WO2022188336A1 - 一种幅相误差情况下基于稀疏重构的波达方向估计方法 - Google Patents

一种幅相误差情况下基于稀疏重构的波达方向估计方法 Download PDF

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WO2022188336A1
WO2022188336A1 PCT/CN2021/109106 CN2021109106W WO2022188336A1 WO 2022188336 A1 WO2022188336 A1 WO 2022188336A1 CN 2021109106 W CN2021109106 W CN 2021109106W WO 2022188336 A1 WO2022188336 A1 WO 2022188336A1
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matrix
angle
sparse
equation
covariance matrix
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PCT/CN2021/109106
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French (fr)
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宋春毅
俞鼎柯
陈钦
席玉章
王昕�
方文巍
徐志伟
李欢
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浙江大学
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Priority to JP2022549527A priority Critical patent/JP7321612B2/ja
Priority to US17/830,258 priority patent/US20220308150A1/en
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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
    • G01S3/16—Systems for determining direction or deviation from predetermined direction using amplitude comparison of signals derived sequentially from receiving antennas or antenna systems having differently-oriented directivity characteristics or from an antenna system having periodically-varied orientation of directivity characteristic
    • G01S3/22—Systems for determining direction or deviation from predetermined direction using amplitude comparison of signals derived sequentially from receiving antennas or antenna systems having differently-oriented directivity characteristics or from an antenna system having periodically-varied orientation of directivity characteristic derived from different combinations of signals from separate antennas, e.g. comparing sum with difference
    • 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
    • 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
    • G01S3/023—Monitoring or calibrating

Definitions

  • the invention relates to the field of array signal processing, in particular to a direction-of-arrival (Direction-of-arrival, DOA) estimation method based on sparse reconstruction in the case of a gain-phase error (Gain-phase Error).
  • DOA direction-of-arrival
  • Gain-phase Error gain-phase error
  • Direction of arrival estimation of signals is an important research content in the field of array signal processing, which is widely used in radar, sonar, wireless communication and other fields.
  • MUSIC Multiple Signal Classification
  • ESPRIT Rotational Invariance Techniques
  • Most of these classic high-resolution algorithms are based on the premise that the array manifold is accurately known. In the process of practical engineering applications, the amplifier gain is inconsistent when the signal is transmitted in the channel due to changes in climate, environment and the array components themselves. , which leads to the amplitude and phase errors between the array antenna channels, which will cause the deviation of the actual array manifold, and make the performance of the classical high-resolution signal DOA estimation algorithm drop sharply, and even fail in severe cases.
  • the early array error correction was mainly realized by discrete measurement, interpolation and storage of the array manifold directly. Then, by modeling the array disturbance, the array error correction is gradually transformed into a parameter estimation problem, which can be roughly divided into active correction and self-correction.
  • Active correction requires an external auxiliary source or other auxiliary facilities, which increases the cost of signal direction of arrival estimation equipment to a certain extent, and has strict requirements on hardware and environment, which is not applicable in many cases.
  • Self-calibration is to estimate the signal direction of arrival and the array error parameters according to a certain optimization function. It does not require additional auxiliary sources whose azimuth is accurately known, and can realize online estimation.
  • the present invention provides a method for estimating direction of arrival based on sparse reconstruction in the case of amplitude and phase errors, and the specific technical solutions are as follows:
  • a direction of arrival estimation method based on sparse reconstruction in the case of amplitude and phase errors includes the following steps:
  • the S1 is realized through the following sub-steps:
  • M represents the number of array elements
  • ⁇ m represents the eigenvalues arranged in descending order
  • ⁇ m represents the eigenvector corresponding to the eigenvalue ⁇ m
  • ( ⁇ ) H represents the conjugate transpose
  • K represents the number of sources
  • ⁇ m represents the magnitude error estimate value of the mth array element, r m, m represents the value at the covariance matrix (m, m);
  • IM represents the identity matrix of size M.
