WO2023070994A1 - 一种阵列超分辨波达方向估计方法 - Google Patents

一种阵列超分辨波达方向估计方法 Download PDF

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WO2023070994A1
WO2023070994A1 PCT/CN2022/077069 CN2022077069W WO2023070994A1 WO 2023070994 A1 WO2023070994 A1 WO 2023070994A1 CN 2022077069 W CN2022077069 W CN 2022077069W WO 2023070994 A1 WO2023070994 A1 WO 2023070994A1
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signal
array
neural network
angle
arrival
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黄岩
张彦君
陶俊
张慧
兰吕鸿康
刘江
洪伟
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东南大学
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/78—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using electromagnetic waves other than radio waves
    • G01S3/782—Systems for determining direction or deviation from predetermined direction
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/80—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using ultrasonic, sonic or infrasonic waves
    • G01S3/802—Systems for determining direction or deviation from predetermined direction
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods

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  • the invention belongs to the field of direction-of-arrival estimation, in particular to a super-resolution direction-of-arrival estimation method based on deep learning.
  • DOA estimation is a research problem with a long history in the field of radar and wireless communication, and it is an important branch of array signal processing.
  • the commonly used classic super-resolution DOA estimation algorithm is the multiple signal classification (Multiple Signal Classification, MUSIC) algorithm proposed by the American scholar Schmidt in 1986, which constructs a spatial pseudospectrum by separating the signal subspace and noise subspace, and realizes the traditional estimation method to super Resolving leaps in DOA estimates.
  • ESPRIT Rotational Invariance
  • Deep learning is a popular nonlinear algorithm. With the recent rapid development of computer technology, the powerful ability of deep learning to solve problems has become increasingly prominent, and its application has also been expanded to various fields, including DOA estimation of array signals. Compared with traditional methods, DOA estimation with deep learning has many unique advantages, and it has become a very attractive research direction for researchers in recent years. First of all, the deep learning method only takes a long time during network training, and the result can be obtained quickly when it is applied to DOA estimation after training.
  • the deep learning network can also perform well in environments with lower signal-to-noise ratios and fewer snapshots It extracts data features accurately and shows good robustness.
  • the grids with an interval of 1° are used for training, and the actual signals are usually off-grid.
  • the signal, that is, the target angle contains a decimal part.
  • the off-grid signal angle is accurate to two decimal places, and the decimal part is ignored at this time.
  • the true angle of the signal is not exactly on the grid, a large estimation error will occur, and the performance of DOA estimation will also decrease, so a higher angular resolution needs to be considered.
  • using grid training with a large interval will ignore a lot of information about the fractional part of the signal angle.
  • Purpose of the invention Aiming at the limitations of angular resolution encountered in the existing DOA estimation method based on deep learning, mainly off-grid signals in the general sense, a new two-stage neural network structure is proposed for the direction of arrival Estimation, after using the grid with an interval of 1° to accurately estimate the target angle, further realize the estimation of the fractional part of the signal angle, reaching a resolution of 0.01°.
  • step (7) Apply the DOA estimation neural network model obtained in step (6) to estimate DOA, randomly generate incident signals at a certain angle within the angle range, repeat the data processing process in step (4) to obtain the input of the neural network, and use The trained neural network model estimates the angle, obtains the output vector and processes it, calculates the estimated angle and outputs it.
  • the angle search range is from ⁇ min to ⁇ max
  • the estimated value of the received signal autocorrelation matrix is Preprocessing is to take the real value of the diagonal and the real and imaginary parts of the lower half triangle to combine into a real vector:
  • ⁇ i,j ,i,j ⁇ 1,2,...,N ⁇ represents the elements of the i-th row and j-th column of the matrix R y , and represent the real part and the imaginary part respectively, and the obtained vector x is the input of the model, and its length is N 2 .
