A high order fd-pade method for simulating radio wave propagation

Abstract

The application discloses a high-order FD-Pade method for simulating radio wave propagation, and comprises the following steps: constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates about radio wave propagation according to the cross-sectional direction, the propagation direction, the scalar component of the magnetic field or the electric field, the refractive index and the wave number of the radio wave propagation; introducing an auxiliary function to the three-dimensional parabolic equation, simplifying to obtain a forward 3D-PE, so as to derive a solving form; performing Pade approximation on the solving form to obtain a three-dimensional high-order Pade equation in the propagation direction; performing a fourth-order FD approximation on the three-dimensional high-order Pade equation to deduce a high-order FD-Pade approximate 3D-PE; and completing the simulation of the radio wave propagation according to the obtained 3D-PE.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of radio technology, more particularly, to a high-order FD-Pade method for simulating wave propagation. BACKGROUND

[0002] In the field of radio technology, the methods for simulating wave propagation mainly include parabolic equation (PE) method, finite difference time domain (FDTD) method, method of moments (MoM) and ray tracing method. Among them, the parabolic equation (PE) method has the advantage of being able to simulate the refraction and diffraction problems of wave propagation on complex terrain at the same time, and has been applied in various wave propagation calculation scenarios in recent years. It mainly focuses on applying PE to the prediction of related parameters of the guidance system, wave analysis of phased array antennas under complex conditions based on PE, analysis and application of hybrid PE and other wave propagation simulation methods, processing of wave propagation problems under complex boundary conditions, simulation of irregular terrain problems, proposing new PE solving methods, and error and comparison analysis.

[0003] At present, the overall research mainly focuses on the problem of two-dimensional PE, and there are few studies on three-dimensional PE. Finite difference (FD) method is an effective solution method for solving complex terrain boundary wave propagation problems, which is mainly applied to the approximate processing in the cross-sectional direction perpendicular to the direction of wave propagation. In the direction of wave propagation, Taylor expansion is a common approximate processing method for solving PE. Pade approximation is an approximate method that matches Taylor expansion, and its high-order approximation has certain numerical stability. In the latest research on three-dimensional PE (3D-PE) for simulating wave propagation, low-order FD-Pade approximation is mainly considered. In order to improve the accuracy of wave propagation simulation, it is necessary to study the high-order 3D-PE problem. SUMMARY

[0004] The present application provides a high-order FD-Pade method for simulating wave propagation, which effectively improves the accuracy of wave propagation simulation and has the characteristics of fast calculation efficiency, in order to solve the problem of low accuracy of wave propagation simulation in the prior art.

[0005] To achieve the above-mentioned purposes of the present application, the technical solutions adopted are as follows:

[0006] A high-order FD-Pade method for simulating wave propagation, the method comprising the following steps:

[0007] S1: constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates according to the cross-sectional direction of wave propagation, the direction of wave propagation, the scalar component of magnetic field or electric field, the refractive index and the wave number;

[0008] S2: introducing an auxiliary function to the three-dimensional parabolic equation, simplifying to obtain a forward 3D-PE, thereby deriving a solution form;

[0009] S3: Pade approximation to the solution form, obtaining a three-dimensional high-order Pade equation in the propagation direction;

[0010] S4: fourth-order FD approximation of the differential operator to the three-dimensional high-order Pade equation, deriving a high-order FD-Pade approximation 3D-PE;

[0011] S5: completing the radio wave propagation simulation according to the 3D-PE obtained in step S4.

[0012] Preferably, the three-dimensional parabolic equation in three-dimensional rectangular coordinates about radio wave propagation is constructed as follows:

[0013]

[0014] Wherein, y and z represent the cross-sectional direction of radio wave propagation, x represents the propagation direction, ψ represents the scalar component of the horizontal or vertical polarized magnetic field or electric field, n represents the refractive index, and k represents the wave number.

[0015] Further, the expression of the auxiliary function is as follows:

[0016] u(x, z) = e -ikx ψ(x, z). (2)

[0017] Further, the forward 3D-PE is:

[0018]

[0019] Wherein:

[0020]

[0021] A solution form in the propagation direction is:

[0022]

[0023] Wherein, Δx is the grid size in the propagation direction.

