BER Reduction Using Partial-Elements Selection in IRS-UAV Communications With Imperfect Phase Compensation

This article considers minimizing the communications bit error rate (BER) of unmanned aerial vehicles when assisted by intelligent reflecting surfaces. By noting that increasing the number of IRS elements in the presence of phase errors does not necessarily improve the system’s BER, it is crucial to use only the elements that contribute to reducing such a parameter. To this end, we propose an efficient algorithm to select the elements that can improve BER. The proposed algorithm has lower complexity and comparable BER to the optimum selection process, which is an NP-hard problem. The accuracy of the estimated phase is evaluated by deriving the probability distribution function (PDF) of the least-square channel estimator, and showing that the PDF can be closely approximated by the von Mises distribution at high signal-to-noise ratios. The obtained analytical and simulation results show that using all the available reflectors can significantly deteriorate the BER, and thus, partial element selection is necessary. It is shown that, in some scenarios, using about 26% of the reflectors provides more than tenfold BER reduction. The number of selected reflectors may drop to only 10% of the total elements. As such, the unassigned 90% of the elements can be allocated to serve other users, and the overhead associated with phase information is significantly reduced.


I. INTRODUCTION
Unmanned aerial vehicles (UAVs)-based communications have been on a rise with the aim of providing higher transmission rates and better link reliability for wireless communications systems [1]- [5]. However, the promised reliability and transmission rates may degrade if a line-ofsight (LoS) link is not guaranteed. To ensure auspicious reliability, intelligent reflecting surface (IRS)-assisted UAVs can be a promising solution as IRS can be designed to provide high quality communications links [1], [6]- [10]. To achieve this goal, the reflected signals from each IRS element should be added coherently by adjusting the phase of each IRS element. Optimization of resources for UAVbased communication has been discussed in [4] and [5]. Most of the work related to IRS reported in the literature demonstrates that increasing the number of reflecting elements improves the signal quality with perfect phase estimation and compensation [7], or when all reflectors have equal phase errors statistics [8]- [10]. However, when the statistical properties of the phase error for reflectors are not identical, then increasing the number of reflecting elements does not necessarily improve the link quality [7].
Generally speaking, the phase estimation and compensation of IRS cascaded channels are challenging processes and cannot be performed perfectly. Therefore, the impact of phase alignment on IRS performance has been considered widely in the literature [8], [9], [11]- [15]. For example, bit error rate (BER) and outage probability expressions have been derived in the presence of phase errors for a singleuser IRS under different channel models including LoS [8] and Nakagami-m fading [15]. The achievable capacity for IRS-assisted UAV communications under LoS assumption is analyzed in [9]. The impact of the phase quantization errors on the achievable capacity is discussed in [11], where it is shown that phase quantization and phase noise can severely degrade the achievable rate.

A. Motivation and Contributions
As can be noted from the cited literature and references listed therein, the IRS system error performance is typically analyzed while considering that all reflecting elements have the same phase error statistics. For example, Al-Jarrah et al. [7]- [9] modeled the phase error of each reflecting element as a von Mises random variable with a particular concentration factor κ, but the performance is evaluated, while assuming that the κ value is equal for all reflecting elements. Therefore, increasing the number of elements always improves the BER. However, practically this may not be the case because phase estimation accuracy can vary for different reflecting elements based on the link quality between the base station (BS), reflecting element, and destination receiver [16]. Consequently, the contribution of each reflecting element to the overall signal quality depends on its phase estimation accuracy. Therefore, in this article we capitalize on our work [8] by considering a more general and realistic model where different reflectors may have different phase-error characteristics. We demonstrate that reflecting elements with low phase estimates accuracy can degrade the overall signal quality [9] and increase the BER. As such, only a subset of elements with a particular phase accuracy should be activated, which can be performed through an optimization process. To clarify the concept even further, consider a simple case where an IRS panel has only two elements. If the phase error between the two elements is uniformly distributed in [−π, π ), then the overall channel becomes hyper-Rayleigh [17], and the performance will be worse than the case if only one element is used. In other words, the link between the BS and receiver becomes equivalent to a multipath channel with two multipath components [17].
To the best of the authors' knowledge, there is no work in literature that considers the impact of the individual reflecting elements or applies elements' selection based on their contribution to the BER improvement. Therefore, this work considers optimizing the IRS elements' selection process to reduce the average BER. In addition to BER reduction, reducing the number of elements reduces the overhead associated with phase information feedback to the IRS panel, and allows serving other users in the network. Moreover, the phase error distribution is derived for the case of LoS communications in closed-form, and it is shown that it can be closely approximated to the von Mises distribution, which has been widely used to model the phase error, but without proper justification. The optimum solution of the considered problem is initially obtained using brute-force search, then a lower complexity optimum solution is developed by noting the relation between the BER and link quality for different IRS elements. Finally, an efficient selection rule called sort-and-drop selection (SDS) is proposed to provide near-optimum results at low complexity. The SDS results are compared to the optimum solution and to the results obtained and using genetic algorithm (GA) [18].

