A robust adaptive beamforming method based on dimension reduction space

Abstract

The application provides a robust adaptive beamforming method based on a dimension-reduced space. The method first decomposes an optimal beam into characteristic beams with different directivity and robustness, converts a design space of a beamformer from an array element domain to a characteristic beam domain, linearly represents an array response through a set of orthogonal characteristic beam base functions, and converts a weight optimization problem into flexible configuration of modal coefficients. On this basis, a low-dimensional robust subspace is constructed through modal truncation, and optimal modal coefficients are adaptively solved in the dimension-reduced space according to a minimum variance distortionless response criterion, so that an effective balance between directivity, robustness and interference suppression is achieved. The method disclosed by the application can output a higher signal-to-interference-and-noise ratio and has a stronger error resistance while keeping the robustness controllable, compared with the method disclosed in the prior art.

Description

Technical Field

[0001] This invention pertains to the fields of acoustics, array signal processing, and sonar technology, specifically a robust adaptive beamforming method based on a reduced-dimensional space. Background Technology

[0002] Beamforming is a technique that weights the data received by a sensor array to generate array gain, thus suppressing noise and interference while maintaining the desired signal. Superdirective beamformers can achieve higher array gain compared to conventional beamforming methods. However, superdirective methods are sensitive to random errors in sensor amplitude, phase, and position, making it difficult to achieve good theoretical performance. Therefore, a trade-off between array gain and robustness is usually necessary. Reference 1, "Y. Wang, Y. Yang, Y. MA, Z. He. Robust high-order superdirectivity of circular sensor arrays. J. Acoust. Soc. Am. 2014, 136, 1712–1724," discloses a robust superdirective beamforming method based on eigenvalue decomposition and synthesis. This method decomposes the optimal weight vector into orthogonal components with different robustness and directivity through eigenvalue decomposition, achieving a good balance between robustness and directivity. However, this method is essentially a "data-independent" fixed beamformer design, with its beam pattern predetermined. It cannot adaptively optimize for the presence of non-uniform noise and strong interference in the actual acoustic environment, thus its performance is not optimal in real-world conditions. Reference 2, "F. Zhang, C. Pan, J. Chen, J. Benesty. MPPCAD: Minimum PowerPattern Constrained Adaptive Differential Beamforming. IEEE Signal Process. Lett. 2025, 32, 2099-2103," proposes an adaptive differential beamforming method. This method dynamically shapes the beam pattern to suppress interference by adaptively optimizing polynomial coefficients. However, to ensure that the maximum beam response direction is always precisely aligned with the desired azimuth, the MPPCAD method must impose strict and non-convex constraints on all its coefficients. These constraints limit the design freedom of the beamformer, preventing it from fully utilizing higher-order modal degrees of freedom to form better beam shapes. Furthermore, this method is only applicable to linear arrays, limiting its applicability.Reference 3, "Somasundaram SD, Parsons N H, Li P, et al. Reduced-dimension robust capon beamforming using Krylov-subspace techniques [J]. IEEE Transactions on Aerospace and Electronic Systems, 2015, 51 (1): 270-289," proposes a dimension-reduced robust Capon beamforming (IRDRCB method) based on Krylov subspace techniques. This method reduces the problem dimension through data-dependent subspace projection. However, this data-dependent dimension reduction process amplifies the estimation error and model mismatch in the covariance matrix, limiting its application in real-world scenarios. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a robust adaptive beamforming method based on a reduced-dimensional space. For interference and non-uniform noise, the mode weighting coefficients are optimized in a low-dimensional robust space based on the minimum variance distortion-free criterion, thereby improving the output signal-to-interference-plus-noise ratio of the robust beamformer.

[0004] To address the aforementioned technical problems, embodiments of the present invention provide the following technical solution: a robust adaptive beamforming method based on dimensionality reduction space, comprising the following steps: The projection of the received data from the computational array into a reduced-dimensional space is used to calculate the L snapshots received by the M-element array. Projected onto a reduced-dimensional space spanned by the first N characteristic beams; The optimal N-dimensional modal weighting coefficients are calculated by solving the projection of the data covariance matrix into the reduced-dimensional space and the projection of the steering vector into the reduced-dimensional space. A robust adaptive beammap of order N is synthesized by linearly combining the characteristic beams of each order with the corresponding optimal mode weighting coefficients. The output signal-to-interference-plus-noise ratio, directivity index, and error sensitivity index of the robust adaptive beammap of order N are then calculated.

