Correction of antenna far-field image error data based on spherical scanning

The antenna pattern is an important graph for measuring antenna performance, and various parameters of the antenna can be observed from the antenna pattern. The method of obtaining antenna far-field pattern through near-field measurement technology and near-to-far-field transformation algorithm has the advantages of being free from environmental interference, low cost, and fast speed, making it an effective research method. In the process of spherical near-field measurement, it is necessary to ensure the stable operation of the probe used for measurement. If one of the probe channels is abnormal, it will have a serious impact on the measurement of the antenna pattern. This article proposes an algorithm for repairing near-field measurement data through Newton interpolation based on spherical scanning. The feasibility of the proposed method is verified by using the HFSS simulation data and the measured data of the microwave anechoic chamber. It is proved that the algorithm proposed in this paper has obvious effect on improving the accuracy of antenna pattern.


Introduction
When designing antennas and modifying their performance, we often focus on the far-field characteristics of the antenna, so it becomes an important tool for exploring antenna performance.With the increasingly widespread application of high-performance multifunctional composite antennas, accurate measurement of antennas and drawing of directional patterns have become particularly important.Outdoor antenna far-field measurement has undergone years of development, and various technologies have become mature.However, when measuring outdoors, it is necessary to ensure that the distance between the tested antenna and the receiving antenna is large enough.On the other hand, outdoor environments such as weather, building obstruction, and electromagnetic interference can also cause interference to the measurement.Not only will confidentiality be challenged, but the measurement efficiency and accuracy of the antenna cannot be guaranteed.By utilizing accurate antenna data, the performance of the antenna can be improved and a more practical antenna can be designed.So for the measurement of large-sized antennas, near-field measurement is an effective and crucial method.It can transform near-field data to far-field data through algorithms.Due to the absence of distance factors for near-field testing and the influence of external environment, it has the advantages of high accuracy, safety, confidentiality, and can work 24/7.It can also conveniently simulate and control various electromagnetic environments.According to references [1] - [4], nearfield measurement can be divided into planar measurement, cylindrical measurement, and spherical measurement based on different scanning surfaces.Spherical measurement is widely used because it does not produce truncated data, as shown in Figure 1.

Figure 1. Schematic diagram of spherical scanning
There are three main theoretical foundations for spherical near-field measurement [5], namely the uniqueness theorem of electromagnetic fields, the mode expansion theory of electromagnetic waves, and the scattering matrix theory.Reference [6] introduces antenna near-to-far-field transformation based on scattering matrix theory.The scattering matrix theory of antennas was first proposed by Dick [7], and then it has been widely applied to the study of the radiation, scattering, and coupling characteristics of any lossless and irregularly shaped antenna.Literature [8] introduced spherical nearfield measurement based on electromagnetic wave mode expansion theory, which is the key to realizing spherical near-to-far-field transformation algorithm.In the 1990s, Chinese scientists proposed this transformation method [9], which expanded the field established by the antenna to be tested in space into the sum of spherical wave function, calculated the weighting coefficient through the orthogonal relationship between modes, and derived the near-to-far-field transformation formula.
With the development of antenna near-field measurement technology, in order to improve the accuracy and efficiency of scanning, multi probe measurement has been introduced into spherical near-field measurement.The first company to propose a multi probe spherical near-field measurement system was SATIMO in France [10], which is a major innovation in the implementation of spherical near-field measurement technology.The working principle of a multi probe system is still based on the basic spherical near-field measurement theory, but the scanning method is updated from mechanical scanning to electrical scanning, which greatly improves the scanning speed.However, multi probe measurement systems face an issue that cannot be ignored ---single or even multiple probes problems affect the accuracy of measurement.The probe used for antenna scanning needs to be calibrated to avoid the influence of the probe itself on the measurement data.However, in practical applications, there are often situations such as insufficient probe calibration and temporary failures of the probe.The measurement error caused by problem probes can cause the error to be amplified during the conversion process.In order to meet the accuracy of measurement, it is necessary to replace the probe and perform channel calibration again, but this also increases the time and expense costs.At present, the research on spherical scanning is mainly to improve the scanning speed, and most of the methods proposed for the wrong probe are calibration algorithms.But this paper is based on the correction of antenna far-field pattern data under the condition of saving time and cost as much as possible or when the conditions do not meet the calibration or even replacing the probe.This article will perform near-to-far-field transformation of horn antenna data based on spherical mode wave expansion method, and simulate the occurrence of probe failure by replacing some data in a set of data.The algorithm proposed in this article will be applied for correction to restore the correct far-field pattern.In order to verify the practical operability of the algorithm, the data measured in the microwave anechoic chamber will also be simulated and verified.The conclusion is drawn that this algorithm has rigor and feasibility in both theoretical derivation and practical operation.

