CN112379327A - Two-dimensional DOA estimation and cross coupling correction method based on rank loss estimation - Google Patents

Abstract

The invention discloses a two-dimensional DOA estimation and cross coupling correction method based on rank loss estimation, and belongs to the technical field of array signal processing. Firstly, constructing a signal receiving model of a uniform planar antenna array under a mutual coupling error by constructing a mutual coupling error matrix; then, separating the information source data from the cross-coupling error vector by using a transform domain relation of parameter mapping, and constructing a corresponding spectral peak search function by using a rank loss estimation criterion to realize the estimation of the two-dimensional orientation parameters; and finally, constructing a decoupling matrix by using the obtained estimated orientation parameters to realize the estimation of the mutual coupling matrix, and further carrying out mutual coupling correction according to the estimation. On the basis, the estimation result of the method is verified by using proper simulation, and compared with the existing estimation algorithm, the DOA estimation result of the method has smaller root mean square error and higher estimation success rate, which indicates that the method has better performance.

Description

A 2D DOA Estimation and Mutual Coupling Correction Method Based on Rank Loss Estimation

technical field

The invention relates to the related field of array signal processing, can be applied to wireless communication, radar positioning and the like, and is especially suitable for solving the problems of two-dimensional DOA estimation and mutual coupling correction under the condition of mutual coupling error of antenna array.

Background technique

Array signal processing has always been one of the hot research directions of many scholars. Because of its many applications in satellite communication, radar detection, underwater positioning, etc., related technologies have been widely and rapidly developed in recent years. . The purpose of the array signal processing is to sample the data received by the array, and then perform the calculation through the analysis algorithm, so as to obtain the parameter information of the desired signal.

Direction of arrival (DOA) estimation is an important branch of array signal processing. The main purpose of DOA estimation is to determine the spatial position of multiple signals of interest located in a certain area in space, that is, the azimuth and elevation angles of each signal reaching the reference array element of the array. Because two-dimensional DOA estimation can estimate the source position in three-dimensional space, which meets the requirements of practical engineering, it has been widely studied by many scholars. The two-dimensional MUSIC algorithm is a typical two-dimensional DOA estimation algorithm. This algorithm can realize asymptotic and unbiased estimation of two-dimensional angles, but its realization process requires spectral peak search in two-dimensional space, resulting in excessive calculation. In the follow-up studies, some scholars applied the ESPRIT algorithm to the two-dimensional DOA estimation algorithm. This algorithm does not require spectral peak search, so the calculation amount is small, and it has high application value.

The above algorithms all require a known precise array flow pattern. However, in the process of practical application, there is an inherent electromagnetic effect between each antenna. The signal energy received by one antenna may be radiated to other nearby antennas, especially When the distance between the array elements is small, the electromagnetic radiation effect is more serious, and the interaction between the antennas is usually regarded as a mutual coupling effect. For a two-dimensional planar array, the number of array elements near each array element is large, and the mutual coupling effect is particularly serious. At this time, the ideal array flow pattern will change due to this electromagnetic disturbance, which can no longer meet the actual requirements. Engineering requirements, if this disturbance is not processed, the performance of many existing antenna signal processing technologies will deteriorate, and the practicability of the algorithm will also be greatly reduced. Most of the current research on solving the mutual coupling problem is based on the one-dimensional DOA estimation problem. Aiming at the mutual coupling problem of 2D DOA estimation under the planar array, it is mainly realized by setting auxiliary array elements outside the array. When the antenna array is uniformly coupled, the MUSIC algorithm is used to estimate the 2D DOA. This algorithm will cause loss of array aperture, and the amount of calculation of spectral peak search is relatively large. In response to such problems, the present invention proposes a two-dimensional DOA estimation algorithm under mutual coupling conditions based on a uniform planar array to obtain a more accurate source orientation and facilitate mutual coupling error correction. The system block diagram is shown in Figure 1.

