Two-step fitting method for 2D DOA estimation based on planar symmetric sparse array

Abstract

The invention discloses a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array, which mainly solves the problem that the existing method based on the sparse array cannot process mixed circular and non-circular signals, and belongs to the technical field of array signal processing. The implementation steps are: constructing a sparse symmetric planar array; modeling the sparse symmetric planar array signal; first-step conjugate augmented covariance matrix fitting preprocessing; first-step fitting conjugate augmented covariance matrix data transformation; last-step conjugate augmented covariance matrix fitting preprocessing; sequential decoupling postprocessing to realize multi-dimensional parameter estimation of mixed signals. The invention is based on two-step fitting conjugate augmented covariance matrix data, and through two-step decoupling sequential estimation of the two-dimensional direction of arrival, it can effectively avoid multi-dimensional spectrum peak search with high computational complexity, and maintain high recognizability, and can be used for perception and positioning in the fields of communication, radar, etc.

Description

Technical Field

[0001] The invention belongs to the technical field of signal processing, and in particular relates to a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array. Background Art

[0002] The direction of arrival indicates the direction of the signal when it reaches the antenna array at the receiving end. The receiving end can determine the location of the sending end based on the direction of arrival of the signal. When multiple signals arrive at the receiving end at the same time, they will be superimposed to form a signal. The receiving end can determine the direction of arrival of each original signal based on the received signal, and then determine the location of each sending end based on the direction of arrival of each original signal.

[0003] In the prior art, sparse arrays are often used to achieve high-degree-of-freedom estimation of the signal arrival direction. Sparse arrays can reduce the number of sensors, thereby reducing the cost of use, and can reduce the mutual coupling between array elements and reduce the cost of hardware equipment;

[0004] However, most of the current sparse array designs are for non-circular signals, mainly based on the construction of continuous difference arrays and sum arrays. The difference arrays or sum arrays may have holes, and cannot achieve the joint estimation of the arrival direction of mixed circular and non-circular signals. Therefore, it is urgent to explore sparse arrays that can process mixed circular and non-circular signals and the corresponding arrival direction estimation method. Summary of the invention

[0005] The purpose of the present invention is to provide a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array, through which only the number of circular signals, the number of non-circular signals, the sparse symmetric planar array configuration and the sparse symmetric planar array output need to be obtained, and a sparse array of mixed circular and non-circular signal directions of arrival can be jointly estimated; and a two-step conjugate augmented covariance matrix data fitting and sequential decoupling technology are used to achieve circular and non-circular signal direction of arrival estimation with low computational complexity;

[0006] To achieve the above object, the present invention provides a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array, comprising the following steps:

[0007] Step 1: Construct a sparse symmetric planar array: Set the plane coordinate system based on the geometric centroid of the sparse symmetric planar array, establish the Cartesian origin at the first antenna, and define the sensor position vector along the x-axis as n x d x , and the position vector of the sensor along the y-axis is defined as n y d y , the arrays along the x and y axes are symmetrical line arrays arranged in a cascade nested array, n x and n y represents a vector consisting of symmetric integers, d x and d ydenote the minimum inter-sensor distances along the x and y axes of the sparse symmetric planar array, respectively;

[0008] The incident signal parameters are expressed as follows: α m , β m are the angles between the arrival direction of the mth signal and the y-axis and x-axis, θ m ,φ m Respectively represent the azimuth and elevation angles of the mth signal; ψ m is the non-circular phase of the mth non-circular signal;

[0009] Step 2: Signal modeling for sparse symmetric planar array. For sparse symmetric planar array, consider the incident situation of M circular and non-circular mixed signals. The array output formula is as follows:

[0010]

[0011] In the above formula, represents the steering vector corresponding to the mth incident signal, where a x,m and a y,m are the steering vectors corresponding to the x-axis subarray and the y-axis subarray of the m-th signal, respectively. represents the Kronecker product, s m (t) is the complex envelope of the mth incident signal, n(t) is zero-mean Gaussian white noise;

