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

Abstract

The invention discloses a satellite radiation pattern reconstruction method and system based on low-dimensional representation, and relates to the technical field of satellite antenna radiation pattern measurement and reconstruction, and the method comprises the steps: representing a complete radiation pattern of a to-be-measured satellite as a linear combination of a preset basis matrix and a coefficient vector; based on a D optimal sampling criterion, selecting target sampling points from the M candidate sampling points to form a sampling set, and selecting a Fisher information matrix determinant for maximizing a coefficient vector estimation process as a target; performing measurement on N target sampling points in the sampling set to obtain an observation vector, and solving an estimated value of a coefficient vector by using a maximum likelihood estimation method; and reconstructing a complete radiation pattern by using the estimated value and a preset basis matrix. The method has the advantages that high-precision radiation pattern reconstruction is realized under limited sampling points, and the measurement efficiency and reliability are remarkably improved.

Description

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: 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and ; S2, based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process; S3, in the sampling set S N Measurements are taken at each target sampling point 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. .

[0007] In one alternative implementation, in S1, the preset basis matrix The discrete cosine transform basis function matrix is ​​used; The first of the discrete cosine transform basis function matrix m Line 1 k Column elements The expression is: in, m For the preset basis matrix row index, ; For the preset basis matrix Column indexes.

[0008] 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 is... K It is determined based on preset reconstruction accuracy indicators, specifically through the following methods: Based on historical radiation patterns Perform the following steps: S11. Regarding the historical radiation pattern Perform a discrete cosine transform and calculate all discrete cosine transform coefficients. : in, -1; Historical radiation pattern The m One element; The normalization coefficient is defined as: S12. Based on S11, calculate the historical radiation pattern. h Total energy The calculation formula is: S23. Based on S11 and S12, determine the preset basis matrix. The number of pre-defined basis functions K : in, The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold.

[0009] In one alternative implementation, S2 employs a greedy D-optimal sampling algorithm from... M Select from candidate sampling points One sampling point; The greedy D-optimal sampling algorithm includes the following steps: S21. Initialize the sampling set S to be empty and initialize the information matrix. A for ;in, >0 For regularization parameters, for The identity matrix; S22. For each sample set that does not belong to the sampling set S Candidate sampling points i Calculate candidate sampling points i Corresponding information gain ;in, For the preset basis matrix Middle and candidate sampling points i The corresponding row vector; 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, The information gain corresponding to the target sampling point; S24, Target sampling point sampling set S In the middle, update the sampling 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 sample set is reached. Include One target sampling point.

[0010] 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: ; in, For the preset basis matrix The central bank index belongs to the sample set. S submatrix, In the sampling set S The observation vector.

[0011] 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: 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and ; The sampling strategy planning module is used for sampling based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process; Measurement and estimation module, used in the sample set S N Measurements are taken at each target sampling point 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 pattern using the estimates obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. .

[0012] As can be seen from the above, the satellite radiation pattern reconstruction method and system based on low-dimensional representation provided in this application represents the radiation pattern as a linear combination of a low-dimensional basis matrix and a coefficient vector, and is based on... D The optimal sampling criterion optimizes the selection of sampling points, enabling high-precision radiation pattern reconstruction with a limited number of sampling points. This has the advantages of significantly improving measurement efficiency and reliability. Attached Figure Description

[0013] 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.

[0014] Figure 1 This is a flowchart illustrating a satellite radiation pattern reconstruction method based on low-dimensional representation according to an embodiment of the present invention. Figure 2 The preset number of basis functions in this embodiment of the invention. K The determined result diagram; Figure 3 This is a schematic diagram of a satellite radiation pattern reconstruction method based on low-dimensional representation according to an embodiment of the present invention. Figure 4 This is 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. Figure 5 The figure shows a comparison of the empirical cumulative distribution function of the reconstructed mean square error between the optimal sampling method of this invention and traditional uniform sampling and random sampling methods. Detailed Implementation

[0015] 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.

[0016] 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.

[0017] In this regard, such as Figure 1 As shown, this application proposes a satellite radiation pattern reconstruction method based on low-dimensional representation, which specifically includes the following steps: 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and .

[0018] Step S2, based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, ; Choose to maximize the coefficient vector The determinant of the Fisher information matrix in the estimation process is the objective.

[0019] Step S3, in the sampling set S N Measurements are taken at each target sampling point to obtain the observation vector. And the coefficient vector is solved using the maximum likelihood estimation method. The estimated value .

[0020] Step S4: Use the estimated value obtained in step S3 and the preset basis matrix The complete radiation pattern was reconstructed. .

[0021] 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.

[0022] 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.