SUMMARY OF THE INVENTION

The main technical problem solved by the present invention is that, considering the mutual coupling effect existing between the antenna arrays in the process of receiving the transmitted signal by the uniform planar antenna array, the rank loss estimation criterion is used to design the algorithm, so that the dimensionality reduction operation of the two-dimensional DOA estimation can be achieved. Then, the estimated value of the two-dimensional DOA parameter is obtained, and then the estimated value of the mutual coupling coefficient is obtained through the operation of eigenvalue decomposition, so as to realize the correction of the mutual coupling error.

In order to solve the above problems, the present invention adopts the following technical scheme:

A two-dimensional DOA estimation and mutual coupling correction method based on rank loss estimation. The specific contents of the method are as follows:

Step 1: Model and simulate the antenna array system, and the uniform planar array used is shown in Figure 2. Considering the mutual coupling signal model, each array element will not only receive the incident part of the incoming wave signal, but also receive the radiation signal of other surrounding elements, so the signal actually received by the array element is actually the incident wave and the coupled component. superimposed. Therefore, the array flow pattern under ideal conditions is corrected to obtain the response model of the array output;

Step 2: Obtain the covariance matrix R _x of the array output by using the corrected array output response, and perform eigenvalue decomposition on it to obtain the noise subspace E _noise and the signal subspace E _source . And then based on the subspace theory to solve the problem.

Step 3: In practice, the coupling strength between the array elements will also attenuate with the increase of the distance between the array elements, and the effect between the farther apart array elements can be ignored. According to the symmetrical structure and reciprocity relationship of the uniform plane array, we model the mutual coupling matrix C to express the mutual coupling effect between the array elements;

的参数变换可以得到一对映射关系

以此实现数据的降维处理，再利用Toeplitz矩阵及Kronecker积的性质，实现阵元信息与互耦系数的独立，根据秩损估计准则构造两次谱峰搜索函数，通过两次一维谱峰搜索代替二维谱峰搜索，从而计算得到方位参数估计值

Step 4: Use the pair of K pitch and azimuth angles

The parameter transformation of can get a pair of mapping relations

In this way, the dimensionality reduction processing of the data is realized, and the properties of the Toeplitz matrix and the Kronecker product are used to realize the independence of the array element information and the mutual coupling coefficient. According to the rank loss estimation criterion, two spectral peak search functions are constructed. The search replaces the two-dimensional spectral peak search, thereby calculating the estimated value of the orientation parameter

恢复出相对应的方位参数估计值

Step 5: Leverage Mapping Relationships

The corresponding azimuth parameter estimates are recovered

构造一个解耦合矩阵T，并利用特征值分解法得到所有的耦合系数，以此重构出互耦矩阵的估计值

进而可以据此进行互耦校正。Step 6: Estimate the mutual coupling coefficient. According to the obtained parameter estimates

Construct a decoupling matrix T, and use the eigenvalue decomposition method to get all the coupling coefficients, so as to reconstruct the estimated value of the mutual coupling matrix

Further, mutual coupling correction can be performed accordingly.

The features of the present invention are as follows:

(1) Considering the mutual coupling effect between the array elements in the uniform planar antenna array, the mutual coupling matrix is modeled, and a signal model that is more suitable for practical applications is established.

(2) The source data and the mutual coupling error vector are separated by the transform domain relationship of the parameter mapping, and then the corresponding spectral peak search function is constructed by using the rank loss estimation criterion to realize the estimation of the two-dimensional orientation parameters.

(3) Using the obtained estimated parameters, construct the decoupling matrix, realize the estimation of the mutual coupling matrix, and then perform the mutual coupling correction accordingly.

