CN109407047B - Amplitude-phase error calibration and direction-of-arrival estimation method based on rank loss root finding - Google Patents

Amplitude-phase error calibration and direction-of-arrival estimation method based on rank loss root finding Download PDF

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CN109407047B
CN109407047B CN201811094657.3A CN201811094657A CN109407047B CN 109407047 B CN109407047 B CN 109407047B CN 201811094657 A CN201811094657 A CN 201811094657A CN 109407047 B CN109407047 B CN 109407047B
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戴继生
方忠驰
王兰
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Dragon Totem Technology Hefei Co ltd
Shanghai Fuyuan Zhiye Forge Co ltd
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Abstract

本发明公开了一种基于秩损求根的幅相误差校准和波达方向估计方法，包括步骤1：接收系统接收到的雷达信号经过匹配滤波后，在接收机处的输出表示为x(t)。步骤2：求矩阵x(t)的协方差矩阵R，并对其进行特征值分解，得到噪声子空间矩阵EN。步骤3：利用噪声子空间矩阵EN，构造矩阵U。步骤4：利用U，构造一个关于变量x的多项式f(x)，并令f(x)＝0，求得模最接近于1的N个根

Figure DDA0001805256110000011
步骤5：利用
Figure DDA0001805256110000012
求得N个DOA的估计值。本发明利用求多项式根的方法代替求解矩阵行列式的值，大大降低了计算量，节省运算时间。

Figure 201811094657

The invention discloses an amplitude-phase error calibration and direction-of-arrival estimation method based on rank loss root-finding. ). Step 2: Find the covariance matrix R of the matrix x(t), and perform eigenvalue decomposition on it to obtain the noise subspace matrix E N . Step 3: Construct the matrix U using the noise subspace matrix E N . Step 4: Using U, construct a polynomial f(x) about the variable x, and let f(x)=0, find the N roots whose modulus is closest to 1

Figure DDA0001805256110000011
Step 5: Utilize
Figure DDA0001805256110000012
Find estimates of N DOAs. The invention uses the method of finding polynomial roots instead of finding the value of the matrix determinant, which greatly reduces the amount of calculation and saves the operation time.

Figure 201811094657

Description

一种基于秩损求根的幅相误差校准和波达方向估计方法An Amplitude and Phase Error Calibration and Direction of Arrival Estimation Method Based on Rank Loss Root

技术领域technical field

本发明属于雷达信号处理领域，涉及阵列传感器幅度和相位误差的校准，具体地说是一种适用于均匀线性阵列的幅相误差自动校准以及波达方向估计的方法。The invention belongs to the field of radar signal processing, and relates to the calibration of the amplitude and phase errors of array sensors, in particular to a method for automatic calibration of amplitude and phase errors and estimation of direction of arrival for uniform linear arrays.

背景技术Background technique

近几十年来，波达方向(Direction of Arrival,DOA)估计一直是阵列信号处理的热点问题，被广泛应用于雷达、声呐、无源定位、无线通信等诸多领域。DOA估计的主要目的是在噪声环境下，检测和估计多个信号的方位。针对DOA估计问题，人们已尝试提出一些新的DOA估计方法用于阵列传感器幅相误差的校准。例如在文献：C.M.S.See andA.B.Gershman,Direction-of-arrival estimation in partly calibrated subarray-based sensor arrays,IEEE Transactions on Signal Processing,52(2):329-338中，提出了一种秩损(rank-reduction,RARE)算法来校准传感器幅相误差和估计DOA，但是此算法的搜索区间为

Figure BDA0001805256090000011
并需要求解每个角度对应矩阵的行列式的值，因此原始RARE算法计算量较大，搜索时间较长。In recent decades, Direction of Arrival (DOA) estimation has always been a hot issue in array signal processing, and has been widely used in radar, sonar, passive positioning, wireless communications and many other fields. The main purpose of DOA estimation is to detect and estimate the orientation of multiple signals in a noisy environment. For the DOA estimation problem, people have tried to propose some new DOA estimation methods for the calibration of the amplitude and phase errors of the array sensor. For example, in the literature: CMSSee and A.B. Gershman, Direction-of-arrival estimation in partly calibrated subarray-based sensor arrays, IEEE Transactions on Signal Processing, 52(2):329-338, a rank loss (rank loss) is proposed. -reduction, RARE) algorithm to calibrate the sensor amplitude and phase error and estimate DOA, but the search interval of this algorithm is
Figure BDA0001805256090000011
And it is necessary to solve the value of the determinant of the matrix corresponding to each angle, so the original RARE algorithm has a large amount of calculation and a long search time.

