CN110176953B - A Planar Frequency Controlled Array Beamforming Method Based on Generalized Eigenvalue Decomposition - Google Patents

Abstract

The present invention provides a plane frequency-controlled array beam forming method based on generalized eigenvalue decomposition, and the realization method is as follows: The distance between the array elements constructs a plane frequency-controlled array; constructs its corresponding far-field transmit beam pattern according to the plane frequency-controlled array; uses the energy gathering algorithm BCE to solve the eigenvalues and eigenvectors of the far-field transmit beam pattern respectively , the optimal weight matrix of the far-field transmit beam pattern is obtained, thereby forming the beam of the planar frequency-controlled array. The invention utilizes generalized eigenvalue decomposition to solve the optimal transmission weight, can form a focused spot beam in a desired area, and solves the problem that a linear frequency-controlled array cannot form a good spot beam.

Description

A Planar Frequency Controlled Array Beamforming Method Based on Generalized Eigenvalue Decomposition

technical field

The invention belongs to the technical field of frequency-controlled array radar, and in particular relates to a plane frequency-controlled array beam forming method based on generalized eigenvalue decomposition.

Background technique

Due to its unique range-angle dependence, frequency-controlled array radar has a very wide range of applications in radar systems, wireless communications, radar imaging, target estimation and tracking, jamming and anti-jamming. Although the uniform linear frequency-controlled array can form a transmission beam with angle-distance dependence, the distance and azimuth angle coupling of its beam pattern are S-shaped peaks and ridges in the distance-angle two-dimensional plane, which is not an ideal single-peak beam. That is spot beam. In order to form a spot beam, decoupling of the angle dimension and the distance dimension is required, and the linear frequency-controlled array cannot generate spot beams in the true sense. , but at the cost of a loss of beam resolution. At present, the beamforming method for planar frequency-controlled array can use ant colony algorithm to optimize the frequency offset, but the calculation process of this method is relatively complicated.

SUMMARY OF THE INVENTION

In view of the above deficiencies in the prior art, the present invention proposes a plane frequency-controlled array beamforming method based on generalized eigenvalue decomposition, which utilizes generalized eigenvalue decomposition to solve the optimal transmission weight, and can form a focused spot beam in a desired area, solving the problem of This solves the problem that linear frequency-controlled arrays cannot form good spot beams.

In order to achieve the above purpose, the technical scheme adopted in the present invention is:

This solution provides a planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition, including the following steps:

S1. Construct a planar frequency-controlled array according to the number of array elements of the antenna, the carrier frequency, the frequency offset value between adjacent antenna array elements and the spacing between adjacent antenna array elements;

S2, constructing its corresponding far-field transmit beam pattern according to the planar frequency-controlled array;

S3. Use the energy aggregation algorithm BCE to solve the eigenvalues and eigenvectors of the far-field transmit beam pattern respectively to obtain an optimal weight matrix of the far-field transmit beam pattern, thereby forming a plane frequency-controlled array beam.

Still further, the antenna in the step S1 is an omnidirectional antenna.

Still further, the planar frequency-controlled array is composed of M×N antenna array elements, and the carrier frequency f ₀ of the first antenna is 10 GHz, wherein,

The x direction of the antenna array element is distributed with M antenna array elements at equal intervals d _x , and the frequency offset value Δf _x between adjacent antenna array elements is linearly increased in sequence;

In the y direction of the antenna array elements, N antenna array elements are distributed at equal intervals _{dy, and the frequency offset value Δf y between} adjacent antenna array elements increases linearly in sequence.

Still further, the frequency offset values of the adjacent antennas in the x-direction and the y-direction are both 30KHz.

Still further, the distances between adjacent antennas in the x-direction and the y-direction are both 0.03m.

