Recently, unmanned aerial vehicles (UAVs) have been considered as a promising solution for a variety of critical applications such as environmental surveillance, public safety, disaster relief, search and rescue, and purchase delivery . Considering relaying as one of the most elegant data transmission techniques in wireless communications [3, 4, 5], one of the recent applications of UAVs is utilizing them as relays in wireless networks [6, 7, 8]. Constructing a UAV communication network for such applications is a non-trivial task since there is no regulatory and pre-allocated spectrum band for the UAVs. As a result, this network usually coexists with other communication networks, e.g., cellular networks [9, 10]. Thus, studying the problem of interference avoidance/mitigation for the UAV communication network is critical, where the inherent mobility feature of the UAVs can be deployed as an interference evasion mechanism. This fact is the main motivation behind this work.
In most of the related literature, the position planning for a single UAV, which is considered either as a gateway between a set of sensors and a ground node or as a relay node between a pair of transmitter and receiver, is developed [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. In , the optimal position of a set of UAV relays is studied to improve the network connectivity and communication performance of a team of ground nodes/vehicles, where there is no communications among the UAVs themselves. In , a UAV is employed as a mobile relay to ferry data between two disconnected ground nodes. This work aims to maximize the end-to-end throughput of the system by optimizing the source/relay power allocation and the UAV’s trajectory. In , UAV-assisted relay networks are studied in the context of cyber-physical systems, where a relay-based secret-key generation technique between two UAVs are proposed. In , optimal deployment of a UAV in a wireless relay communication system is obtained in order to improve the quality of communications between two obstructed access points, while the symbol error rate is kept below a certain threshold. In 
, UAVs are utilized as moving relays among the ground stations with disconnected communication links in the event of disasters, where a variable-rate relaying approach is proposed to optimize the outage probability and information rate. In, UAV-enabled mobile relaying in the context of the wiretap channel is proposed to facilitate secure wireless communications, the goal of which is to maximize the secrecy rate of the system. In , considering the usage of a UAV as relay between a pair of transmitter and receiver, an end-to-end throughput maximization problem is formulated to optimize the relay trajectory and the source/relay power allocations in a finite time horizon. In , a UAV works as an amplify-and-forward relay between a base station and a mobile device. The trajectory and the transmit power of the UAV and the transmit power of the mobile device are obtained aiming to minimize the outage probability of the system. In , the optimum placement of a UAV in both static and mobile relaying schemes is considered so as to maximize the reliability of the network, for which the total power loss, the overall outage, and the overall bit error rate are used as reliability measures. Also, it is shown that that decode-and-forward relaying is better than amplify-and-forward relaying in terms of reliability. In , position planning of a UAV relay is studied to provide connectivity or a capacity boost for the ground users in a dense urban area, where a nested segmented propagation model is proposed to model the propagation from the UAV to the ground user that might be blocked by obstacles. In , the optimization of both propulsion and transmission energies for a UAV relay is considered, where the problem is studied as an optimal control problem for energy minimization based on dynamic models for both transmission and mobility. Studying the UAV placement planning in the multi-hop relay communication context, in which multiple UAVs can be utilized between the transmitter and the receiver, is a new topic studied in [22, 23, 24, 25]. The aim of these works is similar to the aforementioned literature; however, data transmission through multiple UAVs makes their methodology different. Moreover, there are some similar works in the literature of sensor networks, among which the most relevant ones are [26, 27]. In , the two-dimensional (2-D) placement of relays is investigated aiming to increase the achievable transmission rate. In , the impromptu
(as-you-go) placement of the relay nodes between a pair of source and sink node is addressed considering the distance between those nodes as a random variable, where the space is restricted to be one-dimensional (1-D).
