A novel hierarchical classification framework for estimating the DOA of coprime arrays under sensor position errors.

Abstract

The DOA estimation algorithm for coprime arrays under the presence of sensor position errors, based on a novel hierarchical classification framework, is a fundamental problem in array signal processing, primarily applied in radar and wireless communication. This invention proposes a novel hierarchical classification framework for DOA estimation of coprime arrays, achieving good adaptability and high accuracy even with array defects. Unlike existing deep learning methods, this invention uses a hierarchical classifier to address the non-uniformity between categories caused by angular spacing. The DOA estimation process is divided into multiple levels from general to fine, thus decomposing the complex classification task into simpler hierarchical subtasks. First, a preprocessing scheme is designed to improve model performance. Second, a label tree is constructed as prior information for the model. Finally, by learning the semantic relationships between categories, the hierarchical classification model directly outputs the DOA of the signal without requiring spectral peak search.

Description

Technical Field

[0001] This invention presents a direction-of-arrival (DOA) estimation method applicable to coprime array systems. As a fundamental problem in array signal processing, DOA has applications in radar, sonar, and wireless communication. This invention can achieve DOA estimation under multiple signal incidence conditions and exhibits good adaptability to array defects. Background Technology

[0002] Common DOA estimation methods include: traditional methods (CBF method, MVDR method), maximum likelihood method, and subspace methods (MUSIC and ESPRIT). These methods essentially model the mapping relationship between array output and signal direction, collectively known as parameter-based methods. Subspace methods exhibit superior performance in common scenarios, including good resolution and computational complexity, and are therefore studied more extensively than other methods. Coprime arrays, as a type of sparse linear array, have received widespread attention and research due to their advantages over uniform linear arrays. Sparse linear arrays can achieve high-resolution and accurate DOA estimation through optimized design of element positions and weights. Compared to traditional uniform linear arrays, they can better eliminate mutual interference between signals, improving estimation accuracy. Furthermore, due to the adaptability of sparse linear arrays, they exhibit better robustness against some common interferences. In contrast, traditional uniform linear arrays may perform poorly in these situations. Moreover, with the same number of sensors as uniform linear arrays, sparse linear arrays can increase degrees of freedom, allowing for a reduction in the number of array elements in practical deployments. In DOA estimation based on coprime arrays, a virtual uniform linear array is obtained by reconstructing the difference array after virtualizing the coprime array. Then, classical parametric methods are applied to the corresponding extended covariance matrix to complete the DOA estimation. However, since the accuracy of DOA estimation is actually related to the closeness of the modeling mapping, and the complexity of the mapping relationship depends on the actual scenario, it is difficult to accurately express the mapping relationship when various array defects and other disturbances occur.

[0003] With the resurgence of machine learning and the inherent limitations of parametric methods in complex scenarios, machine learning methods have begun to be applied to DOA estimation. The most common methods are RBF-based and SVR-based methods. These methods enable the model to spontaneously learn the mapping relationship through training, overcoming the limitations of traditional parametric methods. However, array output models vary across different real-world scenarios, making it difficult to establish a comprehensive training set. When the distribution of the test set differs from the training set, the estimation performance suffers significantly.

[0004] Deep learning, as a special type of machine learning, has enabled researchers to construct deeper learning models to achieve general multi-DOA estimation due to its powerful capabilities and flexibility. This typically involves reconstructing the spectrum of the incident signal using a learning model, followed by linear interpolation to obtain the DOA of the target signal. In complex scenarios with array defects, these methods have demonstrated superior performance compared to classical parametric methods. Some researchers treat DOA estimation as a multi-label binary classification problem, where each node in the classifier's output represents a predefined orientation grid. Other papers consider treating the spectrum reconstruction process as a regression problem. Summary of the Invention

[0005] The invention problem to be solved

[0006] In complex scenarios, deep learning-based DOA estimation methods are more robust and adaptive than classical parametric methods. These methods can be categorized into multi-label classification-based and regression-based methods. However, existing deep learning methods struggle to balance good adaptability with high estimation accuracy in unknown scenarios. Regression networks are more susceptible to noise than classification networks and lack good generalization ability. Most multi-label binary classification networks typically employ planar N-way classifiers. They treat all categories as equally indistinguishable, ignoring the non-uniformity between categories caused by the magnitude of the incident angle interval, which affects the accuracy of DOA estimation. These methods also require peak searching of the reconstructed spectrum of the output, increasing computational time. Furthermore, most existing deep learning-based methods are based on uniform linear arrays (ULAs) for DOA estimation, and are rarely applicable to coprime array systems.

