A fast high-precision array mutual coupling self-correction method based on central conjugate zeroization

Abstract

The application relates to a fast high-precision array mutual coupling self-correction method based on central conjugate zeroization, which is used for improving the mutual coupling correction precision of an array and improving the execution speed of an algorithm. The method of the application firstly performs array arrangement and signal modeling when mutual coupling exists, then performs data dimension compression based on singular value decomposition, establishes an optimization problem based on central conjugate zeroization, solves the optimization problem based on a real SQP optimization method, and finally estimates the wave arrival direction and the array mutual coupling. The method of the application can improve the mutual coupling correction precision by endowing the zeroization filter with the central conjugate property, so that the zero points of the filter are located on the unit circle, the established model is consistent with the array signal processing model, and the execution speed of the algorithm can be improved without secondary iteration calculation.

Description

TECHNICAL FIELD

[0001] The present application relates to a fast and high-precision array mutual coupling self-correction method based on central conjugate zeroization, in particular to array signal processing in the presence of array element mutual coupling, and belongs to the technical field of electronic information, and particularly to the technical field of array signal processing. BACKGROUND

[0002] One branch of array signal processing technology is to estimate the direction of arrival (DOA) of signals using sensor arrays, which is widely used in military and civilian fields such as radar, communication, sonar and medical diagnosis. The DOA estimation method based on parameter modeling is more popular due to its high resolution. However, the performance of this kind of method is easily affected by the array's own errors, such as element amplitude and phase errors, element position errors and mutual coupling. Array mutual coupling is derived from the propagation characteristics of waves, and is generally difficult to avoid. Therefore, the mutual coupling correction problem of the array has attracted widespread attention.

[0003] Array mutual coupling correction techniques are divided into offline correction and online self-correction. Offline correction is generally carried out in a darkroom. By measuring the array response in multiple directions in a single-path environment offline, the mutual coupling correction coefficient is calculated, so that the mutual coupling coefficient can be compensated when the array is actually applied to achieve mutual coupling correction. This method has high reliability, but also has high equipment cost and time cost. Online self-correction, also known as self-correction, is to estimate the array mutual coupling coefficient simultaneously when the array performs DOA estimation. This method has lower cost and can track the changes of array errors caused by external environment. Therefore, this kind of method has been widely studied in recent years, and many new self-correction algorithms have emerged.

[0004] Document 1 (Pan YJ, et al., A method for joint estimation of DOA and mutual coupling based on annihilation constraint [P], 202111172456.2, 2021.12.03) proposes a mutual coupling correction method based on annihilation constraint. This method does not require prior information of mutual coupling degrees of freedom, and has low computational complexity. However, since the zero point of the filter is not limited to be located on the unit circle, there is a deviation between the model built and the actual model, which makes the estimation accuracy of the mutual coupling coefficient low. In order to improve the estimation accuracy of the mutual coupling, document 2 (Y. Pan, et al., Joint estimation of DOA and mutual coupling via imposing an annihilation constraint [J], Digital Signal Processing, 130, 103726, 2022) proposes that after zero filtering, a post-processing method is used to iteratively estimate the mutual coupling coefficient twice. This method improves the estimation accuracy of the mutual coupling coefficient to some extent. However, since this method requires two iteration processes, it will accumulate errors on the one hand, and increase the computational complexity on the other hand, prolonging the algorithm processing time. In addition, document 2 also does not limit the zero point of the filter to be located on the unit circle, so the parameter estimation accuracy still has room for improvement. SUMMARY

[0005] In order to overcome the shortcomings of the existing research, the present application provides a fast and high-precision array mutual coupling self-correction method based on central conjugate annihilation.

