# Intelligent Spectrum Management and Trajectory Design for UAV-Assisted Cognitive Ambient Backscatter Networks

Authors: Jiazhou Liu; Sa Xiao; Huayan Guo; Xiangwei Zhou; Shixin He
Journal: Wireless Communications and Mobile Computing
 (2021)

---

## Abstract

In this paper, we consider a novelInternet of Things (IoT) system in smart city called unmanned aerial vehicle- (UAV-) assisted cognitive backscatter network, where a UAV is employed as both a relay and a radio frequency source to help the data transmission between ground IoT backscatter devices (BDs) and a remote data center (DC). However, since the IoT applications are usually not assigned dedicated spectrum resource in smart cities, these data transmissions from BDs to the DC should share the licensed spectrum of cellular users (CUs). Therefore, we aim to maximize the minimum uplink throughput among all BDs while avoiding severe interference to CUs via joint spectrum management and UAV trajectory design. To solve the problem, we propose an iterative method utilizing block coordinated decent to partition the variables into two blocks. For the spectrum management problem, we first prove its convexity with the transmit power and time scheduling and then propose a two-step method to solve the two variables sequentially. For the UAV trajectory design problem, we resort to the fractional programming method to handle it. Simulation results demonstrate that the proposed algorithm can significantly increase the average max-min rate of the BDs while guaranteeing the acceptable interference to CUs with a fast convergence speed.


---

## Body

## 1. Introduction

TheInternet of Things (IoT) is one of the most important application scenarios for the fifth-generation (5G) wireless communications. It enables different smart devices to connect with each other via wireless networks, thus has a great potential to remould vertical industries such as manufacturing, agriculture, urban-construction, and transportation [1, 2]. Since the IoT allows massive devices to access wireless networks, however, it also faces various practical challenges in its deployment [3]. For one thing, the IoT devices are usually too small to equip with high-capacity batteries. For another, massive deployed IoT devices may also exacerbate spectrum scarcity problems [4]. As a result, future widespread IoT applications highly require energy- and spectrum-efficient wireless communications techniques [5, 6].In recent years,ambient backscatter communications (AmBC) has emerged as a cutting-edge technique in IoT networks [7–9]. With it, IoT devices are able to transmit by intentionally changing their own antenna impedances to reflect surrounding ambient radio frequency (RF) signals, e.g., TV or WiFi signals, instead of self-generated RF signals. Therefore, it neither requires any power-hungry circuit components such as the oscillators and analog-to-digital converters, nor additional spectrum resource for data transmission in AmBC [10]. As a result, AmBC has been considered as a promising candidate to support future IoT applications with stringent spectrum and energy constraints [11]. Due to the double fading effect with backscatter links and direct interference from ambient RF signals, however, AmBC is usually reliable at distances of tens of meters. Moreover, the transmission length becomes much shorter if the backscatter devices (BDs) are buried under other objects. Therefore, increasing the coverage range as well as improving the link quality is the critical issues for the application of AmBC in large-scale IoT scenarios, e.g., smart agriculture or smart city [12, 13].Mounted with miniaturized communication transceivers,unmanned aerial vehicles (UAVs) can work as mobile relays to provide reliable wireless connectivity between distant ground devices. Compared with traditional terrestrial relays, UAV relays are in general more flexible and cost-effective to deploy and likely to have better communication links due to the high chance of line-of-sight (LoS) links [14, 15]. Therefore, exploiting the agility of UAVs to build dual-hop LoS communication links is an efficient way to enhance the coverage of AmBC.There has been some preliminary work regarding UAV-assisted relaying communications. In [16], an energy-efficient cooperative relaying scheme has been proposed for a UAV-assisted relaying network, while in [17], joint power allocation and trajectory design have been investigated to maximize the end-to-end throughput of UAV-assisted relaying transmission. Note that both studies focus on UAV communications in traditional cellular systems. In backscatter-enabled systems, the UAV needs to work as a RF source for the BDs in addition to being a relay. Therefore, novel trajectory design and resource scheduling schemes are required to deal with the coexistence of cellular and backscatter communications. Although the hardware prototype of UAV-assisted backscatter communications has been realized in [18], it still lacks theoretical analysis and optimization for the whole system.In this paper, we consider a UAV-assisted cognitive backscatter network, where a UAV is not only employed as a relay to provide wireless connectivity between a remotedata center (DC) and BDs but also works as the RF source for these BDs. However, since the IoT applications are usually not assigned dedicated spectrum resource in smart cities, the data transmission from BDs to the DC should share the spectrum licensed to cellular users (CUs). Therefore, we aim to maximize the minimum uplink rate among all BDs while avoiding severe interference to adjacent CUs via joint trajectory design and spectrum management. To achieve this objective, we propose an iterative method utilizing block coordinated decent (BCD) [19] to partition the variables into the time scheduling and power control variable and the trajectory design variable. For the former one, a two-step method is proposed to solve the transmit power and time scheduling sequentially, while for the latter one, we resort to the fractional programming (FP) method [20]. Both the convergence and complexity of the proposed algorithm are demonstrated by simulation results.The rest of the paper is organized as follows. In Section 2, we describe the system model and formulate the optimization problem. To solve the problem, we develop a BCD and FP based iterative method in Section 3. Then, we present simulation results in Section 4. Finally, conclusions are drawn in Section 5.