[0007] Summarize the motivation for the invention

[0008] Deep learning methods overcome the limitations of traditional parameter-based methods in complex scenarios. However, existing deep learning methods using multi-label classification or regression cannot simultaneously achieve high accuracy and good adaptability. To achieve high-accuracy DOA estimation while maintaining good adaptability, we aim to use a novel classification method that improves accuracy by considering class non-uniformity, thus maintaining both robustness and adaptability. A very natural and intuitive alternative classification method is to organize classifiers hierarchically using a divide-and-conquer strategy. Hierarchical classification networks (HCNs) are currently used to solve classification problems in computer vision, natural language processing, and other related fields. We hope that this hierarchical classification method can be applied to general DOA estimation to address existing problems.

[0009] ● Compared to traditional ULAs, coprime arrays offer the advantage of high degrees of freedom. However, most existing deep learning-based methods rely on uniform linear arrays for DOA estimation and are rarely applicable to coprime array systems. Considering this, we aim to utilize coprime array systems in our deep framework, which will help improve estimation performance.

[0010] ● Proper preprocessing of the received signal is one of the important factors affecting estimation performance. We aim to design a preprocessing scheme based on the output of a coprime array. The preprocessed output data can describe more complex hierarchical relationships to solve hierarchical classification problems.

[0011] Technical solution adopted in this invention

[0012] By treating DOA estimation as a hierarchical classification problem, this invention proposes a deep hierarchical classification framework to address multi-DOA estimation in the presence of array sensor position errors, with a coprime array system used to receive the incident signal. Compared to advanced learning models with deep hierarchical architectures, the hierarchical classification model constructed in this invention, namely the Multi-Output Hierarchical Convolutional Network (MO-HCN), can further improve estimation accuracy while maintaining superior adaptability and robustness. Unlike existing deep learning classification methods, we employ a hierarchical classifier to address the class non-uniformity caused by the angular spacing of the incident direction. By dividing the DOA estimation process into multiple levels according to the concept of general direction to specific direction of arrival, the originally complex classification task can be decomposed into hierarchical simple classification subtasks.

[0013] First, we designed a novel data preprocessing method called Joint Sparse Covariance Matrix (JSCM). Second, we artificially constructed a tree structure to represent the hierarchical relationships between categories, serving as prior information for MO-HCN. Subsequently, the trained MO-HCN progressively performs the classification task according to the general direction of the incident direction towards the specific DOA. Finally, we compared the performance of our method with other state-of-the-art methods under ideal array systems and in cases with array defects. To our knowledge, applying hierarchical classification methods to multi-DOA estimation is still a novel approach.

[0014] Related steps of the present invention

[0015] 1) A coprime array system is used to receive multiple incident signals to obtain the array output signal x(t), whose coprime array output covariance matrix R x It was then calculated.

[0016] 2) By analyzing R x Vectorization is performed, and the vectorized Rx This can be viewed as the received signal of an equivalent virtual array. Subsequently, by removing the virtual array S... d The difference coma is obtained by using repeated elements in the array. This array is a partially augmented array with several missing elements, resulting in a non-uniform array geometry. This non-uniformity makes subsequent statistical signal processing difficult. Therefore, we construct a virtual uniform linear array by removing the non-uniform parts from the virtual sensor. The corresponding augmented covariance matrix R u It can be reconstructed using spatial smoothing techniques.

[0017] 3) The JSCM preprocessing scheme is adopted, with R... x R u The input object is preprocessed, and the resulting multi-channel matrix is ​​used as input for subsequent deep modeling.

[0018] 4) The established multi-directional label tree serves as prior information for the deep model, indicating the hierarchical relationships between categories. The hierarchical label corresponding to each group of incident signals can be determined using the multi-directional label tree.

[0019] 5) Create training datasets, validation datasets, and test datasets.

[0020] 6) Train MO-HCN. A branch training strategy is adopted. During training, the backbone network of the model can extract features hierarchically according to a predefined structure.

[0021] 7) After training, the model outputs hierarchical predictions of the direction information of the incident signal from multiple output networks. The result output by the final output network is the fine class of the target, i.e., the DOA estimation result of the incident signal.

[0022] 4. Beneficial effects

[0023] ● A multi-output hierarchical convolutional network (MO-HCN) is proposed to solve the hierarchical multi-class classification problem with multi-DoA estimation. It directly outputs the DOA of the incident signal without performing spectral peak search. It reduces the difficulty of the classification task by decomposing the complex classification task into simpler hierarchical subtasks. Finally, compared with existing deep learning-based methods, it achieves higher estimation accuracy while maintaining good adaptability.

[0024] ● A multi-directional label tree is constructed as prior information representing hierarchical category relationships. This category hierarchy structure possesses the interpretability of deep convolutional networks. During training, the classifier can utilize the semantic relationships between categories to further aid classification learning and prediction. By learning the hierarchical relationships between categories, MO-HCN will sequentially predict the general direction of the incident signal to its specific direction.

