Energy Constrained Sum-Rate Maximization in IRS-Assisted UAV Networks With Imperfect Channel Information

The focus of this article is maximizing the sum-rate of wireless unmanned aerial vehicle (UAV) networks with intelligent reflecting surfaces (IRS) in the presence of system practical limitations. More specifically, we consider that the phase compensation at the IRS is imperfect due to various factors such as device imperfections and channel estimation errors. Moreover, we consider that the IRS elements have limited switching frequency, which limits the possibility of being allocated to different UAVs over consecutive time slots when time-division multiple access is considered. To this end, we formulate an optimization problem, where the objective is to maximize the network sum-rate subject to total energy and quality-of-service constraints by optimizing the number of IRS elements and power allocated to each UAV. To solve the optimization problem, a low-complexity heuristic algorithm is proposed based on the quality of the estimated phase for each IRS element. The proposed approach is compared to benchmark techniques such as the uniform allocation process and genetic algorithm. The obtained results show that a significant sum-rate improvement of up to 45% can be gained using the proposed algorithm.

Mean of the received signal envelope. ω The radian frequency. φ i The phase of data symbol s i (.) The digamma function. ψ 0, j Phase of CFR from BS to jth element. Phase of the CFR between E j and UAV i A i, j Cascaded CFR between E j and UAV i B i The received signal envelope.

E j
The jth reflecting element. h j,i Magnitude of CFR from IRS element j to UAV i I n (.) Modified Bessel function of the first kind and order n L The total number of reflectors in the IRS panel.

L i
The number of reflectors allocated to the ith UAV. m i, j Mean of phase error. N The number of served UAVs. n i (t ) The AWGN. N s Number of satisfied users. p i Power allocated to ith UAV. P max Maximum power budget. R t i Rate achieved by ith UAV in iteration t R i The achievable normalized rate of UAV i s i The data symbol. T TDMA frame duration. t Iteration index.

T F
Total transmission time duration.

T i
The time slot allocated to the ith UAV. y i, j (t ) The passband received signal.

I. INTRODUCTION
Unmanned aerial vehicles (UAVs) networks are becoming an integral part of several applications such as data collection from distributed Internet of thing (IoT) nodes in smart cities [1], [2], [3]. Therefore, optimizing UAVs' performance in various aspects has received increasing attention in the recent literature [2], [4], [5], [6]. For integrated IoT-UAV applications, one of the main challenges is that most IoT nodes have small transmission power, which limits their transmission range. In such scenarios, UAVs have to fly at low altitudes (LAs) to provide reliable data communications with the IoT nodes. However, the link quality between LA-UAVs and the base station (BS) may deteriorate significantly due to the shadowing caused by high-rise buildings. To mitigate such limitation, intelligent reflective surfaces (IRSs), which is an emerging sixth generation (6 G) technology, is widely proposed to establish a virtual lineof-sight (LoS) link between the LA-UAVs and BS [7], [8], [9], [10]. Although IRS deployment can play a pivotal role in wireless networks, the gain achieved can be hindered by various system limitations and imperfections, particularly the imperfect phase estimation and compensation at the IRS panel as reported in [9], [10], [11].

