Co-prime array satellite communication non-circular interference suppression method in non-uniform noise environment

Abstract

The invention discloses a non-circular interference suppression method for co-prime array satellite communication in a non-uniform noise environment. The non-circular interference suppression method comprises the following steps: 1, removing diagonal elements of a non-circular signal covariance matrix of co-prime array satellite communication; 2, constructing a differential common array and interpolating the differential common array into a differential virtual uniform linear array; 3, restoring a non-circular signal vector of the interpolation difference virtual uniform linear array; 4, a summation common array is constructed and interpolated into a summation virtual uniform linear array; 5, restoring interpolation and summing the non-circular signal vector of the virtual uniform linear array; 6, estimating a satellite communication non-circular signal direction; 7, estimating a signal plus interference covariance matrix, a non-uniform noise covariance matrix and a non-circular coefficient of a non-circular signal; 8, estimating satellite communication non-circular signal power; 9, reconstructing a co-prime array interference and noise covariance matrix; and 10, calculating a satellite communication co-prime array amplification weighting vector. And the non-circular interference suppression performance of satellite communication in a non-uniform noise environment is improved.

Description

Technical Field

[0001] The invention relates to the technical field of satellite communication interference suppression, and in particular to a method for suppressing non-circular interference in coprime array satellite communications under a non-uniform noise environment. Background Art

[0002] Modern satellite communication systems can not only provide global coverage of communication services, but also support high-speed data transmission, multimedia transmission, navigation and positioning, and other functions, becoming an indispensable part of human society. With the popularization of satellite communication technology and the expansion of its application scope, satellite communication systems are facing challenges from various interference sources, such as malicious interference, natural interference, spectrum conflicts, etc. These interferences may lead to a decline in communication quality or even communication interruption, seriously affecting the reliability and stability of satellite communication systems. In order to meet these challenges, various satellite communication interference suppression technologies have been proposed, such as spectrum analysis and management, adaptive beamforming technology, interference positioning and elimination technology, etc., to improve the anti-interference ability of satellite communication systems and ensure that they can operate stably and reliably in complex environments. When the satellite communication signal is a non-circular signal, the interference suppression performance of wide linear beamforming is better than that of beamforming. Wide linear beamforming methods include wide linear Capon beamforming method, generalized non-circular robust Capon beamforming method, non-circular coefficient spatial spectrum wide linear beamforming method, projection constrained wide linear beamforming method, iterative quadratic constrained quadratic programming wide linear beamforming method, and augmented covariance matrix reconstruction wide linear beamforming method.

[0003] In real environments, factors such as complex electromagnetic conditions and imperfect receiving channels will lead to more complex noise characteristics, so there is often non-uniform noise in the signals received by the antenna array. When the number of antenna elements is fixed, the array space freedom of the uniform linear array will be limited, while the mutually prime array structure can expand the array aperture and increase the number of array space freedom. All current wide linear beamforming methods assume that the array noise is uniform and the array antenna is a uniform linear array, without considering the case of non-uniform noise or the mutually prime array structure. Therefore, in a non-uniform noise environment and a mutually prime array structure, the non-circular interference suppression performance of all current wide linear beamforming methods will decrease.

[0004] In view of the above analysis, it is very necessary to study the non-circular interference suppression method of coprime array satellite communication in non-uniform noise environment, so as to improve the interference suppression performance of satellite communication in non-uniform noise environment. Summary of the invention

[0005] Aiming at the technical problem that the non-circular interference suppression performance of satellite communication in a non-uniform noise environment is degraded, the present invention provides a method for suppressing non-circular interference of coprime array satellite communication in a non-uniform noise environment.

[0006] In order to solve the above technical problems, the present invention adopts the following technical solution: a method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment, comprising the following steps:

[0007] Step 1: Remove the diagonal elements of the coprime array satellite communication non-circular signal covariance matrix to obtain the coprime array satellite communication non-circular signal covariance matrix after removing the diagonal elements, and proceed to step 2;

[0008] Step 2: Vectorize the covariance matrix of the coprime array satellite communication non-circular signal after removing the diagonal elements, construct a differential common array and interpolate it into a differential virtual uniform linear array, and then go to step 3;

[0009] Step 3: Establish an atomic norm optimization problem about the interpolation difference virtual uniform linear array non-circular signal vector, solve the restored interpolation difference virtual uniform linear array non-circular signal vector, and go to step 4;

[0010] Step 4: vectorize the pseudo covariance matrix of the coprime array satellite communication non-circular signal, construct a summed common array and interpolate it into a summed virtual uniform linear array, and then go to step 5;

[0011] Step 5: Establish an atomic norm optimization problem about the interpolation summation virtual uniform linear array non-circular signal vector, solve the restored interpolation summation virtual uniform linear array non-circular signal vector, and go to step 6;

[0012] Step 6: Establish a spatial spectrum estimator of the interpolation difference virtual uniform linear array to estimate the direction of the non-circular signal, and then proceed to step 7;

[0013] Step 7: Estimate the signal plus interference covariance matrix, the non-uniform noise covariance matrix and the non-circular coefficients of the non-circular signal, and go to step 8;

[0014] Step 8: Establish the least squares optimization problem of the covariance and pseudo-covariance matrices, estimate the non-circular signal power, and proceed to step 9;

[0015] Step 9: Reconstruct the coprime array interference plus noise covariance matrix, estimate the amplified desired satellite signal steering vector, and proceed to step 10;

[0016] Step 10: Calculate the satellite communication coprime array expansion weight vector to suppress non-circular interference.