  • ⁇ k,m represents the delay of the k-th signal at the m-th array element relative to the reference array element
  • B is with The corresponding steering vector matrix formed by extending to ⁇ , p is the Correspondingly extended to the matrix formed on ⁇ ;
  • w [w 1 , w 2 , ..., w M ] T
  • ⁇ 1 represents the regularization constant
  • a q b q H b q
  • b q represents the qth row of B
  • c [c 1 , c 2 ,..., c Q ] T
  • ⁇ 1 represents another regularization constant
  • z represents any matrix with the same specification as p
  • ⁇ 2 represents the regularization constant
  • D ⁇ B′ ⁇ p
  • D represents the intermediate conversion quantity
  • ⁇ represents the deviation angle matrix
  • E q d q H d q
  • d q represents the th q line
  • the index matrix ⁇ has the same dimension as the grid angle matrix ⁇ , the value of ⁇ is 1 at the index of the estimated angle, and the rest is 0, ( ) represents the point multiplication of the matrix, that is, the corresponding elements of the matrix are multiplied.
  • the amplitude and phase error correction and direction of arrival estimation method based on sparse reconstruction of the present invention effectively eliminates the influence of phase error in the estimation of direction of arrival by directly taking the modulo length of each element of the compensation covariance matrix.
  • Figure 1 is a flow chart of a method for estimating direction of arrival based on sparse reconstruction in the case of amplitude and phase errors.
  • FIG. 2 is a schematic diagram of grid division of the array space domain.
  • FIG. 3 is a comparison diagram of the relationship between the root mean square error and the phase error of the direction of arrival estimation performed by the present invention and other algorithms in the same field.
  • FIG. 4 is a comparison diagram of the relationship between the root mean square error and the signal-to-noise ratio of the direction of arrival estimation performed by the present invention and other algorithms in the same field.
  • the method for estimating direction of arrival based on sparse reconstruction in the case of amplitude and phase errors of the present invention includes the following steps:
  • S1 Calculate the covariance matrix through the array received signal, use the eigendecomposition method to estimate the noise power, estimate and compensate the amplitude error according to the noise power and the main diagonal data of the covariance matrix, and obtain the compensated covariance matrix; the S1 is implemented by the following sub-steps:
  • M represents the number of array elements
  • ⁇ m represents the eigenvalues arranged in descending order
  • ⁇ m represents the eigenvector corresponding to the eigenvalue ⁇ m
  • ( ⁇ ) H represents the conjugate transpose
  • K represents the number of sources
  • ⁇ m represents the magnitude error estimate value of the mth array element, r m, m represents the value at the covariance matrix (m, m);
  • IM represents the identity matrix of size M.
  • the DOA estimation problem is transformed into a non-convex optimization problem under a sparse framework by using a sparse reconstruction method; the S2 is implemented through the following sub-steps:
  • ⁇ k,m represents the delay of the k-th signal at the m-th array element relative to the reference array element
  • B is with The corresponding steering vector matrix formed by extending to ⁇ , p is the Correspondingly extended to the matrix formed on ⁇ ;
  • S3 The two-parameter non-convex optimization problem is transformed into a convex optimization problem by the method of alternating optimization, and the grid angle and the deviation angle are obtained by solving the convex optimization problem, and the two are summed to obtain the final source angle estimate; Said S3 is achieved through the following sub-steps:
  • w [w 1 , w 2 , ..., w M ] T
  • ⁇ 1 represents the regularization constant
  • a q b q H b q
  • b q represents the qth row of B
  • c [c 1 , c 2 ,..., c Q ] T
  • ⁇ 1 represents another regularization constant
  • z represents any matrix with the same specification as p
  • ⁇ 2 represents the regularization constant
  • D ⁇ B′ ⁇ p
  • D represents the intermediate conversion quantity
  • ⁇ represents the deviation angle matrix
  • E q d q H d q
  • d q represents the th q line
  • the index matrix ⁇ has the same dimension as the grid angle matrix ⁇ , the value of ⁇ is 1 at the index of the estimated angle, and the rest is 0, ( ) represents the point multiplication of the matrix, that is, the corresponding elements of the matrix are multiplied.
  • Figure 2 is a schematic diagram of the grid division of the array space domain, in which the diamonds represent the array elements, the hollow circles represent the grid points for dividing the space domain, the grid spacing is ⁇ , and the solid circles represent the actual direction of the signal.
