A Joint Estimation Method for DOA and Mutual Coupling Based on Null Constraint

Abstract

The present invention discloses a joint estimation method for DOA and mutual coupling based on nulling constraints, specifically used to improve and enhance the DOA estimation performance in the presence of array element mutual coupling and simultaneously provide an estimation of the mutual coupling. Existing methods require prior information on the mutual coupling degree, and when the estimated mutual coupling degree is greater than the true mutual coupling degree, the performance will decline. The method of the present invention first conducts array layout and modeling of the received signals in the presence of mutual coupling, then performs dimension compression based on singular value decomposition, establishes an optimization problem based on nulling constraints, and uses an iterative method to solve the optimization problem of the nulling constraints, and finally performs DOA and mutual coupling estimation. The method of the present invention can utilize all array data, without aperture loss; it can dispense with prior information on the mutual coupling degree, and even if the mutual coupling degree is overestimated, it will not have a significant impact on the performance. The method of the present invention improves the DOA and mutual coupling estimation accuracy.

Description

Technical Field

[0001] The present invention belongs to the field of information technology, especially the field of array signal processing, and particularly relates to array signal processing in the case of element mutual coupling, and specifically relates to a joint estimation method of DOA and mutual coupling based on nulling constraint. Background Art

[0002] DOA (Direction of Arrival positioning technology) is an industry term in research fields such as electronics, communication, radar, and sonar. By processing the received echo signals, the distance information and azimuth information of the target can be obtained. DOA estimation belongs to the array signal processing technology, which estimates the direction of arrival of signals using a sensor array and is widely applied in military and civilian technology fields such as radar, communication, sonar, and medical diagnosis. The DOA estimation method based on parametric modeling is more favored due to its high resolution. However, the performance of such methods is easily affected by the imperfections of the array, such as element amplitude-phase errors and element mutual coupling. Element mutual coupling results from the propagation characteristics of waves and is difficult to avoid. Therefore, research on how to correct mutual coupling has received extensive attention.

[0003] To correct mutual coupling, the public document 1 (Z. Ye and C. Liu, “On the resiliency of MUSIC direction finding against antenna sensor coupling,” IEEE transactions on antennas and propagation, vol. 56, no. 2, pp. 371–380, 2008) removes the elements at both ends of the array to equivalently eliminate the influence of mutual coupling and only uses the middle sub-array to estimate DOA. The disadvantage of this method is that it does not make full use of the array and there is aperture loss. The public document 2 (F. Sellone and A. Serra, “A novel online mutual coupling compensation algorithm for uniform and linear arrays,” IEEE Transactions on signal processing, vol. 55, no. 2, pp. 560–573, 2007) uses an alternating minimization of a cost function to correct mutual coupling by iterative calculation. The disadvantage of this method is that there are easily false peaks in the spatial spectrum after correction, which affects the DOA estimation performance. In addition, the methods in the public document 1 and the public document 2 both require prior information of the mutual coupling degree. When the estimated mutual coupling degree is greater than the true mutual coupling degree, the performance of both methods will decline. Summary of the Invention

[0004] The object of the present invention is to propose a joint estimation method of DOA and mutual coupling based on null constraint in view of the deficiencies of the above-mentioned existing technologies.

[0005] The method of the present invention includes the following steps:

[0006] Step (1) Modeling of received signals in the presence of array layout and mutual coupling:

[0007] Arrange a uniform linear array with the number of array elements being M and the spacing between adjacent array elements being d; K narrowband signals with a wavelength of λ, coming from θ1, θ2, …, θ K incident on the uniform linear array, and a total of N snapshots are received. The received data of multiple snapshots is modeled as: Y = CAS + E, where Y = [y[1], y[2], …, y[N]], the signal matrix S = [s[1], s[2], …, s[N]], the noise matrix E = [ε[1], ε[2], …, ε[N]], C is the mutual coupling matrix, C = Toeplitz([1, c1, c2, …, c P , 0, …, 0]), Toeplitz(·) represents generating a Toeplitz matrix, and c1, c2, …, c P are mutual coupling parameters, and P represents the degree of mutual coupling;

[0008] Among them, the received data of the array at the nth snapshot is modeled as: n = 1, 2, …, N; where, represents the set of complex numbers, the manifold matrix A = [a(θ1), a(θ2), …, a(θ K ), a(θ1), a(θ2), …, a(θ K ) represents the steering vector, and the mth element of the kth narrowband signal steering vector a(θ k ) is a m (θ k ) = exp[j(m - 1)u k , k = 1, 2, …, K, m = 1, 2, …, M, u k = 2πd cos(θ k ) / λ; s[n] is the signal vector, and ε[n] is the noise vector.