[0005] Furthermore, the L snapshot data received by the M-element array Projecting onto the reduced-dimensional space spanned by the first N characteristic beams, specifically, is expressed as: ; In the formula, Indicates the direction of signal incidence. and These are the azimuth and elevation angles of the signal, respectively; the symbol H represents the conjugate transpose. The matrix representing the first Nth order characteristic beams is expressed as: ; In the formula, The set representing the space of real numbers. Represents weight components of different orders, symbol " "Indicates complex conjugation, The array manifold representing the characteristic beam domain, yes The complex conjugate of the nth element ; In the formula, yes The matrix composed of the first N+1 eigenvectors, , It is a diagonal matrix, and its diagonal elements are... For different eigenvalues, the subscript "-1" indicates that the reciprocal is being calculated, and the eigenvalues ​​are sorted in descending order. This represents the corresponding eigenvector matrix. The normalized isotropic noise covariance matrix is ​​expressed as: ; In the formula: For the first Array element and the first Distance between array elements; In the formula, For the direction received by the array The expression for an incident unit amplitude plane wave signal is: ; In the formula, , , , Indicates the speed of sound. Let be the position vector of the m-th array element, where "T" represents the transpose.

[0006] Furthermore, the calculation of the N-dimensional optimal modal weighting coefficients specifically involves: ; In the formula, , It is the projection of the data covariance matrix into the reduced-dimensional space. It is the projection of the guiding vector into the reduced-dimensional space.

[0007] Furthermore, the synthesized Nth-order robust adaptive beammap is specifically as follows: ; In the formula, Defined as the nth order characteristic beam. .

[0008] Furthermore, the corresponding output signal-to-interference-plus-noise ratio, directivity index, and error sensitivity index are calculated: The output signal-to-interference-plus-noise ratio of an Nth-order robust adaptive beammap is expressed as: ; In the formula , These are the covariance matrices of the signal and the interference plus noise after projection onto the characteristic beam domain, respectively. This represents the interference plus noise covariance matrix of the element domain.

[0009] The directivity factor D of the Nth-order robust adaptive beammap is calculated by the following formula: ; In the formula This represents the normalized isotropic noise covariance matrix projected onto the low-dimensional feature beam domain.

[0010] Directional Index ; The error sensitivity function T of the Nth-order robust adaptive beammap is calculated by the following formula: ; The This represents the 2-norm of a vector, and the error sensitivity index. .

[0011] The beneficial effects of the above-described technical solution of the present invention are as follows: This invention first decomposes the optimal beam into characteristic beams with different directivity and robustness, transforming the beamformer design space from the element domain to the characteristic beam domain. A set of orthogonal characteristic beam basis functions linearly characterizes the array response, transforming the weight optimization problem into a flexible configuration of mode coefficients. Based on this, mode truncation is implemented to construct a low-dimensional robust subspace. In this reduced-dimensional space, the optimal mode coefficients are adaptively solved according to the minimum variance distortion-free response criterion, achieving an effective balance between directivity, robustness, and interference suppression. The method disclosed in this invention maintains controllable robustness while achieving higher signal-to-interference-plus-noise ratio (SNR) output compared to existing methods, and is applicable to any array. The super-directive beammap obtained by this invention has higher SNR output. It is applicable to any array configuration, and the beamformer design has greater freedom and operational flexibility. It also exhibits better robustness in the presence of element position errors. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the array used in the simulation.

[0013] Figure 2This is a performance comparison chart of the EBMT and EBADS methods at different orders.

[0014] Figure 3 These are composite beam diagrams at various frequencies of 200Hz.

[0015] Figure 4 This is a robustness analysis diagram with positional errors. Detailed Implementation

[0016] To make the technical problems, technical solutions and advantages of the present invention clearer, a detailed description will be given below in conjunction with the accompanying drawings and specific embodiments.