  • the hollow circle and the solid circle are coincident, it means that the actual direction of the signal is just above the grid.
  • the grid division model will generate a certain deviation error ⁇ .
  • FIG. 3 is a comparison diagram of the relationship between the root mean square error and the phase error of the present invention and other algorithms in the same field to estimate the direction of arrival. It can be seen from FIG. 3 that with the increase of the initial phase error, the present invention performs the The root mean square error of the direction estimation does not change with it, and this method (the proposed curve in the figure) can effectively eliminate the influence of the phase error in the direction of arrival estimation.
  • Figure 4 is a comparison diagram of the relationship between the root mean square error and the signal-to-noise ratio of the direction of arrival estimation performed by the present invention and other algorithms in the same field. It can be seen from Figure 4 that with the increase of the signal-to-noise ratio, the direction of arrival estimated The root mean square error decreases with the increase, especially when the signal-to-noise ratio is greater than 15dB, the root mean square error of this method (the proposed curve in the figure) is smaller than that of other algorithms, indicating that this method can improve the direction of arrival. Estimated accuracy.

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  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
  • Measurement Of Velocity Or Position Using Acoustic Or Ultrasonic Waves (AREA)
  • Radar Systems Or Details Thereof (AREA)

Abstract

一种幅相误差情况下基于稀疏重构的波达方向估计方法，首先通过阵列接收信号，采用特征分解的方法来估计噪声功率和幅度误差(S1)；然后基于补偿后的协方差矩阵，使用稀疏重构的方法将波达方向估计问题转化为稀疏框架下的非凸优化问题(S2)；最后采用交替优化的方法来估计网格角度和偏离角度(S3)。波达方向估计方法能有效地消除相位误差在波达方向估计时的影响，具有更好的适应度，提高了算法的分辨率和估计的精度。

Description

一种幅相误差情况下基于稀疏重构的波达方向估计方法 技术领域
本发明涉及阵列信号处理领域，尤其涉及一种幅相误差(Gain-phase Error)情况下基于稀疏重构的波达方向(Direction-of-arrival，DOA)估计方法。