的表达式如下：Still further, the far-field transmit beam pattern

The expression is as follows:

and

表示目标点的方位角。Among them, exp{·} represents the exponential function with the natural constant e as the base, j represents the complex unit, f ₀ represents the center frequency of the transmitted signal, t represents the time, r represents the distance from the reference array element to the target point, c represents the speed of light, W represents the transmit weight matrix of the antenna elements, M represents the number of antenna elements in the x direction, N represents the number of antenna elements in the y direction, m represents the serial number of the antenna elements along the x direction, m=0,1,2 ,...M, n denotes the array element number along the y direction, n=0,1,2,...N, W ^* denotes the conjugate of the emission matrix W, ⊙ denotes the Hadamard product, A denotes the array factor, w _{M-1, N-1} represents the transmission matrix element, a _{M-1, N-1} represents the array factor element, a _mn represents the array factor of the mn-th array element, Δf _mn represents the frequency offset value of the mn-th array element , d _mn represents the array element spacing of the mnth array element, θ represents the inversion angle of the target point,

Indicates the azimuth of the target point.

Still further, the expression of the optimal weight matrix vec(W ^opt ) of the far-field transmit beam pattern in the step S3 is as follows:

表示求解最大权向量，vec(W)^H表示发射权重的矩阵向量化的共轭转置，vec(W)表示发射权重矩阵向量化，A表示在期望辐射范围的功率，B表示总辐射范围内的功率。in,

Represents solving the maximum weight vector, vec(W) ^H represents the conjugate transpose of the matrix vectorization of the emission weight, vec(W) represents the vectorization of the transmission weight matrix, A represents the power in the desired radiation range, and B represents the total radiation range. of power.

Beneficial effects of the present invention:

The present invention constructs a plane frequency controlled array according to the number of antenna elements, the carrier frequency, the frequency offset value between adjacent antenna elements and the spacing between adjacent antenna elements; Field emission beam pattern; use the energy aggregation algorithm BCE to solve the eigenvalues and eigenvectors of the far-field emission beam pattern respectively, and obtain the optimal weight matrix of the far-field emission beam pattern, thereby forming the beam of the planar frequency-controlled array. Through the above steps, the invention uses generalized eigenvalue decomposition to solve the optimal transmission weight, which can form a focused spot beam in a desired area, and solves the problem that a linear frequency-controlled array cannot form a good spot beam.

Description of drawings

FIG. 1 is a flow chart of the method of the present invention.

FIG. 2 is a schematic diagram of a planar frequency controlled array model in this embodiment.

FIG. 3 is a diagram of a three-dimensional space emission model of a planar frequency-controlled array in this embodiment.

FIG. 4 does not use the normalized three-dimensional beam pattern of the planar frequency-controlled array corresponding to the optimized weight matrix in this embodiment.

FIG. 5 is a normalized two-dimensional planar beam diagram corresponding to a planar frequency-controlled array without using an optimized weight matrix in this embodiment.

FIG. 6 is a plane frequency-controlled array spot beam diagram of the target area a in this embodiment.

FIG. 7 is a plane frequency-controlled array spot beam diagram of the target area b in this embodiment.

Detailed ways

The specific embodiments of the present invention are described below to facilitate those skilled in the art to understand the present invention, but it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, as long as various changes Such changes are obvious within the spirit and scope of the present invention as defined and determined by the appended claims, and all inventions and creations utilizing the inventive concept are within the scope of protection.

Example

As shown in FIG. 1 , the present invention discloses a planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition, and its implementation method is as follows:

其中，

是目标点的方位角，取值范围为[0,2π)；θ是目标点的俯仰角，取值范围为[0,π)；r为观测点到参考点的距离，取值范围为[R_min,∞)，其中，R_min是使观测点满足远场假设的最小距离，本发明实例中，假定的平面频控阵模型中的天线均采用全向天线，在不考虑能量随距离衰减的情况下，设置全向天线阵元个数为M＝N＝10，第一个全向天线(参考阵元)的载波频率为f₀＝10GHz，相邻全向天线的频偏值为Δf_x＝Δf_y＝30KHz，相邻全向天线的间隔为d_x＝d_y＝0.03m，俯仰角θ＝90°，即阵列所在平面；S1. Build a planar frequency-controlled array according to the number of antenna elements, the carrier frequency of the antenna elements, the frequency offset between adjacent antenna elements, and the spacing between adjacent antenna elements. Specifically, as shown in Figure 2-Figure As shown in 3, a planar frequency-controlled array is composed of M×N array elements. The array elements in the x direction are distributed with M array elements at equal intervals of d _x , and the frequency offset of each array element increases linearly according to Δf _x . N array elements are distributed at equal intervals in d _y , and the frequency offset of each array element increases linearly by Δf _y in turn. The plane frequency-controlled array transmits signals. The spherical coordinates of the observation point at the far field are