[0023] Preset basis matrix P It is a matrix composed of a set of predefined basis functions, which can effectively represent or approximate the radiation pattern of the satellite under test. The number of columns in this matrix... K Much smaller than the number of rows M This reflects the characteristics of low-dimensional representation.

[0024] coefficient vector oh It is with the preset basis matrix P The corresponding weight vector contains elements representing 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.

[0025] 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.

[0026] 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.

[0027] 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.

[0028] Specifically, this method includes the following steps: 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 preset basis matrix PThe radiation pattern can be constructed using various basis functions. For example, Fourier basis functions can be used, approximating the radiation pattern with sine and cosine functions of different frequencies; alternatively, polynomial basis functions can be used, representing the radiation pattern through combinations of polynomials of different orders. Furthermore, wavelet basis functions can be employed, capturing the details of the radiation pattern through functions with local properties. Through this linear combination, the problem of reconstructing a high-dimensional radiation pattern is transformed into reconstructing a low-dimensional coefficient vector. oh This reduces the estimation problem, thereby decreasing computational complexity and the number of required measurement points. M This represents the total number of all candidate sampling points. K This indicates the preset number of basis functions, and K much smaller M This demonstrates the advantages of low-dimensional representation.

[0029] Subsequently, based on the D optimal sampling criterion, from M Select from candidate sampling points N The target sampling points constitute the sampling set S. Among them, N It should be no less than K The selection process aims to maximize the coefficient vector. oh The objective of the estimation process is to minimize the variance of the estimated coefficient vector, using the determinant of the Fisher information matrix. Specifically, this can be achieved through an exhaustive search, traversing all possible... N For each combination of sampling points, calculate the determinant of the Fisher information matrix for each combination, and select the combination with the largest determinant. However, when... M When the size is large, exhaustive search becomes computationally expensive. As an alternative, heuristic search methods can be used, such as randomly selecting... N Using a limited number of points as the initial sampling set, the sampling set is 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 the sampling set is optimized with a limited number of measurement points. N Below, it is possible to obtain the coefficient vector. oh Accurate estimation.

[0030] Next, in the sampling set S In N Actual measurements are performed at each target sampling point 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 coefficient vector is solved using the maximum likelihood estimation method. oh The estimated value 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.

[0031] 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... With the preset basis matrix P Multiplying, we get all M The radiation pattern values ​​at each candidate sampling point. Therefore, even only at... N Measurements were taken at a limited number of sampling points, enabling the reconstruction of high-resolution, full-space radiation patterns. This step is the final output of the entire method, providing the core data needed to evaluate and diagnose satellite antenna performance.

[0032] 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.

[0033] In one optional implementation, in step S1, the preset basis matrix Ψ is a discrete cosine transform basis function matrix; The first of the discrete cosine transform basis function matrices m Line 1 k Column elements The expression is: in, m For the preset basis matrix row index, ; For the preset basis matrix Column indexes.

[0034] 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... P This ensures that the selected basis functions can effectively capture the key features of the radiation pattern, thereby minimizing the number of basis functions required while maintaining reconstruction accuracy. K .

[0035] The above expression defines in detail the composition of the discrete cosine transform basis function matrix. Wherein, m Represents the preset basis matrix P The row index, corresponding to M A point among the candidate sampling points; k Represents the preset basis matrix P The column indexes correspond to different basis function components. When k When = 0, the basis function is a constant term, representing the DC component of the signal; when k When the value is greater than 0, the basis functions are cosine waves of different frequencies, representing the alternating current (AC) components 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.

[0036] The above technical solution uses the discrete cosine transform basis function matrix as the preset basis matrix. P This method can effectively represent satellite radiation patterns in a low dimension. The discrete cosine transform (DCT) has excellent energy concentration properties, 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 required is significantly reduced. K This can be significantly reduced, thereby lowering the coefficient vector. oh The dimension of the coefficient vector. oh Reducing the dimensionality directly decreases the number of sampling points that need to be determined in the subsequent sampling strategy planning (step S2). N And in measurement and estimation (step S3), the estimated value of the coefficient vector is solved. 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.

[0037] In an alternative implementation, this application further proposes that in step S1, when the preset basis matrix... When the basis function matrix is ​​the discrete cosine transform, the preset number of basis functions is... K It is determined based on preset reconstruction accuracy indicators. The determination method is based on historical radiation patterns. conduct.

[0038] The reconstruction accuracy metric 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. (Historical radiation pattern) h Historical radiation pattern data refers to complete radiation pattern data obtained from past measurements of the satellite under test or similar satellites. As a form of 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. Historical radiation patterns h This data can be obtained through multiple comprehensive measurements of similar satellites or through simulation modeling. These data are typically stored as discrete points, forming a... M Dimensional vector.

