A method and system for three-dimensional target positioning based on fully decoupled atomic norm minimization of uniform planar array FDA-MIMO
By employing a fully decoupled atomic norm minimization method, the received signal is reconstructed into a single-parameter steering vector and reshaped into a three-dimensional low-rank tensor. A low-dimensional observation matrix is then constructed and semi-positive definite programming is performed. This solves the problems of high computational complexity and limited accuracy in three-dimensional positioning of the FDA-MIMO system, achieving efficient and accurate three-dimensional target positioning.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- INST OF ACOUSTICS CHINESE ACAD OF SCI
- Filing Date
- 2026-04-01
- Publication Date
- 2026-07-28
AI Technical Summary
Existing FDA-MIMO systems suffer from high computational complexity and limited accuracy in 3D positioning, especially under single-shot and low signal-to-noise ratio conditions, where existing decoupling methods lose coupling information, leading to a decrease in positioning accuracy.
The fully decoupled atomic norm minimization method is adopted to reconstruct the received signal into the Lekroneck product form of a single-parameter steering vector, which is then reshaped into a three-dimensional low-rank tensor. A low-dimensional observation matrix is constructed using a dimensionality reduction sampling operator, and the problem is transformed into a semi-positive definite programming problem through convex relaxation techniques. The optimal Toplitz matrix is obtained, and Vandermonde decomposition is performed to achieve three-dimensional full-space positioning of the target.
It significantly reduces computational complexity, improves positioning accuracy and multi-target resolution, and can adapt to the computational needs of larger arrays. In particular, it outperforms high-complexity algorithms under single-shot conditions and maintains high accuracy under low signal-to-noise ratio.
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Figure CN122469294A_ABST
Abstract
Description
Technical Field
[0001] This application belongs to the field of sonar positioning technology, specifically relating to a three-dimensional target positioning method and system based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization. Background Technology
[0002] Multiple-input multiple-output (MIMO) systems, leveraging the virtual aperture gain provided by waveform diversity techniques, have become crucial in target detection and estimation. To further enhance detection capabilities, frequency diversity array (FDA) technology has been incorporated into the MIMO architecture, forming the FDA-MIMO system. This system generates a range-angle dependent beam pattern by introducing minute frequency increments between transmit elements, thus enabling joint estimation of target range and angle.
[0003] Although there has been extensive research on positioning technology for FDA-MIMO systems, existing work is mostly limited to two-dimensional positioning based on uniform linear arrays (ULAs). However, in practical applications, it is often necessary to obtain the target's azimuth, elevation, and distance information, i.e., to achieve three-dimensional full-space positioning. Therefore, achieving three-dimensional positioning using FDA-MIMO systems based on uniform planar arrays (UPAs) has extremely high research value and application prospects.
[0004] According to current published literature, research on 3D localization for UPA-FDA-MIMO (Uniform Planar Array-Frequency Diversity Array-Multiple Input Multiple Output) systems is relatively limited. The few existing schemes are mainly based on traditional subspace-based methods (such as the MUSIC algorithm). While these methods can achieve high estimation accuracy and resolution under ideal conditions, they heavily rely on accurate estimation of the signal covariance matrix, leading to significant performance degradation under conditions of few snapshots (especially single snapshots) and low signal-to-noise ratio. To address the limitations of these subspace-based methods, gridless compressed sensing (CS) based on atomic norm minimization (ANM) has emerged as a promising alternative. The ANM method directly performs sparse recovery within the continuous parameter domain, effectively eliminating the "grid mismatch" problem in traditional compressed sensing. The existing technical solutions closest to this invention mainly fall into the following three categories: 1. Vectorized ANM (VANM): This method directly vectorizes the three-dimensional signal to construct a multi-level block Toeplitz (MLT) matrix, forming a high-dimensional semidefinite programming (SDP) optimization problem. This method is theoretically complete and has high accuracy.
[0005] 2. Multi-stage ANM: This method decomposes the complex three-dimensional parameter estimation problem into multiple cascaded low-dimensional ANM problems to reduce computational complexity.
[0006] 3. Decoupled ANM: This type of method attempts to decouple the MLT matrix structure in VANM. Existing typical strategies include Partially Decoupled Atom Norm Minimization (PDANM) and Three-Dimensional Decoupled Atom Norm Minimization (3D-DANM).