A. Related Work
The ultimate performance gain of IRS highly depends on the accuracy of the cascaded channel-phase estimation and compensation processes. Consequently, the IRS system performance with imperfect phase has attracted extensive attention [9], [10], [11], [12]. Interestingly, Al-Jarrah et al. [9] analyzed the bit error rate (BER) of IRS systems with imperfect phase and reported counterintuitive results, which shows that increasing the number of IRS elements might increase the BER in the presence of phase compensation errors. Consequently, Jangsher et al. [13] proposed an algorithm to find the optimum number of IRS elements required to minimize the BER in IRS-assisted UAV communications. The capacity analysis in [10] also shows similar trends in the sense that increasing the number of IRS reflectors does not necessarily improve the capacity. Therefore, optimizing the number of IRS reflectors is crucial to avoid performance loss or excess complexity.
Resource allocation for integrated unmanned aerial vehicle (UAV)-IRS networks is considered widely in the literature [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Nevertheless, the reported work considers that the cascaded-channel phase is estimated and compensated perfectly at the IRS. For example, Yang et al. [14] investigated the joint allocation of resource blocks (RBs), power and IRS reflection coefficients, where dynamic passive beamforming is applied to adjust the IRS reflection coefficients, create artificial time-varying channels, and select a subset of users to be served simultaneously. Game theory is adopted in [15] to study a price-based resource allocation for IRS systems, where BS and IRS belong to different network operators. The IRS elements are divided into multiple modules, which are specified based on the advantage they offer to the communication process. A Stackelberg game is developed to study the cooperation between the BS and IRS operators. In addition, reflecting elements' allocation is studied in [16], where the reflectors are divided into sets of modules with a single controller employed for each module. The authors studied the joint design of the reflection module, power allocation (PA), and passive beamforming with the objective of maximizing the minimum signal-to-interference-plusnoise ratio (SINR) for a certain set of source-destination pairs. In [17], joint PA and beamforming design for an aerial IRS is studied with the objective of maximizing energy efficiency. IRS aided multipair communications is investigated in [18], where multiple pairs of UAVs exchange data through a shared IRS panel, where a genetic algorithm (GA) is employed to maximize the sum-rate of users.
In [19], joint resource allocation with trajectory design is studied, where the flight speed and certain UAV constraints are considered with the objective of maximizing the sumrate. In [20], a UAV trajectory design for multiuser and multi-IRS scenario is studied. The objective of the work is to maximize the energy efficiency with practical IRS considerations such as phase-amplitude relation and discrete phase compensation. In [21], a UAV-assisted IRS system is considered with the aim of maximizing the sum-rate by jointly optimizing the IRS phase shift, transmit power, and time allocation for each ground user. Moreover, the communication process at the UAV is assumed to be powered via wireless energy harvesting. Alghamdi et al. [22] provided a detailed survey of IRS design and discuss the joint optimization of phase control for sum-rate maximization and power minimization problems. Similarly, Gong et al. [23] surveyed the optimization frameworks of IRS with different objectives such as energy efficiency, sum-rate, secrecy-rate, and coverage. Generally speaking, none of the surveyed work considers the partial IRS element selection to maximize the sum-rate in the presence of hardware and operations imperfections.
It is worth noting that each IRS element can serve multiple UAVs using orthogonal or nonorthogonal medium access control (MAC) protocols [24]. In orthogonal MACs, such as time division multiple access (TDMA), each UAV can be allocated a specific time slot. However, TDMA-based IRS system would require ultra-fast reconfiguration of the tunable and adaptive electronic components for the IRS control circuity. However, such a control circuitry has a limited switching frequency, which limits the number of UAVs that can be served in a given TDMA frame [25], [26].

B. Motivation and Contribution
As can be noted from the surveyed literature, the references listed therein, and to the best of the authors' knowledge, the resource allocation problem for IRS-assisted UAV networks has never been considered with imperfect channel state information (CSI) at IRS. Therefore, this work considers the problem of joint IRS elements and power allocation for UAV networks in the presence of imperfect phase compensation, energy and hardware constraints. The main contributions of this article can be summarized as follows: 1) The joint element and power allocation problem is studied in the presence of phase estimation and compensation errors. The allocation problem is formulated to maximize the network sum-rate with minimum individual rate and energy constraints. To satisfy the limited phase configuration frequency at the IRS panel, TDMA is considered, where each IRS element is used only once in each TDMA frame.
2) The accuracy of phase estimation is modeled using the von Mises distribution where the concentration parameter κ can have different values for different IRS elements. Consequently, a low-complexity suboptimal power and element allocation algorithm based on κ is developed and evaluated. The proposed algorithm, denoted as κ-element and power allocation (κ-EPA), is compared to the GA solution and uniform allocation approach.
The obtained results show a signal-to-noise ratio (SNR) improvement of about 6 dB as compared to the uniform allocation approach, which is equivalent to 45% of sum-rate improvement. Moreover, κ-EPA consistently satisfies the rate constraints for a larger number of UAVs. In certain scenarios, the κ-EPA managed to satisfy the rate for 50% of the UAVs, while none of them is satisfied using the uniform allocation.

C. Article Organization
The rest of this article is organized as follows. Section II presents the system model. Section III presents the problem formulation and the proposed allocation algorithm. Section IV presents the obtained results using the proposed algorithm for various operating scenarios. Section V concludes this article.

