Spherical near-field antenna measurement method based on cubic spline interpolation and near-near field transformation

Abstract

The invention provides a spherical near-field antenna measurement method based on cubic spline interpolation and near-near field transformation. The method comprises the following implementation steps: performing cubic spline interpolation on spherical near-field undersampling electric field data of an antenna to be measured; optimizing the spherical near-field electric field data after cubic spline interpolation based on near-near field transformation; and calculating a far-field pattern of the antenna. According to the invention, cubic spline interpolation is carried out on spherical near-field under-sampling electric field data to obtain spherical near-field electric field data satisfying a sampling criterion, and near-near field transformation is carried out on the spherical near-field electric field data after cubic spline interpolation. Spherical near-field electric field data of near-near field transformation are replaced through spherical near-field undersampling electric field data, and optimization of the spherical near-field electric field data after cubic spline interpolation is achieved. The influence of local detail loss caused by excessive smoothing of interpolation and reduction of interpolation precision near an end point caused by a boundary effect on the calculation precision of the antenna far-field directional diagram in the prior art is avoided.

Description

Technical Field

[0001] This invention belongs to the field of antenna measurement technology and relates to a spherical near-field antenna measurement method based on cubic spline interpolation and near-near-field transformation, which can be applied to the design, development and maintenance of various antennas under rapid measurement conditions. Background Technology

[0002] Antenna measurement methods can be broadly classified into two categories: direct methods and indirect methods. Direct methods include far-field methods, focusing methods, and compact field methods. However, the testing efficiency and accuracy of direct methods are insufficient for the requirements of modern antennas. Indirect methods, also known as near-field measurement methods, are further divided into planar near-field measurement, cylindrical near-field measurement, and spherical near-field measurement, depending on the shape of the scanning surface of the probe during near-field measurement. The principle of spherical near-field measurement involves using a probe with known characteristics in an anechoic chamber, under computer control, to measure the antenna under test with a radius of [missing information]. On the surface of the sphere, along the latitude angle and longitude angle A scan is performed to measure the amplitude and phase distribution of the electric field on the scanned sphere. Then, the far-field pattern of the antenna is calculated based on the measurement data and probe characteristics. The spherical near-field antenna measurement method has higher accuracy, greater flexibility, and wide bandwidth adaptability. The antenna far-field pattern is an important indicator of the antenna, which can clearly characterize key antenna parameters such as the directivity, main lobe width, and sidelobe level.

[0003] Existing spherical near-field measurements obtain the amplitude and phase distribution of the spherical electric field by using latitude angles. Longitude angle The two-dimensional scanning method results in a spherical scan count that increases with the square of the frequency, potentially leading to a single complete scan taking tens of hours. This inefficiency is particularly pronounced in high-frequency (millimeter-wave / terahertz) antenna measurements, manifesting in three ways: First, the miniaturization trend of high-frequency antennas necessitates sub-millimeter level probe positioning accuracy, significantly increasing the motion control time of the mechanical scanning system; second, high-frequency measurements are more sensitive to environmental disturbances, and prolonged measurements increase the risk of errors introduced by interference factors such as temperature drift and mechanical vibration; finally, in practical engineering applications such as 5G communication base station array antennas and spaceborne multi-beam antennas, excessively long testing cycles will severely impact product development progress.

[0004] Spherical near-field undersampling measurement obtains the values ​​of uncollected points by interpolating the undersampling electric field data, which can improve measurement efficiency while maintaining the accuracy of spherical near-field measurement. For example, in his 2024 master's thesis "Research on Measurement and Diagnosis Methods of Phase-less Spherical Near-field Antennas", Ma Di disclosed a cubic spline interpolation method for spherical near-field antenna measurement. This method first collects the undersampling spherical near-field electric field data of the antenna under test. Then, it uses cubic spline interpolation to interpolate the spherical near-field data, calculates the spherical near-field data corresponding to each interpolation node, and then combines the interpolated spherical near-field data with the undersampling spherical near-field electric field data to obtain the spherical near-field data of the antenna under test that meets the sampling criteria. Finally, the far-field radiation pattern of the antenna under test is obtained. This method uses cubic spline interpolation to recover near-field data that meets the sampling criteria from undersampled near-field data, and then calculates the far-field radiation pattern. Compared with the far-field radiation pattern calculated directly using undersampled electric field data, the accuracy of cubic spline interpolation is higher. However, the accuracy of cubic spline interpolation depends entirely on the cubic spline interpolation method itself. Although the advantage of cubic spline interpolation is its smoothness, in practical applications, excessive smoothing can lead to the loss of local details and a decrease in interpolation accuracy near the endpoints due to boundary effects. This affects the accuracy and stability of the interpolated data, and thus affects the accurate calculation of the antenna far-field radiation pattern. Summary of the Invention

[0005] The purpose of this invention is to overcome the defects of the prior art and propose a spherical near-field antenna measurement method based on cubic spline interpolation and near-near-field transformation, which solves the technical problem of low measurement accuracy of antenna far-field radiation pattern in the prior art.