in,

is the azimuth angle of the target point, the value range is [0, 2π); θ is the pitch angle of the target point, the value range is [0, π); r is the distance from the observation point to the reference point, the value range is [ R _min ,∞), where R _min is the minimum distance for the observation point to satisfy the assumption of the far field. In the example of the present invention, the antennas in the assumed planar frequency controlled array model are omnidirectional antennas, and the attenuation of energy with distance is not considered. In the case of , set the number of omnidirectional antenna array elements to M=N=10, the carrier frequency of the first omnidirectional antenna (reference array element) is f ₀ =10GHz, and the frequency offset value of adjacent omnidirectional antennas is Δf _x = Δf _y = 30KHz, the interval between adjacent omnidirectional antennas is d _x = _dy = 0.03m, and the elevation angle θ = 90°, that is, the plane where the array is located;

S2. Construct its corresponding far-field transmit beam pattern according to the planar frequency-controlled array. In a specific embodiment, it is assumed that the array element located at the origin of the coordinate system is the first omnidirectional antenna (reference array element), denoted as e ₀₀ , then the array elements at other positions can be recorded as e _mn , where m represents the serial number of the antenna array element along the x direction, m=0, 1, 2,...M, n represents the array element along the y direction Serial number, n=0, 1, 2,...N, the single-frequency signal s _mn (t) transmitted by the array element e _mn , as shown in formula (1):

s _mn (t)=exp(j2πf _mn t) (1)

处接收到的信号

如式(2)所示：far-field observation point

signal received at

As shown in formula (2):

表示第mn个阵元的权重共轭，r_mn表示第mn个阵元的倾斜距离，W＝{w_mn}_M×N为天线阵元的发射权重矩阵，

为阵元e_mn的天线方向图,假定阵列天线的一致性较好，且每个阵元天线是各向同性点源，且频域响应在载频f₀附近平坦，则有

exp{·}表示以自然常数e为底的指数函数，阵元e_mn的载频f_mn＝f₀+mΔf_x+nΔf_y，m＝0,1,2,...,M-1,n＝0,1,2,...,N-1，f₀为参考阵元e₀₀的载频，远场观测点到阵元e_mn的距离

m＝0,1,2,...,M-1,n＝0,1,2,...,N-1，简化式(2)为：Among them, exp{·} represents the exponential function with the natural constant e as the base, f _mn represents the operating frequency of the mn-th array element, t represents the time,

represents the weight conjugate of the mn-th array element, r _mn represents the tilt distance of the mn-th array element, W={w _mn } _M×N is the transmit weight matrix of the antenna element,

is the antenna pattern of the array element e _mn , assuming that the consistency of the array antennas is good, and each array element antenna is an isotropic point source, and the frequency domain response is flat near the carrier frequency f ₀ , then there are

exp{·} represents the exponential function with the natural constant e as the base, the carrier frequency of the array element e _mn f _mn =f ₀ +mΔf _x +nΔf _y , m=0,1,2,...,M-1, n=0,1,2,...,N-1, f ₀ is the carrier frequency of the reference array element e ₀₀ , the distance from the far-field observation point to the array element e _mn

m=0,1,2,...,M-1,n=0,1,2,...,N-1, the simplified formula (2) is:

Δf_x表示x方向相邻天线阵元的频偏值，Δf_y表示y方向相邻天线阵元的频偏值，Δf_mn表示第mn个阵元的频偏值，d_mn表示第mn个阵元的阵元间隔，θ表示目标点的倒向角，