[0006] The specific steps of a fast and high-precision array mutual coupling self-correction method based on central conjugate annihilation are as follows:

[0007] Step 1: Array arrangement and signal modeling when mutual coupling exists:

[0008] A uniform linear array is arranged, the number of array elements is M, the distance between adjacent array elements is d; K wavelengths are λ, and N narrowband signals from θ1, θ2, …, θK directions are incident on the uniform linear array, and a total of N shots are received. The multi-shot receiving data modeling is Y = CAS + E, Y = [y[1], y[2], …, y[N]], S = [s[1], s[2], …, s[N]], E = [ε[1], ε[2], …, ε[N]], C is the mutual coupling matrix, C = Toeplitz([1, c1, c2, …, cP-1, 0, …, 0]), Toeplitz(·) represents generating a symmetric Toeplitz matrix, c1, c2, …, cP-1 are mutual coupling coefficients, and P represents the mutual coupling degrees of freedom. K P P

[0009] ​​​where the received data of the array at the nth snapshot is modeled as: where, represents a complex set, the streamer matrix A = [a(θ1), a(θ2), …, a(θ K )], a(θ1), a(θ2), …, a(θ K ) represents a steering vector, 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,

[0010] u k = 2πd cos(θ k ) / λ; s[n] is a signal vector, and ε[n] is a noise vector.

[0011] Step two: data dimension compression based on singular value decomposition:

[0012] Y is singular value decomposed, and Y is obtained after dimension compression V s represents a matrix composed of the right singular vectors corresponding to the K largest singular values of Y.

[0013] Step three: optimization problem establishment based on central conjugate nulling:

[0014] The optimization problem is expressed as: where ||·||2 represents the 2-norm, the vectorization of Z is z = vec(Z), vec(·) represents the vectorization of stacking the matrix by column, I K represents a K-order unit matrix, represents a Kronecker product, the mutual coupling matrix C' = Toeplitz([1, c T , 0, …, 0]), the mutual coupling vector c = [c1, c2, …, c P′ ] T , and P' represents an estimated value of the mutual coupling degree of freedom. The auxiliary parameter η k is a vector composed of the M(k-1)th to Mth elements of γ, k = 1, 2, …, K, and the to-be-optimized vector υ = [c T , γ T , h T ] T ; the multi-snapshot matrix T - (γ) = [T T (η1), T T (η2), …, TT (η K )] T , To construct the operator for the Toeplitz matrix, T(η) k The element in the i-th row and j-th column of ) is η n The (i-j+K+1)th element, i = 1, 2, ..., (MK), j = 1, 2, ..., (K+1), is the nulling filter. ω is a constant vector; st represents the constraint condition, (·) H 、(·) T Let represent taking the conjugate transpose and transpose respectively, where Im(·) takes the imaginary part of the corresponding variable, and G is the transformation matrix. Im(G H h)=0 indicates that the null filter h has the property of central conjugate.

[0015] Step 4: Solving the optimization problem based on the real number SQP optimization method:

[0016] First initialize Where c0 = 0 P′×1 0 P′×1 Let denot be a P′-dimensional zero vector. The initial values ​​of γ0 and h0 are obtained by assuming the arrays are uncoupled and applying a TLS-ESPTRIT-like method to Z. Then, iterative calculations are performed, and υ is obtained in the (q+1)th iteration. q+1 =υ q +Δυ, q≥0, where Δυ is the result obtained from the qth iteration, and Δυ is the search vector. Among them I P′ Let P' be the identity matrix, and so on; Δυ' is a real vector, obtained by solving the following real linear quadratic optimization problem:

[0017] Where, vector f Re(·) takes the real part of the corresponding variable, and the mutual coupling matrix is ​​obtained in the q-th iteration. vector b D matrix

[0018] Among them 0 2MN×(K+1) This represents a zero matrix with 2MN rows and K+1 columns, and so on; matrix B in, To construct another operator for the Toeplitz matrix, R(h q The i-th line of ) J K+1 It is a K+1 order anti-diagonal identity matrix; a multi-fast-shot Q matrix. Right now For γ qThe vector consisting of the M(k-1) to Mkth elements,

[0019] [·] ·,1:P′ This represents taking the first P' columns of a matrix to form a new matrix, known as the Toeplitz matrix. Hankel matrix for The first element, and so on for the rest.