Spherical near-far field transformation
In this section, a near-to-far-field transformation algorithm for simulation will be introduced, providing a basis for the antenna far field image correction algorithm introduced later.According to the previous text, we know that there are two commonly used algorithms for spherical near-to-far-field transformation measurement.Here, we will use the mode expansion theory of electromagnetic waves.The mode refers to the different modes of vibration of waves, which is the way they propagate.The propagation of electromagnetic waves is related to their amplitude and phase, so different amplitudes and phases correspond to different modes.The mode unfolding theory of electromagnetic waves mainly decomposes electromagnetic waves into the sum of different modes of motion composed of appropriate amplitudes and phases.If the coefficients of each mode are calculated, the expression for far-field electromagnetic waves can be obtained.

Theoretical derivation
Let r a  be the smallest sphere surrounding the antenna to be measured, then in the region of r a  , the vector wave equation for the passive region (homogeneous Helmholtz equation) is satisfied: (1) Introduce a coordinate function in a spherical coordinate system   r, , Where is the Bessel function of the third kind, also called Hankel function.
In order to represent the electric field of r a  , we define two vector functions that satisfy the passive field equation and make them all the solutions of the above wave equation: ( The function Where So linear combination can be used to represent the electric field in a region of r a  : When measuring and sampling, the commonly used probe is a linearly polarized probe, and the sampled data is generally the cross-sectional component of the antenna transmission direction.So the r  component in ( 7) can be omitted.To ensure accuracy, when calculating the far-field pattern, it is not assumed that r is infinite and some components are omitted.Instead, let f kr H kr  、 From this, it can be seen that after determining the mode coefficients and , we can obtain the far-field pattern from (8-a).
Assuming that regardless of the polarization direction of the measuring probe, the output voltage is proportional to the component of the electric field to be measured (for example, if the probe is polarized in the  , the output voltage ( , , ) V R    is proportional to the  component of the electric field), the tangential electric field distribution of the spherical surface can be determined by two types of probes with different orientations.Assuming the output voltage of the probe is: Considering the probe compensation for the test probe, when ignoring the multiple reflections between the probe and the antenna under test, we use differential operator and [11] to express the vector form of the probe's receiving field as: (10) In the above equation, the mismatch coefficients ( , , Using the orthogonality of mn M  and mn N   in equations ( 4), ( 5) and ( 6), we can obtain:

Simulation verification
To verify the correctness of the algorithm, a horn antenna operating at 2.4GHz frequency is simulated using the three-dimensional electromagnetic simulation software HFSS.The simulated antenna nearfield data (with a near-field distance of 4  in this article) is used as near-field measurement data, then put it into a near-to-far-field transformation program written in MATLAB to obtain the normalized far-field directional pattern of the antenna.The accuracy of the algorithm is verified by comparing it with the directional pattern obtained from HFSS simulation.
This article designs and establishes a horn antenna, with the antenna dimensions shown in Table 1 and the antenna model shown in Figure 2. The antenna adopts a WR430 (national standard model BJ22) waveguide.The feed point is at the center of the wide side of the waveguide.The distance from the antenna to the short circuit board is a quarter of the wavelength.The antenna is fed by a coaxial line with a characteristic impedance of 50 Ohms.The operating frequency is around 2.4GHZ.The sampling interval of HFSS in the  direction is 1 °, and the sampling interval in the  direction is 3.6 °.Compare the directional map drawn using the far-field analytical formula with the far-field map obtained using HFSS sampling.Here we take the far-field diagram of , as shown in Figure 3.It can be seen that the far-field pattern of the antenna calculated by the pattern expansion method highly overlaps with the ideal far-field pattern simulated by HFSS, indicating the effectiveness of the transformation algorithm.

