Adaptive beam pattern selection and resource allocation for NOMA-based LEO satellite systems

Adaptive Beam Pattern Selection and Resource Allocation for NOMA-Based LEO Satellite Systems Anyue Wang1, Lei Lei2, Xin Hu3, Eva Lagunas1, Ana I. P´ erez-Neira4, and Symeon Chatzinotas1 1Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg 2School of Information and Communications Engineering, Xi’an Jiaotong University, China 3School of Electronic Engineering, Beijing University of Posts and Telecommunications, China 4Centre Tecnol` ogic de Telecomunicacions de Catalunya (CTTC/CERCA) and Universitat Polit` ecnica de Catalunya (UPC), Spain 1,2,3,4Emails: anyue.wang; eva.lagunas; symeon.chatzinotas@uni.lu, [email protected], [email protected], ana.per[email protected] Abstract—The low earth orbit (LEO) satellite system is one of the promising solutions to provide broadband services to a wide-coverage area for future integrated LEO-6G networks, where users’ demands vary with time and geographical locations. Conventional satellites with xed beam pattern and footprint planning may not be capable of meeting such dynamic requests and irregular trafc distributions. As the development of exible satellite payload with beamforming capabilities, spot beams with exible size and shape are considered potential solutions to this issue. As an early investigation, in this paper, we consider the scenarios where satellite payloads are equipped with multiple beam patterns and study the optimal beam pattern selection. We exploit the potential synergies of joint resource optimization between adaptive beam patterns and non-orthogonal multiple access (NOMA) in a LEO satellite system, where NOMA is employed to reduce intra-beam interference and exible beam pattern is adopted to mitigate inter-satellite interference. The formulated problem is to minimize the capacity-demand gap of terminals, which falls into mixed-integer nonconvex programming (MINCP). To tackle the discrete variables and nonconvexity, we design a joint approach to allocate power and select beam patterns. Numerical results show that the proposed scheme achieves capacity-demand gap reduction of 37.8% over conventional orthogonal multiple access (OMA) and 42.5% over the xed-beam-pattern scheme. Index Terms—low earth orbit (LEO) satellite systems, nonorthogonal multiple access (NOMA), adaptive beam patterns I. INTRODUCTION Low earth orbit (LEO) satellite systems are envisioned as one of the promising solutions for coverage extension in the upcoming beyond 5G (B5G) and 6G era [1]. To deliver exponentially growing trafc and support ubiquitous connectivity, the industry is in the process of deploying hundreds or thousands of LEO satellites in the space [2], e.g., SpaceX Starlink. However, with densely deployed LEO satellites, terminals may receive considerable interference from neighboring satellites [3]. Besides, due to the heterogeneity of trafc distribution, resource allocation in LEO satellite systems needs to be more adaptive. The mitigation of interference and provision of highquality services are challenging, calling for the introduction of more exibility in resource allocation [4]. In conventional satellite systems, beam patterns are xed and the projected beam shapes (or footprints) remain constant D E F G Fig. 1: Illustrative examples of adaptive beam patterns in LEO satellite systems. [5]. When LEO satellites are densely deployed and trafc demands are irregularly distributed, this xed beam pattern may limit the performance improvement. With the development of exible payloads, beam shapes can be altered by changing beam patterns via beamforming networks (BFN) [5]. By adjusting beam patterns on a frame basis, satellite transmission can be more adaptive to irregular trafc distribution [6]–[8], with mitigated co-channel interference [9], and with benets of facilitating integration with terrestrial systems [9], [10]. We provide illustrative instances of adaptive beam patterns in Fig. 1. When a terminal is located in the area covered