WO2023279411A1 - 基于耦合张量分解的l型互质阵列波达方向估计方法 - Google Patents
基于耦合张量分解的l型互质阵列波达方向估计方法 Download PDFInfo
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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
- G01S3/74—Multi-channel systems specially adapted for direction-finding, i.e. having a single antenna system capable of giving simultaneous indications of the directions of different signals
-
- 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
- G01S3/02—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
- G01S3/04—Details
- G01S3/043—Receivers
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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
- G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
Definitions
- the invention belongs to the technical field of array signal processing, and in particular relates to a statistical signal processing technology based on multi-dimensional sparse array virtual domain high-order statistics, specifically an L-type coprime array direction-of-arrival estimation method based on coupled tensor decomposition, which can be used for target setting.
- the coprime array As a sparse array with a systematic structure, the coprime array has the advantages of large aperture, high resolution, and high degree of freedom. It can break through the limitation of the Nyquist sampling rate and improve the comprehensive performance of DOA estimation.
- the common practice is to derive the coprime array received signal to a second-order statistical model, and realize the virtual domain-based Direction of arrival estimation for signal processing.
- existing methods usually model the received signal as a vector and derive the virtual domain signal by vectorizing the covariance matrix of the received signal.
- the vectorized signal processing method not only loses the structural information of the received signal of the coprime array, but also the virtual domain signal model derived from the vectorization There are problems such as structural damage and excessive linear scale.
- the virtual domain signal corresponding to the virtual uniform array is a single-snapshot signal, there is a rank deficit problem in the statistics of the virtual domain signal; in order to solve this problem, the traditional method based on spatial smoothing divides the virtual domain signal, and The segmented virtual domain signals are averaged and statistically processed to obtain full-rank virtual domain signal statistics, so as to realize effective DOA estimation.
- this kind of approach often ignores the spatial correlation attributes between the segmented virtual domain signals, and the statistical averaging process causes performance loss.
- tensor in order to preserve the structured information of multi-dimensional received signals, tensor, as a multi-dimensional data type, has been applied in the field of array signal processing to represent received signals covering complex electromagnetic information; Extraction can achieve high-precision DOA estimation.
- the existing tensor signal processing methods are only effective under the premise of matching the Nyquist sampling rate, and have not yet involved the statistical analysis and virtual domain expansion of coprime array sparse signals.
- traditional tensor signal feature extraction methods often decompose a single independent tensor, and when there are multiple tensor signals with spatial correlation properties, there is no effective means of multi-dimensional feature joint extraction. Therefore, how to combine virtual domain tensor modeling and virtual domain signal correlation processing in the multi-dimensional coprime array scenario to achieve high-precision two-dimensional DOA estimation is still an urgent problem to be solved.
- the purpose of the present invention is to propose a method for estimating the direction of arrival of an L-type coprime array based on coupling tensor decomposition for the problems of damaged multi-dimensional signal structure and loss of information associated with virtual domain signals existing in the existing methods, in order to establish an L-type coprime array DOA estimation method.
- the mass array augments the connection between the virtual domain and the tensor signal modeling, and fully mines the correlation information of the tensor statistics in the multi-dimensional virtual domain to achieve high-precision two-dimensional direction of arrival estimation, which provides a feasible idea and an effective solution.
- the object of the present invention is achieved by the following technical solutions: a method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition, the method comprising the following steps:
- the receiving end uses physical antenna array elements to construct an L-shaped coprime array with separate sub-arrays;
- the L-shaped coprime array consists of two coprime linear arrays located on the x-axis and the y-axis Composed of two coprime linear arrays and The first array elements are laid out from the (1,0) and (1,0) positions on the xoy coordinate system;
- the coprime linear array contains array elements, among which, and is a pair of coprime integers,
- s k [s k,1 ,s k,2 ,...,s k,T ] T is the multi-snapshot sampling signal waveform corresponding to the kth incident signal source
- T is the number of sampling snapshots
- o represents a vector outer product
- the steering vector of , corresponding to the incoming wave direction is The signal source of is expressed as:
- Augmented non-continuous virtual linear arrays on the x-axis and y-axis are constructed by forming difference arrays on the exponent items respectively, Represents the Kronecker product.
- Corresponding to a two-dimensional discontinuous virtual cross array contains a virtual uniform cross array in and are virtual uniform linear arrays on the x-axis and y-axis, respectively; and The position of each virtual array element in is expressed as and in and
- the factor matrix of Re represents the tensor superposition operation on the a-th dimension, Canonical polyadic models representing tensors;
- the method for estimating the direction of arrival of L-shaped coprime arrays based on coupled tensor decomposition is characterized in that the structure of the L-shaped coprime array with separate sub-arrays in step (1) is specifically described as: forming an L-shaped coprime array coprime linear array of arrays Consists of a pair of sparse uniform linear subarrays, the two sparse uniform linear subarrays contain and antenna elements, the distance between the elements is and in, and is a pair of coprime integers; In the two sparse linear uniform sub-arrays, the sub-arrays are combined according to the overlap of the first array elements to obtain coprime linear array of array elements
- the described L-type coprime array DOA estimation method based on coupled tensor decomposition is characterized in that the fourth-order statistic derivation described in step (2), in practice, for the received signal of T sampling snapshots and By finding their fourth-order statistics, the sampling-based fourth-order covariance tensor is obtained
- the L-type coprime array DOA estimation method based on coupling tensor decomposition is characterized in that, the coupling virtual domain tensor construction described in step (5) obtains P x virtual domain tensors Represent the same spatial information in the second and third dimensions, but represent different spatial information in the first dimension, P x virtual domain tensors has a coupled relationship in the second and third dimensions, the first dimension characterizing a virtual uniform linear subarray The angle information, the second dimension characterizes the translation window The angle information of the third dimension represents the translation information in the y-axis direction.
