Beam arrangement method based on spherical coordinate system

Abstract

The invention belongs to the technical field of mobile communication, and particularly relates to a beam arrangement method based on a spherical coordinate system. The method comprises the following steps: carrying out beam arrangement in a spherical coordinate system action range by adopting a low-loss dot matrix beam to obtain a beam initial position; calculating the angle distance of each pair of beams; according to the angle distance of the beam pair, judging whether a beam gap meets a large gap requirement; calculating the offset directions of all beam pairs meeting the large gap requirement; adjusting the beam position according to the offset direction of the beam pair; judging whether all the beam pairs meet a convergence condition or not, and if the convergence condition is met or the maximum number of iterations is reached, stopping iteration to complete beam arrangement; otherwise, continuing iterative computation; according to the invention, by optimizing the beam arrangement mode, the overlapping among the beams is reduced, so that the interference is minimized, and the system capacity is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of mobile communications, and in particular relates to a beam arrangement method based on a spherical coordinate system. Background Art

[0002] In radar systems, beamforming involves arranging antenna beams in a specific order or spatial layout to cover a target area based on a predetermined arrangement strategy. Proper beamforming design can effectively improve radar coverage efficiency, reduce search time, and increase target detection probability. Beamforming requires a balance between overlap and gaps in beam coverage, ensuring seamless coverage (avoiding gaps) while minimizing excessive overlap (reducing redundancy and interference).

[0003] Currently, there are three basic styles of radar beam arrangement, each with distinct characteristics and application scenarios: Columnar beam arrangement typically arranges beams in a straight line or grid at fixed spacing, achieving linear coverage of the entire area. This method is simple to implement, but may suffer from insufficient overlap or gaps at the edges of coverage. Staggered beam arrangement staggers the centers of adjacent beams, creating cross-coverage between them and improving coverage continuity across the entire area. This method achieves more uniform coverage with a relatively small number of beams, alleviating the problem of insufficient overlap to a certain extent. Low-loss point beam arrangement is a more complex arrangement that optimizes the distribution and spacing of beam centers to ensure seamless coverage while minimizing ineffective overlap between beams. This method typically requires precise calculation of beam parameters such as half-power width, coverage area, and angular distance, and is suitable for scenarios requiring high coverage accuracy.

[0004] In existing beam arrangement, sinusoidal space is often used for beam design. However, when beam positions are mapped from sinusoidal space back to a spherical coordinate system, coverage gaps may occur. Therefore, a beam arrangement method based on a spherical coordinate system is urgently needed to ensure a minimum overlap ratio for radar beam arrangement, reduce interference between beams, and ensure that every user has adequate signal coverage. Summary of the Invention

[0005] In view of the shortcomings of the existing technology, the present invention proposes a beam arrangement method based on a spherical coordinate system, which includes:

[0006] S1: Use low-loss lattice beams to arrange beams within the spherical coordinate system to obtain the initial beam position;

[0007] S2: Calculate each pair of beams (C i,n ,C i+1,n+a ) angular distance; where C i,n represents the beam in row i and column n in the spherical coordinate system, C i+1,n+a represents the beam at row i+1 and column n+a in the spherical coordinate system, where a=0 when i is odd and a=1 when i is even;

[0008] S3: Determine whether the beam gap meets the large gap requirement based on the angular distance of the beam pair;

[0009] S4: Calculate the offset directions of all beam pairs that meet the large gap requirement;

[0010] S5: Adjust the beam position according to the offset direction of the beam pair;

[0011] S6: Determine whether all beam pairs meet the convergence condition. If the convergence condition is met or the maximum number of iterations is reached, stop the iteration and complete the beam arrangement; otherwise, return to step S2.

