Satellite radiation pattern reconstruction method and system based on low-dimensional representation

Abstract

This application discloses a satellite radiation pattern reconstruction method and system based on low-dimensional representation, relating to the field of satellite antenna radiation pattern measurement and reconstruction technology. The method includes: representing the complete radiation pattern of the satellite under test as a linear combination of a preset basis matrix and a coefficient vector; based on... D Optimal sampling criteria, from M Select from candidate sampling points A sampling set is formed by several target sampling points, with the objective being to maximize the determinant of the Fisher information matrix in the coefficient vector estimation process; within the sampling set... N Measurements are taken at each target sampling point to obtain the observation vector, and the coefficient vector is estimated using the maximum likelihood estimation method. Using the estimated values ​​and a preset basis matrix, a complete radiation pattern is reconstructed. This invention has the advantages of achieving high-precision radiation pattern reconstruction with a limited number of sampling points, significantly improving measurement efficiency and reliability.

Description

A method and system for reconstructing satellite radiation patterns based on low-dimensional representation Technical Field

[0001] This invention relates to the field of satellite antenna radiation pattern measurement and reconstruction technology, and more specifically, to a satellite radiation pattern reconstruction method and system based on low-dimensional representation. Background Technology

[0002] Satellite antenna radiation patterns, as core parameters characterizing the distribution of radiated power in various directions in space, are indispensable for evaluating antenna performance, analyzing signal coverage, formulating communication link budgets, and performing on-orbit performance monitoring. Complete and accurate radiation pattern data plays a decisive role in satellite mission planning, on-orbit fault diagnosis, and system optimization, directly affecting communication quality and mission success rate.

[0003] Traditional acquisition methods primarily rely on two approaches: ground testing and on-orbit measurement. Ground testing is typically conducted in anechoic chambers or large measurement sites, acquiring full-space radiation pattern data through mechanical scanning or phased array technology. However, these methods struggle to accurately reproduce the vacuum and microgravity environments of a satellite in orbit, leading to systematic deviations between test results and actual operating conditions. Furthermore, for large antennas or complex payloads, testing processes face challenges such as high equipment costs, lengthy cycles, and susceptibility to external electromagnetic interference and weather conditions, making them ill-suited to the rapid iterative development needs of modern satellites. On-orbit measurement utilizes calibration satellites, calibration beacons, or ground receiving station networks to invert radiation characteristics by receiving known reference signals. However, this method is limited by the orbital geometry of the measurement platform, resulting in a narrow range of observable angles; it is constrained by limited communication windows and link bandwidth, leading to short data acquisition times; and it is limited by onboard processing capabilities and data transmission rates, resulting in an extremely limited number of sample points. Therefore, on-orbit measurement typically provides only sparse and incomplete sampled data, unable to directly support the construction of high-resolution full-space radiation patterns. Under these constraints, achieving high-fidelity radiation pattern reconstruction using a limited number of sampling points has become a core challenge in satellite on-orbit calibration. While uniform sampling strategies are simple to operate, they inevitably lead to aliasing when the sampling rate is below the Nyquist frequency, resulting in main lobe distortion, side lobe distortion, and loss of detail information, significantly reducing reconstruction accuracy. Random sampling methods can partially alleviate aliasing, but because they do not consider the inherent structural regularity of the radiation pattern, the reconstruction results fluctuate wildly at low sampling rates, exhibiting poor consistency between different measurement batches, making it difficult to meet the stability requirements of engineering applications. Furthermore, mainstream reconstruction algorithms often employ traditional methods such as polynomial interpolation, least squares fitting, or discrete Fourier transform, failing to effectively utilize the inherent sparsity, low rank, and spatial smoothness of radiation patterns. This results in an inability to balance reconstruction accuracy and computational efficiency when sampling resources are limited, hindering the development of satellite on-orbit autonomous calibration capabilities. In summary, under strictly limited sampling conditions, it is urgent to overcome the bottleneck of synergistic optimization between sampling strategies and reconstruction algorithms to achieve high-precision and high-reliability radiation pattern recovery.

[0004] To address the aforementioned issues, existing technologies urgently need improvement. Summary of the Invention

[0005] In view of this, the present invention provides a satellite radiation pattern reconstruction method and system based on low-dimensional representation, which has the advantages of achieving high-precision radiation pattern reconstruction with limited sampling points and significantly improving measurement efficiency and reliability.

