A DOA super-resolution estimation method and product based on toeplitz covariance enhancement and neural network

Abstract

This invention discloses a DOA super-resolution estimation method and product based on Toeplitz covariance enhancement and neural networks. The method includes: acquiring received data from a uniform linear array, calculating the sample covariance matrix, and performing trace normalization; performing Toeplitz prior correction by averaging the elements of each diagonal and backfilling to construct the Toeplitz prior covariance, and performing Hermitization; based on the sample covariance matrix and the Toeplitz prior covariance, adopting a linear shrinkage model and determining the optimal shrinkage coefficient using the Frobenius norm mean square error as the criterion to obtain the shrinkage covariance matrix; performing vectorized projection onto an overcomplete array dictionary to obtain complex projection features; splitting the features into real and imaginary parts to form a dual channel, inputting it into a convolutional neural network to obtain a spatial spectral vector; and performing spectral peak search to extract the angles corresponding to the K peaks with the largest amplitudes to obtain the DOA estimation result.

Description

Technical Field

[0001] This invention belongs to the fields of radio communication, array signal processing, and intelligent signal processing technology, and particularly relates to a DOA super-resolution estimation method and product based on Toeplitz covariance enhancement and neural networks. Background Technology

[0002] Direction of Arrival (DOA) estimation is a critical issue in array signal processing. Traditional subspace methods such as Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) rely on accurate estimation and eigenvalue decomposition of the sample covariance matrix. Under conditions of insufficient snapshots and low signal-to-noise ratio, these methods are prone to subspace separation instability, spectral peak broadening, and spurious peaks due to large sample covariance variance and deterioration of the condition number, leading to a decrease in resolution. In multipath or strong interference scenarios, source coherence / high correlation can cause covariance rank deficiency, further degrading traditional methods. For the Toeplitz structure of covariance in uniform linear arrays (ULAs), techniques such as Toeplitz correction / structured reconstruction and shrinkage covariance estimation have been developed to improve the stability of small-sample estimation. However, existing deep learning-based solutions often lack a unified front-end that integrates Toeplitz structure priors, closed-loop shrinkage robustness, and coherent source enhancement mechanisms in an analytical and reproducible manner. Furthermore, directly using the sample covariance matrix (or its separated real / imaginary parts) as neural network input without structured statistical preprocessing suffers from high dimensionality, heavy training burden, and limited generalization. These technical bottlenecks make it difficult for DOA estimation to balance estimation accuracy, real-time response speed, and adaptability to complex environments in practical scenarios such as radar detection, wireless communication positioning, and sonar target tracking, severely restricting the deployment efficiency and application expansion of related systems. Summary of the Invention

[0003] To address the problems of sample covariance distortion caused by limited snapshots and low signal-to-noise ratio, underutilization of array structure priors, and rank deficiency in coherent signal source scenarios, this invention aims to overcome the aforementioned shortcomings of existing technologies by proposing a DOA super-resolution estimation method and product based on Toeplitz covariance enhancement and neural networks.

[0004] In view of this, the present invention proposes a DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural networks, comprising: Step 1: Acquire the received data from the uniform linear array, calculate the sample covariance matrix, and perform trace normalization. Step 2: Perform Toeplitz prior correction by averaging the elements of each diagonal and backfilling to construct the Toeplitz prior covariance, and then Hermitize it. Step 3: Based on the sample covariance matrix and the Toeplitz prior covariance, a linear shrinkage model is adopted, and the optimal shrinkage coefficient is determined by the Frobenius norm mean square error criterion, thus obtaining the shrinkage covariance matrix; Step 4: Vectorize the contracted covariance matrix and project it onto the overcomplete array dictionary to obtain the complex projection features; Step 5: Decompose the complex projection features into real and imaginary parts, form a dual-channel input to the convolutional neural network, and obtain the spatial spectral vector; Step 6: Perform a spectral peak search on the spatial spectral vector, extract the angles corresponding to the K peaks with the largest amplitudes, and obtain the DOA estimation results.

[0005] As an improvement to the above method, the sample covariance matrix calculated in step 1 for:

[0006] in, For the number of snapshots, For a uniform linear array receiving snapshot vector, n represents the nth snapshot, and the superscript H represents the conjugate transpose; Sample covariance matrix after trace normalization for:

[0007] in, To take the real part of a complex number, Represents the trace of a matrix.