B. Article Organization
The rest of the article is organized as follows. The system and channel models are presented in Section II. The phase error distribution is derived in Section III. The problem formulation, proposed solution, and complexity evaluation are presented in Section IV. Section V presents the numerical results. Finally, Section VI concludes this article.

II. SYSTEM AND CHANNEL MODELS
This article considers IRS-enabled high-frequency UAV communications, as shown in Fig. 1. The BS is transmitting to a low altitude UAV (LAUAV) with the assistance of an IRS panel attached to a relatively higher altitude platform (HAP). The IRS panel consists of L reflecting elements R 1 , R 2 , . . . , R L , and their indices form the set L = {1, 2, . . . , L}. A microcontroller is employed to control the phase shifts of the IRS elements. The direct link is assumed to be blocked between the BS and LAUAV due to large obstacles, such as high-rise buildings [1]. Therefore, the signal is transmitted from the BS to the LAUAV with the assistance of an IRS panel attached to HAP. The reflecting elements of the IRS panel introduce phase shifts to the signals of the BS and reflect them to the LAUAV. The passband BS signal arriving at the ith IRS reflecting element can be expressed as where a is the amplitude and ϕ is the phase of the transmitted information symbol, f c is the carrier frequency, h i is the channel gain from the BS to the ith reflecting element, and ψ i is the phase shift introduced by the channel. The signal reflected by each IRS element is attenuated by a factor g i ∈ [0, 1] and its phase is shifted by θ i ∈ [−π, π ). The signal received at the destination LAUAV from the L reflecting elements of the HAP is given as (2) where n(t ) is CN ∼ (0, σ 2 n ), i and φ i are, respectively, the gain and phase of IRS to LAUAV channel. The values of θ i ∀i should be selected such that the phases of all reflected signals are aligned at the receiver, i.e., θ i = ψ i + φ i . However, the estimated cascaded channel phaseθ i cannot be estimated and compensated perfectly, which leads to phase alignment errors that can be defined as i θ i −θ i , i ∈ [−π, π ). The received signal with imperfect phase compensation can be expressed as where A i = i h i g i ∈ (−∞, ∞). For phase shift keying (PSK) modulation, the information symbol amplitude a = 1. After down conversion to baseband, and considering the sinusoidal addition theorem [19], [20], the received signal inphase and quadrature components can be written as and where B L is the signal envelope, n I and n Q are the inphase and quadrature components of the additive noise, respectively.
In UAV communications, ground-to-air (G2A) and airto-air (A2A) channels, i and h i , typically have a dominant LoS component, and thus, can be modeled using the Rician distribution with a large K factor [21]. It was experimentally found that the Rician factor for A2A and G2A can be about 12 dB for C-band and L-band signaling [22], [23]. Consequently, free space pathloss dominates the channel gain A i [8], [9]. Moreover, the transmitted signals may experience different atmospheric distortions caused by the spatial temperature variations given that the distance between the IRS elements is larger than the coherence distance, which is typically less than one centimeter for high frequencies [24]. Therefore, the channels' gains between the BS and each IRS element may experience different attenuation such that h i = h j ∀ i = j, {i, j} ∈ L. The same argument also applies to IRS-LAUAV channels. Although the system model considers LoS links for mathematical tractability, it can be used to closely approximate links with Rician and Nakagami-m fading for G2A and A2A channels [7].
Channel estimation in IRS can be performed by dividing the time interval T into two time slots T 1 and T 2 . The time interval T 1 T 2 is allocated for phase estimation, whereθ i ∀i is estimated and fed to the corresponding IRS element controller. After phase estimation and compensation, the data transmission is performed in T 2 . For channel estimation, we consider the protocol proposed in [25]- [27]. Therefore, T 1 is divided into L subslots each of which has a duration of T 1 /L, where only one reflector is switched on in each subslot, while all other IRS elements are switched OFF. Therefore, the received complex baseband signal during the channel estimation of the lth IRS element is y l = A l s l exp( jθ ) + n, where s l is pilot symbol. For least square estimation (LSE), the channel can be estimated aŝ H l = y l /s l , and thus,θ i = arctan(Ĥ Q,l H I,l ), whereĤ I andĤ Q are the in-phase and quadrature components ofĤ l .