[0009] Step (2) Dimension compression based on singular value decomposition:

[0010] Perform singular value decomposition on Y, and then after dimension compression of Y, V s represents the matrix composed of the right singular vectors corresponding to the K largest singular values of Y.

[0011] Step (3) Establish an optimization problem based on null constraint:

[0012] The optimization problem is expressed as: where, ||·||2 represents the 2-norm, the vectorization of Z is z = vec(Z), and vec(·) represents vectorizing a matrix by stacking its columns, and I K represents the K-order identity matrix, represents the Kronecker product, the mutual coupling matrix C′ = Toeplitz([1, c T , 0,..., 0]), the mutual coupling vector c = [c1, c2,..., c P′ T , P′ represents the estimated value of the mutual coupling degree, and the auxiliary parameter η k is the vector composed of the (M(k - 1))-th to the Mk-th elements of γ, k = 1, 2,..., K, and the vector to be optimized υ = [c T , γ T , h T T ; the multi-block snapshot matrix is the operator for constructing the Toeplitz matrix, the element in the i-th row and j-th column of n is the (i - j + K + 1)-th element of η ω is a constant vector; s.t. represents the constraint condition, (·) H , (·) T respectively represent taking the conjugate transpose and the transpose.

[0013] In step (4), an iterative method is used to solve the optimization problem with the nulling constraint:

[0014] First, initialize where c0 = 0 P′ , 0 P′ represents the zero vector of P′×1, and the initial value γ0 of γ and the initial value h0 of h are obtained by assuming no mutual coupling in the array and using the TLS-ESPTRIT-like method for Z; then, iterative calculations are performed. In the (q + 1)-th iteration process, υ q+1 = υ q + Δυ, q ≥ 0, where is the result obtained from the q-th iterative calculation, and Δυ is the search vector, which is obtained by solving the following linear quadratic optimization problem:

[0015] where, the f vector the mutual coupling matrix obtained from the q-th iteration the g vector the Jacobian matrix of the f vector ​​g - vector Jacobian matrix Multi - block snapshot Q matrix That is is the vector composed of the \(M(k - 1)\) - th to the \(Mk\) - th elements of \(\gamma\) q [·] .,1:P′ denotes taking the first \(P'\) columns of the matrix to form a new matrix, Toeplitz matrix Hankel matrix is the first element of and so on; is another operator for constructing the Toeplitz matrix the \(i\) - th row of

[0016] Step (5) DOA and mutual coupling estimation:

[0017] After the iteration converges, the convergence solution is the estimated value of the mutual - coupling vector is the estimated value of \(\gamma\) is the estimated value of \(h\) Substitute into to obtain a polynomial, and solve the \(K\) roots of this polynomial to get \(x_1,x_2,\cdots,x\) K , then the DOA estimation is \(k = 1,2,\cdots,K\); where, \(\angle(\cdot)\) takes the principal value of the argument of the complex number; According to the estimated mutual - coupling matrix

[0018] Preferably, the value of \(\omega\) in step (3) is the initial value of \(h\) in the iterative calculation of step (4), that is, \(\omega=h_0\).

[0019] The method of the present invention has the following beneficial effects compared with the prior art:

[0020] First, the method of the present invention can utilize all array data without aperture loss; Second, the method of the present invention does not require prior information of the mutual - coupling degree, and even if the mutual - coupling degree is over - estimated, it has no significant impact on the performance; Third, the DOA and mutual - coupling estimation accuracy of the method of the present invention is high. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 is the general flow chart of the method of the present invention;

[0022] Figure 2 is the schematic diagram of the comparison of the DOA estimation performance between the method of the present invention and other methods under different estimated values of the mutual - coupling degree;​