[0006] To achieve the above objectives, the technical solution adopted by the present invention includes the following steps:

[0007] (1) Undersample the spherical near-field electric field data of the antenna under test:

[0008] Initialization is based on the center of the antenna under test as the center of the sphere, with a radius of... The sampling sphere, and respectively with , For intervals along latitude angles Longitude angle The direction is used to divide the grid, and then the probe is used to... , The polarization direction is related to the formation The electric field data at the grid points were respectively... Multiple undersampling yields a dimension of of , Spherical near-field undersampled electric field data in polarization direction , ;

[0009] (2) Perform cubic spline interpolation on the undersampled electric field data of the spherical near field:

[0010] Near-field undersampled electric field data of sphere , conduct The cubic spline interpolation, multiplied by a factor of 1, yields a cubic spline interpolation with dimensions of 1. Spherical near-field electric field data , ,in , ;

[0011] (3) Optimize the near-field electric field data of the spherical surface after cubic spline interpolation based on near-near field transformation:

[0012] Near-field electric field data of a sphere after cubic spline interpolation , Perform near-near field transformation and use the spherical near-field undersampled electric field data , The spherical near-field electric field data obtained by near-near-field transformation is replaced, and multiple iterations are performed to obtain the optimized cubic spline interpolated spherical near-field electric field data. , ;

[0013] (4) Calculate the far-field radiation pattern of the antenna:

[0014] spherical near-field electric field data obtained through optimized cubic spline interpolation , Calculate the TE and TM mode expansion coefficients of spherical waves , and through , Calculate the far-field radiation pattern of the antenna .

[0015] Compared with the prior art, the present invention has the following advantages:

[0016] This invention optimizes the near-field electric field data after cubic spline interpolation by performing cubic spline interpolation on the undersampled near-field electric field data of a spherical surface to obtain near-field electric field data that meets the sampling criteria. After performing near-near-field transformation on the near-field electric field data after cubic spline interpolation, the near-field electric field data after near-near-field transformation is replaced by the near-field undersampled near-field electric field data. This avoids the impact of excessive smoothing of interpolation in existing technologies, which leads to the loss of local details, and the decrease in interpolation accuracy near the endpoints caused by boundary effects, on the accuracy of far-field radiation pattern calculation of the antenna. Attached Figure Description

[0017] Figure 1 This is a flowchart illustrating the implementation of the present invention.

[0018] Figure 2 This is a comparison of simulation results of the E-plane radiation pattern of the antenna under test obtained by the present invention and existing technologies when used for undersampling at different multiples. Detailed Implementation

[0019] The present invention will now be described in further detail with reference to the accompanying drawings and specific implementation steps.

[0020] Reference Figure 1 The present invention includes the following steps:

[0021] Step 1) Undersample the spherical near-field electric field data of the antenna under test:

[0022] Initialization is based on the center of the antenna under test as the center of the sphere, with a radius of... The sampling sphere, and respectively with , For intervals along latitude angles Longitude angle The direction is used to divide the grid, and then the probe is used to... , The polarization direction is related to the formation The electric field data at the grid points were respectively... Multiple undersampling yields a dimension of of , Spherical near-field undersampled electric field data in polarization direction , .

[0023] radius of the sampling sphere and sampling interval and Need to meet , , , , This represents the circumferential angle of the probe rotation during near-field electric field sampling on a spherical surface. Indicates the wavelength of electromagnetic waves. This represents the minimum spherical radius surrounding the antenna under test. For latitude angles... Longitude angle The grid formed by directional division Grid points, among which and The calculation formulas are as follows:

[0024] ;

[0025] .

[0026] Step 2) Perform cubic spline interpolation on the near-field undersampled electric field data of the sphere:

[0027] Near-field undersampled electric field data of sphere , conduct The cubic spline interpolation, multiplied by a factor of 1, yields a cubic spline interpolation with dimensions of 1. Spherical near-field electric field data , ,in , By analyzing the undersampled electric field data of the spherical near field The cubic spline interpolation of times ensures that the interpolated spherical near-field data meets the sampling interval required by the Nyquist sampling criterion, thus enabling the antenna far-field radiation pattern to be obtained with a certain accuracy requirement.