[0008] As an improvement to the above method, step 2 specifically includes: Offset for each diagonal Find the mean ,in, The number of elements in a uniform linear array; Extraction operator for diagonal; Construct the Toeplitz prior covariance matrix , No. Set all elements on the diagonal to ; right Hermitization: .

[0009] As an improvement to the above method, the optimal shrinkage coefficient in step 3 The following results were obtained using closed-form estimation: when season ; when hour,

[0010] in, To truncate the operator, , ; Output shrinkage covariance matrix for: .

[0011] As an improvement to the above method, the method further includes performing multi-Toeplitz enhancement preprocessing before step 3 to address coherent signal sources, specifically including: Let the time shift be τ, where τ is an integer and satisfies... ; Corresponding time-shifted sample covariance for: ; right Toeplitz correction and shrinkage were performed sequentially to obtain Then based on Constructing multiple Toeplitz matrices: make Define an m×m flip matrix J, whose elements satisfy: If and only if p + q = m + 1, otherwise ; For any matrix C, the forward and backward averaging yields the enhancement matrix. ,in Indicates conjugation; when constructing a Toeplitz matrix from the first column vector c and the first row vector r. When T satisfies: if , ;otherwise, ; This is the sample covariance matrix input for subsequent steps.

[0012] As an improvement to the above method, step 4 specifically includes: In the preset angle grid { ,…, On}, for each grid point Calculate the steering vector ; exist Construct dictionary column vectors above: ; Construct an overcomplete array dictionary by superimposing column vectors. for: ; Vectorize the shrinkage covariance: And project: ; The complex projection feature is split into real and imaginary input channels: .

[0013] As an improvement to the above method, the convolutional neural network is composed of three cascaded one-dimensional convolutional layers, including: First layer: kernel size 11, input channels 2, output channels 12, batch normalization, ReLU activation; Second layer: kernel size 5, input channels 12, output channels 5, batch normalization, ReLU activation; The third layer has a kernel size of 3, 5 input channels, 1 output channel, batch normalization, and the output is flattened to obtain a spatial spectral vector of length L.

[0014] As an improvement to the above method, the spectral peak search in step 6 specifically involves: given the number of information sources K, selecting the network output spectrum... Take the K indices with the largest amplitude values Output the corresponding angle , and output in ascending order.

[0015] On the other hand, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the above-described method.

[0016] Compared with the prior art, the advantages of the present invention are: The key point of this invention is to use the Toeplitz translation-invariant structure of ULA as a prior explicit embedding covariance estimation. By enhancing the resolvable information of coherent scenes through multiple Toeplitz enhancements, the network no longer relies on self-learning to compensate for statistical instability. This significantly improves estimation accuracy, resolution and robustness under conditions of low snapshot, low SNR and coherent multi-source, while also possessing reproducible and easy-to-deploy engineering capabilities.

[0017] The super-resolution estimation method of this invention can be applied to scenarios requiring high-resolution direction finding, such as beamforming and interference suppression of wireless communication base stations / terminals, direction finding of satellite communication links, azimuth measurement of radar / sonar targets, radio monitoring and positioning, and radio astronomy observation. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the main process of the DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network; Figure 2 This is a schematic diagram of Toeplitz prior correction and shrinkage coefficient calculation; Figure 3 This is a schematic diagram of overcomplete dictionary construction and projection features; Figure 4 This is a block diagram of the Toeplitz-enhanced coherent signal processing branch; Figure 5 These are RMSE performance graphs under different signal-to-noise ratios; Figure 6 These are performance graphs of resolution accuracy under different signal-to-noise ratios; Figure 7 These are RMSE performance graphs at different angle intervals; Figure 8 This is a graph showing the resolution accuracy performance at different angular intervals; Figure 9 This is a graph showing the RMSE performance at different snapshot counts; Figure 10 This is a graph showing the resolution accuracy performance at different snapshot counts. Detailed Implementation