[0012] Preferably, the initial beam position is expressed as:

[0013]

[0014] Among them, (u i,n ,v i,n ) represents the polar coordinates of the beam in row i and column n, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i,n It represents the polar coordinate angle of the beam in row i and column n, az_min represents the minimum azimuth angle of the scope, Indicates that the horizontal direction of the beam is at a pitch angle of The actual beam width in spherical coordinates during programming, Indicates the actual beam width of the beam in the spherical coordinate system when the vertical direction of the beam is arranged at an azimuth angle of az_rad.

[0015] Preferably, the formula for calculating the angular distance of the beam pair is:

[0016]

[0017] Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, u i+1,n The polar radius of the polar coordinates of the beam in the i+1th row and nth column, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i+1,n represents the polar coordinate angle of the beam in row i+1 and column n, vi,n The polar coordinate angle of the beam in row i and column n, u i+1,n+1 represents the polar diameter of the polar coordinates of the beam in the i+1th row and n+1th column, v i+1,n+1 The angle of the polar coordinates of the beam in the i+1th row and n+1th column.

[0018] Preferably, the large gap requirement is expressed as:

[0019] D i,n >0.866·θ el_rad

[0020] Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, θ el_rad Indicates the actual beam width in the spherical coordinate system when the horizontal direction of the beam is arranged at an elevation angle of el_rad.

[0021] Preferably, the formula for calculating the offset direction of the beam pair is:

[0022]

[0023] Among them, s i,n Indicates that the beam C i,n to C i+1,n+a The unit vector of the offset direction is s i,n The opposite direction of u i+1,n+a The polar radius of the polar coordinates of the beam in the i+1th row and n+ath column, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i+1,n+a represents the polar coordinate angle of the beam in row i+1 and column n+a, v i,n Indicates the polar coordinate angle of the beam at row i and column n.

[0024] Preferably, adjusting the beam position is expressed as:

[0025] (u′ i,n ,v′ i,n )=(u i,n ,v i,n )+α·s i,n

[0026] (u′ i+1,n+a ,v′ i+1,n+a )=(u i+1,n+a ,v i+1,n+a )-α·s i,n

[0027] Among them, (u′ i,n ,v′ i,n ) represents the polar coordinates of the beam position after adjustment in row i and column n, (u i,n,v i,n ) represents the polar coordinates of the beam in row i and column n, (u′ i+1,n+a ,v′ i+1,n+a ) represents the polar coordinates of the beam position after adjustment at row i+1 and column n+a, (u i+1,n+a ,v i+1,n+a ) represents the polar coordinates of the beam d in the i+1th row and n+ath column, s i,n Indicates that the beam C i,n to C i+1,n+a The unit vector of , α represents the offset factor.

[0028] Preferably, the convergence condition is expressed as:

[0029]

[0030] d max ≤ε

[0031] Among them, d max Denotes the maximum gap between all beam pairs, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, θ el_rad It represents the actual beam width in the spherical coordinate system when the horizontal direction of the beam is arranged at an elevation angle of el_rad. ε represents the convergence threshold.

[0032] The present invention has the following beneficial effects: It focuses on optimizing and achieving the minimum overlap ratio in a beam arrangement algorithm based on a spherical coordinate system. By optimizing the beam arrangement method based on a spherical coordinate system, it reduces beam overlap, thereby minimizing interference and improving system capacity. This reduces the overlap area of beams, minimizes interference, and simultaneously meets the dynamic coverage requirements brought about by high-speed vehicle movement in the Internet of Vehicles scenario. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 Flowchart of the beam arrangement method based on the spherical coordinate system in the present invention;

[0034] Figure 2 Schematic diagram of the scope of the present invention in the spherical coordinate system;

[0035] Figure 3 The beam arrangement of the low-loss lattice beam in the spherical coordinate system of the present invention;

[0036] Figure 4 The partial spatial beam arrangement of the low-loss lattice beam in the present invention after optimization in the spherical coordinate system;

[0037] Figure 5 The hemispherical spatial beam arrangement of the low-loss lattice beam in the present invention after optimization in the spherical coordinate system;

[0038] Figure 6 This is the global spatial beam arrangement after the low-loss lattice beam in the present invention is optimized in the spherical coordinate system. DETAILED DESCRIPTION

[0039] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0040] The present invention proposes a beam arrangement method based on a spherical coordinate system, such as Figure 1 As shown, the method includes the following contents:

[0041] S1: Use low-loss lattice beams to perform beam arrangement within the scope of the spherical coordinate system to obtain the initial beam position.

