Method for maximizing minimum beamforming gain of SU based on UAV-MA

Abstract

The invention provides a method for maximizing the minimum beamforming gain of SU based on UAV-MA, and the method comprises the steps: 1, building a UMA system which comprises a UAV, K primary users PU and L secondary users SU; an antenna array is integrated on the UAV, and the antenna array comprises N MAs. Step 2, based on a UMA system, obtaining guide angles theta k and phi l, APV and AWV, and finally obtaining beamforming gains of the guide angles theta k and phi l; and step 3, considering the spacing between adjacent antennas, interference to PUs, total available power and constraint of the minimum hovering height of the UAV, and jointly optimizing the height h of the UAV, the APV and the AWV to maximize the minimum beamforming gain of the secondary user. According to the invention, the minimum beamforming gain of the SU can be maximized while the interference to the PU is ensured to be lower than the preset threshold value.

Description

Background Art

[0001] In recent years, moveable antennas (MA), also known as fluid antennas, have attracted much attention due to their ability to create favorable channel conditions through the flexible movement of antennas. This flexibility gives MAs a variety of significant advantages, including improved sensing accuracy, enhanced system capacity, and reduced signal interference. Therefore, research on MAs can fully reveal the full potential of future communication systems, especially in dynamic and unpredictable environments.

[0002] Preliminary studies have demonstrated the superiority of MA from multiple perspectives. 1. Prior Art 1 studies MA-enhanced multi-user communication, aiming to minimize the total transmit power of users by jointly optimizing the location of MA, the transmit power of users, and the receive combination matrix of the base station while meeting the minimum achievable rate requirements of each user. 2. Prior Art 2 uses the amplitude, phase, and angle information of the arrival / departure angle to develop a field response model for MA-based multipath channels. Compared with the traditional fixed-position antenna (FPA), this model can significantly improve the maximum achievable channel gain. 3. Prior Art 3 studies MA-assisted spectrum sharing, which maximizes the received signal power of secondary users (SU) under the co-channel interference power constraints of multiple primary users (PU) by jointly optimizing the beamforming design and MA location. 4. Prior Art 4 MA-based multi-beamforming focuses on maximizing the minimum beamforming gain in multiple desired directions by collaboratively optimizing the antenna position vector (APV) and the antenna weight vector (AWV), while considering the maximum interference power limit in the undesired direction. The above studies mainly utilize the local mobility of MA to construct more favorable channel conditions. Summary of the invention

[0003] The purpose of the present invention is to provide a method based on UAV-MA to maximize the minimum beamforming gain of SU, and the linear MA array based on UAV is deeply studied, so as to introduce a new degree of freedom in the vertical dimension for MA in addition to horizontal movement. The goal of the present invention is to maximize the minimum beamforming gain of SU while ensuring that the interference to PU is lower than a predetermined threshold, which requires the joint optimization of the height h of UAV and the AWV and APV of MA.

[0004] However, the optimization problem with respect to these variables constitutes a challenging non-convex optimization problem, making it very difficult to solve directly.

[0005] To solve this problem, this paper proposes an alternating optimization algorithm to iteratively optimize the height h of the UAV and the APV and AWV of the MA to obtain a near-optimal solution. The relevant numerical results demonstrate the superiority of the proposed algorithm over the benchmark method, indicating that it can provide full beamforming gain for the SU with lower computational complexity.

[0006] The present invention integrates the MA array into an unmanned aerial vehicle (UAV). The maneuverability of the UAV can provide additional degrees of freedom for dynamically adjusting the relative position between the MA and the user.

[0007] The technical solutions adopted are:

[0008] A method for maximizing the minimum beamforming gain of a SU based on a UAV-MA comprises the following steps:

[0009] Step 1: Establish a UMA system, which includes a UAV, K primary users PU and L secondary users SU; an antenna array including N MAs is integrated on the UAV;

[0010] Step 2: Based on the UMA system, obtain the guidance angle θ k and φ l , APV and AWV, and finally obtain the guidance angle θ k and φ l The beamforming gain of

[0011] Step 3: Considering the spacing between adjacent antennas, interference to PUs, total available power, and the constraints of the UAV’s minimum hovering height, the UAV’s height h, APV, and AWV are jointly optimized to maximize the minimum beamforming gain of the secondary user.