II. SYSTEM MODEL AND PROBLEM FORMULATION
This article considers the IRS-assisted UAV network shown in Fig. 1. BS transmits data to multiple LA-UAVs with the assistance of an IRS panel attached to a relatively high-altitude platform (HAP). The direct link between BS and UAVs is considered to be obstructed due to severe shadowing caused by the high-rise buildings. BS communicates with the IRS elements through a dedicated control channel to control the phase shift matrix of the IRS elements [14], [27]. The distance between HAP and UAVs is considered random, which implies that the link strength between each UAV and HAP is random as well. Therefore, the phase estimation accuracy may vary for different UAVs. The distance between BS and HAP is considered fixed since HAP position is typically fixed during the communication process. TDMA is adopted to serve the UAVs with a uniform time allocation, and the IRS elements assigned in a particular time slot cannot be assigned again in any other time slot until the next TDMA frame. The reuse constraint is used to avoid the need for ultra-fast reconfiguration of the IRS elements, which can be infeasible for practical systems [25], [26], [28], [29]. Moreover, as reported in [10], allocating all reflectors to a single user does not necessarily provide tangible rate improvement while it increases the feedback overhead significantly. It is worth noting that if multiple users transmit simultaneously using other multiple access protocols such as frequency division multiple access (FDMA), then the elements that are not assigned to a particular user may cause self-interference to that user. In such scenarios, the phases of all users should be jointly selected to maximize the sum-rate [30]. The set of all IRS elements is denoted as L with cardinality L, and the set of UAVs is denoted as N with cardinality N. Therefore, the set L can be partitioned into N subsets such that L = L 1 ∪ L 2 , . . . , ∪L N , L 1 ∩ L 2 , . . . , ∩L N = ∅, and the cardinality of L i is L i . In each time slot, only one subset can be activated to serve a certain UAV. Fig. 2 shows an example of an IRS panel with L = 16, in which the elements are allocated to three UAVs, i.e., N = 3. The subsets of elements allocated to the three UAVs are L 1 = {3, 5, 11, 15}, L 2 = {1, 4, 7, 8, 9, 16} and L 3 = {2, 6, 10, 12, 13, 14}, and the subsets are activated sequentially in time slots T 1 , T 2 and T 3 , respectively.
The channel estimation is performed using the leastsquares (LS) algorithm, where the overall process follows the protocol given in [31]. Therefore, the transmission frame with a total time T F is divided into two sub-frames τ and T . During τ , the phase estimation is performed and fed back to the IRS controller. The phase estimation process is carried out for all channels in the system in a time-division manner. Additionally, the resource allocation algorithm is executed during this sub-frame. Thereafter, during the sub-frame T , the data transmission takes place utilizing the resources allocated during τ . For TDMA, T is further divided into N uniform time slots, where each slot is allocated to a particular UAV. The subsets of elements allocated to the UAVs are nonoverlapping due to the phase switching frequency constraint.
BS transmits the data symbol s i = √ p i e jφ i to the ith UAV (UAV i ) with the assistance of IRS subset L i , where p i is the allocated power and φ i is the symbol phase. The passband BS signal intended for UAV i arriving at IRS element j (E j ) is expressed as where ω is the radian carrier frequency, 0, j and ψ 0, j are, respectively, the magnitude and phase of the channel frequency response (CFR) between BS and E j .
Given that UAV i is assigned IRS set L i , then the received signal at UAV i can be written as where γ i, j is the reflection coefficient of E j with respect to UAV i , n i (t ) ∼ CN (0, σ 2 n i ) is the additive white Gaussian noise (AWGN), h j,i is CFR magnitude between E j and UAV i . The phase error i, j θ i, j − θ i, j , where θ i, j = ψ 0, j + ϕ i, j , ϕ i, j is the phase of the CFR between E j and UAV i , and θ i, j is the estimated value of θ i, j . The cascaded CFR can be written as 0, j h j,i A i, j , which is modeled as described in [9] and [10]. The results reported in [9] show that this channel model can be used to accurately approximate UAV channels due to the direct LoS environment while making the mathematical analysis tractable.
As shown in [13], the phase estimation error using LS can be modeled as a von Mises distribution with a probability density function given by [10], [11], [13] where m i, j is the mean of the phase error, m i, j = 0 for unbiased estimators, I 0 (.) is the modified Bessel function of the first kind and zero order and κ i, j is the concentration parameter of the distribution, which captures the accuracy of estimated phase. The achievable normalized rate of UAV i with imperfect phase compensation can be closely approximated as [10] where is the gamma function, and (z) is the derivative of (z). Furthermore, μ i and σ 2 i are the mean and variance of the received signal envelope [10], respectively, where μ i = √ p iμi, j and where E[·] is the expectation operator, B i is the envelope of the received signal at UAV i [10, Eq. (3)], [32], and where I n (.) is the modified Bessel function of the first kind and order n. The energy allocated for the ith user is E i = p i × T i , and thus the total energy consumption of the network will be