[0025] ● A novel data preprocessing method called JSCM is designed. It improves upon existing methods specific to uniform linear array systems, making them applicable to coprime array systems. The output of JSCM provides multi-channel input to MO-HCN. It contains hierarchical orientation information, compensating for some of the phase information loss caused by complex-valued decomposition.

[0026] ● To verify the accuracy and adaptability of the proposed method, we conducted simulation experiments considering both ideal array systems and array systems with array defects. For array defects, we considered sensor position errors, the most common type of array defect, existing in a two-dimensional plane. Experimental results show that the hierarchical classification network designed in this invention has better accuracy and adaptability. Attached Figure Description

[0027] 1) Figure 1 Multi-DOA estimation process based on hierarchical classification framework

[0028] 2) Figure 2 Data preprocessing workflow

[0029] 3) Figure 3 Basic tag tree structure

[0030] 4) Figure 4 Structure of MO-HCN for hierarchical classification

[0031] 5) Figure 5 The accuracy performance of each output network of MO-HCN on the validation set during training.

[0032] 6) Figure 6 Accuracy comparison of MO-HCN and its baseline model on the validation set.

[0033] 7) Figure 7 Comparison of RMSE of MO-HCN based on JSCM with other DOA estimation methods. Figure 7 Figure (a) and Figure (b) show the results in low signal-to-noise ratio (SNR) and high SNR environments, respectively, with snapshots = 300.

[0034] 8) Figure 8 Comparison of RMSE and number of snapshots (SNR = 10dB)

[0035] 9) Figure 9 A comparison of the success rate of MO-HCN based on JSCM with other deep learning methods for DOA estimation relative to the number of incident signals. Figure 9 (a) Low signal-to-noise ratio environment; (b) High signal-to-noise ratio environment; (c) Limited snapshots; (d) Sufficient snapshots.

[0036] 10) Figure 10 Relationship between RMSE and perturbation level of array position error (SNR=10dB and snapshot=300).

[0037] 11) Figure 11 Relationship between RMSE and SNR of MO-HCN based on JSCM under different perturbation levels Specific Implementation

[0038] The flowchart for multi-DOA estimation of coprime arrays based on a hierarchical classification framework is as follows: Figure 1 As shown, it can be divided into the following stages: signal model of coprime array system, preprocessing stage of array output signal x(t), construction of multi-directional label tree, construction of hierarchical classification model, construction of dataset, and training strategy and related configuration.

[0039] 1) Signal model of coprime array system

[0040] Consider K directions θ = [θ1, θ2, ..., θ3] K The far-field narrowband uncorrelated signal is obtained from the coprime array system, and the array output signal x(t) is obtained.

[0041] ●Ideal Array System

[0042] The coprime array consists of two sub-linear arrays placed collinearly at the same origin. The number of sensors in the two sub-arrays is an integer pair (N, 2M), and the corresponding sensor spacing is (M, N)d, where d is half the wavelength. Considering the array is placed on a two-dimensional plane, the positions of the total number of sensors (2M+N-1) are represented as follows:

[0043]

[0044] The received signal of a coprime array can be represented as

[0045] x(t)=A(θ)s(t)+n(t)

[0046] Where A(θ) represents the turning matrix of the coprime array; n(t) represents additive white Gaussian noise. The k-th column of A represents the turning vector of the k-th signal source, as shown below:

[0047]

[0048] ● System models with deterministic errors

[0049] To obtain a more general model of the perturbation system, we consider that the sensor position error exists in a two-dimensional plane. We use... and This represents the sensor position error along the x and y axes. The disturbance in the sensor position... It is given by the following formula:

[0050]

[0051] For the perturbation array system to be meaningful, we assume that the sensor position error is much smaller than d. Let the parameter δ represent the sensor position error, and the received signal of the perturbation coprime array can be expressed as:

[0052]

[0053] in, This represents the perturbation steering matrix. The element in the l-th row and k-th column is represented as

[0054]

[0055] 2) Pretreatment stage

[0056] For the array output signal x(t) Perform the following preprocessing steps, such as Figure 2 As shown, the multi-channel matrix is ​​obtained and used as the model input.

[0057] In the first step of JSCM, R is calculated using the array output recorded after T snapshots. In practical scenarios, the formula... It is typically used to obtain M o The number of physical sensors in a coprime array.

[0058] The second step is to vectorize R. The vectorized result is considered as the received signal of the equivalent virtual array. The vectorization operation can be represented as follows:

[0059]

[0060] in, A x The steering matrix of an augmented array, corresponding to a virtual sensor. Location is

[0061]

[0062] We can remove The difference comatrix is ​​obtained by repeating elements in the array. The resulting difference comatrix is ​​a partially augmented array, and the array geometry is non-uniform due to the absence of several elements. This non-uniformity makes subsequent statistical signal processing difficult. Therefore, we will construct a virtual uniform linear array by removing the non-uniform parts from the virtual sensor. By using spatial smoothing techniques, an M-order array can be reconstructed. u ×M u The corresponding augmented covariance matrix R u R u Written as

[0063]

[0064] Among them, Γ p =[0 P×(p-1) I P×P 0 P×(P-1) ], x u =Fr x The received signal is represented as a virtual uniform linear array.