[0017] This invention proposes a robust adaptive beamforming method based on dimensionality reduction space, comprising the following steps: The projection of the received data from the computational array into a reduced-dimensional space is used to calculate the L snapshots received by the M-element array. Projected onto a reduced-dimensional space spanned by the first N characteristic beams; The optimal N-dimensional modal weighting coefficients are calculated by solving the projection of the data covariance matrix into the reduced-dimensional space and the projection of the steering vector into the reduced-dimensional space. A robust adaptive beammap of order N is synthesized by linearly combining the characteristic beams of each order with the corresponding optimal mode weighting coefficients. Calculate the output signal-to-interference-plus-noise ratio, directivity index, and error sensitivity index corresponding to the Nth-order robust adaptive beammap.

[0018] In this embodiment, the L snapshot data received by the M-element array are... Projecting onto the reduced-dimensional space spanned by the first N characteristic beams, specifically, is expressed as: ; In the formula, Indicates the direction of signal incidence. and These are the azimuth and elevation angles of the signal, respectively; the symbol H represents the conjugate transpose. The matrix representing the first Nth order characteristic beams is expressed as: ; In the formula, The set representing the space of real numbers. Represents weight components of different orders, symbol " "Indicates complex conjugation, The array manifold representing the characteristic beam domain, yes The complex conjugate of the nth element ; In the formula, yes The matrix composed of the first N+1 eigenvectors, , It is a diagonal matrix, and its diagonal elements are... For different eigenvalues, the subscript "-1" indicates that the reciprocal is being calculated, and the eigenvalues ​​are sorted in descending order. This represents the corresponding eigenvector matrix. The normalized isotropic noise covariance matrix is ​​expressed as: ; In the formula: For the first Array element and the first Distance between array elements; In the formula, For the direction received by the array The expression for an incident unit amplitude plane wave signal is: ; In the formula, , , , Indicates the speed of sound. Let be the position vector of the m-th array element, where "T" represents the transpose.

[0019] In this embodiment, the calculation of the N-dimensional optimal modal weighting coefficients is as follows: ; In the formula, , It is the projection of the data covariance matrix into the reduced-dimensional space. It is the projection of the guiding vector into the reduced-dimensional space.

[0020] In this embodiment, the synthesis of an Nth-order robust adaptive beammap is specifically as follows: ; In the formula, Defined as the nth order characteristic beam. .

[0021] In this embodiment, the corresponding output signal-to-interference-plus-noise ratio, directivity index, and error sensitivity index are calculated: The output signal-to-interference-plus-noise ratio of an Nth-order robust adaptive beammap is expressed as: ; In the formula , These are the covariance matrices of the signal and the interference plus noise after projection onto the characteristic beam domain, respectively. This represents the interference plus noise covariance matrix of the element domain.

[0022] The directivity factor D of the Nth-order robust adaptive beammap is calculated by the following formula: ; In the formula This represents the normalized isotropic noise covariance matrix projected onto the low-dimensional feature beam domain.

[0023] Directional Index ; The error sensitivity function T of the Nth-order robust adaptive beammap is calculated by the following formula: ; The This represents the 2-norm of a vector, and the error sensitivity index. .

[0024] The following specific embodiments illustrate the principles and technical effects of the present invention: A two-layer circular array consisting of 16 elements is employed, with each layer containing 8 elements and diameters of 5 meters (outer layer) and 3 meters (inner layer). In the simulation, one sensor is removed, resulting in an irregular array, to verify the generality of the proposed method. The element positions are as follows... Figure 1 As shown in the figure. The speed of sound is set to 1500 m / s and the signal frequency to 200 Hz in the simulation. The desired signal originates from the (90°, 237°) direction, while two strong directional interferences are introduced from the (90°, 67°) and (90°, 331°) directions. Spherical isotropic noise is considered, with a signal-to-noise ratio of 0 dB, a signal strength of 10 dB, and an interference strength of 20 dB. The simulation compares conventional beamforming (CBF), data-independent integer-order eigenbeam decomposition and synthesis (EBMT), and the IRDRCB method based on Krylov subspace technology.