by two adjacent satellite beams, e.g., in Fig. 1(a), the terminal may suffer from strong inter-satellite interference. To reduce the interference, both satellites could select patterns with smaller but more concentrated beams to cover the associated terminals, as depicted in Fig. 1(b). Another typical example is to change patterns with larger beams to cover terminals beyond the beam edge but with high demand, e.g., from Fig. 1(c) to Fig. 1(d). Considering the advantages in trafc adaptation and interference mitigation, adaptive beam pattern has received signicant attention. The authors in [6] jointly optimized beamwidth and transmit power to adapt to varying trafc. In [7], more factors affecting the beam shape, e.g., beam center and rotation angle, were taken into account and the footprint planning was designed to match irregular trafc distribution. In both works, the expressions of transmit antenna gain w.r.t. conguration parameters (e.g., beamwidth, beam center, rotation angle) were 978-1-6654-3540-6/22 c ©2022 IEEE © 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. DOI 10.1109/GLOBECOM48099.2022.10000986 derived. But the relationship between transmit antenna gain and conguration parameters is complex in general. Directly optimizing footprints is difcult in practical implementation [5]. Alternatively, multiple candidate beam patterns can be designed in advance for various possible scenarios and a suitable beam pattern can be selected depending on trafc distribution and interference. Beam pattern selection can be applied to both regular and irregular footprints and candidate patterns can be easily updated when necessary. In [8], the authors optimized beam pattern selection to meet irregular trafc distribution and discussed its practical implementation. The above works focused on optimizing rate matching to trafc distribution over an area instead of trafc demands of specic terminals. In this paper, we study beam pattern selection to alleviate capacaity-demand mismatch of terminals and investigate the implicit coupling between beam pattern selection and resource allocation for multiple terminals in each beam. To facilitate the accommodation of multiple terminals in each beam, non-orthogonal multiple access (NOMA) is applied. By superposition coding and successive interference cancellation (SIC), spectral efciency can be enhanced compared to orthogonal multiple access (OMA) [11]. NOMA has proven its advantages in LEO satellite systems, e.g., supporting massive connectivity [12] and assisting le delivery [13]. In this paper, we study resource allocation in LEO satellite systems where NOMA is applied in each satellite beam to reduce intra-beam interference whereas adaptive beam patterns are adopted to mitigate inter-satellite interference. As an early-stage study, it is necessary to outline the optimal strategies of beam pattern selection according to heterogeneous trafc requests and characterize the mutual impacts between NOMA and adaptive beam patterns. Motivated by this, in this paper, we investigate the potential synergy of NOMA and adaptive beam patterns in LEO satellite systems. We formulate a resource allocation problem to jointly optimize power allocation and beam pattern selection, which is mixedinteger nonconvex programming (MINCP). Considering the difculty of solving the MINCP problem, we convert the nonconvexity and binary variables into a solvable way and design a scheme to jointly optimize power and beam patterns. Numerical results verify the advantages of the joint optimization over conventional schemes, e.g., OMA and xed beam pattern. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model We consider downlink transmission in a LEO satellite system, as depicted in Fig. 2. Ssatellites y over the area of interest, where the coverages may overlap partially with neighboring satellite beams. Denote Sas the set of the satellites and Ksas the set of the terminals associated to the s-th satellite. Each satellite generates one beam and all the satellites occupy the same frequency band. The gateway delivers data from the core networks to satellites and collects information from terminals, e.g., channel status, trafc &RUHQHWZRUNV 5HVRXUFHPDQDJHU 'DWD *DWHZD\ )HHGEDFNV 2SWLPL]HG GHFLVLRQ 'HVLUHGVLJQDO ,QWHUVDWHOOLWHLQWHUIHUHQFH ,QWUDEHDPLQWHUIHUHQFH ,QWHUVDWHOOLWHOLQN 2SWLPL]HGGHFLVLRQ 'DWD 'DWD )HHGEDFNV Fig. 2: An illustrative scenario of the considered LEO satellite system. Adaptive beam patterns are adopted to adjust the beam shapes and NOMA is applied in each beam to serve multiple terminals. demand, terminals’ positions, etc. The resource manager, colocated with the gateway, is in charge of executing resource allocation algorithms based on the feedbacks. Denote Nsas the set of the candidate beam patterns of satellite s. Each pattern refers to one specic beam shape. With optimized decisions informed by the gateway, one beam pattern is selected out of Ns. Through altering phases and amplitudes by the BFN on the satellite payload, each feed element generates an elementary beam and the shaped beam is constructed by these elementary beams [5]. The resulted beam shape is dened as the coverage within the ϕ-dB contour, which can be regular (e.g., circular or elliptical beams) or irregular [5]. The relationship between the transmit antenna gain of a beam pattern and the corresponding conguration parameters is sophisticated and may not be captured by explicit expressions, especially for patterns with irregular beam shapes [5], [7], [8]. For better illustration, we use circular beam as an example to abstract the beam pattern model. For the n-th beam pattern of the s-th satellite, the peak gain (regarding the beam center) is expressed as [5], [6], Gtx sn,max =ξϕ 370π θsn,ϕdB 2 ,(1) where ξis the antenna efciency and θsn,ϕdB is the ϕdB angular beamwidth. From (1), we observe that, as the beamwidth increases, the peak gain decreases to disperse outwards to enlarge the coverage. On the other hand, as the beam shrinks, the peak gain increases and the directivity is more concentrated on the beam center. The transmit antenna gain of the s-th satellite to terminal kunder the n-th pattern is [5], [Gtx skn]=[Gtx sn,max]−3Gtx sn,max ξθsk 70π2 ,(2) where θsk is the off-axis angle between the s-th satellite’s beam center and the k-th terminal. The operator [·]converts the value into dB. For other types of beam shapes, the expression would be more complicated, e.g., the transmit antenna gain for elliptical beams depends on beam center position, beamwidth (w.r.t. the major and minor axis), and tilt angle [5], [7]. The channel loss from satellite sto terminal kis, [Ltotal sk ] = [Lba sk]+[Lga sk]+[Lsc sk]+[Lra sk],(3) where Lga sk,Lsc sk, and Lra sk are gaseous, scintillation, and rain attenuation exceeded for ρ%of an average year, respectively. The corresponding models follow 3GPP TR 38.811 [14]. Lba sk is the basic path loss, which is derived by, [Lba sk]=[Lsk]+[Lsf sk],(4) where Lsf sk denotes shadow fading following log-normal distribution. Lsk denotes the free-space path loss, expressed by, Lsk = 3245 + 20 log10(ffreq) + 20 log10(dsk),(5) where dsk is the distance from satellite sto terminal k, and ffreq is the frequency. The channel gain from satellite sto terminal kunder beam pattern nis expressed as, hskn2=Gtx sknGrx skLtotal sk (6) The channel gain from satellite sto terminal kis derived as, hsk2=∑ n∈Ns ysnhskn2,(7) where ysn ∈ 0,1indicates beam pattern selection, where satellite sselects the n-th pattern if ysn = 1 and not otherwise. By employing NOMA, each satellite can multiplex at most ¯ Kterminals with different power levels at the same frequency band. We use φslk to indicate the decoding order between terminal kand lserved by satellite s. If φslk = 0, terminal k performs SIC to decode l’s signal and remove it. If φslk = 1, terminal kviews l’s signal as noise. Denote psk as the transmit power of terminal kassigned by satellite s. We apply one of the most widely-adopted rules to decide decoding orders, where the decoding order is