[0017] In step 1, the diagonal elements of the coprime array satellite communication non-circular signal covariance matrix are removed to obtain the coprime array satellite communication non-circular signal covariance matrix after removing the diagonal elements, which is specifically as follows:

[0018] Suppose a coprime array consists of two subarrays, one of which has N elements and the spacing between adjacent elements is Md, and the other has 2M elements and the spacing between adjacent elements is Nd, d is the unit element spacing, M and N are both coprime integers and M is less than N, the entire coprime array is composed of L = 2M + N-1 elements, where L represents the total number of elements, the first element is the reference element, let represents the set of mutually prime array element positions, z l d represents the position of the lth array element, where l = 1, 2, L, L and z1 = 0; when G satellite non-circular signals are incident on the coprime array, the satellite non-circular signal vector x(i) at the i-th snapshot is expressed as

[0019]

[0020] Among them, a(θ g ) represents the coprime array steering vector of the g-th non-circular signal, s g (i) represents the g-th non-circular signal beam, n(i) represents the coprime array non-uniform noise, the first signal with direction θ1 is regarded as the desired signal, and the other signals are regarded as interference. Let Represents the global interference plus noise vector, the covariance matrix R of the satellite non-circular signal vector x Expressed as

[0021]

[0022] Where E{g} represents the statistical expectation operation, (·) H represents the conjugate transpose operation, p g represents the g-th non-circular signal power, R n Represents the non-uniform noise covariance matrix. In the non-uniform noise background, the noise power of different array elements is different and unknown. At this time, the non-uniform noise covariance matrix R n Expressed as

[0023] R n =diag{p n1 ,p n2 ,L,p nL} (3)

[0024] Where diag{·} represents the diagonalization operation, p nl represents the noise power of the lth array element and l=1,L,L; the pseudo-covariance matrix Q of the satellite non-circular signal vector x Expressed as

[0025]

[0026] Among them, ξ g represents the non-circular coefficient of the g-th non-circular signal;

[0027] Cascade x(i) with its conjugate to obtain the amplified satellite non-circular signal vector for

[0028]

[0029] Among them, s1(i) represents the expected signal, represents the desired signal amplification steering vector, ξ1 represents the desired signal non-circular coefficient, a(θ1) represents the desired signal steering vector, represents the global noise vector, the satellite non-circular signal amplification covariance matrix Expressed as

[0030]

[0031] in,(·) * Represents the conjugate operation.

[0032] In step 2, the covariance matrix of the coprime array satellite communication non-circular signal after removing the diagonal elements is vectorized, and the differential common array is constructed and interpolated into a differential virtual uniform linear array, as follows:

[0033] To process R n The non-uniform diagonal elements in R x Remove the diagonal elements in and get the matrix R after removing the diagonal elements x1 for

[0034]

[0035] Among them, the matrix The diagonal elements of are all 0 and the rest of the elements are all 1. e represents the Hadamard product operation. After removing the diagonal elements, the matrix R x1 Not affected by non-uniform noise.

[0036] R x1 Vectorization is performed to obtain a differential common array virtual signal for

[0037]

[0038] in, represents the Kronecker product operation, vec(g) represents the vectorization operation of converting the matrix into a column vector, and the differential common matrix set Expressed as According to the difference matrix set Rearrange the virtual array element positions in The elements in the , get the rearranged differential common matrix virtual signal In order to make full use of all the information, the collection Therefore, we define for

[0039] Interpolation difference virtual uniform straight line array collection, is a collection that can contain The minimum uniform linear array set of The interpolated differential virtual uniform linear array signal vector is expressed as Elements in Expressed as

[0040]

[0041] in Indicates the virtual array element position v c The virtual signal at d, is composed of and does not belong to the set The set of elements in the interpolated difference virtual uniform linear array signal autocorrelation vector Expressed as

[0042]

[0043] in represents the interpolated differential virtual uniform linear array steering vector of the g-th non-circular signal, The elements in are represented as Indicates the virtual array element position z c Virtual signal at d.

[0044] In step 3, an atomic norm optimization problem about the interpolation difference virtual uniform linear array non-circular signal vector is established to solve the restored interpolation difference virtual uniform linear array non-circular signal vector, as follows:

[0045] Autocorrelation vector of virtual uniform linear array signal from interpolated differences In the expression of

[0046]

[0047] Where span{g} represents the subspace spanned by the vectors in the curly brackets. The linear subspace divided in equation (11) is called the correlation subspace, which is represented by the correlation subspace matrix Construct, for the relevant subspace matrix S d Perform feature decomposition, that is where Λ d1 By S dThe diagonal matrix of the positive eigenvalues ​​in descending order, U d1 By S d The matrix of eigenvectors corresponding to the positive eigenvalues ​​of d2 By S d The remaining eigenvalues ​​of d2 By S d The matrix U is composed of the eigenvectors corresponding to the remaining eigenvalues ​​of d1 The column space of is the relevant subspace, and U d1 The complementary subspaces of the column space are orthogonal, that is

[0048]

[0049] in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Toeplitz structure. In order to recover the interpolated difference virtual uniform linear array signal Construct the following atomic norm minimization problem

[0050]

[0051] in represents the atomic norm, P·P2 represents the l2 norm of the vector, δ represents the threshold, represents a collection of atoms in a continuous domain, Represented by the optimization variable As the first column of the Toeplitz matrix, the constraint

[0052] express

[0053] is a positive semidefinite matrix, and the optimization problem (13) is further expressed as

[0054]

[0055] Where γ represents the regularization parameter and is defined as The atomic norm of for

[0056]

[0057] Using Vandermonde decomposition, the matrix Decompose into matrix The trace of Where Tr(·) represents the trace operation of the matrix, combined with and The expression is derived from

[0058]

[0059] Substituting equation (16) into equation (14), the optimization problem (14) is equivalently transformed into the following form:

[0060]

[0061] in Since the optimization problem (17) is a convex optimization problem, it is solved using CVX software and the solution is expressed as

[0062] In step 4, the pseudo covariance matrix of the coprime array satellite communication non-circular signal is vectorized, and a summed common array is constructed and interpolated into a summed virtual uniform linear array, as follows:

