Aperture expansion method of moving synthetic array based on generalized coprime arrays

Abstract

The present invention discloses a motion synthetic array aperture expansion method based on a generalized coprime array, which belongs to the field of radar signal processing. Specifically, when the conventional coprime array is in a stationary state, a virtual array element S is inserted at the hole in the virtual domain. v , vectorize the covariance matrix of the received signal and remove duplicate elements to obtain the virtual array S I , and calculate the equivalent virtual signal y I ; Then, the temporal and spatial signal source invariance of the coprime array is maintained in the static state, and a synthetic coprime array model in the moving state is established; and the moving synthetic coprime array model is solved to obtain the array element position set in the moving synthetic coprime array; finally, the array element positions in the moving synthetic coprime array are combined with the grid deviation of the spatial domain division to obtain a hierarchical Bayesian model; the signal amplitude sparse vector p and the grid deviation β in the hierarchical Bayesian model are solved using the off-grid sparse Bayesian DOA estimation algorithm; the present invention improves the accuracy of the angle estimation result.

Description

Technical Field

[0001] The present invention belongs to the field of radar signal processing, and in particular is a motion synthetic array aperture expansion method based on a generalized coprime array, which is used to improve the accuracy of angle estimation results. Background Art

[0002] The essence of expanding the degrees of freedom of a sparse array lies in obtaining the corresponding equivalent virtual signal by vectorizing the covariance matrix, and then obtaining the corresponding equivalent virtual array based on the differential commonality array. Spatial smoothing, a common sparse array DOA (Direction of Arrival) algorithm, uses spatial smoothing to achieve signal decorrelation, then uses traditional DOA algorithms to obtain signal angle estimates. The introduction of spatial smoothing reduces the spatial degrees of freedom of the array to a certain extent.

[0003] A coprime array is a classic partially augmentable array. This means holes exist in the virtual equivalent array of a coprime array, leading to model mismatch issues when using spatial smoothing techniques. Existing processing methods mostly discard non-continuous elements on either side and only extract a continuous linear subarray from the equivalent virtual array for DOA estimation. This results in a reduction in available degrees of freedom, a smaller virtual aperture, and degraded DOA estimation performance. To increase the number of elements in the extracted virtual subarray, an extended coprime array is commonly used. Compared to a standard coprime array, an extended coprime array can extract more virtual uniform linear subarrays from the equivalent virtual array.

[0004] Coprime arrays can maintain their angular measurement performance while achieving a sparser element arrangement and a larger virtual aperture. However, the drawback is that the holes limit aperture expansion. In practice, arrays are often in constant motion, and the received signals from consecutive moving elements can be combined to achieve higher measurement performance. For coprime arrays, array motion can fill the holes, thereby achieving a higher degree of freedom.

[0005] Sparse reconstruction algorithms based on spatial grid points face the contradiction between grid division and computational complexity. The denser the grid division, the higher the estimation accuracy, but the computational time increases significantly. Summary of the Invention

[0006] Aiming at the problem of small virtual aperture in traditional coprime array methods, the present invention proposes a motion synthesis array aperture expansion method based on generalized coprime arrays, which improves the accuracy of angle measurement results by adopting array element interpolation technology in the virtual domain.

[0007] The method for expanding the aperture of a moving synthetic array based on a generalized coprime array comprises the following specific steps:

[0008] The specific steps are as follows:

[0009] Step 1: When the conventional coprime array is in a static state, insert a virtual array element S at the hole in the virtual domain. v , vectorize the received signal covariance matrix and remove duplicate elements to obtain the virtual array S I ;

[0010] The calculation formula for removing duplicate elements is:

[0011]

[0012] in, The array flow matrix corresponding to non-repeated virtual array elements, p is the integer compression factor used to change the array element spacing, is the array element noise power, is the unit vector corresponding to the non-repeating virtual array.

[0013] Step 2: Calculate the virtual array S I The equivalent virtual signal

[0014] Expressed as:

[0015]

[0016] in, C represents a set of complex numbers, [·] i Represents the signal corresponding to the virtual array element at the i-th position.

[0017] Interpolated virtual array S I There are two types of array elements: one is the virtual array element existing in the differential common array, and the other is the virtual array element introduced by interpolation.

