CN110109051B - Frequency control array-based cross coupling array DOA estimation method - Google Patents

Abstract

The invention relates to the field of array signal processing, and aims to propose a super-resolution DOA estimation algorithm for a far-field narrowband signal source. For this reason, the technical solution adopted by the present invention is, based on the DOA estimation method of the frequency-controlled array, firstly, the frequency-controlled array signal with uniform frequency difference is generated based on the uniform linear array, and then the mutual coupling coefficient matrix MCM (Mutual Coupling Matrix ) symmetric Toeplitz structure, constructing a selection matrix to truncate the received data, so that the remaining data has a cyclic mutual coupling coefficient; then, by including the unknown mutual coupling coefficient into the source part, a new steering vector of the truncated data is obtained; Then, the received signal is subjected to singular value SVD decomposition; finally, the sparse complete dictionary and the convex optimization solution function are constructed by using the parameterization of the steering vector and the sparse reconstruction theory to improve the accuracy of the estimation result. The invention is mainly applied to the design and manufacture of the frequency control array radar.

Description

DOA estimation method of mutual coupling array based on frequency-controlled array

Technical Field

The present invention relates to the field of array signal processing, and in particular to a mutual coupling array DOA estimation method based on a frequency controlled array.

Background Art

DOA estimation has a wide range of applications in radar, sonar, channel detection, wireless communications, and other scenarios ^[1] . The super-resolution problem of the direction of arrival of two or more adjacent sources has always been a hot topic in array signal processing. In addition, how to overcome the mutual coupling effect between antenna units in an array in actual situations is another hot topic in DOA estimation of sources.

With the introduction of compressed sensing theory ^[2,3] , the sparse representation parameter estimation method based on grid partitioning ^[4,5] has surpassed the performance of traditional DOA estimation algorithms ^[1] such as the MUSIC (Multiple Signal Classification) algorithm and the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithm. In the sparse representation parameter estimation method based on grid partitioning, the estimated parameters are discretized into a series of grids, which form an over-complete dictionary. Assuming that the parameters to be estimated only exist on a few of these grids, the estimation problem of the parameters to be estimated is transformed into a sparse recovery model of the parameters.

FDA (Frequency diverse array) has a similar transmission structure to the traditional phased array. Unlike the phased array that transmits signals of the same frequency, each antenna of the frequency diverse array can transmit signals of different frequencies ^[6,7] . The mutual coupling effect between array elements may affect the accuracy of DOA estimation, because the influence between antennas is mutual and consistent, so the mutual coupling coefficients between antennas are equal. The matrix established by the mutual coupling coefficients between antennas can be represented by the Toeplitz matrix ^[8] .

In the mutual coupling DOA estimation problem, due to the influence of mutual coupling, the traditional MUSIC algorithm and ESPRIT algorithm will fail. Considering the structural characteristics of the mutual coupling coefficient matrix and the development of sparse representation algorithms based on grid partitioning, the reference [12] truncated the received signal and then adopted the L1_SVD algorithm ^[4] for DOA estimation; the reference [13] proposed a DOA estimation algorithm based on block sparseness.

In the mutually coupled one-dimensional uniform linear array, the L1_SVD algorithm and the method of guiding vector parameterization can solve the Toeplitz structure and the block sparsity. The technical route of DOA estimation of mutually coupled array based on frequency-controlled array includes four steps: 1) Establishing the signal model of the frequency-controlled array; 2) Multiplying the signal matrix by the mutual coupling matrix; 3) Converting the original problem into the corresponding L1_SVD problem; 4) Complete the final parameter estimation based on the optimized result in 3). Because the angular distance correlation of the frequency-controlled array has higher resolution characteristics in parameter estimation than the estimation of the phased array under the same conditions.

[1]H.Krim and M.Viberg, "Two decades of array signal processing research: The parametric approach," IEEE Signal Process.Mag., vol.13, no.4, pp.67-94, Jul.1996.

[2] D.Donoho, "Superresolution via sparsity constraints," SIAM Journalon Mathematical Analysis, vol.23, pp.1309-1331, Sep.1992.