[0023] Figure 3 Schematic diagram for comparing the DOA estimation performance of the method of the present invention with other methods at different signal-to-noise ratios;

[0024] Figure 4 Schematic diagram for comparing the mutual coupling estimation performance of the method of the present invention with other methods at different signal-to-noise ratios;

[0025] Figure 5 Schematic diagram for comparing the DOA estimation performance of the method of the present invention with other methods at different numbers of snapshots;

[0026] Figure 6 Schematic diagram for comparing the mutual coupling estimation performance of the method of the present invention with other methods at different numbers of snapshots. Detailed implementation manner

[0027] The following further elaborates on the specific technical solutions and effects of the invention with reference to the accompanying drawings.

[0028] As Figure 1 shown, a joint estimation method for DOA and mutual coupling based on nulling constraint, the specific steps are as follows:

[0029] Step (1) Array layout and modeling of received signals in the presence of mutual coupling:

[0030] Deploy a uniform linear array with the number of array elements being M and the spacing between adjacent array elements being d; K narrowband signals with a wavelength of λ incident on the uniform linear array from directions θ1, θ2,..., θ K The received data of N snapshots is modeled as: Y = CAS + E, where Y = [y[1], y[2],..., y[N]], signal matrix S = [s[1], s[2],..., s[N]], noise matrix E = [ε[1], ε[2],..., ε[N]], C is the mutual coupling matrix. For a uniform linear array, C has the symmetric Toeplitz property, and C = Toeplitz([1, c1, c2,..., c P , 0,..., 0]), Toeplitz(·) represents generating a Toeplitz matrix, and c1, c2,..., c P are mutual coupling parameters, and P represents the degree of mutual coupling.

[0031] Among them, the received data of the array at the nth snapshot is modeled as: n = 1, 2,..., N; where, represents the set of complex numbers, the steering matrix A = [a(θ1), a(θ2),..., a(θ K ), a(θ1), a(θ2),..., a(θ K ) represents the steering vector, and the steering vector a(θ kThe m-th element of () is a m (θ k ) = exp[j(m - 1)u k , k = 1, 2, …, K, m = 1, 2, …, M, u k = 2πd cos(θ k ) / λ; s[n] is the signal vector, ε[n] is the noise vector, and the noise is assumed to be zero-mean Gaussian white noise.

[0032] Step (2) performs dimensionality compression based on singular value decomposition:

[0033] Perform singular value decomposition on Y, then after dimensionality compression, Y obtains V s which represents the matrix composed of the right singular vectors corresponding to the K largest singular values of Y.

[0034] Step (3) establishes an optimization problem based on nulling constraints:

[0035] The optimization problem is expressed as: where, ||·||2 represents the 2-norm, the vectorization of Z is z = vec(Z), vec(·) represents vectorizing the matrix by stacking columns, I K represents the K-order identity matrix, represents the Kronecker product, the mutual coupling matrix C′ = Toeplitz([1, c T , 0,..., 0]), the mutual coupling vector c = [c1, c2,..., c P′ T , P′ represents the estimated value of the mutual coupling degree, the auxiliary parameter η k is the vector composed of the elements from the M(k - 1)-th to the Mk-th of γ, k = 1, 2,..., K, the vector to be optimized υ = [c T , γ T , h T T ; the multi-block tap matrix is the operator for constructing the Toeplitz matrix, the element in the i-th row and j-th column of is the (i - j + K + 1)-th element of η n , i = 1, 2, …, (M - K), j = 1, 2, …, (K + 1), the nulling filter ω is a constant vector, taking the initial value h0 of h in the iterative calculation; s.t. represents the constraint condition, (·) H , (·) T represent taking the conjugate transpose and transpose respectively.