[0006] In a first aspect, the present invention provides a satellite radiation pattern reconstruction method based on low-dimensional representation, comprising:

[0007] S1. Obtain the complete radiation pattern of the satellite to be measured. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.:

[0008]

[0009] Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and ;

[0010] S2. Based on the optimal sampling criterion D, select from M candidate sampling points. The target sampling points constitute the sampling set S; where... The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process;

[0011] S3. Measure the N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value ;

[0012] S4. Using the estimated value obtained in S3 and the preset basis matrix The complete radiation pattern was reconstructed. .

[0013] In one alternative implementation, in S1, the preset basis matrix The discrete cosine transform basis function matrix is ​​used;

[0014] The element in the m-th row and k-th column of the discrete cosine transform basis function matrix The expression is:

[0015]

[0016] Where m is a preset basis matrix row index, ; For the preset basis matrix Column indexes.

[0017] In one alternative implementation, in S1, when the preset basis matrix... When the basis function matrix is ​​the discrete cosine transform, the preset number of basis functions K is determined based on the preset reconstruction accuracy index, specifically through the following method:

[0018] Based on historical radiation patterns Perform the following steps:

[0019] S11. Regarding the historical radiation pattern Perform a discrete cosine transform and calculate all discrete cosine transform coefficients. :

[0020]

[0021] in, -1; Historical radiation pattern The m-th element;

[0022] The normalization coefficient is defined as:

[0023]

[0024] S12. Based on S11, calculate the total energy of the historical radiation pattern h. The calculation formula is:

[0025]

[0026] S23. Based on S11 and S12, determine the preset basis matrix. The preset number of basis functions K:

[0027]

[0028] in, The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold.

[0029] In one optional implementation, S2 uses a greedy D-optimal sampling algorithm to select from M candidate sampling points. One sampling point;

[0030] The greedy D-optimal sampling algorithm includes the following steps:

[0031] S21. Initialize the sampling set S to be empty, and initialize the information matrix A as follows: ;in, >0 is the regularization parameter. for The identity matrix;

[0032] S22. For each candidate sampling point i that does not belong to the sampling set S, calculate the information gain corresponding to candidate sampling point i. ;in, For the preset basis matrix The row vector corresponding to candidate sampling point i in the middle;

[0033] S23. Based on S22, compare the information gains of all candidate sampling points and select the candidate sampling point with the largest information gain as the target sampling point. Its expression is:

[0034]

[0035] in, For the target sampling point, The information gain corresponding to the target sampling point;

[0036] S24, Target sampling point In sample set S, update sample set to and update the information matrix. ;in, For the preset basis matrix Center and target sampling point The corresponding row vector;

[0037] S25. Repeat steps S22 to S24 until the sample set is reached. Include One target sampling point.

[0038] In an optional implementation, in step S3, the coefficient vector is solved using the maximum likelihood estimation method. The estimated value The calculation formula is:

[0039] ;

[0040] in, For the preset basis matrix The row index belongs to a submatrix of the sample set S. Let be the observation vector in the sampling set S.

[0041] Secondly, the present invention provides a satellite radiation pattern reconstruction system based on low-dimensional representation, used to implement the method described in the present invention, the system comprising:

[0042] The modeling module is used to generate the complete radiation pattern of the satellite under test. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.:

[0043]

[0044] Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and ;

[0045] The sampling strategy planning module is used to select from M candidate sampling points based on the D optimal sampling criterion. The target sampling points constitute the sampling set S; where... The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process;

[0046] The measurement and estimation module is used to perform measurements at N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value ;

[0047] The pattern reconstruction module is used to reconstruct the pattern using the estimates obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. .

[0048] As can be seen from the above, the satellite radiation pattern reconstruction method and system based on low-dimensional representation provided in this application, by representing the radiation pattern as a linear combination of a low-dimensional basis matrix and a coefficient vector, and optimizing the selection of sampling points based on the D optimal sampling criterion, can achieve high-precision radiation pattern reconstruction with a limited number of sampling points. It has the advantages of achieving high-precision radiation pattern reconstruction with a limited number of sampling points and significantly improving measurement efficiency and reliability. Attached Figure Description

[0049] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0050] Figure 1 is a flowchart illustrating a satellite radiation pattern reconstruction method based on low-dimensional representation according to an embodiment of the present invention.