[0007] Although existing meshless atomic norm minimization (ANM) techniques have shown some potential in 3D localization of UPA-FDA-MIMO systems, there is still an irreconcilable contradiction between computational complexity and estimation accuracy in various methods. The specific drawbacks are as follows: 1. Vectorized ANM (VANM) has excessively high computational complexity, making it difficult to implement in engineering. The VANM method directly constructs the atomic norm based on the three-dimensional signal, and its optimization model includes a huge three-level block Toeplitz (MLT) matrix. The dimension of this matrix increases exponentially with the array aperture, resulting in an extremely heavy computational load for SDP solution and huge memory consumption, making it almost impossible to run in real-time in practical array systems.
[0008] 2. Multi-stage ANM loses joint information, limiting accuracy. To avoid high-dimensional computation, some methods employ a multi-stage estimation strategy, decomposing the 3D problem into multiple cascaded low-dimensional estimation problems. While this reduces complexity, it severs the joint information between distance and angle, causing errors to propagate step by step, ultimately resulting in a significant decrease in the final 3D positioning accuracy.
[0009] 3. Existing decoupled ANM methods (PDANM and 3D-DANM) still suffer from structural or performance bottlenecks. While the partially decoupled method (PDANM) partially decomposes the three-level block Toplitz matrix, its optimization model still retains a two-dimensional block Toplitz matrix. This reduces computational complexity, but for slightly larger planar arrays (UPAs), the computational load remains enormous, limiting its application in large-aperture arrays. The three-dimensional fully decoupled method (3D-DANM) further decomposes the high-dimensional structure into three independent one-dimensional Toplitz matrices, significantly reducing computation. However, this method has a major structural flaw in constructing SDP constraints: to achieve decoupling, it forces the second diagonal blocks connecting different parameter dimensions to be zero matrices. This mathematically discards the coupling correlation between distance and angle, leading to excessive model relaxation. Under low signal-to-noise ratio or single-shot conditions, this loss of structural information directly results in a significant decrease in positioning accuracy and severely weakens multi-target resolution capabilities. Summary of the Invention
[0010] To overcome the above-mentioned shortcomings, this application proposes a three-dimensional target localization method based on the minimization of the atomic norm of a uniform planar array FDA-MIMO, including: Step S1: Establish a signal model for the uniform planar array FDA-MIMO system. Reconstruct the received signal into the Lekronek product of a single-parameter steering vector through linear mapping. Then, reshape the received signal into a three-dimensional low-rank tensor. Use a dimensionality reduction sampling operator to extract feature slices and construct a low-dimensional observation matrix. Step S2: Construct a fully decoupled set of atoms and atomic norm for the low-dimensional observation matrix, and transform it into a semi-positive definite programming problem using convex relaxation techniques; solve the optimization problem using the interior point method to obtain the optimal Toplitz matrix corresponding to each dimension after decoupling; S3. Based on the optimal Toplitz matrix obtained by the solution, Vandermonde decomposition is performed to extract the target's distance, pitch angle and azimuth angle feature information, and the parameters of each dimension are jointly paired to finally achieve the three-dimensional full-space positioning of the target.
[0011] As an improvement to the above method, the establishment of the received signal model of the uniform planar array FDA-MIMO system includes: Received signal vector of a single snapshot Represented as: ; in, For the target quantity; It is a vector of zero-mean complex Gaussian white noise. and Let f be the number of rows and columns of a uniform planar matrix; The complex reflection coefficient includes the propagation phase. For the first The complex reflection coefficient of a target, The carrier frequency of the reference array element At the speed of light, For the first Radial distance of each target For complex units; For Kronecker product; and The first The transmit and receive steering vectors of each target are represented as follows: ; ; in, For the first The target vision and the goal The included angle of the axis; For the first The target vision and the goal The included angle of the axis; Represents the cross product of vectors; , , and As an auxiliary vector, it is represented as: ; ; ; ; in, is the frequency step size; T represents the matrix transpose.
[0012] As an improvement to the above method, the step of reconstructing the received signal into the Lekronek product form of a single-parameter steering vector through linear mapping includes: The received signal is reconstructed into corresponding... and The Lekronek product form of the single-parameter steering vector: ; Among them, parameters ; Represents the Khatri-Rao product; , and The signal parameter steering vector matrix is represented as: ; matrix A linear operator for mapping three-dimensional tensor signals to vectorized observations: ; in, Represents the face-splitting product; For length is A vector of all 1s; Let be the permutation matrix, for any vector and It satisfies the following properties: ; parameter Represented as: ; Among them, matrix The ( p,q ) elements are 1, where , , The remaining elements are 0. and for or .