[0028] Near-field undersampled electric field data of sphere , conduct The basic principle of cubic spline interpolation is:

[0029] Define function And in each small interval The above is a cubic polynomial, where If it is a given node, then it is called It is a node A cubic spline function on [a node]. If at [a node]... Given function value , ,and If established, it is called Let be the cubic spline interpolation function, and its expression is:

[0030] ;

[0031] From the definition requirements In each small interval The above requires determining four undetermined coefficients, and there are a total of A small interval, therefore it should be determined One parameter. According to exist The second derivative is continuous at the nodes. , The continuity condition should be met. , , ,in and Let represent the first and second derivatives with respect to , respectively. There are a total of . One condition, plus Total conditions Therefore, two more conditions need to be added to determine the result. .

[0032] Typically, it can be found in the range. endpoints , Add a boundary condition to each of the above. Common boundary conditions include the following three types:

[0033] (1) Given the values ​​of the first derivatives at both ends, i.e.:

[0034] ;

[0035] ;

[0036] (2) The second derivatives at both ends are known, that is:

[0037] ;

[0038] ;

[0039] (3) When Therefore When the function is a periodic function with periodicity, then it is required that... It is also a periodic function, and the boundary conditions should satisfy:

[0040] ;

[0041] ;

[0042] ;

[0043] Solving the equation using the aforementioned boundary and continuity conditions yields the values ​​for each interval. This yields the interpolation function for any known node.

[0044] Based on the above-mentioned cubic spline interpolation principle, by... , Within the interval, for the near-field undersampled electric field data of the sphere , conduct The near-field electric field data of the sphere are obtained by cubic spline interpolation. , .

[0045] Step 3) The implementation steps for optimizing the near-field electric field data of the sphere after cubic spline interpolation based on near-near-field transformation are as follows:

[0046] (3a) Initialize parameters:

[0047] Initialize the number of iterations to The maximum number of iterations is , and order ;

[0048] (3b) Near-field electric field data of the sphere after cubic spline interpolation , To perform a near-near field transformation, the steps are as follows:

[0049] (3b1) Spherical near-field electric field data after cubic spline interpolation , Calculate the first The TE and TM mode expansion coefficients of the spherical wave of the antenna under test in the next iteration , , of which Line number Expanding coefficients of TE and TM patterns of columns , The calculation formulas are as follows:

[0050] ;

[0051] ;

[0052] ;

[0053] ;

[0054] ;

[0055] ;

[0056] ;

[0057] in, express index, , This represents the highest order of the spherical wave mode expansion coefficients. and , , This represents the wave number of the electromagnetic wave emitted by the antenna under test. , To indicate a variable, , express index, , This represents the maximum angular modulus of the spherical wave mode expansion coefficients. , express Hankel function of the second kind of sphere, Indicates to exist , Perform double integral operations on it. and They represent respectively to and Differential operations, The imaginary unit, This represents the base of the exponential function. express Step The first kind of associated Legendre function To represent factorial operation, Indicates to The derivative;

[0058] (3b2) The TE and TM mode expansion coefficients of the spherical wave of the antenna under test in this iteration , No. Line number Columns , calculate and The calculation formulas are as follows:

[0059] ;

[0060] ;

[0061] (3c) Obtain the near-field electric field data of the sphere after optimization by cubic spline interpolation:

[0062] Using spherical near-field undersampled electric field data , Replace the same in this iteration , Near-field electric field data of the sphere obtained by near-near field transformation at grid coordinate points , and judge Is it true? If so, then... Replacement result of the next iteration , The near-field electric field data of the sphere is optimized from the electric field data of cubic spline interpolation; otherwise, let , , Then proceed with step (3b).

[0063] Since the spherical wave mode expansion coefficients of the antenna under test calculated using spherical near-field data at different radii remain unchanged, it is possible to use... Using the spherical wave mode expansion coefficients obtained from the near-field electric field data of the spherical surface acquired at the sampling radius as a bridge, the near-field electric field data of the spherical surface for any radius is calculated. This step uses a near-near-field transformation to recover only the near-field electric field data for radii of [missing information]. The spherical near-field electric field data is obtained. The spherical near-field electric field data, after cubic spline interpolation, is corrected through a near-near-field transformation. Then, the undersampled spherical near-field electric field data is used to replace the data from the same iteration. , The spherical near-field electric field data obtained by near-near-field transformation at the grid coordinate points is used to eliminate the error introduced by the near-near-field transformation at the original undersampled near-field electric field data. Through multiple iterative cycles, the error of the interpolated near-field electric field data is minimized, thus optimizing the interpolated data.