[0051] Figure 2 shows the result of determining the preset number of basis functions K in an embodiment of the present invention;

[0052] Figure 3 is a schematic diagram of a satellite radiation pattern reconstruction method based on low-dimensional representation according to an embodiment of the present invention;

[0053] Figure 4 shows a comparison of the reconstructed normalized power of the optimal sampling method in this embodiment of the invention with that of traditional uniform sampling and random sampling methods.

[0054] Figure 5 shows a comparison of the empirical cumulative distribution function of the reconstructed mean square error between the optimal sampling method of this invention and the traditional uniform sampling and random sampling methods. Detailed Implementation

[0055] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0056] Traditionally, acquiring satellite antenna radiation patterns faces numerous challenges, including difficulties in simulating the on-orbit environment through ground testing, high costs, long cycles, and limitations of on-orbit measurements to a finite number of sampling points. Existing sampling strategies are prone to aliasing effects or unstable reconstruction results at low sampling rates, and traditional reconstruction algorithms fail to fully utilize the structural characteristics of the radiation pattern, making it difficult to balance high accuracy with low cost. This results in technical challenges for on-orbit antenna performance evaluation and calibration.

[0057] As shown in Figure 1, this application proposes a satellite radiation pattern reconstruction method based on low-dimensional representation, which specifically includes the following steps:

[0058] Step S1: Obtain the complete radiation pattern of the satellite under test. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.:

[0059]

[0060] Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and .

[0061] Step S2: Based on the D optimal sampling criterion, select from M candidate sampling points. The target sampling points constitute the sampling set S; where... ; Choose to maximize the coefficient vector The determinant of the Fisher information matrix in the estimation process is the objective.

[0062] Step S3: Perform measurements at the N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value .

[0063] Step S4: Use the estimated value obtained in step S3 and the preset basis matrix The complete radiation pattern was reconstructed. .

[0064] Specifically, a satellite radiation pattern refers to a characteristic parameter that describes the distribution of radiated power of a satellite antenna in different spatial directions. Its complete and accurate acquisition is of great significance for satellite mission planning, on-orbit diagnostics, and system optimization.

[0065] Low-dimensional representation is a technique that maps high-dimensional data to a low-dimensional space through mathematical transformations, aiming to capture the core structure and information of the data while reducing its dimensionality. In this method, the complete radiation pattern is represented as a linear combination of a predefined basis matrix and a coefficient vector, thus transforming the high-dimensional reconstruction problem into a low-dimensional coefficient estimation problem.

[0066] The predefined basis matrix Ψ is a matrix composed of a set of predefined basis functions that can effectively represent or approximate the radiation pattern of the satellite under test. The number of columns K of this matrix is ​​much smaller than the number of rows M, reflecting the characteristics of low-dimensional representation.

[0067] The coefficient vector ω is a weight vector corresponding to the predefined basis matrix Ψ, and its elements represent the contribution of each basis function to the linear combination of the radiation pattern. By accurately estimating this coefficient vector, the complete radiation pattern can be reconstructed.

[0068] The D-optimal sampling criterion is an optimal sampling strategy in experimental design theory. Its goal is to select specific sampling points to minimize the variance of the model parameter estimates, which is usually achieved by maximizing the determinant of the Fisher information matrix.

[0069] The Fisher information matrix is ​​a statistical measure of the amount of information contained in observed data about unknown parameters. The larger the determinant, the more accurate the estimate of the parameters and the smaller the uncertainty.

[0070] Maximum likelihood estimation is a commonly used parameter estimation method that finds the parameter value that maximizes the probability of the observed data occurring. This method can provide efficient parameter estimates even with noisy observation data.

[0071] Specifically, this method includes the following steps:

[0072] First, the complete radiation pattern of the satellite under test. Represented as a predefined basis matrix With coefficient vector A linear combination, i.e. The predefined basis matrix Ψ can be composed of various basis functions. For example, Fourier basis functions can be used to approximate the radiation pattern using sine and cosine functions of different frequencies; alternatively, polynomial basis functions can be used to represent the radiation pattern through combinations of polynomials of different orders. Furthermore, wavelet basis functions can be used to capture the details of the radiation pattern through functions with local properties. This linear combination transforms the high-dimensional radiation pattern reconstruction problem into an estimation problem of the low-dimensional coefficient vector ω, thereby reducing computational complexity and the number of required measurement points. Here, M represents the total number of all candidate sampling points, and K represents the number of predefined basis functions, with K being much smaller than M, reflecting the advantages of low-dimensional representation.