[0039] Specifically, the method includes the following steps: Step S11: Analyze historical radiation patterns. h Perform a discrete cosine transform and calculate all discrete cosine transform coefficients. : in, -1; Historical radiation pattern The m One element; The normalization coefficient is defined as: Specifically, Discrete Cosine Transform (DCT) is a commonly used signal processing technique that can convert time-domain or spatial-domain signals to the frequency domain, and it has good energy concentration characteristics. This can be achieved by analyzing historical radiation patterns. h By performing a DCT, 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, Historical radiation pattern h The m There are 1 element, bi, which is a normalization coefficient used to ensure energy conservation or specific properties of the transformation.

[0040] S12. Based on step S11, calculate the historical radiation pattern. h Total energy The calculation formula is: Specifically, total energy Represents historical radiation patterns h The total energy 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.

[0041] Step S13: Based on steps S11 and S12, determine the preset basis matrix. The number of pre-defined basis functions K : in, The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold.

[0042] 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 through accumulation... K The energy of each coefficient, and the total energy. By comparison, we can find the preset reconstruction accuracy index (from 1- (Definition) The minimum number of basis functions required. This means that even if only using... K Even with a single basis function, most of the energy of the original signal can be captured, thus ensuring the accuracy of the reconstruction. K Determining this is an iterative or search process. From K Start with 1, and gradually increase... K The value before calculation K The energy accumulation of each coefficient is calculated until the accumulation reaches or exceeds [a certain value]. The first one that meets the conditions. K The value is the preset number of basis functions. The preset reconstruction accuracy index corresponds to the reconstruction error energy ratio threshold, for example, such as... Figure 2 As shown, if ϵ=0.01, it means that 1% energy loss is allowed, which means that the energy of the reconstructed signal must reach at least 99% of the total energy of the original signal.

[0043] Through the above technical solution, this application provides a method based on historical radiation patterns. h To objectively and quantitatively determine the preset basis matrix P The number of pre-defined basis functions K The method involves analyzing historical radiation patterns. h Performing the discrete cosine transform (DCT) can effectively analyze the energy distribution across different basis functions and calculate the total energy. Based on this, and combined with a preset reconstruction accuracy index ϵ, a minimum number of basis functions can be precisely found, such that these basis functions carry the majority of the total energy. This method avoids... K The arbitrary selection of basis functions ensures that the number of basis functions is minimized while meeting the reconstruction accuracy requirements. This effectively reduces the computational complexity of subsequent sampling point selection and coefficient estimation, and reduces the number of sampling points required for actual measurement, thereby improving the efficiency and resource utilization of radiation pattern reconstruction.

[0044] 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: S21. Initialize the sampling set S to be empty and initialize the information matrix. A for ;in, >0 For regularization parameters, for The identity matrix; In this step, the sampling set S Setting it to empty indicates that no sampling points have been selected. Information Matrix A It 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 issues when there are no sampling points, thus ensuring the stability of subsequent calculations. The dimension of the identity matrix I... With coefficient vector oh The dimensions correspond to each other.

[0045] S22. For each sample set that does not belong to the sampling set S Candidate sampling points i Calculate candidate sampling points i Corresponding information gain ;in, For the preset basis matrix Middle and candidate sampling points i The corresponding row vector; In this step, in each iteration, the algorithm iterates through all samples that have not yet been selected into the sample set. S The candidate sampling points. For each candidate sampling point i Calculate the information gain that it can bring after being added to the current sampling set. q i The formula for calculating information gain measures the impact of a candidate point on the estimated coefficient vector. oh The degree of contribution. Among them, It is a preset basis matrix P The middle corresponds to the candidate sampling point i A row vector that contains the responses of the 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.

[0046] 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: in, For the target sampling point, The information gain corresponding to the target sampling point; 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.

[0047] S24, Target sampling point sampling set S In the middle, update the sampling set to and update the information matrix. ;in, For the preset basis matrix Center and target sampling point The corresponding row vector; Specifically, once the target sampling points are determined... It will then be added to the sampling set. S In the middle, it indicates that the point has been selected for measurement. Simultaneously, the information matrix... A It also needs to be updated to reflect the information brought by the newly added sampling points. 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.

[0048] S25. Repeat steps S22 to S24 until the sample set is obtained. Include One target sampling point.

[0049] Steps S22 to S24 described above are executed cyclically. In each iteration, a sampling point with the highest information gain is selected and added to the sampling set, until the sampling set S contains a preset number N target sampling points. This repetitive process ensures that the final selected... N The sampling points are obtained by progressive optimization using a greedy strategy, aiming to maximize the determinant of the final Fisher information matrix.