- the method for estimating DOA of L-shaped coprime arrays based on coupled tensor decomposition is characterized in that, in the coupling virtual domain tensor construction process described in step (5), by superimposing translational virtual domain signals in the x-axis direction Constructs coupled virtual domain tensors, specifically, for P x virtual uniform subarrays with the same p y subscripts They cover the same angle information in the y-axis direction, and have a spatial translation relationship in the x-axis direction, and their corresponding virtual domain signals Superposition in the third dimension yields P y virtual domain tensors
- the method for estimating direction of arrival of L-shaped coprime arrays based on coupled tensor decomposition is characterized in that, in the coupling virtual domain tensor decomposition described in step (6), the constructed P x three-dimensional virtual domain tensors are utilized.
- the coupling relationship, through the joint least squares optimization problem pair Perform a coupled canonical polyadic decomposition:
- ⁇ F represents the Frobenius norm
- the described L-type coprime array DOA estimation method based on coupled tensor decomposition is characterized in that, in step (6), the estimated spatial factor Extract parameters from and
- the closed-form solution of is:
- the present invention has the following advantages:
- the present invention expresses the actual received signal of the L-shaped coprime array by tensor, explores the derivation form of the multi-dimensional virtual domain signal on the basis of tensorized signal modeling, fully retains and utilizes the original structured information of the received signal;
- the present invention structurally derives a plurality of virtual domain tensors with spatial correlation attributes, which provides a technical premise for fully utilizing the virtual domain signal correlation information to realize the direction of arrival estimation;
- the present invention proposes a coupling processing method for multiple virtual domain tensors, and designs a coupling virtual domain tensor decomposition optimization method based on joint least squares. Under the premise of fully considering the spatial correlation properties of virtual domain tensors, Accurate joint estimation of two-dimensional direction of arrival is realized.
- Fig. 1 is the overall flow chart of the present invention.
- Fig. 2 is a schematic structural diagram of an L-shaped coprime array with sub-arrays arranged in the present invention.
- Fig. 3 is a schematic diagram of a virtual uniform cross array and a virtual uniform sub-array constructed in the present invention.
- Fig. 4 is a performance comparison diagram of the direction of arrival estimation accuracy of the method proposed in the present invention under different signal-to-noise ratio conditions.
- Fig. 5 is a comparison diagram of the direction of arrival estimation accuracy performance of the method proposed in the present invention under the condition of different sampling snapshot numbers.
- the present invention proposes an L-type coprime array DOA estimation method based on coupling tensor decomposition, and by deriving a method based on tensor model
- the L-shaped coprime array virtual domain signal, and the coupling idea of virtual domain tensor is constructed to realize high-precision two-dimensional direction of arrival estimation by using the virtual domain tensor correlation information.
- the realization steps of the present invention are as follows:
- Step 1 Construct an L-shaped coprime array with separated sub-arrays and perform received signal modeling. use on the receiving end
- a physical antenna element constructs an L-shaped coprime array with separate sub-arrays, as shown in Figure 2: a coprime linear array is constructed on the x-axis and y-axis respectively contains antenna elements, where, and is a pair of coprime integers,
- s k [s k,1 ,s k,2 ,...,s k,T ] T is the multi-snapshot sampling signal waveform corresponding to the kth incident signal source
- T is the number of sampling snapshots
- o represents a vector outer product
- the steering vector of , corresponding to the incoming wave direction is The signal source of is expressed as:
- Step 2 Deduce the fourth-order covariance tensor of the received signal of the L-shaped coprime array. coprime linear array and The sampling signal and By calculating their cross-correlation statistics, the second-order cross-correlation matrix is obtained
- Step 3 Deduce the fourth-order virtual domain signal corresponding to the augmented virtual uniform cross array.
- fourth-order covariance tensors In the dimension that represents the spatial information in the same direction, it can correspond to two coprime linear arrays and The conjugate steering vector of and Form a difference array on the exponent item, thereby constructing a non-continuous augmented virtual linear array on the x-axis and y-axis respectively, and correspondingly obtain a two-dimensional non-continuous virtual cross array
- the fourth-order covariance tensor The 1st and 3rd dimensions represent the spatial information in the x-axis direction, and the 2nd and 4th dimensions represent the spatial information in the y-axis direction; for this purpose, define a set of dimensions Through the fourth-order covariance tensor Perform tensor transformation of merging dimensions to obtain a non-continuous virtual cross array corresponding to The fourth-order virtual domain signal of
- Augmented virtual linear arrays on the x-axis and y-axis are constructed by forming difference arrays on the exponent items respectively, Represents the Kronecker product. contains a virtual uniform cross array As shown in Figure 3, where and are virtual uniform linear arrays on the x-axis and y-axis, respectively. and The positions of each virtual array element in are expressed as and in and
- Step 4 Translate and divide the virtual uniform cross array.