identical to the ascending order of ωsk =∑s′∈S\{s}hs′k2∑k′∈Ks′ps′k′+σ2 hsk 2[15], i.e., φslk ={0,if ωsk ≤ωsl; 1,otherwise(8) The signal-to-interference-plus-noise ratio (SINR) of terminal kassociated to satellite sis derived as, γsk =hsk2psk Iintra sk +Iinter sk +σ2,(9) where Iintra sk =l∈Ks\khsk2φslkpsl and Iinter sk = s′∈S\shs′k2k′∈Ks′ps′k′are intra-beam and intersatellite interference, respectively. σ2is the noise power. The offered capacity of terminal kserved by satellite sis, Rsk =Blog(1 + γsk),(10) where Bis bandwidth. B. Problem Formulation We formulate a resource allocation problem to jointly optimize power allocation and beam pattern selection to minimize the sum of capacity-demand gap1: P0: min p,y,φ∑ s∈S∑ k∈K (Rsk −Dsk)2(11a) s.t. ∑ k∈Ks psk ≤¯ Ps,∀s∈S,(11b) ∑ n∈Ns ysn = 1,∀s∈S,(11c) Rsk ≥Rmin sk ,∀k∈Ks∀s∈S,(11d) ωsk −ωsl ≤Cφslk,∀s∈S,∀k, l ∈Ks, k 6=l, (11e) φskl +φslk = 1,∀s∈S,∀k, l ∈Ks, k 6=l, (11f) where p0,y∈ 0,1,φ∈ 0,1collect all psk,ysn, and φslk, respectively. In (11b), the total tranmit power of the s-th satellite should be no larger than the power budget ¯ Ps. Constraints (11c) restrict that each satellite can only select one beam pattern. In (11d), terminals’ minimum-rate requirements is expressed. Constraints (11e) and (11f) convey the relationship between ωsk and φslk as in (8). Here Cis a large number meeting C≥maxs∈S,k∈Kωsk.Ifφslk = 0, terminal kdecodes l’s signal and thus ωsk ≤ωsl.Ifφslk = 1, terminal ldecodes k’s signal and thus ωsk −ωsl ≤C. Together with constraints (11f) which conne only one decoding order for each satellite beam, we can derive that ωsk >ωsl. Due to the non-convexity of the R-function in p,y, and φand the presence of binary variables yand φ,P0is identied as mixed-integer nonconvex programming (MINCP) [16]. Thus solving P0is a non-trivial task. III. A JOINT APPROACH FOR POWER ALLOCATION AND BEAM-PATTERN SELECTION We provide a joint approach to solve P0.Werst equivalently convert the binary variables into continuous conned by nonlinear equations and then reformulate P0into augmented Lagrangian formulation. Next, we design a penalty dual decomposition (PDD) [17] based approach to jointly allocate power and select beam patterns. We use the following equations to equivalently express y: ysn(1 −¯ysn)=0,∀s∈S,∀n∈Ns,(12) ysn −¯ysn = 0,∀s∈S,∀n∈Ns,(13) where y∈ 0,1are relaxed into 0y1.¯ysn are auxiliary variables. Equations (12) and (13) are established if and only if ysn = ¯ysn =0,1. We use ¯ yto collect all ¯ysn. Since φdepends on ωsk, we do not directly optimize φbut decide decoding orders according to (8) when ωsk is updated. We then rewrite P0as the following formulation, P′ 0: min p,y,¯ y∑ s∈S∑ k∈Ks (Rsk −Dsk)2(14a) 1Remark that terminal association is assumed given. The optimization problem involving terminal association will be studied in future works. s.t. (11b),(11c),(11d),(12),(13),(14b) where (12) and (13) are nonlinear equations, leading to the nonconvexity of the problem. Since PDD is designed to tackle the problem with nonlinear equation constraints [17], we propose a PDD-based joint approach. The corresponding augmented Lagrangian function is dened as, L(p,y,¯ y;λ,µ) = ∑ s∈S∑ k∈Ks (Rsk −Dsk)2 +1 2ρ(∑ s∈S∑ n∈Ns (ysn(1 −¯ysn)+ρλsn)2 +∑ s∈S∑ n∈Ns (ysn −¯ysn +ρµsn)2),(15) where λsn and µsn are Lagrangian multipliers for (12) and (13), respectively. λand µcollect all λsn and µsn, respectively. We apply ρ>0to penalize the objective when the solution does not establish the equations. The augmented Lagrangian problem of P′ 0is expressed as, P1: min p,y,¯ y L(p,y,¯ y;λ,µ)s.t. (11b),(11c),(11d)(16) The joint approach is executed in a double-loop way. In the inner loop, we decompose P1into several solvable subproblems, and then sequentially (or parallely) solve them. In the outer loop, the penalty and Lagrangian multipliers are updated. A. Inner Loop At the begining of each iteration, we calculate and sort ωsk to decide decoding orders φslk according to (8). Then we adopt the idea of block coordinate descent (BCD) to decompose P1 into several subproblems with different blocks of variables. 