  • the corresponding label Only the grid point corresponding to round( ⁇ j ) takes a value of 100, and the other positions are 0, and ⁇ j is accurate to two decimal places;
  • the second label is a vector z 2 with a length of 100, representing the first part
  • the determined grid point is the center, and the small grid range with a resolution of 0.01° is taken. Only the grid point corresponding to ⁇ j -round( ⁇ j ) takes a value of 100, and the other positions are 0.
  • data set is
  • the neural network has a two-stage structure and consists of two parts, the first part contains five fully connected layers, the second part contains six fully connected layers, a total of eleven fully connected layers, with two outputs .
  • the output processing is to find the maximum value of the two vectors, and add the corresponding integer angle and decimal angle.
  • the present invention is based on the previous DOA estimation method of deep learning, and trains the integer and fractional parts of the off-network signal with two parts of the network respectively.
  • the first part determines the position of the angle on the grid at 1° intervals, that is, the 1° range where the target angle is located.
  • the second part combines the original input with the output of the first part to get more accurate within the range determined by the first part.
  • the entire network is composed of fully connected layers, with few model parameters, simple and fast calculation, and has good practical application prospects.
  • Figure 1 is a model diagram of a uniform equidistant linear array (ULA);
  • Fig. 2 is the flow chart of the Neural Network Direction of Arrival Estimation designed by the present invention
  • Fig. 3 is the schematic diagram of label setting
  • Fig. 4 is the neural network model figure that the present invention designs
  • Fig. 5 is the loss descending curve during the first part training of the neural network of the present invention.
  • Fig. 6 is the loss descending curve during the second part training of the neural network of the present invention.
  • Figures 7a-7d show the output of the signal data with the incident angles of -27.43° and 13.29° when the signal-to-noise ratio is 5dB respectively after passing through the network.
  • Figure 7a and Figure 7c are the output of the first part of the network
  • Figure 7b and Figure 7d are The output of the second part of the network;
  • Figures 8a-8d show the output of the signal data with the incident angles of -27.43° and 13.29° when the signal-to-noise ratio is 20dB respectively after passing through the network, where Figure 8a and Figure 8c are the output of the first part of the network, and Figure 8b and Figure 8d are The output of the second part of the network;
  • FIG. 9 is an RMSE diagram of direction of arrival estimation by the neural network of the present invention under different signal-to-noise ratios.
  • the length of the DNN input vector is 144.
  • the DNN network of the present invention adopts the mode of off-line training, and data set is divided into training set (90%) and verification set (10%), uses Adaptive moment estimation (Adaptive moment estimation, Adam) to carry out the update/optimization of parameter,
  • the initial learning rate is set to 0.0001, and the loss weights of the two parts of the network are 1 and 0.1, respectively.
  • the training batch size is 1000 and trained for 500 epochs.
  • the network is built with Keras, the operating system is Windows, the processor is Intel i7-9750H, and the GPU is NVIDIA GeForce RTX 2060.
  • Figure 4 Build the neural network model shown in Figure 4 and use the data set for training.
  • the loss curves of the two outputs of the network are shown in Figure 5 and Figure 6 respectively.
  • the output of the model with fixed angles under different SNR conditions is tested, and the output and functions of the two parts of the model are roughly observed.
  • the two target angles used in the test are -27.43° and 13.29° respectively.
  • Figure 7 is the estimated results of two angles when the signal-to-noise ratio is 5dB and the number of snapshots is 100.
  • Figure 7(a) and (c) are the output results of the first part of the model.
  • Figure 8 shows the estimation results of the two angles when the signal-to-noise ratio is 20dB and the number of snapshots is 100.
  • the first part of the model in Figure 8(a) and (c) obtains the candidate area very accurately, and the The second part outputs more accurate results.
  • the output in Figure 8(b) is -0.44°
  • the output in Figure 8(d) is 0.25°
  • the results estimated by the DNN are -27.44° and 13.25°, which differ from the true values by 0.01° and 0.04°, respectively.