Nevertheless, none of the aforementioned works consider the placement of UAV(s) in the presence of interference in the environment. This work can be broken down into two main parts. In the first part, we aim to go one step beyond the current literature and investigate the UAV-assisted wireless communication paradigm in the presence of a major source of interference (MSI), which refers to the source of interference with the dominant effect in the environment. Considering different interpretations for the MSI, e.g., a primary transmitter in UAV cognitive radio networks [9, 28], an eNodeB in UAV-assisted LTE-U/WiFi public safety networks , a malicious user in drone delivery application, or a base station in surveillance application, our paper can be adapted to multiple real-world scenarios. Given the intractability of direct analysis upon having multiple sources of interference in the network, we later show that the interference caused by multiple sources of interference with known locations can be modeled as the interference of a single hypothetical MSI, making our framework and analysis applicable to a wider range of applications. In the second part, we consider a distinct scenario, in which, due to the limited knowledge of the positions of the sources of interference or the time varying nature of the environment, we model the interference as a stochastic phenomenon. For each part, we study the optimal placement planning upon having a single UAV, i.e., dual-hop single link scheme, and multiple UAVs, i.e., multi-hop single link scheme, acting as relays between the transmitter and the receiver. The existence of interference renders our methodology different compared to the current literature; however, the previously derived results can be considered as especial cases in our model by assuming that the MSI is located too far away or it possesses an insignificant transmitting power. Hence, the methodology proposed in this work can motivate multiple follow up works revisiting the previously studied problems considering the presence of interference in their models. Moreover, compared with the relevant literature on multi-hop UAV-assisted relay communication, e.g., [22, 23, 24, 25], mostly focused on obtaining the optimal location/trajectory of the UAVs, in addition to incorporating the interference into our model, we introduce and investigate two new problems: i) determining the minimum required number of UAVs and their locations so as to satisfy a desired SIR, or equivalently data rate, of the system; ii) developing a distributed placement algorithm, which requires message passing only among adjacent UAVs to maximize the (average) SIR of the system.
1) We investigate the problem of optimal UAV position planning considering the effect of interference in the environment in the decode-and-froward relay communication context for both the dual-hop and the multi-hop relay settings. We pursue the problem considering i) the existence of an MSI in the network, and ii) stochastic interference. Moreover, we propose and investigate two novel problems in the multi-hop relay setting: i) determining the minimum required number of UAVs and their optimal positions, and ii) developing an optimal fully distributed position alignment algorithm.
2) Considering a single UAV and an MSI, we develop a theoretical approach to identify the optimal position of the UAV so as to maximize the signal-to-interference ratio (SIR) of the system. Furthermore, the position planning for a single UAV upon having stochastic interference is also addressed.
3) In the multi-hop relay context, considering the existence of an MSI, we develop a theoretical framework that simultaneously determines the minimum required number of UAVs and their optimal positions so as to satisfy a predetermined/desired SIR of the system. We also develop a similar framework considering the stochastic interference in the environment and investigate the optimally of our approach upon having independent and identically distributed (i.i.d.) and non-i.i.d interference along the horizontal axis.
4) In the multi-hop relay context, considering the existence of an MSI and given the number of UAVs, we propose an optimal fully distributed algorithm attaining the maximum attainable SIR of the system, which only requires message exchange among the adjacent UAVs achieved by forward and backward propagations. We also propose a fully distributed position planning considering stochastic interference and investigate its optimally upon having i.i.d. and non-i.i.d interference along the horizontal axis.
We consider data transmission between a pair of transmitter (Tx) and receiver (Rx) co-existing with a major source of interference (MSI). We consider a left-handed coordination system , where the Tx, the Rx, and the MSI are assumed to be on the ground plane defined as . The location of the Tx, the Rx, and the MSI is assumed to be , , and , respectively. We assume for simplicity, which can be readily generalized with minor modification. The transmission powers of the Tx, the UAV, and the MSI are denoted by , , and , respectively. To improve the transmission data rate, it is desired to place a UAV or a set of UAVs, each of which acting as a relay, between the Tx and the Rx. To have tractable solutions, we assume that the UAVs are placed at plane. While such a constraint impose certain limitations to our study, it allows us to obtain some first analytical results that provide insightful guidance for practical design in general and also are meaningful for some specific application scenarios. Also, considering legal regulations, we confine the altitude of the UAVs to .