表示目标点的方位角；where Δf _mn =mΔf _x +nΔf _y ,

Δf _x represents the frequency offset value of the adjacent antenna element in the x direction, Δf _y represents the frequency offset value of the adjacent antenna element in the y direction, Δf _mn represents the frequency offset value of the mnth array element, and _dmn represents the mnth array element. element spacing, θ represents the reversal angle of the target point,

represents the azimuth of the target point;

Assuming the frequency offset values of adjacent antennas Δf _x and Δf _y , the distances d _x and _dy between adjacent antenna elements satisfy the following conditions:

Then formula (4) is further simplified as:

Among them, AF _x is the linear frequency-controlled array array factor on the x-axis, and AF _y is the linear frequency-controlled array array factor on the y-axis, which can be expressed as:

表示x方向上的权值共轭，

表示y方向上的权值共轭，x轴上的频控阵权重w_x＝{w_xm}_M×1和y轴上的频控阵权重w_y＝{w_yn}_N×1表示为：in,

represents the weight conjugate in the x direction,

Represents the weight conjugate in the y-direction, the frequency-controlled array weight w _x ={w _xm } _M×1 on the x-axis and the frequency-controlled array weight w _y ={w _yn } _N×1 on the y-axis are expressed as:

W=w _x ^T w _y ={w _xm ·w _yn } _M×N (8)

Among them, w _x ^T represents the weight vector transposition in the x direction, w _y represents the weight vector in the y direction, w _xm represents the weight value in the x direction, w _yn represents the weight value in the y direction, and _M×N represents the weight vector in the y direction. For a planar frequency controlled array model, the transmit weight matrix of the planar frequency controlled array is decomposed into two linear array frequency controlled array transmit weight vectors, then the transmit beam pattern of the planar frequency controlled array can be written in the following matrix form:

in:

and

表示目标点的方位角；Among them, exp{·} represents the exponential function with the natural constant e as the base, j represents the complex unit, f ₀ represents the center frequency of the transmitted signal, t represents the time, r represents the distance from the reference array element to the target point, c represents the speed of light, W represents the transmit weight matrix of the antenna elements, M represents the number of antenna elements in the x direction, N represents the number of antenna elements in the y direction, m represents the serial number of the antenna elements along the x direction, m=0,1,2 ,...M, n denotes the array element number along the y direction, n=0,1,2,...N, W ^* denotes the conjugate of the emission matrix W, ⊙ denotes the Hadamard product, A denotes the array factor, w _{M-1, N-1} represents the transmission matrix element, a _{M-1, N-1} represents the array factor element, a _mn represents the array factor of the mn-th array element, Δf _mn represents the frequency offset value of the mn-th array element , d _mn represents the array element spacing of the mnth array element, θ represents the inversion angle of the target point,

represents the azimuth of the target point;

S3, using the energy aggregation algorithm BCE to solve the eigenvalues and eigenvectors of the far-field transmit beam pattern respectively, to obtain the optimal weight matrix of the far-field transmit beam pattern, thereby forming the beam of the planar frequency-controlled array, in a specific embodiment In , a planar frequency-controlled array spot beamforming method based on generalized eigenvalue decomposition is proposed by combining the far-field transmit beam pattern with the energy gathering algorithm BCE:

表示为：In a specific embodiment, the transmit beam pattern of the planar frequency controlled array

Expressed as:

It is further derived as:

Among them, A represents the array factor, w _x ^H represents the conjugate transpose of the weight vector in the x direction, a _x ( ) represents the array factor in the x direction, and w _y ^H represents the conjugate transpose of the weight vector in the y direction set, a _y ( ) represents the weight vector in the y direction, W represents the emission weight matrix, H represents the conjugate transpose, and vec( ) represents the vectorization of the matrix;

与总发射功率P_Ω的比值,其表达式为：Define the energy aggregation algorithm BCE as the transmit power within the desired range

The ratio to the total transmit power P _Ω , its expression is:

Ω₀和Ω分别表示期望的波束辐射范围和总的波束辐射范围，

表示期望辐射范围内的积分，∫_Ω表示总辐射范围内的积分，d表示微分符号，θ表示目标点的倒向角，

表示目标点的方位角，r表示参考阵元到目标点的距离，则in,

Ω ₀ and Ω denote the desired beam radiation range and the total beam radiation range, respectively,

represents the integral in the expected radiation range, ∫ _Ω represents the integral in the total radiation range, d represents the differential sign, θ represents the reversal angle of the target point,

represents the azimuth of the target point, r represents the distance from the reference array element to the target point, then

in,

上式中，

表示期望辐射范围内的三重积分，∫∫∫_Ω表示总辐射范围内的三重积分，θ₁和θ₂均表示期望范围的倒向角范围边界，

和

均表示期望范围的方位角范围边界，对于距离维，由于其周期性，r取一个距离周期的范围内，针对任意发射孔径和目标区域波束形成问题，可以构造成广义特征值分解问题，对所述远场发射波束图的特征值和特征向量进行求解，得到关于最大能量聚集算法BCE的最优发射阵列权值的显式表达式：

In the above formula,

represents the triple integral over the desired radiation range, ∫∫∫ _Ω denotes the triple integral over the total radiation range, both θ ₁ and θ ₂ represent the reversal angle range boundary of the desired range,

and

Both represent the azimuth range boundary of the desired range. For the range dimension, due to its periodicity, r is within the range of a range period. For the beamforming problem of any transmit aperture and target area, it can be constructed as a generalized eigenvalue decomposition problem. The eigenvalues and eigenvectors of the far-field transmit beam pattern are solved, and the explicit expression of the optimal transmit array weights for the maximum energy gathering algorithm BCE is obtained:

表示求解最大权向量，vec(W)^H表示发射权重的矩阵向量化的共轭转置，vec(W)表示发射权重矩阵向量化，A表示在期望辐射范围的功率，B表示总辐射范围内的功率，若A和B是厄米特Hermitian矩阵且具有正定特性，则使最大能量聚集算法BCE最大的权向量vec(W^opt)与广义特征值问题的最大特征值对应的特征向量相等，确定A和B项后，通过计算式(14)中最大的特征值和对应的特征向量得到最优解vec(W^opt)，将式(14)中的结果带入代入式(13)，最终得到期望范围内的发射功率与总发射功率的比值：in,

Represents solving the maximum weight vector, vec(W) ^H represents the conjugate transpose of the matrix vectorization of the emission weight, vec(W) represents the vectorization of the transmission weight matrix, A represents the power in the desired radiation range, and B represents the total radiation range. If A and B are Hermitian matrices and have positive definite characteristics, then make the maximum weight vector vec(W ^opt ) of the maximum energy aggregation algorithm BCE equal to the eigenvector corresponding to the maximum eigenvalue of the generalized eigenvalue problem, determine After A and B terms, the optimal solution vec(W ^opt ) is obtained by calculating the largest eigenvalue and corresponding eigenvector in equation (14), and the result in equation (14) is substituted into equation (13), and finally we get Ratio of transmit power in desired range to total transmit power:

In this embodiment, as shown in FIG. 6 and FIG. 7 , according to the results based on generalized eigenvalue decomposition, the target area aΩ ₁ : [70°, 90°], [29km, 32km] and the target area bΩ ₂ : [30 °, 50°], [39km, 42km] The corresponding 10×10 weight matrix elements are 0.5274, 0.5972, 0.8933, 0.9558, 0.9966, 0.9966, 0.9558, 0.8933, 0.5972, 0.5274 (only the diagonal elements are given).. .and 0.2826, 0.4692, 0.7590, 0.9193, 0.9893, 0.9893, 0.9193, 0.7590, 0.4692, 0.2826 (only diagonal elements are given)..., the corresponding two-dimensional emissions formed by the planar frequency control array at the two target areas Spot beam pattern, where Ω ₁ represents the target area a, Ω ₂ represents the target area b, when the transmit weight matrix is an all-one matrix, that is, the optimized weight matrix is not used, the corresponding normalized three-dimensional beam pattern of the planar frequency control array As shown in Figure 4, the two-dimensional plane diagram is shown in Figure 5. Figure 4 shows the normalized three-dimensional beam diagram of the plane frequency controlled array without using the energy-focusing beamforming algorithm, and Figure 5 shows the corresponding two-dimensional beam diagram. By comparing with Figures 6 and 7 (Figures 6 and 7 use the energy-focusing beamforming algorithm), it can be seen that the energy-focusing beamforming algorithm adopted this time can control the focusing of its spot beam by setting the required focusing range Location.