Far field image recovery
Previously, we mentioned the commonly used scanning method -multi probe spherical scanning, and introduced the basic algorithm for near-to-far-field transformation of spherical surfaces.When performing multi probe spherical scanning, due to the increase in the number of probes, there are often situations where a certain probe is not calibrated or the probe itself is damaged, which undoubtedly poses a challenge to the accuracy of measurement data.Figure 4 applies the transformation algorithm mentioned above to simulate incorrect near-field data, and it can be seen that if the probe obtains incorrect near-field data, the far-field data obtained using it will also have problems.If the measurement data can be processed and corrected, there is no need to calibrate or replace the probe while ensuring the accuracy of the far-field image, which saves time and cost.The directional pattern of the antenna has regularity, and there is also a strong correlation between adjacent data.Therefore, this feature can be utilized to detect problematic data in a timely manner by observing measurement data, repair abnormal data, and then process it through near-to-far-field transformation algorithms to obtain accurate far-field directional maps.The repair process will use mathematical algorithms, and polynomial interpolation is a method of using algebraic interpolation to approximate any function, obtaining the approximate form of a function for nodes with known positions but unknown specific functions.The commonly used interpolation formulas include Lagrange interpolation, Newton interpolation, Emily interpolation, and spline interpolation.

Positioning tables
Newton interpolation introduces the concept of difference quotient, making it easier to calculate when interpolation nodes are added.Compared to cubic spline interpolation [12], the Newton interpolation method can perform data correction when multiple continuous probes have problems.
Based on reference [13], we define the zero order difference quotient of   f x at i x as   i f x , and the first order difference quotient of   f x at i x and j x as: The k-order difference quotient of   x is: , ,..., , ,..., , ,..., x as a combination of function values The Newton interpolation formula can be obtained by separating the deformations of the difference quotients of various orders of   Specifically, when the spacing is taken as h , the equidistant Newton interpolation formula can be obtained: For the Newton interpolation method, the calculation and processing of the difference quotient are the most important.When reproducing its function in MATLAB coding, the main idea is to first construct the difference quotient table, and then calculate the interpolated polynomial.When inputting discrete data, calculate the fitting function value at the interpolation point based on the polynomial.

Positioning tables
After theoretical analysis, we found that the Newton interpolation method is suitable for situations where only discrete data points can be measured (such as when a continuous probe malfunctions, only the remaining probe data can be obtained) to fit the discrete points and obtain more accurate analytical values of the function.In this section, we will conduct simulation to verify the accuracy of the algorithm.We will use HFSS to obtain near-field ( =4 L  ) data of the antenna, with scanning intervals are 1     and 3.6     .Assuming that there is a problem with the probe at =35   , it will result in errors in several nearby data due to probe failures.
According to equation (12), the sampling data of the probe on the antenna are and , which respectively contain amplitude and phase information.To ensure the accuracy of the simulation results, we interpolate the amplitude and phase of the gain in the  and  directions respectively, and compare them with the ideal data, as shown in Table 2,3,4 and 5.In order to present it more clearly, we plotted using table data, as shown in Figure 5, Figure 6, Figure 7, and Figure 8.It can be clearly seen from the data in images and table that for the errors of multiple continuous data, this algorithm can be used to correct them.The comparison of direction map obtained from the corrected data through near-to-far-field transformation、ideal direction map and incorrect direction map is shown in Figure 9.It can be seen that the corrected data can smoothly restore the far-field image, and it basically fits the ideal far-field image, verifying the effectiveness of the algorithm.( , )