III. PHASE ERROR DISTRIBUTION
As discussed in Section II, the channel gain is dominated by an LoS component. Since the derivation of the phase error distribution is similar for all reflectors, we omit the subscript i from the equations in this section to simplify the notation. For a deterministic complex channel, the channel can be represented as H = H I + jH Q βe jθ . To estimate the channel phase, a pilot symbol s is transmitted, and the received signal is given by where w ∼ CN (0, 2σ 2 w ) is the additive white Gaussian noise (AWGN). Given that s = 1, then y = (H I + w I ) + j(H Q + w Q ). Consequently, the phase estimate is obtained aŝ Due to the presence of the AWGN, there will be a phase error =θ − θ. By noting that y I and y Q are independent and identically distributed (i.i.d.) Gaussian random variables, y I ∼ N (H I , σ 2 w ) and y Q ∼ N (H Q , σ 2 w ), then their joint probability distribution function (PDF) is given by The PDF ofθ can be obtained by converting the PDF to polar coordinates y I = r cosθ and y Q = r sinθ to obtain f (r,θ ), and then averaging over r. The joint PDF in polar coordinates can be found as Then, f R,ˆ (r,φ) needs to be averaged over r to obtain the marginal PDF ofθ , fˆ (θ ) (10) where γ = 1/(2σ 2 w ). Using [28, (2.3.15.7), pp 344] and after some manipulations we obtain where z(θ ) = H I cosθ + H Q sinθ and erfc(x) is the complementary error function which is defined as erfc(x) 2 √ π x 0 e −t 2 dt. Finally, since =θ − θ, the phase error distribution f ( ) = fˆ ( + θ ) can be expressed as (11), except that z(θ ) is replaced byz(θ ), wherez(θ ) = H I cos( +θ ) + H Q sin( + θ ). Thus, (12) Interestingly, by using the trigonometric identities cos(a + b) = cos a cos b − sin a sin b and sin(a + b) = sin a cos b + cos a sin b, it can be proven thatz(θ ) = β cos( ). For ease of notation, we use f ( ) as f ( ). Consequently, f ( ) can be expressed as
The accuracy of the approximation can be also concluded from Fig. 2, where exact and approximated PDFs are presented for low and moderate SNRs.

B. Von Mises Distribution
To find the relationship between f ( ) as the standard form of the von Mises distribution, we equate f ( ) and the von Mises given in (23). Thereafter, the approximation used in (18) for I 0 (·) can be applied to obtain Applying the trigonometric identity cos(2 ) = 2 cos 2 ( ) − 1, and then the difference between two squares rule yields Since we aim at equating the two functions in (21), it can be observed that for any value of ∈ ( −π 2 , π 2 ), the equality is satisfied if and only if 4υ = κ. Finally, by setting κ = 4υ = 2γ , f ( ) can be written in the von Mises form as As can be noted from (13), the PDF of phase error has several nonlinear elements, and it does not have an analytical solution due to erfc(.), therefore using it for further analysis would mostly yield intractable solutions and can be closely approximated by the von Mises PDF [9], [15] for a wide range of SNRs. Therefore, we consider that where μ i is the mean of the phase estimation error associated with the ith reflector, μ i = 0 for unbiased estimators, and κ i is the concentration parameter of the von Mises PDF, which captures the phase estimation accuracy. Higher values of κ correspond to more accurate phase estimates. Ultimately, → 0 when κ →∞. Moreover, although the channel estimation and data transmission are typically performed within the channel coherence time, the channel estimation process time may exceed the coherence time. Consequently, the SNR can be different during time slots T 1 and T 2 . It is worthy to notice that κ i is a function of SNR, i.e., κ i = 2γ i , and thus, the value of κ i can be determined at the microcontroller of IRS based on previous knowledge of the value of SNR. Fig. 2 shows the derived distribution in (13), the von Mises approximation, and the simulated phase error at SNR = 15 dB. It can be observed that the von Mises distribution perfectly matches the exact PDF. For the case of SNR = 8 dB, the accuracy of the approximation is still high, but slightly worse than the case of 15 dB.