[0036] ​​Step (4) uses an iterative method to solve the optimization problem of nulling constraints;

[0037] First, initialize where c0 = 0 P′ , 0 P′ represents a P′×1 zero vector. The initial value γ0 of γ and the initial value h0 of h are obtained by assuming no mutual coupling in the array and using a method similar to TLS-ESPTRIT for Z. Then, iterative calculations are performed. In the (q + 1)-th iteration process, υ q+1 = υ q + Δυ, q ≥ 0, where is the result obtained from the q-th iterative calculation, and Δυ is the search vector, which is obtained by solving the following linear quadratic optimization problem:

[0038] where the f vector the mutual coupling matrix obtained from the q-th iteration the g vector the Jacobian matrix of the f vector the Jacobian matrix of the g vector the multi-block snapshot Q matrix That is is the vector composed of the (M(k - 1))-th to the Mk-th elements of γ q , [·] .,1:P′ means taking the first P′ columns of the matrix to form a new matrix. are Toeplitz matrix and Hankel matrix respectively. is the first element of, and so on. is another operator for constructing the Toeplitz matrix. The i-th row of i = 1, 2, …, (M - K), and Γ is an anti-diagonal identity matrix.

[0039] Step (5) DOA and mutual coupling estimation:

[0040] After the iteration converges, the convergence solution is the estimated value of the mutual coupling vector, is the estimated value of γ, is the estimated value of h, Substitute into to obtain a polynomial, and solve the K roots of this polynomial to get x1, x2,..., x K , then the DOA estimation is k = 1, 2,..., K; where, ∠(·) takes the principal value of the argument of the complex number; According to Obtain the estimated mutual coupling matrix

[0041] The effectiveness of the method of the present invention will be verified below in combination with simulation examples.

[0042] Simulation Example 1: Set the number of elements of the uniform linear array to M = 10, d = λ / 2, the mutual coupling degree P of the elements to 2, and the mutual coupling parameters to c1 = 0.5 + 0.4j, c2 = 0.1 - 0.03j. Assume K = 2 incident signals, and the DOAs are θ1 = 89° + Δθ1, θ2 = 105° + Δθ2, where Δθ1 and Δθ2 are uniformly distributed within [-0.5°, 0.5°]. The signal-to-noise ratio is set to 15 dB, and the number of snapshots is set to 100. Scan the estimated value of the mutual coupling degree from 2 to 9. Compare the performance differences in the root mean square error of DOA estimation between the method of the present invention and the methods in Document 1 and Document 2 in the background art. After averaging the comparison results through 500 Monte Carlo experiments, they are shown in Figure 2 . Since the maximum value that the estimated value of the mutual coupling degree of the method in Document 1 can take is 3, there are no results for the part exceeding 3. It can be seen from Figure 2 that in the case of overestimation of the mutual coupling, the performance of the method of the present invention is not significantly affected, and the root mean square error of its DOA estimation is still less than θ1 = 0.1°. The other two methods are more affected.

[0043] Simulation Example 2: Set the number of snapshots to 50, and scan the signal-to-noise ratio from -10 dB to 15 dB, and the remaining simulation conditions are the same as those described above. Compare the performance differences in the estimation of DOA and mutual coupling among the methods. To provide a performance reference, the results of the Cramer-Rao lower bound (CRB) are also provided. The CRB can be derived from the literature: Z.-M. Liu and Y.-Y. Zhou, “A unified framework and sparse Bayesian perspective for direction-of-arrival estimation in the presence of array imperfections,” IEEE Transactions on Signal Processing, vol. 61, no. 15, pp. 3786–3798, 2013. The simulation results are as shown in Figure 3 and Figure 4 . It can be seen from the figure that for the estimation of DOA and mutual coupling, the root mean square error of the method of the present invention at each signal-to-noise ratio reaches the minimum, and the estimation accuracy is the highest. And when the signal-to-noise ratio is 5 dB, the estimation of DOA by the method of the present invention reaches the CRB.

[0044] Simulation Example 3: Set the signal-to-noise ratio to 5 dB, and scan the number of snapshots from 10 to 100. The remaining simulation conditions are the same as those described above. Compare the differences in the estimation performance of DOA and mutual coupling for each method. The simulation results are as Figure 5 and Figure 6 shown. It can be seen from the figure that for the estimation of DOA and mutual coupling, the root mean square error of the method of the present invention reaches the minimum at each number of snapshots, and the estimation accuracy is the highest. And when the number of snapshots is 40, the estimation of DOA by the method of the present invention reaches the CRB.