[3] E. Candes, "Compressive sampling," in Proc. of the International Congress of Mathematicians: Madrid, August 22-30, 2006: invited lectures, 2006, pp. 1433-1452.

[4] D.Malioutov, M.Cetin, and A.S.Willsky, "A sparse signalreconstruction perspective for source localization with sensor arrays," IEEE Transactions on Signal Processing, vol.53, no.8, pp.3010-3022.

[5]M.M.Hyder and K.Mahata, "Direction-of-arrival estimation using amixed l2,0norm approximation," IEEE Transactions on Signal Processing, vol.58, no.9, pp.4646-4655, Sep.2010.

[6] P.Antonik, M.C.Wicks, H.D.Griffiths, and C.J.Baker, "Frequency diverse array radars," in 2006IEEE Conference on Radar, April 2006, pp.3pp.215–217

[7] J.Xu, G.Liao, S.Zhu, L.Huang, and H.C.So, "Joint range and angle estimation using mimo radar with frequency diverse array," IEEE Transactionson Signal Processing, vol.63, no.13, pp.3396–3410,July 2015

[8] T.Svantesson, “Modeling and estimation of mutual coupling in auniform linear array of dipoles,” in 1999IEEE International Conference on Acoustics, Speech, and Signal Processing, vol.5, 1999, pp.2961–2964

[9] J.Dai, D.Zhao, and X.Ji, "A sparse representation method for DOAestimation with unknown mutual coupling," IEEE Antennas and WirelessPropagation Letters, vol.11, pp.1210-1213, 2012.

[10] Q.Wang, T.Dou, H.Chen, W.Yan and W.Liu, "Effective Block SparseRepresentation Algorithm for DOA Estimation With Unknown Mutual Coupling," IEEE Communications Letters, vol.21, no.12, pp .2622-2625,Dec.2017.

[11] L.Hao and W.Ping, "DOA estimation in an antenna array with mutualcoupling based on ESPRIT," Proc.International Workshop on Microwave andMillimeter Wave Circuits and System Technology, pp.86-89, 2013.

Summary of the invention

In order to overcome the shortcomings of the prior art, the present invention aims to study the situation where a uniform linear array transmits a frequency-controlled array signal and there is a mutual coupling effect, and proposes a super-resolution DOA estimation algorithm for a far-field narrowband source. To this end, the technical solution adopted by the present invention is a mutual coupling array DOA estimation method based on a frequency-controlled array. First, a frequency-controlled array signal with a uniform frequency difference is generated based on a uniform linear array. Then, a selection matrix is constructed by the symmetric Toeplitz structure of the mutual coupling coefficient matrix MCM (Mutual Coupling Matrix) to truncate the received data so that the remaining data has a cyclic mutual coupling coefficient; then, by including the unknown mutual coupling coefficient into the source part, a new steering vector of the truncated data is obtained; then, the received signal is subjected to singular value SVD decomposition; finally, a sparse complete dictionary and a convex optimization solution function are constructed by using the parameterization of the steering vector and the sparse reconstruction theory to improve the accuracy of the estimation result.

The specific steps are as follows:

Step 1: Establish a transmission signal with mutual coupling under the frequency control array;

进行奇异值分解；Step 2: Receive the signal

Perform singular value decomposition;

Step 3: Calculate the signal subspace of the received data based on the signal unitary space

Step 4: Construct the sparse signal complete dictionary A _J according to the parameterization J of the steering vector;

Step 5: Construct a convex programming function for sparse signal reconstruction based on the l1 norm;

Step 6: Solve the convex programming function and perform spectrum peak search.

Further:

Step 1: There is a uniformly distributed linear array ULA composed of M isotropic antennas. N far-field narrowband signals s _k (t) are incident on the array at angles θ ₁ ,θ ₂ ,…,θ _N, respectively, where n = 1, 2,…, N, and the difference between the transmit frequencies of the array elements is expressed as △f, that is, the transmit frequency of the nth array element is expressed as f _n = (n-1)*△f+f ₀ , where f ₀ is the initial reference array element transmit frequency;

When the mutual coupling effect between array elements is not considered, the received signal matrix is:

X(t)＝[x ₁ (t),x ₂ (t),…,x _M (t)] ^T represents the output signal matrix of M array elements in this snapshot sampling;

s(t) represents the signal matrix composed of the original N narrowband signals. t represents a snapshot.