[0024] Further, Pade approximation is performed on the solution form, specifically, Pade approximation is performed on the exponential pseudo-differential operator in equation (5), to obtain a solution form:

[0025]

[0026] Wherein, a i , b iare parameters of the Pade approximation, which depend on the wave number and the step size; i is the order number of the Pade approximation, and m and n are the order of the approximation;

[0027] Let u(x+Δx,y,z) = u n (x,y,z), introduce an intermediate function v i , and write equation (6) as

[0028]

[0029] where p is the maximum value between m and n;

[0030] It can be seen that the solution form of each term in equation (7) is as follows:

[0031] (1+bL)v = (1+aL)u (8)

[0032] Further, the fourth-order FD approximation is as follows:

[0033]

[0034] where δ 2 u = u j-1 -2u j +u j+1 , j is the point number in the corresponding direction, and Δz is the horizontal grid size on the cross section.

[0035] Further, the fourth-order FD approximation is as follows:

[0036]

[0037] where Δy is the vertical grid size on the cross section.

[0038] Further, equation (10) can be simplified as follows:

[0039]

[0040] Both sides are multiplied by to obtain

[0041]

[0042] Equation (12) is split to derive the 3D-PE form of the high-order FD-Pade approximation as follows:

[0043]

[0044]

[0045] A computer system comprises a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method when executing the computer program.

[0046] A computer readable storage medium, having stored thereon a computer program, wherein the computer program, when executed by a processor, implements the steps of the method.

[0047] The beneficial effects of the present application are as follows:

[0048] The present application considers high-order Pade approximation in the direction of wave propagation, and considers the fourth-order FD approximation in the plane perpendicular to the direction of propagation, which is not considered in the existing method, effectively improving the solving accuracy of PE for simulating wave propagation, thereby effectively improving the simulation accuracy. At the same time, the present application solves the problem of calculation efficiency in solving the problem of three-dimensional wave propagation at present, and introduces the intermediate surface for splitting through the derivation process, thereby effectively improving the solving calculation speed. BRIEF DESCRIPTION OF DRAWINGS

[0049] Figure 1 is a flowchart of the method of embodiment 1.

[0050] Figure 2 is a calculation schematic diagram of the 3D-PE method of high-order FD-Pade approximation.

[0051] Figure 3 is a schematic diagram of the parallel computing method of the present application.

[0052] Figure 4 is a specific example analysis result graph of the method of the present application. DETAILED DESCRIPTION

[0053] The present application will be described in detail below in combination with the drawings and specific embodiments.

[0054] Embodiment 1

[0055] A high-order FD-Pade method for simulating wave propagation, the method comprising the following steps:

[0056] S1: constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates with respect to wave propagation according to the cross-sectional direction of wave propagation, the propagation direction, the scalar component of the magnetic field or the electric field, the refractive index, and the wave number;

[0057] S2: introducing an auxiliary function into the three-dimensional parabolic equation, and simplifying to obtain a forward 3D-PE, thereby deriving a solving form;

[0058] S3: performing Pade approximation on the solving form to obtain a three-dimensional high-order Pade equation in the direction of propagation;

[0059] S4: FD approximation of fourth order differential operator is performed on the three-dimensional high-order Pade equation to derive a high-order FD-Pade approximation 3D-PE;

[0060] S5: The 3D-PE obtained in step S4 is used to complete the simulation of the radio wave propagation.

[0061] In one specific embodiment, the three-dimensional parabolic equation in three-dimensional rectangular coordinates for the radio wave propagation is constructed as follows:

[0062]

[0063] where y and z represent the cross-sectional directions of the radio wave propagation, x represents the propagation direction, ψ represents the scalar component of the magnetic field or the electric field of the horizontal or vertical polarization, n represents the refractive index, and k represents the wave number.

[0064] In one specific embodiment, the expression of the auxiliary function is as follows:

[0065] u(x, z) = e -ikx ψ(x, z). (2)

[0066] In one specific embodiment, the forward 3D-PE is as follows:

[0067]

[0068] where:

[0069]

[0070] A solution form in the propagation direction is as follows:

[0071]

[0072] where Δx is the grid size in the propagation direction.

[0073] In one specific embodiment, Pade approximation is performed on the solution form, and specifically, Pade approximation is performed on the exponential pseudo-differential operator in equation (5) to obtain a solution form:

[0074]

[0075] where a i and b i are parameters of the Pade approximation, which depend on the wave number and the step size; i is the order number of the Pade approximation; and m and n are the orders of the approximation.