[0073] Subsequently, based on the optimal sampling criterion D, N target sampling points are selected from M candidate sampling points to form a sampling set S. Here, N should not be less than K. This selection process aims to maximize the determinant of the Fisher information matrix in the coefficient vector ω estimation process, striving to minimize the variance of the coefficient vector estimation. Specifically, an exhaustive search approach can be used, traversing all possible combinations of N sampling points, calculating the determinant of the Fisher information matrix for each combination, and selecting the combination with the largest determinant. However, when M is large, the computational cost of exhaustive search is enormous. As an alternative, a heuristic search method can be used. For example, N points can be randomly selected as the initial sampling set, and then the sampling set can be gradually optimized by randomly swapping points inside and outside the sampling set and evaluating the change in the determinant of the Fisher information matrix after each swap. This step ensures that an accurate estimate of the coefficient vector ω can be obtained with a limited number of measurement points N.

[0074] Next, actual measurements are performed at N target sampling points in the sampling set S to obtain the observation vector. These measurements contain the response of the radiation pattern in a specific spatial direction and may be affected by noise. Subsequently, the maximum likelihood estimation method is used to solve for the estimated value of the coefficient vector ω. The core idea of ​​maximum likelihood estimation is that, given the observed data... and model In this case, we seek the coefficient vector most likely to have generated the observed data. Specifically, we can construct a likelihood function that describes the probability of the observed data occurring given the coefficient vector. By optimizing this likelihood function, for example through gradient descent or other numerical optimization algorithms, we can find the value that maximizes the likelihood function, thus obtaining an estimate of the coefficient vector.

[0075] Finally, the estimated value obtained in step S3 is used. Using the preset basis matrix Ψ, a complete radiation pattern is reconstructed. The reconstruction process is a direct linear combination, that is, the estimated coefficient vector... Multiplying the values ​​by the preset basis matrix Ψ yields the radiation pattern values ​​at all M candidate sampling points. Thus, even with measurements taken at only a finite number of N sampling points, a high-resolution, full-space radiation pattern can be reconstructed. This step is the final output of the entire method, providing the core data needed for evaluating and diagnosing satellite antenna performance.

[0076] This method effectively reduces the dimensionality of the reconstruction problem by representing the satellite radiation pattern in a low dimension. Combined with the D-optimal sampling criterion, this method strategically selects the sampling positions with the highest information content with a limited number of measurement points, thereby improving the accuracy of coefficient vector estimation. Thus, even under on-orbit measurement conditions with limited sampling times, it can achieve high-precision and high-stability reconstruction of the satellite radiation pattern, overcoming the limitations of traditional methods that suffer from large reconstruction errors and unstable results at low sampling rates, and providing a reliable basis for on-orbit performance evaluation of satellite antennas.

[0077] In one optional implementation, in step S1, the preset basis matrix Ψ is a discrete cosine transform basis function matrix;

[0078] The elements in the m-th row and k-th column of the discrete cosine transform basis function matrix The expression is:

[0079]

[0080] Where m is a preset basis matrix row index, ; For the preset basis matrix Column indexes.

[0081] Specifically, the Discrete Cosine Transform (DCT) is a mathematical transformation that decomposes a signal into cosine components of different frequencies. Its basis functions exhibit good energy concentration characteristics; that is, for most real-world signals, the main energy is concentrated in a few low-frequency coefficients. This makes DCT well-suited for efficient low-dimensional representation of signals with smooth or slowly varying characteristics (such as satellite radiation patterns). By selecting the Discrete Cosine Transform as the preset basis matrix Ψ, it can be ensured that the selected basis functions effectively capture the key features of the radiation pattern, thereby minimizing the required number of basis functions K while maintaining reconstruction accuracy.