We consider the line-of-sight (LoS) and the non-line-of-sight (NLoS) channel models, for which the path-loss is given by:
where , , () is the excessive path loss factor incurred by shadowing, scattering, etc., in the LoS (NLoS) link, is the carrier frequency, is the speed of light, is the path-loss exponent111The LOS model is used for the air-to-air channel between the UAVs, for which is a well-known choice. In some scenarios, the value of for the NLOS link is assumed to be greater than . This leads to straightforward modifications in the derived results., and is the Euclidean distance between node and node . The link between two UAVs (air-to-air) is modeled using the LoS model, while the link between the MSI and the Rx (ground-to-ground) is modeled based on the NLoS model. To model the link between a UAV and the Rx/Tx/MSI (air-to-ground and ground-to-air) either the LoS or the NLoS model [25, 23] or a weighted average between the LoS model and the NLoS model [30, 31, 32] can be used. In this paper, we consider a general case and denote the path loss between a UAV and node located on the ground by . We assume that is constant in the range , and thus , where is a function. Further discussions on obtaining the in different environments can be found in [30, 31, 32]. Due to the geographical limitations, direct communication between the Tx and Rx is not considered, which is a valid assumption especially when the Tx and the Rx are far away or there are obstacles between them [25, 23].
Iii Position Planning for a Single UAV Considering an MSI
Let , denote the SIR at the UAV located at and the SIR at the Rx, respectively (see Fig. 1), which are given by:
Considering the conventional decode-and-forward relay mode, the SIR of the system is given by :
Assuming equal bandwidths for both links, maximizing the data rate between the Tx and the Rx is equivalent to maximizing the by tuning the location of the UAV described as:
The presence of an MSI renders our approach different from most of the works mentioned in Section I mainly due to its effect on the SIR expressions making them non-convex with respect to (w.r.t) the position of the UAV(s), which leads to the inapplicability of the conventional optimization techniques. In this work, we exploit geometry and functional analysis to obtain the subsequent derivations. In the following, we propose two lemmas, which are later used to derive the main results.
In geometry, a locus is the set of all points satisfying the same conditions or possessing the same properties.
The locus of the points satisfying is given by the following expression222In this work, and superscripts always denote the larger and the smaller solution, respectively.:
with , where , , and are given by (6).
The proof can be carried out using algebraic manipulations, which is omitted due to the limited space. ∎
For , the stationary points  with respect to , , is given by:
Also, has no stationary point with respect to when . With , we have
On the other hand, has no stationary points when and
The proof can be carried out by taking the following steps: (i) Analysis of to obtain the stationary points. (ii) Examining the signs of , , and at the stationary points. (iii) Inspecting the behavior of the SIR expressions at the boundary points. ∎
In practice, one of the following scenarios may occur: (i) The UAV position is vertically fixed and horizontally adjustable [34, 35]. This may arise in urban applications, in which there is a desired altitude for the UAVs to avoid collision with other flying objects. (ii) The UAV position is horizontally fixed and vertically adjustable. This happens specially in the surveillance and information gathering applications, in which the position of the UAV is fixed in the desired horizontal position and only the altitude can be tuned . (iii) The UAV position is neither vertically nor horizontally fixed, which is practical in non-urban areas with a few flying objects. In the following, we tackle these scenarios in order. Henceforth, whenever we refer to the roots of an equation or the points in the locus, the feasible space is confined to and .
Iii-1 Finding the optimal horizontal position of the UAV for a given altitude
Given a fixed altitude , the horizontal positions satisfying (5) can be obtained by solving the quartic equation given in (12), where the characteristic of this equation considering is described as follows:
Case 1) : In this case, the quartic equation has no solution. With some algebraic manipulations, this case can be represented as the following constraint:
Therefore, the necessary condition to have at least a feasible solution for (12) is , which can be represented as .
Case 2) and and : In this case, the quartic equation has one solution , which can be numerically obtained. This case can be represented by the following conditions:
Case 3) and and : In this case, the quartic equation has no solution. This case can be represented by the following conditions:
Case 4) and and : In this case, the quartic equation has at least a feasible solution. This condition can be represented as follows:
where is defined in (17).