[0063] Since the pseudo-covariance matrix Q x Does not contain non-uniform noise components, Q x Will not be affected by non-uniform noise, directly vectorize Q x Get the summed common array virtual signal for

[0064]

[0065] Among them, the sum of the common matrix set Expressed as According to the sum of the matrix set Rearrangement of virtual array element positions elements, and obtain the rearranged summed common matrix virtual signal Supplementary summation matrix set The missing element in , defined A collection of virtual uniform straight line arrays for interpolation and summation, is a The minimum uniform linear array set of The interpolated and summed virtual uniform linear array signal is expressed as Elements Expressed as

[0066]

[0067] in Is caused by and does not belong to The set of elements of which interpolation and summation of the autocorrelation vector of the virtual uniform linear array signal Expressed as

[0068]

[0069] in, represents the interpolated summed virtual uniform linear array steering vector of the g-th non-circular signal, The elements of

[0070] In step 5, an atomic norm optimization problem about the interpolation summation virtual uniform linear array non-circular signal vector is established to solve the restored interpolation summation virtual uniform linear array non-circular signal vector, as follows:

[0071] Autocorrelation vector of virtual uniform linear array signal from interpolation and summation In the expression (20), we can further deduce

[0072]

[0073] The linear subspace divided in formula (21) is also a related subspace, which is represented by the related subspace matrix Construct the relevant subspace matrix S s The feature decomposition is where Λ s1 By S s The diagonal matrix of the positive eigenvalues ​​in descending order, U s1 By S s The matrix of eigenvectors corresponding to the positive eigenvalues ​​of s2 By S s The remaining eigenvalues ​​of s2 By S s The matrix composed of the eigenvectors corresponding to the remaining eigenvalues ​​of is obviously, and U s1 The complementary subspaces of the column space are orthogonal, that is

[0074]

[0075] in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Hankel structure. In order to recover the interpolated and summed virtual uniform linear array signal, the following Hankel optimization problem is constructed:

[0076]

[0077] where μ′ represents the regularization parameter, represents the Hankel matrix and As the first column of this matrix, the constraint express is a positive semidefinite matrix. Since the optimization problem (23) is a convex optimization problem, CVX software is used to solve it. The solution is expressed as

[0078] In step 6, a spatial spectrum estimator of an interpolated difference virtual uniform linear array is established to estimate the direction of non-circular signals, as follows:

[0079] Since already obtained and Constructing the Toeplitz matrix of interpolation difference virtual uniform linear array and interpolation and summation of virtual uniform linear array Hankel matrix The traditional azimuth estimation technology can be applied to the interpolation difference virtual uniform linear array or the interpolation sum virtual uniform linear array to estimate the direction of non-circular signals, and the interpolation difference virtual uniform linear array spatial spectrum f is constructed using the classic multiple signal classification algorithm. MUSIC (θ) is

[0080]

[0081] Among them, Φ n Representation Matrix The noise subspace of f MUSIC The angles corresponding to the G peaks of (θ) are used to estimate the direction of the non-circular signal. The final estimated direction of the non-circular signal is expressed as L,

[0082] In step 7, the signal plus interference covariance matrix, the non-uniform noise covariance matrix and the non-circular coefficients of the non-circular signal are estimated as follows:

[0083] Estimate the non-uniform noise covariance matrix R n The basic idea is to get the covariance matrix R from x Remove the signal plus interference covariance matrix R si , in order to estimate the signal plus interference covariance matrix R si , establish the following optimization problem

[0084]

[0085] Among them, P·P F represents the Frobenius norm of the matrix, I L represents an L×L dimensional identity matrix, ∈ represents a constant and ∈>0, and the solution of the optimization problem (25) is expressed as The estimated inhomogeneous noise covariance matrix is

[0086]

[0087] in, Represents a diagonal matrix consisting of the main diagonal elements of the matrix within the curly brackets.

[0088] By using the estimated signal direction Its corresponding non-circular coefficient Estimated

[0089]

[0090] in, τ represents the satellite non-circular signal amplification covariance matrix The minimum eigenvalue of .

[0091] In step 8, the least squares optimization problem of the covariance and pseudo-covariance matrices is established to estimate the non-circular signal power, as follows:

[0092] Since the power of non-circular signals is unknown, we can use the matrix R x and Q x Extract the non-circular signal power value and estimate the power diagonal matrix. In order to estimate the power diagonal matrix P, construct the following optimization problem

[0093]

[0094] in, represents the estimated coprime array steering matrix, P = diag{p1,p2,L,p G} represents the power diagonal matrix, 0 L represents an L×L dimensional zero matrix, represents the non-circular coefficient diagonal matrix, the constraint P ≥ 0 means that P is a positive semidefinite matrix, and the solution of the optimization problem (28) is expressed as

[0095]

[0096] in

[0097]

[0098] In step 9, the coprime array interference plus noise covariance matrix is ​​reconstructed to estimate the amplified desired satellite signal steering vector, as follows:

[0099] Reconstructing the covariance matrix of the amplified interference plus noise in coprime array satellite communication

[0100]

[0101] in represents the estimated interference plus noise covariance matrix, represents the estimated pseudo-interference plus noise covariance matrix;

[0102] Using the estimated desired signal direction and the estimated expected signal non-circular coefficient Amplify the desired signal of the coprime array to guide the vector Estimated

[0103]

[0104] In step 10, the satellite communication coprime array expansion weight vector is calculated to suppress non-circular interference, as follows:

[0105] Calculation of the expanded weight vector for wide linear beamforming of coprime array satellite communications for

[0106]

[0107] Compared with the prior art, the present invention has the following beneficial effects:

[0108] (1) The influence of non-uniform noise is eliminated by removing the diagonal elements of the covariance matrix of the coprime array satellite communication non-circular signal and directly using the pseudo-covariance matrix of the coprime array satellite communication non-circular signal.