[0020] Step 5: Direction of Arrival and Mutual Coupling Estimation:

[0021] After iterative convergence, a convergent solution is obtained. These are estimates of the mutually coupled vectors. This is an estimate of γ. This is an estimate of h. Will Substitute We obtain a polynomial, and solve for its K roots to get x1, x2, ..., x. K Then the DOA estimate is k = 1, 2, ..., K; where ∠(·) takes the principal argument of the complex number; according to The estimated mutual coupling matrix is ​​obtained.

[0022] Preferably, the value of ω in step three is... Where h0 is the initial value of h in the iterative calculation of step four. It is the complex conjugate of h0.

[0023] Preferably, P′ in step three can be equal to or less than P in step one.

[0024] Preferably, the transformation matrix G in step three is set as follows: when K is odd, let n = (K+1) / 2, then... Among them, I n J represents an n-order identity matrix. n Let K be an n-order anti-diagonal identity matrix; when K is even, let n = (K-1) / 2, then...

[0025] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0026] This invention imparts the property of central conjugate to the null filter, thereby placing the zero point of the filter on the unit circle, making the established model consistent with the array signal processing model, which can improve the accuracy of mutual coupling correction; secondly, the method of this invention does not require secondary iteration, thus improving the algorithm execution speed. Attached Figure Description

[0027] The technical solutions in the embodiments of the present application or the prior art will be described below more clearly with the help of the accompanying drawings needed in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description only represent some embodiments of the present application, and all other embodiments obtained by a person of ordinary skill in the art without any creative effort based on these drawings also belong to the protection scope of the present application.

[0028] Figure 1 Flow chart of the fast high-precision array mutual coupling self-correction method based on center conjugate zeroization of the present application;

[0029] Figure 2 Comparison diagram of the present application method and other methods in different signal-to-noise ratios for direction of arrival estimation;

[0030] Figure 3 Comparison diagram of the present application method and other methods in different signal-to-noise ratios for mutual coupling estimation performance;

[0031] Figure 4 Comparison diagram of the present application method and other methods for algorithm execution time;

[0032] Figure 5 Comparison diagram of the present application method and other methods for filter zero point position. DETAILED DESCRIPTION

[0033] The technical solutions in the embodiments of the present application will be described clearly and completely below with the help of the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments only represent some embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without any creative effort also belong to the protection scope of the present application.

[0034] As shown in Figure 1 a fast high-precision array mutual coupling self-correction method based on center conjugate zeroization, the specific steps are as follows:

[0035] Step 1: Array arrangement and signal modeling when mutual coupling exists:

[0036] A uniform linear array is arranged, the number of array elements is M, the distance between adjacent array elements is d; K narrowband signals with a wavelength of λ are incident on the uniform linear array from θ1, θ2, …, θK directions, and a total of N shots are received. The multi-shot receiving data modeling is Y = CAS + E, Y = [y[1], y[2], …, y[N]], S = [s[1], s[2], …, s[N]], E = [ε[1], ε[2], …, ε[N]], C is the mutual coupling matrix, C = Toeplitz([1, c1, c2, …, cM-1]), A is the array steering matrix, A = [a[1], a[2], …, a[M]], a[m] = [1, e-j2πd(sin(θ1) / λ)m, …, e-j2πd(sin(θK) / λ)(M-1)m]T, m = 1, 2, …, M, S is the signal matrix, E is the noise matrix, and Y is the received data matrix. K A uniform linear array is arranged, the number of array elements is M, the distance between adjacent array elements is d; K narrowband signals with a wavelength of λ are incident on the uniform linear array from θ1, θ2, …, θK directions, and a total of N shots are received. The multi-shot receiving data modeling is Y = CAS + E, Y = [y[1], y[2], …, y[N]], S = [s[1], s[2], …, s[N]], E = [ε[1], ε[2], …, ε[N]], C is the mutual coupling matrix, C = Toeplitz([1, c1, c2, …, cM-1]), A is the array steering matrix, A = [a[1], a[2], …, a[M]], a[m] = [1, e-j2πd(sin(θ1) / λ)m, …, e-j2πd(sin(θK) / λ)(M-1)m]T, m = 1, 2, …, M, S is the signal matrix, E is the noise matrix, and Y is the received data matrix.P ,0,…,0]), Toeplitz(·) represents generating a symmetric Toeplitz matrix, c1, c2, …, c P are mutual coupling coefficients, and P represents mutual coupling degrees of freedom;

[0037] wherein the received data of the array at the nth snapshot is modeled as: wherein, represents a complex number set, a stream matrix A = [a(θ1), a(θ2), …, a(θ K )], a(θ1), a(θ2), …, a(θ K ) represent a 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,

[0038] u k = 2πd cos(θ k ) / λ; s[n] is a signal vector, and ε[n] is a noise vector, and the noise is set as zero-mean Gaussian white noise.