Case 5) and and : In this case, the quartic equation may or may not have a feasible solution. This condition can be expressed as follows:
For a fixed altitude, according to Lemma 2: (i) is a monotone increasing function w.r.t , and (ii) depending on the value of the stationary points of , is a monotone decreasing function w.r.t in the interval and a non-decreasing function w.r.t in the interval . This corollary is a result of these two facts combined with the usage of functional analysis. ∎
In the following theorem, we use the results of Corollary 1 to determine the optimal position of the UAV. However, the above corollary also provides a practical guide to design the and w.r.t the position of the MSI, which can be obtained through calculation of through , and the conditions given on the ratio of these two variables in (13)-(20). Similarly, it discloses useful guides for the malicious user to effectively place the MSI. Nonetheless, we leave these interpretations as future work since they are not the focus of this paper.
Iii-2 Finding the optimal vertical position of the UAV for a given horizontal position
In this case, the vertical positions (altitudes) satisfying (5) can be easily derived since on the right hand side of the equation is known. Using Lemma 2, we obtain the following theorem to identify the optimal position of the UAV.
Given a fixed horizontal position , the optimal altitude of the UAV is given by:
Case 1) : .
Case 3) and (5) has no feasible solution: can be derived by solely inspecting the boundary positions:
The proof is an immediate result of studying the behaviors of the SIR expressions given in Lemma 2. ∎
Iii-3 Finding the optimal position when both and of the UAV are adjustable
In the previous scenarios, the locus defined in (5) reduces to an equation since one variable (either or ) is given, which is not the case here. In this case, the optimal position of the UAV is identified in the following theorem.
Let denote the set of all the feasible solutions of the locus described in (5). The optimal position of the UAV is given by:
Case 1) If the Locus has no solution, the optimal position can be derived by solely examining the boundary positions:
The proof is an immediate result of studying the behaviors of the SIR expressions given in Lemma 2. ∎
Iii-a Special Case
Our derived expressions can be simplified to provide insights for various special situations. For example, suppose that the MSI is located on the segment between the Tx and the Rx (), , and . Considering (5), we get:
Normalizing the to , the defined in (5) is given by:
The existence of a solution for (5) requires or , which is equivalent to:
It can be seen that the position of the MSI has a significant impact on these intervals and subsequently the placement of the UAV, especially if , it imposes , and subsequently , which implies no feasible/practical solution for (5).333The notations and are used to denote approaching the limiting value from the left and the right, respectively. Consequently, the optimal position is identified based on Case 1 of Theorem 3.
Also, assuming that the source of interference is placed far away or it has a negligible transmission power, the interference will not play a key role in the design anymore. In this case, the SIR expressions in (2) will be replaced with signal-to-noise-ratio (SNR) expressions, which are much easier to handle compared to SIR expressions since they are monotone functions w.r.t both and . In this case, a similar approach can be followed to obtain the optimal position of the UAV, which will result in simplified versions of Theorem 1, 2, 3. The same philosophy holds for the following discussion on position planning for multiple UAVs.
Iv Position Planning for Multiple UAVs Considering an MSI
We investigate the placement planning upon utilizing multiple UAVs from two different points of view. First, we consider a cost effective design, in which the network designer aims to identify the minimum required number of utilized UAVs and determine their positions so as to satisfy a predetermined SIR of the system. Second, we assume that the network designer is provided with a set of UAVs, and endeavors to configure their positions so as to maximize the SIR of the system.
Iv-a Network Design to Achieve a Desired SIR
Let denote the desired SIR of the system and assume that is the minimum number of UAVs needed to satisfy the SIR constraint, which will be derived later. We index the Tx node by , the UAVs between the Tx and the Rx from to , and the Rx node by . We denote the horizontal distance between two consecutive nodes and by , , and consider . To have tractable derivations, we assume that all the UAVs have the same altitude . It can be verified that this assumption maximizes the SIR between two adjacent UAVs for a given horizontal distance. The model is depicted in Fig. 2. Let denote the SIR at the node, which can be obtained as:
Similar to the single UAV scenario, is given by:
Iv-A1 The SIR expressions and the feasibility constraints
From (26), it can be observed that achieving any desired () may not be feasible. To derive the feasibility conditions for the , we need to analyze the links between the Tx and , among the adjacent UAVs, and from to the Rx.