[0109] (2) By interpolating the difference common array into a difference virtual uniform linear array and interpolating the sum common array into a sum virtual uniform linear array, the spatial freedom of the array is increased.

[0110] (3) The accuracy of reconstructing the interference plus noise covariance matrix of the coprime array is improved by estimating the non-uniform noise covariance matrix, non-circular signal coefficients, non-circular signal direction and non-circular signal power. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0112] Figure 1 It is a schematic diagram of the process of the present invention.

[0113] Figure 2 It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the input signal-to-noise ratio under the condition that the array information is accurately known in an embodiment of the present invention.

[0114] Figure 3 It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the number of sampling snapshots under the condition that the array information is accurately known in an embodiment of the present invention.

[0115] Figure 4It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the input signal-to-noise ratio under the condition of a fixed signal direction error in an embodiment of the present invention.

[0116] Figure 5 It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the number of sampling snapshots under the condition of fixed signal directional error in an embodiment of the present invention.

[0117] Figure 6 It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the input signal-to-noise ratio under the condition of random signal directional error in an embodiment of the present invention.

[0118] Figure 7 It is a schematic diagram of the relationship between the output signal-to-interference-noise ratio and the number of sampling snapshots under the condition of random directional error of the signal in an embodiment of the present invention. DETAILED DESCRIPTION

[0119] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0120] Example 1

[0121] like Figures 1 to 7 As shown, this embodiment provides a method for suppressing interference in mutual prime array satellite communication in a non-uniform noise environment, comprising the following steps:

[0122] Step 1: Remove the diagonal elements of the coprime array satellite communication non-circular signal covariance matrix to obtain the coprime array satellite communication non-circular signal covariance matrix after removing the diagonal elements, as follows:

[0123] Suppose a coprime array consists of two subarrays, one of which has N elements and the spacing between adjacent elements is Md, and the other has 2M elements and the spacing between adjacent elements is Nd, d is the unit element spacing, M and N are both coprime integers and M is less than N, the entire coprime array is composed of L = 2M + N-1 elements, where L represents the total number of elements, the first element is the reference element, let represents the set of mutually prime array element positions, z l d represents the position of the lth array element, where l = 1, 2, L, L and z1 = 0; when G satellite non-circular signals are incident on the coprime array, the satellite non-circular signal vector x(i) at the i-th snapshot is expressed as

[0124]

[0125] Among them, a(θ g ) represents the coprime array steering vector of the g-th non-circular signal, s g(i) represents the g-th non-circular signal beam, n(i) represents the coprime array non-uniform noise, the first signal with direction θ1 is regarded as the desired signal, and the other signals are regarded as interference. Let Represents the global interference plus noise vector, the covariance matrix R of the satellite non-circular signal vector x Expressed as

[0126]

[0127] Where E{g} represents the statistical expectation operation, (·) H represents the conjugate transpose operation, p g represents the g-th non-circular signal power, R n Represents the non-uniform noise covariance matrix. In the non-uniform noise background, the noise power of different array elements is different and unknown. At this time, the non-uniform noise covariance matrix R n Expressed as

[0128] R n =diag{p n1 ,p n2 ,L,p nL} (3)

[0129] Among them, diag{·} represents the diagonalization operation, p nl represents the noise power of the lth array element and l=1,L,L; the pseudo-covariance matrix Q of the satellite non-circular signal vector x Expressed as

[0130]

[0131] Among them, ξ g represents the non-circular coefficient of the g-th non-circular signal;

[0132] Cascade x(i) with its conjugate to obtain the amplified satellite non-circular signal vector for

[0133]

[0134] Among them, s1(i) represents the expected signal, represents the desired signal amplification steering vector, ξ1 represents the desired signal non-circular coefficient, a(θ1) represents the desired signal steering vector, represents the global noise vector, the satellite non-circular signal amplification covariance matrix Expressed as

[0135]

[0136] in,(·) * Represents the conjugate operation.

[0137] Step 2: Vectorize the covariance matrix of the coprime array satellite communication non-circular signal after removing the diagonal elements, construct a differential common array and interpolate it into a differential virtual uniform linear array, as follows:

[0138] To process R n The non-uniform diagonal elements in R x Remove the diagonal elements in and get the matrix R after removing the diagonal elements x1 for

[0139]

[0140] Among them, the matrix The diagonal elements of are all 0 and the rest of the elements are all 1. e represents the Hadamard product operation. After removing the diagonal elements, the matrix R x1 Not affected by non-uniform noise.

[0141] R x1 Vectorization is performed to obtain a differential common array virtual signal for

[0142]

[0143] in, represents the Kronecker product operation, vec(g) represents the vectorization operation of converting the matrix into a column vector, and the differential common matrix set Expressed as According to the difference matrix set Rearrange the virtual array element positions in The elements in the , get the rearranged differential common matrix virtual signal In order to make full use of all the information, the collection Therefore, we define for

[0144] Interpolation difference virtual uniform straight line array collection, is a collection that can contain The minimum uniform linear array set of The interpolated differential virtual uniform linear array signal vector is expressed as Elements in Expressed as

[0145]

[0146] in Indicates the virtual array element position v c The virtual signal at d, is composed of and does not belong to the set The set of elements in the interpolated difference virtual uniform linear array signal autocorrelation vector Expressed as

[0147]

[0148] in represents the interpolated differential virtual uniform linear array steering vector of the g-th non-circular signal, The elements in are represented as Indicates the virtual array element position z c Virtual signal at d.