[0013] As an improvement to the above method, the step of reshaping the received signal into a three-dimensional low-rank tensor includes: The dimension of the three-dimensional low-rank tensor is , The three-dimensional low-rank tensor is represented as: ; in, It is a three-dimensional low-rank tensor; This represents the outer product of vectors.
[0014] As an improvement to the above method, the step of extracting feature slices using a dimensionality reduction sampling operator to construct a low-dimensional observation matrix includes: From the three-dimensional low-rank tensor Extract the first frontal slice and the first side slice Constructing a low-dimensional observation matrix : ; in, This is a dimensionality reduction sampling operator.
[0015] As an improvement to the above method, the construction of a completely decoupled set of atoms and atomic norm for the low-dimensional observation matrix is expressed as follows: Atom set Represented as: ; The atomic norm is expressed as: .
[0016] As an improvement to the above method, the feature is that the problem is transformed into a semi-positive definite programming problem using convex relaxation techniques, expressed as: The semidefinite programming form is expressed as: ; in, For the reason The generated Toplitz matrix; Represents the trace of a matrix; For auxiliary variables; H represents the conjugate matrix; These are the variables to be optimized.
[0017] As an improvement to the above method, step S2 further includes: when the signal-to-noise ratio is lower than a set threshold, using tensor redundant observation information in conjunction with a multi-slice constraint mechanism, a new semidefinite programming problem is formed, expressed as: The semidefinite programming form is expressed as: ; in, and Represents signal tensor The k One front slice and one side slice; index set and The slices selected along each dimension are specified.
[0018] This application also provides a three-dimensional target localization system based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization, implemented using the above method. The system includes: A low-dimensional observation matrix module is constructed to establish a received signal model for a uniform planar array FDA-MIMO system. The received signal is reconstructed into the Lekronek product of a single-parameter steering vector through linear mapping, and then the received signal is reshaped into a three-dimensional low-rank tensor. The feature slices are extracted using the dimensionality reduction sampling operator to construct the low-dimensional observation matrix. The module for obtaining the optimal Toplitz matrix is used to construct a fully decoupled set of atoms and atomic norm for a low-dimensional observation matrix, and transform it into a semi-positive definite programming problem using convex relaxation techniques; the optimization problem is solved by the interior point method to obtain the optimal Toplitz matrix for each dimension after decoupling. The target's three-dimensional full-space positioning module is used to perform Vandermonde decomposition based on the solved optimal Toplitz matrix to extract the target's distance, pitch angle, and azimuth angle features, and to perform joint pairing processing on the parameters of each dimension to finally achieve the target's three-dimensional full-space positioning.
[0019] Compared with existing technologies, the advantages of this application are: 1. Compared to high-complexity atomic norm algorithms (VANM, PDANM), the CDANM algorithm framework proposed in this invention decomposes the originally complex three-level block Toplitz matrix into three independent first-level Toplitz matrices, significantly reducing the computational complexity of the optimization problem and enabling it to adapt to the computational needs of larger-scale arrays. The multi-slice constraint mechanism (MS-CDANM) flexibly introduces observation information from redundant slices in the signal tensor to construct additional constraints, effectively improving the algorithm's positioning accuracy under single-shot conditions, reaching a level comparable to the high-complexity algorithms VANM and PDANM. By controlling the number of added slices, a solution that flexibly balances computational efficiency and performance is provided for practical applications, facilitating the implementation of UPA-FDA-MIMO in actual 3D positioning.
[0020] 2. Compared with the low-complexity fully decoupled 3D-DANM, the CDANM algorithm framework introduces an auxiliary optimization variable matrix located at the connection between the distance and angle Toplitz blocks in the decoupled framework. This variable can implicitly represent and recover the joint coupling structure of distance and angle, thereby significantly improving computational efficiency while ensuring that the algorithm has a rigorous structure. It effectively avoids the performance loss caused by excessive relaxation in existing decoupled models, and improves estimation accuracy and multi-target resolution compared with 3D-DANM. Attached Figure Description
[0021] Figure 1 The diagram shown is a geometric structure of the UPA-FDA-MIMO system. Figure 2(a) shows the target localization results (VANM) of 100 independent Monte Carlo trials using four methods in a noise-free scenario. Figure 2(b) shows the target localization results (PDANM) of 100 independent Monte Carlo trials using four methods in a noise-free scenario. Figure 2(c) shows the target localization results of 100 independent Monte Carlo trials of four methods in a noise-free scene (3D-DANM). Figure 2(d) shows the target localization results (CDANM) of 100 independent Monte Carlo trials using four methods in a noise-free scenario. Figure 3(a) shows the RMSE performance (distance) of different algorithms. ); Figure 3(b) shows the RMSE performance (angle) of different algorithms. ); Figure 3(c) shows the RMSE performance (angle) of different algorithms. ); Figure 4(a) shows the planar array size. M ( M = N ) and target number K Phase transition diagram (3D-DANM); Figure 4(b) shows the planar array size. M ( M = N ) and target number K Phase transition diagram (CDANM); Figure 4(c) shows the planar array size. M ( M = N ) and target number K Phase transition diagram (MS-CDANM (Half)); Figure 4(d) shows the planar array size. M ( M = N ) and target number K Phase transition diagram (S-CDANM (Full)); Figure 5 The diagram shows the CPU execution time for different algorithms. Figure 6 The diagram shows the flowchart of the FDA-MIMO three-dimensional target localization method based on the minimization of the uniform planar array with fully decoupled atomic norm. Detailed Implementation
[0022] The technical solution of this application will be described in detail below with reference to the accompanying drawings.