[0012] For a sparse symmetric planar array, the relationship in the two-dimensional direction of arrival is as follows:

[0013] sinφ m sinθ m =cosα m ,sinφ m cosθ m = cosβ m (2)

[0014] In the above formula, θ m ∈[0,2π) and φ m ∈[0,π]; finally we get L x Dimensional guide vector a x,m and L y Dimensional guide vector a y,m They are as follows:

[0015]

[0016] In the above formula, [·] T represents transpose, λ represents wavelength, Indicates the first x The distance between the sensor and the coordinate origin along the x-axis, Indicates the first y The distance between a sensor and the origin of the coordinate system along the y-axis. The coordinate of the origin of the coordinate system is x1=y1=0;

[0017] The number of non-circular and circular signals in the incident signal is M respectively. r and M c =MM r , then the array conjugate augmentation output The corresponding conjugate augmented covariance matrix is ​​expressed as:

[0018]

[0019] in, and σ 2 Corresponding to the power of the mth signal and noise, respectively, (·) H represents the conjugate transpose operation;

[0020]

[0021] Indicated by a m and The block diagonal matrix composed of (·) * represents conjugation, Indicates 2L x L y ×2L x L y -dimensional identity matrix, the corresponding subscripts represent the row and column dimensions of the matrix;

[0022] Step 3: First step conjugate augmented covariance matrix fitting preprocessing;

[0023] For formula (5), rewrite the conjugate augmented covariance matrix as follows:

[0024]

[0025] In the above formula, is the array output covariance matrix, is the array output conjugate covariance matrix; for After fitting data preprocessing, the final results are as follows:

[0026]

[0027] In the above formula, in is the virtual array steering vector corresponding to the mth signal;

[0028] Step 4: First step is to fit the conjugate augmented covariance matrix data transformation;

[0029] Pair Matrix Reorganize the data as shown below:

[0030]

[0031] in

[0032]

[0033] J contains L xv L y Subarray Each sub-matrix J p,q There is only one element equal to 1, and

[0034] J p,q (q,p)=1, (27)

[0035] And J p,q All other elements in are 0; the above data reorganization will make the initial matrix Transform to a new matrix as follows:

[0036]

[0037] In the above formula, and is the virtual sparse array steering vector corresponding to the mth signal;

[0038] Step 5: Tail-step conjugate augmented covariance matrix fitting preprocessing;

[0039] The conjugate augmented covariance matrix data of the first step fitting is processed by the last step fitting to obtain the final conjugate augmented covariance matrix corresponding to the virtual uniform rectangular array as follows:

[0040]

[0041] In the above formula,

[0042]

[0043] represents the virtual uniform rectangular array steering vector corresponding to the mth signal;

[0044] Step 6: Sequential decoupling post-processing realizes multi-dimensional parameter estimation of mixed signals and obtains multi-dimensional parameter estimation results of incident signals.

[0045] Preferably, in step 3, the derivation process of preprocessing is as follows:

[0046] For formula (6), replace and Decompose into Lx ×L x of sub-matrices, the formula is as follows:

[0047]

[0048] Then the sub-matrices in the above two equations are expressed as follows:

[0049]

[0050] The above two equations can also be expressed as:

[0051]

[0052] in, represents the lth y1 The output of the sub-array is represents the lth y2 Output of sub-arrays;

[0053] Respectively and Vectorization:

[0054]

[0055] In the above formula, It is Khatri-Rao. represents the steering matrix corresponding to the x-axis subarray of the sparse symmetric planar array,

[0056]

[0057] The angle parameter between the element and the M signals and power parameters Related,

[0058]

[0059] Elements and angle parameters M r Non-circular signal power parameters and noncircular phase parameters Related;

[0060] When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in

[0061]

[0062] and

[0063]