A. BER Analysis
Given that the microcontroller can switch OFF the lth IRS element by setting g l = 0, then the BER at the LAUAV can be approximated as [8] dy L (24) where M is the modulation order, ω L is the truncated PDF normalization factor, which can be approximated by 1 for large number of reflectors, α = [α 1 , α 2 , . . . , α L ], α l ∈ {0, 1} ∀l, Q(X ) = 1/2 erfc(X/ √ 2), X = C 2 y L /σ 2 n , C 1 and C 2 are constants that depend on the modulation scheme and order [30,  L , which can approximated as a Gaussian distribution using the central limit theorem (CLT) for large L values, μ Y L (α) and σ 2 Y L (α) are the mean and variance of Y L , respectively, which are functions of α. Henceforth, we will omit (α) from the notation and denote P E (α), μ Y L (α), and σ 2 Y L (α) as P E , μ Y L , and σ 2 Y L , respectively, to simplify the notation. The squared signal envelope is The mean μ Y L and variance σ 2 Y L can be represented as [8] and where E[·]is the expectation operation, and E[Y 2 L ] is given by  (27) can be found as whereθ q = I 1 (κ q ) I 0 (κ q ) . On the other hand, the term T 1 can be derived as [8] By observing that φ j = φ k and φ i = φ l , taking into account all possible scenarios for the equality between φ j and {φ i , φ l }, and between φ k and {φ i , φ l }, the value E k, j,i,l can be derived as Based on the BER in (24), it can be numerically demonstrated that P E is not strictly decreasing versus L when the values of κ i are not equal for all values of i. Consequently, only a subset of elements should be selected, while all others should be switched OFF.
It is worth noting that the BER in (24) can be evaluated in closed-form in terms of the parabolic cylindrical function (PCF), as described in [8]. However, evaluating the exact value of the PCF is infeasible because it is generally represented using an infinite series. To obtain an accurate evaluation, two different series representations are used based on the argument value. The closed-form expression of P E is given by [8] where and D (.) (.) is the PCF expressed in terms of Whittaker's function D −a− 1 2 (x). To numerically evaluate (31), we used n A = 30. For computing the PCF, the implementation given in [31] is used. More specifically, when {a, x} < 5 routine pu [31, Table I] is used, whereas for |x| a routine pulx is used.  (1) E (α) using (24) 6: for l = 2 : L 7: Set: α l = 1 8: Compute P (l ) E (α) using (24) 9: if P (l ) E > P (l−1) Because the objective function is nonconvex and the solution requires integer programming, then the optimization problem is NP-hard [32]. To solve the problem, the following approaches are considered.
1) Exhaustive Search: By noting that the optimization problem in (33) is an integer selection problem, then the optimum solution can be obtained using exhaustive search. Therefore, P E (α) is computed for all possible elementsselection combinations, and the set of elements that provide the minimum P E (α) is selected. The complexity of this approach is generally very high because there is a total of 2 L − 1 combinations that have to be evaluated. Consequently, this approach can be prohibitively complex, particularly for a large number of elements.