[0065] Third step, we examine R and R u Sparsification is performed to obtain covariance matrices with the same dimensions. and The specific steps are as follows:

[0066]

[0067] in, At this time, it was generated and The dimensions remain unchanged, having M o >M u To ensure that the two matrices have the same dimensions without losing information, The following is a reconstruction:

[0068]

[0069] Fourth, we divide the elements of the covariance matrix into real and imaginary parts to simplify the features, thus obtaining four dimensions M. u ×M u Matrix. They are integrated into

[0070]

[0071] Wherein, the dimension of Input is M. u ×M u ×4. Imag{·} and Real{·} represent the imaginary and real components of the complex-valued matrix, respectively.

[0072] 3) Construction of a multi-directional tag tree

[0073] We need to construct a multi-directional label tree to determine the hierarchical labels for each group of multiple incident signals. The multi-directional label tree is constructed considering three levels. During construction, a base label tree is first built, representing the class hierarchy within a single signal scene. Subsequently, the multi-directional label tree is expanded from the base label tree based on the directional relationships of the incident signals.

[0074] Basic tag tree structure as follows Figure 3 As shown. Assume the spatial range of the incident direction is [θ]. min θ max The entire spatial range is composed of equally spaced Δ (i) By dividing the region, sub-regions can be obtained. Where N (i) The calculation is as follows:

[0075]

[0076] Each spatial sub-region This corresponds to the coarse category at level i in the label tree. The hierarchy of coarse categories is determined by the size of the partitioning interval, i.e.

[0077] i < j, Δ (i) >Δ (j)

[0078] Each category in level i can be accessed through N in level j. ij The clustering is obtained from N categories. ij The following calculation was performed:

[0079]

[0080] Where i < j. If Clustering yields categories Then it exists in for The corresponding spatial sub-region. Specifically, the highest-level category in the label tree is the fine-grained category, which is provided by the classification task as the target category, i.e., the actual DOA of the signal. The partitioning interval Δ (3) It is uniquely determined by the orientation grid. If there exists a direction from θ... (1) to θ (K) The incident signal then has

[0081]

[0082] The expansion process of the multi-directional tag tree is shown below. Consider a signal incident range identical to the base tag tree. For a set of incident signals... First, arrange them in order according to the magnitude of their incident angles:

[0083]

[0084] Secondly, the category of each incident signal at the corresponding level is determined based on the basic label tree. For the incident signal... Its hierarchical labels are represented as

[0085]

[0086] Finally, all the incident signals in this set The categories at level i are combined to form a new category, which serves as the basis for this set of incident signals. The coarse category at the same level i can be represented as:

[0087]

[0088] In particular, The fine category can be represented as:

[0089]

[0090] Where, Δ ij =θ j -θ i , i < j. Multiple incident signals The hierarchical tags can be obtained as follows:

[0091]

[0092] 4) Hierarchical classification model

[0093] The hierarchical classification model uses a preprocessed multi-channel matrix as input to perform hierarchical classification. The structure of this model is as follows: Figure 4 As shown below, the MO-HCN performs classification tasks and its corresponding structural functions are described.

[0094] MO-HCN is a hierarchical deep framework consisting of a deep convolutional neural network as the backbone and multiple output networks. The backbone network extracts interpretable hierarchical features from the input signal. Shallow features containing general directional information are extracted by shallow networks, while deep networks extract deep features with precise directional information. Each output network is connected to convolutional layers of different depths within the backbone network, sequentially outputting hierarchical categories of multiple input signals along the main convolutional flow. Clearly, because the intervals between the divisions producing fine-grained categories are smaller, the resulting closely spaced sub-regions are more difficult to distinguish, thus requiring deeper networks to perform more specific feature extraction. Therefore, the output targets of the multiple output networks satisfy the conditions for hierarchical feature extraction by the convolutional layers, demonstrating the interpretability of the hierarchical network relationships.

[0095] When predicting hierarchical categories of a set of multi-input signals, MB-DCN extracts features sequentially by first determining the general direction of the input signals and then focusing on a smaller range. The prediction for each category is accomplished by a classification model composed of a corresponding portion of the backbone network and the output network. This method of learning hierarchical relationships for classification effectively improves the performance of DOA estimation compared to traditional classification models, especially under conditions of low signal-to-noise ratio or very small incident signal intervals. Furthermore, this global hierarchical classification method simplifies the classification process and avoids the propagation of misclassifications from upper-level nodes to their descendant nodes, effectively reducing estimation errors.

