerial links has been addressed in this paper in terms of the error analysis, and OP.The system under investigation involves three UAVs, where two of them are exchanged data while the third UAV acts as a relay.The performance has been analyzed while considering the impact of the fluctuations in the antenna orientation of all aerial nodes that are caused by the vibration of the hovering UAVs.Furthermore, comprehensive analysis has been conducted, successfully yielding closed-form expressions for the average BER and OP.These analytical findings have been shown to closely align with simulation results.Through extensive investigation, the performance of the TWR system is assessed across various factors, including the distance between nodes, the number of antenna elements, UAV vibration intensity and modulation order.

FIGURE 1 .
FIGURE 1.The considered system of TWR mmWave-based aerial link.

2 ,
x kR ∈X y kR − P × ρ kR G kR η kR x kR(14) where X corresponds to the potential set of x kR , and | • | denotes the norm operator.The resulting bit vectors obtained from the detected symbols at R are represented by â and b from nodes A and B, respectively.