A Phase-Free Plane Near-Field Measurement Method Based on the Conjugate Gradient Method

Abstract

This invention discloses a phase-free planar near-field measurement method based on the conjugate gradient method. The steps include: performing planar near-field measurements on a two-dimensional narrow-beam antenna; acquiring the amplitude information of the near-field electric field on two planes; iteratively recovering the near-field phase of the two-dimensional narrow-beam antenna using the electric field on the two measurement planes; generating global error parameters; iteratively recovering the phase of the near-field electric field on the two planes; compensating for the propagation electric field using the conjugate gradient method; and terminating the iteration when the convergence condition is met. This invention has the advantages of fast convergence and high operating efficiency, making it suitable for engineering implementation.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, and more specifically relates to a phase-free planar near-field antenna measurement method based on the conjugate gradient method within the field of antenna radiation pattern technology. This invention can be used to measure the amplitude and phase information of an antenna to obtain the radiation pattern of a millimeter-wave antenna. Background Technology

[0002] Traditional millimeter-wave antenna pattern measurement involves measuring the antenna's amplitude and phase information, and then using a near-field to far-field transformation algorithm to reconstruct the far-field pattern. However, for the millimeter-wave band, due to probe positioning errors and the accuracy of the vector network analyzer, the antenna measurement system cannot obtain accurate phase information. Therefore, a phase retrieval algorithm is needed to recover the phase information of the antenna under test. Currently, there are two main types of phase retrieval algorithms.

[0003] The first method is a phase-free near-field antenna measurement method based on an iterative Fourier transform algorithm. For example, Xi'an University of Electronic Science and Technology disclosed a phase retrieval method based on iterative Fourier transform in its patent application "A Phase-Free Near-Field Antenna Measurement Method Based on Iterative Fourier Transform Algorithm" (Application No. 201910632593.6, Publication No. CN 110470914 A). The implementation steps of this method are as follows: Iterating between the two planes using amplitude information, and applying amplitude constraints on each plane. An error functional is minimized using an error attenuation algorithm. The iteration terminates when the error reaches a convergence value. The recovered phase information is combined with the measured amplitude information to obtain a complex field, and then a plane near-field to far-field transform algorithm is used to obtain the far-field radiation pattern. However, this method still has a drawback: since phase retrieval is a non-convex problem, it may converge to a local minimum after applying amplitude constraints, thus converging to a false solution. This results in low accuracy of the recovered phase pattern. Furthermore, since no optimization algorithm is used to determine the optimal direction after each iteration, the convergence time is very long and the number of iterations is very large, which prevents this method from being widely used in practical engineering applications.

[0004] The second method employs phase retrieval techniques using the Iterative Fourier Transform Algorithm (IFTA) and the Differential Evolution Algorithm (DEA), for example:

[0005] In his paper "Phase-Free Near-Field Measurement Technology for Scanning Beam Antennas" (Journal of Radio Science, Xi'an University of Electronic Science and Technology, 2016), Zuo Yanchun proposed a phase-free measurement method for scanning beam antennas. This method uses DEA to generate an initial iterative phase, combines the initial phase with the measured amplitude information, and uses IFTA to reconstruct the phase of the scanning beam. Although this method can recover the antenna's phase information, it still has shortcomings. Because the method uses the DEA algorithm to generate the initial phase, the DEA algorithm has a long running time. Furthermore, due to the low robustness of IFTA, even though the DEA algorithm is used to avoid local minima, the algorithm may still fail to converge to the desired result during the iteration process due to the lack of a search algorithm. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of the prior art by proposing a phase-free planar near-field antenna measurement method based on the conjugate gradient method. This method is used to solve the problems of inaccurate phase measurement caused by probe positioning errors in millimeter-wave antenna measurement, and the accuracy problems caused by local minima in the phase recovery algorithm. Since the iterative Fourier algorithm iterates on two planes without using a search algorithm, it results in excessive iterations and excessive time consumption.

[0007] The purpose and idea of ​​this invention is to address the accuracy problem caused by local minima in phase retrieval algorithms. This invention employs the conjugate gradient method. The uniqueness of the solution to the phase problem and the quadraticity of the B operator provide a very useful property: each segment in the unknown space maps to a parabolic arc in the measurement data. Therefore, as long as there are no errors in the measured amplitude information, and provided that a sufficient number of functionals are chosen and the measurement data is correct, the functional will not have local minima.