Compared with the prior art, the present invention has the following beneficial effects:

The invention proposes a two-dimensional DOA estimation and mutual coupling correction method based on rank loss estimation. The signal receiving model of the uniform planar antenna array under the mutual coupling error is constructed by constructing the mutual coupling error matrix; the source data and the mutual coupling error vector are separated by using the transform domain relationship of parameter mapping, and then the corresponding spectrum is constructed by using the rank loss estimation criterion The peak search function is used to estimate the two-dimensional azimuth parameters; finally, the decoupling matrix is constructed by using the estimated azimuth parameters that have been obtained to realize the estimation of the mutual coupling matrix, and then the mutual coupling can be corrected accordingly. The estimation results of the proposed method are verified by suitable simulations. The root mean square error is very sensitive to very large or small errors in a set of measurements, so the root mean square error can well reflect the precision of the measurement. Compared with the current estimation algorithm, the method of the present invention has smaller root mean square error of the DOA estimation result and higher estimation success rate, which means that the method proposed in this paper has better effect.

Description of drawings

Fig. 1 is the flow chart of the method involved in the present invention;

Figure 2 is a uniform planar antenna array model;

Fig. 3 Variation curve of root mean square error with the change of signal-to-noise ratio;

Figure 4. The change curve of the algorithm success rate when the signal-to-noise ratio changes;

Fig. 5 Variation curve of root mean square error when the number of snapshots changes;

Figure 6. The change curve of the algorithm success rate error when the number of snapshots changes.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and implementation steps.

The flowchart of the two-dimensional DOA estimation and mutual coupling correction method based on rank loss estimation is shown in Figure 1, which includes the following steps:

Figure 1 is a flow chart of the present invention.

的角度入射到阵列上，

表示第k个信号的方位角和俯仰角，则阵列的输出响应被表示为The first step: consider the existence of mutual coupling error, study the uniform area array model, the number of array elements is M×N, and the distances between the array elements are d _x and d _y respectively; at a certain time t, consider K arrays far-field narrowband signals with

incident on the array at an angle,

represents the azimuth and elevation angles of the k-th signal, then the output response of the array is expressed as

x(t)=CAs(t)+n(t) (1)

表示理想的阵列响应矩阵，其中

s(t)表示源信号向量，n(t)为不相关的噪声矢量，其均值为零，方差为σ²；其中j表示复数的虚数单位，

表示Kronecker积，λ表示信号的波长；where C represents the mutual coupling matrix,

represents the ideal array response matrix, where

s(t) represents the source signal vector, n(t) is the uncorrelated noise vector with zero mean and variance σ ² ; where j represents the complex imaginary unit,

represents the Kronecker product, and λ represents the wavelength of the signal;

Step 2: Model the mutual coupling matrix C; since the uniform planar array is considered to be composed of N uniform linear arrays, there is also a mutual coupling effect between different sub-arrays; according to the symmetry of the uniform planar antenna array and reciprocity properties, the mutual coupling matrix C is expressed as

为由耦合向量

所构造的Toeplitz矩阵，

表示与x轴平行放置的第n个均匀线阵的第m个阵元的耦合系数；考虑每个阵元与其四周的P个阵元产生互耦效应，得到耦合向量

c_n中的零元素表示远处的互耦作用忽略不计，则当n>P时，子矩阵C_n为零矩阵；in,

coupled vector

The constructed Toeplitz matrix,

Represents the coupling coefficient of the mth element of the nth uniform linear array placed parallel to the x-axis; considering the mutual coupling effect of each array element and its surrounding P array elements, the coupling vector is obtained

The zero element in c _n means that the mutual coupling in the distance is negligible, then when n>P, the sub-matrix C _n is a zero matrix;

Step 3: Through the output response of the array, the covariance matrix response of the array output is obtained as

R _x =E{x(t) ^xH (t)}=CAR _s A ^H C ^H +σ ² I (3)

等效估计出来，同理得到相应的子空间的估计值

和

In the formula, R _s =E{s(t)s ^H (t)} is the source covariance matrix, E{·} is the expected operation of the matrix, (·) ^H is the conjugate transpose operation, and I is the MN× The identity matrix of MN dimension; the eigenvalue decomposition of R _x is performed to obtain the noise subspace E _noise and the signal subspace E _source ; when the L number of snapshot samples are received, the covariance matrix passes through