[0023] The purpose of this invention is to provide a three-dimensional target localization method based on the minimization of the uniform planar array FDA-MIMO with complete decoupling atomic norm, aiming to achieve more efficient and accurate three-dimensional full-space target localization, so as to solve the problems of large computational load of existing high-dimensional parameter estimation and loss of accuracy of decoupling algorithms.
[0024] Specifically, addressing the issues of excessive computational complexity in traditional Vectorized ANM (VANM) and decreased accuracy in existing fully decoupled methods (such as 3D-DANM) due to the forced neglect of coupling information, this invention proposes a structure-preserving fully decoupled strategy. This strategy, on the one hand, decomposes the originally complex three-level block Toplitz matrix into three independent first-level Toplitz matrices through mathematical reconstruction, significantly reducing the computational complexity of the optimization problem and enabling it to adapt to the computational needs of larger-scale arrays. On the other hand, an auxiliary optimization variable matrix located at the connection point of the distance and angle Toplitz blocks is introduced into the decoupled framework. This variable implicitly represents and recovers the joint coupling structure of distance and angle, thereby significantly improving computational efficiency while ensuring the algorithm has a rigorous structure, effectively avoiding the performance loss caused by excessive relaxation in existing decoupled models.
[0025] Furthermore, addressing the potential limitations of basic decoupling schemes that rely solely on tensor skeleton information, this invention proposes a multi-slice constraint mechanism (MS-CDANM). This mechanism constructs additional constraints by flexibly introducing observation information from redundant slices in the signal tensor. This mechanism can suppress noise effects by utilizing redundancy in the data structure without significantly increasing computational costs. It not only effectively improves the algorithm's positioning accuracy under single-shot conditions, reaching a level comparable to high-complexity algorithms like VANM and PDANM, but also enhances multi-target resolution, providing a flexible solution for balancing computational efficiency and performance in practical applications.
[0026] Example 1 like Figure 6 As shown, the 3D target localization method based on the uniform planar array FDA-MIMO of fully decoupled atomic norm minimization provided in this application includes: Step S1: Establish a signal receiving model for a uniform planar array FDA-MIMO system. Reconstruct the received signal into the Lekronek product of a single-parameter steering vector through linear mapping to achieve parameter decoupling. Then, reshape it into a three-dimensional low-rank tensor and use a dimensionality reduction sampling operator to extract feature slices to construct a low-dimensional observation matrix. Step S2: Construct a fully decoupled set of atoms and atomic norms for the low-dimensional observation matrix. Utilize convex relaxation techniques to transform it into a positive semidefinite programming problem, converting the original three-layer Toplitz matrix into three independent Toplitz matrices, significantly reducing the dimensionality of the semidefinite programming problem. When the signal-to-noise ratio is low (below a set threshold), a multi-slice constraint mechanism can be combined to utilize tensor redundant observation information to form a new positive semidefinite programming problem. Solve the optimization problem using the interior-point method to obtain the optimal Toplitz matrix corresponding to each dimension after decoupling. S3. Based on the three independent optimal Toplitz matrices obtained by the solution, Vandermonde decomposition is performed to extract the target's distance, pitch angle and azimuth angle feature information, and the parameters of each dimension are jointly paired to finally achieve high-precision and efficient three-dimensional full-space positioning of the target.