[0042] In radar communications, radar beams scan the coverage area according to a specific beam sequence. This ensures coverage while using the minimum number of beams, significantly improving network efficiency and user experience. Therefore, beam position arrangement design is crucial for base stations to perform signal coverage and multi-user communication tasks. Beam arrangement refers to the division of antenna beam positions within the coverage area of the base station antenna according to a specific arrangement and scanning order. Different beam position arrangement schemes can be designed based on different network requirements and scenarios. Currently, the most common beam arrangement methods include column beams, staggered beams, and low-loss dot array beams.

[0043] When analyzing the radar beam, the spherical coordinate system is usually used as a reference. However, when the beam pointing deviates from the normal by a large angle, beam broadening will occur in the spherical coordinate system, resulting in increased beam arrangement and computational complexity. In contrast, the sinusoidal coordinate system can avoid beam broadening due to its projection characteristics, thereby simplifying the beam design. However, when the arrangement results in the sinusoidal coordinate system are mapped back to the spherical coordinate system, uneven coverage or loopholes will be introduced due to the nonlinear characteristics of the coordinate transformation. After comprehensive consideration. The present invention chooses to perform beam arrangement directly in the spherical coordinate system, assuming that the main lobe direction of the beam is aligned with the spherical normal (i.e., perpendicular to the spherical radiation), which can effectively constrain the degree of broadening while avoiding the risk of coverage distortion caused by the coordinate system conversion.

[0044] Low-loss lattice beams are used to perform beam arrangement within the scope of the spherical coordinate system. Specifically:

[0045] Calculate the azimuthal beam width θ of the beam in the spherical coordinate system el_rad and elevation beamwidth

[0046]

[0047] Among them, θ el_rad Indicates the actual beam width of the beam in the spherical coordinate system when the horizontal direction is arranged at an elevation angle of el_rad. Indicates the actual beam width of the beam in the spherical coordinate system when the vertical direction is arranged at an azimuth angle of az_rad, θ B Indicates the horizontal beam width in radians. R is the radius of the sphere in spherical coordinates. el_rad is the elevation angle in radians. az_rad is the azimuth angle in radians. It represents the width of the vertical beam in radians.

[0048] The range of the demarcated area is converted from radar rectangular coordinates to radar spherical coordinates. The range area demarcated in the radar spherical coordinates is shown as follows: Figure 2 As shown, the diagram is drawn with the azimuth scanning range of -50° to 50° and the elevation scanning range of 0° to 50° as an example.

[0049] The coordinates of a point in the spherical coordinate system are defined as (u i,0 ,v i,0 )~(u i,n ,v i,n ), where i represents the row and n represents the column, assuming the sphere radius is 1. The spherical coordinate system calculates the number of beams that need to be arranged in the pitch dimension, and uses the coordinate points 0 and n to calculate the number of beams M that need to be arranged in the i dimension in the spherical coordinate system. i :

[0050] i is an even number

[0051] Get the initial position of the beam and arrange the coordinates of the beam as follows:

[0052]

[0053] Among them, (u i,n ,v i,n ) represents the polar coordinates of the beam in row i and column n, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i,n It represents the polar coordinate angle of the beam in row i and column n, az_min represents the minimum azimuth angle of the scope, Indicates that the horizontal direction of the beam is at a pitch angle of The actual beam width in spherical coordinates during programming, Indicates the actual beam width of the beam in the spherical coordinate system when the vertical direction of the beam is arranged at an azimuth angle of az_rad.