III. IRS ELEMENT AND POWER ALLOCATION
This section presents the formulated problem and proposed solution for the system model discussed in Section II. The problem formulation is motivated by the fact that in the presence of phase errors, the sum-rate becomes highly dependent on the IRS elements and UAV association. More specifically, assigning an IRS element to a particular UAV might increase the sum-rate substantially, while it is not the case if the IRS element is assigned to any other UAV.

A. Optimization Problem Formulation
The problem is formulated for a time sub-frame of duration T , which is divided into N slots, T 1 , T 2 , . . . , T N . In each time slot, BS sends data to one UAV with the assistance of the allocated IRS elements. Without loss of generality, the time slot for each UAV is normalized to one second for notational simplicity, and thus, the power and energy can be used interchangeably. In the case that other slot durations is desired, the energy has to be scaled by the slot duration. The sum-rate optimization problem can be formulated as subject to: where E th in the energy budget of BS, R i is the minimum rate required for UAV i , γ = [γ i, j ] N×L is a matrix of all reflection coefficients, and p [p i ] 1×N is a vector for all power assignments. The factor γ i, j is defined as The objective function in (9) is used to maximize the achievable sum-rate of all UAVs in the network. Constraint (10a) is the quality of service (QoS) constraint, which ensures that each UAV should have at least a rate of R i bps/Hz using the allocated subset of reflectors and energy. Constraint (10b) ensures that no reflector is allocated to more than one UAV during the transmission sub-frame T . Constraint (10c) is used to ensure that the total allocated energy is less than a particular predefined threshold E th . For TDMA, only UAV i may receive during T i with p i ≤ P max . In the Algorithm 1: κ-EPA Algorithm. 1×N 3. Initialize: p i = P max ,N = N, N s = ∅, t = 1, 4. do % IRS element allocation for a certain PA: 5. for = 1 : L 6. for i ∈ N 7.
Compute R t i using (4) and verify R t end % PA for a given reflectors allocation: absence of an energy constraint, each UAV will be allocated the maximum power, and thus the total energy to be consumed is T × P max E max , which implies that no power optimization is needed. Nevertheless, if BS has a limited energy budget for transmission, then power optimization to satisfy the energy constraint is crucial. For example, in green communications, BS might use solar cells to charge the batteries during the daytime and rely on the charged batteries during nighttime [33]. Consequently, the values of p i ∀i should be optimized to maximize the sum-rate while E th < E max . Apparently, the formulated problem is a mixed-integer nonlinear programming problem (MINLP) which is NPhard problem and cannot be solved in polynomial time [34]. Therefore, a heuristic algorithm is proposed to provide a low complexity solution.