[0045] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A joint estimation method for DOA and mutual coupling based on nulling constraint, characterized in that The details are as follows: Step (1) Modeling of received signals when array arrangement and mutual coupling exist: Deploy a uniform linear array with the number of array elements being M and the spacing between adjacent array elements being d; K narrowband signals with a wavelength of λ and coming from directions θ1, θ2, …, θ K are incident on the uniform linear array, and a total of N snapshots are received. The received data of multiple snapshots is modeled as: Y = CAS + E, where Y = [y[1], y[2], …, y[N]], the signal matrix S = [s[1], s[2], …, s[N]], the noise matrix E = [ε[1], ε[2], …, ε[N]], C is the mutual coupling matrix, C = Toeplitz([1, c1, c2, …, c P , 0, …, 0]), Toeplitz(·) represents generating a Toeplitz matrix, and c1, c2, …, c P are the mutual coupling parameters, and P represents the degree of mutual coupling; Among them, the received data of the array at the nth snapshot is modeled as: Among them, represents the set of complex numbers, the manifold matrix A = [a(θ1), a(θ2), …, a(θ K )], a(θ1), a(θ2), …, a(θ K ) represents the steering vector, and the mth element of the steering vector a(θ k ) of the kth narrowband signal is a m (θ k ) = exp[j(m - 1)u k , k = 1, 2, …, K, m = 1, 2, …, M, u k = 2πdcos(θ k ) / λ; s[n] is the signal vector, and ε[n] is the noise vector; Step (2) performs dimension compression based on singular value decomposition: Perform singular value decomposition on Y, then after dimensionality compression of Y, we get V s a matrix formed by the right singular vectors corresponding to the K largest singular values of Y; Step (3) establishes an optimization problem based on zeroing constraints: The optimization problem is expressed as: where, ||·||2 represents the 2-norm, the vectorization of Z is z = vec(Z), and vec(·) represents vectorizing a matrix by stacking its columns, and I K represents the K-order identity matrix, represents the Kronecker product, the mutual coupling matrix C′ = Toeplitz([1, c T , 0,..., 0]), the mutual coupling vector c = [c1, c2,..., c P′ T , P′ represents the estimated value of the mutual coupling degree, and the auxiliary parameter η k is the vector composed of the (M(k - 1))-th to the Mk-th elements of γ, k = 1, 2,..., K, and the vector to be optimized υ = [c T , γ T , h T T ; the multi-snapshot matrix is the operator for constructing the Toeplitz matrix, the element in the i-th row and j-th column of n is the (i - j + K + 1)-th element of η ω is a constant vector; s.t. represents the constraint condition, (·) H and (·) T represent taking the conjugate transpose and transpose respectively;​​ Step (4) uses an iterative method to solve the optimization problem of zeroing constraints: Initialize first where c0 = 0 P′ , 0 P′ represents a zero vector of P′×1, and the initial value γ0 of γ and the initial value h0 of h are obtained by assuming no mutual coupling in the array and using the TLS-ESPTRIT-like method for Z; then iterative calculations are performed, and υ is obtained in the (q + 1)-th iteration process q+1 = υ q + Δυ, q ≥ 0, where is the result obtained from the q-th iterative calculation, and Δυ is the search vector, which is obtained by solving the following linear quadratic optimization problem: Among them, the f vector The mutual coupling matrix obtained in the q-th iteration The g vector The Jacobian matrix of the f vector The Jacobian matrix of the g vector The multi-snapshot Q matrix That is Is γ q The vector formed by the M(k - 1)-th to the Mk-th elements of [·] ·,1:P′ Denotes taking the first P′ columns of the matrix to form a new matrix, the Toeplitz matrix The Hankel matrix Is The first element of , and so on; Is another operator for constructing the Toeplitz matrix The i-th row of is Γ is the anti-diagonal identity matrix; Step (5) DOA and mutual coupling estimation: After iteration convergence, a convergent solution is obtained is the estimated value of the mutual coupling vector, is the estimated value of γ, is the estimated value of h, Substitute into to obtain a polynomial, and solve the K roots of this polynomial to get x1, x2,..., x K , then the DOA estimation is where, ∠(·) takes the principal value of the argument of the complex number; According to the estimated mutual coupling matrix is obtained 2. The joint estimation method of DOA and mutual coupling based on nulling constraint according to claim 1, characterized in that: The value of ω in step (3) is the initial value of h in the iterative calculation of step (4), that is, ω = h0.

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