## 2. System Model and Problem Formulation

### 2.1. System Model

We consider a UAV-assisted cognitive backscatter network as shown in Figure1, where M BDs are fixed within an area without terrestrial networks and a UAV is employed as a relay for the data transmission from the BDs to a remote DC. The BDs are able to access the UAV via reflecting the relaying signals transmitted from it in a cyclical time-division multiple access (TDMA). The UAV works in full-duplex (FD) mode and can adjust its trajectory to improve the connectivity quality. Since the IoT applications are usually not assigned dedicated spectrum resource, the data transmission from the BDs to the DC cannot severely interfere the data reception at the adjacent CU. For analysis simplicity, we assume that the whole period is equally divided into K time slots, where K is sufficiently large so that the UAV is approximately stationary within each time slot. In addition, the location information of the UAV, BDs, and the CU is assumed to be available at the DC. The frame structure is shown in Figure 2. Specially, βm,k is denoted as the portion of time assigned to BD m in time slot k.Figure 1
System model for a UAV-assisted backscatter network.Figure 2
Frame structure for a UAV-assisted backscatter network.In this paper, we adopt 3D Cartesian coordinate as in [21]. Specially, the DC is set as the origin. The horizontal coordinates of BD m and the CU are fixed as lm=xm,ymT and l0=x0,y0T, respectively, and the corresponding altitudes are denoted as Hm and H0, respectively. The UAV flies at a fixed altitude H, and its horizontal coordinate at time slot k is denoted as qk=xuk,yukT. The cellular links from the UAV to the DC, and the CU and the dyadic backscatter link between BD m and the UAV all follow the free-space path loss model. Then, the corresponding channel power gain is given by(1)gk=γ2H2+qk2,g0k=γ2H−H02+qk−l02,hmk=γ1H−Hm2+qk−lm22,respectively, where γ2 and γ1 are the corresponding channel powers at distance of 1 m. Additionally, the channel power gain of the self-interference (SI) channel is denoted as ζ, whose value is determined by the adopted SI cancellation scheme.Letsmk;n denote the signal transmitted from BD m during the n-th symbol period in time slot k. Assume the decoded-and-forward (DF) mode is adopted by the UAV. Then, the corresponding relay signal transmitted by the UAV is smk;n−τ, where τ is the processing delay. As mentioned before, since the BDs do not have any active circuit to generate RF signals, they need to reflect the received relay signals for data transmission (for the initial τ symbol reception, the UAV can transmit any RF waveforms to the BDs). As a result, the received signals at the UAV and DC can be expressed as(2)ym1k;n=hmkPkαsmk;n−τsmk;n+ζPksmk;n−τ⏟SI+z1k;n,ym2k;n=gkPksmk;n−τ+z2k;n,respectively, where Pk denotes the transmit power of the UAV at slot k, 0≤α≤1 is the reflection coefficient (RC) at the BDs, and z1k;n and z2k;n denote the additive Gaussian noise. It should be noted integrated circuits in BDs do not include any active radio frequency components such as oscillators, analog-to-digital converters, and power amplifiers. Therefore, the noise at the BDs is very low and usually assumed to be neglected as in [11].The desired signals at the UAV and DC aresmk;n and smk;n−τ, respectively. Meanwhile, τ is large enough so that smk;n and smk;n−τ are independent. Thus, the signal-to-interference-plus-noise ratio (SINR) of the first and second hop is given by(3)SINRm,k1=hmkαPkδ2+ζPk,(4)SINRm,k2=gkPkδ2,respectively. It should be noted that the interference from the BD to the DC is ignored in (4), due to the long distance in between. Then, the transmit rate from BD m to the DC in time slot k is(5)Rm,k=minRm,k1,Rm,k2,where Rm,k1=log1+SINRm,k1 and Rm,k2=log1+SINRm,k2.