- the virtual uniform subarray The virtual domain signal of can be expressed as:
- Step 5 Construct the coupled virtual domain tensor by superimposing the shifted virtual domain signals.
- Virtual uniform subarray due to translational partitioning There is a spatial translation relationship between them, and the virtual domain signals corresponding to these virtual uniform subarrays are structurally superimposed to obtain several virtual domain tensors with coupling relationships.
- P y virtual uniform subarrays with the same P x subscript They cover the same angle information in the x-axis direction, and have a spatial translation relationship in the y-axis direction. Therefore, their corresponding virtual domain signals Superimpose on the third dimension to get P x three-dimensional coupled virtual domain tensors
- the factor matrix of Re represents the tensor superposition operation on the a-th dimension, Represents the canonical polyadic model of a tensor; the constructed P x three-dimensional virtual domain tensors In the second dimension (the translation window Angle information) and the third dimension (translation information in the y-axis direction) represent the same spatial information, while in the first dimension (virtual uniform linear subarray Different spatial information is represented on the angle information), for this reason, the virtual domain tensor There is a coupling relationship in the second dimension and the third dimension.
- the coupled virtual domain tensor can be constructed by superimposing translational virtual domain signals in the x-axis direction.
- P x virtual uniform subarrays with the same p y subscript They cover the same angle information in the y-axis direction, and have a spatial translation relationship in the x-axis direction, and their corresponding virtual domain signals can be Superimpose on the third dimension to get P y three-dimensional virtual domain tensors
- the constructed P y three-dimensional virtual domain tensors In the first dimension (the translation window Angle information) and the third dimension (translation information in the x-axis direction) represent the same spatial information, while in the second dimension (virtual uniform linear subarray Different spatial information is represented on the angle information), for this reason, the virtual domain tensor Has a coupling relationship on the first dimension and the third dimension;
- Step 6 Obtain the direction of arrival estimation results by coupling virtual domain tensor decomposition. Utilize the constructed P x virtual field tensors The coupling relationship, through the joint least squares optimization problem pair Perform a coupled canonical polyadic decomposition:
- L-shaped coprime array is used to receive the incident signal, and its parameters are selected as That is, the L-shaped coprime array of architectures contains antenna elements. Assume that there are two incident narrowband signals, and the azimuth and elevation angles of the incident direction are [20.5°, 30.5°] and [45.6°, 40.6°] respectively.
- the method proposed in the present invention makes full use of the structured information of the received signal of the L-shaped coprime array by constructing a virtual domain tensor, thereby having a more superior direction of arrival Estimation accuracy; on the other hand, compared to the TensorMUSIC method based on traditional tensor decomposition, the performance advantage of the proposed method of the present invention comes from the spatial correlation properties of multi-dimensional virtual domain signals fully utilized by coupling virtual domain tensor processing, while traditional The tensor decomposition method only processes the tensor of a single virtual domain after spatial smoothing, resulting in the loss of the associated information of the virtual domain signal.
- the present invention constructs the relationship between the L-shaped coprime array multi-dimensional virtual domain and the tensor signal modeling, deduces the sparse tensor signal to the virtual domain tensor model, and digs deeply into the received signal of the L-shaped coprime array. and the multi-dimensional features of the virtual domain; moreover, the spatial superposition mechanism of the virtual domain signal is established, and the virtual domain tensor with the spatial coupling relationship is constructed without introducing spatial smoothness; finally, the present invention uses the virtual domain tensor The coupled decomposition of , realizes the precise estimation of two-dimensional direction of arrival, and gives its closed-form solution.