其中

即表示第i个信号的角度θ_i到达第n个阵元是以第一个为参考相位的相位差，其中d表示均匀阵列相邻两个阵元之间的距离,

c为光速，

是噪声信号矩阵，其中的每一列代表每个阵元接收的噪声信号，在考虑互耦因素的情况下，单快拍即单次采样接收信号：A is the array manifold matrix composed of the steering vector a(θ),

in

That is, the angle _θi of the i-th signal reaching the n-th array element is the phase difference with the first one as the reference phase, where d represents the distance between two adjacent array elements of the uniform array,

c is the speed of light,

is the noise signal matrix, where each column represents the noise signal received by each array element. Considering the mutual coupling factor, a single snapshot is a single sampling received signal:

Each column of the matrix represents the coupling coefficient between the array element and the array element at other positions. 1 represents the array element being compared. The entire matrix C represents the coupling coefficient matrix composed of the coupling coefficients between the array elements. The received signal model under L snapshots is:

表示多个快拍下噪声信号组成的矩阵；S represents the matrix composed of narrowband signals under multiple snapshots,

Represents the matrix composed of noise signals under multiple snapshots;

其中L是快拍数，X(t)是接收信号矩阵，其中第n个信号的发射频率为f_n＝f₀+(n-1)△f，总的导向矢量按照与距离相关参数和与角度有关分开成两个矩阵，表示为：Step 2: Calculate the covariance matrix of the received signal:

Where L is the number of snapshots, X(t) is the received signal matrix, where the transmission frequency of the nth signal is f _n =f ₀ +(n-1)Δf, and the total steering vector is divided into two matrices according to the distance-related parameters and the angle-related parameters, expressed as:

The symbol ⊙ represents the multiplication operation of corresponding positions, a _Θ is a matrix composed of angle-related items, a _r represents a matrix composed of distance-related items, θ represents the incident angle when the signal reaches the array element. Because the signal is a far-field signal, the original θ ₁ ,θ ₂ ,…,θ _N can be regarded as the same incident angle; r represents the distance from the signal to the array element;

Step 3: Perform singular value decomposition on the covariance matrix R: R=UΛV, where U and V are M×M and N×N unitary matrices respectively (their sizes depend on the number of array elements M and the number of signals N), Λ is an M×N singular value diagonal matrix, and the corresponding signal and noise parts are also singular value decomposed: U=[U _S U _N ], U _s represents the U matrix of the signal, and U _N represents the U matrix of the noise; similarly, V=[V _S V _N ] ^T , V _S represents the V matrix of the signal, V _N represents the V matrix of the noise, T represents the transposition of the matrix, and Λ=diag[Λ _S Λ _N ];

Step 4: For the mutual coupling of matrices, construct the selection matrix D _K = [I _K 0], where I _K is a unit matrix of size K × K, and K is the number of pre-selected signal sources. Calculate the signal subspace _RS = RVD _K ;

Step 5: Based on the parameterization of the steering vector,

Construct a sparse complete dictionary A _J = [J(θ ₁ ), J(θ ₂ ), …, J(θ _N )];

Step 6: Use the l1 norm to constrain the spatial sparse features of the signal. The constraints are the time domain sparseness of the l2 norm and the suppression of noise, that is, construct a convex programming function

Step 7: Use l1-SVD theory to automatically select the regularization parameter ξ with a 99% confidence interval suppression. According to the orthogonality between the signal subspace and the noise subspace, solve the angle θ corresponding to the noise subspace U _n = 0. Use the convex optimization toolkit CVX to estimate the sparse signal space spectrum, and finally perform a one-dimensional spectrum peak search.

Step 8: Repeat steps 5 and 6 to improve the estimation accuracy.

The characteristics and beneficial effects of the present invention are:

The main advantage of the present invention is that it has a high angle measurement accuracy, and the DOA estimation still maintains excellent performance when the array mutual coupling phenomenon is obvious. At the same time, the estimation performance is better than that of the phased array under the same conditions.