[0076] Let u(x+Δx, y, z) = u n (y, z), and introduce an intermediate function v iEquation (6) can be written as

[0077]

[0078] where p is the maximum value between m and n;

[0079] It can be seen that the solution form of each term in equation (7) is as follows:

[0080] (1 + bL)v = (1 + aL)u (8)

[0081] In a specific embodiment, the fourth-order FD is approximated as:

[0082]

[0083] where δ 2 u = u j-1 - 2u j + u j+1 , j is the point sequence number in the corresponding direction, and Δz is the horizontal grid size on the cross section.

[0084] In a specific embodiment, the fourth-order FD is approximated for y and z directions, and equation (4) and equation (9) are substituted into equation (8) to obtain

[0085]

[0086] where Δy is the vertical grid size on the cross section.

[0087] In a specific embodiment, equation (10) can be simplified as:

[0088]

[0089] Both sides are multiplied by to obtain

[0090]

[0091] where the relevant abbreviation mode is as follows

[0092]

[0093]

[0094] Equation (12) is split to derive the 3D-PE form of high-order FD-Pade approximation as follows:

[0095]

[0096]

[0097] The method described in the embodiment first constructs the equation form and solving form of the parabolic equation, performs Pade approximation on the solving form, and obtains a three-dimensional high-order Pade equation in the propagation direction. Secondly, the three-dimensional high-order Pade equation is subjected to fourth-order FD approximation of differential operator, and a 3D-PE method of high-order FD-Pade approximation is derived. The specific calculation mode of the method is as follows Figure 2 It can be seen that u is the field strength value of the previous advancing surface, V m is the field strength value of the intermediate auxiliary calculation surface, and V is the auxiliary next advancing surface field strength value required for equation calculation. The black point corresponds to the field value point in the z direction, and the white point corresponds to the field value point in the y direction. Step 1 represents that the field strength value in the z direction of the intermediate auxiliary calculation surface is calculated from the field value in the z direction of the previous advancing surface, and Step 2 represents that the auxiliary next advancing surface y field strength value required for equation calculation is calculated from the field strength value in the y direction of the intermediate auxiliary calculation surface.

[0098] The 3D-PE of high-order FD-Pade approximation derived by the application has parallel computing characteristics, is more efficient in calculation, and has higher calculation precision than low-order calculation. The parallel solving mode is as shown in Figure 3 The u1-u6 are the field strength values of each vertical column in the z direction of the current calculation surface, each column is simultaneously allocated to each calculation core for calculation and processing, so that the calculation purpose of the next field strength surface is achieved by single-surface multi-column parallelization, and the calculation efficiency is doubled. By introducing the intermediate calculation surface, the three-dimensional calculation problem is reduced in dimension, so that parallel processing is realized.

[0099] A specific example is considered, and the parameters are set as follows: a Gaussian transmitting source of 300 MHZ, a half-power lobe width of 15 degrees, an inclination angle of 0 degrees, a calculation area of 1000 m*1000 m*6000 m, and an approximate order of Pade(2,3).

[0100] Taking the double-ray method as a theoretical reference, the field strength values at x=4000 m in the propagation direction on the cross-sectional direction y=0 m are taken, and the corresponding propagation factors are calculated. It can be seen from Figure 4 that the 3D-PE method of high-order FD-Pade approximation has a higher propagation elevation angle than the 3D-PE method of low-order FD-Pade approximation, the calculation precision of the high-order method is higher, and the method has the characteristics of high-accuracy simulation of radio wave propagation.

[0101] Embodiment 2

[0102] A computer system comprises a memory, a processor, and a computer program stored on the memory and executable on the processor, and when the processor executes the computer program, the method steps are as follows:

[0103] S1: Constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates about the wave propagation according to the cross-sectional direction, the propagation direction, the scalar component of the magnetic field or the electric field, the refractive index, and the wave number of the wave propagation;

[0104] S2: Introducing an auxiliary function to the three-dimensional parabolic equation, simplifying to obtain a forward 3D-PE, thereby leading to a solution form;

[0105] S3: Performing Pade approximation on the solution form to obtain a three-dimensional high-order Pade equation in the propagation direction;

[0106] S4: Performing fourth-order FD approximation of the differential operator on the three-dimensional high-order Pade equation to derive a high-order FD-Pade approximate 3D-PE;

[0107] S5: Completing the wave propagation simulation according to the 3D-PE obtained in step S4.