发明内容SUMMARY OF THE INVENTION

针对现有方法的不足，本发明提出了一种基于秩损(rank-reduction,RARE)求根(ROOT)的幅相误差校准和DOA估计方法，该方法在原始RARE算法上进行改进，通过构造多项式，然后求出多项式的根来减少运算量，缩短运算时间。In view of the shortcomings of the existing methods, the present invention proposes an amplitude and phase error calibration and DOA estimation method based on rank-reduction (RARE) root-finding (ROOT). The method is improved on the original RARE algorithm. Polynomial, and then find the roots of the polynomial to reduce the amount of operation and shorten the operation time.

用于实现本发明的技术解决方案包括如下步骤：The technical solution for realizing the present invention comprises the following steps:

步骤1：接收系统接收到的雷达信号经过匹配滤波后，在接收机处的输出表示为x(t)。Step 1: After the radar signal received by the receiving system is matched and filtered, the output at the receiver is expressed as x(t).

步骤2：求矩阵x(t)的协方差矩阵R，并对其进行特征值分解，得到噪声子空间矩阵EN。Step 2: Find the covariance matrix R of the matrix x(t), and perform eigenvalue decomposition on it to obtain the noise subspace matrix E N .

步骤3：利用噪声子空间矩阵EN，构造矩阵U。Step 3: Construct the matrix U using the noise subspace matrix E N .

步骤4：利用U，构造一个关于变量x的多项式f(x)，并令f(x)＝0，求得模最接近于1的N个根

Figure BDA0001805256090000012
Step 4: Using U, construct a polynomial f(x) about the variable x, and let f(x)=0, find the N roots whose modulus is closest to 1
Figure BDA0001805256090000012

步骤5：利用

Figure BDA0001805256090000013
求得N个DOA的估计值。Step 5: Utilize
Figure BDA0001805256090000013
Find estimates of N DOAs.

本发明的有益效果：Beneficial effects of the present invention:

与现有的RARE算法相比，本发明利用求多项式根的方法代替求解矩阵行列式的值，大大降低了计算量，节省运算时间。Compared with the existing RARE algorithm, the present invention uses the method of finding polynomial roots instead of finding the value of the matrix determinant, which greatly reduces the amount of calculation and saves the computing time.

附图说明Description of drawings

图1是本发明实施流程图。Figure 1 is a flow chart of the implementation of the present invention.

图2是200次蒙特卡洛实验条件下，信噪比从-10dB到10dB时，本发明与原始RARE方法分别估计信道的均方根误差比较。Figure 2 is a comparison of the root mean square error of the channel estimated by the present invention and the original RARE method when the signal-to-noise ratio is from -10dB to 10dB under the condition of 200 Monte Carlo experiments.

图3是200次蒙特卡洛实验条件下，信噪比从-10dB到10dB时，本发明与原始RARE方法分别估计信道所用时间的比较。Figure 3 is a comparison of the time spent in channel estimation between the present invention and the original RARE method when the signal-to-noise ratio is from -10dB to 10dB under the condition of 200 Monte Carlo experiments.

具体实施方式Detailed ways

下面结合附图对本发明作进一步说明。The present invention will be further described below in conjunction with the accompanying drawings.

如图1所示，本发明的实施方法包括如下步骤(1)至(5)：As shown in Figure 1, the implementation method of the present invention comprises the following steps (1) to (5):

(1)系统接收到的信号经过匹配滤波后，在接收机得到的均匀线阵在t时刻的输出数据为x(t)＝ΓAs(t)+n(t)，其中：(1) After the signal received by the system is matched and filtered, the output data of the uniform linear array obtained by the receiver at time t is x(t)=ΓAs(t)+n(t), where:

Figure BDA0001805256090000021
Γ表示幅相误差矩阵，它定义为矩阵diag{τ1,τ2,...,τM}，其中M表示阵元个数，diag{·}代表对角矩阵，此对角阵内的对角元素为τ1,τ2,...,τM，幅相误差系数τm＝ρmexp(jδm),m＝1,2,...M，ρm和δm分别表示第m个阵元的幅值误差和相位误差。假设总共M个阵元中有Mc个阵元未校准，不失一般性，记其序号为{1,2,...,Mc}，则其它阵元的幅相误差系数可记为τm＝1,m∈{Mc+1,Mc+2,...,M}。
Figure BDA0001805256090000021
Γ represents the magnitude and phase error matrix, which is defined as a matrix diag{τ 1 ,τ 2 ,...,τ M }, where M represents the number of array elements, and diag{·} represents a diagonal matrix. The diagonal elements are τ 1 , τ 2 ,...,τ M , the amplitude and phase error coefficients τ m =ρ m exp(jδ m ), m=1,2,...M, ρ m and δ m respectively represent The amplitude error and phase error of the mth element. Assuming that there are M c array elements in the total M array elements that are not calibrated, without loss of generality, record their serial numbers as {1,2,...,M c }, then the amplitude and phase error coefficients of other array elements can be recorded as τ m =1, m∈{M c +1,M c +2,...,M}.

Figure BDA0001805256090000022
A表示阵列流型矩阵，它的定义为[a(θ1),a(θ2),...,a(θN)]，其中N为入射信号个数，θn,n＝1,2,...,N，表示第n个真实DOA值，
Figure BDA0001805256090000023
Figure BDA0001805256090000024
其中j表示虚数，
Figure BDA0001805256090000025
d表示阵元间距，λ表示电磁波的波长，(·)T表示矩阵转置。
Figure BDA0001805256090000022
A represents the array manifold matrix, which is defined as [a(θ 1 ),a(θ 2 ),...,a(θ N )], where N is the number of incident signals, θ n ,n=1, 2,...,N, representing the nth real DOA value,
Figure BDA0001805256090000023
Figure BDA0001805256090000024
where j is an imaginary number,
Figure BDA0001805256090000025
d represents the spacing of the array elements, λ represents the wavelength of the electromagnetic wave, and (·) T represents the matrix transposition.

Figure BDA0001805256090000026
s(t)表示t时刻一个N维入射信号向量，n(t)表示t时刻一个M维零均值高斯白噪声向量。
Figure BDA0001805256090000026
s(t) represents an N-dimensional incident signal vector at time t, and n(t) represents an M-dimensional zero-mean Gaussian white noise vector at time t.

(2)利用阵列输出数据，求得阵列数据的协方差矩阵：(2) Use the array output data to obtain the covariance matrix of the array data:

R＝E{x(t)xH(t)}＝ΓARSAHΓH+σ2IM R=E{x(t) xH (t)}=ΓAR S A H Γ H +σ 2 I M

其中σ2和IM分别表示噪声方差和M维单位矩阵，(·)H表示共轭转置，RS代表信号协方差矩阵，定义为RS＝E{s(t)sH(t)}，E{·}代表期望，然后将R特征分解得到

Figure BDA0001805256090000031
其中ΛS是协方差矩阵的N个大特征值构成的对角矩阵，ES是与其对应的特征向量；ΛN为其余的M-N个小特征值构成的对角矩阵，与其对应的特征向量为EN，这里称为噪声子空间矩阵。where σ 2 and IM represent the noise variance and M-dimensional identity matrix, respectively, ( ) H represents the conjugate transpose, and R S represents the signal covariance matrix, defined as R S =E{s(t) sH (t) }, E{·} represents the expectation, and then the R feature is decomposed to get
Figure BDA0001805256090000031
where Λ S is the diagonal matrix composed of N large eigenvalues of the covariance matrix, and E S is the corresponding eigenvector; Λ N is the diagonal matrix composed of the remaining MN small eigenvalues, and the corresponding eigenvectors are EN , here called the noise subspace matrix.

(3)利用噪声子空间矩阵，构造

Figure BDA0001805256090000032
然后将U划分成四个子矩阵，即：(3) Using the noise subspace matrix, construct
Figure BDA0001805256090000032
Then divide U into four sub-matrices, namely:

Figure BDA0001805256090000033
Figure BDA0001805256090000033

U1的维度是Mc×Mc，U2的维度是Mc×(M-Mc)，U3的维度是(M-Mc)×Mc，U4的维度是(M-Mc)×(M-Mc)。The dimension of U 1 is M c ×M c , the dimension of U 2 is M c ×(MM c ), the dimension of U 3 is (MM c )×M c , and the dimension of U 4 is (MM c )×(MM c ).