[0012] Preferably, the beamforming gain in step 2 is:

[0013]

[0014] APV is expressed as

[0015] AWV is defined as

[0016] Preferably, the optimization problem in step 3 can be modeled as:

[0017]

[0018] ‖w‖ 2 ≤1,(4d)

[0019]

[0020] h≥H 0 (4f)

[0021] in, is the position of the nth antenna;

[0022] D 0 is the minimum distance between adjacent antennas;

[0023] η is a predetermined threshold of the beamforming gain of the SU;

[0024] H 0 is the minimum hovering altitude of the UAV.

[0025] Preferably, when the APV and AWV are given and the height h is optimized, the model in step 3 is:

[0026]

[0027] Preferably, when APV and h are given and AWV is optimized, the model in step 3 is:

[0028]

[0029] Preferably, when AWV and h are given and APV is optimized, the model in step 3 is:

[0030]

[0031] st(4a),(4f),(17),(19),(20),

[0032]

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044] Compared with the prior art, the advantages of the present invention are:

[0045] 1. Joint optimization of integrated UAV and MA: It is proposed for the first time to integrate the MA array into the UAV, and jointly optimize the communication performance through the vertical movement of the UAV and the local movement of the MA.

[0046] 2. Alternating optimization algorithm for non-convex problems: For complex non-convex optimization problems, a low-complexity alternating optimization algorithm is designed, which effectively solves the joint optimization problem of UAV hovering height, APV and AWV.

[0047] 3. Efficient beamforming gain and interference management: Through the alternating optimization algorithm, it is possible to ensure that SUs obtain the maximum beamforming gain while ensuring that PUs interference is limited, thereby improving the overall system performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 for a UAV-based movable antenna array;

[0049] Figure 2 is the relationship between beamforming gain and UAV altitude;

[0050] Figure 3 is the relationship between the maximum-minimum beamforming gain and the number of iterations;

[0051] Figure 4 Comparison of beam patterns of different schemes. DETAILED DESCRIPTION

[0052] 1. System model and problem construction

[0053] The present invention considers a UAV-based MA system, hereinafter referred to as UMA (UAV-mounted moveable antenna), which has N antennas, such as Figure 1 shown.

[0054] The antennas are distributed in a linear region of length D along the x-axis.

[0055] definition For the antenna set, is the position of the nth antenna.

[0056] Therefore, APV can be expressed as

[0057] In addition, it is assumed that the UAV is hovering at the origin of the x-axis, i.e., x=0, and its height h can be dynamically adjusted along the z-axis.

[0058] The present invention assumes that there are K PUs and L SUs, whose positions are fixed and can be expressed as and

[0059] Therefore, the steering angle θ between the kth PU and the lth SU is k ∈[0,π], and φ l ∈[0,π], They can be expressed as:

[0060]

[0061] Therefore, the array steering vector can be further expressed as:

[0062]

[0063] Here, λ represents the wavelength.

[0064] The present invention defines AWV as

[0065] Therefore, for the conduction angle θ k and φ l The beamforming gains of can be expressed as:

[0066]

[0067] The goal of the present invention is to maximize the minimum beamforming gain δ of the secondary user by jointly optimizing the height h of the UAV, APV(x), and AWV(w), while taking into account the constraints of the spacing between adjacent antennas, interference to PUs, total available power, and the minimum hovering height of the UAV.

[0068] Therefore, the optimization problem can be modeled as:

[0069]

[0070] ‖w‖ 2 ≤1,(4d)

[0071]

[0072] h≥H 0 (4f)

[0073] Among them, D 0 is the minimum distance between adjacent antennas,

[0074] η is the predetermined threshold of the beamforming gain of SU,

[0075] H 0 is the minimum hovering altitude of the UAV.

[0076] In particular, equation (4a) ensures that there is no coupling between antennas;

[0077] Formula (4b) ensures that the beamforming gain of any SU is not less than δ;

[0078] Formula (4c) limits the interference to any PU to not exceed a predetermined threshold;

[0079] Formula (4d) stipulates that the normalized power of the antenna does not exceed 1;

[0080] Formula (4e) ensures that the antenna is adjusted within a limited area;

[0081] Formula (4f) ensures that the UAV hovers above the minimum safe altitude.

[0082] Problem (P1) is a complex optimization problem, mainly because its constraints are non-convex constraints on h, w or x, and there are complex interdependencies between these variables, which significantly increases the complexity of solving the problem.