[0018] Step 3: Maintain the invariance of the temporal and spatial signal sources of the coprime array in a stationary state, and establish a synthetic coprime array model in a moving state;

[0019] For coprime arrays with M and N sub-elements respectively, the motion synthesis array model is expressed as:

[0020]

[0021] Where x(t) is the output signal of the receiving array at time t, and its expression is:

[0022]

[0023] s k (t) is the incident angle θ at time t k The source signal of θk is the angle corresponding to the kth source signal; K is the total number of source signals; v is the speed of the coprime array moving uniformly in the horizontal direction; λ is the signal wavelength; a(θ k ) is the incident angle θ k The steering vector of the source signal; n(t) is the Gaussian white noise at time t; A is the array flow type of the signal; s(t) is the source signal at time t; The received signal is synthesized after equivalent phase synchronization.

[0024] For a near-field narrowband signal, at time t+τ, the output signal of the receiving array is:

[0025]

[0026] in is the shift array manifold matrix,

[0027] For a far-field narrowband signal, at time t+τ, the output signal of the receiving array is:

[0028] x(t+τ)=exp(j2πfτ)Ψs(t)+n(t+τ)

[0029] f is the signal frequency.

[0030] Select vτ = d, which is half the wavelength, and the total number of array elements is M + N - 1. Then the element steering vector at time t + τ is:

[0031]

[0032]

[0033] The noise signal in the received signal is synthesized after equivalent phase synchronization.

[0034] Step 4: Solve the motion synthesis coprime array model to obtain the array element position set in the motion synthesis coprime array;

[0035] The array element position set in the coprime array includes the following two forms:

[0036] 1) When the coprime array moves half a wavelength, the moving composite array is constructed, and the array element position set after the movement is:

[0037] P c1 ={(Mn+1)d,0≤n≤N-1}∪{(Nm+1)d,0≤m≤M-1}

[0038] The position of the synthesized array element after movement is:

[0039] Pc =P c0 ∪P c1

[0040] P c0 is the initial array element position set before moving.

[0041] 2) Since the coprime array contains pairs of array elements separated by only a single array element spacing, some array elements overlap;

[0042] The cross-correlation set in the differential commonality set of coprime arrays is:

[0043] S 12 ={Mk1-Nk2}∪{Nk2-Mk1}

[0044] S 1′2′ ={Mk′1-Nk′2}∪{Nk′2-Mk′1}

[0045] Among them, 1 and 2 represent the two sub-arrays of the original coprime array, 1′ and 2′ represent the two sub-arrays of the coprime array after shifting, 0≤k1,k′1≤N-1,0≤k2,k′2≤M-1, S 12 =S 1′2′ .

[0046] The set of cross-correlations between the original array and the shifted array is:

[0047] S 11′ ={Mk1-Mk′1-1}∪{Mk′-Mk1+1}

[0048] S 22′ ={Nk2-Nk′2-1}∪{Nk′2-Nk2+1}

[0049] S 12′ ={Mk1-Nk′2-1}∪{Nk′2-Mk1+1}

[0050] S 21′ ={Nk2-Mk′1-1}∪{Mk′1-Nk2+1}

[0051] Among them, S 11′ 、S 22′ Corresponding to the cross-correlation statistics of the same sub-array at different times, S 12′ 、S 21′ The cross-correlation statistics corresponding to different sub-arrays at different times, S 11′ ∪S 22′ ∪S 12′ ∪S 21′ =S 12′ ∪S 21′ .

[0052] Finally, the element position set of the composite array is expressed as:

[0053] S ca =S 12 ∪S 12′ ∪S 21′

[0054] Step 5: For the array element positions in the motion synthetic coprime array, combined with the grid deviation of the spatial domain division, a hierarchical Bayesian model is obtained.

[0055] When the real target is not exactly located at the predefined spatial grid point, it is necessary to reconstruct the array receiving model taking into account the grid deviation: the spatial domain is divided into grids, the corresponding angle overcomplete set is is uniformly distributed, that is Is a constant. If the incident angle of some signal source is For these angles θ k′ , find the corresponding closest grid point in the angle set Therefore, the corresponding linearized steering vector is expressed as: in, If stipulated By transferring the approximation grid bias into the measurement noise, the hierarchical Bayesian model is expressed as:

[0056]

[0057] Φ(β) is the matrix representation of the steering vector; represents the noise signal of each grid area at time t; β represents the grid deviation; Represents a sparse vector of signal amplitudes.