(4)令

Figure BDA0001805256090000034
构造一个(2M-2Mc-1)×1维的向量v，它的第i个元素vi定义为矩阵S的第k个对角线上所有元素之和。构造一个关于变量x的多项式：(4) Order
Figure BDA0001805256090000034
Construct a (2M-2M c -1)×1-dimensional vector v, whose i -th element vi is defined as the sum of all elements on the k-th diagonal of matrix S. Construct a polynomial in the variable x:

Figure BDA0001805256090000035
Figure BDA0001805256090000035

令函数f(x)＝0，求得模最接近于1的N个根

Figure BDA0001805256090000036
Let the function f(x)=0, find the N roots whose modulus is closest to 1
Figure BDA0001805256090000036

(5)利用所得的

Figure BDA0001805256090000037
求得N个DOA的估计值：(5) Utilize the income
Figure BDA0001805256090000037
Find estimates of N DOAs:

Figure BDA0001805256090000038
Figure BDA0001805256090000038

其中：

Figure BDA0001805256090000039
表示
Figure BDA00018052560900000310
的相位角，θn取值范围是
Figure BDA00018052560900000311
in:
Figure BDA0001805256090000039
express
Figure BDA00018052560900000310
The phase angle of θ n is in the range of
Figure BDA00018052560900000311

下面结合仿真实验对本发明的效果做进一步说明。The effect of the present invention will be further described below in conjunction with simulation experiments.

为了评估本方法的性能，考虑系统，发射阵列的阵元间距为电磁波半波长的均匀线阵，发射阵列的阵元个数M＝8，发射阵列阵元之间未校准的幅相系数的个数Mc＝4，分别为

Figure BDA00018052560900000312
假设远场有两个相互独立的目标信号源，分别位于θ1＝-15°,θ2＝20°。在所有实验中，假设噪声为零均值高斯白噪声，快拍数为L＝400。In order to evaluate the performance of this method, considering the system, the element spacing of the transmitting array is a uniform linear array with half-wavelength electromagnetic waves, the number of elements in the transmitting array is M=8, and the number of uncalibrated amplitude and phase coefficients between the elements of the transmitting array is M=8. The number Mc = 4, respectively
Figure BDA00018052560900000312
It is assumed that there are two mutually independent target signal sources in the far field, located at θ 1 =-15° and θ 2 =20° respectively. In all experiments, the noise is assumed to be zero mean Gaussian white noise and the number of snapshots is L=400.

实验条件Experimental conditions

采用本发明在信噪比从-10dB到10dB时对目标角度进行200次角度估计，仿真结果如图2和图3所示。Using the present invention, the target angle is estimated 200 times when the signal-to-noise ratio is from -10dB to 10dB, and the simulation results are shown in Fig. 2 and Fig. 3 .

实验分析experiment analysis

从图2可以看出，本发明能精确地估计出真实的DOA值，并且其性能略优于原始RARE方法。It can be seen from Figure 2 that the present invention can accurately estimate the real DOA value, and its performance is slightly better than the original RARE method.

从图3可以看出，本发明所使用的时间明显小于原始RARE所使用的时间。It can be seen from FIG. 3 that the time used by the present invention is significantly smaller than the time used by the original RARE.

上文所列出的一系列的详细说明仅仅是针对本发明的可行性实施方式的具体说明，它们并非用以限制本发明的保护范围，凡未脱离本发明技艺精神所作的等效实施方式或变更均应包含在本发明的保护范围之内。The series of detailed descriptions listed above are only specific descriptions for the feasible embodiments of the present invention, and they are not used to limit the protection scope of the present invention. Changes should all be included within the protection scope of the present invention.