[0019] This invention proposes a Toeplitz Enhanced Neural Network for DOA estimation (TENN-DOA) super-resolution estimation method. The method includes: Step 1: Acquire the received data from the uniform linear array, calculate the sample covariance matrix, and perform trace normalization. Step 2: Perform Toeplitz prior correction by averaging the elements of each diagonal and backfilling to construct the Toeplitz prior covariance, and then Hermitize it. Step 3: Based on the sample covariance matrix and the Toeplitz prior covariance, a linear shrinkage model is adopted, and the optimal shrinkage coefficient is determined by the Frobenius norm mean square error criterion, thus obtaining the shrinkage covariance matrix; Step 4: Vectorize the contracted covariance matrix and project it onto the overcomplete array dictionary to obtain the complex projection features; Step 5: Decompose the complex projection features into real and imaginary parts, form a dual-channel input to the convolutional neural network, and obtain the spatial spectral vector; Step 6: Perform a spectral peak search on the spatial spectral vector, extract the angles corresponding to the K peaks with the largest amplitudes, and obtain the DOA estimation results.

[0020] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0021] Example 1 Embodiment 1 of the present invention proposes a DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural networks, such as... Figure 1 The diagram shown is a flowchart of the method implementation.

[0022] (1) Array model and parameter definition The number of ULA array elements is The number of quick shots is The number of information sources is The guiding vector is:

[0023] in For carrier wavelength, The element spacing is used. In this embodiment, the carrier frequency is employed. , And take half wavelength spacing .

[0024] Angular grid is In the example implementation , .

[0025] (2) Standard SNR with added noise Observations of noiseless array The average signal power can be defined as: When X is the array noiseless observation matrix (M is the number of array elements, N is the number of snapshots), the above formula is equivalent to taking the arithmetic mean of the modulus square of all observation samples.

[0026] Let the target signal-to-noise ratio be Corresponding linear ratio The noise level meets the following requirements:

[0027] Generate zero mean and variance White noise n and obtain .

[0028] (3) Sample covariance and scaling normalization From (2), we can obtain the array receiver. A quick snapshot vector Its true covariance matrix is The sample covariance matrix is ​​defined as follows:

[0029] To enhance scale robustness, trace normalization is performed:

[0030] (4) Toeplitz prior correction (diagonal average backfill + Hermitization) like Figure 2 The diagram shown illustrates the Toeplitz prior correction and the calculation of the shrinkage coefficient.

[0031] To facilitate implementation and reproduction, a diagonal extraction operator is defined. When k≥0, When k < 0, .make for The arithmetic mean of the elements is the Toeplitz prior covariance. satisfy .

[0032] For a uniform linear array, the ideal covariance has a Toeplitz structure, therefore... Toeplitz prior covariance constructed by backfilling the diagonal mean Offset for each diagonal line Find the mean And the Toeplitz matrix Set all diagonal elements to The mathematical representation of the formula and its Hermitianization are as follows:

[0033] To reduce the impact of sample covariance estimation error on subsequent eigenvalue decomposition and spectral estimation under finite snapshot conditions.

[0034] (5) Deterministic linear contraction coefficient and contraction covariance output To reduce the impact of limited sample snapshots on covariance error, a linear shrinkage covariance estimation method is proposed by introducing a Toeplitz structure prior.

[0035]

[0036] The shrinkage coefficient is selected based on the mean square error of the Frobenius norm:

[0037] Define the difference matrix: , .but Substitute it into And unfolded to:

[0038] right Taking the derivative and setting it to zero yields the optimal shrinkage coefficient: right Taking the derivative and setting it to zero yields the optimal shrinkage coefficient:

[0039] because Cannot be observed directly. The expected value term is estimated using a closed-form method with a sample computable statistic:

[0040] To ensure that the shrinkage coefficient satisfies 0 ≤ α ≤ 1, its estimated value is truncated: α = clip(α ,0,1)=min(max(α ,0),1). Where clip(·) is the truncation operator, used to restrict the input to a given closed interval.

[0041] in To The closed-form estimator, This is the numerical stability constant. It is obtained from the above equation. Then, output linear shrinkage covariance. This serves as the subsequent dictionary projection and input to the convolutional neural network.