[0058] Step 6: Use the off-grid sparse Bayesian DOA estimation algorithm to solve the signal amplitude sparse vector in the hierarchical Bayesian model and grid deviation β.

[0059] The posterior distribution probability density function of the signal amplitude is:

[0060]

[0061] Among them, α is the parameter describing the variance; L is the number of snapshots; is the signal amplitude in the Bayesian model at time t. The corresponding mean μ(t) and variance Σ are as follows:

[0062]

[0063] Φ is the matrix representation of the steering vector; z(t) is the result of the off-grid sparse Bayesian model based on the sparse representation at time t; Γ is the diagonal matrix corresponding to the signal accuracy that obeys the gamma distribution.

[0064] The joint probability density function is:

[0065]

[0066] Will As a hidden variable, maximize the expected value

[0067] The estimation of the grid deviation value β is equivalent to maximizing the following formula:

[0068]

[0069]

[0070] in,

[0071]

[0072]

[0073] ⊙ represents the Schur product; C is the error coefficient matrix in the matrixization of the corresponding steering vector in the off-grid sparse Bayesian posterior. is a mathematical symbol, which means taking the real part of a complex number, μ is the mean matrix of the sparse signal, μ * Take the conjugate of the sparse signal mean matrix.

[0074] The grid deviation estimate at each iteration is:

[0075]

[0076] r is a constant.

[0077] The advantages of the present invention are:

[0078] 1) A motion synthesis array aperture expansion method based on a generalized coprime array. To address the hole problem existing in the equivalent virtual array of the generalized coprime array, the present invention uses virtual array interpolation technology to avoid direct information discarding and make full use of the information of the echo signal, thereby significantly increasing the number of estimable signal sources and reducing angle measurement errors.

[0079] 2) A motion synthetic array aperture expansion method based on a generalized coprime array. Compared with the traditional assumption that the array is in a static state, the present invention introduces a motion synthetic coprime array under the premise that the array is in a continuous motion state, and uses motion to compensate for the holes in the coprime array, thereby significantly improving the array's degree of freedom while ensuring the system's angle measurement accuracy.

[0080] 3) A motion synthetic array aperture expansion method based on a generalized coprime array. Compared with the traditional sparse Bayesian learning algorithm based on predefined spatial grid points, the present invention addresses the contradiction between the grid deviation in spatial grid division and the sparse Bayesian computational complexity. It uses a de-gridding model to enable the angle measurement results to break through the grid limitations and reduce the angle measurement error. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 Flowchart of the motion synthetic array aperture expansion method based on generalized coprime arrays of the present invention;

[0082] Figure 2 Schematic diagram of virtual array interpolation of the motion synthetic array aperture expansion method based on generalized coprime arrays of the present invention;

[0083] Figure 3 This is a schematic diagram of the motion synthesis array structure of the present invention using motion to compensate for holes in a coprime array;

[0084] Figure 4 This is a schematic diagram of the off-grid sparse Bayesian model that uses an off-grid model to break through grid limitations. DETAILED DESCRIPTION

[0085] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0086] Based on array signal processing theory, the present invention proposes a motion synthesis array aperture expansion method based on a generalized coprime array. The DOA estimation algorithm for the generalized coprime array is optimized and improved. Virtual array interpolation and motion synthesis array techniques are used to compensate for information loss in the coprime array, respectively. The non-uniform virtual array is converted into a uniform virtual array, thereby avoiding information loss and waste. For the grid deviation problem in sparse Bayesian algorithms, a de-gridding sparse Bayesian algorithm is used to reduce the grid deviation value, enabling target angle estimation for targets not located at grid points, improving the accuracy of DOA estimation while reducing computational complexity.

[0087] like Figure 1 The specific steps are as follows:

[0088] Step 1: When the conventional coprime array is in a static state, insert a virtual array element S at the hole in the virtual domain. v , vectorize the received signal covariance matrix and remove duplicate elements to obtain the virtual array S I ;

[0089] Assume that the received signal covariance matrix is vectorized to obtain:

[0090]

[0091] Among them, A v The flow matrix corresponding to the original virtual array, since the array redundancy is not 0, the signal obtained after removing duplicate elements is:

[0092]

[0093] in, The array flow matrix corresponding to the non-repeated virtual array elements, the corresponding array element position set S v is a non-contiguous array. p is an integer compression factor used to change the spacing between array elements. is the array element noise power, is the unit vector corresponding to the non-repeating virtual array.