Traditional array mutual coupling self-correction algorithms mostly discard the array elements at both ends of the entire array and only use the received information of the middle array element, which will inevitably affect the measurement accuracy. Unlike other algorithms, this algorithm makes full use of the received information of all array elements in the uniform array, and uses the parameterized operation of the steering vector to reorganize the received data model under the mutual coupling condition, thereby constructing a new complete dictionary for sparse reconstruction. In the solution process, singular value decomposition is used to reduce the dimension of the data, thereby reducing the computational complexity and reducing noise.

In terms of DOA estimation performance, the present invention is compared with the algorithms in the references under different signal-to-noise ratios and different snapshot numbers, using the root mean square error (RMSE) as a performance measurement indicator, with the number of signals set to 2. The results are shown in the following figure. Figure 1: When the signal-to-noise ratio is 5dB and the number of snapshots is 400, the performance of the provided method is the best. As can be seen from Figures 2 and 3, when the number of snapshots is 400, as the signal-to-noise ratio increases, the root mean square error of the algorithm is smaller than that of other algorithms in the references, and when the signal-to-noise ratio is 20dB, as the number of snapshots increases, the algorithm performance gradually improves and outperforms other algorithms.

Description of the drawings:

Fig. 1 DOA estimation accuracy of several methods.

Fig. 2 Relationship between DOA estimation accuracy and signal-to-noise ratio.

Fig. 3 Relationship between DOA estimation accuracy and snapshot number.

Fig. 4 Array model with uniformly distributed elements.

Fig. 5 is a flow chart of the method.

DETAILED DESCRIPTION

The present invention belongs to the field of array signal processing. According to the characteristics of angle distance correlation of frequency-controlled array, the output signal of a uniform linear array under mutual coupling is analyzed and reconstructed, and a sparse reconstruction parameter estimation framework is applied to mutual coupling DOA (Direction of Arrival) estimation, thereby completing DOA estimation of far-field narrowband signals.

First, a frequency-controlled array signal with uniform frequency difference is generated based on a uniform linear array. Then, a selection matrix is constructed by the symmetric Toeplitz structure of the mutual coupling matrix (MCM) to truncate the received data so that the remaining data has a cyclic mutual coupling coefficient. Then, by including the unknown mutual coupling coefficient into the source part, a new steering vector for the truncated data is obtained. Then, the received signal is subjected to singular value decomposition (SVD decomposition) to reduce the amount of calculation and denoise. Finally, the parameterization of the steering vector and the sparse reconstruction theory are used to construct a sparse complete dictionary and a convex optimization solution function to improve the accuracy of the estimation result. The specific scheme is as follows:

Mutually coupled array DOA estimation method based on frequency-controlled array:

Step 1: As shown in Figure (1), there is a uniformly distributed linear array (ULA) composed of M isotropic antennas. At a distance, N far-field narrowband signals s _k (t), (k = 1, 2, ..., N) are incident on the array at angles θ ₁ , θ ₂ , ..., θ _N. The transmit frequencies of the array elements differ by △f, that is, the transmit frequency of the nth array element is f _n = (n-1) * △f + f ₀ . _{f 0} is the initial reference array element transmit frequency.

When the mutual coupling effect between array elements is not considered, the received signal matrix is:

X(t)＝[x ₁ (t),x ₂ (t),…,x _M (t)] ^T represents the output signal matrix of M array elements in this snapshot sampling.

s(t) represents the signal matrix composed of the original N narrowband signals. t represents a snapshot.

其中

即表示第i个信号的角度θ_i到达第n个阵元是以第一个为参考相位的相位差，其中d表示均匀阵列相邻两个阵元之间的距离，

c为光速。

是噪声信号矩阵，其中的每一列代表噪声信号。在考虑互耦因素的情况下，单快拍接收信号：A is the array manifold matrix composed of the steering vector a(θ),

in

That is, the angle _θi of the i-th signal reaching the n-th array element is the phase difference with the first one as the reference phase, where d represents the distance between two adjacent array elements of the uniform array,

c is the speed of light.