2) Low-Complexity Optimum Selection: To explain the low-complexity optimum selection, consider that the reflecting elements are sorted in a descending order based on their κ values. Therefore, reflecting element E i in the sorted set will have concentration coefficient κ i , where κ i > κ j ∀ i < j. Therefore, the sorted set of IRS elements can be written as E = [E 1 , E 2 , . . . , E L ]. In the brute-force approach, all possible combinations, without repetition, that be formed from E should be evaluated, and the one that minimizes (33) should be selected, which implies that 2 L − 1 operations are needed. However, it can be observed from (24) that P E (1, 1, . . . , 0) < P E (1, 0, 1, 0, . . . , 0) P E (1, 0, 1, 0, . . . , 0) < P E (1, 0, 0, 1, 0, . . . , 0) . . . Although in (34) we considered the case where only two elements are selected out of the available L elements, by induction, the same process can be generalized for more than two. Consequently, if we evaluate α = [1, 1, 0, . . . , 0], then there is no need to evaluate other cases of α which have only two activated elements. The same argument applies to any other number of activated elements. Consequently, we need only to test the cases sequentially starting from the minimum number of elements, and adding the element with the next lowest κ in the subsequent step. Therefore, the total number of trials required to find the optimum set of reflectors is L, which is much less than 2 L − 1 in the case of brute force search. Because κ-II is generally worse than κ-I, the BER does not decrease steeply, as in the case of Fig. 3. Moreover, the figure shows that the objective function might have multiple local minima at high SNRs. For example, for SNR of = 0 dB, there are  two local minima values when the number of reflectors is 4 and 29. Fig. 5 shows the BER using a squared quadrature amplitude modulation (QAM) with modulation order 16, and hence, C 1 = 3 and C 2 = 1/5. As can be noted from the figure, the BER with the 16-QAM generally follows those of the BPSK, but with some differences. For example, at low SNRs of −20 and −15, the AWGN becomes more impactful on the BER, as compared to the phase error. Consequently, the entire set of reflecting elements was used, and the minimum BER is obtained at L = 50. Moreover, for the case of SNR = 0 dB, there is only one minimum, which actually corresponds to the optimum value.
3) Suboptimum Sort-and-Drop Selection (SDS) Algorithm: The suboptimum selection algorithm can be directly derived from the low-complexity optimum algorithm, where the search process stops when the first minimum, local, or global, point is found. Therefore, the SDS algorithm initially sorts all IRS elements based on their phase error statistics, which is indicated by κ, and then evaluates the impact of including each IRS element on the BER. The algorithm stops when adding an element increases the BER, which indicates the first minimum point. Algorithm 1 summarizes the proposed SDS algorithm.
As described in Algorithm 1, all reflectors are sorted in a descending order based on their κ values. Then the first element is selected from the sorted set and P E ([1, 0, 0, . . . , 0]) is computed. In the second iteration, the second element in the sorted set is included and P E ([1, 1, 0, . . . , 0]) is computed. Then, P E in the second iteration P (2) E is compared with P E that was obtained in the first iteration P (1) E . If P (2) E < P (1) E the algorithm proceeds to the third iteration, where α = [1, 1, 1, 0, 0, . . . , 0]. If P (3) E < P (2) E then the algorithm starts another iteration, otherwise it stops, and so forth. Although the SDS algorithm may not give the optimum solution, it offers a reasonable BER performance with low complexity and less number of reflectors.
It should be noted that (24) is generally accurate for large values of L, roughly L > 3, because it is based on the CLT. Therefore, Algorithm 1 should start at l = 4. In the case that a small number of reflectors should be considered, then Algorithm 1 remains unchanged, except that the BER for l < 4 should be computed, as reported in [8].