背景技术
信号的波达方向估计是阵列信号处理领域的一个重要研究内容，被广泛应用于雷达、声呐、无线通信等领域。关于信号的波达方向估计有着许多经典的高分辨率算法，其中包括多重信号分类(Multiple Signal Classification，MUSIC)算法和旋转不变子空间(Estimation of Signal Parameters via Rotational Invariance Techniques，ESPRIT)算法等。这些经典的高分辨率算法大多都以阵列流形精确已知为前提的，在实际工程应用过程中，由于气候、环境及阵元器件本身等因素的变化产生信号在信道中传输时放大器增益不一致，导致了阵列天线通道间的幅度和相位误差，这会造成实际阵列流形发生偏差，使得经典的高分辨率信号波达方向估计算法的性能急剧下降，严重时甚至失效。
早期的阵列误差校正主要是通过对阵列流形直接进行离散测量、内插、存储来实现的。而后人们通过对阵列扰动进行建模，将阵列误差校正逐渐转化为参数估计问题，大体可以分为有源校正和自校正。有源校正需要外置辅助源或其他辅助设施，在一定程度上增加了信号波达方向估计设备的成本，且对硬件和环境的要求严格，在很多情况下并不适用。自校正则是根据某种优化函数对信号波达方向和阵列误差参数进行估计，它无需额外的方位精确已知的辅助源，且可以实现在线估计。随着现代信息技术的飞速发展，信号环境正朝着低信噪比、小快拍数等条件转变，在这样的条件下，现有基于子空间的校正算法性能表现不尽如人意，这给需要大量接收数据的幅相误差自校正算法带来了极大的挑战。
近年来，稀疏重构技术和压缩感知理论的兴起和发展吸引了大量学者的研究，基于稀疏重构的波达方向估计和幅相误差校正方法为现代信号环境下的校正算法提供了新的思路，其对任意阵列形状的适应性更好，需求数据量更少。用稀疏的形式表示阵列数据模型，然后通过求解优化问题从而获取原始信号进而得到波达方向角，可以很大程度地提高估计算法的准确度，从而弥补传统算法的不足。在实际实验过程中，这类方法需要对整个空间域进行网格划分，而网络划分的粗细程度会直接影响算法计算复杂度和波达方向的估计精度。当信号方向未严格落在划分的网格上(Off-grid)时会引入偏离误差，从而导致估计精度随着真实信号 与网格间偏移量的增加而降低。
发明内容
针对现有技术的不足，本发明提供一种幅相误差情况下基于稀疏重构的波达方向估计方法，具体技术方案如下：
一种幅相误差情况下基于稀疏重构的波达方向估计方法，该方法包括以下步骤：
S1：通过阵列接收信号计算协方差矩阵，采用特征分解的方法来估计噪声功率，根据所述噪声功率和协方差矩阵的主对角线数据进行幅度误差的估计和补偿，得到补偿后的协方差矩阵；
S2：根据S1得到的补偿后的协方差矩阵，采用稀疏重构的方法将波达方向估计问题转化为稀疏框架下的非凸优化问题；
S3：采用交替优化的方法将双参数的非凸优化问题转化成凸优化问题，求解凸优化问题得到网格角度和偏离角度，得到最终的信源角度估计值。
进一步地，所述S1通过以下子步骤来实现：
S1.1：计算阵列接收信号X(t)的协方差矩阵R，而后采用下式对其进行特征值分解，从中获得降序排列的特征值λ m
Figure PCTCN2021109106-appb-000001
其中，M表示阵元个数，λ m表示降序排列的特征值，ν m表示为与特征值λ m相对应的特征向量，(·) H表示共轭转置；
S1.2：根据S1.1所得的特征值λ m，利用下式估计噪声功率
Figure PCTCN2021109106-appb-000002
Figure PCTCN2021109106-appb-000003
其中，K表示信源个数；
S1.3：根据所得协方差矩阵R和噪声功率估计值
Figure PCTCN2021109106-appb-000004
采用下式对幅度误差进行估计
Figure PCTCN2021109106-appb-000005
其中，ρ m表示第m个阵元的幅度误差估计值，r m，m表示协方差矩阵(m，m)处的值；
S1.4：利用下式将估计所得的幅度误差矩阵ρ m在协方差矩阵R中进行补偿，消去幅度误差的影响，得到补偿后的协方差矩阵R 1
Figure PCTCN2021109106-appb-000006
其中，G＝diag{[ρ 1，ρ 2，...，ρ M]}表示幅度误差估计矩阵,I M表示大小为M的单位矩阵。
进一步地，所述S2通过以下子步骤来实现：
S2.1：根据S1.4所得补偿后的协方差矩阵R 1，对其矩阵元素进行取模操作得|R 1|，取其上三角区域的元素，并消去主对角线中重复的相同大小的元素，然后按下式进行重新排列
Figure PCTCN2021109106-appb-000007