Equivalent estimation is obtained, and the estimated value of the corresponding subspace is obtained in the same way

and

Step 4: Perform parameter transformation, and separate the two-dimensional DOA estimation from the mutual coupling error with the help of the transform domain; define

则

其中then get a pair of mapping relations

but

in

Then the equivalent array steering vector is written as

Among them, a _yn represents the nth element of a _y (v _k );

A theorem is introduced below to handle the subsequent steps;

Theorem 1: For a banded Toeplitz matrix A∈C ^M×M and a vector x∈C ^M×1 , the following matrix operation relations hold

Ax=Q(x)a

Among them, a=A _1p (p=1,2,..,P)∈C ^P×1 , where A _1p represents the element of the first row and the pth column of matrix A, and P is the number of non-zero elements in the first row of matrix A number, and Q(x) is expressed as

Q(x)=Q ₁ (x)+Q ₂ (x)

[ ] _ih represents the element of the i-th row and h-column of the matrix;

则运用定理1得出define vector

Then, using Theorem 1, we get

则将式(6)整理为where n=1,2,...,P; defines the vector

Then formula (6) can be sorted into

Among them, Q _y (v _k ) is an N×P-dimensional matrix, and Q _x (u _k ) is an M×P-dimensional matrix. According to Theorem 1, the two have the same matrix structure. According to the subspace principle, the array manifold and noise subspace Spaces are mutually orthogonal, so

定义

则Using the properties of the Kronecker product

definition

but

Among them, _IM and IP represent the identity matrix of M order and _P order, respectively;

则式(10)仅在G(v_k)缺秩的条件下是成立的，则det{G(v_k)}＝0；根据第二步，在接收到L次快拍数的条件下，G(v_k)等效为Step 5: According to the principle of rank loss estimation, since MN-K≥MP, G(v _k ) is full rank. From equation (10), it can be seen that due to

Then Equation (10) is only valid under the condition that G(v _k ) lacks rank, then det{G(v _k )}=0; according to the second step, under the condition of receiving L snapshots, G(v _k ) is equivalent to

So construct the following spectral function

将秩损估计扩展到二维情况下使用，定义一个新的P²×P²维的矩阵In the formula, det{·} represents the determinant operation of the matrix, and tr{·} represents the trace operation of the matrix; based on the above formula, the spectral peak search is performed to obtain the estimated value of the orientation parameter v _k

Extend the rank loss estimation to the two-dimensional case and define a new P ² ×P ² -dimensional matrix

代入式(13)，得到

满足秩损原则的条件，即MN-K≥P²,于是构造如下谱函数The estimated value that will be obtained

Substituting into equation (13), we get

Satisfy the condition of the rank loss principle, that is, MN-K≥P ² , so the following spectral function is constructed

Similarly, by searching for spectral peaks, the estimated value of the orientation parameter u _k is obtained.

恢复出对应的俯仰角和方位角参数

Step 6: According to the mapping relationship

Recover the corresponding pitch and azimuth parameters

上的谱峰搜索是等价的；It can be seen that the peak search results in the (u _k ,v _k ) domain are the same as

The peak search on is equivalent;

Step 7: After obtaining the estimated values of all azimuth parameters, estimate the mutual coupling coefficient; construct the following matrix using the obtained estimated values

的第一个值为自耦系数，所以取值为1，那么得到互耦系数的估计值The eigenvalue decomposition operation is performed on T, due to the mutual coupling vector

The first value of the autocoupling coefficient, so the value is 1, then the estimated value of the mutual coupling coefficient is obtained

就重构出互耦矩阵

从而实现对天线阵列的互耦校正。where e _min [ ] represents the operator for finding the eigenvector corresponding to the minimum eigenvalue; and then according to the obtained estimated value of the mutual coupling vector

to reconstruct the mutual coupling matrix

Thereby, the mutual coupling correction to the antenna array is realized.