Claims (4)

1.一种基于秩损求根的幅相误差校准和波达方向估计方法，其特征在于，包括如下步骤：1. Amplitude and phase error calibration and direction of arrival estimation method based on rank loss seeking root, is characterized in that, comprises the steps: 步骤1：接收系统接收到的雷达信号经过匹配滤波后，在接收机处的输出表示为x(t)；x(t)指在接收机得到的均匀线阵在t时刻的输出数据；所述x(t)的表达式为：Step 1: After the radar signal received by the receiving system is matched and filtered, the output at the receiver is expressed as x(t); x(t) refers to the output data of the uniform linear array obtained at the receiver at time t; the The expression for x(t) is: x(t)＝ΓAs(t)+n(t)，x(t)=ΓAs(t)+n(t), 其中：in: Γ表示幅相误差矩阵，A表示阵列流型矩阵，Γ represents the magnitude and phase error matrix, A represents the array manifold matrix, s(t)表示t时刻一个N维入射信号向量，n(t)表示t时刻一个M维零均值高斯白噪声向量；s(t) represents an N-dimensional incident signal vector at time t, and n(t) represents an M-dimensional zero-mean Gaussian white noise vector at time t; 所述幅相误差矩阵Γ定义为矩阵diag{τ1,τ2,...,τM}，其中M表示阵元个数，diag{·}代表对角矩阵，此对角阵内的对角元素为τ1,τ2,...,τM，幅相误差系数τm＝ρmexp(jδm),m＝1,2,...M，ρm和δm分别表示第m个阵元的幅值误差和相位误差；假设总共M个阵元中有Mc个阵元未校准，记其序号为{1,2,...,Mc}，则其它阵元的幅相误差系数记为τm＝1,m∈{Mc+1,Mc+2,...,M}；The amplitude and phase error matrix Γ is defined as a matrix diag{τ 1 ,τ 2 ,...,τ M }, where M represents the number of array elements, and diag{·} represents a diagonal matrix. The angle elements are τ 1 , τ 2 ,...,τ M , the amplitude and phase error coefficients τ m =ρ m exp(jδ m ), m=1,2,...M, where ρ m and δ m represent the first Amplitude error and phase error of m array elements; assuming that M c array elements in the total M array elements are not calibrated, record their serial numbers as {1, 2,...,M c }, then the other array elements The amplitude-phase error coefficient is recorded as τ m =1,m∈{M c +1,M c +2,...,M}; 步骤2：求矩阵x(t)的协方差矩阵R，并对其进行特征值分解，得到噪声子空间矩阵EN；Step 2: Find the covariance matrix R of the matrix x(t), and perform eigenvalue decomposition on it to obtain the noise subspace matrix E N ; 步骤3：利用噪声子空间矩阵EN，构造矩阵U；具体实现包括：Step 3: Use the noise subspace matrix E N to construct the matrix U; the specific implementation includes: 利用噪声子空间矩阵EN构造矩阵
Figure FDA0003684083190000011
然后将U划分成四个子矩阵，即：Constructing Matrix Using Noise Subspace Matrix EN
Figure FDA0003684083190000011
Then divide U into four sub-matrices, namely:
Figure FDA0003684083190000012
Figure FDA0003684083190000012
 U1的维度是Mc×Mc，U2的维度是Mc×(M-Mc)，U3的维度是(M-Mc)×Mc，U4的维度是(M-Mc)×(M-Mc)；M表示阵元个数，Mc表示未校准阵元个数；The dimension of U 1 is M c ×M c , the dimension of U 2 is M c ×(MM c ), the dimension of U 3 is (MM c )×M c , and the dimension of U 4 is (MM c )×(MM c ); M represents the number of array elements, and M c represents the number of uncalibrated array elements; 步骤4：利用U构造一个关于变量x的多项式f(x)，并令f(x)＝0，求得模最接近于1的N个根
Figure FDA0003684083190000013
具体实现包括：Step 4: Use U to construct a polynomial f(x) about the variable x, and let f(x)=0, find the N roots whose modulus is closest to 1
Figure FDA0003684083190000013
Specific implementations include: 令
Figure FDA0003684083190000014
构造一个(2M-2Mc-1)×1维的向量v，它的第i个元素vi定义为矩阵S的第k个对角线上所有元素之和；构造一个关于变量x的多项式：make
Figure FDA0003684083190000014
Construct a (2M-2M c -1)×1-dimensional vector v, whose i -th element vi is defined as the sum of all elements on the k-th diagonal of matrix S; construct a polynomial about variable x:
Figure FDA0003684083190000021