Figure PCTCN2021109106-appb-000008
Figure PCTCN2021109106-appb-000009
其中，
Figure PCTCN2021109106-appb-000010
是新定义的由角度θ k构成的导向矢量矩阵，
Figure PCTCN2021109106-appb-000011
是新定义的由K个信号的功率构成的矩阵，σ k 2表示第k个信号的功率，(·) T表示转置，b(θ k)表示对应于角度θ k的导向矢量，其取值如下式所示
Figure PCTCN2021109106-appb-000012
其中，τ k，m表示第k个信号在第m个阵元相对于参考阵元的延迟；
S2.2：设定空间网格间距Δ，构造超完备的角度集合Θ＝{-90°，-90°+Δ，...，90°-Δ}，从而将式(5)扩展到Θ上得到下式的超完备输出模型
x＝|Bp|           (9)
B＝[b(-90°)，b(-90°+Δ)，...，b(90°-Δ)]       (10)
Figure PCTCN2021109106-appb-000013
其中，B是与
Figure PCTCN2021109106-appb-000014
相对应的扩展到Θ上所构成的导向矢量矩阵，p是与
Figure PCTCN2021109106-appb-000015
相对应的扩展到Θ上所构成的矩阵；
S2.3：当实际信源方向
Figure PCTCN2021109106-appb-000016
未能严格落在所构造的网格上时存在偏离角度δ，采用一阶泰勒展开将导向矢量B(θ)修正为
Figure PCTCN2021109106-appb-000017
其中，
Figure PCTCN2021109106-appb-000018
为修正后的导向矢量；
S2.4：利用优化理论，将S2.3所得修正后的超完备输出模型转化为下式的非凸优化问题
min p，δ||x-|Bp+B′δp||| 2 2。       (13)
进一步地，所述S3通过以下子步骤来实现：
S3.1：初始化偏离角度矩阵δ＝O l，优化式(13)的问题，将其转化为下式问题
Figure PCTCN2021109106-appb-000019
其中，w＝[w 1，w 2，...，w M] T，γ 1表示规则化常数，A q＝b q Hb q，b q表示B的第q行；
S3.2：采用可行点追踪算法的思想将式(14)转化为下式的凸优化问题，并求解式(15)得稀疏矩阵p，求得稀疏矩阵p中非零项对应角度θ；
Figure PCTCN2021109106-appb-000020
其中，c＝[c 1，c 2，...，c Q] T，μ 1表示另一个规则化常数，z表示任意的和p同规格的矩阵；
S3.3：根据S3.2所得稀疏矩阵p来求解式(13)的问题，将其转化为下式问题
Figure PCTCN2021109106-appb-000021
其中，γ 2表示规则化常数，C＝Bp表示已知量，Dδ＝B′δp，D表示中间转换量，δ表示偏离角度矩阵，E q＝d q Hd q，d q表示D的第q行；
S3.4：采用可行点跟踪算法的思想将式(16)转化为下式的凸优化问题，求解公式(17)，得到偏离角度估计矩阵δ
Figure PCTCN2021109106-appb-000022
S3.5：获取S3.2所得的网格角度矩阵θ对应的索引矩阵β，将网格角度矩阵θ和S3.4所得的偏离角度矩阵δ加和的结果与索引矩阵β进行点乘，得最终的信源角度估计值为
Figure PCTCN2021109106-appb-000023
其中，索引矩阵β与网格角度矩阵θ维度相同，β在估计角度的索引处值为1，其余为0，(·)代表矩阵的点乘，即矩阵对应元素相乘。
本发明的有益效果如下：
本发明的基于稀疏重构的幅相误差校正及波达方向估计方法，通过直接取补偿协方差矩 阵各元素的模长，有效地消除了相位误差在波达方向估计时的影响，采用稀疏重构的技术，着眼于补偿信号未能严格落在划分的网格上时产生的偏离误差的情况，提升了波达方向估计的精度。
附图说明
图1是幅相误差情况下基于稀疏重构的波达方向估计方法流程图。
图2是阵列空间域网格划分示意图。
图3是本发明与同领域其他算法进行波达方向估计的均方根误差与相位误差的关系对比图。
图4是本发明与同领域其他算法进行波达方向估计的均方根误差与信噪比的关系对比图。
具体实施方式
下面根据附图和优选实施例详细描述本发明，本发明的目的和效果将变得更加明白，应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。
如图1所示，本发明的幅相误差情况下基于稀疏重构的波达方向估计方法，包括如下步骤：