Is the noise signal matrix, each column of which represents the noise signal. Considering the mutual coupling factor, the single snapshot received signal is:

Each column of the matrix represents the coupling coefficient between the array element and the array element at other positions, and 1 represents the array element being compared. The entire matrix C represents the coupling coefficient matrix formed by the mutual coupling between the array elements. The received signal model under L snapshots is:

其中L是快拍数，X(t)是接收信号矩阵。其中第n个信号的发射频率为f_n＝f₀+(n-1)△f，总的导向矢量可以按照与距离相关参数和与角度有关分开成两个矩阵，表示为Step 2: Calculate the covariance matrix of the received signal:

Where L is the number of snapshots, and X(t) is the received signal matrix. The transmission frequency of the nth signal is f _n = f ₀ + (n-1) △ f. The total steering vector can be divided into two matrices according to the distance-related parameters and the angle-related parameters, expressed as

The ⊙ symbol indicates the multiplication operation of the corresponding positions. _{a Θ} is a matrix composed of angle-related items, and a _r is a matrix composed of distance-related items. θ represents the incident angle when the signal reaches the array element. Because the signal is a far-field signal, the original θ ₁ ,θ ₂ ,…,θ _N can be regarded as the same incident angle; r represents the distance from the signal to the array element, d is the uniform array element spacing, and c is the speed of light.

Step 3: Perform singular value decomposition on the covariance matrix R, R = UΛV, where U and V are M×M and N×N unitary matrices respectively, and Λ is a singular value diagonal matrix of size M×N. (Its size depends on the number of array elements M and the number of signals N). The corresponding signal and noise parts are also decomposed into singular values U = [U _S U _N ], U _s represents the U matrix of the signal, and U _N represents the U matrix of the noise; similarly V = [V _S V _N ] ^T (the brackets represent the V matrix of the signal and the V matrix of the noise respectively), T represents the transposition of the matrix, and Λ = diag [Λ _S Λ _N ]. (The singular value diagonal matrix is also divided into the singular value diagonal matrix of the signal and the noise, and the minimum singular value in the singular value matrix of the signal is larger than the maximum singular value in the singular value matrix of the noise.)

Step 4: For the mutual coupling of matrices, construct a selection matrix D _K = [I _K 0], where I _K is a unit matrix of size K×K, and K is the number of pre-selected signal sources. Calculate the signal subspace _RS = RVD _K .

Step 5: Based on the parameterization of the steering vector,

Construct a sparse complete dictionary A _J =[J(θ ₁ ),J(θ ₂ ),…,J(θ _N )].

Step 6: Use the l1 norm to constrain the spatial sparse features of the signal. The constraints are the time domain sparseness of the l2 norm and the suppression of noise, that is, construct a convex programming function

Step 7: Use l1-SVD theory to automatically select the regularization parameter ξ with a 99% confidence interval suppression. According to the orthogonality of the signal subspace and the noise subspace, solve the angle θ corresponding to Un = 0. Use the convex optimization toolkit CVX to estimate the sparse signal space spectrum, and finally perform a one-dimensional spectrum peak search.

Step 8: Repeat steps 5 and 6 to improve the estimation accuracy.

The process is shown in Figure 5.

Claims (3)

1. A frequency-controlled array DOA estimation method based on a mutual coupling array, characterized in that, first, a frequency-controlled array signal with uniform frequency difference is generated based on a uniform linear array, and then a selection matrix is constructed to truncate the received data by using the symmetric Toeplitz structure of the mutual coupling coefficient matrix MCM (Mutual Coupling Matrix) so that the remaining data has a cyclic mutual coupling coefficient; then, a new steering vector of the truncated data is obtained by including the unknown mutual coupling coefficient into the source part; then, the received signal is subjected to singular value SVD decomposition; finally, a sparse complete dictionary and a convex optimization solution function are constructed by using the parameterization of the steering vector and the sparse reconstruction theory to improve the accuracy of the estimation result. 2. The method for estimating DOA of a frequency-controlled array-based mutual coupling array according to claim 1, wherein the specific steps are as follows: Step 1: Establish a transmission signal with mutual coupling under the frequency control array; Step 2: Receive the signal

Perform singular value decomposition; Step 3: Calculate the signal subspace of the received data based on the signal unitary space