[0064] Step 4) Calculate the far-field radiation pattern of the antenna:

[0065] spherical near-field electric field data obtained through optimized cubic spline interpolation , Calculate the TE and TM mode expansion coefficients of spherical waves , and through , Calculate the far-field radiation pattern of the antenna The calculation formula is:

[0066] ;

[0067] ;

[0068] ;

[0069] in, , They represent , The Middle Line number The expansion coefficients of the TE and TM patterns of the column. and They represent and The far-field pattern function of the component. and They represent and The unit vector of direction.

[0070] The technical effects of the present invention will be further explained below with reference to simulation results:

[0071] 1. Experimental conditions and contents:

[0072] The E-plane radiation patterns of the antenna under test obtained by the present invention and the existing cubic spline interpolation spherical near-field antenna measurement method under 1.5x and 2x undersampling were simulated using MATLAB 2022a software. The results are as follows: Figure 2 As shown in (a) and (b).

[0073] 2. Analysis of experimental results:

[0074] Reference Figure 2 (a) and (b), the horizontal axis represents the far field of the antenna under test. The vertical axis represents the amplitude value of the E-plane radiation pattern of the antenna under test. The black dashed line in the figure represents the amplitude value of the theoretical far-field E-plane radiation pattern of the antenna under test. The red solid line represents the amplitude value of the E-plane radiation pattern of the antenna under test calculated by the prior art cubic spline interpolation after interpolating the undersampled near-field electric field data. The green solid line represents the amplitude value of the E-plane radiation pattern of the antenna under test calculated by the present invention after interpolating and near-near-field transforming the undersampled near-field electric field data.

[0075] pass Figure 2 (a) It can be seen that, with 1.5 times undersampled spherical near-field electric field data, using existing techniques for interpolation, the far-field E-plane radiation pattern of the antenna under test calculated from the interpolated near-field data has an error of 2 dB at the first null and first sidelobe on both the left and right sides, and an amplitude error of 4.7 dB at the second null and second sidelobe on both the left and right sides, compared to the theoretical far-field E-plane radiation pattern. However, using the present invention to interpolate and perform near-near-field transformation processing on the 1.5 times undersampled spherical near-field electric field data, the far-field E-plane radiation pattern of the antenna under test calculated has an amplitude error of only 0.2 dB at the first and second nulls on the right side, and only 0.5 dB at the first sidelobe on both the left and right sides, compared to the theoretical far-field E-plane radiation pattern. Overall, the present invention provides a 2dB improvement over the theoretical far-field pattern of the antenna under test obtained by calculating the near-field electric field data of the 1.5x undersampled spherical surface using existing technologies.

[0076] pass Figure 2(b) It can be seen that, with 2x undersampled spherical near-field electric field data, the far-field radiation patterns obtained by the present invention and the prior art only have a small error compared to the theoretical far-field radiation pattern at the main lobe. At large angles, they can only recover the general trend of the far-field radiation pattern of the antenna under test. However, it is obvious that the far-field E-plane radiation pattern obtained by the present invention has a 2dB improvement in accuracy compared to the far-field E-plane radiation pattern obtained by the prior art. Specifically, when interpolating the 2x undersampled spherical near-field electric field data using the prior art, the far-field E-plane radiation pattern of the antenna under test calculated by the interpolated near-field data has a 2dB amplitude error at the first null point on both the left and right sides, a 6dB amplitude error at the second null point on both the left and right sides, and a 1.5dB amplitude error at the second sidelobe on both the left and right sides. The far-field E-plane radiation pattern of the antenna under test calculated using this invention, after interpolation and near-near-field transformation of the 2x undersampled spherical near-field electric field data, shows a 1dB amplitude error at the first null point on both the left and right sides, only a 0.3dB amplitude error at the first sidelobe on the right, and only a 0.5dB amplitude error at the second null depth and second sidelobe on both the left and right sides, compared to the theoretical far-field E-plane radiation pattern. Overall, this invention provides a 4dB improvement over existing techniques for calculating the far-field E-plane radiation pattern of the antenna under test using 2x undersampled spherical near-field electric field data compared to the theoretical far-field radiation pattern.

[0077] The simulation results above fully demonstrate that, for spherical near-field antenna measurements, in the interpolation recovery of undersampled near-field electric field data, the present invention significantly improves accuracy and stability compared to existing technologies in calculating the far-field radiation pattern of the antenna under test.