[0008] The implementation steps of this invention are as follows:

[0009] Step 1: Perform planar near-field measurements on the two-dimensional narrow-beam antenna;

[0010] Step 2: Iterate the electric field on two measurement planes to recover the near-field phase of the two-dimensional narrow-beam antenna:

[0011] Step 2.1: Use the random function to generate a phase matrix on the measurement surface. The size of this matrix is ​​N. s ×M s The elements of this matrix are randomly selected within the range of [-180°, 180°]; the matrix amplitude is used as the modulus, and the random phase matrix is ​​used as the phase angle to form an iterative field. The iterative field on the first plane If the number of rows and columns of a matrix are each increased by a factor of 4, the size of the expanded matrix will be 4N. s×4M s The filling matrix is ​​filled with zeros, and then a fast Fourier transform is performed on the matrix to obtain the first plane spectrum of the antenna under test.

[0012] Step 2.2: Multiply the spectrum of the first plane by the phase shift factor. Obtain the plane spectrum on the second plane Among them, e (·) This indicates an exponential operation with the natural constant e as the base, j represents the imaginary unit sign, ω represents the angular velocity of the electromagnetic wave, and k represents the angular velocity of the electromagnetic wave. x k represents the wavenumber component of the electromagnetic field along the x-direction. y This represents the wavenumber component of the electromagnetic field along the y-direction;

[0013] Step 3, Generate the reconstructed electric field matrix:

[0014] Plane spectrum on the second plane Performing the inverse Fourier transform yields the propagating electric field matrix on the second plane. The electric field matrix propagating on the second plane Perform amplitude constraint operation, that is, discard the electric field matrix propagating on the second plane. The amplitude components are used to obtain the propagation electric field matrix. Phase components, propagation electric field matrix The phase components and the measured electric field matrix on the second plane The amplitudes constitute the reconstructed electric field matrix;

[0015] Step 4, generate global error parameters:

[0016] Step 4.1: Calculate the root mean square values ​​of the measurement matrix and the iterative field according to the following formula to obtain the global error parameters:

[0017]

[0018] Where, φ (k) (E) represents the root mean square value of the measurement matrix and the iteration field at the k-th iteration, ||·|| 2 To represent a quadratic operator, The symbol |·| represents the real part of a complex number, and |·| represents the absolute value operation. Indicates the k-th measurement surface S k The iterative field obtained by the upsampled electric field and the iterative Fourier algorithm, k = 1, 2, Let T represent the measurement amplitude matrix of the k-th measurement plane of the two-dimensional narrow-beam antenna, where k = 1, 2, and T represents the plane spectrum propagation operator. e denotes an exponential operation with the natural constant e as the base. This represents the Fourier operator. This represents the inverse Fourier operator. The square distribution is chosen to filter out interference outside the bandwidth being considered.

[0019] Step 4.2, replace the error parameters with the reconstructed electric field matrix. Update the global error parameter φ(E);

[0020] Step 5: Determine whether the root mean square value ε at the current iteration is less than or equal to -30dB. If yes, proceed to step 7; otherwise, proceed to step 6.

[0021] Step 6: Iterate the electric field using the gradient of the error parameters;

[0022] Step 6.1, calculate the gradient of the global cost function according to the following formula.

[0023]

[0024] Among them, T + The adjoint matrix represents the plane wave spectrum propagation operator.

[0025] Step 6.2: Compensate for the propagating electric field using the conjugate gradient method:

[0026] Step 6.2.1, calculate the optimized direction of the electric field for each iteration according to the following formula:

[0027]

[0028] Where, d (k+1) d represents the direction of electric field optimization in the (k+1)th iteration. (k) β represents the direction of electric field optimization in the k-th iteration. (k) This represents a scalar parameter; the larger the value, the stronger the effect of the previous descent direction. Its calculation formula is as follows:

[0029]

[0030] in, This represents the global error parameter after the k-th iteration. Let φ represent the global error parameter after the (k+1)th iteration. (k-1) This represents the global error parameter after the (k-1)th iteration;

[0031] Step 6.2.2: Compensate the electric field after each iteration according to the following formula:

[0032] E (k+1) =E (k) +α (k) d (k)

[0033] Among them, E (k+1) Let E represent the iterative electric field after compensation in the (k+1)th iteration. (k) Let represent the iterative electric field after the k-th iteration compensation, and α represent the step size parameter;

[0034] Step 7: Determine whether the global error parameter value of the current iteration is less than or equal to -30dB. If yes, proceed to step 8; otherwise, proceed to step 3.