  • the signal-to-noise ratio is improved, the performance of DNN is also improved, and the estimated value of the model is more accurate.
  • the present invention divides DOA estimation into two steps, and designs a two-stage network.
  • the first part of the network can accurately estimate the target angle at the level of 1°, with an accuracy rate close to 100%; the second part is based on this Further estimation of the target angle with a resolution of 0.01° can control the error within a very limited range, which can realize accurate estimation of off-grid signals.

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Abstract

一种阵列超高分辨波达方向估计方法，基于接收信号自相关矩阵中的信息，采用两段式的深度神经网络结构，通过第一部分确定信号角度所处的1°区间，通过第二部分在更高分辨率的网格上对信号角度进行具体估计，二者结合实现对信号入射角度的精确估计，包括一般意义上的离网信号，可以达到0.01°级别的分辨率。使用两段式的神经网络结构，在实现超高分辨波达方向估计的同时，有效避免了神经网络参数过多、训练时间过长的问题，并且计算简单、反应快速，符合实际应用的需求。

Description

一种阵列超分辨波达方向估计方法 技术领域
本发明属于波达方向估计领域，尤其涉及一种基于深度学习的超分辨波达方向估计方法。
背景技术
波达方向(Direction of Arrival，DOA)估计是雷达和无线通信领域中有着悠久历史的一个研究问题，是阵列信号处理的重要分支。常用的经典超分辨DOA估计算法有美国学者Schmidt在1986年提出的多重信号分类(Multiple Signal Classification,MUSIC)算法，通过分离信号子空间和噪声子空间构造出空间伪谱，实现传统估计方法到超分辨DOA估计的飞跃。同样成功的还有旋转不变子空间(Estimation of Signal Parameters via Rotational Invariance,ESPRIT)算法，通过阵列不同子阵间的旋转不变性实现超分辨DOA估计。这两个种算法之后也产生了很多变体，得到了进一步的研究和发展。
深度学习是一种热门的非线性算法，随着近些计算机技术的快速发展，深度学习解决问题的强大能力愈发凸显，其应用也被拓展到了各个领域，包括阵列信号的DOA估计。由于相对于传统的方法，用深度学习进行DOA估计有着很多独特的优越性，近些年也成为了对科研人员非常有吸引力的研究方向。首先，深度学习的方法仅在网络训练时耗时较长，训练完成后应用于DOA估计时很快能得到结果，其快速高效的特点对于实际应用有着重要的意义；其次，网络训练完成后不需要对输入参数进行复杂的操作，通过简单的加减法即可得到估计结果，更符合实际应用的需求；最后，深度学习网络在信噪比更低、快拍数更少的也能很好地提取数据特征，表现出很好的鲁棒性。
目前基于深度学习的波达方向估计方法也有一些，但在以往的基于深度学习的DOA估计的研究中，都是采取间隔为1°的网格来训练的，而实际的信号通常都是离网信号，即目标角度含有小数部分，一般意义上的离网信号角度精确到小数点后两位，此时小数部分便被忽略了。当信号的真实角度不是正好位于网格上时，会产生较大的估计误差，DOA估计的性能也会下降，因此需要考虑更高的角度分辨率。另外，使用间隔较大的网格训练会忽略掉很多信号角度小数部分的信息。减小网格的间距，或者说增大角度分辨率，是一种提升估计精度的有效方法，但是在深度学习的方法中这就 意味着增加输出向量的长度，随之而来的问题是网络的参数会大大增加。因此，需要设计一种合理的网络模型，在提高估计精度的同时，避免网络参数过多的问题。