1) Optimize p:We rst optimize pby solving the following subproblem, min p F1(p) = ∑ s∈S∑ k∈Ks (Rsk −Dsk)2s.t. (11b),(11d)(17) We can observe that constraints (11d) are linear as, hsk2psk ≥(2Rmin sk −1) Iintra sk +Iinter sk +σ2(18) But F1(p)is nonconvex. At the i-th iteration, we approximate F1(p)at around p=p(i−1) to a surrogate function ˜ F1(p), which should satisfy the following two conditions: •˜ F1(p)is strictly convex in p; •O˜ F1(p(i−1)) = OF1(p(i−1)), where OF1(·)is the gradient. Based on the conditions, we apply the following surrogate function, ˜ F1(p)=(p−p(i−1))OF1(p(i−1))+ 1 2α‖p−p(i−1)‖2,(19) where α>0. The gradient is computed by deriving the partial derivatives as in (20). Then we solve the following problem, min p˜ F1(p)s.t. (11b),(11d),(21) which is quadratic convex programming. The optimal solution p∗can be solved by interior-point method [16]. Then we update pby the following rule, p(i)=p(i−1) +δ(i) 1(p∗−p(i−1)),(22) where δ(i) 1is the stepsize for updating pat the i-th iteration. 2) Optimize y:Then, we solve the following subproblem to optimize y, min y F2(y) = ∑ s∈S∑ k∈Ks (Rsk −Dsk)2 +1 2ρ(∑ s∈S∑ n∈Ns (ysn(1 −¯ysn)+ρλsn)2 +∑ s∈S∑ n∈N (ysn −¯ysn +ρµsn)2)s.t. (11c),(11d),(23) which is nonconvex due to ysn appearing at both numerator and denominator of the SINR funtion in (9). We adopt the following surrogate function to approximate F2(y)at y= y(i−1), ˜ F2(y) = (y−y(i−1))OF2(y(i−1))+ 1 2βy−y(i−1)2,(24) where OF2collects the partial derivatives, which is derived in (25). Then the problem is converted into, min y˜ F2(y)s.t. (11c),(11d)(26) By obtaining the optimal y∗via interior-point method, we can update yby, y(i)=y(i−1) +δ(i) 2(y∗−y(i−1)),(27) where δ(i) 2is the stepsize for updating yat the i-th iteration. 3) Optimize ¯ y:We optimize ¯ yby solving the following subproblem, min ¯ y∑ s∈S∑ n∈N (ysn(1−¯ysn)+ρλsn)2+(ysn −¯ysn+ρµsn)2 ,(28) which can be decomposed into SN convex subproblems: min ¯ysn (ysn(1 −¯ysn)+ρλsn)2+ (ysn −¯ysn +ρµsn)2(29) By letting the rst-order partial derivative equal to zero, the optimal ¯ysn can be obtained by, ¯ysn =ysn +y2 sn +ρµsn +ρλsnysn 1 + y2 sn (30) B. Outer Loop In the outer loop, we update the Lagrangian multipliers as, λsn =λsn +1 ρysn(1 −¯ysn), µsn =µsn +1 ρ(ysn −¯ysn)(31) The the penalty is updated by, ρ=ρη,(32) where 0<η≤1. ∂F1 ∂psk =2B(Rsk −Dsk )hsk 2 hsk 2psk +Iintra sk +Iinter sk +σ2 +∑ l∈Ks\{k} 2B(Rsl −Dsl)hsl2φslk hsl2psl +Iintra sl +Iinter sl +σ2−hsl2φslk Iintra sl +Iinter sl +σ2 +∑ s′∈S\{s}∑ k′∈Ks′ 2B(Rs′k′−Ds′k′)hsk′2 hs′k′2ps′k′+Iintra s′k′+Iinter s′k′+σ2−hsk′2 Iintra s′k′+Iinter s′k′+σ2(20) ∂F2 ∂ysn =∑ k∈Ks 2B(Rsk −Dsk )       hskn 2(psk + l∈Ks\{k} φslk psl) hsk 2psk +Iintra sk +Iinter sk +σ2− hskn 2 l∈Ks\{k} φslk psl Iintra sk +Iinter sk +σ2       +∑ s′∈S\{s}∑ k′∈Ks′ 2B(Rs′k′−Ds′k′)   hsk′n2 k∈Ks psk hs′k′2ps′k′+Iintra s′k′+Iinter s′k′+σ2− hsk′n2 k∈Ks psk Iintra s′k′+Iinter s′k′+σ2  (25) Algorithm 1 Joint Power Allocation and Pattern Selection Input: Initialized p,y,¯ y. 1: repeat 2: repeat 3: Calculate ωsk and update φby (8). 4: Solve problem (21) and update pby (22). 5: Solve problem (26) and update yby (27). 6: Update ¯ yby (30). 7: until Convergence or reaching Iiterations 8: Update λand µby (31) and ρby (32). 9: until Convergence or reaching Jiterations Output: Optimized p,y,φ. The procedure of the joint scheme is summarized in Alg. 1. The inner loop (line 2 to line 8) iterates until it converges or reaches the maximum number of iteration I. The outer loop (line 1 to line 10) terminates when convergence or reaching Jiterations. The complexity of Alg. 1 mainly falls into solving convex quadratic programming problems in (21) and (26). Assume K= maxs∈SKsand N= maxs∈SNs. The complexity of solving problem (21), which contains SK variables and S+SK linear constraints, is O(S3K3(S+SK)) [18]. Similarly, problem (26) is with SN variables and S+SK linear constraints, thus the corresponding complexity is O(S3N3(S+SK)). The overall complexity of Alg. 