[0083] 2. Solution Algorithm

[0084] The present invention decomposes the original optimization problem (P1) into three sub-problems and solves each sub-problem sequentially in an iterative manner, wherein each sub-problem focuses on optimizing one of the UAV's height h, antenna weight vector AWV w, or antenna position vector APV x, while keeping the other variables fixed.

[0085] A. Optimize h given w and x

[0086] The goal of this part is to maximize δ by optimizing h given w and x, which can be reformulated as the following subproblems:

[0087]

[0088] st(4b),(4c),(4f),

[0089] Among them, equation (4b) and equation (4c) are non-convex functions with respect to h.

[0090] Therefore, the present invention relaxes equations (4b) and (4c) into convex constraints by adopting the successive convex approximation (SCA) technique.

[0091] For the sake of convenience, the present invention uses represents the nth element of w, where |w n | is the amplitude, ∠w n For phase.

[0092] In addition, the present invention also defines the following variables:

[0093]

[0094] Therefore, G(w,x,φ l ) can be further expressed as:

[0095]

[0096] in, as well as

[0097] Since G(w,x,φ l ) is neither convex nor concave with respect to h.

[0098] Therefore, the present invention constructs a proxy function based on the second-order Taylor expansion to perform local approximation.

[0099] Specifically, for a given point The second-order Taylor expansion of cos(f(l)) can be expressed as:

[0100]

[0101] According to the Cauchy inequality, we can get:

[0102] (ll 0 ) 2 ≥0 and

[0103]

[0104] Therefore, the present invention can construct a concave proxy function as shown below To approximate cos(f(l)):

[0105]

[0106] in,

[0107] Then, in the i-th SCA iteration, for a given h i (the initial value of the i-th SCA iteration process), by setting and You can get:

[0108] and

[0109]

[0110] Therefore, G(w,x,φ l)The proxy function that provides the global lower bound can be constructed as:

[0111]

[0112] in, Given by:

[0113]

[0114] in,

[0115]

[0116] Coming soon and Substituting into formula (7), we can obtain formula (8).

[0117] In addition, since the structure of equation (4c) is similar to equation (4b), equation (4c) can be relaxed by modifying the relaxation process of constructing equation (4b).

[0118] Specifically, according to the Cauchy inequality, we have:

[0119] (ll 0 ) 2 ≥0 and

[0120]

[0121] Therefore G(w,x,θ k ) can be approximated as:

[0122]

[0123] in, Given by:

[0124]

[0125] Among them, the present invention defines:

[0126]

[0127] Therefore, during the i-th SCA iteration, h can be optimized by solving the following optimization problem.

[0128]

[0129] Since Equations (10a) and (10b) are convex quadratic constraints on h, and Equation (4g) is a linear constraint on h, (P1.2.1) is a convex optimization problem that can be efficiently solved using existing solvers (such as CVX).

[0130] B. Optimize w given h and x

[0131] The goal of this part is to maximize δ by optimizing w in (P1) given h and x, which can be restated as the following subproblem:

[0132]

[0133] st(4b),(4c),(4d),

[0134] Wherein, formula (4b) is a non-convex function with respect to w, so the present invention intends to use SCA technology to relax formula (4b).

[0135] Specifically, during the i-th SCA iteration, for a given

[0136] (initial value of the i-th SCA iteration process), since G(w,x,φ l ) is a convex function about w, and the present invention can be achieved by i Apply the first-order Taylor expansion to G(w,x,φ l ) is globally approximated by the following linear proxy function

[0137]

[0138] Wherein, Re{} represents the real part of the complex number.

[0139] Therefore, in the i-th SCA iteration process, for a given The optimization of w can be transformed into the following optimization problem:

[0140]

[0141] Among them, equation (12a) is a linear constraint condition on w, and equations (4c) and (4d) are convex quadratic constraints on w.

[0142] Therefore, (P1.3.1) is a convex optimization problem that can be solved by existing solvers such as CVX.

[0143] C. Optimize x given h and w

[0144] The goal of this part is to maximize δ by optimizing x given h and w, which can be reformulated as the following subproblems:

[0145]

[0146] st(4a),(4b),(4c),(4f),

[0147] Among them, equation (4b) and equation (4c) are non-convex constraints on x.