发明内容
发明目的：针对现有基于深度学习的DOA估计方法中遇到的角度分辨率的局限性，主要是一般意义上的离网信号，提出一种新的两段式的神经网络结构用于波达方向估计，在使用间隔为1°的网格对目标角度进行准确估计后，更进一步地实现对信号角度小数部分的估计，达到0.01°级别的分辨率。
技术方案：一种超高分辨的阵列信号波达方向估计方法，包括如下步骤：
(1)确定入射信号的数量K，角度范围，待估角度空间，角度分辨率；
(2)确定接收阵列模型，阵元数N，波长λ与阵元间距d，噪声类型；
(3)确定接收信号的快拍数L，根据待估角度空间和信噪比情况随机产生入射信号并计算得到接收信号的快拍；
(4)由接收信号的快拍估计其自相关矩阵，进行预处理，得到实数向量作为神经网络的训练输入；
(5)对入射信号的实际角度做预处理并进行向量编码，得到两个向量，作为神经网络的训练标签；
(6)将输入和标签整合为数据集，划分训练集和验证集，搭建神经网络并进行训练，根据训练过程中训练集和验证集的loss曲线对网络参数和结构进行调整，得到高分辨率的波达方向估计神经网络模型，使估计误差降低到0.01°级别；
(7)应用步骤(6)得到的波达方向估计神经网络模型来进行DOA估计，随机生成角度范围内某个角度的入射信号，重复步骤(4)的数据处理流程得到神经网络的输入，使用训练好的神经网络模型进行角度估计，得到输出向量并进行处理，计算出所估计的角度并输出。
所述步骤(1)中，入射信号数量K＝1，信号为高斯信号，角度搜索范围是φ min到φ max，角度分辨率为Δφ＝0.01°。
所述步骤(2)中，阵元间距是波长的一半，即d＝λ/2，噪声为加性高斯白噪声，与信号不相关。
所述步骤(3)中，接收信号为y(t)＝A(θ)s(t)+n(t),t＝1,...,L；s(t)为入射信号，n(t)为加性高斯白噪声，A(θ)为导向矢量矩阵，y(t)表示接收信号在t时刻的采样。
所述步骤(4)中，接收信号自相关矩阵的估计值为
Figure PCTCN2022077069-appb-000001
预处理是取对角线的实数值以及下半三角的实部和虚部组合成一个实数向量：
Figure PCTCN2022077069-appb-000002
其中，τ i,j,i,j∈{1,2,...,N}表示矩阵R y第i行第j列的元素，
Figure PCTCN2022077069-appb-000003
和
Figure PCTCN2022077069-appb-000004
分别表示实部和虚部，所得向量x即模型的输入，长度为N 2。
所述步骤(5)中，第一个标签是长度为φ max-φ min+1的向量z 1，代表了分辨率为Δφ＝1°的大网格，对于目标角度θ j，对应的标签
Figure PCTCN2022077069-appb-000005
仅在round(θ j)所对应的格点上取值为100，其余位置为0，θ j精确到小数点后两位；第二个标签是长度为100的向量z 2，代表了以第一部分确定的格点为中心，所取的分辨率为0.01°的小网格范围，仅在θ j-round(θ j)所对应的格点上取值为100，其余位置为0。
所述步骤(6)中，数据集为
Figure PCTCN2022077069-appb-000006
所述神经网络为两段式的结构，包含两个部分，第一个部分包含五个全连接层，第二个部分包含六个全连接层，总共十一个全连接层，有两个输出。
所述步骤(7)中，对输出的处理是寻找两个向量的最大值，并将所对应的整数角度和小数角度相加。
有益效果：本发明基于以往的深度学习的DOA估计方法，通过将离网信号的整数和小数部分分别用两部分网络进行训练。第一部分确定角度在1°间隔的网格上的位置，即目标角度所在的1°区间范围，第二部分由原本的输入结合上第一部分的输出，在第一部分所确定的范围内得到更精确的小数部分的角度，从而在不用过多地增加训练参数和输出向量长度的情况下，从数据中提取有用的信息并实现更加精确、超高分辨的DOA估计。整个网络都由全连接层组成，模型参数少、计算简单快速，具有良好的实际应用前景。
附图说明
图1为均匀等距线阵(ULA)的模型图；
图2为本发明设计的神经网络波达方向估计的流程图；
图3为标签设置的原理图；