[0035] Step 8: Calculate the element values ​​in the far-field radiation pattern corresponding to each coordinate point in the two-dimensional narrow-beam antenna spherical coordinate system according to the following formula:

[0036]

[0037]

[0038] in, In the spherical coordinate system representing a two-dimensional narrow-beam antenna The element values ​​in the far-field pattern corresponding to the coordinate point in the θ direction. Represents a two-dimensional narrow-beam antenna in spherical coordinates. Corresponding coordinates Element values ​​in the far-field radiation pattern of the direction; A x Let A represent the plane spectral components of a two-dimensional narrow-beam antenna along the x-direction in spherical coordinates. y This represents the plane spectral component of a two-dimensional narrow-beam antenna along the y-direction in spherical coordinates.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] First, because this invention selects a globally minimizing functional and solves for the minimization direction after each iteration to recover the phase of the antenna measurement, it overcomes the accuracy problem caused by local minima in the phase recovery algorithm of existing phase-free near-field antenna measurement methods that cannot overcome the problem of phase recovery algorithms due to local minima. This makes the invention highly accurate and improves the accuracy of pattern recovery when applied to antenna far-field pattern calculation.

[0041] Secondly, because this invention employs the conjugate gradient method, it compensates for the phase of the near-field electric field after each iteration based on the minimum direction obtained from the solution. This overcomes the problem of long execution time of the phase recovery algorithm (DEA) in existing phase-free measurement methods for scanning beam antennas. This results in the invention having the characteristics of fast convergence and high operating efficiency. Attached image description:

[0042] Figure 1 This is a flowchart of the present invention;

[0043] Figure 2This is a schematic diagram of the measurement used in the simulation experiment of this invention;

[0044] Figure 3 The simulation experiments of this invention use the E-plane and H-plane radiation patterns recovered from the phase-free plane near-field measurement method based on conjugate gradients.

[0045] Figure 4 This is the convergence curve of the global error parameters of the present invention;

[0046] Figure 5 This is a comparison diagram of the recovered phase and the actual phase of the present invention. Figure 5 (a) is the recovered phase. Figure 5 (b) is the actual phase. Detailed Implementation

[0047] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0048] Reference Figure 1 The implementation steps of the embodiments of the present invention will be further described below.

[0049] Step 1: Perform planar near-field measurements on the two-dimensional narrow-beam antenna:

[0050] Step 1.1: Select two measurement surfaces within a range of [3,10] wavelengths from the aperture of the two-dimensional narrow-beam antenna. The first measurement surface is S1, and the second measurement surface is S2. d1 and d2 are the distances between measurement surfaces S1 and S2 and the antenna aperture, respectively, and d3 is the distance between d1 and d2. Generally, 2 to 5 wavelengths are selected. In this embodiment, d1 = 4λ and d2 = 6λ, where λ is the wavelength of the two-dimensional narrow-beam antenna. Perform a pre-scan along the x-direction on the two measurement surfaces S1 and S2 to obtain the level difference between the center point of the sampling surface and the edge point of the sampling surface, thus determining the width L of the scanning surface. x In an embodiment of the present invention, a pre-scan is performed along the x-direction on two measurement surfaces S1 and S2 to obtain a level difference of -40dB between the level at the center point of the sampling surface and the level at the edge point of the sampling surface, thereby determining the width L of the scanning surface. x .

[0051] Step 1.2: Using the following formula, calculate the maximum reliable angular domain θ of the two-dimensional narrow-beam antenna on the first measurement surface S1. x :

[0052]

[0053] Where arctg(·) represents the arctangent function, d iThe distance i = 1, 2 represents the distance between the i-th measurement plane and the scanning plane of the two-dimensional narrow beam antenna. a represents the length of the aperture of the two-dimensional narrow beam antenna, and b represents the width of the aperture of the two-dimensional narrow beam antenna. In the embodiment of the present invention, since the length of the sampling plane is equal to the width, only one pre-scan is needed for the two measurement planes S1.