[0027] Example 2 like Figure 1 As shown, consider a single-base UPA-FDA-MIMO system with co-located transmit and receive arrays, where the transmit and receive arrays share a common size. A uniform array. Let the distance between adjacent array elements be... And using the element in the upper left corner as the reference element. Assume the far field exists. The first goal, of which the second One goal Coordinates are , representing the radial distance of the target and the line of sight of the target, respectively. Shaft and The included angle of the axes. Define the first... line, number The array element of the column is the first Each transmitting element has a carrier frequency. for: (1) in, The carrier frequency of the reference array element It is the frequency step size and satisfies To prevent grating lobe effects and spatial blurring, the element spacing... Set to half the wavelength corresponding to the maximum transmission frequency. The transmission signal of each transmitting element It can be represented as: (2) in, It is the total energy emitted. The pulse width. Let be the baseband complex envelope. The complex envelopes of different array elements satisfy orthogonality, that is: (3) in, For any time delay. Similarly, the first... line, number The receiving array element of the column is defined as the first... If there are multiple receiving array elements, then that element receives signals from the target. The echo signal is: (4) In the formula, For the goal The complex reflection coefficient. and These respectively represent the signal from the first... Each launch element to the target And from its return to the The transmission delay of each receiving array element is specifically expressed as: (5) (6) in, The speed is the speed of light. The echo signal from each receiving array element undergoes down-conversion and matched filtering, taking all... After adding targets and noise, and introducing reasonable approximations, the received signal vector of a single snapshot is... It can be represented as: (7) in, For Kronecker product, The complex reflection coefficient includes the propagation phase. It is a vector of zero-mean complex Gaussian white noise. and The target The transmit and receive steering vectors can be decomposed into: (8) (9) here , , , It can be further broken down into: (10) in, Represents the cross product of vectors; auxiliary vector The definition is as follows: (11) (12) In the formula Pick or , Pick or Based on the above signal model, this invention aims to obtain the received signal... China and other countries jointly estimated the distance and angle parameters of multiple targets. However, the steering vector in formula (7) has multiple coupled parameters, which inevitably causes the estimation error of the preceding steps to be propagated to the subsequent steps, thus limiting the overall estimation accuracy. Therefore, it is necessary to further derive the decoupled signal model and reconstruct the received signal into corresponding parameters. and The Lekronek product form of the single-parameter steering vector: (13) in, Represents the Khatri-Rao product; The signal parameter steering vector matrix and its dimensions are as follows: (14) matrix A linear operator that maps three-dimensional tensor signals to vectorized observations: (15) in, Represents the face-splitting product; For length is A vector of all 1s Let be the permutation matrix, for any vector and It satisfies the following properties: (16) In addition, for the sake of simplicity Define matrix The ( p,q ) elements are 1, where , , All other elements are 0; These are all corresponding vector dimensions. Based on this, we have: (17) Based on the aforementioned distance-angle decoupled signal model (13), the recovered virtual signal vector Remodeled into a three-dimensional low-rank tensor The tensor The dimension is Its mathematical expression is: (18) in, Represents the cross product of vectors. These correspond to the dimensions of the distance and the two angles, respectively. To avoid the enormous computational cost of directly processing high-dimensional tensors, a dimensionality reduction sampling operator is introduced. tensor Feature extraction is performed. Specifically, from tensors... Extract the first frontal slice and the first side slice To construct a low-dimensional observation matrix And they are cascaded horizontally as follows: (19) Next, we construct an optimization problem based on minimizing the atomic norm, to obtain information from the observation matrix. To restore the target parameters, a new set of atoms is first constructed: (20) Based on this set of atoms, we can define The fully decoupled atomic norm (CDAN) is: (twenty one) The CDANM problem has been theoretically proven to be solvable using the following new semidefinite programming (SDP) form. (twenty two) in, For the reason The generated toplitz matrix, Represents the trace of a matrix; Let these be the variables to be optimized. After solving the optimization problem, the optimal solution can be obtained. , and Then, by using Vandermonde decomposition and pairing, the three-dimensional coordinates of the target can be obtained to achieve the positioning effect.
[0028] Here, it is worth noting that, compared to 3D-DANM, equation (22) has an additional auxiliary variable on this diagonal block. The existence of this variable implicitly preserves complete distance-angle structural information. Subsequent simulations have also verified this.
[0029] To address the potential limitations of CDANM which only utilizes tensor skeleton information, this invention proposes a multi-slice constraint mechanism (MS-CDANM). This mechanism flexibly introduces observation information from redundant slices in the signal tensor to construct additional constraints, thereby improving accuracy in noisy conditions. Its SDP form is as follows: (twenty three) in, and Represents signal tensor The k A front slice and a side slice. Index set. and This specifies the slices selected along each dimension. (Compare to the reference slice...) k =1) The relevant constraints are omitted because they are consistent with the constraints in equation (22). Based on the number of slices used, there are two commonly used modes: MS-CDANM (Full) and MS-CDANM (Half), which provide a trade-off between computational complexity and estimation accuracy.