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Abstract
一种基于耦合张量分解的L型互质阵列波达方向估计方法，主要解决现有方法中多维信号结构受损和虚拟域信号关联信息丢失的问题，其实现步骤是：构建子阵分置的L型互质阵列并进行接收信号建模；推导L型互质阵列接收信号的四阶协方差张量；推导对应增广虚拟均匀十字阵列的四阶虚拟域信号；平移分割虚拟均匀十字阵列；通过叠加平移虚拟域信号构造耦合虚拟域张量；通过耦合虚拟域张量分解获得波达方向估计结果。充分利用所构建子阵分置的L型互质阵列虚拟域张量统计量的空间关联属性，通过耦合虚拟域张量处理实现了高精度的二维波达方向估计，可用于目标定位。
Description
本发明属于阵列信号处理技术领域，尤其涉及基于多维稀疏阵列虚拟域高阶统计量的统计信号处理技术，具体是一种基于耦合张量分解的L型互质阵列波达方向估计方法，可用于目标定位。
互质阵列作为一种具有系统化结构的稀疏阵列，具备大孔径、高分辨率、高自由度的优势，能够突破奈奎斯特采样速率的限制，实现波达方向估计综合性能的提升。为了在互质阵列场景下实现匹配奈奎斯特采样速率的波达方向估计，常用做法是将互质阵列接收信号推导至二阶统计量模型，通过构造增广的虚拟均匀阵列实现基于虚拟域信号处理的波达方向估计。然而，现有方法通常将接收信号建模成矢量，并通过矢量化接收信号协方差矩阵推导虚拟域信号。在部署多维互质阵列的场景中，由于接收信号涵盖多维度的时空信息，矢量化信号的处理方法不仅损失了互质阵列接收信号的结构化信息，且由矢量化推导得到的虚拟域信号模型存在结构受损、线性尺度过大等问题。另一方面，由于对应虚拟均匀阵列的虚拟域信号是单快拍信号，因此虚拟域信号统计量存在秩亏问题；为了解决该问题，传统基于空间平滑的方法将虚拟域信号进行分割，并对分割后的虚拟域信号进行平均统计处理以得到满秩的虚拟域信号统计量，从而实现有效的波达方向估计。然而，这类做法往往忽略了被分割虚拟域信号之间的空间关联属性，统计平均的处理过程造成了性能损失。
针对以上问题，为了保留多维接收信号的结构化信息，张量作为一种多维的数据类型，开始被应用于阵列信号处理领域，用于表征涵盖复杂电磁信息的接收信号；通过对其进行多维特征提取，可实现高精度的波达方向估计。然而，现有张量信号处理方法仅仅在匹配奈奎斯特采样速率的前提下有效，尚未涉及到互质阵列稀疏信号的统计分析及其虚拟域拓展。另一方面，传统的张量信号特征提取方法往往是针对单个独立张量进行分解，而当存在多个具备空间关联属性的张量 信号时，缺乏有效的多维特征联合提取手段。为此，如何在多维互质阵列的场景下结合虚拟域张量建模和虚拟域信号关联处理，实现高精度的二维波达方向估计，仍然是一个亟待解决的问题。
发明内容
本发明的目的在于针对现有方法存在的多维信号结构受损和虚拟域信号关联信息丢失问题，提出一种基于耦合张量分解的L型互质阵列波达方向估计方法，为建立L型互质阵列增广虚拟域与张量信号建模的联系，充分挖掘多维虚拟域张量统计量的关联信息，以实现高精度的二维波达方向估计提供了可行的思路和有效的解决方案。
本发明的目的是通过以下技术方案来实现的：一种基于耦合张量分解的L型互质阵列波达方向估计方法，该方法包含以下步骤：
(1)接收端使用
个物理天线阵元，构建一个子阵分置的L型互质阵列；该L型互质阵列由位于x轴和y轴上的两个互质线性阵列
组成，两个互质线性阵列
和
的首阵元分别从xoy坐标系上(1,0)和(1,0)位置开始布设；互质线性阵列
中包含
个阵元，其中，
和
为一对互质整数，|·|表示集合的势；分别用
和
表示L型互质阵列中各阵元在x轴和y轴上的位置，其中，
单位间隔d取为入射窄带信号波长的一半；
假设有K个来自
方向的远场窄带非相干信号源，则组成L型互质阵列的互质线性阵列
接收信号建模为：
其中，s
k＝[s
k,1,s
k,2,…,s
k,T]
T为对应第k个入射信号源的多快拍采样信号波形，T为采样快拍数，ο表示矢量外积，
为与各信号源相互独立的噪声，
为
的导引矢量，对应于来波方向为
的信号源，表示为：
其中，
[·]
T表示转置操作；
(2)通过求
和
的互相关统计量，得到二阶互相关矩阵
其中，
表示第k个入射信号源的功率，E{·}表示取数学期望操作，(·)
H表示共轭转置操作，(·)
*表示共轭操作，在二阶互相关矩阵的基础上，推导子阵分置L型互质阵列的四阶统计量，即通过计算二阶互相关矩阵
的自相关得到四阶协方差张量
(3)定义维度集合
通过对四阶协方差张量
进行维度合并的张量变换，得到一个四阶虚拟域信号
其中，
和
分别通过在指数项上形成差集数组，构造出x轴和y轴上的增广非连续虚拟线性阵列，
表示Kronecker积。
对应一个二维非连续虚拟十字阵列
中包含一个虚拟均匀十字阵列
其中
和
分别为x轴和y轴上的虚拟均匀线性阵列；
和
中各虚拟阵元的位置表示为
和
其中
且
从非连续虚拟十字阵列
的虚拟域信号
中提取对应于虚拟均匀十字阵列
中各虚拟阵元位置的元素，得到
所对应的虚拟域信号
建模为：
其中，
分别为
和