[0054] Amplitude measurements were performed on the two-dimensional narrow-beam antenna at two sampling planes, with the measurement points spaced at λ / 4. The amplitudes measured at all sampling points were then combined into an N-value. s ×M s The magnitude matrix, N s M represents the number of sampling points in the x-direction. s This represents the number of sampling points in the y-direction. In embodiments of the present invention, M... s =N s =128.

[0055] Step 2: Iterate the electric field on the two measurement planes to recover the near-field phase of the two-dimensional narrow-beam antenna.

[0056] Step 2.1: Use the `random` function to generate a phase matrix on the measurement surface. This matrix has a size of 128×128, and its elements are randomly selected values ​​within the range [-180°, 180°]. Use the matrix amplitude as the magnitude and the random phase matrix as the phase angle to form the iterative field. To improve the resolution of the recovered phase, the iterative field on the first plane is first... The number of rows and columns of the matrix is ​​increased by a factor of 4, resulting in a matrix size of 512×512. The padded matrix is ​​then filled with zeros. Finally, a fast Fourier transform is performed on the matrix to obtain the first plane spectrum of the antenna under test.

[0057] Step 3: Propagate the planar spectrum of the first plane to the second plane to obtain the reconstructed electric field matrix.

[0058] Multiply the spectrum of the first plane by the phase shift factor Obtain the plane spectrum on the second plane Among them, e (·) This indicates an exponential operation with the natural constant e as the base, j represents the imaginary unit sign, ω represents the angular velocity of the electromagnetic wave, and k represents the angular velocity of the electromagnetic wave. x k represents the wavenumber component of the electromagnetic field along the x-direction. y This represents the wavenumber component of the electromagnetic field along the y-direction.

[0059] Plane spectrum on the second plane Performing the inverse Fourier transform yields the propagating electric field matrix on the second plane. The electric field matrix propagating on the second plane Perform amplitude constraint operation, that is, discard the electric field matrix propagating on the second plane. The amplitude components are used to obtain the propagation electric field matrix. Phase components, propagation electric field matrix The phase components and the measured electric field matrix on the second plane The amplitudes form the reconstructed electric field matrix on the second plane.

[0060] Step 4: Propagate the reconstructed electric field matrix of the second plane to the first plane.

[0061] The second-plane spectrum of the antenna under test is obtained by performing a fast Fourier transform on the reconstructed electric field matrix in the second plane. Using the spectrum of the second plane Multiply by phase shift factor Obtain the plane spectrum on the first plane Plane spectrum on the first plane Performing the inverse Fourier transform yields the electric field matrix propagating on the first plane. The electric field matrix propagating on the second plane Perform amplitude constraint operation, that is, discard the electric field matrix propagating on the first plane. The amplitude components are used to obtain the propagation electric field matrix. Phase components, propagation electric field matrix The phase components and the measured electric field matrix on the first plane The amplitudes form the reconstructed electric field matrix on the first plane.

[0062] Step 5: Generate global error parameters.

[0063] Step 5.1: Calculate the root mean square values ​​of the measurement matrix and the iterative field according to the following formula to obtain the global error parameters:

[0064]

[0065] Where, φ (k) (E) represents the root mean square value of the measurement matrix and the iteration field at the k-th iteration, ||·|| 2 To represent a quadratic operator, The symbol |·| represents the real part of a complex number, and |·| represents the absolute value operation. Indicates the k-th measurement surface S k The iterative field obtained by the upsampled electric field and the iterative Fourier algorithm, k = 1, 2, Let T represent the measurement amplitude matrix of the k-th measurement plane of the two-dimensional narrow-beam antenna, where k = 1, 2, and T represents the plane spectrum propagation operator. e denotes an exponential operation with the natural constant e as the base. This represents the Fourier operator. This represents the inverse Fourier operator. The square distribution is chosen to filter out interference beyond the bandwidth being considered.

[0066] Step 5.2, replace the error parameters with the reconstructed electric field matrix. Update the global error parameter φ(E).

