WO2022016796A1 - 一种线列阵阵列扩展方法 - Google Patents

一种线列阵阵列扩展方法 Download PDF

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WO2022016796A1
WO2022016796A1 PCT/CN2020/137877 CN2020137877W WO2022016796A1 WO 2022016796 A1 WO2022016796 A1 WO 2022016796A1 CN 2020137877 W CN2020137877 W CN 2020137877W WO 2022016796 A1 WO2022016796 A1 WO 2022016796A1
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arrays
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毛卫宁
钱进
陈建润
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Southeast University
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    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
    • H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
    • H04B7/0842—Weighted combining
    • H04B7/086—Weighted combining using weights depending on external parameters, e.g. direction of arrival [DOA], predetermined weights or beamforming

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  • the invention relates to an array signal processing method, in particular to a line array array expansion method.
  • Array signal processing technology is widely used in many military and civilian fields such as radar, communication, sonar, etc., and is a focal issue in these fields.
  • the signal-to-noise ratio is a key factor affecting the signal processing performance of an array.
  • the number of physical array elements and the array aperture are determined. How to improve the detection performance of weak targets under low signal-to-noise ratio is an urgent problem to be solved in engineering applications.
  • Using the array expansion technology it is an effective way to realize the expansion of the array aperture and improve the detection performance of weak targets under low signal-to-noise ratio by virtually increasing the number of array elements.
  • the commonly used array expansion techniques mainly include: based on high-order cumulant method, based on interpolation transformation method, reconstructed data method based on source and received signal characteristics, and line array based on broadband signal delay characteristics and signal non-circular symmetry Array extension methods, etc.
  • the high-order cumulant method has good and stable array expansion characteristics, and according to the property that the high-order cumulant of the Gaussian signal is zero, it can suppress the Gaussian noise in the system, so that it can be used in different Gaussian noise environments. Good estimation performance, but the method has a huge amount of calculation and there is a lot of redundant information. When the number of array elements increases, these redundant information may cause coupling between the array elements; the interpolation transformation method realizes the array by increasing the number of array elements.
  • the present invention provides a linear array array expansion method with good robustness and small computation load under the condition of low signal-to-noise ratio.
  • a line array array expansion method comprising the following steps:
  • the receiving array is a uniform linear array
  • the number of array elements is 2M
  • the K far-field signal sources are incident on the receiving array as plane waves
  • the array receiving signals are:
  • x(t) is the received signal vector of the 2M ⁇ 1-dimensional array:
  • x(t) [x 1 (t),x 2 (t),...,x 2M (t)] T
  • s(t) and n(t) are the K ⁇ 1-dimensional source signal vector and the 2M ⁇ 1-dimensional noise vector, respectively:
  • n(t) [n 1 (t),n 2 (t),...,n 2M (t)] T
  • A is a 2M ⁇ K-dimensional array manifold matrix:
  • A [a( ⁇ 1 ),a( ⁇ 2 ),...,a( ⁇ K )]
  • ⁇ is the signal wavelength
  • d is the array element spacing
  • ⁇ k is the azimuth of the kth signal source
  • k 1,2,...,K.
  • the linear array is divided into two sub-arrays according to odd and even array elements, and the received signals of the two sub-arrays are respectively:
  • x o (t) [x 1 (t),x 3 (t),...,x 2M-1 (t)] T
  • x i (t) is the received signal of the i-th array element
  • x o (t) is the received signal of the sub-array composed of odd-numbered array elements
  • x e (t) is the sub-array composed of even-numbered array elements. receive signal.
  • the received signal covariance matrix of the two sub-arrays formed by odd and even array elements is:
  • a o and A e are the array manifold matrix of two sub-arrays composed of odd and even array elements, respectively, obtained from the array manifold matrix A, and R ss is the source signal covariance matrix.
  • the constructed extended receiving array signal is:
  • the signal covariance matrix of the extended receiving array is:
  • step (6) utilize conventional beamforming CBF or minimum variance undistorted response MVDR beamforming method to carry out beamforming and target detection to the extended array, and the beam output power of CBF and MVDR beamforming is:
  • w [(a o ) T , (a o b -1 ) T , (a o b) T , (a o ) T ] T
  • b e j2 ⁇ 2dcos ⁇ / ⁇
  • a o is an odd-numbered array element
  • the steering vector of the formed sub-array, a o [1 e j2 ⁇ 2dcos ⁇ / ⁇ ... e j2 ⁇ (2M-2)dcos ⁇ / ⁇ ] T
  • ⁇ is the signal wavelength
  • d is the array element spacing
  • ⁇ is the beam scanning azimuth in space.