### 2.2. Problem Formulation

We aim to maximize the minimum average rate among all BDs while avoiding severe interference to the CU via joint time scheduling, transmit power control, and UAV trajectory design over all time slots. DefineB=βm,k,∀m,k, Q=qk,∀k, and P=Pk,∀k. Then, the optimization problem can be mathematically formulated as(6)maxB,Q,PGB,QΔ=minm1K∑k=1Kβm,kRm,k,(7a)s.t.∑m=1Mβm,k≤1,∀k∈1,⋯,K,(7b)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(7c)q0=qK,(7d)g0kPk≤λ,∀k∈1,⋯,K,(7e)Pk≤Pmax,∀k∈1,⋯,K,where vmax is the maximum speed of the UAV, q0 is the initial location of the UAV, λ is the interference threshold at the CU, and Pmax is the maximum transmit power of the UAV. Constraint (7a) is originated from the fact that the overall length of assigned time should not exceed the total time slot length. Constraint (7b) is the flight speed constraint, and constraint (7c) implies that the UAV will return to the initial point at the end of the whole time period. Constraints (7d) and (7e) define the feasible transmit power of the UAV at any time slot. It should be noted that the interference from BDs to the CU is ignored in (7d), because the power of backscatter signals are much smaller than the original incident signal in general.The above problem is challenging to solve directly, because the objective function in (6) is nonconvex with respect to B, Q, and P. In addition, B, Q, and P are coupled in the constraints. Therefore, instead of solving it directly, we use the BCD method to deal with it in the following section.


## 2.1. System Model

We consider a UAV-assisted cognitive backscatter network as shown in Figure1, where M BDs are fixed within an area without terrestrial networks and a UAV is employed as a relay for the data transmission from the BDs to a remote DC. The BDs are able to access the UAV via reflecting the relaying signals transmitted from it in a cyclical time-division multiple access (TDMA). The UAV works in full-duplex (FD) mode and can adjust its trajectory to improve the connectivity quality. Since the IoT applications are usually not assigned dedicated spectrum resource, the data transmission from the BDs to the DC cannot severely interfere the data reception at the adjacent CU. For analysis simplicity, we assume that the whole period is equally divided into K time slots, where K is sufficiently large so that the UAV is approximately stationary within each time slot. In addition, the location information of the UAV, BDs, and the CU is assumed to be available at the DC. The frame structure is shown in Figure 2. Specially, βm,k is denoted as the portion of time assigned to BD m in time slot k.Figure 1
System model for a UAV-assisted backscatter network.Figure 2
Frame structure for a UAV-assisted backscatter network.In this paper, we adopt 3D Cartesian coordinate as in [21]. Specially, the DC is set as the origin. The horizontal coordinates of BD m and the CU are fixed as lm=xm,ymT and l0=x0,y0T, respectively, and the corresponding altitudes are denoted as Hm and H0, respectively. The UAV flies at a fixed altitude H, and its horizontal coordinate at time slot k is denoted as qk=xuk,yukT. The cellular links from the UAV to the DC, and the CU and the dyadic backscatter link between BD m and the UAV all follow the free-space path loss model. Then, the corresponding channel power gain is given by(1)gk=γ2H2+qk2,g0k=γ2H−H02+qk−l02,hmk=γ1H−Hm2+qk−lm22,respectively, where γ2 and γ1 are the corresponding channel powers at distance of 1 m. Additionally, the channel power gain of the self-interference (SI) channel is denoted as ζ, whose value is determined by the adopted SI cancellation scheme.Letsmk;n denote the signal transmitted from BD m during the n-th symbol period in time slot k. Assume the decoded-and-forward (DF) mode is adopted by the UAV. Then, the corresponding relay signal transmitted by the UAV is smk;n−τ, where τ is the processing delay. As mentioned before, since the BDs do not have any active circuit to generate RF signals, they need to reflect the received relay signals for data transmission (for the initial τ symbol reception, the UAV can transmit any RF waveforms to the BDs). As a result, the received signals at the UAV and DC can be expressed as(2)ym1k;n=hmkPkαsmk;n−τsmk;n+ζPksmk;n−τ⏟SI+z1k;n,ym2k;n=gkPksmk;n−τ+z2k;n,respectively, where Pk denotes the transmit power of the UAV at slot k, 0≤α≤1 is the reflection coefficient (RC) at the BDs, and z1k;n and z2k;n denote the additive Gaussian noise. It should be noted integrated circuits in BDs do not include any active radio frequency components such as oscillators, analog-to-digital converters, and power amplifiers. Therefore, the noise at the BDs is very low and usually assumed to be neglected as in [11].The desired signals at the UAV and DC aresmk;n and smk;n−τ, respectively. Meanwhile, τ is large enough so that smk;n and smk;n−τ are independent. Thus, the signal-to-interference-plus-noise ratio (SINR) of the first and second hop is given by(3)SINRm,k1=hmkαPkδ2+ζPk,(4)SINRm,k2=gkPkδ2,respectively. It should be noted that the interference from the BD to the DC is ignored in (4), due to the long distance in between. Then, the transmit rate from BD m to the DC in time slot k is(5)Rm,k=minRm,k1,Rm,k2,where Rm,k1=log1+SINRm,k1 and Rm,k2=log1+SINRm,k2.