[0030] Here, we conducted simulation experiments to evaluate the localization accuracy, multi-target identification capability, and runtime of the proposed algorithm. Three targets were randomly selected for the experiment, and the results were compared with state-of-the-art decoupled ANM algorithms. Figures 2(a)-2(d) show the results of 100 independent Monte Carlo trials in a noise-free scene, where black circles and red dots represent the true values and estimated values for each trial, respectively. As shown in Figures 2(a), 2(b), and 2(d), the estimated values of VANM, PDANM, and CDANM closely cluster around the true values. This confirms the structural effectiveness of CDANM, demonstrating its ability to successfully recover accurate target locations without mismatch. In contrast, Figure 2(c) shows that even in the absence of noise, 3D-DANM fails to provide reliable estimation results; its generated point cloud distribution is scattered, and there are numerous outliers, indicating a loss of structural information.
[0031] Next, we evaluate the estimation performance under different signal-to-noise ratio (SNR) conditions. Figures 3(a)–3(c) compare the root mean square error (RMSE) performance of the proposed CDANM and MS-CDANM with the baseline method at nine SNR levels (from -10 dB to 30 dB), with 50 independent Monte Carlo trials performed for each setting. As consistently observed across all three parameter dimensions, CDANM exhibits sensitivity to noise and a high RMSE because its minimum-slice sampling discards the noise-averaged advantage of the full observation tensor. Nevertheless, it still outperforms 3D-DANM at SNR ≥ 0 dB, confirming its significant information loss. To mitigate this sensitivity, a multi-slice strategy is employed: MS-CDANM (Full) performs close to the computationally expensive VANM and PDANM, while MS-CDANM (Half) achieves significantly better noise robustness compared to CDANM.
[0032] In addition to positioning accuracy, the experiment also investigated the multi-target identifiability of the proposed CDANM. The phase transition diagrams in Figures 4(a)-4(d) illustrate the changes in array size. M (in N = M ) and target quantity KThe empirical probability of successful localization when changes occur. Specifically, M The range is 3 to 11, while K The number of targets increased from 2 to 12. In this simulation, regardless of the array size, a minimum spacing of Δmin ≥ 0.06 was applied to the randomly generated targets. For each ( M , K Yes, the success probability was calculated using 20 Monte Carlo trials, employing the same success criteria as before. In these figures, grayscale intensity corresponds to the success probability, with white areas (lower right) representing successful localization and black areas (upper left) representing failure. Comparing Figure 4(a) with Figure 4(b) reveals that CDANM significantly outperforms 3D-DANM because it successfully resolves more targets within the same array size. Furthermore, Figures 4(b), 4(c), and 4(d) show that the successfully recovered area significantly expands as MS-CDANM uses more slices, indicating that utilizing multiple slices effectively increases the maximum number of identifiable targets, thereby improving overall capacity.
[0033] Finally, the efficiency of the proposed CDANM and MS-CDANM algorithms is verified by comparing the CPU runtime of different algorithms. Figure 5 As shown, VANM almost quickly becomes computationally intractable (when M When the value is greater than 4, this is due to memory exhaustion (marked as Out of Memory, OOM), while PDAMN exhibits a steep complexity curve, and in M The operation failed due to insufficient memory when the value was 12. Despite their high precision, these limitations indicate that they are not suitable for large-scale arrays.
[0034] In contrast, the proposed CDANM exhibits a more gradual increase in runtime, closely resembling the trend of the most efficient benchmark model, 3D-DANM. Furthermore, the MS-CDANM framework offers a flexible trade-off between complexity and performance. Although MS-CDANM (Full) is more expensive due to additional slicing constraints, it... M Before reaching the memory limit at 14, it remains significantly more efficient than PDANM while achieving almost the same accuracy. Meanwhile, MS-CDANM (Half) strikes a balance, offering a significant speed improvement over VANM and PDANM without compromising high-fidelity positioning.