[0064] The elements in are the same; among them, Indicates the lth parallel to the x-axis y1 Virtual output of subarrays, Indicates the lth parallel to the x-axis y2 The virtual output of the sub-array, L xv Indicates the number of virtual array elements of each sub-array;

[0065] Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L xv +1)-dimensional virtual difference matrix and (2L xv +1)-dimensional virtual and combined arrays, and then derive and As shown below:

[0066]

[0067] in, is the virtual sub-array steering vector on the x-axis corresponding to the m-th signal, and the formula is as follows:

[0068]

[0069] Reconstructing the covariance matrix and the conjugate covariance matrix The formula is as follows:

[0070]

[0071] Then the conjugate augmented covariance matrix of the first-step reconstruction of the virtual sparse plane array is expressed as follows:

[0072]

[0073] In the above formula, in represents the first-step reconstructed virtual sparse planar array steering vector corresponding to the m-th signal.

[0074] Preferably, in step 5, the specific process of pretreatment is as follows:

[0075] Transform the data of formula (28), because the sensor arrangement in step 1 is different, then First, Divide into Covariance submatrices:

[0076]

[0077] as well as Conjugate covariance submatrices:

[0078]

[0079]

[0080] In the above formula, Indicates the lth parallel to the y-axis xv1 Virtual output of sub-arrays, It indicates that the lth xv2 Virtual output of sub-arrays; and Also expressed as:

[0081]

[0082] The following are and Vectorization:

[0083]

[0084] in, The first step is to reconstruct the steering matrix corresponding to the sub-matrix along the y-axis direction of the virtual sparse planar array;

[0085]

[0086] Elements and angle parameters Related to the power parameters of the M signals;

[0087]

[0088] Elements and angles M r The power of a non-circular signal is related to the non-circular phase parameter;

[0089] When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in

[0090]

[0091] and

[0092]

[0093] The elements in are the same; in the above formula, is parallel to the y-axis xv1 sub-array virtual outputs, is parallel to the y-axis xv2 The virtual output of the sub-array, L yv is the number of virtual array elements of each sub-array;

[0094] Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L yv +1)-dimensional virtual difference matrix and (2L yv +1)-dimensional virtual and combined arrays, and then derive and as follows:

[0095]

[0096] In the above formula, Represents the y-axis sub-array steering vector of the virtual uniform rectangular array corresponding to the m-th signal:

[0097]

[0098] Among them l yv =1,2,…,L yv ;

[0099] Covariance matrix

[0100]

[0101] and the conjugate covariance matrix

[0102]

[0103] After reorganization, 2L was finally obtained xv L yv ×2L xv L yv The conjugate augmented covariance matrix of the dimensional virtual uniform rectangular array is shown in formula (46).

[0104] Preferably, in step six, the corresponding parameter estimation process is specifically as follows:

[0105] By using the multiple signal classification method, according to formula (46), we get the following formula:

[0106]

[0107] To M r non-circular signals are established, among which U n Is 2L xv L yv-M r -2M c The noise subspace spanned by the eigenvectors corresponding to the smaller eigenvalues;

[0108] h m ≠02 and Under these conditions, the following formula is obtained according to the rank deficiency principle:

[0109]

[0110] The above formula can be re-expressed as follows:

[0111]

[0112] in

[0113]

[0114] According to formula (47), (49) and (50) also apply to M c circular signal, then according to the rank deficiency principle, we get the formula:

[0115] det(F m )=0 (52)

[0116] The angles corresponding to the M signals are both true; and the angle Estimated by one-dimensional spectrum peak search;

[0117] Then, based on the estimated angle and the corresponding Estimated angle β m ,m=1,2,…,M; specifically, β m ,m=1,2,…,M by The eigenvector corresponding to the minimum eigenvalue Make an estimate:

[0118]

[0119] Among them, f m is a non-zero constant, ∠(·) represents the phase angle of the complex number in the brackets; by estimating (α m ,β m ),m=1,2,…,M r get The estimation of the non-circular phase of the non-circular signal ψ is finally obtained. m ,m=1,2,…,M r Estimates:

[0120]

[0121] Among them, gm ≈g m h m for The eigenvector corresponding to the minimum eigenvalue, g m is a non-zero constant. According to the multiple signal classification method, we get the formula

[0122]

[0123] This is true for every circular signal; then use the upper or lower half of the noise subspace to estimate M c The two-dimensional direction of arrival of the circular signal is is a non-zero vector, and

[0124]

[0125] For any vector All of them hold, then we have the following formula:

[0126]

[0127] For circular signals, α m ,m=M r +1,M r +2,…,M are estimated by one-dimensional spectrum peak search;

[0128] Angle β m ,m=M r +1,M r +2,…,M by The eigenvector corresponding to the minimum eigenvalue Make an estimate:

[0129]

[0130] in, yes The estimate of m is a non-zero constant;

[0131] The two-dimensional direction of arrival and non-circular phase of the non-circular signal are estimated by formulas (49)-(54), and the two-dimensional direction of arrival of the circular signal is estimated by formulas (55)-(58).

[0132] The present invention adopts the above-mentioned two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array, which has the following advantages:

[0133] (1) The present invention uses symmetric sparse linear arrays such as cascaded nested arrays to construct a planar symmetric sparse array with a systematic array design structure. Based on the array data, the conjugate augmented covariance matrix corresponding to the virtual uniform rectangular array can be quickly fitted to achieve two-dimensional direction of arrival estimation of mixed circular and non-circular signals;

[0134] (2) The present invention is based on sequential decoupling processing to obtain a sequential estimation of the two-dimensional wave arrival direction. The proposed method does not require multi-dimensional spectrum peak scanning and pairing process, which effectively improves the calculation efficiency.

[0135] (3) The order of the two-dimensional angle estimation process of the present invention is interchangeable, and the reconstructed conjugate augmented covariance matrix is ​​used to increase the array degree of freedom, thereby having higher identifiability.

[0136] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0137] Figure 1 It is a flow chart of the two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array of the present invention;

[0138] Figure 2 It is a schematic diagram of angle parameters in a sparse symmetric planar array in a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array according to the present invention;

[0139] Figure 3 In the two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array of the present invention, (a) is a first step of reconstructing a virtual sparse planar array, and (b) is a virtual uniform rectangular array;

[0140] Figure 4 It is a scatter plot of multi-parameter estimation of a determined mixed signal in a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array according to the present invention;

[0141] Figure 5 The two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array of the present invention is always the two-dimensional direction of arrival of the mixed signal in the presence of mixed circular and non-circular signals;

[0142] Figure 6 The two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array of the present invention is always the two-dimensional direction of arrival of the circular signal in the presence of mixed circular and non-circular signals;

[0143] Figure 7 A graph showing the relationship between the detection success probability and the signal-to-noise ratio in the two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array according to the present invention;

[0144] Figure 8This is a relationship diagram between the detection success probability and the number of snapshots in the two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array of the present invention. DETAILED DESCRIPTION

[0145] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention described and shown in the drawings here can be arranged and designed in various different configurations. The specific model specifications need to be selected and determined according to the actual specifications of the device, and the specific selection calculation method adopts the existing technology in the field, so it will not be described in detail.