[0094] In order to make full use of the array element position set S v Contains all the array element information, and realizes signal processing that satisfies the Nyquist sampling theorem in the virtual domain, and the array element position set S v Insert virtual array elements into the holes in the array to form Figure 2 The virtual array S shown I .

[0095] Step 2: Calculate the interpolated virtual array S I The equivalent virtual signal

[0096] Compute equivalent virtual signals Increasing the number of array elements in the interceptable virtual linear subarray can fundamentally solve the problem of information loss.

[0097] Interpolation is performed in the virtual domain. There are no real array elements or real received signals at the virtual array elements, which only exist in a mathematical sense. Therefore, the interpolated virtual array elements can be regarded as actual antenna array elements that exist but are turned off. The corresponding received signals are set to zero initial value, which can be expressed as:

[0098]

[0099] in, C represents a set of complex numbers, [·] i Represents the signal corresponding to the virtual array element at the i-th position.

[0100] Interpolated virtual array S I There are two types of array elements: one is the virtual array element existing in the differential commonality array, and the other is the virtual array element introduced by interpolation. By this method, the corresponding interpolated virtual array S I The equivalent virtual signal This signal contains all the information in the original signal, and there is no model mismatch problem when applying spatial smoothing technology to DOA estimation, which improves the DOA estimation performance.

[0101] Step 3: Maintain the invariance of the temporal and spatial signal sources of the coprime array in a stationary state, and establish a synthetic coprime array model in a moving state;

[0102] The aforementioned research assumes a stationary array. However, in practical applications of sparse arrays based on millimeter-wave radar, the array is often in constant motion, such as on aircraft, vehicles, and ships. Short-term platform movement can be considered as being in the same sensing environment, including target position, directional angle, and short-term signal structure. In this case, based on the aforementioned theory, the array can combine and process continuously received data while in motion to achieve higher measurement performance. For coprime arrays, array motion can fill in holes, thereby achieving a higher degree of freedom.

[0103] Since it is unreasonable to assume that the signal environment is stationary over relatively long periods of time, especially when the array aperture is large, and fast data acquisition requires processing short observation periods, this example only considers the case where the array translation motion is one-half wavelength, that is, maintaining the invariance of the time and space signal sources, and the motion synthesis coprime array model and solution problem.

[0104] Consider a moving coprime array, which consists of a linear array of M elements and a linear array of N elements (M and N are coprime integers), and the array moves along the x-axis at a constant speed v. The moving composite array model is expressed as:

[0105]

[0106] Assuming that the direction of the signal source relative to the array can be considered unchanged between two adjacent sampling points, since the distance the array moves within a sampling time is very small, the output signal x(t) received from the array at time t is expressed as:

[0107]

[0108] s k (t) is the incident angle θ at time t k The source signal of θ k is the angle corresponding to the kth source signal; K is the total number of source signals; v is the speed of the coprime array moving uniformly in the horizontal direction; λ is the signal wavelength; a(θ k ) is the incident angle θ k The steering vector of the source signal; n(t) is the Gaussian white noise at time t; A is the array flow type of the signal; s(t) is the source signal at time t; The received signal is synthesized after equivalent phase synchronization.

[0109] For a near-field narrowband signal, at time t+τ, the output signal of the receiving array is:

[0110]

[0111] in is the shift array manifold matrix,

[0112] For a far-field narrowband signal, at time t+τ, the output signal of the receiving array is:

[0113] x(t+τ)=exp(j2πfτ)Ψs(t)+n(t+τ)

[0114] f is the signal frequency.

[0115] Select vτ = d, which is half the wavelength, and the total number of array elements is M + N - 1. Then the element steering vector at time t + τ is:

[0116]

[0117]

[0118] The noise signal in the received signal is synthesized after equivalent phase synchronization.