[0096] The backbone deep convolutional network consists of 20 layers, which can be viewed as four similar modules, excluding input and output layers. The entire backbone deep network structure can be represented as a non-linear mapping function, which can be written as:

[0097]

[0098] in, This represents the output value of the k-th layer in the MO-HCN backbone network. The four modules are identical, each with five layers. For the i-th module, we have:

[0099]

[0100] in, It is a convolutional layer. This is the set of weights and biases. The k-th layer... It can be written as:

[0101]

[0102] Where * represents a convolution operation, and ReLU represents an activation function; Indicates the regularization layer; For a max pooling layer, it can be defined as in, Indicates a rectangular area Maximum output value, Represents a rectangular area The element located at (r, s).

[0103] The multi-output network consists of three modules with similar structures. The feedforward transform of the q-th output network consists of three fully connected layers, including hidden layers and output layers, calculated as follows:

[0104]

[0105] in, This is the set of weights and biases for the fully connected layer. This represents the output value of the l-th layer in the q-th output network. The calculation is feedforward:

[0106]

[0107] Here, RELU represents the RELU activation function.

[0108] Each output network and its connected portions of the backbone network together form a classification model that predicts the corresponding category. This is used to predict the first-level coarse class. The output network is located after the second module of the backbone network. Therefore, the output of the tenth layer in the backbone network serves as the input of the first output network. Depend on Input to first network output The mapping relationship can be represented as:

[0109]

[0110] Used to predict the second-level coarse class The output network is located after the third module of the backbone network. Therefore, the output of layer 15 in the backbone network serves as the input of the second output network. Depend on Input to the second network output The mapping relationship can be represented as:

[0111]

[0112] Used for predicting fine-grained target classes The output network is located at the end of the backbone network; therefore, the output at the end of the backbone network serves as the input to the third output network. Depend on Input to third network output The mapping relationship can be represented as:

[0113]

[0114] 5) Training strategies and related configurations

[0115] After building an effective deep model, an appropriate training strategy is needed to train it, which plays a crucial role in the performance of multi-DOA estimation. This invention employs a multi-branch training strategy for training MO-HCN. This strategy allows for adjusting the emphasis on different levels of the classification model at different training stages, ensuring that relevant layers are trained with greater focus. By applying the multi-branch training strategy, gradient vanishing can be effectively prevented, thus avoiding the problem of stopping the update of relevant parameters at lower levels of the model.

[0116] The degree of emphasis during training is determined by the loss weights [w1w2w3] assigned to each output network, where w k This represents the loss weight of the output network in MO-HCN corresponding to the label of the k-th layer. For standardization purposes, we define w as... k ∈[0,1],

[0117] The loss weight distribution is adjusted in stages during training by setting different iteration conditions. Since the initial training phase aims to prioritize low-level feature extraction while later focusing on deeper features, the initial weight distribution can be set to [0.98, 0.01, 0.01]. After several iterations, the loss weight distribution is adjusted to [0.10, 0.80, 0.10]. The total loss function of MO-HCN is defined as the weighted sum of the prediction losses of all branches in the model, specifically:

[0118]

[0119] Where i represents the sampling of the i-th minibatch; k represents the corresponding level in the label tree; w k The loss weights are for the k-th layer. The log term represents the cross-entropy loss of the i-th sample on the k-th layer output network; f j This represents the j-th element in the class score vector f, output by the last layer of the output network, i.e., the model's understanding of a sample in its true class y. i The prediction score is used to measure the model's confidence in the correct category.

[0120] 7. Results Analysis

[0121] To fully demonstrate the effectiveness and superiority of this invention, we conducted simulation experiments from multiple perspectives and compared it with other advanced methods.

[0122] First, we introduce the simulation setup. Then, we illustrate the characteristics of MO-HCN by demonstrating its performance during training. Finally, we present simulation results under ideal array system and existing sensor position error conditions, and compare MO-HCN with other state-of-the-art solutions.

[0123] 1) Simulation settings

[0124] In the simulation experiment, the coprime array we used consisted of a pair of subarrays with a sensor spacing of (M, N)d0 = (3, 5)d0, where d0 represents half the wavelength. This coprime array had 10 sensors located at [0, 3, 5, 6, 9, 10, 12, 15, 20, 25]d0. The entire incident space range we considered was [-60°, 60°]. The hierarchical spatial sub-regions were divided by angular spacing {40°, 10°}, corresponding to the first and second level coarse categories in the basic label tree, respectively. The spatial spectrum was constructed using a 1° grid, from which directions θ1 = -60°, θ2 = -59°, ..., θ 121 =60° directional grids, which are used as fine-grained classes of targets.