[0054] The calculated beam coordinates are plotted into the radar spherical coordinate system as Figure 3 As shown.. Figure 3 As can be seen from the figure, since spherical coordinates are a curved surface, the low-loss point array beam will have obvious loopholes in the beam arrangement of the spherical coordinate system, and seamless coverage cannot be achieved. The following will optimize the beam arrangement to achieve seamless coverage to address this problem.

[0055] Assume that the coverage area of each beam is a circular area, and the center point of the beam is (u i,n ,v i,n ), the coverage radius is r. The coverage area of the beam can be expressed as C i,n :

[0056] C i,n ={(u,v)|d((u,v),(u i,n ,v i,n ))≤r}

[0057] Among them, C i,n represents the beam coverage of the i-th row and n-th column, and d(·,·) represents the angular distance between two points. The goal of seamless coverage is to ensure that the entire target area Ω is covered by the beam with no gaps between beams, which can be expressed as:

[0058]

[0059] Where Ω represents the target coverage area and N represents the total number of beams.

[0060] In order to achieve seamless coverage, the gaps between beams need to be minimized. An optimization objective of minimum overlap is constructed:

[0061]

[0062] Among them, D i,n Indicates the angular distance between the beam in the i-th row and the n-th column and the diagonal beam, where el_rad is 0°.

[0063] The following steps are used to achieve the optimization goal of minimum overlap rate.

[0064] S2: Calculate each pair of beams (C i,n ,C i+1,n+a ) angular distance; where C i,n represents the beam coverage of the i-th row and n-th column in the spherical coordinate system, C i+1,n+a It represents the beam coverage of the i+1th row and n+ath column in the spherical coordinate system. When i is an odd number, a=0; when i is an even number, a=1.

[0065] Set the maximum number of iterations T, the convergence threshold ε and the offset factor α; calculate the angular distance between the two beams as D i,n :

[0066]

[0067] Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, u i+1,n The polar radius of the polar coordinates of the beam in the i+1th row and nth column, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i+1,n represents the polar coordinate angle of the beam in row i+1 and column n, v i,n The polar coordinate angle of the beam in row i and column n, u i+1,n+1 represents the polar diameter of the polar coordinates of the beam in the i+1th row and n+1th column, v i+1,n+1 Indicates the polar coordinate angle of the beam at row (i+1) and column (n+1).

[0068] S3: Determine whether the beam gap meets the large gap requirement based on the angular distance of the beam pair.

[0069] The large gap requirement is expressed as:

[0070] D i,n >0.866·θ el_rad

[0071] Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, θ el_rad Indicates the actual beam width in the spherical coordinate system when the horizontal direction of the beam is arranged at an elevation angle of el_rad.

[0072] S4: Calculate the offset directions of all beam pairs that meet the large gap requirement.

[0073] For beam pairs with gaps (C i,n ,C i+1,n+a ), calculate the offset direction:

[0074] Calculate from beam C i,n to C i+1,n+a The unit vector of :

[0075]

[0076] The offset direction is s i,n in the opposite direction (i.e. moving towards the gap).

[0077] S5: Adjust the beam position according to the offset direction of the beam pair.

[0078] For the beam C with gap i,n and C i+1,n+a , adjust their positions respectively:

[0079] (u′ i,n ,v′ i,n )=(u i,n ,v i,n )+α·s i,n

[0080] (u′ i+1,n+a ,v′ i+1,n+a )=(u i+1,n+a ,v i+1,n+a )-α·s i,n

[0081] Among them, (u′ i,n ,v′ i,n ) represents the polar coordinates of the beam position after adjustment in row i and column n, (u i,n ,v i,n ) represents the polar coordinates of the beam in row i and column n, (u′ i+1,n+a ,v′ i+1,n+a ) represents the polar coordinates of the beam position after adjustment at row i+1 and column n+a, (u i+1,n+a ,v i+1,n+a ) represents the polar coordinates of the beam d in the i+1th row and n+ath column, s i,n Indicates that the beam C i,n to C i+1,n+a The unit vector of , α represents the offset factor.