1 is O(JIS3(S+SK)(K3+N3)). IV. NUMERICAL RESULTS In simulation, we consider a square area with 500 ×500 km2. Denote ¯ Kas the number of multiplexed terminals. Each satellite chooses ¯ Kterminals with the best channel gains. The optimization involving terminal association will be discussed in future works. The parameters are summarized in Table I, unless otherwise stated. We consider beam patterns which generate circular beams with 4.3-dB beamwidths 12◦,18◦,24◦,48◦,96◦. Beams with 12◦ TABLE I: Simulation parameters Parameter Value Frequency, ffreq 20 GHz (Ka band) Bandwidth, W400 MHz Satellite height 600 km Number of satellites, S5 Power budget, ¯ Ps43 dBm Receive antenna gain 42, 36, 32 dBi Noise power, σ2-126.47 dBW Minimum association rate, Rmin k500 kbps Number of iteration, I,J200, 20 12345678910 (a) Iteration index 8 10 12 14 16 18 20 22 24 26 Capacity-demand gap ( 104) 1 2 3 4 5 6 7 8 9 10 (b) Iteration index 10-6 10-5 10-4 10-3 10-2 10-1 100 101 Fig. 3: Convergence performance of Alg. 1: (a) Evolution of capacity-demand gap; (b) Constraint violation. beamwidth have the largest directivity to the beam center and smallest inter-satellite interference whereas those with 96◦beamwidth generate the widest service range but large interference to terminals associated to adjacent satellites. For pracitcal consideration, we assume the decoding error ratio due to SIC imperfection is 10−3. The results are averaged over 500 instances. In Fig. 3, we present the convergence performance of Alg. 1. The capacity-demand gap increases as the algorithm evolves, and eventually converges at the 7-th iteration. At the end of the iterations, the constraint violation maintains at below 10−5, which guarantees the establishment of the equality constraints (12) and (13). We evaluate the gap performance of the proposed scheme w.r.t. the number of multiplexed terminals ¯ Kin Fig. 4(a). Note that the case of ¯ K= 1 refers to the conventional 200 400 600 800 1000 1200 (b) Average traffic demand (Mbps) 0 20 40 60 80 100 120 140 Capacity-demand gap ( 104) 1 2 3 4 5 40 60 80 100 120 Capacity-demand gap ( 104) Fig. 4: Gap performance w.r.t. (a) different ¯ K; (b) average trafc demand (¯ K= 3). 0 1 2 3 4 5 Number of satellites with multi p le beam p atterns 0 5 10 15 20 25 30 Capacity-demand gap ( 104) flexible (5 patterns) + conventional flexible (3 patterns) + conventional Fig. 5: Performance evaluation in the scenarios with the coexistence of satellites with single pattern (beamwidth 2.4◦) and multiple patterns. OMA scheme. Compared to the OMA scheme, NOMA has performance gain from 37.8% ( ¯ K= 2) to 72.2% ( ¯ K= 5) in minimizing the capacity-demand gap. In Fig. 4(b), we compare the performance of the proposed scheme between multiple beam patterns and xed single beam pattern. Among the three xed-pattern schemes, the one with 24◦beamwidth outperforms the other two, demonstrating the necessity of appropriate selection of beam patterns. The performance is largely improved by introducing spatial-domain exibility compared to conventional single beam pattern schemes. Compared to the scheme with xed 24◦beamwidth, the schemes with 3 patterns and 5 patterns have reduction of 42.5% and 60.9% in capacity-demand gap. In practice, LEO satellites functioned with adaptive beam patterns would coexist with those with conventional payloads in the system. We evaluate the scenarios with the coexistence of different types of LEO satellites in Fig. 5. By introducing satellites with adaptive beam patterns, the capacity-demand gap can be largely reduced, from 26.5×104to 15.1×104and 18.5×104when introducing 5 and 3 beam patterns, respectively. The more satellites with spatial-domain exibility, the more performance gain is obtained. V. CONCLUSION In this paper, we have investigated potential synergies of adaptive beam patterns and NOMA in a LEO satellite system. We have formulated an resource problem of jointly optimizing power allocation and beam pattern selection. 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