[0039] Specifically, the mutual coupling matrix C can be written as

[0040]

[0041] Step two: data dimension compression based on singular value decomposition:

[0042] singular value decomposition is performed on Y, and Y is obtained after dimension compression V s represents a matrix composed of right singular vectors corresponding to K largest singular values of Y.

[0043] Step three: optimization problem establishment based on central conjugate nullification:

[0044] The optimization problem is represented as: wherein ||·||2 represents a 2-norm, z = vec(Z) of vectorization of Z, vec(·) represents vectorization of stacking matrices by columns, I K represents a K-order unit matrix, represents a Kronecker product, a mutual coupling matrix C′ = Toeplitz([1, c T , 0, …, 0]), and a mutual coupling vector c = [c1, c2, …, c P′ ] TP' represents the estimated value of the mutual coupling degree of freedom, P' can be equal to or less than P, that is, no accurate prior information of the mutual coupling degree of freedom is required; the auxiliary parameter η k is a vector composed of the M(k-1)th to the Mth elements of γ, k = 1, 2,..., K, the to-be-optimized vector υ = [c T , γ T , h T ] T ; the multi-shot matrix T - (γ) = [T T (η1), T T (η2),..., T T (η K )] T , is an operator for constructing a Toeplitz matrix, the i-th row and the j-th column element of T(η k ) is the i-j+K+1th element of η n , i = 1, 2,..., (M-K), j = 1, 2,..., (K+1), the zeroing filter ω is a constant vector, and the value is where h0 is the initial value of h in the next iteration calculation, is the complex conjugate of h0; s.t. represents a constraint condition, (·) H , (·) T respectively represent taking the conjugate transpose and the transpose, Im(·) takes the imaginary part of the corresponding variable, G is a transformation matrix. Im(G H h) = 0 represents that the zeroing filter h has the property of central conjugate. The so-called vector has the property of central conjugate is that the elements of the vector are conjugate symmetric relative to the center point. That is, for h, there is h = J K+1 h * . The central conjugate symmetry of the filter coefficients is a necessary condition to ensure that the filter zero point is located on the unit circle, so that the built model is consistent with the real array signal processing model. The setting method of the transformation matrix G is as follows: when K is odd, let n = (K+1) / 2, and have where I n represents an n-order unit matrix, and J n represents an n-order anti-diagonal unit matrix; when K is even, let n = (K-1) / 2, and have

[0045] Step four: solving the optimization problem based on the real SQP optimization method:

[0046] First, initialize where c0 = 0 P′×1 , 0 P′×1denotes a P'-dimensional zero vector, the initial value of γ, γ0, and the initial value of h, h0, are obtained by assuming that the array is uncoupled and performing a TLS-ESPRIT-like method on Z; then iterative calculation is performed, and in the q+1th iteration, we obtain q+1 = υ q + Δυ, q ≥ 0, where is the result obtained in the qth iteration calculation, and Δυ is the search vector. where I P′ denotes a P'-order identity matrix, and the rest is similar; Δυ' is a real vector, which is obtained by solving the following real linear quadratic optimization problem:

[0047] where f is a vector Re(·) takes the real part of the corresponding variable, the mutual coupling matrix b is a vector D is a matrix

[0048] where 0 2MN×(K+1) denotes a zero matrix of 2MN rows and K+1 columns, and the rest is similar; B is a matrix where, is another operator for constructing Toeplitz matrix, the i-th row of R(h q ) is J K+1 is a K+1-order anti-diagonal identity matrix; the multi-shot Q matrix i.e. is the vector composed of the M(k-1)th to the Mkth elements of γ q ,

[0049] [·] ·,1:P′ denotes taking the first P' columns of the matrix to form a new matrix, the Toeplitz matrix Hankel matrix is the 1st element of h , and the rest is similar.