Analysis of the links between the Tx and () and between and the Rx () is similar to the discussion provided in Section III (see Lemma 2). Hence, we skip them and consider the SIR at , . For this UAV, the stationary point of the SIR expression is given by:
using which it can be validated that:
where is the minimum feasible distance between two UAVs considering the mechanical constrains. Combining these derivations with those in Section III, we obtain the feasibility condition declared in (31).
Iv-A2 Design Methodology
To derive the minimum number of needed UAVs and their optimal positions so as to satisfy a desired , we pursue the following three main steps: (i) Considering , for , we obtain the maximum distance from the Tx (toward the Rx) that satisfies the SIR constraint. (ii) Considering , for , we obtain the maximum distance from the Rx (toward the Tx) that satisfies the desired SIR. (iii) Consider the segment between and with length , we use the SIR expressions of the remaining UAVs to minimize the number of UAVs required to cover the distance while satisfying the desired . In the following, we explain these steps in more detail.
Considering , we solve , the answer of which is given by (32).
Then, using Lemma 2, is given by:
where are given in (36).
Finally, the minimum number of required UAVs is given by:
Note that according to (35) and (36), calculation of each only requires the knowledge of , . Hence, the solution of (38) can be easily obtained by initially assuming and increasing the value of by until the constraint in the right hand side of the equation is met.
Iv-B Position Planning for a Given Number of UAVs
In this case, there exist multiple UAVs dedicated as relays to the network, which are expected to be positioned to maximize the SIR of the system. To this end, an algorithm can be immediately proposed based on our results in the previous subsection, which considers the number of UAVs as given and slowly increases the SIR () starting from to find the maximum value of for which in (38) becomes equal to the number of given UAVs. Afterward, the positions of the UAVs can be obtained as discussed before. Nevertheless, this is a centralized approach. In the following, we propose a distributed algorithm for the same purpose, where the UAVs locally compute their positions based on the knowledge of the positions of their adjacent neighbors, which can be obtained through simple message passing. Considering the SIR expressions in (26), for to , we express the SIRs w.r.t the positions of the UAVs located after them (closer to Rx) as:
The following facts are immediate consequences of examining (39) and (36): (i) With a known and a (hypothetically) given value for (), starting with the distance between the subsequent UAVs can be locally obtained up to using a backward propagation, by which each UAV transmits its position rearward to the adjacent UAV located toward the Tx (see (39)), where
(ii) With a known and a (hypothetically) given value for the (), starting with , the distance between the subsequent UAVs can be obtained up to in a forward propagation, by which each UAV transmits its position to the adjacent UAV located toward the Rx (see (35), (36)). Hence, to obtain the positions three parameters are needed: , and . Note that in the mentioned propagations, no message is exchanged between the two UAVs in the middle (), and thus the SIR at might be less than . Given these facts, we propose a distributed algorithm for position planning of multiple UAVs, the pseudo code of which is given in Algorithm 1. In this algorithm, we first locate the first UAV above the Tx and the last UAV above the Rx and derive the initial desired (); subsequently, we set the position of these UAVs to have as the SIR of the first link and the last link of the network (lines 1-1). Afterward, using forward and backward propagation, the UAVs locally obtain their positions w.r.t the position of their adjacent UAVs (lines 1-1) so as to satisfy the desired . Then, the SIR at is inspected (line 1). If this SIR satisfies the desired SIR of the system at the current iteration (i.e., at iteration : ), the algorithm stops; otherwise, it moves the first and the last UAVs and starts over with a new desired value for for the next iteration (lines 1-1). Note that simultaneous identification of the positions achieved through using forward and backward propagations leads to a faster convergence since at each time instant two distances are calculated in parallel.