的导引矢量；
(4)从
和
中分别提取子阵列
作为平移窗口；分别将平移窗口
和
沿着x轴和y轴的负半轴方向逐次平移一个虚拟阵元间隔，得到P
x个虚拟均匀线性子阵列
和P
y个虚拟均匀线性子阵列
其中
则虚拟均匀子阵列
的虚拟域信号可表示为：
其中，
分别为
和
的导引矢量；
(5)对于具有相同p
x下标的P个虚拟均匀子阵列
将它们对应的虚拟域信号
在第三维度上进行叠加，得到P
x个三维的耦合虚拟域张量
其中，
为平移窗口
的导引矢量，
表示沿着y轴方向的平移因子，
和Q
y＝[q
y(1),q
y(2),…,q
y(K)]为
的因子矩阵，
表示在第a维度上的张量叠加操作，
表示张量的canonical polyadic模型；
(6)对所构造的P
x个耦合虚拟域张量
进行耦合canonical polyadic分解，得到因子矩阵
的估计值
其中包含空间因子
的估计值
随后，从空间因子的估计值
中提取二维波达方向估计结果
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(1)所述子阵分置的L型互质阵列结构具体描述为：组成L型互质阵列的互质线性阵列
由一对稀疏均匀线性子阵列构成，两个稀疏均匀线性子阵列分别包含
和
个天线阵元，阵元间距分别为
和
其中，
和
为一对互质整数；
中两个稀疏线性均匀子阵列按照首阵元重叠的方式进行子阵列组合，获得包含
个阵元的互质线性阵列
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(2)所述的四阶统计量推导，在实际中，对T个采样快拍的接收信号
和
通过求它们的四阶统计量，得到基于采样的四阶协方差张量
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(5)所述的耦合虚拟域张量构造，得到的P
x个虚拟域张量
在第二维度和第三维度上表征相同的空间信息，而在第一维度上表征不同的空间信息，P
x个虚拟 域张量
在第二维度和第三维度上具有耦合关系，所述第一维度表征虚拟均匀线性子阵列
的角度信息，所述第二维度表征平移窗口
的角度信息，所述第三维度表征y轴方向的平移信息。
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(5)所述的耦合虚拟域张量构造过程，通过在x轴方向上叠加平移虚拟域信号构造耦合虚拟域张量，具体为，对于具有相同p
y下标的P
x个虚拟均匀子阵列
它们在y轴方向上涵盖相同的角度信息，在x轴方向上则具备空间平移关系，将它们对应的虚拟域信号
在第三维度上进行叠加，得到P
y个虚拟域张量
其中，
为平移窗口
的导引矢量，
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(6)所述的耦合虚拟域张量分解，利用所构造的P
x个三维虚拟域张量
的耦合关系，通过联合最小二乘优化问题对
进行耦合canonical polyadic分解：
其中，‖·‖
F表示Frobenius范数；求解该联合最小二乘优化问题，得到因子矩阵
的估计值
在耦合虚拟域张量分解问题中，可辨识目标数K的最大值为
超过所构建子阵分置L型互质阵列的实际物理阵元个数。
所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，在步骤(6)中，对估计得到的空间因子
从中提取参数
和
其中，
中虚拟阵元的位置索引，
为
中虚拟阵元的位置索引，z＝[0,1,…,P
y-1]
T表示平移步长，∠(·)表示一个复数的取幅角操作，
表示伪逆操作；最后，根据{μ
1(k),μ
2(k)}与二维波达方向
的关系，即
和
得到二维波达方向估计
的闭式解为：
本发明与现有技术相比具有以下优点：
(1)本发明通过张量表示L型互质阵列实际接收信号，在张量化信号建模的基础上，探究多维虚拟域信号的推导形式，充分保留并利用了接收信号的原始结构化信息；
(2)本发明基于多维虚拟域信号的平移增广，结构化推导了多个具备空间关联属性的虚拟域张量，为充分利用虚拟域信号关联信息实现波达方向估计提供了技 术前提；
(3)本发明提出了面向多个虚拟域张量的耦合处理手段，设计了基于联合最小二乘的耦合虚拟域张量分解优化方法，在充分考虑虚拟域张量空间关联属性的前提下，实现了二维波达方向的精确联合估计。
图1是本发明的总体流程框图。
图2是本发明所提子阵分置L型互质阵列的结构示意图。
图3是本发明所构造虚拟均匀十字阵列和虚拟均匀子阵列示意图。
图4是本发明所提方法在不同信噪比条件下的波达方向估计精度性能比较图。
图5是本发明所提方法在不同采样快拍数条件下的波达方向估计精度性能比较图。
以下参照附图，对本发明的技术方案作进一步的详细说明。
为了解决现有方法存在的多维信号结构受损和虚拟域信号关联信息丢失问题，本发明提出了一种基于耦合张量分解的L型互质阵列波达方向估计方法，通过推导基于张量模型的L型互质阵列虚拟域信号，并构建虚拟域张量的耦合思路，以利用虚拟域张量关联信息实现高精度的二维波达方向估计。参照图1，本发明的实现步骤如下：