[0125] Table 1 Parameter Settings

[0126]

[0127] We construct a multi-directional label tree considering a dual-signal incidence case. The spatial range of both incident signals is θ∈[-60°, 60°], and the angular interval between the two signals is considered to be {2°, 4°, ..., 40°}. When the angular interval is Δ... i At that time, the direction θ1 of signal 1 changes from -60° to 60°-Δ i If sampling is performed at 1°, then the direction θ2 of signal 2 is θ1 + Δ i Ultimately, we can determine the number of classes in each level of the multi-directional label tree. When constructing the training dataset, we used a snapshot of T=300 and a signal-to-noise ratio of SNR=10dB to collect the coprime array output information. The number of classes in the label tree and the relevant settings for network training are shown in Table 1. The dataset was shuffled before each iteration during training. The software platform we used was PyCharm 2021.3.1, with Python 3.6.13, Keras 2.2.4, and CUDA 9.0. MATLAB R2022a was used when constructing the coprime array output data.

[0128] 2) Training performance of MO-HCN

[0129] In this section, we demonstrate the behavior of each output network in MO-HCN during training to illustrate the hierarchical relationships between them. We also compare MO-HCN with its corresponding baseline model, providing an initial indication of the superiority of hierarchical classification networks over traditional DCNs. The training setup in this section is not identical to the final setup used for DOA estimation, which is intended to fully showcase the characteristics of MO-HCN.

[0130] like Figure 5 As shown, we first present the DOA estimation accuracy of each output network on the validation set as iterates. The results indicate that output network 1 has the highest accuracy and fastest convergence speed, starting to converge early in training, followed by output networks 2 and 3. These performance results are consistent with expectations that using only simple features extracted from shallow networks to predict coarse categories, and that the number of categories in coarse classification is less than that in fine classification, simplifies classification decisions, making it easier to achieve high accuracy and faster convergence. We also note that output networks with higher branches exhibit a steeper growth trend after fluctuations, due to the multi-branch training strategy employed in hierarchical training.

[0131] like Figure 6 As shown, we present a performance comparison with the MO-HCN baseline model. The baseline model is a traditional DCN obtained by removing the internal output network. The results initially demonstrate that MO-HCN outperforms the traditional DCN. It can be observed that the baseline model corresponds to a stationary curve with an accuracy of 86.65% at iteration 45. The growth trend of MO-HCN gradually steepens with the number of iterations, reaching an accuracy of 98.71% at 45 iterations. This indicates that when coarse classification training has converged, MO-HCN devotes more effort to learning more specific features. Furthermore, accurate learning of low-level mappings contributes to subsequent high-level classification. With increasing training epochs, MO-HCN surpasses the baseline model with even higher accuracy.

[0132] 3) Performance analysis under ideal array system

[0133] Under ideal array systems, we compare the performance of our invention with baseline models and other state-of-the-art DoA estimation methods, such as Deep Convolutional Networks (DCNs) and SS-based multi-signal classification (SS-MUSIC), and introduce CRB for comparison. To demonstrate that the proposed JSCM contributes to improving model performance, we compare it with commonly used input methods in ULA systems, where the ULA output covariance vector is replaced by a virtual ULA covariance vector (VUCV) of coprime arrays to ensure the reasonableness of the comparison. We maximize the DOA estimation performance of MO-HCN using the training settings in Table 1. In the following... Figure 7 and Figure 8 In the following section, we compared the RMSE performance. For fairness, DCN used the same training settings as MO-HCN. We considered the case with two incident angles {-57°, -51°} and Trials = 4000, and the RMSE metric used is expressed as follows:

[0134]

[0135] We are Figure 9 The previous section presented a performance comparison of DOA estimation success rates. We will compare this with deep learning-based methods to further demonstrate the superiority of MO-HCN over traditional DCN. In our experiments, we consider all signal occurrence scenarios, with a single-signal test set size of 484 = 121 × 4 and a dual-signal test set size of 8000 = 2000 × 4.

[0136] Figure 7 RMSE performance was compared under both high and low signal-to-noise ratio (SNR) environments. MO-HCN exhibited the best performance in both environments. Particularly in high-noise environments, MO-HCN demonstrated significantly stronger robustness compared to other methods. Furthermore, JSCM effectively enhanced the performance of MO-HCN. Figure 7 In (a), when the signal-to-noise ratio (SNR) is greater than -8 dB, the estimation performance of all methods steadily improves with the influence of noise. When the SNR is less than -8 dB, the difference between these methods and CRB increases sharply. Figure 7 (b) shows that in high-noise environments, the MO-HCN with JSCM has a lower RMSE than the MO-HCN with VUCV. The performance of the baseline model is close to that of the MO-HCN.

[0137] Figure 8 The impact of different numbers of snapshots on the RMSE of different methods is shown. MO-HCN exhibits better overall performance than other deep learning-based methods. We observed that MO-HCN has the lowest RMSE when the number of snapshots is greater than 250. When the number of snapshots is less than 250, SS-MUSIC significantly outperforms the other methods. Considering that the RMSE of small snapshots is too high for practical application, we chose to use a sufficiently large number of snapshots in practice.