[0067] Step 6: Determine whether the root mean square value ε at the current iteration is less than or equal to -30dB. If yes, proceed to step 7; otherwise, proceed to step 6.

[0068] Step 7: Iterate the electric field using the gradient of the error parameters.

[0069] Step 7.1: Calculate the gradient of the global cost function according to the following formula.

[0070]

[0071] Among them, T + The adjoint matrix represents the plane wave spectrum propagation operator.

[0072] Step 7.2: Compensate the propagating electric field using the conjugate gradient method.

[0073] Step 7.2.1, calculate the optimized direction of the electric field for each iteration according to the following formula:

[0074]

[0075] Where, d (k+1) d represents the direction of electric field optimization in the (k+1)th iteration. (k) β represents the direction of electric field optimization in the k-th iteration. (k) This represents a scalar parameter; the larger the value, the greater the influence of the previous descent direction. Its calculation formula is as follows.

[0076]

[0077] in, This represents the global error parameter after the k-th iteration. This represents the global error parameter after the (k+1)th iteration. This represents the global error parameter after the (k-1)th iteration.

[0078] Step 7.2.2: Compensate the electric field after each iteration according to the following formula:

[0079] E (k+1) =E (k) +α (k) d (k)

[0080] Among them, E (k+1) Let E represent the iterative electric field after compensation in the (k+1)th iteration. (k) Let represent the iterative electric field after the k-th iteration compensation, and α represent the step size parameter. In this embodiment of the invention, α = 0.1.

[0081] Step 8: Determine whether the global error parameter value of the current iteration is less than or equal to -30dB. If yes, proceed to step 8; otherwise, proceed to step 3.

[0082] Step 9: Calculate the element values ​​in the far-field radiation pattern corresponding to each coordinate point in the spherical coordinate system of the two-dimensional narrow-beam antenna according to the following formula:

[0083]

[0084]

[0085] in, In the spherical coordinate system representing a two-dimensional narrow-beam antenna The element values ​​in the far-field pattern corresponding to the coordinate point in the θ direction. Represents a two-dimensional narrow-beam antenna in spherical coordinates. Corresponding coordinates The element values ​​in the far-field radiation pattern of the direction. In this embodiment of the invention, along θ in the two-dimensional narrow-beam antenna spherical coordinate system, respectively, 256 coordinate points were selected for each direction at equal intervals. A x Let A represent the plane spectral components of a two-dimensional narrow-beam antenna along the x-direction in spherical coordinates. y This represents the plane spectral component of a two-dimensional narrow-beam antenna along the y-direction in spherical coordinates.

[0086] The effects of this invention will be further illustrated below with simulation experiments:

[0087] 1. Simulation experimental conditions:

[0088] The hardware platform for the simulation experiment of this invention is: AMD Ryzen 7 5800H processor with a main frequency of 3.5GHz and 16GB of memory.

[0089] The software platform for the simulation experiment of this invention is: Windows 10 operating system and Matlab R2022a.

[0090] 2. Simulation content and result analysis:

[0091] In the simulation experiment of this invention, a two-dimensional narrow beam antenna is adopted with a symmetrical dipole array, and the near-field phase of the two-dimensional narrow beam antenna is recovered by a phase recovery algorithm based on conjugate gradient.

[0092] The parameter settings of the symmetrical oscillator array used in the simulation experiment of this invention are shown in Table 1.

[0093] Table 1. Parameter settings for symmetrical oscillator arrays

[0094]

[0095] The simulation experiment of this invention uses the method of this invention to sample the amplitude of the electric field in the two parallel planes in step 1. Figure (1) is a flowchart based on the conjugate gradient method. Figure 2 (a) is a layout diagram of a two-dimensional narrow-beam antenna and two parallel planes. Figure 2 (b) shows the symmetrical oscillator array model. The simulation results are as follows: Figure 3 As shown. Figure 3 (a) A comparison diagram showing the theoretical E-plane far-field pattern and the E-plane far-field pattern obtained through the phase retrieval algorithm, from... Figure 3 As shown in (a), the E-plane far-field patterns of the two methods basically overlap at the antenna main lobe. After the 10th sidelobe, the H-plane far-field pattern recovered using the iterative Fourier algorithm based on the conjugate gradient algorithm has a slightly higher level than the theoretical H-plane far-field pattern, with a maximum deviation of 4dB. Overall, the pattern recovery is good. The E-plane far-field pattern is recovered using the iterative Fourier method based on the conjugate gradient. Figure 3 (b) shows a comparison between the theoretical H-plane far-field pattern and the H-plane far-field pattern obtained through near-far-field transformation. From Figure 3 As can be seen in (b), the H-plane far-field radiation patterns of the two methods basically overlap within the range of ±45°, with a deviation of about 4° after the fourth sidelobe, and the overall radiation pattern is well recovered. Figure 4 The iterative curve represents the error parameter. It can be seen that the iterative algorithm converges after 1600 iterations. Figure 5 (a) represents the theoretical near-field phase. Figure 5 (b) represents the recovered near-field phase.