[0082] The above expression defines in detail the composition of the discrete cosine transform basis function matrix. Here, m represents the row index of the preset basis matrix Ψ, corresponding to a point among the M candidate sampling points; k represents the column index of the preset basis matrix Ψ, corresponding to different basis function components. When k=0, the basis functions are constant terms, representing the DC component of the signal; when k>0, the basis functions are cosine waves of different frequencies, representing the AC component of the signal. Normalization coefficients. and This ensures the orthogonality of the basis vectors and appropriate amplitude scaling, thereby guaranteeing the invertibility and numerical stability of the transformation. This structure allows each basis function to contribute independently to the reconstruction of the radiation pattern and enables efficient representation of the signal's frequency content.

[0083] By employing the discrete cosine transform (DCT) basis function matrix as the preset basis matrix Ψ, a low-dimensional representation of the satellite radiation pattern can be effectively achieved. The DCT possesses excellent energy concentration characteristics, allowing the main information of the radiation pattern to be concentrated in a few low-frequency coefficients. This means that, under the same reconstruction accuracy requirements, the number of basis functions K required can be significantly reduced, thereby lowering the dimension of the coefficient vector ω. The reduced dimension of the coefficient vector ω directly reduces the number of sampling points N that need to be determined in the subsequent sampling strategy planning (step S2), and the calculation of the coefficient vector estimate in measurement and estimation (step S3). The computational complexity is reduced. Furthermore, because the discrete cosine transform basis functions can better capture the smoothness of the radiation pattern, the reconstructed complete radiation pattern... It has higher accuracy, especially with limited sampling points, and can more accurately recover the details of the original radiation pattern, thereby improving the efficiency and accuracy of the entire reconstruction method.

[0084] In an alternative implementation, this application further proposes that in step S1, when the preset basis matrix... When determining the discrete cosine transform basis function matrix, the preset number of basis functions K is determined based on a preset reconstruction accuracy index. The determination method is based on historical radiation patterns. conduct.

[0085] The reconstruction accuracy index is a quantitative standard used to measure the difference between the reconstructed radiation pattern and the true radiation pattern. For example, it could be a reconstruction error energy ratio threshold, used to guide the determination of the preset number of basis functions. The historical radiation pattern *h* refers to complete radiation pattern data obtained from past measurements of the satellite under test or similar satellites. As prior knowledge, it reflects the typical characteristics and energy distribution patterns of the radiation pattern of this type of satellite. By analyzing historical data, the selection of the number of basis functions for the current satellite under test can be effectively guided, making it more consistent with actual physical characteristics and improving the accuracy and efficiency of reconstruction. The historical radiation pattern *h* can be obtained through multiple omnidirectional measurements of similar satellites or through simulation modeling. This data is usually stored in the form of discrete points, forming an M-dimensional vector.

[0086] Specifically, the method includes the following steps:

[0087] Step S11: Perform discrete cosine transform on the historical radiation pattern h and calculate all discrete cosine transform coefficients. :

[0088]

[0089] in, -1; Historical radiation pattern The m-th element;

[0090] The normalization coefficient is defined as:

[0091]

[0092] Specifically, Discrete Cosine Transform (DCT) is a commonly used signal processing technique that can convert signals from the time or spatial domain to the frequency domain, and it possesses good energy concentration characteristics. By performing DCT on the historical radiation pattern h, it can be decomposed into a series of basis function coefficients with different frequency components. The magnitudes of these coefficients reflect the importance of the corresponding frequency components in the original signal; the more concentrated the energy of the coefficient, the greater the contribution of its corresponding basis function to the signal. DCT calculations can employ fast algorithms, such as variants of the Fast Fourier Transform (FFT), to improve computational efficiency. During the calculation process, The m-th element of the historical radiation pattern h is represented by bi, which is a normalization coefficient used to ensure energy conservation or specific properties of the transformation.

[0093] S12. Based on step S11, calculate the total energy of the historical radiation pattern h. The calculation formula is:

[0094]

[0095] Specifically, total energy This represents the total energy of the historical radiation pattern h in the frequency domain. In signal processing, the energy of a signal is often related to its importance or information content. By calculating the total energy, subsequent determinations can be made. A benchmark is provided, whereby the reconstruction accuracy metric will be measured as a percentage of total energy.

[0096] Step S13: Based on steps S11 and S12, determine the preset basis matrix. The preset number of basis functions K:

[0097]

[0098] in, The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold.