[0138] Figure 9 The estimated success rate is shown for different numbers of incident signals. Figure 9 (a) and Figure 9 (b) Simulation results are presented for low and high signal-to-noise ratio (SNR) environments, respectively. We randomly collected data within the SNR (dB) ranges [-12, 0) and (0, 12] to represent the low and high SNR environments. Next, Figure 9(c) and Figure 9 (d) illustrates the impact of snapshots on the estimation success rate. We randomly collected data from snapshot ranges of (0, 400] and (400, 800] to obtain finite and sufficient snapshots, respectively. The results show that the JSCM-based MO-HCN method outperforms other deep learning-based methods in various scenarios. The performance comparison among these deep learning-based methods is consistent under different numbers of incident signals. Although the performance of the baseline model approaches that of MO-HCN when there are sufficient snapshots and a high signal-to-noise ratio, the performance of the baseline model drops sharply under interference environments.

[0139] 4) Performance analysis under array sensor position error

[0140] In this section, we will consider the defect of sensor position error in array systems. Error is caused by factors with a covariance matrix. It is generated by a zero-mean Gaussian distribution. The magnitude of the sensor position error can be used... Adjustments were made. We compared the performance of RMSE with advanced methods, and the results are as follows: Figure 10 and Figure 11 As shown.

[0141] exist Figure 10 In the middle, we showed RMSE and The relationship diagram shows that the MO-HCN with JSCM has the best adaptability to sensor position errors. We observed that the RMSE performance of the MO-HCN based on VUCV is not as good as that using JSCM, indicating that JSCM is more effective than conventional VUCV. Furthermore, the estimation performance of SS-MUSIC degrades significantly in the presence of array position errors, demonstrating the inapplicability of parameter-based methods to array defects.

[0142] exist Figure 11 In this paper, we demonstrate how variations in sensor position error affect the RMSE of MO-HCN at different signal-to-noise ratios (dB). The results show that the negative impact of array defects on DOA estimation is unavoidable. The RMSE is close to that of an ideal array system when the sensor position error is small. As the sensor position error increases, the root mean square error (RMSE) increases significantly at the same SNR. Furthermore, the overall trend of the RMSE curves with different position errors is the same: the RMSE decreases with increasing SNR, and then the trend stabilizes.

Claims

1. A novel hierarchical classification framework for estimating the DOA of coprime arrays under sensor position errors, characterized in that... The method includes: a) A hierarchical classification deep model, namely a multi-output hierarchical convolutional network (MO-HCN), is applicable to ideal coprime array systems and coprime array systems with array sensor position error defects. Its composition and functional implementation include: MO-HCN is a hierarchical deep framework consisting of a deep convolutional neural network as the backbone and multiple output networks. Each output network is connected to convolutional layers of different depths within the backbone network to obtain classification models for different level categories. When predicting the hierarchical category of a set of multiple signal incident directions, the backbone network extracts features sequentially by first determining the general direction of the incident signal and then focusing on a smaller range. The output networks output the hierarchical categories of multiple input signals sequentially along the main convolutional flow, including the hierarchical spatial sub-region to which the incident signal belongs and the specific signal DOA. The backbone deep convolutional network consists of 20 layers, which can be viewed as four similar modules. It does not contain input or output layers. The entire backbone deep network structure can be represented as a non-linear mapping function, which can be written as: in, This represents the output value of the k-th layer in the MO-HCN backbone network. The four modules are identical, and each module has five layers. For the i-th module, we have: in, It is a convolutional layer. The set of weights and biases, for the k-th layer It can be written as: Where * represents a convolution operation, and ReLU represents the activation function. Indicates the regularization layer. For a max pooling layer, it can be defined as in, Indicates a rectangular area Maximum output value, Represents a rectangular area The element located at (r, s); The multi-output network consists of three modules with similar structures. The feedforward transform of the q-th output network consists of three fully connected layers, including hidden layers and output layers, calculated as follows: in, It is the set of weights and biases for the fully connected layer. This represents the output value of the l-th layer in the q-th output network. The calculation is feedforward: Where RELU represents the RELU activation function; Each output network and its connected portion of the backbone network together form a classification model that predicts the corresponding category, used to predict the first-level coarse class. The output network is located after the second module of the backbone network. Therefore, the output of the tenth layer in the backbone network serves as the input of the first output network. Depend on Input to first network output The mapping relationship can be represented as: Used to predict the second-level coarse class The output network is located after the third module of the backbone network. Therefore, the output of layer 15 in the backbone network serves as the input of the second output network. Depend on Input to the second network output The mapping relationship can be represented as: Used for predicting fine-grained target classes The output network is located at the end of the backbone network; therefore, the output at the end of the backbone network serves as the input to the third output network. Depend on Input to third network output The mapping relationship can be represented as: b) Data preprocessing scheme for coprime array output; c) The multi-directional label tree, used to provide prior information for deep models, is a class hierarchy structure that can represent the hierarchical relationship between categories; d) Dataset and training strategy used for model training.