## 2.2. Problem Formulation

We aim to maximize the minimum average rate among all BDs while avoiding severe interference to the CU via joint time scheduling, transmit power control, and UAV trajectory design over all time slots. DefineB=βm,k,∀m,k, Q=qk,∀k, and P=Pk,∀k. Then, the optimization problem can be mathematically formulated as(6)maxB,Q,PGB,QΔ=minm1K∑k=1Kβm,kRm,k,(7a)s.t.∑m=1Mβm,k≤1,∀k∈1,⋯,K,(7b)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(7c)q0=qK,(7d)g0kPk≤λ,∀k∈1,⋯,K,(7e)Pk≤Pmax,∀k∈1,⋯,K,where vmax is the maximum speed of the UAV, q0 is the initial location of the UAV, λ is the interference threshold at the CU, and Pmax is the maximum transmit power of the UAV. Constraint (7a) is originated from the fact that the overall length of assigned time should not exceed the total time slot length. Constraint (7b) is the flight speed constraint, and constraint (7c) implies that the UAV will return to the initial point at the end of the whole time period. Constraints (7d) and (7e) define the feasible transmit power of the UAV at any time slot. It should be noted that the interference from BDs to the CU is ignored in (7d), because the power of backscatter signals are much smaller than the original incident signal in general.The above problem is challenging to solve directly, because the objective function in (6) is nonconvex with respect to B, Q, and P. In addition, B, Q, and P are coupled in the constraints. Therefore, instead of solving it directly, we use the BCD method to deal with it in the following section.


## 3. Proposed Algorithm

To solve problem (4), we propose the following iterative algorithm, which applies the BCD to partitioning the variables into two blocks, i.e., the time scheduling and power control variable B,P and the UAV trajectory variable Q(8)Q0⏟Initialization→⋯→Bn−1,Pn−1→Qn−1⏟n−1−thiteration→Bn,Pn→Qn⏟n−thiteration→⋯→B∗,P∗→Q∗⏟optimalsolution.Specially, the algorithm starts by finding an initial UAV trajectoryQ0. At each iteration n, it first derives Bn,Pn for given Qn−1 from the last iteration and then finds Qn for the fixed Bn,Pn. The process is repeated until no further improvement is obtained. In the following, we first present the solution for B,P and Q, respectively. Then, we propose a novel method to find the initial trajectory of the UAV. Finally, we give the whole procedure of the proposed algorithm and prove its convergence.


### 3.1. Optimal Time Scheduling and Power Control

Given the UAV trajectoryQn−1, the optimal time scheduling and power control can be obtained by solving the following problem:(9)maxB,P,ηη,(10a)s.t.1K∑k=1Kβm,kRm,k≥η,∀m,(10b)∑m=1Mβm,k≤1,∀m,(10c)g0kPk≤λ,∀k∈1,⋯,K,(10d)Pk≤Pmax,∀k∈1,⋯,K,where η is the auxiliary variable. This problem is convex with B and P, and a two-step method is propose to deal with the transmit power control and time scheduling sequentially.Since the value ofg0k is known when Qn−1 is given, the optimal transmit power to the above problem is(11)Pk=maxPmax,λg0k.Then, withPk given in (11), Rm,k is known, and problem (9) becomes a standard linear programming problem in B. Therefore, we can adopt the interior-point method to deal the time scheduling in polynomial time [22].