[0146] Example

[0147] like Figure 1-Figure 3 As shown, the present invention provides a two-step fitting two-dimensional direction of arrival estimation method based on a planar symmetric sparse array, comprising the following steps:

[0148] Step 1: Construct a sparse symmetric planar array: Set the plane coordinate system based on the geometric centroid of the sparse symmetric planar array, establish the Cartesian origin at the first antenna, and define the sensor position vector along the x-axis as n x d x , and the position vector of the sensor along the y-axis is defined as n y d y , where the common linear arrangement along the x and y axes is a symmetric linear array such as a cascade nested array, n x and n y represents a vector consisting of symmetric integers, d x and d y denote the minimum inter-sensor distances along the x and y axes of the sparse symmetric planar array, respectively;

[0149] The incident signal parameters are expressed as follows: α m , β m are the angles between the arrival direction of the mth signal and the y-axis and x-axis, θ m ,φ m Respectively represent the azimuth and elevation angles of the mth signal; ψ m is the non-circular phase of the mth non-circular signal;

[0150] Step 2: Signal modeling for sparse symmetric planar array. For sparse symmetric planar array, consider the incident situation of M circular and non-circular mixed signals. The array output formula is as follows:

[0151]

[0152] In the above formula, represents the steering vector corresponding to the mth incident signal, where a x,m and a y,m are the steering vectors corresponding to the x-axis subarray and the y-axis subarray of the m-th signal, respectively. represents the Kronecker product, s m (t) is the complex envelope of the mth incident signal, n(t) is zero-mean Gaussian white noise;

[0153] For a sparse symmetric planar array, the relationship in the two-dimensional direction of arrival is as follows:

[0154] sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m (2)

[0155] In the above formula, θ m ∈[0,2π) and φ m ∈[0,π]; finally we get L x Dimensional guide vector a x,m and L y Dimensional guide vector a y,m They are as follows:

[0156]

[0157] In the above formula, [·] T represents transpose, λ represents wavelength, Indicates the first x The distance between the sensor and the coordinate origin along the x-axis, Indicates the first y The distance between a sensor and the origin of the coordinate system along the y-axis. The coordinate of the origin of the coordinate system is x1=y1=0;

[0158] The number of non-circular and circular signals in the incident signal is M respectively. r and M c =MM r , then the array conjugate augmentation output The corresponding conjugate augmented covariance matrix is ​​expressed as:

[0159]

[0160] in, and σ 2 Corresponding to the power of the mth signal and noise, respectively, (·) H represents the conjugate transpose operation;

[0161]

[0162] Indicated by a m and The block diagonal matrix composed of (·) * represents conjugation, Indicates 2L x L y ×2L x L y -dimensional identity matrix, the corresponding subscripts represent the row and column dimensions of the matrix;

[0163] Step 3: First step conjugate augmented covariance matrix fitting preprocessing;

[0164] For formula (5), rewrite the conjugate augmented covariance matrix as follows:

[0165]

[0166] In the above formula, is the array output covariance matrix, is the array output conjugate covariance matrix, for Perform fitting data preprocessing, the derivation process of preprocessing is as follows:

[0167] For formula (6), replace and Decompose into L x ×L x of sub-matrices, the formula is as follows:

[0168]

[0169] Then the sub-matrices in the above two equations are expressed as follows:

[0170]

[0171] The above two equations can also be expressed as:

[0172]

[0173] in, represents the lth y1 The output of the sub-array is represents the lth y2 Output of sub-arrays;

[0174] Respectively and Vectorization:

[0175]

[0176] In the above formula, It is Khatri-Rao. represents the steering matrix corresponding to the x-axis subarray of the sparse symmetric planar array,

[0177]

[0178] The angle between the element and the M signals and power Related,

[0179]

[0180] Elements and angle parameters M r Non-circular signal power parameters and noncircular phase parameters Related;

[0181] When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in

[0182]

[0183] and

[0184]

[0185] The elements in are the same; among them, Indicates the lth parallel to the x-axis y1 Virtual output of subarrays, Indicates the lth parallel to the x-axis y2 The virtual output of the sub-array, L xv Indicates the number of virtual array elements of each sub-array;

[0186] Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L xv +1)-dimensional virtual difference matrix and (2L xv +1)-dimensional virtual and combined arrays, and then derive and As shown below:

[0187]

[0188] in, is the virtual subarray steering vector on the x-axis corresponding to the mth signal, and the formula is as follows:

[0189]

[0190] Reconstructing the covariance matrix and the conjugate covariance matrix The formula is as follows:

[0191]

[0192] Then the conjugate augmented covariance matrix of the first-step reconstruction of the virtual sparse plane array is expressed as follows:

[0193]

[0194] In the above formula, in represents the first-step reconstructed virtual sparse planar array steering vector corresponding to the m-th signal.