[0148] Therefore, the present invention proposes to relax equations (4b) and (4c) by adopting the SCA technology.

[0149] For the sake of convenience, the present invention defines

[0150] Therefore, G(w,x,φ l ) can be further expressed as:

[0151]

[0152] Among them, the present invention defines

[0153] Since G(w,x,φ l ) is neither about f l (x n ,x m ) is not a convex function of f l (x n ,x m ) is a concave function.

[0154] Therefore, the present invention intends to construct a proxy function based on the second-order Taylor expansion, so as to perform local approximation thereof.

[0155] Specifically, for a given The second-order Taylor expansion of cos(l) can be expressed as:

[0156]

[0157] Since cos(l 0 )≤1 and (ll 0 ) 2 ≥0, the present invention can construct a concave proxy function as shown below To approximate cos(l):

[0158]

[0159] Since G(w,x,θ k ) is neither a convex function nor a concave function about x. The present invention intends to follow Part A and construct a convex proxy function based on the second-order Taylor expansion to perform local approximation on it.

[0160] Therefore, in the i-th SCA iteration, for a given (initial value of the i-th SCA iteration process), let l←f l (x n ,x m )and

[0161] Then G(w,x,φ l )The proxy function that provides the global lower bound can be constructed as:

[0162]

[0163] in, and Given by:

[0164]

[0165] Among them, the present invention defines and

[0166] b l [n] represents vector b l The nth element of . is the vector b l The transpose of .

[0167] Please note that A l can be proved to be a negative semidefinite matrix.

[0168] Therefore, equation (4b) can be relaxed to the following convex function with respect to x, that is,

[0169]

[0170] On the other hand, since the structure of equation (4c) is similar to that of equation (4b), the present invention can relax it by modifying the process of constructing a relaxed convex constraint given in equation (17).

[0171] The modification process is as follows: Let us construct a Function, l←f k (x n ,x m )and Substituting into it we get expression.

[0172] Therefore, G(w,x,θ k ) The proxy function that provides the global upper bound can be constructed as:

[0173]

[0174] in, and Given by:

[0175]

[0176] in, Representation vector The nth element of . For vector The transpose of .

[0177] Please note, It can be strictly proved to be a positive semidefinite matrix.

[0178] Therefore, equation (4c) can be relaxed into the following convex constraint condition on x:

[0179]

[0180] Note that G(w,x,θ k ) must be greater than 0. However, G(w,x,θ) given in Eq. (19) k ) cannot guarantee this requirement.

[0181] Therefore, the following constraints should be met:

[0182]

[0183] in and Given by the following formula

[0184]

[0185] in, Representation vector The nth element of . For vector The transpose of .

[0186] The solution process is similar to that of Formula 15.

[0187] Please note, It can also be shown to be similar to A l Therefore, during the i-th SCA iteration, x can be optimized by solving the following optimization problem:

[0188]

[0189] st(4a),(4f),(17),(19),(20).

[0190] Since (4a) and (4f) are linear constraints, and (17), (19), and (20) are convex quadratic constraints on x, (P1.4.1) is a convex problem that can be solved efficiently using CVX.

[0191] D. Overall algorithm and complexity analysis

[0192] Algorithm 1 summarizes the overall algorithm for solving (P1).

[0193] Hypothesis I h ,I w and I x denote the number of iterations to solve (P1.2.1), (P1.3.1) and (P1.4.1) respectively. In each iteration, h, w and x are optimized alternately using the interior point method, so their individual complexities can be expressed as and

[0194] Therefore, the total computational complexity is Where I represents the number of iterations of the iterative solution (P1.2.1), (P1.3.1) and (P1.4.1).

[0195]

[0196] E. Simulation Results

[0197] In the simulation, unless otherwise specified, the present invention sets the following parameters as: N=8, K=2, L=2, η=0.1, H 0 =10m,λ=0.1m,D 0 =λ / 2, D = 8D 0 , and σ=10 -3 The positions of SU and PU are set to s = [-11.91, 5.77] T m and p = [-56.71, 17.32] T m.h 0 Can be To be specific, It can be obtained by solving the following equation:

[0198]

[0199] in,

[0200] In addition, the proposed algorithm is compared with three benchmark schemes: UMA-AH, FPA, and MA. The detailed information is as follows:

[0201] 1) UMA-AH: The initialization height of the proposed algorithm is arbitrarily set, i.e., h 0 =H 0 ;

[0202] 2) FPA: the altitude of the drone and the position of the MAs are fixed;

[0203] 3)MA: The altitude of the drone is fixed, while AWV and APV are optimized alternately.