### 3.2. UAV Trajectory Design

For the given transmit power and time schedulingBn,Pn, problem (4) is simplified as the UAV trajectory design problem as follows(12)maxQ,ηη,(13a)s.t.1K∑k=1Kβm,kR¯m,k≥η,∀m,(13b)R¯m,k≤Rm,k1,∀m,(13c)R¯m,k≤Rm,k2,∀m,(13d)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(13e)q0=qK,(13f)g0kPk≤λ,∀k∈1,⋯,K,where both η and R¯m,k are the auxiliary variables.It is obvious that bothRm,k1 and Rm,k2 are nonconcave with respect to Q. To solve the problem, we first need to reshape Rm,k1 and Rm,k2 into more trackable forms.SetCm,k1qk=τP+δ2H−Hm2+qk−lm22, Cm,k2qk=δ2H2+qk2, A1=γ1αP, and A2=γ2P. Then, we have(14)Rm,k1=log1+A1Cm,k1qk,(15)Rm,k2=log1+A2Cm,k2qk.Apparently, bothA1 and A2 are constant, while Cm,k1 and Cm,k2 are convex with respect to qk. Therefore, the above problem is a concave-convex FP problem, which can be efficiently solved by the FP method proposed in [20].The main idea of the proposed FP method is to transform problem (12) into a sequence of convex optimization problems by reshaping the fractional terms within Rm,k1 and Rm,k2 into quadratic structures. Denote Q¯l=q¯l,∀k as the fixed point at the l-th iteration in the FP method. Then, problem (12) can be solved by the following sequence programming(16)Q¯l+1,η∗,R¯m,k∗=argmaxQ,η,R¯m,kη,(17a)s.t.1K∑k=1Kβm,kR¯m,k≥η,∀m,(17b)R¯m,k≤R^m,k1,∀m,(17c)R¯m,k≤R^m,k2,∀m,(17d)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(17e)q0=qK,where R^m,k1 and R^m,k2 are given by(18)R^m,k1=log1+2ym,klA1−ym,kl2Cm,k1qk,(19)R^m,k2=log1+2zm,klA2−zm,kl2Cm,k2qk,respectively, ym,kl=A1/Cm,k1q¯kl, and zm,kl=A2/Cm,k2q¯kl.In general, the FP method can start with any feasibleQ. The BCD is adopted as an outer loop for the FP method in this paper; however, we set Q¯0=Qn−1 to guarantee the convergence of the proposed algorithm. It is apparent that problem (16) is a standard convex optimization problem at each iteration l. Thus, we adopt CVX to solve it.The whole procedure of the proposed FP method is summarized as Algorithm1, and its effectiveness is shown in the following theorem, which is proved in Appendix A.Theorem 1.
With sequential programming given in (16), Algorithm 1 generates a sequence Q¯l rendering a increasing value of η, which finally converges to a local optimum η∗.Algorithm 1: FP algorithm. FP method for UAV trajectory design.
1Initialize:2l=0, Q¯0=Qn−1, tolerance σ>0;3Repeat:4  Updateym,kl=A1/Cm,k1q¯kl and zm,kl=A2/Cm,k2q¯kl;5  ReshapeR^m,k1 and R^m,k2 by (18) and (19), respectively;6  Solve the reformulated convex problem in (16) by CVX;7l=l+1;8Until:Q¯l−Q¯l−1≤σ;9  SetQn=Q¯l.


### 3.3. Trajectory Initialization Method

In this subsection, we use the simple circular trajectory to initialize the UAV trajectoryQ0. That is, the initial trajectory of the UAV is set to be a circular with a constant speed of v, where 0≤v≤vmax.Definewm as the optimal relay location for BD m. Then, wm can be obtained from the following theorem, which is proved in Appendix B.Theorem 2.
DefineD=lm. Then, wm is given by(20)wm=lm,ifFD≥0,0,0T,ifFD≤0,μlm,otherwise,where(21)Fx=γ2τPmax+δ2H−Hm2+D−x22−γ1αδ2H2+x2,
andμ=F−10/D. F−1x is the inverse function of Fx.Intuitively, the UAV’s trajectory should cover as manywm as possible. Therefore, the center of the circular trajectory is the geometric center of all wm, i.e., c=∑m=1Mwm/M, and the radius of the circular trajectory should satisfy(22)r=minmaxmwm−c,vmaxT2π,where the first term is the minimum radius to cover all wm, and the second term is the maximum radius due to the UAV speed constraint. Based on c and r, the initial trajectory of the UAV is given by(23)qk0=c+rcosθk,rsinθkT,∀k,where θk=2πk−1/K−1.