[0119] The sampled signals at time t and t+τ have a fixed phase difference j2πfτ, which is the phase correction factor. After eliminating the influence of the array element phase correction factor at time t+τ, the equivalent sampled data of the synthesized array element at time t can be obtained, thus filling the holes in the original array and expanding the array aperture. The equivalent phase-synchronized synthesized received signal is:

[0120]

[0121] In the present invention, the influence of phase noise is studied, the influence of model mismatch is analyzed, and a method for solving the DOA problem in a moving sparse array is proposed.

[0122] The assumed reception model is always different from the actual model, and model mismatch will lead to performance degradation. Consider an M-element coprime array with uncertain phase noise. The signal received by the m-th element at time t+τ is:

[0123]

[0124] Among them, k It represents the uncertain phase value between the signals received at time t and t+τ due to phase noise. Before forming the extended aperture array, it is necessary to use Ψ kEstimate the phase correction factor. Here we define a phase correction matrix as:

[0125]

[0126] Among them, β k =2πf k τ+Ψ k Therefore, the received signal of the motion synthesis array can be expressed as:

[0127]

[0128] The covariance matrix of the received signal is:

[0129]

[0130] Define E1 = AT, E2 = AΦT, where T is a non-singular matrix. So the expression is:

[0131] E2=ATT -1 ΦT=E1T -1 ΦT

[0132] Definition Ψ=T -1 ΦT, so Since Ψ and Φ have the same eigenvalues, they are diagonal elements of Φ. Decompose the matrix Ψ and obtain the eigenvalue λ k , we can get the phase correction factor estimate β k ,k=1,...,K, and satisfy:

[0133] β k =angle(λ k )

[0134] Then, by studying the influence of array position error, the impact of model mismatch is analyzed, and a method for solving the DOA problem in moving sparse arrays is proposed.

[0135] During array element motion, the array position will fluctuate and deviate from the expected position, thus affecting the DOA estimation result. Therefore, it is necessary to analyze the impact of array position error on DOA estimation.

[0136] Consider an array of M elements, (x m ,y m ) is the coordinate of the mth array element. Due to the random perturbations of the array element positions, it is usually assumed that (x m ,y m) is unknown. If the array shape changes with motion, the performance of the DOA estimation method will degrade rapidly if not handled properly. In this invention, the main goal is to jointly estimate the uncertainty of the shape of a moving sparse array and the DOA. In practical scenarios, what people generally know is the nominal array element position. The position error can be defined as Since the array flow vector depends on the position of the array, for the disturbed array element m, the delay error generated by the kth signal can be expressed as

[0137]

[0138] The modified steering vector can be expressed as the product of the azimuth-dependent diagonal matrix Φ(θ) and the nominal flow pattern vector.

[0139]

[0140] Therefore, the received signal model can be expressed as:

[0141]

[0142] in

[0143]

[0144]

[0145]

[0146] Assuming the position error of the array element is small enough, the linear approximation can be used to obtain:

[0147]

[0148] because It depends on the array element position error, so it is necessary to estimate before estimating the MUSIC spectrum The problem can be formulated as maximizing the following function:

[0149]

[0150] The above formula means that the The phase is used to select a vector subspace with the best orthogonality to the noise as the actual a(θ). Assume that the two targets are located at θ a and θ b =θ a +Δθ, corresponding to and Where Δθ is small enough. So

[0151]

[0152] At this time, through the first-order Taylor expansion, It can be approximated as:

[0153]

[0154] in, So we can get:

[0155]

[0156] Omit include and j(2π / λ), we get

[0157]

[0158] if If established, Also holds true. A sufficient condition for So you can add constraints Right now Therefore, the optimal problem can be written as:

[0159]

[0160] In theory, depends on the azimuth angle and the position error of the array element, while g(θ) depends on the azimuth angle and the nominal element position. Assuming that the position error of the array element is independent of the nominal element position, then It is usually close to 0. The objective function is:

[0161]

[0162] in, Thus, the Lagrangian cost function is derived as:

[0163]

[0164] because is a real vector, then is a real function. Define B(θ)=Re(D H (θ)D(θ)), z(θ)=Re(D H (θ)d(θ)), then the cost function can be written as:

[0165]

[0166] right The partial derivative of is:

[0167]

[0168] Afterwards you can get The estimate of can be expressed as:

[0169]