[0042] when season .when At that time, take:

[0043]

[0044] Output shrinkage covariance:

[0045] (6) Multi-Toeplitz enhanced pretreatment To address coherent source signals and further enhance information, multi-Toeplitz enhancement preprocessing is performed. Let the time shift (Delay) be denoted as τ, where τ is an integer and satisfies... The corresponding time-shifted sample covariance can be estimated as follows: .in This indicates the conjugate transpose. (For) Toeplitz correction and shrinkage were performed sequentially to obtain Then based on Constructing multiple Toeplitz matrices: make Define an m×m flip matrix J whose elements satisfy: If and only if p + q = m + 1, otherwise For any matrix C, averaging forward and backward yields the enhanced matrix. ,in This represents conjugation. When constructing a Toeplitz matrix from the first column vector c and the first row vector r... When T satisfies and .

[0046] (7) Overcomplete array dictionary projection and dual-channel feature construction like Figure 3 The diagram shown illustrates the construction of an overcomplete dictionary and projected features. Figure 4 This is a block diagram of the Toeplitz-enhanced coherent signal processing branch.

[0047] In the angle grid Construct dictionary column vectors above:

[0048] dictionary matrix: Vectorize the contraction covariance: And project:

[0049] The complex projection feature is split into real and imaginary input channels:

[0050] (8) Spatial Spectrum Regression of Convolutional Neural Networks Network input is The output is The spatial spectral vector. The network consists of three concatenated one-dimensional convolutional layers, with the parameters of each layer fixed as follows: Conv1d:2→12,kernel=11,padding=5,BatchNorm,ReLU; Conv1d:12→5,kernel=5,padding=2,BatchNorm,ReLU; Conv1d:5→1,kernel=3,padding=1,Flatten outputs a spectral vector of length L.

[0051] Training employs mean squared error loss and the Adam optimizer, with a learning rate of [missing information]. The iteration rounds are 200, and the weights of the best validation set are saved.

[0052] (9) DOA output (spectral peak to angle) Given the number of information sources K, the final output spectrum of the network is... Take the K indices with the largest amplitude values Output the corresponding angle The results are sorted in ascending order as the DOA estimation results. The training labels are one-hot encoded: each true angle is mapped to the nearest grid point and set to 1.

[0053] Example 2 Embodiments of the present invention may also provide a computer program product, including a computer program. When the computer program is executed by a processor, it can implement the various steps in the above method embodiments.

[0054] Experimental data: Figure 5 , Figure 6 The RMSE and PoR curves of various algorithms are presented under the settings of 200 snapshots and angle sets {-5°, 4°, 20°}, with SNR ranging from -15dB to 10dB (26 values ​​in total). The curve trends in the figure show that in the low SNR range (e.g., around -15dB to -10dB), traditional subspace methods, even with spatial smoothing, still exhibit higher RMSE, lower resolution probability, and tend to broaden spectral peaks. In contrast, the method of this invention achieves lower RMSE and higher PoR under the same conditions. This demonstrates that by suppressing the covariance instability caused by small samples / low SNR through time-shifted covariance, Toeplitz prior correction, and closed-loop linear contraction, it can obtain more stable subspace information and more reliable angle estimation output.

[0055] Figure 7 , Figure 8 The curves showing the changes in RMSE and PoR of various algorithms with angle intervals gradually increased from 0.2° to 18° under the conditions of two coherent sources, 64 snapshots, and SNR=10dB, with a fixed angle center of 0°. The curves show that when the angle interval is small, traditional methods are more prone to misjudgments due to peak merging or spurious peaks, leading to a significant increase in RMSE and a significant decrease in PoR. The method of this invention achieves the PoR≥50% criterion even with a smaller angle interval (0.4°), and enters the stable region of PoR≈1 more quickly as the angle interval gradually increases, demonstrating stronger near-interval resolution and super-resolution characteristics.

[0056] Figure 9 , Figure 10The graphs show the changes in RMSE and PoR of various algorithms with angle intervals, under the setting of SNR=-10dB and angle set {-5°, 4°, 20°}, as the number of snapshots increases from 10 to 150 (15 possible values). The trends in the graphs show that in regions with fewer snapshots, traditional methods are more susceptible to excessively large sample covariance variance and unstable subspace separation, resulting in slower RMSE decreases and lagging PoR improvement. The method of this invention achieves faster RMSE convergence and PoR improvement even with a smaller number of snapshots, indicating that it is more effective in suppressing small sample covariance distortion and is suitable for scenarios with limited snapshots.