### 3.4. Overall Algorithm and Convergence

We summarize the proposed iterative algorithm as Algorithm2, which handles joint time scheduling, transmit power control, and UAV trajectory design based on the BCD and FP methods. Specially, the algorithm starts by designing the initial UAV trajectory Q0 as in Section 3.3. Given a fixed Qn−1 in the n−1-th iteration, Pn is obtained from (11), and Bn is determined by using the interior-point method given in Section 3.1. Then, for a fixed Bn,Pn, Qn is obtained by applying the FP method in Algorithm 1. The process repeats until the convergence condition is satisfied. The convergence of the algorithm is analyzed in the following theorem, which is proved in Appendix C.Theorem 3.
For problem (6), the proposed iterative algorithm in Algorithm 2 generates a sequence of Bn,Pn,Qn rendering GBn,Pn,Qn≥GBn−1,Pn−1,Qn−1, which finally converges to a local optimal solution B∗,P∗,Q∗.Although the convergence speed of the FP and BCD methods cannot be analyzed theoretically [20, 23], both methods converge typically in a few iterations for a moderate number of BDs as will be numerically shown in Section 4. Meanwhile, the trajectory design problem in each iteration of the FP algorithm and the time scheduling and transmit power control problem in each iteration of the BCD method are all convex optimization problems, which can be solved in polynomial time. Therefore, the time complexity of the proposed iterative method is polynomial, and it can be practically implemented with fast convergence for a UAV-assisted cognitive AmBC system with a moderate number of BDs.Algorithm 2: The iterative algorithm to problem (6). The procedure of proposed iterative algorithm.
1Initialize:2  Setn=0, tolerance σ>0;3  SetQ0 according to Section III-C;4Repeat:5  Setn=n+1;6  For fixedQn−1, Pn is obtained from (11), and Bn is obtained by the interior-point method;7  For fixedBn,Pn, Qn is found by the FP method in Algorithm 1;8Until:GBn,Pn,Qn−GBn−1,Pn−1,Qn−1≤σ;9  SetB∗,P∗,Q∗=Bn,Pn,Qn;10Return:B∗,P∗,Q∗.


## 3.1. Optimal Time Scheduling and Power Control

Given the UAV trajectoryQn−1, the optimal time scheduling and power control can be obtained by solving the following problem:(9)maxB,P,ηη,(10a)s.t.1K∑k=1Kβm,kRm,k≥η,∀m,(10b)∑m=1Mβm,k≤1,∀m,(10c)g0kPk≤λ,∀k∈1,⋯,K,(10d)Pk≤Pmax,∀k∈1,⋯,K,where η is the auxiliary variable. This problem is convex with B and P, and a two-step method is propose to deal with the transmit power control and time scheduling sequentially.Since the value ofg0k is known when Qn−1 is given, the optimal transmit power to the above problem is(11)Pk=maxPmax,λg0k.Then, withPk given in (11), Rm,k is known, and problem (9) becomes a standard linear programming problem in B. Therefore, we can adopt the interior-point method to deal the time scheduling in polynomial time [22].