[0204] exist Figure 2 In FIG. 1 , the present invention shows the relationship between the beamforming gain (ordinate) of the first SU of the FPA and MA references and the UAV height (abscissa) in the two cases of N=6 and N=10.

[0205] In the FPA scheme, the beamforming gain fluctuates sharply with the UAV altitude, while in the MA scheme, this fluctuation is much more gradual. This is mainly because the change in the UAV altitude causes the antenna's steering angle to change to the user, and when the AWV and APV remain unchanged, the beamforming gain will fluctuate dramatically. This shows that the UAV altitude does have a significant impact on the system performance, which verifies the research motivation of this invention. Specifically, the MA scheme can achieve full beamforming gain within a specific altitude range, while the FPA scheme can only achieve beamforming gain at a certain point. This shows that it is challenging to obtain the optimal UAV altitude for the proposed UMA scheme.

[0206] exist Figure 3 In FIG. 1 , the present invention shows the relationship between the maximum and minimum beamforming gains (ordinate) and the number of iterations (abscissa) of the proposed scheme and its variants.

[0207] Here, "AH" and "AW" refer to random height initialization and random AWV initialization, respectively. It is worth noting that the proposed UMA scheme shows fast convergence to full beamforming gain at different η values. The maximum and minimum beamforming gains of the UMA-AH method are slightly reduced when η = 0.05, and the convergence period is slightly prolonged, but it is still better than the UMA-AW and UMA-AHAW methods in terms of convergence speed and overall system performance. Therefore, the present invention can be used from Figure 3 It is inferred that the proposed high initialization technique contributes significantly to improving the convergence speed and the maximum and minimum beamforming gains.

[0208] Figure 4 A comparison of the beam patterns of different baseline schemes is shown. Figure 4 It can be seen that all four schemes can suppress the interference between the two PUs below the predetermined threshold. Figure 4It can also be seen that the proposed scheme provides full beamforming gain for two SUs (i.e., N = 8), while UMA-AH, MA, and FPA achieve beamforming gains of 7.75, 7.57, and 4.47, respectively. This is because the proposed scheme can flexibly adjust the steering vector by exploiting the new degree of freedom provided by the UAV's maneuverability.

Claims

1. A method based on UAV-MA to maximize the minimum beamforming gain of SU, It is characterized in that The following steps are involved: Step 1: Establish a UMA system, which includes a UAV, K primary users PU and L secondary users SU; an antenna array including N MAs is integrated on the UAV; Step 2: Based on the UMA system, obtain the guidance angle θ k and φ l , APV and AWV, and finally obtain the guidance angle θ k and φ l The beamforming gain of Step 3: Considering the spacing between adjacent antennas, interference to PUs, total available power, and the constraints of the UAV’s minimum hovering height, the UAV’s height h, APV, and AWV are jointly optimized to maximize the minimum beamforming gain of the secondary user.

2. The method for maximizing the minimum beamforming gain of SU based on UAV-MA according to claim 1, It is characterized in that The beamforming gain in step 2 is: APV is expressed as AWV is defined as 3. The method for maximizing the minimum beamforming gain of SU based on UAV-MA according to claim 2, It is characterized in that The optimization problem in step 3 can be modeled as: ‖w‖ 2 ≤1,(4d) h≥H 0 (4f) in, is the position of the nth antenna; D 0 is the minimum distance between adjacent antennas; η is a predetermined threshold of the beamforming gain of the SU; H 0 is the minimum hovering altitude of the UAV.

4. The method for maximizing the minimum beamforming gain of SU based on UAV-MA according to claim 3, It is characterized in that When the APV and AWV are given and the height h is optimized, the model in step 3 is:

5. The method for maximizing the minimum beamforming gain of SU based on UAV-MA according to claim 3, It is characterized in that When APV and h are given and AWV is optimized, the model in step 3 is: (4c),(4d).

6. The method for maximizing the minimum beamforming gain of SU based on UAV-MA according to claim 3, It is characterized in that When optimizing APV given AWV and h, the model in step 3 is: st(4a),(4f),(17),(19),(20),

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