[0082] By introducing an offset factor, the beam positions are dynamically adjusted to fill the gaps between beams, ensuring seamless coverage within the target area. At the same time, the beam distribution is optimized to reduce overlapping areas and improve coverage efficiency.

[0083] S6: Determine whether all beam pairs meet the convergence condition. If the convergence condition is met or the maximum number of iterations is reached, stop the iteration and complete the beam arrangement; otherwise, return to step S2.

[0084]

[0085] d max ≤ε

[0086] Among them, d max represents the maximum gap between all beam pairs, and ε represents the convergence threshold.

[0087] If d max If ≤ε or the maximum number of iterations T is reached, the algorithm terminates and beam arrangement is completed; otherwise, return to step 2.

[0088] The present invention is simulated and verified:

[0089] Overlap ratio calculation:

[0090] (1) Discretization scope and construction of area elements

[0091] Assume that the scope Ω is on the sphere with the azimuth angle φ at [φ min ,φ max ] and the pitch angle θ in [θ min ,θ max ] definition. For the convenience of calculation, we discretize this region into N φ ×N θ grid points, and its specific construction is:

[0092]

[0093] The area element corresponding to each grid unit on the sphere is (unit sphere):

[0094] dA jk =cos(θ k )ΔθΔφ,

[0095] in:

[0096]

[0097] The total area of the scope:

[0098]

[0099] (2) Determine whether the grid point is covered by the beam

[0100] A single beam B is defined as: the beam center is located at (φ0, θ0) (unit: radian) and the half angle is δ. Any point on the sphere (φ j ,θ k ) to the spherical great circle distance d from the beam center jk Calculated according to the spherical great circle distance formula:

[0101] cos d jk = sinθ k sinθ0+cosθ k cosθ0cos(φ j -φ0)

[0102] thereby:

[0103] d jk =arccos(sinθ k sinθ0+cosθ k cosθ0cos(φ j -φ0))

[0104] Define the coverage indicator function:

[0105]

[0106] For each grid point (φ j ,θ k ), count how many beams it is covered by, that is, calculate the coverage count:

[0107]

[0108] The coverage indicator function is related to the area of each grid cell dA jk After multiplying and summing, we get the effective area A of the beam within the scope B :

[0109]

[0110] Consider N beams, each beam B i The areas within the scope Ω are A i , when different beams overlap, the overlapping coverage area will only be calculated once. Let the total coverage area of multiple beams within the scope be:

[0111]

[0112] When R=1, it means that each unit area in the scope is covered by exactly one beam; if R>1, it means that there is an overlapping area; if R<1, it means that there is an uncovered area in the scope.

[0113] If you want to calculate this value via a grid method, you can:

[0114] The repetition rate R is:

[0115]

[0116] Simulation analysis:

[0117] The present invention is based on a low-loss beam arrangement algorithm in sinusoidal space, applies the low-loss beam arrangement algorithm to the radar's spherical coordinate system, and optimizes the beam arrangement algorithm on this basis to ensure complete coverage without loopholes, ultimately achieving beam coverage with minimum overlap. Figure 4 It can be seen intuitively that there are no coverage holes and the overlap rate is around 1.306. Figure 5 , Figure 6 These are the hemispherical and global spatial beam arrangements of the low-loss dot matrix beam after optimization in the spherical coordinate system. It can be seen that there are no coverage gaps.

[0118] In summary, this invention focuses on optimizing and achieving the minimum overlap ratio in a spherical coordinate-based beam arrangement algorithm. By optimizing the beam arrangement method based on spherical coordinates, this method reduces beam overlap, thereby minimizing interference and improving system capacity. This reduces beam overlap, minimizing interference, and simultaneously meeting the dynamic coverage requirements of high-speed vehicle movement in connected vehicle scenarios.