[0170] Based on the above DOA and steering vector, the position of the array element can be estimated. Assume that the deviation of the direction estimation is Manifold vector a(θ k ) can be represented as:

[0171]

[0172] Since Δθ k is extremely small, so sinΔθ k ≈Δθ k ,cosΔθ k ≈1. The above formula can be expressed as:

[0173]

[0174] in,

[0175] definition Can get

[0176]

[0177] m ranges from 2 to M, and k ranges from 1 to K, so there are (M-1)K equations. The above formula can be expressed as:

[0178] Wβ=η

[0179]

[0180]

[0181] η=[Δ 2,1 ,...,Δ M,1 ,...,Δ 2,K ,...,Δ M,K ] T

[0182] Among them, c .,k =[c 2,k ,...,c M,k ] T ,Γ=[0 (M-2)×1 I M-2 ]. Its least squares solution is Therefore, the estimated array element position can be obtained as:

[0183]

[0184] Step 4: Solve the motion synthesis coprime array model to obtain the array element position set in the motion synthesis coprime array;

[0185] like Figure 3 As shown; the array element position set in the coprime array includes the following two forms:

[0186] 1) When the coprime array moves half a wavelength, the moving composite array is constructed, and the array element position set after the movement is:

[0187] P c1 ={(Mn+1)d,0≤n≤N-1}∪{(Nm+1)d,0≤m≤M-1}

[0188] The position of the synthesized array element after movement is:

[0189] P c =P c0 ∪P c1

[0190] P c0 is the initial array element position set before moving.

[0191] 2) Since the coprime array contains pairs of array elements separated by only a single array element spacing, some array elements overlap;

[0192] The number of elements in the synthesized array is less than twice the number of elements in the original array due to overlap in some elements. This overlap occurs because coprime arrays contain pairs of elements separated by only a single element spacing. The differential commonality set of coprime arrays can be divided into two subsets: one derived from the autocorrelation statistics of the received signals, and one derived from the cross-correlation statistics of the received signals. Since the virtual element positions formed by the autocorrelation are a set of cross-correlations, only the cross-correlation set is used when calculating the degrees of freedom of the array.

[0193] S 12 ={Mk1-Nk2}∪{Nk2-Mk1}

[0194] S 1′2′ ={Mk′1-Nk′2}∪{Nk′2-Mk′1}

[0195] Among them, 1 and 2 represent the two sub-arrays of the original coprime array, 1′ and 2′ represent the two sub-arrays of the coprime array after shifting, 0≤k1,k′1≤N-1,0≤k2,k′2≤M-1, S 12 =S 1′2′ .

[0196] The set of cross-correlations between the original array and the shifted array is:

[0197] S11′ ={Mk1-Mk′1-1}∪{Mk′-Mk1+1}

[0198] S 22′ ={Nk2-Nk′2-1}∪{Nk′2-Nk2+1}

[0199] S 12′ ={Mk1-Nk′2-1}∪{Nk′2-Mk1+1}

[0200] S 21′ ={Nk2-Mk′1-1}∪{Mk′1-Nk2+1}

[0201] Among them, S 11′ 、S 22′ Corresponding to the cross-correlation statistics of the same sub-array at different times, S 12′ 、S 21′ The cross-correlation statistics corresponding to different sub-arrays at different times, S 11′ ∪S 22′ ∪S 12′ ∪S 21′ =S 12′ ∪S 21′ .

[0202] Finally, the element position set of the composite array is expressed as:

[0203] S ca =S 12 ∪S 12′ ∪S 21′

[0204] Step 5: For the array element positions in the motion synthetic coprime array, combined with the grid deviation of the spatial domain division, a hierarchical Bayesian model is obtained.

[0205] For the SBL algorithm based on predefined spatial grid points, in practice, it is often encountered that the real target is not exactly located on the grid point. To solve this problem, it is necessary to rebuild the array receiving model considering the grid deviation. Assume that the spatial domain is divided into grids, the corresponding angle overcomplete set is Also assuming is uniformly distributed, that is is a constant, the number of sources is K, and the number of array elements is M. If the incident angle of some sources is For these angles θ k′ , the corresponding nearest grid point can be found in the angle set Therefore, the corresponding linearized steering vector is expressed as:

[0206] in, If stipulated By transferring the approximate grid bias into the measurement noise, the off-grid sparse Bayesian model based on the sparsified representation is expressed as:

[0207]

[0208] Φ(β) is the matrix representation of the steering vector; represents the noise signal of each grid area at time t; β represents the grid deviation; Represents a sparse vector of signal amplitudes.