## 3.2. UAV Trajectory Design

For the given transmit power and time schedulingBn,Pn, problem (4) is simplified as the UAV trajectory design problem as follows(12)maxQ,ηη,(13a)s.t.1K∑k=1Kβm,kR¯m,k≥η,∀m,(13b)R¯m,k≤Rm,k1,∀m,(13c)R¯m,k≤Rm,k2,∀m,(13d)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(13e)q0=qK,(13f)g0kPk≤λ,∀k∈1,⋯,K,where both η and R¯m,k are the auxiliary variables.It is obvious that bothRm,k1 and Rm,k2 are nonconcave with respect to Q. To solve the problem, we first need to reshape Rm,k1 and Rm,k2 into more trackable forms.SetCm,k1qk=τP+δ2H−Hm2+qk−lm22, Cm,k2qk=δ2H2+qk2, A1=γ1αP, and A2=γ2P. Then, we have(14)Rm,k1=log1+A1Cm,k1qk,(15)Rm,k2=log1+A2Cm,k2qk.Apparently, bothA1 and A2 are constant, while Cm,k1 and Cm,k2 are convex with respect to qk. Therefore, the above problem is a concave-convex FP problem, which can be efficiently solved by the FP method proposed in [20].The main idea of the proposed FP method is to transform problem (12) into a sequence of convex optimization problems by reshaping the fractional terms within Rm,k1 and Rm,k2 into quadratic structures. Denote Q¯l=q¯l,∀k as the fixed point at the l-th iteration in the FP method. Then, problem (12) can be solved by the following sequence programming(16)Q¯l+1,η∗,R¯m,k∗=argmaxQ,η,R¯m,kη,(17a)s.t.1K∑k=1Kβm,kR¯m,k≥η,∀m,(17b)R¯m,k≤R^m,k1,∀m,(17c)R¯m,k≤R^m,k2,∀m,(17d)qk+1−qk≤vmaxTK,∀k∈0,⋯,K−1,(17e)q0=qK,where R^m,k1 and R^m,k2 are given by(18)R^m,k1=log1+2ym,klA1−ym,kl2Cm,k1qk,(19)R^m,k2=log1+2zm,klA2−zm,kl2Cm,k2qk,respectively, ym,kl=A1/Cm,k1q¯kl, and zm,kl=A2/Cm,k2q¯kl.In general, the FP method can start with any feasibleQ. The BCD is adopted as an outer loop for the FP method in this paper; however, we set Q¯0=Qn−1 to guarantee the convergence of the proposed algorithm. It is apparent that problem (16) is a standard convex optimization problem at each iteration l. Thus, we adopt CVX to solve it.The whole procedure of the proposed FP method is summarized as Algorithm1, and its effectiveness is shown in the following theorem, which is proved in Appendix A.Theorem 1.
With sequential programming given in (16), Algorithm 1 generates a sequence Q¯l rendering a increasing value of η, which finally converges to a local optimum η∗.Algorithm 1: FP algorithm. FP method for UAV trajectory design.
1Initialize:2l=0, Q¯0=Qn−1, tolerance σ>0;3Repeat:4  Updateym,kl=A1/Cm,k1q¯kl and zm,kl=A2/Cm,k2q¯kl;5  ReshapeR^m,k1 and R^m,k2 by (18) and (19), respectively;6  Solve the reformulated convex problem in (16) by CVX;7l=l+1;8Until:Q¯l−Q¯l−1≤σ;9  SetQn=Q¯l.


## 3.3. Trajectory Initialization Method

In this subsection, we use the simple circular trajectory to initialize the UAV trajectoryQ0. That is, the initial trajectory of the UAV is set to be a circular with a constant speed of v, where 0≤v≤vmax.Definewm as the optimal relay location for BD m. Then, wm can be obtained from the following theorem, which is proved in Appendix B.Theorem 2.
DefineD=lm. Then, wm is given by(20)wm=lm,ifFD≥0,0,0T,ifFD≤0,μlm,otherwise,where(21)Fx=γ2τPmax+δ2H−Hm2+D−x22−γ1αδ2H2+x2,
andμ=F−10/D. F−1x is the inverse function of Fx.Intuitively, the UAV’s trajectory should cover as manywm as possible. Therefore, the center of the circular trajectory is the geometric center of all wm, i.e., c=∑m=1Mwm/M, and the radius of the circular trajectory should satisfy(22)r=minmaxmwm−c,vmaxT2π,where the first term is the minimum radius to cover all wm, and the second term is the maximum radius due to the UAV speed constraint. Based on c and r, the initial trajectory of the UAV is given by(23)qk0=c+rcosθk,rsinθkT,∀k,where θk=2πk−1/K−1.