[0035] Example 3 This application also provides a three-dimensional target localization system based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization, implemented using the above method. The system includes: A low-dimensional observation matrix module is constructed to establish a received signal model for a uniform planar array FDA-MIMO system. The received signal is reconstructed into the Lekronek product of a single-parameter steering vector through linear mapping, and then the received signal is reshaped into a three-dimensional low-rank tensor. The feature slices are extracted using the dimensionality reduction sampling operator to construct the low-dimensional observation matrix. The module for obtaining the optimal Toplitz matrix is used to construct a fully decoupled set of atoms and atomic norm for a low-dimensional observation matrix, and transform it into a semi-positive definite programming problem using convex relaxation techniques; the optimization problem is solved by the interior point method to obtain the optimal Toplitz matrix for each dimension after decoupling. The target's three-dimensional full-space positioning module is used to perform Vandermonde decomposition based on the solved optimal Toplitz matrix to extract the target's distance, pitch angle, and azimuth angle features, and to perform joint pairing processing on the parameters of each dimension to finally achieve the target's three-dimensional full-space positioning.
[0036] This application may also provide a computer device, including: at least one processor, memory, at least one network interface, and a user interface. The various components in this device are coupled together via a bus system. It is understood that the bus system is used to implement communication between these components. In addition to a data bus, the bus system also includes a power bus, a control bus, and a status signal bus.
[0037] The user interface can include a display, keyboard, or clicking device. Examples include a mouse, trackball, touchpad, or touchscreen.
[0038] It is understood that the memory in the embodiments disclosed in this application may be volatile memory or non-volatile memory, or may include both volatile and non-volatile memory. The non-volatile memory may be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory may be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as Static Random Access Memory (SRAM), Dynamic Random Access Memory (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced Synchronous DRAM (ESDRAM), Synchlink DRAM (SLDRAM), and Direct Rambus RAM (DRRAM). The memories described herein are intended to include, but are not limited to, these and any other suitable types of memory.
[0039] In some implementations, the memory stores elements such as executable modules or data structures, or subsets thereof, or extended sets thereof: operating systems and applications.
[0040] The operating system includes various system programs, such as the framework layer, core library layer, and driver layer, used to implement various basic business functions and handle hardware-based tasks. The application programs include various applications, such as media players and browsers, used to implement various application functions. Programs implementing the methods of the embodiments of this disclosure can be included in the application programs.
[0041] In the above embodiments, the processor can also invoke programs or instructions stored in memory, specifically programs or instructions stored in an application program, for the following purposes: Follow the steps described above.
[0042] The above methods can be applied to or implemented by a processor. The processor may be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above methods can be completed by integrated logic circuits in the processor's hardware or by software instructions. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the methods, steps, and logic diagrams disclosed above. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the disclosed methods can be directly implemented by a hardware decoding processor, or by a combination of hardware and software modules in the decoding processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory; the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above methods.
[0043] It is understood that the embodiments described in this application can be implemented using hardware, software, firmware, middleware, microcode, or a combination thereof. For hardware implementation, the processing unit can be implemented in one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), general-purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in this application, or combinations thereof.
[0044] For software implementation, the technology of this application can be implemented by executing the functional modules (e.g., procedures, functions, etc.) of this application. The software code can be stored in memory and executed by a processor. The memory can be implemented in the processor or outside the processor.
[0045] This application may also provide a non-volatile storage medium for storing a computer program. When the computer program is executed by a processor, it can implement the steps in the above method embodiments.
[0046] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application and are not intended to limit it. Although this application has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of this application do not depart from the spirit and scope of the technical solutions of this application, and should all be covered within the scope of the claims of this application.
Claims
1. A three-dimensional target localization method based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization, comprising: Step S1: Establish a signal model for the uniform planar array FDA-MIMO system. Reconstruct the received signal into the Lekronek product of a single-parameter steering vector through linear mapping. Then, reshape the received signal into a three-dimensional low-rank tensor. Use a dimensionality reduction sampling operator to extract feature slices and construct a low-dimensional observation matrix. Step S2: Construct a fully decoupled set of atoms and atomic norm for the low-dimensional observation matrix, and transform it into a semi-positive definite programming problem using convex relaxation techniques; solve the optimization problem using the interior point method to obtain the optimal Toplitz matrix corresponding to each dimension after decoupling; S3. Based on the optimal Toplitz matrix obtained by the solution, Vandermonde decomposition is performed to extract the target's distance, pitch angle and azimuth angle feature information, and the parameters of each dimension are jointly paired to finally achieve the three-dimensional full-space positioning of the target.