[0057] The comprehensive comparative experiments across three categories demonstrate that, under typical DOA scenarios such as coherent sources, low SNR, insufficient snapshots, and small angular intervals, the method of this invention achieves superior results in both RMSE and resolution probability. Compared to spectral search methods such as MUSIC (including spatial smoothing), it significantly improves real-time processing capabilities. This performance advantage directly corresponds to the covariance preprocessing chain proposed in this invention, which combines "time-shifted multi-Toeplitz structure enhancement + Toeplitz prior correction + closed-loop linear contraction robustness." This chain effectively alleviates the instability in subspace separation caused by large sample covariance estimation errors, thereby improving DOA estimation accuracy, resolution, and robustness. Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention 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 the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural networks, comprising: Step 1: Acquire the received data from the uniform linear array, calculate the sample covariance matrix, and perform trace normalization. Step 2: Perform Toeplitz prior correction by averaging the elements of each diagonal and backfilling to construct the Toeplitz prior covariance, and then Hermitize it. Step 3: Based on the sample covariance matrix and the Toeplitz prior covariance, a linear shrinkage model is adopted, and the optimal shrinkage coefficient is determined by the Frobenius norm mean square error criterion, thus obtaining the shrinkage covariance matrix; Step 4: Vectorize the contracted covariance matrix and project it onto the overcomplete array dictionary to obtain the complex projection features; Step 5: Decompose the complex projection features into real and imaginary parts, form a dual-channel input to the convolutional neural network, and obtain the spatial spectral vector; Step 6: Perform a spectral peak search on the spatial spectral vector, extract the angles corresponding to the K peaks with the largest amplitudes, and obtain the DOA estimation results.

2. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network as described in claim 1, characterized in that, The sample covariance matrix calculated in step 1 for: in, For the number of snapshots, For a uniform linear array receiving snapshot vector, n represents the nth snapshot, and the superscript H represents the conjugate transpose; Sample covariance matrix after trace normalization for: in, To take the real part of a complex number, Represents the trace of a matrix.

3. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network according to claim 2, characterized in that, Step 2 specifically includes: Offset for each diagonal Find the mean ,in, The number of elements in a uniform linear array; Extraction operator for diagonal; Construct the Toeplitz prior covariance matrix , No. Set all elements on the diagonal to ; right Hermitization: 。 4. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network as described in claim 3, characterized in that, The optimal shrinkage coefficient in step 3 The following results were obtained using closed-form estimation: when season ; when hour, in, To truncate the operator, , ; Output shrinkage covariance matrix for: 。 5. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network according to claim 4, characterized in that, The method further includes performing multi-Toeplitz enhancement preprocessing before step 3 to address coherent signal sources, specifically including: Let the time shift be τ, where τ is an integer and satisfies... ; Corresponding time-shifted sample covariance for: ; right Toeplitz correction and shrinkage were performed sequentially to obtain Then based on Constructing multiple Toeplitz matrices: make Define an m×m flip matrix J, whose elements satisfy: If and only if p + q = m + 1, otherwise ; For any matrix C, the forward and backward averaging yields the enhancement matrix. ,in Indicates conjugation; when constructing a Toeplitz matrix from the first column vector c and the first row vector r. When T satisfies: if , ;otherwise, ; This is the sample covariance matrix input for subsequent steps.

6. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network according to claim 4, characterized in that, Step 4 specifically includes: In the preset angle grid { ,…, On}, for each grid point Calculate the steering vector ; exist Construct dictionary column vectors above: ; Construct an overcomplete array dictionary by superimposing column vectors. for: ; Vectorize the shrinkage covariance: And project: ; The complex projection feature is split into real and imaginary input channels: 。 7. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network according to claim 6, characterized in that, The convolutional neural network consists of three concatenated one-dimensional convolutional layers, including: First layer: kernel size 11, input channels 2, output channels 12, batch normalization, ReLU activation; Second layer: kernel size 5, input channels 12, output channels 5, batch normalization, ReLU activation; The third layer has a kernel size of 3, 5 input channels, 1 output channel, batch normalization, and the output is flattened to obtain a spatial spectral vector of length L.

8. The DOA super-resolution estimation method based on Toeplitz covariance enhancement and neural network according to claim 6, characterized in that, The spectral peak search in step 6 specifically involves: given the number of information sources K, selecting the network output spectrum... Take the K indices with the largest amplitude values Output the corresponding angle , and output in ascending order.

9. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method of claim 1.

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