步骤1：构建子阵分置的L型互质阵列并进行接收信号建模。在接收端使用
个物理天线阵元构建子阵分置的L型互质阵列，如图2所示：在x轴和y轴上分别构造一个互质线性阵列
中包含
个天线阵元，其中，
和
为一对互质整数，|·|表示集合的势；这两个互质线性阵列
和
的首阵元分别从xoy坐标系上(1,0)和(1,0)位置开始布设，因此，组成L型互质阵列的两个互质线性阵列
和
互不重叠；分别用
和
表示L型互质阵列各阵元在x轴和y轴上的 位置，其中，
单位间隔d取为入射窄带信号波长的一半；组成L型互质阵列的互质线性阵列
由一对稀疏均匀线性子阵列构成，两个稀疏均匀线性子阵列分别包含
和
个天线阵元，阵元间距分别为
和
中两个稀疏均匀线性子阵列按照首阵元重叠的方式进行子阵列组合，获得包含
个阵元的互质线性阵列
假设有K个来自
方向的远场窄带非相干信号源，则将组成L型互质阵列两个互质线性阵列
和
的接收信号建模为：
其中，s
k＝[s
k,1,s
k,2,…,s
k,T]
T为对应第k个入射信号源的多快拍采样信号波形，T为采样快拍数，ο表示矢量外积，
为与各信号源相互独立的噪声，
为
的导引矢量，对应于来波方向为
的信号源，表示为：
其中，
[·]
T表示转置操作；
步骤2：推导L型互质阵列接收信号的四阶协方差张量。利用互质线性阵列
和
的采样信号
和
通过求它们的互相关统计量，得到二阶互相关矩阵
其中，
表示第k个入射信号源的功率，E{·}表示取数学期望操作，(·)
H表示共轭转置操作，(·)
*表示共轭操作；通过计算接收信号的互相关矩阵，将原始接收信号中噪声部分
的影响有效消除。为了实现增广虚拟阵列推导，在二阶互相关统计量的基础上，进一步推导L型互质阵列的四阶统计量。对二阶互相关矩阵
求得其自相关得到四阶协方差张量
在实际中，基于采样的四阶协方差张量
通过求接收信号
和
的四阶统计量得到：
步骤3：推导对应增广虚拟均匀十字阵列的四阶虚拟域信号。通过合并四阶协方差张量
中表征同一方向空间信息的维度，可以使对应两个互质线性阵列
和
的共轭导引矢量
和
在指数项上形成差集数组，从而分别在x轴和y轴上构造一个非连续增广虚拟线性阵列，对应得到一个二维非连续虚拟十字阵列
具体地，四阶协方差张量
的第1、3维度表征x轴方向的空间信息，第2、4维度表征y轴方向的空间信息；为此，定义维度集合
通过对四阶协方差张量
进行维度合并的张量变换，得到一个对应于非连续虚拟十字阵列
的四阶虚拟域信号
其中，
和
分别通过在指数项上形成差集数组，构造出x轴和y轴上的增广虚拟线性阵列，
表示Kronecker积。
中包含一个虚拟均匀十字阵列
如图3所示，其中
和
分别为x轴和y轴上的虚拟均匀线性阵列。
和
中各虚拟阵元的位置分别表示为
和
其中
且
从非连续虚拟十字阵列
的虚拟域信号
中提取对应于虚拟均匀十字阵列
中各虚拟阵元位置的元素，得到
所对应的虚拟域信号
建模为：
其中，
分别表示
和
的导引矢量；
步骤4：平移分割虚拟均匀十字阵列。考虑到组成虚拟均匀十字阵列
的两个虚拟均匀线性阵列
和
分别关于x＝1和y＝1轴对称，从
和
中分别提取子阵列
作为平移窗口；然后，分别将平移窗口
和
沿着x轴和y轴的负半轴方向逐次平移一个虚拟阵元间隔，得到P
x个虚拟均匀线性子阵列
和P
y个虚拟均匀线性子阵列
如图3所示，这里，
则虚拟均匀子阵列
的虚拟域信号可表示为：
其中，
分别为
和
的导引矢量；
步骤5：通过叠加平移虚拟域信号构造耦合虚拟域张量。由于平移分割得到的虚拟均匀子阵列
互相之间存在空间平移关系，对这些虚拟均匀子阵列对 应的虚拟域信号进行结构化叠加得到若干具有耦合关系的虚拟域张量。具体而言，对于具有相同P
x下标的P
y个虚拟均匀子阵列
它们在x轴方向上涵盖相同的角度信息，在y轴方向上则具备空间平移关系，为此，将它们对应的虚拟域信号
在第三维度上进行叠加，得到P
x个三维的耦合虚拟域张量
其中，
为平移窗口
的导引矢量，
表示沿着y轴方向的平移因子，
和Q
y＝[q
y(1),q
y(2),…,q
y(K)]为
的因子矩阵，
表示在第a维度上的张量叠加操作，
表示张量的canonical polyadic模型；所构造的P
x个三维虚拟域张量
在第二维度(平移窗口
的角度信息)和第三维度(y轴方向的平移信息)上表征相同的空间信息，而在第一维度(虚拟均匀线性子阵列
的角度信息)上表征不同的空间信息，为此，虚拟域张量
在第二维度和第三维度上具有耦合关系。
类似地，可以在x轴方向上叠加平移虚拟域信号构造耦合虚拟域张量。具体而言，对于具有相同p
y下标的P
x个虚拟均匀子阵列
它们在y轴方向上涵盖相同的角度信息，在x轴方向上则具备空间平移关系，可以将它们对应的虚拟域信号
在第三维度上进行叠加，得到P
y个三维虚拟域张量
其中，
为平移窗口
的导引矢量，
表示沿着x轴方向的平移因子，
和Q
x＝[q
x(1),q
x(2),…,q
x(K)]为
的因子矩阵；所构造的P
y个三维虚拟域张量
在第一维度(平移窗口