2. The 3D target localization method based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization according to claim 1, characterized in that, The establishment of the received signal model for the uniform planar array FDA-MIMO system includes: Received signal vector of a single snapshot Represented as: ； in, For the target quantity; It is a vector of zero-mean complex Gaussian white noise. and Let f be the number of rows and columns of a uniform planar matrix; The complex reflection coefficient includes the propagation phase. For the first The complex reflection coefficient of a target, The carrier frequency of the reference array element At the speed of light, For the first Radial distance of each target For complex units; For Kronecker product; and The first The transmit and receive steering vectors of each target are represented as follows: ； ； in, For the first The target vision and the goal The included angle of the axis; For the first The target vision and the goal The included angle of the axis; Represents the cross product of vectors; , , and As an auxiliary vector, it is represented as: ； ； ； ； in, is the frequency step size; T represents the matrix transpose.
3. The 3D target localization method based on the uniform planar array FDA-MIMO of fully decoupled atomic norm minimization according to claim 2, characterized in that, The process of reconstructing the received signal into the Lekronek product form of a single-parameter steering vector through linear mapping includes: The received signal is reconstructed into corresponding... and The Lekronek product form of the single-parameter steering vector: ； Among them, parameters ; Represents the Khatri-Rao product; , and The signal parameter steering vector matrix is represented as: ； matrix A linear operator for mapping three-dimensional tensor signals to vectorized observations: ； in, Represents the face-splitting product; For length is A vector of all 1s; Let be the permutation matrix, for any vector and It satisfies the following properties: ； parameter Represented as: ； Among them, matrix The ( p,q ) elements are 1, where , , The remaining elements are 0. and for or .
4. The 3D target localization method based on the uniform planar array FDA-MIMO of fully decoupled atomic norm minimization according to claim 3, characterized in that, The process of reshaping the received signal into a three-dimensional low-rank tensor includes: The dimension of the three-dimensional low-rank tensor is , The three-dimensional low-rank tensor is represented as: ； in, It is a three-dimensional low-rank tensor; This represents the outer product of vectors.
5. The FDA-MIMO three-dimensional target localization method based on fully decoupled atomic norm minimization of uniform planar arrays according to claim 4, characterized in that, The method of extracting feature slices using a dimensionality reduction sampling operator to construct a low-dimensional observation matrix includes: From the three-dimensional low-rank tensor Extract the first frontal slice and the first side slice Constructing a low-dimensional observation matrix : ； in, This is a dimensionality reduction sampling operator.
6. The 3D target localization method based on the minimization of the fully decoupled atomic norm using a uniform planar array FDA-MIMO according to claim 5, characterized in that, The construction of a fully decoupled set of atoms and atomic norm for a low-dimensional observation matrix is expressed as follows: Atom set Represented as: ； The atomic norm is expressed as: 。 7. The FDA-MIMO three-dimensional target localization method based on fully decoupled atomic norm minimization of uniform planar arrays according to claim 6, characterized in that, The problem is transformed into a semidefinite programming problem using convex relaxation techniques, as follows: The semidefinite programming form is expressed as: ； in, For the reason The generated Toplitz matrix; Represents the trace of a matrix; For auxiliary variables; H represents the conjugate matrix; These are the variables to be optimized.
8. The FDA-MIMO three-dimensional target localization method based on fully decoupled atomic norm minimization of uniform planar array according to claim 7, characterized in that, Step S2 further includes: when the signal-to-noise ratio is lower than a set threshold, combining the multi-slice constraint mechanism with tensor redundant observation information to form a new semidefinite programming problem, expressed as: The semidefinite programming form is expressed as: ； in, and Represents signal tensor The k One front slice and one side slice; index set and The slices selected along each dimension are specified.
9. A three-dimensional target localization system based on a uniform planar array FDA-MIMO with fully decoupled atomic norm minimization, implemented according to the method described in any one of claims 1-8, characterized in that, The system includes: A low-dimensional observation matrix module is constructed to establish a received signal model for a uniform planar array FDA-MIMO system. The received signal is reconstructed into the Lekronek product of a single-parameter steering vector through linear mapping, and then the received signal is reshaped into a three-dimensional low-rank tensor. The feature slices are extracted using the dimensionality reduction sampling operator to construct the low-dimensional observation matrix. The module for obtaining the optimal Toplitz matrix is used to construct a fully decoupled set of atoms and atomic norms for a low-dimensional observation matrix, transforming it into a semidefinite programming problem using convex relaxation techniques; the optimization problem is solved using the interior-point method to obtain the optimal Toplitz matrix for each dimension after decoupling; and... The target's three-dimensional full-space positioning module is used to perform Vandermonde decomposition based on the solved optimal Toplitz matrix to extract the target's distance, pitch angle, and azimuth angle features, and to perform joint pairing processing on the parameters of each dimension to finally achieve the target's three-dimensional full-space positioning.