The above embodiments are only exemplary embodiments of the present invention, and are not intended to limit the present invention, and the protection scope of the present invention is defined by the claims. Those skilled in the art can make various modifications or equivalent replacements to the present invention within the spirit and protection scope of the present invention, and such modifications or equivalent replacements should also be regarded as falling within the protection scope of the present invention.

Result analysis

In order to prove the effectiveness of the algorithm of the present invention, we select the MUSIC algorithm with a known error coefficient, and compare the algorithm of the present invention based on the self-correction algorithm of the auxiliary array element.

First, compare the success rate and root mean square error (RMSE) of the three algorithms when the signal-to-noise ratio (SNR) changes. to 25dB in 3dB steps. In order to ensure the accuracy of the test, 200 independent simulation experiments were performed under each signal-to-noise ratio. The experimental results are shown in Figure 3 and Figure 4. From the result graph, it can be seen that when the signal-to-noise ratio increases, the success rate of the three algorithms increases, and the root mean square error decreases. The longitudinal comparison shows that the performance of the algorithm of the present invention is equal to the success rate and the root mean square error. It is better than the self-correction algorithm based on the auxiliary array element, indicating that the estimation result obtained by the algorithm of the present invention is closer to the real value.

Then compare the success rate and root mean square error (RMSE) of the three algorithms when the number of snapshots (Snapshots) changes. In the experiment, the signal-to-noise ratio was set to 10dB and kept constant, and the number of snapshots was changed from 300 to 900 at intervals of 100. In order to ensure the accuracy of the test, 200 independent simulation experiments are performed under each snapshot number, and the experimental results are shown in Figure 5 and Figure 6. From the result graph, it can be seen that when the number of snapshots increases, the success rate of the three algorithms increases, and the root mean square error decreases. The longitudinal comparison shows that the performance of the algorithm of the present invention is equal to the success rate and the root mean square error. It is better than the self-correction algorithm based on the auxiliary array element, indicating that the estimation result obtained by the algorithm of the present invention is closer to the real value.

Claims (2)

1. a two-dimensional DOA estimation and mutual coupling correction method based on rank loss estimation, is characterized in that: (1) Consider the mutual coupling effect between the array elements in the uniform planar antenna array, model the mutual coupling matrix, and establish a signal model that is more in line with practical applications; (2) Use the transform domain relationship of parameter mapping to separate the source data from the mutual coupling error vector, and then use the rank loss estimation criterion to construct the corresponding spectral peak search function to realize the estimation of the two-dimensional orientation parameters and obtain better prediction results. ; (3) Using the obtained estimated parameters, construct a decoupling matrix, realize the estimation of the mutual coupling matrix, and then correct the mutual coupling error accordingly. 2. the method for claim 1 is characterized in that comprising the following steps: The first step: consider the existence of mutual coupling error, study the uniform area array model, the number of array elements is M×N, and the distances between the array elements are d _x and d _y respectively; at a certain time t, consider K arrays far-field narrowband signals with

incident on the array at an angle,

represents the azimuth and elevation angles of the k-th signal, then the output response of the array is expressed as x(t)=CAs(t)+n(t) (1) where C represents the mutual coupling matrix,

represents the ideal array response matrix, where

s(t) represents the source signal vector, n(t) is the uncorrelated noise vector with zero mean and variance σ ² ; where j represents the complex imaginary unit,

represents the Kronecker product, and λ represents the wavelength of the signal; Step 2: Model the mutual coupling matrix C; since the uniform planar array is considered to be composed of N uniform linear arrays, there is also a mutual coupling effect between different sub-arrays; according to the symmetry of the uniform planar antenna array and reciprocity properties, the mutual coupling matrix C is expressed as

in,

coupled vector

The constructed Toeplitz matrix,

Represents the coupling coefficient of the mth element of the nth uniform linear array placed parallel to the x-axis; considering the mutual coupling effect of each array element and its surrounding P array elements, the coupling vector is obtained