B. κ-Element and Power Allocation Algorithm
This section presents the proposed element and PA algorithm, where the phase estimation accuracy is represented by the variable κ. The problem is a mixedinteger problem as the power variable [p i ] 1×N is continuous while the reflectors allocation parameter [γ i, j ] N×L is binary. To make the problem tractable, it can be decomposed into two parts, PA for a given reflectors allocation, and IRS reflectors allocation for a known PA. The κ-EPA is a centralized algorithm that runs at BS and computes [κ i, j ] N×L at the start of each transmission time, after CSI estimation. After computing [κ i, j ] N×L , it is fed to the IRS panel and will be used for transmission during the time of next sub-frame.
Algorithm 1 summarizes the steps of the proposed κ-EPA algorithm. The inputs are the set of UAVs N, set of IRS reflectors L, the matrix κ [ 1×N . The initial rate of each UAV, R 0 1 , R 0 2 , . . . , R 0 N , is computed by randomly allocating four reflectors to each of UAV, and by initially assuming that κ i = ∞ and p i = P max ∀i. The reason for allocating four reflectors is that (4) is accurate for L ≥ 4. The outputs are the IRS reflector allocation matrix, which can be used to generate the subsets L 1 , L 2 , . . . , L N , γ [γ i, j ] N×L and the power allocation vector p [p i ] 1×N . The proposed κ-EPA is an iterative algorithm and it stops when the sum-rate for all users achieved at iteration t is less then the sum-rate achieved in iteration t − 1. Initially, the algorithm sets p i = P max ∀i, based on which, the IRS reflector allocation matrix (Line 6 − 19) and the power allocation vector in Line 20 are computed sequentially.
1) IRS Element Allocation for a Certain PA: For a certain PA, the formulated problem in (9) With Constraints (10a) and (10b), the elements' allocation problem is still NP-hard. The allocation of an IRS element for a certain UAV is performed based on κ i, j . Because high κ values correspond to better phase estimation accuracy, the algorithm tends to allocate E j to UAV i with the highest κ i, j ∀i, and thus the binary matrix γ can be updated. It is worth noting that the jth column of the phase compensation accuracy matrix κ contains the κ values of the links between E j and all UAVs. The element allocation algorithm assigns an element to the UAV whose channel has the highest κ value. The proposed algorithm works column-wise, i.e., for each reflector, it chooses a UAV in a round-robin manner. Once all UAVs are allocated at least one reflector in any round, the algorithm verifies Constraint (10a). If a reflector is allocated, it is removed from the reflectors set to satisfy Constraint (10b), i.e., a reflector cannot be allocated to more than one UAV. The rate of each UAV is computed using (4) and compared to the rate threshold to verify Constraint (10a).
2) PA for a Given Reflectors Allocation: Power is allocated for each UAV with a particular IRS elements allocation, which is the output of the algorithm described in Section III-B1. Thus, the formulated problem in (9)-(10c) is reduced to a continuous variable optimization problem, which is a function of [p i ] 1×N , and the objective function is subject to Constraints (10a) and (10c). The objective function and the constraints are concave with respect to [ The Lagrangian of the problem can be stated as where λ i ,λ ∈ R + are the dual variables. The dual problem is The optimal power allocation is computed using the partial derivative of L(p i , λ i ,λ), i.e., ∂L(p i ,λ i ,λ) ∂ p i = 0, and it can be found as The Lagrange variables can be computed using the subgradient algorithm. Therefore, the problem defined in (13) can be solved using standard algorithms such as the subgradient algorithm for convex/concave problems to find the optimal solution [35]. In this work, the disciplined convex optimization tool CVX is adopted to solve the problem, where CVX employs semidefinite programming (SDP) solver SDPT3 [35].

C. Complexity of the Proposed Allocation Algorithm
The computational complexity of the κ-EPA algorithm is mainly dominated by three operations, the computation of (4), finding the maximum element in κ j , and performing the subgradient algorithm. Moreover, the κ-EPA is an iterative algorithm, and hence, the number of iterations is another key factor that affects the overall complexity. In each iteration, the rate in (4) will be computed L × N times, finding the maximum κ j will be performed L times, and the subgradient algorithm will be applied one time. Therefore, for typical values of N and L, the computational complexity is generally low. The number of iterations depends on the rate threshold R and SNR, nevertheless, the number of iterations for all the considered scenarios in this work did not exceed 3. Moreover, the obtained results in Section IV show that the optimal power converges to the uniform allocation algorithm for a wide range of SNRs. Consequently, additional complexity reduction can be achieved by applying the subgradient algorithm only for the SNRs, where the allocation deviates significantly from the uniform allocation algorithm.
As can be noted from the problem formulation in Section III-A, the considered optimization problem is NP hard, and hence, the proposed algorithm is heuristic. Consequently, the optimal solution for such problems cannot be TABLE I Values of R i and L i Using κ-EPA, N = 10, E th = 10 J, L = 140, and κ ∼ H (6) guaranteed. As for the decomposed parts of the algorithm, the PA problem is concave and the optimal solution for a given element allocation can be computed. However, the elements' allocation algorithm is heuristic, and thus does not guarantee optimal allocation. Exhaustive search methods can be used to find the optimal elements' allocation, but with high complexity where the search space is equal to 2 N×L − 1 combination. Although some of these combinations may not require all operations because certain constraints cannot be satisfied, the feasibility is still a burden. The complexity of the GA and κ-EPA algorithms is evaluated in terms of simulation time, and the obtained results indicate that the GA algorithm requires significant additional time compared to the κ-EPA algorithm as discussed in Section IV.