Claims

1. A phase-free planar near-field measurement method based on the conjugate gradient method, characterized in that, For the near-field amplitude measurement of a two-dimensional narrow-beam antenna on two planes, an iterative Fourier method based on conjugate gradients is used to iterate on both planes to minimize an error parameter. Once the error parameter converges, the far-field radiation pattern of the antenna is recovered using the recovered iterative field. The specific steps of this method are as follows: Step 1: Perform planar near-field measurements on the two-dimensional narrow-beam antenna; Step 2: Iterate the electric field on two measurement planes to recover the near-field phase of the two-dimensional narrow-beam antenna: Step 2.1: Use the random function to generate a phase matrix on the measurement surface. The size of this matrix is ​​N. s ×M s The elements of this matrix are randomly selected within the range of [-180°, 180°]; the matrix amplitude is used as the modulus, and the random phase matrix is ​​used as the phase angle to form an iterative field. The iterative field on the first plane If the number of rows and columns of a matrix are each increased by a factor of 4, the size of the expanded matrix will be 4N. s ×4M s The filling matrix is ​​filled with zeros, and then a fast Fourier transform is performed on the matrix to obtain the first plane spectrum of the antenna under test. Step 2.2: Multiply the spectrum of the first plane by the phase shift factor. Obtain the plane spectrum on the second plane Among them, e (·) This indicates an exponential operation with the natural constant e as the base, j represents the imaginary unit sign, ω represents the angular velocity of the electromagnetic wave, and k represents the angular velocity of the electromagnetic wave. x k represents the wavenumber component of the electromagnetic field along the x-direction. y This represents the wavenumber component of the electromagnetic field along the y-direction; Step 3, Generate the reconstructed electric field matrix: Plane spectrum on the second plane Performing the inverse Fourier transform yields the propagating electric field matrix on the second plane. The electric field matrix propagating on the second plane Perform amplitude constraint operation, that is, discard the electric field matrix propagating on the second plane. The amplitude components are used to obtain the propagation electric field matrix. Phase components, propagation electric field matrix The phase components and the measured electric field matrix on the second plane The amplitudes constitute the reconstructed electric field matrix; Step 4: Propagate the reconstructed electric field matrix of the second plane to the first plane: The second-plane spectrum of the antenna under test is obtained by performing a fast Fourier transform on the reconstructed electric field matrix in the second plane. Using the spectrum of the second plane Multiply by phase shift factor Obtain the plane spectrum on the first plane Plane spectrum on the first plane Performing the inverse Fourier transform yields the electric field matrix propagating on the first plane. The electric field matrix propagating on the second plane Perform amplitude constraint operation, that is, discard the electric field matrix propagating on the first plane. The amplitude components are used to obtain the propagation electric field matrix. Phase components, propagation electric field matrix The phase components and the measured electric field matrix on the first plane The amplitudes form the reconstructed electric field matrix on the first plane; Step 5, generate global error parameters: Step 5.1: Calculate the root mean square values ​​of the measurement matrix and the iterative field according to the following formula to obtain the global error parameters: Where, φ (k) (E) represents the root mean square value of the measurement matrix and the iteration field at the k-th iteration, ||·|| 2 To represent a quadratic operator, The symbol |·| represents the real part of a complex number, and |·| represents the absolute value operation. Indicates the k-th measurement surface S k The iterative field obtained by the upsampled electric field and the iterative Fourier algorithm, k = 1, 2, Let T represent the measurement amplitude matrix of the k-th measurement plane of the two-dimensional narrow-beam antenna, where k = 1, 2, and T represents the plane spectrum propagation operator. e denotes an exponential operation with the natural constant e as the base. This represents the Fourier operator. This represents the inverse Fourier operator. The square distribution is chosen to filter out interference outside the bandwidth being considered. Step 5.2, replace the error parameters with the reconstructed electric field matrix. Update the global error parameter φ(E); Step 6: Determine whether the root mean square value ε at the current iteration is less than or equal to -30dB. If yes, proceed to step 7; otherwise, proceed to step 6. Step 7: Iterate the electric field using the gradient of the error parameters; Step 7.1: Calculate the gradient of the global cost function according to the following formula. Among them, T + The adjoint matrix represents the plane wave spectrum propagation operator. Step 7.2: Compensate for the propagating electric field using the conjugate gradient method: Step 7.2.1, calculate the optimized direction of the electric field for each iteration according to the following formula: Where, d (k+1) d represents the direction of electric field optimization in the (k+1)th iteration. (k) β represents the direction of electric field optimization in the k-th iteration. (k) This represents a scalar parameter; the larger the value, the stronger the effect of the previous descent direction. Its calculation formula is as follows: in, This represents the global error parameter after the k-th iteration. This represents the global error parameter after the (k+1)th iteration. This represents the global error parameter after the (k-1)th iteration; Step 7.2.2: Compensate the electric field after each iteration according to the following formula: AND (k+1) =E (k) +α (k) d (k) Among them, E (k+1) Let E represent the iterative electric field after compensation in the (k+1)th iteration. (k) Let represent the iterative electric field after the k-th iteration compensation, and α represent the step size parameter; Step 8: Determine whether the global error parameter value of the current iteration is less than or equal to -30dB. If so, proceed to step 8; otherwise, propagate it to the first plane using the above method and then perform amplitude constraint operation. Perform step 3; Step 9: Calculate the element values ​​in the far-field radiation pattern corresponding to each coordinate point in the spherical coordinate system of the two-dimensional narrow-beam antenna according to the following formula: in, In the spherical coordinate system representing a two-dimensional narrow-beam antenna The element values ​​in the far-field pattern corresponding to the coordinate point in the θ direction. Represents a two-dimensional narrow-beam antenna in spherical coordinates. Corresponding coordinates Element values ​​in the far-field radiation pattern of the direction; A x Let A represent the plane spectral components of a two-dimensional narrow-beam antenna along the x-direction in spherical coordinates. y This represents the plane spectral component of a two-dimensional narrow-beam antenna along the y-direction in spherical coordinates.