The zero element in c _n means that the mutual coupling in the distance is negligible, then when n>P, the sub-matrix C _n is a zero matrix; Step 3: Through the output response of the array, the covariance matrix response of the array output is obtained as R _x =E{x(t) ^xH (t)}=CAR _s A ^H C ^H +σ ² I (3) In the formula, R _s =E{s(t)s ^H (t)} is the source covariance matrix, E{·} is the expected operation of the matrix, (·) ^H is the conjugate transpose operation, and I is the MN× The identity matrix of MN dimension; the eigenvalue decomposition of R _x is performed to obtain the noise subspace E _noise and the signal subspace E _source ; when the L number of snapshot samples are received, the covariance matrix passes through

Equivalent estimation is obtained, and the estimated value of the corresponding subspace is obtained in the same way

and

Step 4: Perform parameter transformation, and separate the two-dimensional DOA estimation from the mutual coupling error with the help of the transform domain; define

Then get a pair of mapping relationship

but

in

Then the equivalent array steering vector is written as

Among them, a _yn represents the nth element of a _y (v _k ); A theorem is introduced below to handle the subsequent steps; Theorem 1: For a banded Toeplitz matrix A∈C ^M×M and a vector x∈C ^M×1 , the following matrix operation relations hold Ax=Q(x)a Among them, a=A _1p (p=1,2,..,P)∈C ^P×1 , where A _1p represents the element of the first row and the pth column of matrix A, and P is the number of non-zero elements in the first row of matrix A number, and Q(x) is expressed as Q(x)=Q ₁ (x)+Q ₂ (x)

[ ] _ih represents the element of the i-th row and h-column of the matrix; define vector

Then, using Theorem 1, we get

where n=1,2,...,P; defines the vector

Then formula (6) can be sorted into

Among them, Q _y (v _k ) is an N×P-dimensional matrix, and Q _x (u _k ) is an M×P-dimensional matrix. According to Theorem 1, the two have the same matrix structure. According to the subspace principle, the array manifold and noise subspace Spaces are mutually orthogonal, so

Using the properties of the Kronecker product

definition

but

Among them, _IM and IP represent the identity matrix of M order and _P order, respectively; Step 5: According to the principle of rank loss estimation, since MN-K≥MP, G(v _k ) is full rank. From equation (10), it can be seen that due to

Then Equation (10) is only valid under the condition that G(v _k ) lacks rank, then det{G(v _k )}=0; according to the second step, under the condition of receiving L snapshots, G(v _k ) is equivalent to

So construct the following spectral function

In the formula, det{·} represents the determinant operation of the matrix, and tr{·} represents the trace operation of the matrix; based on the above formula, the spectral peak search is performed to obtain the estimated value of the orientation parameter v _k

Extend the rank loss estimation to the two-dimensional case and define a new P ² ×P ² -dimensional matrix

The estimated value that will be obtained

Substituting into equation (13), we get

Satisfy the condition of the rank loss principle, that is, MN-K≥P ² , so the following spectral function is constructed

Similarly, by searching for spectral peaks, the estimated value of the orientation parameter u _k is obtained.

Step 6: According to the mapping relationship

Recover the corresponding pitch and azimuth parameters

It can be seen that the peak search results in the (u _k ,v _k ) domain are the same as

The peak search on is equivalent; Step 7: After obtaining the estimated values of all azimuth parameters, estimate the mutual coupling coefficient; construct the following matrix using the obtained estimated values

The eigenvalue decomposition operation is performed on T, due to the mutual coupling vector

The first value of the autocoupling coefficient, so the value is 1, then the estimated value of the mutual coupling coefficient is obtained

where e _min [ ] represents the operator for finding the eigenvector corresponding to the minimum eigenvalue; and then according to the obtained estimated value of the mutual coupling vector

to reconstruct the mutual coupling matrix

Thereby, the mutual coupling correction to the antenna array is realized.

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