[0025] Figure 2 This is a performance comparison chart of the EBMT and EBADS methods at different orders. Where: Figure 2 Sub-figure (a) is the EBMT beammap. Figure 2 Sub-figure (b) is the EBADS beammap. Figure 2 Subgraph (c) is the modal excitation coefficient diagram of EBADS. Figure 2 Subgraph (d) is the DIs diagram. Figure 2 Subgraph (e) is the SIs diagram. Figure 2 Subplot (f) is the SINRs plot. Figure 2 The performance of EBMT and EBADS methods was analyzed as the synthesis order varied from 0 to the highest order. Figure 2 Subgraph (a) and Figure 2The beamplot comparison in sub-figure (b) shows that the main lobe of both methods gradually narrows with increasing synthesis order, reflecting the contribution of higher-order modes to improved directivity. The core difference lies in null characteristics: EBMT, as a data-independent method, predetermines null positions based on array geometry and truncation order; while the EBADS method exhibits clear data-driven characteristics. Although some nulls vary with order in the beamplot, starting from order 3, it can generate stable nulls at 67° and 331° in two fixed interference directions. However, when N=0, 1, 2, the null characteristics are not obvious due to the severely insufficient spatial degrees of freedom provided by lower-order modes. Figure 2 The modal coefficient distribution in subgraph (c) reveals the intrinsic mechanism by which this method achieves flexible beamforming: as the order increases, highly directive higher-order characteristic beams are effectively utilized to synthesize beams with narrower main lobes. Both methods show a synchronous increase in DI and SI with increasing order, indicating that achieving high directivity comes at the cost of reduced robustness, and the performance of the two methods is similar. However, the SINR of the two methods exhibits significant differences, such as... Figure 2 As shown in sub-figure (f), the EBADS method can maintain a high and stable SINR at different orders, demonstrating its effective adaptability to interference environments; while the SINR of the EBMT method fluctuates greatly, and its performance is highly dependent on whether the null point of the beam pattern at a specific order is exactly aligned with the direction of interference.

[0026] Figure 3 These are the composite beamforms at various frequencies of 200Hz. Figure 3 Subgraph (a) N=5. Figure 3 Subgraph (b) N=8. Figure 3 Subgraph (c) N=11. Figure 3 The synthesized beam plots shown and the quantitative indicators provided in Table 1 reveal the performance trade-offs of the two methods at different degrees of freedom.

[0027] Table 1 At order N=11, the EBMT method achieved the highest directivity DI (12.86 dB), but also the worst robustness (SI = 16.59 dB) and the lowest SINR among all orders. In contrast, the EBADS method exhibited good overall performance at N=11, with a DI (12.47 dB) very close to that of EBMT, while achieving a much higher SINR (13.51 dB), directly benefiting from the stable null formed in the interference direction by its data-driven characteristics. However, EBADS also showed poor robustness at this order (15.50 dB), indicating that both methods sacrificed robustness for high directivity in higher-order modes. At N=8, EBMT achieved the narrowest main lobe (39.89°) and a relatively high DI (11.46 dB), but its robustness remained poor. The EBADS method achieved the highest SINR (13.24 dB) at N=8, while maintaining almost the same main lobe width (38.89°) and DI (11.10 dB) as EBMT. More importantly, its robustness (SI = -1.1 dB) is also far superior to EBMT of the same order. At lower orders (N=5), the differences in strategy between the two methods are most significant. EBMT prioritizes directivity and main lobe width using limited degrees of freedom, but its SINR improvement is limited. EBADS, on the other hand, prioritizes interference immunity and robustness at low degrees of freedom, exhibiting a better SINR (12.26 dB) and the best robustness (SI = -6.21 dB), at the cost of significant main lobe widening and reduced DI. These quantitative comparisons for specific orders demonstrate the effectiveness of the EBADS method.