Claims (6)

1. a plane frequency-controlled array beamforming method based on generalized eigenvalue decomposition, is characterized in that, comprises the steps: S1. Construct a planar frequency-controlled array according to the number of array elements of the antenna, the carrier frequency, the frequency offset value between adjacent antenna array elements and the spacing between adjacent antenna array elements; S2, constructing its corresponding far-field transmit beam pattern according to the planar frequency-controlled array; S3, using the energy aggregation algorithm BCE to solve the eigenvalues and eigenvectors of the far-field transmit beam pattern respectively, to obtain the optimal weight matrix of the far-field transmit beam pattern, thereby forming the beam of the planar frequency-controlled array; The expression of the energy aggregation algorithm BCE in the step S3 is:

where BCE ^opt represents the energy focusing algorithm BCE, vec(W ^opt ) ^H represents the conjugate transpose of the optimal weight matrix vec(W ^opt ) of the far-field transmit beam pattern, A represents the power in the desired radiation range, and B represents the total power in the radiation range; The expression of the optimal weight matrix vec(W ^opt ) of the far-field transmit beam pattern in the step S3 is as follows:

in,

Represents solving the maximum weight vector, vec(W) ^H represents the conjugate transpose of the matrix vectorization of the transmit weights, and vec(W) represents the vectorization of the transmit weight matrix. 2 . The planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition according to claim 1 , wherein the antenna in the step S1 is an omnidirectional antenna. 3 . 3. The planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition according to claim 1, wherein the planar frequency-controlled array is composed of M×N number of antenna array elements, and the first antenna The carrier frequency f ₀ is 10GHz, where M represents the number of antenna elements in the x direction, and N represents the number of antenna elements in the y direction; The x direction of the antenna array element is distributed with M antenna array elements at equal intervals d _x , and the frequency offset value Δf _x between adjacent antenna array elements is linearly increased in sequence; In the y direction of the antenna array elements, N antenna array elements are distributed at equal intervals _{dy, and the frequency offset value Δf y between} adjacent antenna array elements increases linearly in sequence. 4 . The planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition according to claim 3 , wherein the frequency offset values of the adjacent antennas in the x-direction and the y-direction are both 30KHz. 5 . 5 . The planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition according to claim 3 , wherein the distances between adjacent antennas in the x-direction and the y-direction are both 0.03m. 6 . 6. The planar frequency-controlled array beamforming method based on generalized eigenvalue decomposition according to claim 1, wherein the far-field transmit beam pattern

The expression is as follows:

and

Among them, exp{·} represents the exponential function with the natural constant e as the base, j represents the complex unit, f ₀ represents the center frequency of the transmitted signal, t represents the time, r represents the distance from the reference array element to the target point, c represents the speed of light, W represents the transmit weight matrix of the antenna elements, M represents the number of antenna elements in the x direction, N represents the number of antenna elements in the y direction, m represents the serial number of the antenna elements along the x direction, m=0,1,2 ,...M-1, n represents the serial number of the array element along the y direction, n=0,1,2,...N-1, W ^* represents the conjugate of the emission matrix W, ⊙ represents the Hadamard product, Y represents the array factor, w _{M-1, N-1} represents the element in the emission weight matrix, a _{M-1, N-1} represents the array factor element, a _mn represents the array factor of the mnth array element, θ represents the target point inverted angle,

Represents the azimuth of the target point, Δf _x represents the frequency offset between adjacent antenna elements along the x direction, Δf _y represents the frequency offset between adjacent antenna elements along the y direction; d _x represents the adjacent antenna along the x direction The spacing between array elements, dy represents the spacing between adjacent antenna array elements along the _y direction.

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