  • the present invention reconstructs the array data by utilizing the irrelevance of the noise of each array element and the rotational invariance of the line array to realize the expansion of the one-dimensional line array. Improving the detection performance of weak targets through array expansion can greatly reduce the hardware cost of the detection system. Compared with the prior art, the present invention further reduces the influence of noise while expanding the array, reduces the beam output side lobes, is beneficial to the detection of weak targets under low signal-to-noise ratio, and improves the robustness; The method is simple, the calculation amount is small, the non-circular symmetry of the source signal is not required, and the practicability is improved.
  • FIG. 1 is a flowchart of a method for expanding a linear array array provided by an embodiment of the present invention
  • FIG. 2 is a schematic diagram of a variation curve of the output power of an extended array conventional beamforming beam with azimuth obtained according to an embodiment of the present invention
  • FIG. 3 is a schematic diagram of a curve diagram of the variation curve of the output power of the expanded array minimum variance undistorted response beam-forming beam with azimuth obtained according to an embodiment of the present invention
  • FIG. 4 is a schematic diagram of a curve of the variation of the detection probability of the expanded array with the signal-to-noise ratio obtained according to an embodiment of the present invention.
  • an array expansion method for a one-dimensional line array includes the following steps:
  • Step (1) obtain the signal received by the line array:
  • the given target source signal is band-limited noise, the frequency band is 2000Hz-2400Hz, the sampling frequency is 16000Hz, the uniform line array with the number of array elements is 6, the speed of sound is 1500m/s, and the distance between the array elements is half of the wavelength of the center frequency.
  • the number of beats is 1024, the number of sources is 1, the target azimuth is 60°, and the signal-to-noise ratio is -10dB.
  • the array received signal is:
  • x(t) is the received signal vector of the 2M ⁇ 1-dimensional array:
  • s(t) and n(t) are the K ⁇ 1-dimensional signal vector (sent from the signal source) and the 2M ⁇ 1-dimensional noise vector, respectively:
  • n(t) [n 1 (t),n 2 (t),...,n 2M (t)] T (4)
  • the noise is Gaussian white noise with zero mean and variance ⁇ 2
  • A is a 2M ⁇ K-dimensional array manifold matrix:
  • ⁇ is the signal wavelength
  • d is the array element spacing
  • j is the complex factor
  • Step (2) divide the linear array into two sub-arrays according to odd and even array elements:
  • the received signals of the two sub-arrays composed of odd and even array elements are:
  • x i (t) is the received signal of the i-th array element
  • x o (t) is the received signal of the sub-array composed of odd-numbered array elements
  • x e (t) is the sub-array composed of even-numbered array elements. receive signal.
  • Step (3) calculate the received signal covariance matrix of the two subarrays formed by odd and even array elements:
  • a o and A e are the array manifold matrix of two sub-arrays composed of odd and even array elements respectively, which can be obtained from the array manifold matrix A.
  • A is a 2M ⁇ K-dimensional matrix, and the odd numbers in A are extracted.
  • a o is obtained by row ⁇ K dimension
  • a e is obtained by taking the even row ⁇ K dimension in A
  • R ss is the source signal covariance matrix.
  • the superscript T means transpose
  • the superscript H means conjugation. Since the noise of each array element is uncorrelated, the received signal covariance matrix R xx of the sub-array composed of odd and even array elements removes the influence of noise.
  • Step (4) construct the extended receiving array signal:
  • Step (5) calculate the signal covariance matrix of the extended receiving array:
  • Step (6) calculate the beam output power of conventional beamforming (Conventional Beamforming, CBF) and minimum variance distortion-free response beamforming (Minimum Variance Distortionless Response, MVDR):
  • w [(a o ) T , (a o b -1 ) T , (a o b) T , (a o ) T ] T
  • b e j2 ⁇ 2dcos ⁇ / ⁇
  • a o is composed of odd array elements
  • the steering vector of the sub-array, a o [1 e j2 ⁇ 2dcos ⁇ / ⁇ ... e j2 ⁇ (2M-2)dcos ⁇ / ⁇ ] T
  • ⁇ is the signal wavelength
  • d is the array element spacing
  • ⁇ is the beam scanning azimuth in space.