S1：通过阵列接收信号计算协方差矩阵，采用特征分解的方法来估计噪声功率，根据所述噪声功率和协方差矩阵的主对角线数据进行幅度误差的估计和补偿，得到补偿后的协方差矩阵；所述S1通过以下子步骤来实现：
S1.1：计算阵列接收信号X(t)的协方差矩阵R，而后采用下式对其进行特征值分解，从中获得降序排列的特征值λ m
Figure PCTCN2021109106-appb-000024
其中，M表示阵元个数，λ m表示降序排列的特征值，ν m表示为与特征值λ m相对应的特征向量，(·) H表示共轭转置；
S1.2：根据S1.1所得的特征值λ m，利用下式估计噪声功率
Figure PCTCN2021109106-appb-000025
Figure PCTCN2021109106-appb-000026
其中，K表示信源个数；
S1.3：根据所得协方差矩阵R和噪声功率估计值
Figure PCTCN2021109106-appb-000027
采用下式对幅度误差进行估计
Figure PCTCN2021109106-appb-000028
其中，ρ m表示第m个阵元的幅度误差估计值，r m，m表示协方差矩阵(m，m)处的值；
S1.4：利用下式将估计所得的幅度误差矩阵ρ m在协方差矩阵R中进行补偿，消去幅度误差的影响，得到补偿后的协方差矩阵R 1
Figure PCTCN2021109106-appb-000029
其中，G＝diag{[ρ 1，ρ 2，...，ρ M]}表示幅度误差估计矩阵,I M表示大小为M的单位矩阵。
S2：根据S1得到的补偿后的协方差矩阵，采用稀疏重构的方法将波达方向估计问题转化为稀疏框架下的非凸优化问题；所述S2通过以下子步骤来实现：
S2.1：根据S1.4所得补偿后的协方差矩阵R 1，对其矩阵元素进行取模操作得|R 1|，取其上三角区域的元素，并消去主对角线中重复的相同大小的元素，然后按下式进行重新排列
Figure PCTCN2021109106-appb-000030
Figure PCTCN2021109106-appb-000031
Figure PCTCN2021109106-appb-000032
其中，
Figure PCTCN2021109106-appb-000033
是新定义的由角度θ k构成的导向矢量矩阵，
Figure PCTCN2021109106-appb-000034
是新定义的由K个信号的功率构成的矩阵，σ k 2表示第k个信号的功率，(·) T表示转置，b(θ k)表示对应于角度θ k的导向矢量，其取值如下式所示
Figure PCTCN2021109106-appb-000035
其中，τ k，m表示第k个信号在第m个阵元相对于参考阵元的延迟；
S2.2：设定波达方向角域空间范围网格间距Δ，由稀疏重构的方法构造超完备的角度集合Θ＝{-90°，-90°+Δ，...，90°-Δ}，从而将式(5)扩展到Θ上得到下式的超完备输出模型
x＝|Bp|           (9)
B＝[b(-90°)，b(-90°+Δ)，...，b(90°-Δ)]          (10)
Figure PCTCN2021109106-appb-000036
其中，B是与
Figure PCTCN2021109106-appb-000037
相对应的扩展到Θ上所构成的导向矢量矩阵，p是与
Figure PCTCN2021109106-appb-000038
相对应的扩展到Θ上所构成的矩阵；
S2.3：当实际信源方向
Figure PCTCN2021109106-appb-000039
未能严格落在所构造的网格上时存在偏离角度δ，采用一阶泰勒展开将导向矢量B(θ)修正为
Figure PCTCN2021109106-appb-000040
其中，
Figure PCTCN2021109106-appb-000041
为修正后的导向矢量；
S2.4：利用优化理论，将S2.3所得修正后的超完备输出模型转化为下式的非凸优化问题
min p，δ||x-|Bp+B′δp||| 2 2。         (13)
S3：采用交替优化的方法将双参数的非凸优化问题转化成凸优化问题，求解凸优化问题得到网格角度和偏离角度，并将两者求和，得到最终的信源角度估计值；所述S3通过以下子步骤来实现：
S3.1：初始化偏离角度矩阵δ＝O l，优化式(13)的问题，将其转化为下式问题
Figure PCTCN2021109106-appb-000042
其中，w＝[w 1，w 2，...，w M] T，γ 1表示规则化常数，A q＝b q Hb q，b q表示B的第q行；
S3.2：采用可行点追踪算法的思想将式(14)转化为下式的凸优化问题，并求解式(15)得稀疏矩阵p，求得稀疏矩阵p中非零项对应角度θ；