C. Complexity
To find the set of reflecting elements that should be activated, the computation of (24) is required for each trial set of reflectors. Therefore, the exhaustive search would require a total of 2 L − 1 times computation of (24). For the SDS, the worst case scenario would require L times computation of (24), which occurs when all κ values are roughly equal for all reflectors. However, although performing the linear search is significantly less complex than the brute force search, significant additional complexity reduction can be achieved using the Bisection method, where the number of computations can be expressed as [33, eq. (35)] where δ is the line search accuracy. Because the number of reflectors is an integer, we set δ = 1. Therefore, I < L ∀L.
For the case of L = 50 the number of computations is 8.

V. NUMERICAL RESULTS
This section presents the performance of the proposed SDS algorithm and compares its performance to the optimum solution as well as to heuristic GA. The optimum solution uses exhaustive search over 2 L − 1 combinations.  For these numerical results, we considered A i = 1, C 1 = 1, and C 2 = 2 unless specified. In the heuristic GA, the BER is used as the fitness function, which is computed for each selected individual set of reflectors, and the next population is based on the current generation with the best fitness. For the GA implementation, a total of 20 generations and eight initial solutions are considered. Table I shows the SDS BER and selected number of reflectors, where all reflectors have the same value of κ. The table shows that the total number of selected elements L SDS is generally inversely proportional to SNR and directly proportional to κ. Such performance is obtained because the BER at low SNRs is dominated by the AWGN, and hence, increasing L SDS reduces the BER. When the SNR is increased to −6 dB, L SDS increases with the increase of κ. In this case, the BER depends on both the phase noise and SNR. Therefore, L SDS increases when the phase error decreases. Fig. 6 compares the BER using SDS, the optimum solution, and GA for L = 15. Moreover, the BER is presented when all reflectors are used, and when the best for reflectors are selected. The used sets of κ are, κ-III = [20, 20, 20, 20, U (0, 1), U (0, 1), . . .] and κ-IV = [20,20,20,20,20,9,8,7,6,5,4,3,2,1,0]. Both Fig. 6(a) and (b) show that the SDS algorithm perfectly matches the optimum. Fig. 6(a) shows that a noticeable improvement is obtained by the SDS algorithm when compared to the case of L = 15 for SNR > −10 dB. Fig. 6(b) shows that the BER using the SDS algorithm provides comparable BER to the all reflectors selection scenario over the considered SNR range. This behavior confirms the fact that the selection of a few reflectors with accurate phase compensation can provide equal or lower BER. It can be seen also that the performance of GA approaches the other algorithms at certain SNR values, while it deviates at other SNR values. Interestingly, the selection of the best four reflectors provides worse BER compared to SDS, GA, and the optimum solution.The performance of GA depends on the number of considered initial solutions, which is generally a small fraction of the complete solution set. Therefore, some of the initial solutions may be significantly far from the optimal solution, and the GA might stop before reaching the minimum point. The SDS is based on sorting, and in most scenarios, it converges to the minimum. Consequently, the SDS offers better performance than GA. Fig. 7 shows the distribution of the number of reflectors selected using the SDS algorithm for low, moderate, and high SNRs. The total number of reflecting elements L ∈ {15, 50}, and κ = [20, 20, 20, U (0, 3), U (0, 3), . . .]. The figure also compares the impact of varying g i on the number of selected reflectors, where g i ∼ U (0.6, 1) ∀i [34]. As can be noted from the figure, the number of selected reflectors is inversely proportional to SNR. Additionally, it can be noted that more reflectors are selected when g i < 1. Such behavior is obtained because using g i < 1 is equivalent to reducing the SNR, which is inversely proportional to the number of selected reflectors. For L = 50, the average number of selected elements at SNR=−10 dB is 27, whereas the average is 16.5 and 4 reflectors for SNR=0 and 10 dB, respectively. A similar trend can be noted for L = 15, where the average number of reflectors is 11, 7, and 4, respectively. It is also worth noting that the average number of selected elements for the two cases converges to the same value at high SNRs. The case of e max = 0 corresponds to the perfect estimation scenario. As can be noted from the figure, the SDS demonstrates high tolerance when e max = 1 because the κ is generally low for a large number of reflectors. Consequently, the selected number of reflectors marginally changes at certain SNRs. When e max = 10, certain reflectors will erroneously have a high κ, and thus, will be allocated reflection elements, which may cause severe BER degradation at high SNRs. At low SNRs, the performance is mostly dominated by the AWGN, which makes the impact of the estimation error less significant. Nevertheless, the SDS always provides performance improvement because selecting all elements actually corresponds to the worst-case scenario.

VI. CONCLUSION
This article presented a reflector-selection algorithm for UAV-IRS network with imperfect phase estimation and compensation, where the aim is to minimize the BER. Additionally, the distribution of the phase error is derived in a closed-form, and an accurate approximation was provided using the von Mises PDF with parameter κ. The proposed selection algorithm depends on the SNR and κ value of each reflector. The obtained BER results show that the proposed SDS algorithm may provide significant BER enhancement as compared to the all-reflectors case, and a comparable performance with the optimum algorithm. In addition to the BER reduction, the partial element selection allows for increasing the number of UAVs served by the IRS panel. In particular scenarios, the obtained results show that assigning only 10% of the total reflectors can minimize the BER and allow allocating the remaining 90% of the elements to other users.