[0149] Step 3: Establish an atomic norm optimization problem about the interpolation difference virtual uniform linear array non-circular signal vector, and solve the restored interpolation difference virtual uniform linear array non-circular signal vector, as follows:

[0150] Autocorrelation vector of virtual uniform linear array signal from interpolated differences In the expression of

[0151]

[0152] Where span{g} represents the subspace spanned by the vectors in the curly brackets. The linear subspace divided in equation (11) is called the correlation subspace, which is represented by the correlation subspace matrix Construct, for the relevant subspace matrix S d Perform feature decomposition, that is where Λ d1 By S d The diagonal matrix of the positive eigenvalues ​​in descending order, U d1 By S d The matrix of eigenvectors corresponding to the positive eigenvalues ​​of d2 By S d The remaining eigenvalues ​​of d2 By S d The matrix U is composed of the eigenvectors corresponding to the remaining eigenvalues ​​of d1 The column space of is the relevant subspace, and U d1 The complementary subspaces of the column space are orthogonal, that is

[0153]

[0154] in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Toeplitz structure. In order to recover the interpolated difference virtual uniform linear array signal Construct the following atomic norm minimization problem

[0155]

[0156] in represents the atomic norm, P·P2 represents the l2 norm of the vector, δ represents the threshold, represents a collection of atoms in a continuous domain, Represented by the optimization variable As the first column of the Toeplitz matrix, the constraint express is a positive semidefinite matrix, and the optimization problem (13) is further expressed as

[0157]

[0158] Where γ represents the regularization parameter and is defined as The atomic norm of for

[0159] for

[0160]

[0161] Using Vandermonde decomposition, the matrix Decompose into matrix The trace of Where Tr(·) represents the trace operation of the matrix, combined with and The expression is derived from

[0162]

[0163] Substituting equation (16) into equation (14), the optimization problem (14) is equivalently transformed into the following form:

[0164]

[0165] in Since the optimization problem (17) is a convex optimization problem, it is solved using CVX software and the solution is expressed as

[0166] Step 4: Vectorize the pseudo-covariance matrix of the coprime array satellite communication non-circular signal, construct a summed common array and interpolate it into a summed virtual uniform linear array, as follows:

[0167] Since the pseudo-covariance matrix Q xDoes not contain non-uniform noise components, Q x Will not be affected by non-uniform noise, directly vectorize Q x Get the summed common array virtual signal for

[0168]

[0169] Among them, the sum of the common matrix set Expressed as According to the sum of the matrix set Rearrangement of virtual array element positions elements, and obtain the rearranged summed common matrix virtual signal Supplementary summation matrix set The missing element in , defined A collection of virtual uniform straight line arrays for interpolation and summation, is a The minimum uniform linear array set of The interpolated and summed virtual uniform linear array signal is expressed as Elements Expressed as

[0170]

[0171] in Is caused by and does not belong to The set of elements of which interpolation and summation of the autocorrelation vector of the virtual uniform linear array signal Expressed as

[0172]

[0173] in, represents the interpolated summed virtual uniform linear array steering vector of the g-th non-circular signal, The elements of

[0174] Step 5: Establish an atomic norm optimization problem about the interpolation summation virtual uniform linear array non-circular signal vector, and solve the restored interpolation summation virtual uniform linear array non-circular signal vector, as follows:

[0175] Autocorrelation vector of virtual uniform linear array signal from interpolation and summation In the expression (20), we can further deduce

[0176]

[0177] The linear subspace divided in formula (21) is also a related subspace, which is represented by the related subspace matrix Construct the relevant subspace matrix S s The feature decomposition is where Λ s1 By S s The diagonal matrix of the positive eigenvalues ​​in descending order, U s1 By S s The matrix of eigenvectors corresponding to the positive eigenvalues ​​of s2 By S s The remaining eigenvalues ​​of s2 By S s The matrix composed of the eigenvectors corresponding to the remaining eigenvalues ​​of is obviously, and U s1 The complementary subspaces of the column space are orthogonal, that is

[0178]

[0179] in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Hankel structure. In order to recover the interpolated and summed virtual uniform linear array signal, the following Hankel optimization problem is constructed:

[0180]

[0181] where μ′ represents the regularization parameter, represents the Hankel matrix and As the first column of this matrix, the constraint express is a positive semidefinite matrix. Since the optimization problem (23) is a convex optimization problem, CVX software is used to solve it. The solution is expressed as

[0182] Step 6: Establish a spatial spectrum estimator of the interpolated difference virtual uniform linear array to estimate the direction of the non-circular signal, as follows:

[0183] Since already obtained and Constructing the Toeplitz matrix of interpolation difference virtual uniform linear array and interpolation and summation of virtual uniform linear array Hankel matrix The traditional azimuth estimation technology can be applied to the interpolation difference virtual uniform linear array or the interpolation sum virtual uniform linear array to estimate the direction of non-circular signals, and the interpolation difference virtual uniform linear array spatial spectrum f is constructed using the classic multiple signal classification algorithm. MUSIC (θ) is

[0184]

[0185] Among them, Φ n Representation Matrix The noise subspace of f MUSIC The angles corresponding to the G peaks of (θ) are used to estimate the direction of the non-circular signal. The final estimated direction of the non-circular signal is expressed as L,

[0186] Step 7: Estimate the signal plus interference covariance matrix, the non-uniform noise covariance matrix and the non-circular coefficients of the non-circular signal as follows:

[0187] Estimate the non-uniform noise covariance matrix R n The basic idea is to get the covariance matrix R from x Remove the signal plus interference covariance matrix R si , in order to estimate the signal plus interference covariance matrix R si , establish the following optimization problem

[0188]

[0189] Among them, P·P F represents the Frobenius norm of the matrix, I L represents an L×L dimensional identity matrix, ∈ represents a constant and ∈>0, and the solution of the optimization problem (25) is expressed as The estimated inhomogeneous noise covariance matrix is

[0190]

[0191] in, Represents a diagonal matrix consisting of the main diagonal elements of the matrix within the curly brackets.