的角度信息)和第三维度(x轴方向的平移信息)上表征相同的空间信息，而在第二维度(虚拟均匀线性子阵列
的角度信息)上表征不同的空间信息，为此，虚拟域张量
在第一维度和第三维度上具有耦合关系；
步骤6：通过耦合虚拟域张量分解获得波达方向估计结果。利用所构造的P
x个虚拟域张量
的耦合关系，通过联合最小二乘优化问题对
进行耦合canonical polyadic分解：
其中，
表示因子矩阵
的估计值，由空间因子
的估计值
组成，‖·‖
F表示Frobenius范数；求解该联合最小二乘优化问题，得到
在该问题中，可辨识目标数K的最大值为
超过所构建子阵分置的L型互质阵列的实际物理阵元个数。同样地，可以对所构造P
y个三维虚拟域张量
进行耦合canonical polyadic分解，估计其因子矩阵
从空间因子的估计值
中提取参数
和
其中，
表示
中虚拟阵元的位置索 引，
表示
中虚拟阵元的位置索引，z＝[0,1,…,P
y-1]
T表示平移步长，∠(·)表示一个复数的取幅角操作，
表示伪逆操作。最后，根据{μ
1(k),μ
2(k)}与二维波达方向
的关系，即
和
得到二维波达方向估计
的闭式解为：
下面结合仿真实例对本发明的效果做进一步的描述。
仿真实例：采用L型互质阵列接收入射信号，其参数选取为
即架构的L型互质阵列共包含
个天线阵元。假定有2个入射窄带信号，入射方向方位角和俯仰角分别是[20.5°,30.5°]和[45.6°,40.6°]。将本发明所提基于耦合张量分解的L型互质阵列波达方向估计方法与传统基于矢量化虚拟域信号处理的Estimation of Signal Parameters via Rotational Invariant Techniques(ESPRIT)方法，以及基于传统张量分解的TensorMultipleSignal Classification(Tensor MUSIC)方法进行对比，分别在图4和图5中对比上述方法在不同信噪比和不同采样快拍数条件下的二维波达方向估计精度性能。
在采样快拍数为T＝300条件下，绘制波达方向估计均方根误差随信噪比变化的性能对比曲线，如图4所示；在信噪比SNR＝0dB条件下，绘制波达方向估计均方根误差随采样快拍数变化的性能对比曲线，如图5所示。从图4和图5的对比结果可以看出，无论是在不同的信噪比场景，还是在不同的采样快拍数场景下，本发明所提方法在波达方向估计精度上均存在性能优势。相比于基于矢量化虚拟域信号处理的ESPRIT方法，本发明所提方法通过构建虚拟域张量，充分利用了L型互质阵列接收信号的结构化信息，从而具备更为优越的波达方向估计精度；另一方面，相比于基于传统张量分解的TensorMUSIC方法，本发明所提方法的性能优势来源于通过耦合虚拟域张量处理充分利用了多维虚拟域信号的空间关联属性，而传统张量分解方法只针对空间平滑后的单一虚拟域张量进行处理，造 成了虚拟域信号关联信息的丢失。
综上所述，本发明构建L型互质阵列多维虚拟域与张量信号建模之间的关联，将稀疏张量信号推导至虚拟域张量模型，深入挖掘了L型互质阵列接收信号和虚拟域的多维特征；再者，建立起虚拟域信号的空间叠加机理，在无需引入空间平滑的前提下，构造出具备空间耦合关系的虚拟域张量；最后，本发明通过虚拟域张量的耦合分解，实现了二维波达方向的精确估计，并给出了其闭式解。
以上所述仅是本发明的优选实施方式，虽然本发明已以较佳实施例披露如上，然而并非用以限定本发明。任何熟悉本领域的技术人员，在不脱离本发明技术方案范围情况下，都可利用上述揭示的方法和技术内容对本发明技术方案做出许多可能的变动和修饰，或修改为等同变化的等效实施例。因此，凡是未脱离本发明技术方案的内容，依据本发明的技术实质对以上实施例所做的任何的简单修改、等同变化及修饰，均仍属于本发明技术方案保护的范围内。
Claims (7)
- 一种基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，包含以下步骤：(1)接收端使用个物理天线阵元，构建一个子阵分置的L型互质阵列；该L型互质阵列由位于x轴和y轴上的两个互质线性阵列
组成，两个互质线性阵列
和
的首阵元分别从xoy坐标系上(1,0)和(1,0)位置开始布设；互质线性阵列
中包含
个阵元，其中，
和
为一对互质整数，|·|表示集合的势；分别用
和
表示L型互质阵列中各阵元在x轴和y轴上的位置，其中，
单位间隔d取为入射窄带信号波长的一半；
假设有K个来自方向的远场窄带非相干信号源，则组成L型互质阵列的互质线性阵列
接收信号建模为：
其中，s k＝[s k,1,s k,2,…,s k,T] T为对应第k个入射信号源的多快拍采样信号波形，T为采样快拍数，表示矢量外积，
为与各信号源相互独立的噪声，
为
的导引矢量，对应于来波方向为
的信号源，表示为：
其中，[·] T表示转置操作；
(2)通过求和
的互相关统计量，得到二阶互相关矩阵
其中，表示第k个入射信号源的功率，E{·}表示取数学期 望操作，(·) H表示共轭转置操作，(·) *表示共轭操作，在二阶互相关矩阵的基础上，推导子阵分置L型互质阵列的四阶统计量，即通过计算二阶互相关矩阵