IV. NUMERICAL RESULTS
This section presents the performance evaluation of the proposed κ-EPA algorithm in terms of the sum-rate, user satisfaction, and energy allocated per user. The proposed algorithm is compared to the GA and the uniform allocation. The GA computes the IRS element allocation based on the mutation process, and in each iteration, the allocation that provides the best fitness is selected. For the uniform allocation, the IRS elements and power are allocated to all UAVs uniformly. The values of A i, j = 1, ∀{i, j} and κ i, j ∀{i, j} are selected randomly from a half-normal distribution with parameter σ 2 κ , which represents the variance of the ordinary normal distribution, i.e., κ ∼ H(σ 2 κ ). The results are averaged over 2 × 10 3 simulation iterations, and the value of σ 2 κ is specified for each scenario. The value of P max is considered as 10 W. Table I shows the achieved individual rates for all UAVs, R i ∀i, and the assigned number of elements L i when N = 10, E th = 10 J, L = 140, and κ ∼ H (6). In this scenario, different UAVs may have different R i values. As can be noted from the results, more reflectors are allocated to UAVs with higher R i . However, if the number of elements is not sufficient, and the total energy is equal to E th , then the solution is infeasible. For example, when SNR ≤ −3 dB, the problem has no feasible solution because UAV 5 and UAV 9 rate requirements cannot be satisfied. On the other hand, if all users' rates are satisfied and some elements remain unallocated, then the remaining reflectors are allocated to the UAVs based on the κ value between the element and UAVs. If κ i, j > κ v, j ∀v = i, then element j is allocated to UAV i . The table also shows the impact of varying the SNR on the individual and sum rates. For the individual rates, it can be noted that increasing SNR does not necessarily increase R i , given that R i ≥ R i . On the contrary, increasing SNR consistently improves the sum-rate. Fig. 3 compares the sum-rate of the κ-EPA with the uniform allocation for L = 100 and 500 with N = 10, and with the GA for L = 30 and N = 3. The results are obtained for E th = 3 J, R i = 1 bps/Hz ∀i, and κ i, j ∼ H (1). It can be observed from the figure that an SNR improvement of about 6 and 3 dB can be obtained using the κ-EPA as compared to the uniform allocation in the cases of L = 500 and L = 100, respectively. For the {L = 30, N = 3} case, the achieved improvement using the κ-EPA is generally small as compared to the uniform distribution scenario because the freedom to allocate a small number of IRS elements to the UAVs is limited. It is worth noting that small values of L and N are considered in this case because the complexity of the GA is massive when such values are large. For the case of GA,  the population size is 2500 and the maximum number of generations is 180. The proposed κ-EPA outperforms GA by about 3 dB. Moreover, the measured simulation time of the κ-EPA algorithm is 7 × 10 −2 s. and of GA is 187 s. Fig. 4 shows the sum-rate offered by the κ-EPA and uniform allocation algorithms for two different κ distributions and two different energy thresholds E th , where R i = 1 bps/Hz ∀i, N = 10, and L = 100. The energy thresholds considered are E th = 3 J and E th = 1 J. The values of σ 2 κ are 1 and 20, where σ 2 κ = 1 indicates that the phase error of the IRS elements is high, whereas σ 2 κ = 20 indicates that the phase error is relatively low. It can be observed that for the case of κ i, j ∼ H (20), the improvement gained by the κ-EPA algorithm is marginal for both energy thresholds. Such performance is achieved because when the phase error is low, the elements' allocation will affect the sum-rate only marginally because the IRS elements and power will be allocated nearly uniformly for all UAVs. Therefore, the gain obtained by the optimization process is insignificant when compared to the uniform allocation. On the contrary, when certain links have high phase errors, the optimization process offers a significant gain. In the figure, R i = 1 bps/Hz ∀i is considered, and the results obtained show that R i ≥ R i ∀i. Fig. 5 shows the sum-rate for κ-EPA compared with the optimal solution that is obtained using exhaustive