图4为本发明设计的神经网络模型图；
图5为本发明神经网络的第一部分训练时的loss下降曲线；
图6为本发明神经网络的第二部分训练时的loss下降曲线；
图7a-图7d为信噪比5dB时入射角度为-27.43°和13.29°的信号数据分别通过网络后的输出，其中，图7a和图7c为网络第一部分的输出，图7b和图7d为网络第二部分的输出；
图8a-图8d为信噪比20dB时入射角度为-27.43°和13.29°的信号数据分别通过网络后的输出，其中，图8a和图8c为网络第一部分的输出，图8b和图8d为网络第二部分的输出；
图9为本发明神经网络在不同信噪比情况下波达方向估计的RMSE图。
具体实施方式
下面结合附图对本发明做更进一步的解释。
采用了以下的一些参数和设置。采用阵元数N＝12，阵元间隔为半波长的均匀等距线阵(ULA)，信号源的个数K＝1。根据阵元数量，DNN输入向量的长度为144。DOA范围是-60°到60°，考虑角度分辨率为0.01°，总共有12001种入射角度情况，在每种信噪比情况下随机产生12001×5＝60005个数据，训练和测试的输入都是自相关矩阵的估计值预处理后得到的实数向量，采用接收信号的L＝100个快拍来计算。为了训练所提出的模型，我们使用了高信噪比{25，26，27，28，29，30}dB情况下的数据，总的样本数量是D＝6×60005＝360030。
本发明的DNN网络采用离线训练的方式，数据集被划分为训练集(90％)和验证集(10％),用自适应矩估计(Adaptive moment estimation，Adam)来进行参数的更新/优化，初始学习率设置为0.0001，两部分网络的loss权重分别为1和0.1。训练批量大小为1000，训练了500个epoch。网络用Keras搭建，运行的操作系统是Windows，处理器是Intel i7-9750H，GPU是NVIDIA GeForce RTX 2060。
搭建如图4所示的神经网络模型，并用数据集进行训练，训练过程中网络两个输出的loss曲线分别如图5和图6所示。训练完成后，测试了在不同信噪比条件下固定角 度的模型输出情况，粗略地观察模型两个部分的输出以及作用，测试用到的两个目标角度分别为-27.43°和13.29°。图7是信噪比为5dB，快拍数为100时两个角度的估计结果，其中图7(a)和(c)是模型第一部分的输出结果，可以看出输出的峰值和真实的标签完全重合，对应的结果分别为-27°和13°，确定了一个非常精确的候选区域，把待估的角度分别限定在了区间[-27.5,-26.5)和[12.5,13.5)内。之后，图7(b)和(d)是模型第二部分的始出结果，在第一部分的所确定的区间内得到更为精确的角度。(b)中的输出结果为-0.36°，(d)中的输出结果为0.19°，因此估计得到的角度分别为-27°+(-0.36°)＝-27.36°和13°+0.19°＝13.19°，与实际结果分别相差0.07°和0.1°。显然，估计结果与真实值之间的误差已经非常小了，被降低到了0.1°的级别，甚至更低。
图8是信噪比为20dB，快拍数为100时两个角度的估计结果，同样的，图8(a)和(c)中模型的第一部分很准确地得到了候选区域，并且在第二部分输出了更准确的结果。图8(b)中的输出为-0.44°，图8(d)中的输出为0.25°，因此DNN估计的结果分别为-27.44°和13.25°，与真实值分别相差0.01°和0.04°。显然，当信噪比提升时，DNN的表现也得到了提升，模型的估计值更加准确。
如图9所示是本发明的神经网络在不同信噪比情况下DOA估计的RMSE，可以看出即便是在较低的信噪比情况下，估计误差也被降到了0.1°左右，在高信噪比情况下甚至将误差控制在了0.02～0.03°左右。
本发明将DOA估计分为两步，设计了一个两段式的网络，第一部分的网络可以对目标角度在1°级别上的准确估计，接近100％的准确率；第二部分在此基础上对目标角度做0.01°分辨率的进一步估计，将误差控制在非常有限的范围内，可以实现对离网信号的精确估计。
以上所述仅是本发明的优选实施方式，应当指出，对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。

Claims (8)

  1. 一种阵列信号超分辨波达方向估计方法，其特征在于，包括如下步骤：