  • Fig. 2 shows the change of the output power of the conventional beamforming beam with the azimuth of the extended line array and the unextended line array
  • Fig. 3 is the change of the output power of the MVDR beamforming beam of the extended line array and the unextended line array with the azimuth
  • Figure 4 shows the change of the detection probability of the extended line array and the unextended line array with the signal-to-noise ratio. 100 times of Monte Carlo experiments are obtained. It can be seen from Figure 4 that the detection probability of the extended line array under low signal-to-noise ratio is significantly higher than that of the unextended line array, which is conducive to the detection of weak targets.

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Abstract

一种线性列阵阵列扩展方法，属于列阵波束形成技术领域，包括：（1）获取线列阵接收信号；（2）按奇、偶阵元将线列阵划分为两个子阵，分别得到两个子阵的接收信号；（3）计算奇、偶阵元构成的两个子阵的接收信号协方差矩阵；（4）根据两个子阵接收信号的协方差矩阵以及线阵列的旋转不变性构造扩展接收阵列信号；（5）计算扩展接收阵列的信号协方差矩阵；（6）根据扩展接收阵列信号Y的协方差矩阵，利用波束形成方法对扩展阵列进行波束形成和目标检测。解决了现有线阵列扩展方法低信噪比下性能下降，鲁棒性差；要求信源信号具有非圆对称性，应用有局限性；阵列扩展方法复杂，计算量大等问题。

Description

一种线列阵阵列扩展方法 技术领域
本发明涉及阵列信号处理方法，具体涉及一种线列阵阵列扩展方法。
背景技术
阵列信号处理技术广泛应用于雷达、通信、声呐等众多军用和民用领域，是这些领域中的一个焦点问题。信噪比是影响阵列信号处理性能的一个关键因素。对于给定的阵列配置，物理阵元数和阵列孔径是确定的，如何提高低信噪比下弱目标的探测性能是工程应用中亟待解决的问题。利用阵列扩展技术，通过虚拟增加阵元数目，实现阵列孔径扩展，提高低信噪比下弱目标的探测性能不失为一种有效途径。目前常用的阵列扩展技术主要有：基于高阶累积量法、基于内插变换法、基于信源和接收信号特性的重构数据法和基于宽带信号时延特性和信号非圆对称性的线列阵扩展方法等。
高阶累积量法有良好稳定的阵列扩展特性，且根据高斯信号的高阶累积量为零的性质，能够对系统中的高斯噪声起到抑制作用，使得其能够在不同的高斯噪声环境下具有良好的估计性能，但该方法计算量庞大，存在很多的冗余信息，当阵元数增加时，这些冗余信息可能使阵元之间发生耦合；内插变换法通过增加阵元数实现阵列扩展，但子区域的长度和内插变换角度的步长难以确定，如何兼顾算法计算量和计算精度是内插变换法的一大难点；基于信源和接收信号特性重构数据法的提出旨在解决内插变换法中存在的“角度敏感”、“插值步长敏感”以及运算量大等问题，但主要应用于L型阵列等特殊的二维阵列，如何将该方法运用到一维阵列模型有待研究；目前常用的一维线列阵扩展方法有基于线列阵时延特性的扩展方法和基于接收信号非圆特性的线列阵扩展方法，前者可以在阵列扩展的同时，实现相干信源的解相干，但在低信噪比条件下鲁棒性较差，后者要求入射信号具有非圆对称性，应用中有局限性。
因此，有必要提出一种新的线列阵的阵列扩展方法。