[0108] Embodiment 3

[0109] A computer readable storage medium has a computer program stored thereon, and the computer program, when executed by a processor, implements the following method steps:

[0110] S1: Constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates about the wave propagation according to the cross-sectional direction, the propagation direction, the scalar component of the magnetic field or the electric field, the refractive index, and the wave number of the wave propagation;

[0111] S2: Introducing an auxiliary function to the three-dimensional parabolic equation, simplifying to obtain a forward 3D-PE, thereby leading to a solution form;

[0112] S3: Performing Pade approximation on the solution form to obtain a three-dimensional high-order Pade equation in the propagation direction;

[0113] S4: Performing fourth-order FD approximation of the differential operator on the three-dimensional high-order Pade equation to derive a high-order FD-Pade approximate 3D-PE;

[0114] S5: Completing the wave propagation simulation according to the 3D-PE obtained in step S4.

[0115] Obviously, the above embodiments of the present application are only examples for clearly illustrating the present application, and are not intended to limit the implementation modes of the present application. Any modification, equivalent replacement, and improvement within the spirit and principle of the present application shall be included in the protection scope of the claims of the present application.

Claims

1. A high order FD-Pade method for simulating the propagation of an electric wave, characterized in that: The method carries out fourth-order FD approximation of differential operator for the three-dimensional high-order Pade equation and completes the simulation of the radio wave propagation according to the obtained 3D-PE; The method comprises the following steps: S1: constructing a three-dimensional parabolic equation in three-dimensional rectangular coordinates according to the cross-sectional direction, the propagation direction, the scalar component of the magnetic field or the electric field, the refractive index and the wave number of the radio wave propagation, and the equation is in the following form: ；（1） S2: introducing an auxiliary function into the three-dimensional parabolic equation and simplifying to obtain a forward 3D-PE, thereby introducing a solving form, and the 3D-PE obtained by simplifying is as follows: （2） The solving form is as follows: （3） wherein, is the grid size in the direction of propagation; S3: carrying out Pade approximation on the solving form to obtain a three-dimensional high-order Pade equation in the propagation direction as follows: The Pade approximation is carried out on the solving form, and specifically, the exponential pseudo-differential operator in formula (3) is subjected to Pade approximation to obtain the solving form: （4） wherein a i , b i are parameters of the Pade approximation, depending on the wave number and the step size; i is the order number of the Pade approximation, m and n is the order of the approximation, make u ( x +∆ x , y , z ) = u n ( y , z ), introducing intermediate functions v i Equation (4) can be written as （5） wherein p is m with n the largest value between Therefore, the solving form of each term in formula (5) is as follows: ；（6） S4: carrying out fourth-order FD approximation of differential operator for the three-dimensional high-order Pade equation to derive a high-order FD-Pade approximation 3D-PE equation as follows: （7） wherein, , j is the point number in the corresponding direction, is the horizontal grid size on the cross section, Fourth-order FD approximation is carried out on the y and z directions, and formula (6) and (7) are obtained together with the auxiliary function: （8） wherein is the vertical grid size in cross section, Simplifying formula (8) can obtain: （9） Both sides are multiplied simultaneously resulting in （10） Formula (10) is split to derive a high-order FD-Pade approximation 3D-PE form as follows: (10a) (10b) S5: completing the simulation of the radio wave propagation according to the 3D-PE obtained in step S4.

2. The higher order FD-Pade method for simulating the propagation of an electromagnetic wave according to claim 1, wherein, In the step S1, the three-dimensional parabolic equation includes: y and z represents a cross-sectional direction of the wave propagation, x represents a propagation direction, and ψ represents a scalar component of a magnetic field or an electric field of horizontal or vertical polarization, n represents a refractive index, k represents a wave number.

3. The higher order FD-Pade method for simulating the propagation of an electromagnetic wave according to claim 1, wherein, In the step S2, the expression of the auxiliary function is as follows: （11）。 4. The higher order FD-Pade method for simulating the propagation of an electromagnetic wave according to claim 3, wherein, In the step S3, the operator definition is as follows: （12）。 5. A computer system comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that: When the processor executes the computer program, the steps of the method according to any one of claims 1-4 are realized.

6. A computer readable storage medium having stored thereon a computer program, characterized in that: When the computer program is executed by the processor, the steps of the method according to any one of claims 1-4 are realized.

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