    (1)确定入射信号的数量K，角度范围，待估角度空间，角度分辨率；
    (2)确定接收阵列模型，阵元数N，波长λ与阵元间距d，噪声类型；
    (3)确定接收信号的快拍数L，根据待估角度空间和信噪比情况随机产生入射信号并计算得到接收信号的快拍；
    (4)由接收信号的快拍估计其自相关矩阵，进行预处理，得到实数向量作为神经网络的训练输入；
    (5)对入射信号的实际角度做预处理并进行向量编码，得到两个向量，作为神经网络的训练标签；
    (6)将输入和标签整合为数据集，划分训练集和验证集，搭建神经网络并进行训练，根据训练过程中训练集和验证集的loss曲线对网络参数和结构进行调整，得到高分辨率的波达方向估计神经网络模型，使估计误差降低到0.01°级别；
    (7)应用步骤(6)得到的波达方向估计神经 网络模型来进行DOA估计，随机生成角度范围内某个角度的入射信号，重复步骤(4)的数据处理流程得到神经网络的输入，使用训练好的神经网络模型进行角度估计，得到输出向量并进行处理，计算出所估计的角度并输出。
  2. 根据权利要求1所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(1)中，入射信号数量K＝1，信号为高斯信号，角度搜索范围是φ min到φ max，角度分辨率为Δφ＝0.01°。
  3. 根据权利要求1所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(2)中，阵元间距是波长的一半，即d＝λ/2，噪声为加性高斯白噪声，与信号不相关。
  4. 根据权利要求1所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(3)中，接收信号为y(t)＝A(θ)s(t)+n(t),t＝1,...,L；s(t)为入射信号，n(t)为加性高斯白噪声，A(θ)为导向矢量矩阵，y(t)表示接收信号在t时刻的采样。
  5. 根据权利要求1所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(4)中，接收信号自相关矩阵的估计值为
    Figure PCTCN2022077069-appb-100001
    预处理是取对角线的实数值以及下半三角的实部和虚部组合成一个实数向量：
    Figure PCTCN2022077069-appb-100002
    其中，τ i,j,i,j∈{1,2,...,N}表示矩阵R y第i行第j列的元素，
    Figure PCTCN2022077069-appb-100003
    和
    Figure PCTCN2022077069-appb-100004
    分别表示实部和虚部，所得向量x即模型的输入，长度为N 2。
  6. 根据权利要求2所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(5)中，第一个标签是长度为φ max-φ min+1的向量z 1，代表了分辨率为Δφ＝1°的大网格，对于目标角度θ j，对应的标签
    Figure PCTCN2022077069-appb-100005
    仅在round(θ j)所对应的格点上取值为100，其余位置为0，θ j精确到小数点后两位；第二个标签是长度为100的向量z 2，代表了以第一部分确定的格点为中心，所取的分辨率为0.01°的小网格范围，仅在θ j-round(θ j)所对应的格点上取值为100，其余位置为0。
  7. 根据权利要求6所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(6)中，数据集为
    Figure PCTCN2022077069-appb-100006
    所述神经网络为两段式的结构，包含两个部分，第一个部分包含五个全连接层，第二个部分包含六个全连接层，总共十一个全连接层，有两个输出。
  8. 根据权利要求1所述的超高分辨的阵列信号波达方向估计方法，其特征在于：步骤(7)中，对输出的处理是寻找两个向量的最大值，并将所对应的整数角度和小数角度相加。
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