[0209] The off-grid model can be regarded as a first-order approximation of the true observation model, while the gridded model can be regarded as a zero-order approximation of the true observation model; the hierarchical Bayesian model such as Figure 4 shown.

[0210] Step 6: Use the off-grid sparse Bayesian DOA estimation algorithm to solve the signal amplitude sparse vector in the hierarchical Bayesian model and grid deviation β.

[0211] The posterior distribution probability density function of the signal amplitude is:

[0212]

[0213] Where z is the result of the off-grid sparse Bayesian model based on sparse representation; α is the parameter describing the variance; is the array element noise variance; L is the number of snapshots; is the signal amplitude in the Bayesian model at time t. The corresponding mean μ(t) and variance Σ are as follows:

[0214]

[0215] Φ is the matrix representation of the steering vector; z(t) is the result of the off-grid sparse Bayesian model based on the sparse representation at time t; Γ is the diagonal matrix corresponding to the signal accuracy that obeys the gamma distribution.

[0216] The joint probability density function is:

[0217]

[0218] Will As a hidden variable, maximize the expected value

[0219] Then the hyperparameter update rule is:

[0220]

[0221]

[0222] in, L is the number of snapshots, ρ = b / a,

[0223] The estimation of the grid deviation value β is equivalent to maximizing the following formula:

[0224]

[0225]

[0226]

[0227] where ⊙ represents the Schur product; C is the error coefficient matrix in the matrixization of the corresponding steering vector in the off-grid sparse Bayesian posterior. is a mathematical symbol, which means taking the real part of a complex number, μ is the mean matrix of the sparse signal, μ * Take the conjugate of the sparse signal mean matrix.

[0228] The grid deviation estimate at each iteration is:

[0229]

[0230] r is a constant.