2. The method according to claim 1, characterized in that, The data preprocessing scheme described above uses the output of a coprime array as the target for preprocessing operations, and finally obtains a multi-channel matrix. The operations include: In the first step of JSCM, R is calculated using the array output recorded after T snapshots. In practical scenarios, the formula... It is typically used to obtain M o The number of physical sensors in a coprime array; The second step is to vectorize R. The vectorized result is considered as the received signal of the equivalent virtual array. The vectorization operation can be represented as follows: in, A x The steering matrix of an augmented array, corresponding to the virtual sensor S. d Location is S d ={(s i d-s j d,0)|i,j=1,…,2M+N-1} By removing S d Repeating elements in the array can yield a difference comatrix, which is a partially augmented array. The lack of several elements results in a non-uniform array geometry, making subsequent statistical signal processing difficult. Therefore, we will construct a virtual uniform linear array by removing the non-uniform parts from a virtual sensor. Using spatial smoothing techniques, we can reconstruct an M-order array. u ×M u The corresponding augmented covariance matrix R u R u Written as in, x u =Fr x The received signal is represented as a virtual uniform linear array; Third step, we examine R and R u Sparsification is performed to obtain covariance matrices with the same dimensions. and The specific steps are as follows: in, At this time, it was generated and The dimensions remain unchanged, having M o >M u In order to ensure that the two matrices have the same dimensions without losing information, The following is a reconstruction: Fourth, we divide the elements of the covariance matrix into real and imaginary parts to simplify the features, thus obtaining four dimensions M. u ×M u The matrices are integrated into Wherein, the dimension of Input is M. u ×M u ×4, where Imag{·} and Real{·} represent the imaginary and real components of the complex-valued matrix, respectively.

3. The method according to claim 1, characterized in that, The multi-directional label tree is constructed considering the dual-signal incidence case. First, a basic label tree is constructed for the single-signal case. Then, the basic label tree is expanded according to the directional relationship between the signals to obtain the multi-directional label tree. The operation includes: The multi-directional label tree is constructed using three levels. When constructing the basic label tree, it is assumed that the spatial range of the incident direction is [θ]. min θ max The entire spatial range is composed of equally spaced Δ (i) By dividing the region, sub-regions can be obtained. Where N (i) The calculation is as follows: Each spatial sub-region This corresponds to the coarse category at level i in the label tree. The hierarchy of coarse categories is determined by the size of the partitioning interval, i.e. i<j,Δ (i) >D (j) Each category in level i can be accessed through N in level j. ij Clustering of N categories yields N ij The following calculation was performed: Where i < j, if Clustering yields categories Then it exists in for In the corresponding spatial sub-region, the highest-level category in the label tree is the fine-grained category, which is provided by the classification task as the target category, i.e., the actual DOA of the signal, with a division interval Δ. (3) Uniquely determined by the orientation grid, if there exists from θ (1) to θ (K) The incident signal then has The expansion process of the multi-directional tag tree is shown below. Considering a signal incident range with the same as the base tag tree, for a set of incident signals... First, arrange them in order according to the magnitude of their incident angles: Secondly, the category of each incident signal at the corresponding level is determined based on the basic label tree. Its hierarchical labels are represented as Finally, all the incident signals in this set The categories at level i are combined to form a new category, which serves as the basis for this set of incident signals. The coarse category at the same level i can be represented as: The fine category can be represented as: Where, Δ ij =θ j -θ i If i < j, multiple incident signals The hierarchical tags can be obtained as follows:

4. The method according to claim 1, characterized in that, The dataset includes the output of the data preprocessing scheme, namely a multi-channel matrix and its corresponding dual-signal direction hierarchical labels.

5. The method according to claim 1, characterized in that, The training strategy described is a multi-branch training strategy, which can adjust the emphasis on training different parts of the deep model at different stages of training. The operation includes: The degree of emphasis during training is determined by the loss weights [w1w2w3] assigned to each output network, where w k This represents the loss weight of the output network in MO-HCN corresponding to the label of the k-th layer. For standardization purposes, we define w as... k ∈[0,1], The phased adjustment of the loss weight distribution during training is accomplished by setting different iteration conditions. The total loss function of MO-HCN is defined as the weighted sum of the prediction losses of all branches in the model, specifically: Where i represents the sampling of the i-th minibatch, k represents the corresponding level in the label tree, and w k Let f be the loss weight corresponding to the k-th layer, and let the log term represent the cross-entropy loss of the i-th sample on the k-th layer output network. j This represents the j-th element in the class score vector f, output by the last layer of the output network, i.e., the model's understanding of a sample in its true class y. i The prediction score is used to measure the model's confidence in the correct category.

Images