[0099] Specifically, it utilizes the energy concentration characteristic of DCT coefficients, where most of the signal energy is concentrated in a few low-frequency coefficients. This is achieved by accumulating the energy of the first K coefficients and then combining it with the total energy. By comparison, we can find the preset reconstruction accuracy index (from 1- The minimum number of basis functions required (defined). This means that even using only K basis functions, most of the energy of the original signal can be captured, thus ensuring the accuracy of the reconstruction. Determining K is an iterative or search process. Starting with K=1, the value of K is gradually increased, and the cumulative sum of the energy of the first K coefficients is calculated until the cumulative sum reaches or exceeds [the specified value]. The first K value that satisfies the condition is the preset number of basis functions. The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold. For example, as shown in Figure 2, if ϵ=0.01, it means that 1% energy loss is allowed, that is, the energy of the reconstructed signal is required to reach at least 99% of the total energy of the original signal.

[0100] Through the above technical solution, this application provides a method for objectively and quantitatively determining the number K of preset basis functions in a preset basis matrix Ψ based on a historical radiation pattern h. By performing a discrete cosine transform on the historical radiation pattern h, the energy distribution on different basis functions can be effectively analyzed, and the total energy can be calculated. Based on this, and combined with a preset reconstruction accuracy index ϵ, a minimum number of basis functions can be precisely found, ensuring that these basis functions carry the majority of the total energy. This method avoids the arbitrary selection of K, ensuring that the number of basis functions is minimized while meeting reconstruction accuracy requirements. This effectively reduces the computational complexity of subsequent sampling point selection and coefficient estimation, and decreases the number of sampling points required for actual measurements, thereby improving the efficiency and resource utilization of radiation pattern reconstruction.

[0101] In an optional implementation, this application further proposes that, in S2 above, a greedy D-optimal sampling algorithm be used to select N sampling points from M candidate sampling points. This greedy D-optimal sampling algorithm includes the following steps:

[0102] S21. Initialize the sampling set S to be empty, and initialize the information matrix A as follows: ;in, >0 is the regularization parameter. for The identity matrix;

[0103] In this step, the sampling set S is set to empty, indicating that no sampling points have been selected. The information matrix A is initialized as a regularized identity matrix, where the regularization parameter ε is a small positive number. Its purpose is to ensure that the information matrix A is invertible in the early stages of the sampling process, avoiding singularity problems when there are no sampling points, thereby ensuring the stability of subsequent calculations. The dimension of the identity matrix I... It corresponds to the dimension of the coefficient vector ω.

[0104] S22. For each candidate sampling point i that does not belong to the sampling set S, calculate the information gain corresponding to candidate sampling point i. ;in, For the preset basis matrix The row vector corresponding to candidate sampling point i in the middle;

[0105] In this step, in each iteration, the algorithm iterates through all candidate sampling points that have not yet been selected into the sampling set S. For each candidate sampling point i, it calculates the information gain q that would be gained by adding it to the current sampling set. i The formula for calculating information gain measures the contribution of a candidate point to the estimated coefficient vector ω. Wherein, A is the row vector in the predefined basis matrix Ψ corresponding to the candidate sampling point i, which contains the response of that sampling point on each basis function. -1 This is the inverse of the current information matrix A, reflecting the amount of information provided by the currently selected sampling points. By calculating the information gain, the potential value of each unselected point in improving estimation accuracy can be quantified.

[0106] S23. Based on step S22, compare the information gain of all candidate sampling points and select the candidate sampling point with the largest information gain as the target sampling point. Its expression is:

[0107]

[0108] in, For the target sampling point, The information gain corresponding to the target sampling point;

[0109] Specifically, the information gain of all candidate sampling points is compared, and the candidate sampling point with the largest information gain is selected as the target sampling point. Its expression is: ,in, For the target sampling point, Let be the information gain corresponding to the target sampling point. After calculating the information gain of all unselected candidate sampling points, the algorithm compares these gain values ​​and selects the candidate sampling point with the largest information gain as the target sampling point j* for this iteration. This selection is the core of the "greedy" strategy, that is, choosing the local solution that is currently optimal at each step. By selecting the point with the largest information gain, it can be ensured that each added sampling point can maximize the estimation accuracy.

[0110] S24, Target sampling point In sample set S, update sample set to and update the information matrix. ;in, For the preset basis matrix Center and target sampling point The corresponding row vector;

[0111] Specifically, once the target sampling points are determined... This point will be added to the sampling set S, indicating that it has been selected for measurement. Simultaneously, the information matrix A also needs to be updated to reflect the information brought by the newly added sampling point. Update formula The information contribution of new sampling points is efficiently accumulated into the information matrix to prepare for the calculation of information gain in the next iteration.