Figure FDA0003684083190000021
 令函数f(x)＝0，求得模最接近于1的N个根
Figure FDA0003684083190000022
vn为向量v的第n个元素；Let the function f(x)=0, find the N roots whose modulus is closest to 1
Figure FDA0003684083190000022
v n is the nth element of the vector v; 步骤5：利用
Figure FDA0003684083190000023
求得N个DOA的估计值。Step 5: Utilize
Figure FDA0003684083190000023
Find estimates of N DOAs. 2.根据权利要求1所述的一种基于秩损求根的幅相误差校准和波达方向估计方法，其特征在于，所述阵列流型矩阵A定义为[a(θ1),a(θ2),...,a(θN)]，其中N为入射信号个数，θn,n＝1,2,...,N，表示第n个真实DOA值，
Figure FDA0003684083190000024
Figure FDA0003684083190000025
其中j表示虚数，
Figure FDA0003684083190000026
d表示阵元间距，λ表示电磁波的波长，(·)T表示矩阵转置。2. A rank-loss root-based amplitude-phase error calibration and DOA estimation method according to claim 1, wherein the array flow pattern matrix A is defined as [a(θ 1 ), a( θ 2 ),...,a(θ N )], where N is the number of incident signals, θ n , n=1,2,...,N, represents the nth real DOA value,
Figure FDA0003684083190000024
Figure FDA0003684083190000025
where j is an imaginary number,
Figure FDA0003684083190000026
d represents the spacing of the array elements, λ represents the wavelength of the electromagnetic wave, and (·) T represents the matrix transposition. 3.根据权利要求1所述的一种基于秩损求根的幅相误差校准和波达方向估计方法，其特征在于，步骤2的具体实现包括：3. a kind of amplitude and phase error calibration and direction of arrival estimation method based on rank loss root-seeking according to claim 1, is characterized in that, the concrete realization of step 2 comprises: 利用阵列输出数据，求得阵列数据的协方差矩阵：Using the array output data, find the covariance matrix of the array data: R＝E{x(t)xH(t)}＝ΓARSAHΓH+σ2IM R=E{x(t) xH (t)}=ΓAR S A H Γ H +σ 2 I M 其中σ2和IM分别表示噪声方差和M维单位矩阵，(·)H表示共轭转置，RS代表信号协方差矩阵，定义为RS＝E{s(t)sH(t)}，E{·}代表期望，然后将R特征分解得到
Figure FDA0003684083190000027
其中ΛS是协方差矩阵的N个大特征值构成的对角矩阵，ES是与其对应的特征向量；ΛN为其余的M-N个小特征值构成的对角矩阵，与其对应的特征向量为EN，称为噪声子空间矩阵。where σ 2 and IM represent the noise variance and M-dimensional identity matrix, respectively, ( ) H represents the conjugate transpose, and R S represents the signal covariance matrix, defined as R S =E{s(t) sH (t) }, E{·} represents the expectation, and then the R feature is decomposed to get
Figure FDA0003684083190000027
where Λ S is the diagonal matrix composed of N large eigenvalues of the covariance matrix, and E S is the corresponding eigenvector; Λ N is the diagonal matrix composed of the remaining MN small eigenvalues, and the corresponding eigenvectors are EN , called the noise subspace matrix. 4.根据权利要求1所述的一种基于秩损求根的幅相误差校准和波达方向估计方法，其特征在于，步骤5中所述利用
Figure FDA0003684083190000031
求得N个DOA的估计值为：4. a kind of amplitude and phase error calibration and direction of arrival estimation method based on rank loss root-seeking according to claim 1, is characterized in that, described in step 5, utilizes
Figure FDA0003684083190000031
The estimated value of N DOAs obtained is:
Figure FDA0003684083190000032
Figure FDA0003684083190000032
 其中：
Figure FDA0003684083190000033
表示
Figure FDA0003684083190000034
的相位角，θn取值范围是
Figure FDA0003684083190000035
d表示阵元间距，λ表示电磁波的波长。in:
Figure FDA0003684083190000033
express
Figure FDA0003684083190000034
The phase angle of θ n is in the range of
Figure FDA0003684083190000035
d represents the spacing of the array elements, and λ represents the wavelength of the electromagnetic wave.
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