Figure PCTCN2021109106-appb-000043
其中，c＝[c 1，c 2，...，c Q] T，μ 1表示另一个规则化常数，z表示任意的和p同规格的矩阵；
S3.3：根据S3.2所得稀疏矩阵p来求解式(13)的问题，将其转化为下式问题
Figure PCTCN2021109106-appb-000044
其中，γ 2表示规则化常数，C＝Bp表示已知量，Dδ＝B′δp，D表示中间转换量，δ表示偏离角度矩阵，E q＝d q Hd q，d q表示D的第q行；
S3.4：采用可行点跟踪算法的思想将式(16)转化为下式的凸优化问题，求解公式(17)，得到偏离角度估计矩阵δ
Figure PCTCN2021109106-appb-000045
S3.5：获取S3.2所得的网格角度矩阵θ对应的索引矩阵β，将网格角度矩阵θ和S3.4所 得的偏离角度矩阵δ加和的结果与索引矩阵β进行点乘，得最终的信源角度估计值为
Figure PCTCN2021109106-appb-000046
其中，索引矩阵β与网格角度矩阵θ维度相同，β在估计角度的索引处值为1，其余为0，(·)代表矩阵的点乘，即矩阵对应元素相乘。
图2是阵列空间域网格划分示意图，其中菱形代表阵元，空心圆代表划分空间域的网格点，网格间距为Δ，实心圆代表信号的实际方向。当空心圆和实心圆重合时表示信号的实际方向正好落在网格之上，反之，网格划分模型将产生一定的偏离误差δ。
图3是本发明与同领域其他算法进行波达方向估计的均方根误差与相位误差的关系对比图，从图3中可以看出，随着初始相位误差的增大，本发明进行波达方向估计的均方根误差并不随之变化，本方法(图中的proposed曲线)能有效地消除相位误差在波达方向估计时的影响。
图4是本发明与同领域其他算法进行波达方向估计的均方根误差与信噪比的关系对比图，从图4中可以看出，随着信噪比的增加，波达方向估计的均方根误差均随着减小，特别是当信噪比大于15dB时，本方法(图中的proposed曲线)的均方根误差相比于其他算法更小，说明本方法能提升波达方向估计的精度。
本领域普通技术人员可以理解，以上所述仅为发明的优选实例而已，并不用于限制发明，尽管参照前述实例对发明进行了详细的说明，对于本领域的技术人员来说，其依然可以对前述各实例记载的技术方案进行修改，或者对其中部分技术特征进行等同替换。凡在发明的精神和原则之内，所做的修改、等同替换等均应包含在发明的保护范围之内。

Claims (3)

  1. 一种幅相误差情况下基于稀疏重构的波达方向估计方法，其特征在于，该方法包括以下步骤：
    S1：通过阵列接收信号X(t)计算协方差矩阵R，采用特征分解的方法来估计噪声功率，根据所述噪声功率和协方差矩阵的主对角线数据进行幅度误差的估计和补偿，得到补偿后的协方差矩阵R 1；
    S2：根据S1得到的补偿后的协方差矩阵R 1，采用稀疏重构的方法将波达方向估计问题转化为稀疏框架下的非凸优化问题；具体通过如下的子步骤来实现：
    S2.1：根据补偿后的协方差矩阵R 1，对其矩阵元素进行取模操作得|R 1|，取其上三角区域的元素，并消去主对角线中重复的相同大小的元素，然后按下式进行重新排列
    Figure PCTCN2021109106-appb-100001
    Figure PCTCN2021109106-appb-100002
    Figure PCTCN2021109106-appb-100003
    其中，
    Figure PCTCN2021109106-appb-100004
    是新定义的由角度θ k构成的导向矢量矩阵，
    Figure PCTCN2021109106-appb-100005
    是新定义的由K个信号的功率构成的矩阵，σ k 2表示第k个信号的功率，(·) T表示转置，b(θ k)表示对应于角度θ k的导向矢量，其取值如下式所示
    Figure PCTCN2021109106-appb-100006
    其中，τ k，m表示第k个信号在第m个阵元相对于参考阵元的延迟；
    S2.2：设定空间网格间距Δ，构造超完备的角度集合Θ＝{-90°，-90°+Δ，...，90°-Δ}，从而将式(1)扩展到Θ上得到下式的超完备输出模型
    x＝|Bp|    (5)
    B＝[b(-90°)，b(-90°+Δ)，...，b(90°-Δ)]    (6)
    Figure PCTCN2021109106-appb-100007
    其中，B是与
    Figure PCTCN2021109106-appb-100008