[0112] S25. Repeat steps S22 to S24 until the sample set is obtained. Include One target sampling point.

[0113] Steps S22 to S24 described above are executed cyclically. In each iteration, a sampling point with the largest information gain is selected and added to the sampling set, until the sampling set S contains a preset number of N target sampling points. This repeated process ensures that the final N selected sampling points are obtained through stepwise optimization using a greedy strategy, aiming to maximize the final determinant of the Fisher information matrix.

[0114] By employing a greedy D-optimal sampling algorithm, this application effectively addresses the problems of high computational complexity and low efficiency encountered when directly solving for the optimal sampling point set under the D-optimal sampling criterion. The algorithm iteratively selects candidate sampling points with the highest information gain, progressively constructing a sampling set S. In each step, the algorithm quantifies the potential contribution of each unselected point to the estimated coefficient vector and selects the most informative point to add to the sampling set. This locally optimal selection strategy enables efficient approximation of the globally optimal sampling point configuration with limited computational resources. Ultimately, the selected N target sampling points maximize the determinant of the Fisher information matrix in the coefficient vector ω estimation process, thereby significantly improving the coefficient vector... The estimation accuracy. More accurate coefficient vector estimates. By combining the preset basis matrix Ψ, a more accurate satellite radiation pattern that is closer to reality can be reconstructed. This improves the accuracy and reliability of the entire radiation pattern reconstruction method.

[0115] In an alternative implementation, in S3 above, the estimated value of the coefficient vector is solved using the maximum likelihood estimation method. The calculation formula is:

[0116]

[0117] in, For the preset basis matrix The row index belongs to a submatrix of the sample set S. Let be the observation vector in the sampling set S.

[0118] Specifically, the calculation formula This is a closed-form solution based on the least squares principle, which is equivalent to maximum likelihood estimation when assuming the measurement noise follows a Gaussian distribution. The formula involves... Projected onto a predefined basis matrix In the space of Zhang Cheng, the coefficient vector ω that best explains the observation data is found. Its role is to provide a direct, efficient, and mathematically optimal method to estimate the low-dimensional representation of the coefficient vector from a limited number of sampled measurements, laying the foundation for subsequent reconstruction of the complete radiation pattern.

[0119] in, It is a submatrix whose row indices belong to the sampling set S in the preset basis matrix Ψ. Specifically, if the sampling set S contains N target sampling points out of M candidate sampling points, then... This involves extracting an N×K dimensional matrix from the original M×K dimensional pre-defined basis matrix Ψ, forming an N×K dimensional matrix composed of the row vectors corresponding to the N target sampling points. This submatrix Its function is to map the low-dimensional coefficient vector ω to the theoretical measured value on the sampling point, and it is a bridge connecting the low-dimensional representation and the actual measurement data.

[0120] at the same time, Let be the observation vector in the sampling set S. This vector is an N×1 dimensional column vector, where each element corresponds to the actual measured radiation intensity value at a target sampling point in the sampling set S. It is real data obtained by the system from the physical world and is the direct basis for estimating the coefficient vector ω.

[0121] By using the specific calculation formulas described above To solve for the estimated value of the coefficient vector ω, this application provides a direct and computationally efficient maximum likelihood estimation method. This formula utilizes the observation vectors on the sampling set S. and the corresponding subset of the basis matrix This method obtains the estimated coefficient vector in one step through matrix operations. This closed-form solution avoids iterative optimization, significantly reducing computational complexity and time consumption, making it particularly suitable for scenarios requiring rapid reconstruction of radiation patterns. Furthermore, when the measurement noise meets certain statistical properties (such as Gaussian white noise), this method provides an unbiased estimate with minimal variance, ensuring the accuracy and reliability of the estimation results and thus guaranteeing the final, complete reconstructed radiation pattern. The accuracy effectively solves the challenge of how to efficiently and accurately estimate coefficient vectors with limited measurement data.