[0119] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention in detail. It should be understood that the above embodiments are only preferred implementation plans of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A beam arrangement method based on a spherical coordinate system, characterized in that: include: S1: Use low-loss lattice beams to arrange beams within the spherical coordinate system to obtain the initial beam position; S2: Calculate each pair of beams (C i,n ,C i+1,n+a ) angular distance; where C i,n represents the beam coverage of the i-th row and n-th column in the spherical coordinate system, C i+1,n+a represents the beam coverage of the i+1th row and n+ath column in the spherical coordinate system. When i is an odd number, a=0; when i is an even number, a=1. S3: Determine whether the beam gap meets the large gap requirement based on the angular distance of the beam pair; S4: Calculate the offset directions of all beam pairs that meet the large gap requirement; S5: Adjust the beam position according to the offset direction of the beam pair; S6: Determine whether all beam pairs meet the convergence condition. If the convergence condition is met or the maximum number of iterations is reached, stop the iteration and complete the beam arrangement. Otherwise, return to step S2.

2. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: The initial beam position is expressed as: Among them, (u i,n ,v i,n ) represents the polar coordinates of the beam in row i and column n, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i,n It represents the polar coordinate angle of the beam in row i and column n, az_min represents the minimum azimuth angle of the scope, Indicates that the horizontal direction of the beam is at a pitch angle of The actual beam width in spherical coordinates during programming, Indicates the actual beam width of the beam in the spherical coordinate system when the vertical direction of the beam is arranged at an azimuth angle of az_rad.

3. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: The formula for calculating the angular distance of a beam pair is: Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, u i+1,n The polar radius of the polar coordinates of the beam in the i+1th row and nth column, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i+1,n represents the polar coordinate angle of the beam in row i+1 and column n, v i,n The polar coordinate angle of the beam in row i and column n, u i+1,n+1 represents the polar diameter of the polar coordinates of the beam in the i+1th row and n+1th column, v i+1,n+1 The angle of the polar coordinates of the beam in the i+1th row and n+1th column.

4. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: The large gap requirement is expressed as: D i,n >0.866·θ el_rad Among them, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, θ el_rad Indicates the actual beam width in the spherical coordinate system when the horizontal direction of the beam is arranged at an elevation angle of el_rad.

5. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: The formula for calculating the offset direction of the beam pair is: Among them, s i,n Indicates that the beam C i,n to C i+1,n+a The unit vector of the offset direction is s i,n The opposite direction of u i+1,n+a The polar radius of the polar coordinates of the beam in the i+1th row and n+ath column, u i,n represents the polar diameter of the polar coordinates of the beam in the i-th row and n-th column, v i+1,n+a represents the polar coordinate angle of the beam in row i+1 and column n+a, v i,n Indicates the polar coordinate angle of the beam at row i and column n.

6. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: Adjusting the beam position is expressed as: (u′ i,n ,v′ i,n )=(u i,n ,v i,n )+α·s i,n (u′ i+1,n+a ,v′ i+1,n+a )=(u i+1,n+a ,v i+1,n+a )-α·s i,n Among them, (u′ i,n ,v′ i,n ) represents the polar coordinates of the beam position after adjustment in row i and column n, (u i,n ,v i,n ) represents the polar coordinates of the beam in row i and column n, (u i + 1,n+a ,v i + 1,n+a ) represents the polar coordinates of the beam position after adjustment at row i+1 and column n+a, (u i+1,n+a ,v i+1,n+a ) represents the polar coordinates of the beam d in the i+1th row and n+ath column, s i,n Indicates that the beam C i,n to C i+1,n+a The unit vector of , α represents the offset factor.

7. The beam arrangement method based on a spherical coordinate system according to claim 1, characterized in that: The convergence condition is expressed as: d max ≤ε Among them, d max Denotes the maximum gap between all beam pairs, D i,n represents the angular distance between the beam in row i and column n and the diagonal beam, θ el_rad It represents the actual beam width in the spherical coordinate system when the horizontal direction of the beam is arranged at an elevation angle of el_rad. ε represents the convergence threshold.

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