发明内容
发明目的：为克服上述现有技术的不足，本发明提供一种低信噪比条件下鲁棒性好、运算量小的线列阵阵列扩展方法。
技术方案：为实现上述目的，本发明采用如下技术方案：
一种线列阵阵列扩展方法，包括以下步骤：
(1)获取线列阵的接收信号x(t)；
(2)将线列阵按奇、偶阵元分成两个子阵，分别得到两个子阵的接收信号x o(t)和x e(t)；
(3)计算奇、偶阵元构成的两个子阵的接收信号协方差矩阵R xx；
(4)根据两个子阵接收信号的协方差矩阵R xx以及线列阵的旋转不变性构造扩展接收阵列信号Y；
(5)计算扩展接收阵列信号Y的协方差矩阵R YY；
(6)根据扩展接收阵列信号Y的协方差矩阵R YY，利用波束形成方法对扩展阵列进行波束形成和目标检测。
进一步的，所述步骤(1)中接收阵为均匀线列阵，阵元数为2M，K个远场信号源以平面波入射到接收阵，阵列接收信号为：
x(t)＝As(t)+n(t)
其中，x(t)为2M×1维阵列接收信号向量：
x(t)＝[x 1(t),x 2(t),…,x 2M(t)] T
s(t)和n(t)分别为K×1维源信号向量和2M×1维噪声向量：
s(t)＝[s 1(t),s 2(t),…,s K(t)] T
n(t)＝[n 1(t),n 2(t),…,n 2M(t)] T
A为2M×K维的阵列流型矩阵：
A＝[a(θ 1),a(θ 2),…,a(θ K)]
其中，
Figure PCTCN2020137877-appb-000001
λ为信号波长，d为阵元间距，θ k为第k个信号源的方位，k＝1,2,…,K。
进一步的，所述步骤(2)中，线列阵按奇、偶阵元分成两个子阵，两个子阵的接收信号分别为：
x o(t)＝[x 1(t),x 3(t),…,x 2M-1(t)] T
x e(t)＝[x 2(t),x 4(t),…,x 2M(t)] T
其中，x i(t)为第i个阵元的接收信号，x o(t)为奇数号阵元构成的子阵的接收信号，x e(t)为偶数号阵元构成的子阵的接收信号。
进一步的，所述步骤(3)中，奇、偶阵元构成的两个子阵的接收信号协方差矩阵为：
R xx＝E{x o(t)·x e Η(t)}＝A oR ssA e Η
其中，A o和A e分别为奇、偶阵元构成的两个子阵的阵列流型矩阵，由阵列流型矩阵A得 到，R ss为源信号协方差矩阵。
进一步的，所述步骤(4)中，构造的扩展接收阵列信号为：
Figure PCTCN2020137877-appb-000002
其中，Y 1＝R xx(:,1:M-1)，Y 2＝R xx(:,2:M)，R xx(:,1:M-1)和R xx(:,2:M)分别为协方差矩阵R xx的前M-1列和后M-1列构成的矩阵，矩阵Y 3为矩阵Y 2的共轭和行、列倒置，矩阵Y 4为矩阵Y 1的共轭和行、列倒置。
进一步的，所述步骤(5)中，扩展接收阵列的信号协方差矩阵为：
R YY＝E(YY Η)
更进一步的，所述步骤(6)中，利用常规波束形成CBF或最小方差无畸变响应MVDR波束形成方法对扩展阵列进行波束形成和目标检测，CBF和MVDR波束形成的波束输出功率为：
P CBF＝w ΗR YYw
Figure PCTCN2020137877-appb-000003
其中，w＝[(a o) T,(a ob -1) T,(a ob) T,(a o) T] T，b＝e j2π2dcosθ/λ，a o为奇数号阵元所构成的子阵的导向向量，a o＝[1 e j2π2dcosθ/λ … e j2π(2M-2)dcosθ/λ] T，λ为信号波长，d为阵元间距，θ为空间的波束扫描方位。
有益效果：本发明利用各阵元噪声的不相关性，以及线列阵的旋转不变性重构阵列数据，实现了一维线列阵的扩展。通过阵列扩展提高弱目标的探测性能，可以大大降低探测系统的硬件成本。与现有技术相比，本发明在阵列扩展的同时，进一步减小了噪声的影响，降低了波束输出旁瓣，有利于低信噪比下弱目标的检测，提高了鲁棒性；阵列扩展方法简单，计算量小，不要求信源信号具有非圆对称性，提高了实用性。
附图说明
图1是本发明实施例提供的线列阵阵列扩展方法流程图；
图2是根据本发明实施例得到的扩展阵常规波束形成波束输出功率随方位变化曲线示意图；
图3是根据本发明实施例得到的扩展阵最小方差无畸变响应波束形成波束输出功率随方位变化曲线示意图；