[0122] Furthermore, as shown in Figure 3, this application embodiment also provides a satellite radiation pattern reconstruction system based on low-dimensional representation, including:

[0123] The modeling module is used to generate the complete radiation pattern of the satellite under test. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.:

[0124]

[0125] Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and ;

[0126] The sampling strategy planning module is used to select from M candidate sampling points based on the D optimal sampling criterion. The target sampling points constitute the sampling set S; where... ; Select to maximize the coefficient vector The objective is to estimate the determinant of the Fisher information matrix in the estimation process;

[0127] The measurement and estimation module is used to perform measurements at N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value ;

[0128] The pattern reconstruction module is used to reconstruct the pattern using the estimates obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. .

[0129] As shown in Figures 4 and 5, simulation results demonstrate that, under the same number of samplings, the method proposed in this invention can significantly reduce the mean square error (MSE) of the reconstructed radiation pattern compared to traditional uniform sampling and random sampling methods.

[0130] For example, in a typical scenario, MSE is composed of Reduce to Even under different signal-to-noise ratios (SNR) and sampling rates, the method of this invention maintains high reconstruction accuracy and stability.

[0131] This embodiment fully verifies the effectiveness and robustness of the method of the present invention under low sampling rate conditions, and can significantly improve the efficiency and accuracy of on-orbit antenna pattern positioning and reconstruction.

[0132] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A method for reconstructing satellite radiation patterns based on low-dimensional representation, characterized in that, include: S1. Obtain the complete radiation pattern of the satellite under test. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.: Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and S2. Based on the optimal sampling criterion D, select from M candidate sampling points. The target sampling points constitute the sampling set S; where... The selection is to maximize the coefficient vector. The determinant of the Fisher information matrix in the estimation process is the target; S3, measurements are performed at the N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value S4. Using the estimated value obtained in S3 and the preset basis matrix The complete radiation pattern was reconstructed. 。 2. The method according to claim 1, characterized in that, In S1, the preset basis matrix The discrete cosine transform basis function matrix is ​​used; the elements in the m-th row and k-th column of the discrete cosine transform basis function matrix are... The expression is: Where m is a preset basis matrix row index, ； For the preset basis matrix Column indexes.

3. The method according to claim 2, characterized in that, In S1, when the preset basis matrix When determining the discrete cosine transform basis function matrix, the preset number of basis functions K is determined based on a preset reconstruction accuracy index, specifically through the following method: based on historical radiation pattern. Perform the following steps: S11, for the historical radiation pattern Perform a discrete cosine transform and calculate all discrete cosine transform coefficients. ： in, -1； Historical radiation pattern The m-th element; The normalization coefficient is defined as: S12. Based on S11, calculate the total energy of the historical radiation pattern h. The calculation formula is: S13. Based on S11 and S12, determine the preset basis matrix. The preset number of basis functions K: in, The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold.

4. The method according to claim 1, characterized in that, S2 uses a greedy D-optimal sampling algorithm to select from M candidate sampling points. The greedy D-optimal sampling algorithm includes the following steps: S21, initializing the sampling set S as an empty set, and initializing the information matrix A as... ;in, >0 is the regularization parameter. for The identity matrix; S22. For each candidate sampling point i that does not belong to the sampling set S, calculate the information gain corresponding to candidate sampling point i. ;in, For the preset basis matrix S23. Based on S22, compare the information gains of all candidate sampling points and select the candidate sampling point with the largest information gain as the target sampling point. Its expression is: in, For the target sampling point, S24. Set the information gain corresponding to the target sampling point; In sample set S, update sample set to and update the information matrix. ;in, For the preset basis matrix Center and target sampling point The corresponding row vector; S25, Repeat steps S22 to S24 until the sampling set is reached. Include One target sampling point.

5. A satellite radiation pattern reconstruction system based on low-dimensional representation, characterized in that, The system for implementing the method of any one of claims 1 to 4 includes: a modeling module for generating a complete radiation pattern of the satellite under test. Represented as a preset basis matrix With coefficient vector A linear combination of, i.e.: Where M is the total number of all candidate sampling points; K is the preset number of basis functions, and The sampling strategy planning module is used to select from M candidate sampling points based on the D optimal sampling criterion. The target sampling points constitute the sampling set S; where... The selection is to maximize the coefficient vector. The determinant of the Fisher information matrix in the estimation process is the target; the measurement and estimation module is used to perform measurements on N target sampling points in the sampling set S to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value The pattern reconstruction module is used to reconstruct the radiation pattern using the estimated values ​​obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. 。

Images