[0050] 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. This algorithm iteratively selects candidate sampling points with the highest information gain, gradually constructing the 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 allows for efficient approximation of the globally optimal sampling point configuration with limited computational resources. Ultimately, the selected... N Each target sampling point can maximize the coefficient vector. oh The determinant of the Fisher information matrix in the estimation process is used to significantly improve the coefficient vector. The estimation accuracy. More accurate coefficient vector estimates. Combined with the preset basis matrix P It can reconstruct a more accurate satellite radiation pattern that is closer to the real situation. This improves the accuracy and reliability of the entire radiation pattern reconstruction method.

[0051] 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: in, For the preset basis matrix The central bank index belongs to the sample set. S submatrix, In the sampling set S The observation vector.

[0052] 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 Zhang Cheng's space, the coefficient vector that best explains the observed data can be found. oh Its role is to provide a direct, efficient, and mathematically optimal method to estimate the coefficient vector of the low-dimensional representation from a limited number of sampled measurements, laying the foundation for subsequent reconstruction of the complete radiation pattern.

[0053] in, The row index of the preset basis matrix Ψ belongs to the sampling set. S The submatrix. Specifically, if the sampling set S contains M Among the candidate sampling points N For each target sampling point, then It is from the original M×K In the pre-defined basis matrix Ψ, extract the corresponding... N The row vectors formed by the target sampling points N×K A dimensional matrix. This submatrix Its function is to transform the low-dimensional coefficient vector oh The theoretical measurements mapped onto the sampling points serve as a bridge connecting the low-dimensional representation with the actual measurement data.

[0054] at the same time, In the sampling set S The observation vector. This vector is a N× A 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 it is used to generate coefficient vectors. oh The direct basis for the estimate.

[0055] By using the specific calculation formulas described above To solve for the coefficient vector oh To estimate the value, this application provides a direct and computationally efficient maximum likelihood estimation method. The formula utilizes a sample set.S Observation vector on 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.

[0056] In addition, such as Figure 3 As shown in the embodiments of this application, a satellite radiation pattern reconstruction system based on low-dimensional representation is also provided, including: 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and ; The sampling strategy planning module is used for sampling based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, ; Select to maximize the coefficient vector The objective is to estimate the determinant of the Fisher information matrix in the estimation process; The measurement and estimation module is used for the measurement of samples in the sample set S. N Measurements are taken at each target sampling point 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 pattern using the estimates obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. .

[0057] like Figure 4 He Ru Figure 5As shown, 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 radiation pattern reconstruction compared to traditional uniform sampling and random sampling methods.

[0058] 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.

[0059] 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.

[0060] 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and ; S2, based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process; S3, in the sampling set S N Measurements are taken at each target sampling point 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 first of the discrete cosine transform basis function matrix m Line 1 k Column elements The expression is: in, m For the 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 the basis function matrix is ​​the discrete cosine transform, the preset number of basis functions is... K It is determined based on preset reconstruction accuracy indicators, specifically through the following methods: Based on historical radiation patterns Perform the following steps: S11. Regarding the historical radiation pattern Perform a discrete cosine transform and calculate all discrete cosine transform coefficients. : in, -1; Historical radiation pattern The m One element; The normalization coefficient is defined as: S12. Based on S11, calculate the historical radiation pattern. h Total energy The calculation formula is: S13. Based on S11 and S12, determine the preset basis matrix. The number of pre-defined 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, The S2 uses a greedy D-optimal sampling algorithm from M Select from candidate sampling points One sampling point; The greedy D-optimal sampling algorithm includes the following steps: S21. Initialize the sampling set S to be empty and initialize the information matrix. A for ;in, >0 For regularization parameters, for The identity matrix; S22. For each sample set that does not belong to the sampling set S Candidate sampling points i Calculate candidate sampling points i Corresponding information gain ;in, For the preset basis matrix Middle and candidate sampling points i The corresponding row vector; 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, The information gain corresponding to the target sampling point; S24, Target sampling point sampling set S In the middle, update the sampling 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 sample set is reached. Include One target sampling point.

5. The method according to claim 1, characterized in that, In step S3, the coefficient vector is solved using the maximum likelihood estimation method. The estimated value The calculation formula is: ； in, For the preset basis matrix The central bank index belongs to the sample set. S submatrix, In the sampling set S The observation vector.

6. 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 5 comprises: 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.: in, M The total number of all candidate sampling points; K The number of basis functions is the preset number, and ; The sampling strategy planning module is used for sampling based on D Optimal sampling criteria, from M Select from candidate sampling points The target sampling points constitute the sampling set. S ;in, The selection is to maximize the coefficient vector. The objective is to estimate the determinant of the Fisher information matrix in the estimation process; Measurement and estimation module, used in the sample set S N Measurements are taken at each target sampling point 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 pattern using the estimates obtained from the measurement and estimation modules. and the preset basis matrix The complete radiation pattern was reconstructed. .

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