[0050] The so-called real SQP optimization method refers to converting the nonlinear complex optimization problem into real iterative solution, and only one real linear quadratic optimization problem is solved each time. The corresponding real linear quadratic optimization problem can be solved by the Lagrange multiplier method.

[0051] Step five; DOA and mutual coupling estimation:

[0052] After the iteration converges, the converged solution is the estimated value of the mutual coupling vector, is the estimated value of γ, an estimate of h, K

[0053] k = 1, 2, …, K; where ∠(·) takes the principal value of the amplitude of a complex number; according to

[0054] The effect of the method of the present application is verified below in combination with a simulation example.

[0055] Simulation Example 1: Set the number of uniform linear array elements M = 8, d = λ / 2, element mutual coupling degree P = 2, mutual coupling parameters c1 = 0.5 + 0.4j, c2 = 0.1 - 0.03j, and P' = P. Assume K = 2 incident signals, DOA θ1 = 89° + Δθ1 and θ2 = 105° + Δθ2, and Δθ1 and Δθ2 are uniformly distributed within [-0.5°, 0.5°]. Set the number of snapshots to 50, and the signal-to-noise ratio is scanned from -10 dB to 10 dB. Compare the estimation performance of the method of the present application and the method of the background art document 2 on the direction of arrival and mutual coupling coefficient. In order to provide a performance reference, the results of the Cramer-Rao lower bound (CRB) are also provided. The CRB can be derived by 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 shown in Figure 2 and Figure 3 From the figures, it can be seen that for the estimation of the direction of arrival and mutual coupling, the estimation accuracy of the method of the present application is higher than that of the comparative algorithm at a low signal-to-noise ratio.

[0056] Simulation Example 2: Set the signal-to-noise ratio to 15 dB, and the rest of the simulation conditions are the same as before. Compare the execution time difference of each method algorithm. The simulation results are shown in Figure 4 From the figures, it can be seen that the algorithm execution speed of the method of the present application is faster than that of the comparative method. The single execution time is less than 1.2 milliseconds.

[0057] ​​​​​Simulation example 3: the signal-to-noise ratio is set to -5dB, and the rest of the simulation conditions are the same as above. The zero point positions of the filters of various methods are compared. The simulation results are shown in Fig. 3. Figure 5 As can be seen from the figure, only the filter zero point in the method of the present application is located on the unit circle, thus more conforming to the real array signal processing model, and higher parameter estimation accuracy can be obtained.

[0058] The embodiments of the present application are described in detail above with reference to the drawings, but the present application is not limited to the described embodiments. Various changes, modifications, replacements and variations of the embodiments can be made by those skilled in the art without departing from the principles and spirits of the present application, and still fall within the protection scope of the present application.