的自相关得到四阶协方差张量
(3)定义维度集合通过对四阶协方差张量
进行维度合并的张量变换，得到一个四阶虚拟域信号
其中，和
分别通过在指数项上形成差集数组，构造出x轴和y轴上的增广非连续虚拟线性阵列，
表示Kronecker积。
对应一个二维非连续虚拟十字阵列
中包含一个虚拟均匀十字阵列
其中
和
分别为x轴和y轴上的虚拟均匀线性阵列；
和
中各虚拟阵元的位置表示为
和
其中
且
从非连续虚拟十字阵列的虚拟域信号
中提取对应于虚拟均匀十字阵列
中各虚拟阵元位置的元素，得到
所对应的虚拟域信号
建模为：
其中，
分别为和
的导引矢量；
(4)从和
中分别提取子阵列
作为平移窗口；分别将平移窗口
和
沿着x轴和y轴的负半轴方向逐次平移一个虚拟阵元间隔，得到P x个虚拟均匀线性子阵列
和P y个虚拟均匀线性子阵列
其中
则虚拟均匀子阵列
的虚拟域信号可表示为：
其中，
分别为和
的导引矢量；
(5)对于具有相同p x下标的P y个虚拟均匀子阵列将它们对应的虚拟域信号
在第三维度上进行叠加，得到P x个三维的耦合虚拟域张量
其中，为平移窗口
的导引矢量，
表示沿着y轴方向的平移因子，
和Q y＝[q y(1),q y(2),…,q y(K)]为
的因子矩阵，
表示在第a维度上的张量 叠加操作，
表示张量的canonical polyadic模型；
(6)对所构造的P x个耦合虚拟域张量进行耦合canonical polyadic分解，得到因子矩阵
的估计值
其中包含空间因子
的估计值
随后，从空间因子的估计值
中提取二维波达方向估计结果
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(1)所述子阵分置的L型互质阵列结构具体描述为：组成L型互质阵列的互质线性阵列由一对稀疏均匀线性子阵列构成，两个稀疏均匀线性子阵列分别包含
和
个天线阵元，阵元间距分别为
和
其中，
和
为一对互质整数；
中两个稀疏线性均匀子阵列按照首阵元重叠的方式进行子阵列组合，获得包含
个阵元的互质线性阵列
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(2)所述的四阶统计量推导，在实际中，对T个采样快拍的接收信号和
通过求它们的四阶统计量，得到基于采样的四阶协方差张量
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(5)所述的耦合虚拟域张量构造，得到的P x个虚拟域张量在第二维度和第三维度上表征相同的空间信息，而在第一维度上表征不同的空间信息，P x个虚拟域张量
在第二维度和第三维度上具有耦合关系，所述第一维度表征虚拟均匀线性子阵列
的角度信息，所述第二维度表征平移窗口
的角度信息，所述第三维度表征y轴方向的平移信息。
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(5)所述的耦合虚拟域张量构造过程，通过在x轴方向上叠加平移虚拟域信号构造耦合虚拟域张量，具体为，对于具有相同p y下标的P x个虚拟均匀子阵列它们在y轴方向上涵盖相同的角度信息，在x轴方向上则 具备空间平移关系，将它们对应的虚拟域信号
在第三维度上进行叠加，得到P y个虚拟域张量
其中，为平移窗口
的导引矢量，
表示沿着x轴方向的平移因子，
和Q x＝[q x(1),q x(2),…,q x(K)]为
的因子矩阵；所构造的P y个三维虚拟域张量
在第一维度和第三维度上表征相同的空间信息，而在第二维度上表征不同的空间信息，为此，虚拟域张量
在第一维度和第三维度上具有耦合关系；对所构造P y个三维虚拟域张量
进行耦合canonical polyadic分解，估计其因子矩阵
所述第一维度表征平移窗口
的角度信息，所述第二维度表征虚拟均匀线性子阵列
的角度信息，所述第三维度表征x轴方向的平移信息。
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方法，其特征在于，步骤(6)所述的耦合虚拟域张量分解，利用所构造的P x个三维虚拟域张量的耦合关系，通过联合最小二乘优化问题对
进行耦合canonical polyadic分解：
其中，‖·‖ F表示Frobenius范数；求解该联合最小二乘优化问题，得到因子矩阵的估计值
在耦合虚拟域张量分解问题中，可辨识目标数K的最大值为
超过所构建子阵分置L型互质阵列的实际物理阵元个数。
- 根据权利要求1所述的基于耦合张量分解的L型互质阵列波达方向估计方 法，其特征在于，在步骤(6)中，对估计得到的空间因子从中提取参数
和
其中，为
中虚拟阵元的位置索引，
为
中虚拟阵元的位置索引，z＝[0,1,…,P y-1] T表示平移步长，∠(·)表示一个复数的取幅角操作，
表示伪逆操作；最后，根据{μ 1(k),μ 2(k)}与二维波达方向
的关系，即
和
得到二维波达方向估计
的闭式解为：
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