Microwave anechoic chamber measurement
To verify the practical operation of the algorithm, we conducted measurement verification using a microwave anechoic chamber.The SATIMO microwave anechoic chamber, as shown in Figure 10, can provide various passive and active tests, with a testing frequency range ranging from 400MHz to 6GHz.SATIMO has more stable working performance, faster testing speed, and more accurate measurement compared to general darkrooms.In addition to traditional antennas, it also has the function of testing WIFI 2.4Ghz/5.8Ghzand OTA, so it has been widely used.The schematic diagram of near-field measurement is shown in Figure 11.The standard horn is fixed on a two-dimensional turntable, and the turntable pitch rotation enables the horn to align with various probes.The turntable tooling needs to be specially designed so that when the horn is fixed on the turntable, the distance from the center of the horn aperture surface to each probe is equal.In this case, we use a microwave anechoic chamber with 23 probes.
To verify the practicality of the algorithm, we will conduct near-field testing of the 2.4GHz horn antenna in a microwave anechoic chamber, simulating the problem with the probe at 30 ° and 60 °, resulting in errors in the far-field antenna pattern, as shown in Figure 12.Next, increase the sampling rate and simulate the situation where one of the probes malfunctioned, resulting in three sets of inaccurate data, as shown in Figure 13.Data and Correct Data When we find obvious dents or protrusions in the far-field image of the antenna, we first consider whether the data measured by the probe is inaccurate.Then, using the algorithm introduced earlier, we take other correct probe sampling data for processing, and try to recover the correct data at the damaged probe as much as possible.In Tables 6 and 7, the comparison of corrected data and correct data clearly shows that the correction algorithm is effective for missing or erroneous data from multiple probes, with an error of less than 5%, which verifies the actual operability of the algorithm.Therefore, we can determine that this method is helpful for repairing and restoring the correct far-field data in the actual antenna measurement.At the same time, it avoids the time cost of replacing probes and rebuilding the test environment.

Conclusion
This article uses polynomial interpolation method to first conduct simulation testing: process the measurement data of abnormal probes to achieve effective repair of abnormal probe data.The repaired data after near-to-far-field transformation can match the theoretical value.By comparing the directional maps obtained from near-far field transformation of unrepaired data with the theoretical directional maps, it can be seen that the erroneous data brings huge errors, indicating that this algorithm has strong practicality and necessity.Subsequently, simulation verification was conducted using measured data from the darkroom, and the recovered probe data had a smaller error compared to the correct data.This verified the effectiveness of this algorithm in correcting antenna far-field image data.


the tangential expression of the electromagnetic field is obtained: (8-a) (8-b)


represent the reflection coefficient of the measuring antenna, and L  represent the reflection coefficient of the measuring probe.Substitute equations (8-a) and (8-b) to obtain:

Figure 3 .
Figure 3. Calculated value and theoretical value at

Figure 4 .
Figure 4. Theoretical FF data and calculated values by abnormal data

Figure 5 .Figure 7 .
Figure 5. Theoretical NF data and repaired data Figure 6.Theoretical NF data and repaired data NearEPhi_ am p litud e [V_ p er_ m eter]Th eta An g el(d eg ree)

Figure 9 .
Figure 9. Theoretical FF data, calculated value by abnormal data and repaired value

Figure 12 .
Figure 12.Comparison of Simulated Error Figure 13.Comparison of Simulated Error Data and Correct DataData and Correct Data When we find obvious dents or protrusions in the far-field image of the antenna, we first consider whether the data measured by the probe is inaccurate.Then, using the algorithm introduced earlier, we take other correct probe sampling data for processing, and try to recover the correct data at the damaged probe as much as possible.In Tables6 and 7, the comparison of corrected data and correct data clearly shows that the correction algorithm is effective for missing or erroneous data from multiple probes, with an error of less than 5%, which verifies the actual operability of the algorithm.Therefore, we can determine that this method is helpful for repairing and restoring the correct far-field data in the actual antenna measurement.At the same time, it avoids the time cost of replacing probes and rebuilding the test environment.

Table 2 .
Amplitude value of .

Table 3 .
Phase value of .

Table 4 .
Amplitude value of .

Table 5 .
Phase value of .

Table 6 .
Comparison of measured and corrected data.

Table 7 .
Comparison of measured and corrected data.