## 3.4. Overall Algorithm and Convergence

We summarize the proposed iterative algorithm as Algorithm2, which handles joint time scheduling, transmit power control, and UAV trajectory design based on the BCD and FP methods. Specially, the algorithm starts by designing the initial UAV trajectory Q0 as in Section 3.3. Given a fixed Qn−1 in the n−1-th iteration, Pn is obtained from (11), and Bn is determined by using the interior-point method given in Section 3.1. Then, for a fixed Bn,Pn, Qn is obtained by applying the FP method in Algorithm 1. The process repeats until the convergence condition is satisfied. The convergence of the algorithm is analyzed in the following theorem, which is proved in Appendix C.Theorem 3.
For problem (6), the proposed iterative algorithm in Algorithm 2 generates a sequence of Bn,Pn,Qn rendering GBn,Pn,Qn≥GBn−1,Pn−1,Qn−1, which finally converges to a local optimal solution B∗,P∗,Q∗.Although the convergence speed of the FP and BCD methods cannot be analyzed theoretically [20, 23], both methods converge typically in a few iterations for a moderate number of BDs as will be numerically shown in Section 4. Meanwhile, the trajectory design problem in each iteration of the FP algorithm and the time scheduling and transmit power control problem in each iteration of the BCD method are all convex optimization problems, which can be solved in polynomial time. Therefore, the time complexity of the proposed iterative method is polynomial, and it can be practically implemented with fast convergence for a UAV-assisted cognitive AmBC system with a moderate number of BDs.Algorithm 2: The iterative algorithm to problem (6). The procedure of proposed iterative algorithm.
1Initialize:2  Setn=0, tolerance σ>0;3  SetQ0 according to Section III-C;4Repeat:5  Setn=n+1;6  For fixedQn−1, Pn is obtained from (11), and Bn is obtained by the interior-point method;7  For fixedBn,Pn, Qn is found by the FP method in Algorithm 1;8Until:GBn,Pn,Qn−GBn−1,Pn−1,Qn−1≤σ;9  SetB∗,P∗,Q∗=Bn,Pn,Qn;10Return:B∗,P∗,Q∗.


## 4. Simulation Results

In this section, we provide simulation results to verify the effectiveness of the proposed algorithm in Algorithm2. We consider a UAV-assisted cognitive backscatter network as shown in Figure 3, where the DC is located in the cell center, and 5 BDs are randomly distributed within the area with the radius of 200 m. The UAV flies at a fixed height of 10 m, and its maximum flight speed is 20 m/s. In the simulation, we set the transmit power of the UAV and the power spectrum density of the noise to be 30 dBm and −139 dBm/Hz, respectively. In addition, other parameters are set as γ1=0.001, γ2=0.01, τ=−110 dB, and α=0.5.Figure 3
The UAV trajectory of the proposed algorithm for different time periods.Figure3 illustrates the UAV trajectory of the proposed algorithm for different time periods T. From the figure, we can observe that the UAV adaptably enlarge and adjust its trajectory to approach the optimal relay locations of BDs when T changes. Specially, when T is sufficiently large, i.e., T=400 s, the UAV is able to visit all these optimal locations one by one. In this case, the trajectory of the UAV becomes a close loop with segments connecting all these points, and the UAV tends to spend more time near these points (flies much slower) to obtain better communication links.Figure4 illustrates the performance of the proposed algorithm for different time periods T. For comparison, we also investigate the circular trajectory (CT) and static location (SL) algorithms [21] in the simulation. Specially, the CT algorithm utilizes the initial circular trajectory, while the SL one requires that the UAV hovers statically over the geometric center of the optimal relay locations. It should be noted that both algorithms adopt the linear programming algorithm proposed in Section 3.1 to solve the time scheduling problem. From the figure, the average max-min rate of the SL method is independent with T since the links of the first and second hops are all time invariant for a static UAV. In comparison, the average max-min rate of the proposed algorithm increases with T and becomes saturated when T is sufficient large. This is expected because the UAV is able to adjust its trajectory and flight speed to spend more time near the locations rendering better communication channels. Thus, the average max-min rate increases with T. For the CT method, however, the UAV cannot change its flight speed. Thus, when T exceeds a certain amount, i.e., T=150 s, increasing T may increase the time spent on those undesirable area. As a result, the average max-min rate of the CT algorithm slightly decreases when T is larger than 150 s.Figure 4
The average max-min rate of the proposed algorithm for different time periods.Figure5 illustrates the complexity of the proposed algorithm for different T. It can be easily observe from the figure that the proposed algorithm at most takes 25 iterations to converge, which demonstrates the effectiveness of the proposed algorithm. Meanwhile, it cannot find any law for the iteration times of the FP algorithm for different T. This also consistent with the fact that the complexity of the FP is difficult to derive theoretically [20].Figure 5
The complexity of the proposed algorithm for different time periods.


## 5. Conclusions

In this paper, we have studied joint trajectory design and spectrum management for a UAV-assisted cognitive backscatter network to optimize the average max-min uplink rate of BDs. To achieve this objective, we have proposed an iterative method utilizing BCD to partition the variables into two blocks. For the time scheduling and power control problem, the interior-point method has been proposed to solve it, while for the trajectory design one, the FP method has been adopted. Both the convergence and complexity of the proposed algorithm have been demonstrated by simulation results.



---

*Source: 5536726-2021-04-07.xml*