图4是根据本发明实施例得到的扩展阵检测概率随信噪比变化的曲线示意图。
具体实施方式
下面结合附图对本发明的技术方案作进一步说明。
参照图1，在一实施例中，针对一维线列阵的阵列扩展方法包括以下步骤：
步骤(1)，获取线列阵接收信号：
给定目标源信号为带限噪声，频带为2000Hz-2400Hz，采样频率16000Hz，采用阵元数为6的均匀线列阵，声速1500m/s，阵元间距为中心频率波长的一半，单次快拍数为1024，信源数为1，目标方位为60°，信噪比为-10dB。
阵列接收信号为：
x(t)＝As(t)+n(t)            (1)
其中，x(t)为2M×1维阵列接收信号向量：
x(t)＝[x 1(t),x 2(t),…,x 2M(t)] T        (2)
s(t)和n(t)分别为K×1维信号向量(信号源发出)和2M×1维噪声向量：
s(t)＝[s 1(t),s 2(t),…,s K(t)] T            (3)
n(t)＝[n 1(t),n 2(t),…,n 2M(t)] T           (4)
噪声是均值为零，方差为σ 2的高斯白噪声，A为2M×K维的阵列流型矩阵：
A＝[a(θ 1),a(θ 2),…,a(θ K)]         (5)
其中，
Figure PCTCN2020137877-appb-000004
λ为信号波长，d为阵元间距，θ k为第k个信号源的方位，k＝1,2,…,K；j表示复数因子；
步骤(2)，按奇、偶阵元将线列阵划分为两个子阵：
奇、偶阵元构成的两个子阵的接收信号为：
x o(t)＝[x 1(t),x 3(t),…,x 2M-1(t)] T     (6)
x e(t)＝[x 2(t),x 4(t),…,x 2M(t)] T       (7)
其中，x i(t)为第i个阵元的接收信号，x o(t)为奇数号阵元构成的子阵的接收信号，x e(t)为偶数号阵元构成的子阵的接收信号。
步骤(3)，计算奇、偶阵元构成的两个子阵的接收信号协方差矩阵：
R xx＝E{x o(t)·x e Η(t)}＝A oR ssA e Η      (8)
其中，A o和A e分别为奇、偶阵元构成的两个子阵的阵列流型矩阵，可由阵列流型矩阵A得到，如上所述，A是2M×K维矩阵，提取A中的奇数行×K维得到A o，将A中的偶数行×K维得到A e，R ss为源信号协方差矩阵。上标T表示转置，上标H表示共轭。由于各阵元的噪声不相关，奇、偶阵元构成的子阵的接收信号协方差矩阵R xx去除了噪声的影响。
步骤(4)，构造扩展接收阵列信号：
Figure PCTCN2020137877-appb-000005
其中，Y 1＝R xx(:,1:M-1)，Y 2＝R xx(:,2:M)，R xx(:,1:M-1)和R xx(:,2:M)分别为协方差矩阵R xx的前M-1列和后M-1列构成的矩阵，前M-1列指的是1到M-1列，后M-1列指的是2到M列，矩阵Y 3、Y 4由线列阵的旋转不变性得到，分别为矩阵Y 2、Y 1的共轭和行、列倒置；具体地，Y 3是由Y 2先共轭再行列倒置得到，Y 4是由Y 1先共轭再行列倒置得到。
步骤(5)，计算扩展接收阵列的信号协方差矩阵：
R YY＝E(YY Η)       (10)
步骤(6)，计算常规波束形成(Conventional Beamforming,CBF)和最小方差无畸变响应波束形成(Minimum Variance Distortionless Response,MVDR)的波束输出功率：
P CBF＝w ΗR YYw        (11)
Figure PCTCN2020137877-appb-000006
其中，w＝[(a o) T,(a ob -1) T,(a ob) T,(a o) T] T，b＝e j2π2dcosθ/λ，a o为奇阵元所构成的子阵的导向向量，a o＝[1 e j2π2dcosθ/λ … e j2π(2M-2)dcosθ/λ] T，λ为信号波长，d为阵元间距，θ为空间的波束扫描方位。
图2示出了扩展线列阵与未扩展线列阵常规波束形成波束输出功率随方位的变化，图3为扩展线列阵与未扩展线列阵MVDR波束形成波束输出功率随方位的变化，由图2和图3可见，扩展线列阵具有更低的旁瓣，表明对噪声的抑制能力增强；图4为扩展线列阵与未扩展线列阵检测概率随信噪比的变化，由100次蒙特卡洛实验得到，由图4可见，低信噪比下扩展线列阵的检测概率明显高于未扩展线列阵，有利于弱目标的检测。