Claims

1. A motion synthetic array aperture expansion method based on generalized coprime arrays, characterized in that: The specific steps are as follows: Step 1: When the conventional coprime array is in a static state, insert a virtual array element S at the hole in the virtual domain. v , vectorize the covariance matrix of the received signal and remove duplicate elements to obtain the virtual array S I ; The calculation formula for removing duplicate elements is: in, The flow matrix corresponding to the non-repeated virtual array, p is the integer compression factor used to change the spacing between array elements, is the array element noise power, is the unit vector corresponding to the non-repeating virtual array; Step 2: Calculate the virtual array S I The equivalent virtual signal Expressed as: in, C represents a set of complex numbers, [·] i represents the signal corresponding to the virtual array element at the i-th position; Step 3: Maintain the invariance of the temporal and spatial signal sources of the coprime array in a stationary state, and establish a synthetic coprime array model in a moving state; For coprime arrays with M and N sub-elements respectively, the motion synthesis array model is expressed as: Where x(t) is the output signal of the receiving array at time t; A is the equivalent phase-synchronized synthesized received signal; s =[a s (θ1),a s (θ2),…,a s (θ K )], a(θ k ) is the incident angle θ k The steering vector of the source signal; v is the speed of the coprime array moving uniformly in the horizontal direction; λ is the signal wavelength; θ k is the angle corresponding to the kth source signal; s(t) is the source signal at time t; n(t) is the Gaussian white noise at time t; The noise signal in the synthesized received signal after equivalent phase synchronization; Step 4: Solve the motion synthesis coprime array model to obtain the array element position set in the motion synthesis coprime array; The array element position set in the coprime array includes the following two forms: 1) When the coprime array moves half a wavelength, the moving composite array is constructed, and the array element position set after the movement is: P c1 ={(Mn+1)d,0≤n≤N-1}∪{(Nm+1)d,0≤m≤M-1} The position of the synthesized array element after movement is: P c =P c0 ∪P c1 P c0 is the initial array element position set before moving; 2) Since the coprime array contains pairs of array elements separated by only a single array element spacing, some array elements overlap; The cross-correlation set in the differential commonality set of coprime arrays is: S 12 ={Mk1-Nk2}∪{Nk2-Mk1} S 1′2′ ={Mk1′-Nk2′}∪{Nk2′-Mk1′} Among them, 1 and 2 represent the two sub-arrays of the original coprime array, 1′ and 2′ represent the two sub-arrays of the coprime array after shifting, 0≤k1,k1′≤N-1,0≤k2,k2′≤M-1, S 12 =S 1′2′ ; The set of cross-correlations between the original array and the shifted array is: S 11′ ={Mk1-Mk1′-1}∪{Mk′-Mk1+1} S 22′ ={Nk2-Nk2′-1}∪{Nk2′-Nk2+1} S 12′ ={Mk1-Nk2′-1}∪{Nk2′-Mk1+1} S 21′ ={Nk2-Mk1′-1}∪{Mk1′-Nk2+1} Among them, S 11′ 、S 22′ Corresponding to the cross-correlation statistics of the same sub-array at different times, S 12′ 、S 21′ The cross-correlation statistics corresponding to different sub-arrays at different times, S 11′ ∪S 22′ ∪S 12′ ∪S 21′ =S 12′ ∪S 21′ ; Finally, the element position set of the composite array is expressed as: S ca =S 12 ∪S 12′ ∪S 21′ Step 5: Based on the positions of the elements in the motion synthesis coprime array and the grid deviation of the spatial domain division, a hierarchical Bayesian model is obtained; The hierarchical Bayesian model is expressed as: Φ(β) is the matrix representation of the steering vector; represents the noise signal of each grid area at time t; β represents the grid deviation; represents the signal amplitude sparse vector; Step 6: Use the off-grid sparse Bayesian DOA estimation algorithm to solve the signal amplitude sparse vector in the hierarchical Bayesian model and grid deviation β; The posterior distribution probability density function of the signal amplitude is: Among them, α is the parameter describing the variance; L is the number of snapshots; is the signal amplitude in the Bayesian model at time t; the corresponding mean μ(t) and variance Σ are as follows: Φ is the matrix representation of the steering vector; z(t) is the result of the off-grid sparse Bayesian model based on the sparse representation at time t; Γ is the diagonal matrix corresponding to the signal accuracy that follows the gamma distribution; The joint probability density function is: Will As a hidden variable, maximize the expected value The estimation of the grid deviation value β is equivalent to maximizing the following formula: in, ⊙ represents the Schur product; C is the error coefficient matrix in the matrixization of the corresponding steering vector after the off-grid sparse Bayesian approach; is a mathematical symbol, which means taking the real part of a complex number, μ is the mean matrix of the sparse signal, μ * is the conjugate of the sparse signal mean matrix; A is the array manifold of the signal; The grid deviation estimate at each iteration is: r is a constant.

2. The method for expanding the aperture of a moving synthetic array based on a generalized coprime array according to claim 1, wherein: The virtual array S I There are two types of array elements: one is the virtual array element existing in the differential common array, and the other is the virtual array element introduced by interpolation.

3. The method for expanding the aperture of a moving synthetic array based on a generalized coprime array according to claim 1, wherein: In step 3, the expression of the output signal x(t) of the receiving array at time t is: s k (t) is the incident angle θ at time t k The source signal; K is the total number of source signals; For a near-field narrowband signal, at time t+τ, the output signal of the receiving array is: in is the shift array flow matrix; For a far-field narrowband signal, at time t+τ, the output signal of the receiving array is: x(t+τ)=exp(j2πfτ)Ψs(t)+n(t+τ) f is the signal frequency; Select vτ = d, which is half the wavelength, and the total number of array elements is M + N - 1. Then the element steering vector at time t + τ is:

4. The method for expanding the aperture of a moving synthetic array based on a generalized coprime array according to claim 1, wherein: The step five is specifically as follows: When the real target is not exactly located at the predefined spatial grid point, it is necessary to reconstruct the array receiving model taking into account the grid deviation: the spatial domain is divided into grids, the corresponding angle overcomplete set is is uniformly distributed, that is is a constant; if the incident angle of some source is For these angles θ k′ , find the corresponding closest grid point in the angle set Therefore, the linearized expression of the corresponding steering vector is: in, If stipulated By transferring the approximate grid bias into the measurement noise, a hierarchical Bayesian model is obtained.

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