Step 4: Construct the sparse signal complete dictionary A _J according to the parameterization J of the steering vector; Step 5: Construct a convex programming function for sparse signal reconstruction based on the l1 norm; Step 6: Solve the convex programming function and perform spectrum peak search. 3. The frequency-controlled array-based mutual coupling array DOA estimation method according to claim 1, further comprising: Step 1: There is a uniformly distributed linear array ULA composed of M isotropic antennas. N far-field narrowband signals s _k (t) are incident on the array at angles θ ₁ ,θ ₂ ,…,θ _N, respectively, where n = 1, 2,…, N, and the difference between the transmit frequencies of the array elements is expressed as △f, that is, the transmit frequency of the nth array element is expressed as f _n = (n-1)*△f+f ₀ , where f ₀ is the initial reference array element transmit frequency; When the mutual coupling effect between array elements is not considered, the received signal matrix is:

X(t)＝[x ₁ (t),x ₂ (t),…,x _M (t)] ^T represents the output signal matrix of M array elements in this snapshot sampling; s(t) represents the signal matrix composed of the original N narrowband signals, and t represents a snapshot; A is the array manifold matrix composed of the steering vector a(θ),

in

That is, the angle _θi of the i-th signal reaching the n-th array element is the phase difference with the first one as the reference phase, where d represents the distance between two adjacent array elements of the uniform array,

c is the speed of light,

is the noise signal matrix, where each column represents the noise signal received by each array element. Considering the mutual coupling factor, a single snapshot is a single sampling received signal:

Each column of the matrix represents the coupling coefficient between the array element and the array element at other positions. 1 represents the array element being compared. The entire matrix C represents the coupling coefficient matrix composed of the coupling coefficients between the array elements. The received signal model under L snapshots is:

S represents the matrix composed of narrowband signals under multiple snapshots,

Represents the matrix composed of noise signals under multiple snapshots; Step 2: Calculate the covariance matrix of the received signal:

Where L is the number of snapshots, X(t) is the received signal matrix, where the transmission frequency of the nth signal is f _n =f ₀ +(n-1)Δf, and the total steering vector is divided into two matrices according to the distance-related parameters and the angle-related parameters, expressed as:

The symbol ⊙ represents the multiplication operation of corresponding positions, a _Θ is a matrix composed of angle-related items, a _r represents a matrix composed of distance-related items, θ represents the incident angle when the signal reaches the array element. Because the signal is a far-field signal, the original θ ₁ ,θ ₂ ,…,θ _N can be regarded as the same incident angle; r represents the distance from the signal to the array element; Step 3: Perform singular value decomposition on the covariance matrix R: R=UΛV, where U and V are M×M and N×N unitary matrices respectively (their sizes depend on the number of array elements M and the number of signals N), Λ is an M×N singular value diagonal matrix, and the corresponding signal and noise parts are also singular value decomposed: U=[U _S U _N ], U _s represents the U matrix of the signal, and U _N represents the U matrix of the noise; similarly, V=[V _S V _N ] ^T , V _S represents the V matrix of the signal, V _N represents the V matrix of the noise, T represents the transposition of the matrix, and Λ=diag[Λ _S Λ _N ]; Step 4: For the mutual coupling of matrices, construct a selection matrix D _K = [I _K 0], where I _K is a unit matrix of size K × K, K is the number of pre-selected signal sources, and calculate the signal subspace _RS = RVD _K ; Step 5: Based on the parameterization of the steering vector,

Construct a sparse complete dictionary A _J = [J(θ ₁ ), J(θ ₂ ), …, J(θ _N )]; Step 6: Use the l1 norm to constrain the spatial sparse features of the signal. The constraints are the time domain sparseness of the l2 norm and the suppression of noise, that is, construct a convex programming function

Step 7: Use l1-SVD theory to automatically select the regularization parameter ξ with a 99% confidence interval suppression. According to the orthogonality between the signal subspace and the noise subspace, solve the angle θ corresponding to the noise subspace U _n = 0. Use the convex optimization toolkit CVX to estimate the sparse signal space spectrum, and finally perform a one-dimensional spectrum peak search. Step 8: Repeat steps 5 and 6 to improve the estimation accuracy.

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