Claims (7)

  1. 一种线列阵阵列扩展方法，其特征在于，包括以下步骤：
    (1)获取线列阵的接收信号x(t)；
    (2)将线列阵按奇、偶阵元分成两个子阵，分别得到两个子阵的接收信号x o(t)和x e(t)；
    (3)计算奇、偶阵元构成的两个子阵的接收信号协方差矩阵R xx；
    (4)根据两个子阵接收信号的协方差矩阵R xx以及线列阵的旋转不变性构造扩展接收阵列信号Y；
    (5)计算扩展接收阵列信号Y的协方差矩阵R YY；
    (6)根据扩展接收阵列信号Y的协方差矩阵R YY，利用波束形成方法对扩展阵列进行波束形成和目标检测。
  2. 根据权利要求1所述的线列阵阵列扩展方法，其特征在于，所述步骤(1)中线列阵的接收信号为：
    x(t)＝As(t)+n(t)
    其中，x(t)为2M×1维阵列接收信号向量：
    x(t)＝[x 1(t),x 2(t),…,x 2M(t)] Τ
    s(t)和n(t)分别为K×1维源信号向量和2M×1维噪声向量：
    s(t)＝[s 1(t),s 2(t),…,s K(t)] Τ
    n(t)＝[n 1(t),n 2(t),…,n 2M(t)] Τ
    A为2M×K维的阵列流型矩阵：
    A＝[a(θ 1),a(θ 2),…,a(θ K)]
    其中，
    Figure PCTCN2020137877-appb-100001
    λ为信号波长，d为阵元间距，θ k为第k个信号源的方位，k＝1,2,…,K，K为信号源个数，2M为线列阵的阵元数。
  3. 根据权利要求1所述的线列阵阵列扩展方法，其特征在于，所述步骤(2)中，线列阵按奇、偶阵元分成两个子阵，两个子阵的接收信号分别为：
    x o(t)＝[x 1(t),x 3(t),…,x 2M-1(t)] Τ
    x e(t)＝[x 2(t),x 4(t),…,x 2M(t)] Τ
    其中x i(t)为第i个阵元的接收信号，x o(t)为奇数号阵元构成的子阵的接收信号，x e(t)为偶数号阵元构成的子阵的接收信号，2M为线列阵的阵元数目。
  4. 根据权利要求1所述的线列阵阵列扩展方法，其特征在于，所述步骤(3)中，奇、偶阵元构成的两个子阵接收信号协方差矩阵为：
    R xx＝E{x o(t)·x e Η(t)}＝A oR ssA e Η
    其中，A o和A e分别为奇、偶阵元构成的两个子阵的阵列流型矩阵，R ss为源信号协方差矩阵。
  5. 根据权利要求1所述的线列阵阵列扩展方法，其特征在于，所述步骤(4)中，构造的扩展接收阵列信号Y为：
    Figure PCTCN2020137877-appb-100002
    其中，Y 1＝R xx(:,1:M-1)，Y 2＝R xx(:,2:M)，R xx(:,1:M-1)和R xx(:,2:M)分别为协方差矩阵R xx的前M-1列和后M-1列构成的矩阵，矩阵Y 3为矩阵Y 2的共轭和行、列倒置，矩阵Y 4为矩阵Y 1的共轭和行、列倒置。
  6. 根据权利要求5所述的线列阵阵列扩展方法，其特征在于，所述步骤(5)中，扩展接收阵列的信号协方差矩阵为：R YY＝E(YY Η)。
  7. 根据权利要求1所述的线列阵阵列扩展方法，其特征在于，所述步骤(6)中，利用常规波束形成CBF或最小方差无畸变响应MVDR波束形成方法对扩展阵列进行波束形成和目标检测，CBF和MVDR波束形成的波束输出功率分别为：
    P CBF＝w ΗR YYw
    Figure PCTCN2020137877-appb-100003
    其中，w＝[(a o) Τ,(a ob -1) Τ,(a ob) Τ,(a o) Τ] Τ，b＝e j2π2dcosθ/λ，a o为奇数号阵元所构成的子阵的导向向量，a o＝[1 e j2π2dcosθ/λ … e j2π(2M-2)dcosθ/λ] T，λ为信号波长，d为阵元间距，θ为空间的波束扫描方位。
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