Claims

1. A fast and high-precision array mutual coupling self-calibration method based on central conjugate nullification, characterized in that: Includes the following steps: Step 1: Array layout and signal modeling with mutual coupling present: A uniform linear array is deployed with M elements and d spacing between adjacent elements; K wavelengths, each λ, originate from θ1, θ2, ..., θ K A narrowband signal in a specific direction is incident on a uniform linear array, and a total of N snapshots are received. The data received from multiple snapshots is modeled as: Y = CAS + E, 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 a mutually coupled matrix, C = Toeplitz([1,c1,c2,…,c...). P The Toeplitz function (·) generates a symmetric Toeplitz matrix, c1, c2, ..., c3. P Here, P represents the mutual coupling coefficient, and P represents the mutual coupling degree of freedom. The received data from the array during the nth snapshot is modeled as follows: n = 1, 2, ..., N; where, Let A represent the set of complex numbers, and let A = [a(θ1), a(θ2), ..., a(θ)] as the manifold matrix. K )],a(θ1),a(θ2),…,a(θ K ) represents the steering vector, and the steering vector of the k-th narrowband signal is a(θ). k The 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 =2πdcos(θ) k ) / λ; s[n] is the signal vector, ε[n] is the noise vector; Step 2: Perform singular value decomposition on Y, then Y is obtained after dimensionality compression. V s This represents the matrix consisting of the right singular vectors corresponding to the K largest singular values ​​of Y. Step 3: Establishing the optimization problem based on central conjugate nullification: The optimization problem is represented as: Where ||·||2 represents the 2-norm, the vectorization of Z is z = vec(Z), vec(·) represents vectorization by stacking the matrices column by column, and I K Describes a K-order identity matrix. Represents the Kronecker product, and the mutual coupling matrix C′ = Toeplitz([1,c)). T The mutual coupling vectors are c = [c1, c2, ..., c0], [c1, c2, ..., c0]). P′ ] T P′ represents the estimated value of the mutually coupled degrees of freedom, and the auxiliary parameter. η k Let γ be a vector consisting of the M(k-1)th to Mkth elements, k = 1, 2, ..., K, and let υ = [c T ,γ T ,h T ] T Multi-shot matrix To construct the operator for the Toeplitz matrix, T(η) k The element in the i-th row and j-th column of ) is η n The (i-j+K+1)th element, i = 1, 2, ..., (MK), j = 1, 2, ..., (K+1), is the nulling filter. ω is a constant vector; st represents the constraint condition, (·) H 、(·) T Represents taking the conjugate transpose and transpose respectively, Im(·) takes the imaginary part of the corresponding variable, and G is the transformation matrix; Im(G H h)=0 indicates that the null filter h has the property of central conjugate; Step 4: Solving the optimization problem based on the real number SQP optimization method: First initialize Where c0 = 0 P′×1 0 P′×1 Let denot be a P′-dimensional zero vector. The initial values ​​of γ0 and h0 are obtained by assuming the arrays are uncoupled and applying a TLS-ESPTRIT-like method to Z. Then, iterative calculations are performed, and υ is obtained in the (q+1)th iteration. q+1 =υ q +Δυ, q≥0, where The result is obtained from the qth iteration, and Δυ is the search vector; Among them I P′ Let P' be the identity matrix, and so on; Δυ' is a real vector, obtained by solving the following real linear quadratic optimization problem: Where, vector f Re(·) takes the real part of the corresponding variable, and the mutual coupling matrix is ​​obtained in the q-th iteration. vector b D matrix Among them 0 2MN×(K+1) This represents a zero matrix with 2MN rows and K+1 columns, and so on; matrix B in, To construct another operator for the Toeplitz matrix, R(h q The i-th line of ) J K+1 It is a K+1 order anti-diagonal identity matrix; a multi-fast-shot Q matrix. Right now For γ q The vector consisting of the M(k-1) to Mkth elements, [·] ·,1:P′ This represents taking the first P' columns of a matrix to form a new matrix, known as the Toeplitz matrix. Hankel matrix for The first element, and so on for the rest; Step 5: Direction of arrival and mutual coupling estimation: After iterative convergence, a convergent solution is obtained. These are estimates of the mutually coupled vectors. This is an estimate of γ. This is an estimate of h. Will Substitute We obtain a polynomial, and solve for its K roots to get x1, x2, ..., x. K Then the DOA estimate is k = 1, 2, ..., K; where ∠(·) takes the principal argument of the complex number; according to The estimated mutual coupling matrix is ​​obtained.

2. The fast, high-precision array mutual coupling self-calibration method based on center conjugate nullification according to claim 1, characterized in that: In step three, ω takes the value of Where h0 is the initial value of h in the iterative calculation of step four. It is the complex conjugate of h0.

3. The fast, high-precision array mutual coupling self-calibration method based on center conjugate nullification according to claim 1, characterized in that: In step three, P′ can be equal to or less than P in step one.

4. The fast, high-precision array mutual coupling self-calibration method based on center conjugate nullification according to claim 1, characterized in that: In step three, the transformation matrix G is set as follows: When K is odd, let n = (K+1) / 2, then... Among them, I n J represents an n-order identity matrix. n Let K be an n-order anti-diagonal identity matrix; when K is even, let n = (K-1) / 2, then...

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