[0028] Figure 4 Subgraph (a) Figure 4 subgraph (b) and Figure 4 Subplot (c) illustrates the performance of the EBADS and IRDRCB methods at different synthesis orders when random errors exist in the element positions. Four error levels were set in the simulation (with standard deviations of 10...). -1 10 -2 10 -3 With 10 -4 (meters), signal-to-noise ratio set to 0 dB, snapshot count set to 1000. Under low-order conditions ( The performance curves of the EBADS method highly overlap at different error levels, indicating its inherent insensitivity to array mismatch at low degrees of freedom. As the order increases, the performance evolution shows a clear dependence on the error level: at smaller errors (e.g., 10⁻⁶), the performance becomes significantly more dependent on the error level. -4 When m), DI and SINR can continuously increase with the order, until The performance values ​​reached 12.87 dB and 13.59 dB respectively; however, as the error increased, the performance curve showed an inflection point that first rose and then fell, and the larger the error, the earlier the inflection point appeared. However, for the EBADS method, SI also showed a dependence on the error level; generally, at the same order, the larger the applied error, the higher the SI value. The trends of DI, SI, and SINR of the IRDRCB method at different orders were basically consistent with those of the EBADS method, indicating that both have similar mechanisms in dealing with array errors. However, since the dimensionality reduction space constructed by IRDRCB depends entirely on the array received data, its performance is more significantly affected by errors in the data. Figure 4 The comparison highlights the advantages of the EBADS method in constructing robust subspaces that are independent of data, enabling it to maintain high performance even in real-world environments with element errors.

[0029] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A robust adaptive beamforming method based on dimensionality reduction space, characterized in that, Includes the following steps: The projection of the received data from the computational array into a reduced-dimensional space is used to calculate the L snapshots received by the M-element array. Projecting onto the reduced-dimensional space spanned by the first N characteristic beams, specifically, is expressed as: ； In the formula, Indicates the direction of signal incidence. and These are the azimuth and elevation angles of the signal, respectively; the symbol H represents the conjugate transpose. The matrix representing the first Nth order characteristic beams is expressed as: ； In the formula, The set representing the space of real numbers. Represents weight components of different orders, symbol " "Indicates complex conjugation, The array manifold representing the characteristic beam domain, yes The complex conjugate of the nth element ; In the formula, yes The matrix composed of the first N+1 eigenvectors, , It is a diagonal matrix, and its diagonal elements are... For different eigenvalues, the subscript "-1" indicates that the reciprocal is being calculated, and the eigenvalues ​​are sorted in descending order. This represents the corresponding eigenvector matrix. The normalized isotropic noise covariance matrix is ​​expressed as: ； In the formula: For the first Array element and the first Distance between array elements; In the formula, For the direction received by the array The expression for an incident unit amplitude plane wave signal is: ； In the formula, , , , Indicates the speed of sound. Let be the position vector of the m-th element, where "T" represents the transpose; The optimal N-dimensional modal weighting coefficients are calculated by solving the projection of the data covariance matrix into the reduced-dimensional space and the projection of the steering vector into the reduced-dimensional space. A robust adaptive beammap of order N is synthesized by linearly combining characteristic beams of each order with their corresponding optimal mode weighting coefficients. The output signal-to-interference-plus-noise ratio (SIR), directivity index, and error sensitivity index of the robust adaptive beammap of order N are then calculated. Specifically, the synthesis of the robust adaptive beammap of order N involves: ； In the formula, Defined as the nth order characteristic beam. ; The calculation of the output signal-to-interference-plus-noise ratio, directivity index, and error sensitivity index corresponding to the Nth-order robust adaptive beammap is specifically as follows: The output signal-to-interference-plus-noise ratio of an Nth-order robust adaptive beammap is expressed as: ； In the formula , These are the covariance matrices of the signal and the interference plus noise after projection onto the characteristic beam domain, respectively. The signal covariance matrix in the element domain. Represents the interference plus noise covariance matrix of the element domain; The directivity factor D of the Nth-order robust adaptive beammap is calculated by the following formula: ； In the formula The normalized isotropic noise covariance matrix projected onto the low-dimensional characteristic beam domain; directivity index. ; The error sensitivity function T of the Nth-order robust adaptive beammap is calculated by the following formula: ； The This represents the 2-norm of a vector, and the error sensitivity index. .

2. The robust adaptive beamforming method based on dimensionality reduction space according to claim 1, characterized in that, The calculation of the N-dimensional optimal modal weighting coefficients is specifically as follows: ； In the formula, , It is the projection of the data covariance matrix into the reduced-dimensional space. It is the projection of the guiding vector into the reduced-dimensional space.

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