[0192] By using the estimated signal direction Its corresponding non-circular coefficient Estimated

[0193]

[0194] in, τ represents the satellite non-circular signal amplification covariance matrix The minimum eigenvalue of .

[0195] Step 8: Establish the least squares optimization problem of the covariance and pseudo-covariance matrices to estimate the non-circular signal power as follows:

[0196] Since the power of non-circular signals is unknown, we can use the matrix R xand Q x Extract the non-circular signal power value and estimate the power diagonal matrix. In order to estimate the power diagonal matrix P, construct the following optimization problem

[0197]

[0198] in, represents the estimated coprime array steering matrix, P = diag{p1,p2,L,p G} represents the power diagonal matrix, 0 L represents an L×L dimensional zero matrix, represents the non-circular coefficient diagonal matrix, the constraint P ≥ 0 means that P is a positive semidefinite matrix, and the solution of the optimization problem (28) is expressed as

[0199]

[0200] in

[0201]

[0202]

[0203] Step 9: Reconstruct the coprime array interference plus noise covariance matrix and estimate the amplified desired satellite signal steering vector, as follows:

[0204] Reconstructing the covariance matrix of the amplified interference plus noise in coprime array satellite communication

[0205]

[0206] in represents the estimated interference plus noise covariance matrix, represents the estimated pseudo-interference plus noise covariance matrix;

[0207] Using the estimated desired signal direction and the estimated expected signal non-circular coefficient Amplify the desired signal of the coprime array to guide the vector Estimated

[0208]

[0209] Step 10: Calculate the satellite communication coprime array expansion weight vector to suppress non-circular interference, as follows:

[0210] Calculation of the expanded weight vector for wide linear beamforming of coprime array satellite communications for

[0211]

[0212] Embodiment 2:

[0213] The above method is used to perform the non-circular interference suppression experiment of coprime array satellite communication in a non-uniform noise environment:

[0214] Suppose a coprime array is composed of coprime integers M=3 and N=5, then the coprime array consists of 10 array elements in total. The set of physical array element positions of the coprime array is Where d represents half the signal wavelength. In this coprime array, the incident direction of a desired non-circular signal is 0° and the non-circular phase is 70°, and the incident directions of the two non-circular interferences are -40° and 50°, respectively, and the non-circular phases are -120° and 80°, respectively. The above signals are all binary phase shift keying modulation signals. The background noise obeys an independent zero-mean Gaussian distribution, and the noise covariance matrix can be expressed as R n =diag{11,5,10.2,3.2,7,9,7.3,6.6,12.3,9.2}.

[0215] Experiment 1: Output signal-to-interference-noise ratio performance when array information is precisely known. Figure 2 The relationship between the output signal-to-noise ratio and the input signal-to-noise ratio of all test methods is shown. Figure 3 The relationship between the output signal-to-noise ratio and the number of sampling snapshots of all the tested methods is shown. These results show that the performance of the method proposed in this embodiment is better than that of other methods, because the amplified interference plus noise covariance matrix and the estimated amplified desired signal steering vector accurately reconstructed by the method proposed in this embodiment can effectively eliminate non-uniform noise.

[0216] Experiment 2: Output signal-to-interference-noise ratio performance under the condition of fixed signal direction error. Assuming that the expected non-circular signal direction is 0°, and the actual expected non-circular signal direction is -2°, the expected non-circular signal direction error is fixed at 2°. Figure 4 The relationship between the output signal-to-noise ratio and the input signal-to-noise ratio of all test methods is shown. Figure 5 The relationship curves of the output signal-to-interference-noise ratio of all test methods versus the number of sampling snapshots are shown. It can be seen that the test performance of the method proposed in this embodiment is better than that of other methods, indicating that the method proposed in this embodiment is robust to fixed direction errors.

[0217] Experiment 3: Output SINR performance under the condition of random signal direction error. Assume that the random direction errors of the desired signal and interference are uniformly distributed in the angle interval [-3°, 3°]. In other words, the direction of the desired signal is randomly generated from the angle interval [-3°, 3°], while the directions of the two interferences are generated from the angle intervals [-43°, -37°] and [47°, 53°] respectively. Figure 6 The relationship curve between the output signal-to-noise ratio and the input signal-to-noise ratio of all test methods is shown. Figure 7The relationship curves between the output signal to noise ratio and the number of sampling snapshots of all the test methods are shown. It is obvious that the method proposed in this embodiment has the best performance among all the test methods, which shows that the method proposed in this embodiment can simultaneously process the non-uniform noise and random directional error of the coprime array.

[0218] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for suppressing non-circular interference in coprime array satellite communication under non-uniform noise environment, characterized in that: The following steps are involved: Step 1: Remove the diagonal elements of the coprime array satellite communication non-circular signal covariance matrix to obtain the coprime array satellite communication non-circular signal covariance matrix after removing the diagonal elements, and proceed to step 2; Step 2: Vectorize the covariance matrix of the coprime array satellite communication non-circular signal after removing the diagonal elements, construct a differential common array and interpolate it into a differential virtual uniform linear array, and then go to step 3; Step 3: Establish an atomic norm optimization problem about the interpolation difference virtual uniform linear array non-circular signal vector, solve the restored interpolation difference virtual uniform linear array non-circular signal vector, and go to step 4; Step 4: vectorize the pseudo covariance matrix of the coprime array satellite communication non-circular signal, construct a summed common array and interpolate it into a summed virtual uniform linear array, and then go to step 5; Step 5: Establish an atomic norm optimization problem about the interpolation summation virtual uniform linear array non-circular signal vector, solve the restored interpolation summation virtual uniform linear array non-circular signal vector, and go to step 6; Step 6: Establish a spatial spectrum estimator of the interpolation difference virtual uniform linear array to estimate the direction of the non-circular signal, and then go to step 7; Step 7: Estimate the signal plus interference covariance matrix, the non-uniform noise covariance matrix and the non-circular coefficients of the non-circular signal, and go to step 8; Step 8: Establish the least squares optimization problem of the covariance and pseudo-covariance matrices, estimate the non-circular signal power, and go to step 9; Step 9: Reconstruct the coprime array interference plus noise covariance matrix, estimate the amplified desired satellite signal steering vector, and proceed to step 10; Step 10: Calculate the satellite communication coprime array expansion weight vector to suppress non-circular interference.

2. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 1, characterized in that: In step 1, the diagonal elements of the coprime array satellite communication non-circular signal covariance matrix are removed to obtain the coprime array satellite communication non-circular signal covariance matrix after removing the diagonal elements, which is specifically as follows: Suppose a coprime array consists of two subarrays, one of which has N elements and the spacing between adjacent elements is Md, and the other has 2M elements and the spacing between adjacent elements is Nd, d is the unit element spacing, M and N are both coprime integers and M is less than N, the entire coprime array is composed of L = 2M + N-1 elements, where L represents the total number of elements, the first element is the reference element, let represents the set of mutually prime array element positions, z l d represents the position of the lth array element, where l = 1, 2, L, L and z1 = 0; when G satellite non-circular signals are incident on the coprime array, the satellite non-circular signal vector x(i) at the i-th snapshot is expressed as Among them, a(θ g ) represents the coprime array steering vector of the g-th non-circular signal, s g (i) represents the g-th non-circular signal beam, n(i) represents the coprime array non-uniform noise, the first signal with direction θ1 is regarded as the desired signal, and the other signals are regarded as interference. Let Represents the global interference plus noise vector, the covariance matrix R of the satellite non-circular signal vector x Expressed as Where E{g} represents the statistical expectation operation, (·) H represents the conjugate transpose operation, p g represents the g-th non-circular signal power, R n Represents the non-uniform noise covariance matrix. In the non-uniform noise background, the noise power of different array elements is different and unknown. At this time, the non-uniform noise covariance matrix R n Expressed as R n diag{p n1 ,p n2 ,L,p nL } (3) Where diag{·} represents the diagonalization operation, p nl represents the noise power of the lth array element and l=1,L,L; the pseudo-covariance matrix Q of the satellite non-circular signal vector x Expressed as Among them, ξ g represents the non-circular coefficient of the g-th non-circular signal; Cascade x(i) with its conjugate to obtain the amplified satellite non-circular signal vector for Among them, s1(i) represents the expected signal, represents the desired signal amplification steering vector, ξ1 represents the desired signal non-circular coefficient, a(θ1) represents the desired signal steering vector, represents the global noise vector, the satellite non-circular signal amplification covariance matrix Expressed as in,(·) * Represents the conjugate operation.

3. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 2, characterized in that: In step 2, the covariance matrix of the coprime array satellite communication non-circular signal after removing the diagonal elements is vectorized, and the differential common array is constructed and interpolated into a differential virtual uniform linear array, as follows: To process R n The non-uniform diagonal elements in R x Remove the diagonal elements in and get the matrix R after removing the diagonal elements x1 for Among them, the matrix The diagonal elements of are all 0 and the rest of the elements are all 1. e represents the Hadamard product operation. After removing the diagonal elements, the matrix R x1 Not affected by non-uniform noise. R x1 Perform vectorization to obtain a differential common array virtual signal for in, represents the Kronecker product operation, vec(g) represents the vectorization operation of converting the matrix into a column vector, and the differential common matrix set Expressed as According to the difference matrix set Rearrange the virtual array element positions in The elements in the , get the rearranged differential common matrix virtual signal In order to make full use of all the information, the collection Therefore, we define is a set of interpolation difference virtual uniform straight line arrays, is a collection that can contain The minimum uniform linear array set of The interpolated differential virtual uniform linear array signal vector is expressed as Elements in Expressed as in Indicates the virtual array element position v c The virtual signal at d, is composed of and does not belong to the set The set of elements in the interpolated difference virtual uniform linear array signal autocorrelation vector Expressed as in represents the interpolated differential virtual uniform linear array steering vector of the g-th non-circular signal, The elements in are represented as Indicates the virtual array element position z c Virtual signal at d.

4. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 3, characterized in that: In step 3, an atomic norm optimization problem about the interpolation difference virtual uniform linear array non-circular signal vector is established to solve the restored interpolation difference virtual uniform linear array non-circular signal vector, as follows: Autocorrelation vector of virtual uniform linear array signal from interpolated differences In the expression of Where span{g} represents the subspace spanned by the vectors in the curly brackets. The linear subspace divided in equation (11) is called the correlation subspace, which is represented by the correlation subspace matrix Construct, for the relevant subspace matrix S d Perform feature decomposition, that is where Λ d1 By S d The diagonal matrix of the positive eigenvalues ​​in descending order, U d1 By S d The matrix of eigenvectors corresponding to the positive eigenvalues ​​of d2 By S d The remaining eigenvalues ​​of d2 By S d The matrix U is composed of the eigenvectors corresponding to the remaining eigenvalues ​​of d1 The column space of is the relevant subspace, and U d1 The complementary subspaces of the column space are orthogonal, that is in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Toeplitz structure. In order to recover the interpolated difference virtual uniform linear array signal Construct the following atomic norm minimization problem in represents the atomic norm, P·P2 represents the l2 norm of the vector, δ represents the threshold, represents a collection of atoms in a continuous domain, Represented by the optimization variable As the first column of the Toeplitz matrix, the constraint express is a positive semidefinite matrix, and the optimization problem (13) is further expressed as Where γ represents the regularization parameter and is defined as The atomic norm of for Using Vandermonde decomposition, the matrix Decompose into matrix The trace of Where Tr(·) represents the trace operation of the matrix, combined with and The expression is derived from Substituting equation (16) into equation (14), the optimization problem (14) is equivalently transformed into the following form: in Since the optimization problem (17) is a convex optimization problem, it is solved using CVX software and the solution is expressed as 5. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 4, characterized in that: In step 4, the pseudo covariance matrix of the coprime array satellite communication non-circular signal is vectorized, and a summed common array is constructed and interpolated into a summed virtual uniform linear array, as follows: Since the pseudo-covariance matrix Q x Does not contain non-uniform noise components, Q x Will not be affected by non-uniform noise, directly vectorize Q x Get the summed common array virtual signal for Among them, the sum of the common matrix set Expressed as According to the sum of the matrix set Rearrangement of virtual array element positions elements, and obtain the rearranged summed common matrix virtual signal Supplementary summation matrix set The missing element in , defined A collection of virtual uniform straight line arrays for interpolation and summation, is a The minimum uniform linear array set of The interpolated and summed virtual uniform linear array signal is expressed as Elements Expressed as in Is caused by and does not belong to The set of elements of which interpolation and summation of the autocorrelation vector of the virtual uniform linear array signal Expressed as in, represents the interpolated summed virtual uniform linear array steering vector of the g-th non-circular signal, The elements of 6. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 5, characterized in that: In step 5, an atomic norm optimization problem about the interpolation summation virtual uniform linear array non-circular signal vector is established to solve the restored interpolation summation virtual uniform linear array non-circular signal vector, as follows: Autocorrelation vector of virtual uniform linear array signal from interpolation and summation In the expression (20), we can further deduce The linear subspace divided in formula (21) is also a related subspace, which is represented by the related subspace matrix Construct the relevant subspace matrix S s The feature decomposition is where Λ s1 By S s The diagonal matrix of the positive eigenvalues ​​in descending order, U s1 By S s The matrix of eigenvectors corresponding to the positive eigenvalues ​​of s2 By S s The remaining eigenvalues ​​of s2 By S s The matrix composed of the eigenvectors corresponding to the remaining eigenvalues ​​of is obviously, and U s1 The complementary subspaces of the column space are orthogonal, that is in Representing a collection The number of elements in , The dimension is The identity matrix of The covariance matrix of has a Hankel structure. In order to recover the interpolated and summed virtual uniform linear array signal, the following Hankel optimization problem is constructed: Where μ′ represents the regularization parameter, represents the Hankel matrix and As the first column of this matrix, the constraint express is a positive semidefinite matrix. Since the optimization problem (23) is a convex optimization problem, CVX software is used to solve it. The solution is expressed as 7. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 6, characterized in that: In step 6, a spatial spectrum estimator of an interpolated difference virtual uniform linear array is established to estimate the direction of non-circular signals, as follows: Since already obtained and Constructing the Toeplitz matrix of interpolation difference virtual uniform linear array and interpolation and summation of virtual uniform linear array Hankel matrix The traditional azimuth estimation technology can be applied to the interpolation difference virtual uniform linear array or the interpolation sum virtual uniform linear array to estimate the direction of non-circular signals, and the interpolation difference virtual uniform linear array spatial spectrum f is constructed using the classic multiple signal classification algorithm. MUSIC (θ) is Among them, Φ n Representation Matrix The noise subspace of f MUSIC The angles corresponding to the G peaks of (θ) are used to estimate the direction of the non-circular signal. The final estimated direction of the non-circular signal is expressed as L, 8. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 7, characterized in that: In step 7, the signal plus interference covariance matrix, the non-uniform noise covariance matrix and the non-circular coefficients of the non-circular signal are estimated as follows: Estimate the non-uniform noise covariance matrix R n The basic idea is to get the covariance matrix R from x Remove the signal plus interference covariance matrix R si , in order to estimate the signal plus interference covariance matrix R si , establish the following optimization problem Among them, P·P F represents the Frobenius norm of the matrix, I L represents an L×L dimensional identity matrix, ∈ represents a constant and ∈>0, and the solution of the optimization problem (25) is expressed as The estimated inhomogeneous noise covariance matrix is in, represents the diagonal matrix consisting of the main diagonal elements of the matrix in the braces; By using the estimated signal direction Its corresponding non-circular coefficient Estimated in, τ represents the satellite non-circular signal amplification covariance matrix The minimum eigenvalue of .

9. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 8, characterized in that: In step 8, the least squares optimization problem of the covariance and pseudo-covariance matrices is established to estimate the non-circular signal power, as follows: Since the power of non-circular signals is unknown, we can use the matrix R x and Q x Extract the non-circular signal power value and estimate the power diagonal matrix. In order to estimate the power diagonal matrix P, construct the following optimization problem in, represents the estimated coprime array steering matrix, P = diag{p1,p2,L,p G } represents the power diagonal matrix, 0 L represents an L×L dimensional zero matrix, represents the non-circular coefficient diagonal matrix, the constraint P ≥ 0 means that P is a positive semidefinite matrix, and the solution of the optimization problem (28) is expressed as in 10. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 9, characterized in that: In step 9, the coprime array interference plus noise covariance matrix is ​​reconstructed to estimate the amplified desired satellite signal steering vector, as follows: Reconstructing the covariance matrix of the amplified interference plus noise in coprime array satellite communication in represents the estimated interference plus noise covariance matrix, represents the estimated pseudo-interference plus noise covariance matrix; Using the estimated desired signal direction and the estimated expected signal non-circular coefficient Amplify the desired signal of the coprime array to guide the vector Estimated 11. The method for suppressing non-circular interference of mutually prime array satellite communication in a non-uniform noise environment according to claim 10, characterized in that: In step 10, the satellite communication coprime array expansion weight vector is calculated to suppress non-circular interference, as follows: Calculation of the expanded weight vector for wide linear beamforming of coprime array satellite communications for

Images