Claims

1. A method for measuring a spherical near-field antenna based on cubic spline interpolation and near-near-field transformation, characterized in that, Includes the following steps: (1) Undersample the spherical near-field electric field data of the antenna under test: Initialization is based on the center of the antenna under test as the center of the sphere, with a radius of... The sampling sphere, and respectively with , For intervals along latitude angles Longitude angle The direction is used to divide the grid, and then the probe is used to... , The polarization direction is related to the formation The electric field data at the grid points were respectively... Multiple undersampling yields a dimension of of , Spherical near-field undersampled electric field data in polarization direction , ; (2) Perform cubic spline interpolation on the undersampled electric field data of the spherical near field: Near-field undersampled electric field data of sphere , conduct The cubic spline interpolation, multiplied by a factor of 1, yields a cubic spline interpolation with dimensions of 1. Spherical near-field electric field data , ,in , ; (3) Optimize the near-field electric field data of the spherical surface after cubic spline interpolation based on near-near field transformation: Near-field electric field data of a sphere after cubic spline interpolation , Perform near-near field transformation and use the spherical near-field undersampled electric field data , The spherical near-field electric field data obtained by near-near-field transformation is replaced, and multiple iterations are performed to obtain the optimized cubic spline interpolated spherical near-field electric field data. , ; (4) Calculate the far-field radiation pattern of the antenna: spherical near-field electric field data obtained through optimized cubic spline interpolation , Calculate the TE and TM mode expansion coefficients of spherical waves , and through , Calculate the far-field radiation pattern of the antenna .

2. The method according to claim 1, characterized in that, The radius of the sampling sphere mentioned in step (1) and sampling interval and satisfy: , , , , This represents the circumferential angle of the probe rotation during near-field electric field sampling on a spherical surface. Indicates the wavelength of electromagnetic waves. This represents the minimum spherical radius that surrounds the antenna under test.

3. The method according to claim 2, characterized in that, The latitude angle mentioned in step (1) Longitude angle The grid formed by directional division Grid points, among which and The calculation formulas are as follows: ； 。 4. The method according to claim 3, characterized in that, The optimization of the near-field electric field data of the sphere after cubic spline interpolation based on near-near-field transformation described in step (3) is achieved as follows: (3a) Initialize parameters: Initialize the number of iterations to The maximum number of iterations is , and order ; (3b) Near-field electric field data of the sphere after cubic spline interpolation , Perform a near-near field transformation to obtain the spherical near-field electric field data of the near-near field transformation in this iteration. , ; (3c) Obtain the near-field electric field data of the sphere after optimization by cubic spline interpolation: Using spherical near-field undersampled electric field data , Replace the same in this iteration , Near-field electric field data of the sphere obtained by near-near field transformation at grid coordinate points , and judge Is it true? If so, then... Replacement result of the next iteration , The near-field electric field data of the sphere is optimized from the electric field data of cubic spline interpolation; otherwise, let , , Then proceed with step (3b).

5. The method according to claim 4, characterized in that, The near-field electric field data of the sphere after cubic spline interpolation described in step (3b) , To perform a near-near field transformation, the steps are as follows: (3b1) Spherical near-field electric field data after cubic spline interpolation , Calculate the first The TE and TM mode expansion coefficients of the spherical wave of the antenna under test in the next iteration , , of which Line number Expanding coefficients of TE and TM patterns of columns , The calculation formulas are as follows: ； ； ； ； ； ； ； in, express index, , This represents the highest order of the spherical wave mode expansion coefficients. and , , This represents the wave number of the electromagnetic wave emitted by the antenna under test. , To indicate a variable, , express index, , This represents the maximum angular modulus of the spherical wave mode expansion coefficients. , express Hankel function of the second kind of sphere, Indicates to exist , Perform double integral operations on it. and They represent respectively to and Differential operations, The imaginary unit, This represents the base of the exponential function. express Step The first kind of associated Legendre function To represent factorial operation, Indicates to The derivative; (3b2) The TE and TM mode expansion coefficients of the spherical wave of the antenna under test in this iteration , No. Line number Columns , calculate and : ； 。 6. The method according to claim 5, characterized in that, The far-field radiation pattern of the antenna described in step (4) The calculation formula is: ； ； ； in, , They represent , The Middle Line number The expansion coefficients of the TE and TM patterns of the column. and They represent and The far-field pattern function of the component. and They represent and The unit vector of direction.

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