search, for a scenario with N = 2, L = 6 and κ ∼ H (1). The optimal solution, which is considered as an upper bound benchmark, is computed based on the exhaustive search of all possible feasible options and the best solution was chosen. A small-size network is considered to ensure that exhaustive search can be performed within a reasonable time. It can be observed that for this scenario the achieved sum-rate of κ-EPA is near optimal. Fig. 6 shows the allocated energy versus SNR for certain UAVs, where N = 10, L = 140, E th = 10 J and R = [1, 2, 3, 4, 10, 1, 2, 3, 10, 5] bps/Hz. In this figure, it is assumed that 40 reflectors have κ ∼ H (20) and the remaining 100 reflectors have κ ∼ H (6). The allocated energy is presented for UAVs 1,2,5,9, and 10. It can be noted that the power is allocated uniformly at low SNRs irrespective of R i . This is mainly because the performance at low SNRs is dominated by the additive noise, and thus, changing the allocated power will not significantly impact the sum-rate. On the other hand, for SNR in the range [−2, 2] dB, a higher power is allocated to the UAVs with a higher R i to enable satisfying their rate requirements. Consequently, lower power is allocated to UAVs with lower R i . At high SNRs, all UAVs have their rates satisfied, nevertheless, the UAVs with low R i will benefit more from increasing  their power. Interestingly, the powers allocated to all UAVs converge to unity. In Fig. 6(b), the rate difference per user R i = R i − R i is presented. A negative rate difference implies that the user did not achieve the required rate, and thus it is in an outage. It is noted that at SNR= 2 dB, a slight drop is observed in the rate difference of the UAVs with low rate requirements, and the UAVs that were previously in an outage are able to satisfy their rate ( R i ≈ 0). Fig. 7 compares the rates achieved per user with a rate threshold R i . It can be noted that UAV 5 and UAV 9 are in an outage at SNR = −4 and −2 dB. The other UAVs managed to achieve the required rate threshold. For the UAVs with the same rate threshold, it can be noted that they have approximately achieved the same rate at the considered SNRs. For example, UAV 1 and UAV 6 have equal achievable rates. Table II presents the number of UAVs for which R i is satisfied using κ-EPA and the uniform allocation algorithms for different energy allocation thresholds. It can be noted that more users are satisfied when E th = 20 J as compared to E th = 10 J. With a higher energy threshold, more energy can be allocated to UAVs which results in improving the average rate of the users. It can be concluded that more UAVs are satisfied with a significant improvement in sum-rate using κ-EPA algorithm. This can be attributed to the fact that κ-EPA algorithm considers the QoS constraint, whereas the uniform allocation distributes the reflectors among UAVs evenly.

V. CONCLUSION
In this article, the joint allocation of IRS elements and transmit power was investigated with imperfect phase compensation for an UAV-IRS network. The joint allocation problem was formulated with the objective of maximizing the sum-rate such that the QoS, transmission power, and element-switching frequency constraints are satisfied. A centralized algorithm called κ-EPA, which uses the phase error information to allocate the IRS elements and power in the network, was proposed. The proposed algorithm was evaluated for various operating scenarios in terms of phase errors, number of reflectors, and SNR. The obtained results showed that an SNR improvement of about 3 and 6 dB when the number of reflectors is 100 and 500, respectively. A significant improvement in terms of complexity is also achieved using κ-EPA as compared to the GA algorithm. It is worth noting that an additional degree of freedom can be gained by incorporating nonorthogonal multiple access (NOMA) to establish a hybrid NOMA-TDMA MAC. Therefore, the number of time slots in the TDMA frame can be reduced at the expense of some interuser interference. Consequently, this problem is worth considering in our future work. The adoption of other MAC protocols such as FDMA would be also interesting for the research community. Sobia Jangsher (Member, IEEE) received the