    相对应的扩展到Θ上所构成的导向矢量矩阵，p是与
    Figure PCTCN2021109106-appb-100009
    相对应的扩展到Θ上所构成的矩阵；
    S2.3：当实际信源方向
    Figure PCTCN2021109106-appb-100010
    未能严格落在所构造的网格上时存在偏离角度δ，采用一阶泰勒展开将导向矢量B(θ)修正为
    Figure PCTCN2021109106-appb-100011
    其中，
    Figure PCTCN2021109106-appb-100012
    为修正后的导向矢量；
    S2.4：利用优化理论，将S2.3所得修正后的超完备输出模型转化为下式的非凸优化问题
    Figure PCTCN2021109106-appb-100013
    S3：采用交替优化的方法将双参数的非凸优化问题转化成凸优化问题，求解凸优化问题得到网格角度和偏离角度，得到最终的信源角度估计值。
  2. 根据权利要求1所述的幅相误差校正及波达方向估计方法，其特征在于，所述S1通过以下子步骤来实现：
    S1.1：计算阵列接收信号X(t)的协方差矩阵R，而后采用下式对其进行特征值分解，从中获得降序排列的特征值λ m
    Figure PCTCN2021109106-appb-100014
    其中，M表示阵元个数，λ m表示降序排列的特征值，v m表示为与特征值λ m相对应的特征向量，(·) H表示共轭转置；
    S1.2：根据S1.1所得的特征值λ m，利用下式估计噪声功率
    Figure PCTCN2021109106-appb-100015
    Figure PCTCN2021109106-appb-100016
    其中，K表示信源个数；
    S1.3：根据所得协方差矩阵R和噪声功率估计值
    Figure PCTCN2021109106-appb-100017
    采用下式对幅度误差进行估计
    Figure PCTCN2021109106-appb-100018
    其中，ρ m表示第m个阵元的幅度误差估计值，r m，m表示协方差矩阵(m，m)处的值；
    S1.4：利用下式将估计所得的幅度误差矩阵ρ m在协方差矩阵R中进行补偿，消去幅度误差的影响，得到补偿后的协方差矩阵R 1
    Figure PCTCN2021109106-appb-100019
    其中，G＝diag{[ρ 1，ρ 2，...，ρ M]}表示幅度误差估计矩阵,I M表示大小为M的单位矩阵。
  3. 根据权利要求1所述的幅相误差校正及波达方向估计方法，其特征在于，所述S3通过以下子步骤来实现：
    S3.1：初始化偏离角度矩阵δ＝0 l，优化式(13)的问题，将其转化为下式问题
    Figure PCTCN2021109106-appb-100020
    其中，w＝[w 1，w 2，...，w M] T，γ 1表示规则化常数，A q＝b q Hb q，b q表示B的第q行；
    S3.2：采用可行点追踪算法的思想将式(14)转化为下式的凸优化问题，并求解式(15)得稀疏矩阵p，求得稀疏矩阵p中非零项对应角度θ；
    Figure PCTCN2021109106-appb-100021
    其中，c＝[c 1，c 2，...，c Q] T，μ 1表示另一个规则化常数，z表示任意的和p同规格的矩阵；
    S3.3：根据S3.2所得稀疏矩阵p来求解式(13)的问题，将其转化为下式问题
    Figure PCTCN2021109106-appb-100022
    其中，γ 2表示规则化常数，C＝Bp表示已知量，Dδ＝B′δp，D表示中间转换量，δ表示偏离角度矩阵，E q＝d q Hd q，d q表示D的第q行；
    S3.4：采用可行点跟踪算法的思想将式(16)转化为下式的凸优化问题，求解公式(17)，得到偏离角度估计矩阵δ
    Figure PCTCN2021109106-appb-100023
    S3.5：获取S3.2所得的网格角度矩阵θ对应的索引矩阵β，将网格角度矩阵θ和S3.4所得的偏离角度矩阵δ加和的结果与索引矩阵β进行点乘，得最终的信源角度估计值为
    Figure PCTCN2021109106-appb-100024
    其中，索引矩阵β与网格角度矩阵θ维度相同，β在估计角度的索引处值为1，其余为0，(·)代表矩阵的点乘，即矩阵对应元素相乘。
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