2. The phase-free planar near-field measurement method based on the conjugate gradient method according to claim 1, characterized in that, The steps for performing planar near-field measurements on the two-dimensional narrow-beam antenna described in step 1 are as follows: The first step is to select two measurement surfaces within a range of [3,10] wavelengths from the aperture of the two-dimensional narrow-beam antenna: the first measurement surface S1 and the second measurement surface S2. d1 and d2 are the distances between measurement surfaces S1 and S2 and the antenna aperture, respectively, and d3 is the distance between d1 and d2. Generally, 2 to 5 wavelengths are selected; λ is the wavelength of the two-dimensional narrow-beam antenna. A pre-scan is performed along the x-direction on the two measurement surfaces S1 and S2 to obtain the level difference between the center point and the edge point of the sampling surface, thus determining the width L of the scanning surface. x The voltage level difference between the center point and the edge point of the sampling surface was found to be -40dB, thus determining the width L of the scanning surface. x ; The second step is to use the following formula to calculate the maximum reliable angular domain θ of the two-dimensional narrow-beam antenna on the first measurement surface S1. x : Where arctg(·) represents the arctangent function, d i Let represent the distance (i=1,2) between the i-th measurement plane and the scanning plane of the two-dimensional narrow-beam antenna, where 'a' represents the length of the antenna aperture and 'b' represents the width of the antenna aperture. Amplitude measurements are performed on the two-dimensional narrow-beam antenna on two sampling planes, with a measurement point interval of λ / 4. The amplitude values ​​measured at all sampling points are combined into an N-value. s ×M s The magnitude matrix, N s M represents the number of sampling points in the x-direction. s This represents the number of sampling points in the y-direction.

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