[0195] Step 4: First step is to fit the conjugate augmented covariance matrix data transformation;

[0196] Pair Matrix Reorganize the data as shown below:

[0197]

[0198] in

[0199]

[0200] J contains L xv L y Subarray Each sub-matrix J p,q There is only one element equal to 1, and

[0201] J p,q (q,p)=1, (27)

[0202] And J p,q All other elements in are 0; the above data reorganization will make the initial matrix Transform to a new matrix as follows:

[0203]

[0204] In the above formula, and is the virtual sparse array steering vector corresponding to the mth signal;

[0205] Step 5: Tail-step conjugate augmented covariance matrix fitting preprocessing;

[0206] The specific process of performing the tail-step fitting on the conjugate augmented covariance matrix data of the first-step fitting is as follows:

[0207] Transform the data of formula (28), because the sensor arrangement in step 1 is different, then First, Divide into Covariance submatrices:

[0208]

[0209] as well as Conjugate covariance submatrices:

[0210]

[0211] In the above formula, Indicates the lth parallel to the y-axis xv1 Virtual output of sub-arrays, It indicates that the lth xv2 Virtual output of sub-arrays; and Also expressed as:

[0212]

[0213] The following are and Vectorization:

[0214]

[0215] in, The first step is to reconstruct the steering matrix corresponding to the sub-matrix along the y-axis direction of the virtual sparse planar array;

[0216]

[0217] Elements and angle parameters Related to the power parameters of the M signals;

[0218]

[0219] Elements and angles M r The power of a non-circular signal is related to the non-circular phase parameter;

[0220] When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in

[0221]

[0222] and

[0223]

[0224] The elements in are the same; in the above formula, is parallel to the y-axis xv1 sub-array virtual outputs, is parallel to the y-axis xv2 The virtual output of the sub-array, L yv is the number of virtual array elements of each sub-array;

[0225] Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L yv +1)-dimensional virtual difference matrix and (2L yv +1)-dimensional virtual and combined arrays, and then derive and as follows:

[0226]

[0227] In the above formula, Represents the y-axis sub-array steering vector of the virtual uniform rectangular array corresponding to the m-th signal:

[0228]

[0229] In the above formula, l yv =1,2,…,L yv ;

[0230] Covariance matrix

[0231]

[0232] and the conjugate covariance matrix

[0233]

[0234] After reorganization, 2L was finally obtained xv L yv ×2L xv L yv The conjugate augmented covariance matrix of the dimensional virtual uniform rectangular array is shown in formula (46).

[0235]

[0236] In the above formula,

[0237]

[0238] represents the virtual uniform rectangular array steering vector corresponding to the mth signal;

[0239] Step 6: Sequential decoupling post-processing realizes the multi-dimensional parameter estimation of the mixed signal, and obtains the estimation result of the multi-dimensional parameters of the incident signal. The estimation process is as follows:

[0240] By using the multiple signal classification method, according to formula (46), we get the following formula:

[0241]

[0242] To M r non-circular signals are established, among which U n Is 2L xv L yv -M r -2M c The noise subspace spanned by the eigenvectors corresponding to the smaller eigenvalues;

[0243] h m ≠02 and Under these conditions, the following formula is obtained according to the rank deficiency principle:

[0244]

[0245] The above formula can be re-expressed as follows:

[0246]

[0247] in

[0248]

[0249] According to formula (47), (49) and (50) also apply to M c circular signal, then according to the above rank deficiency principle, the formula is obtained:

[0250] det(F m )=0(52)

[0251] The angles corresponding to the M signals are both true; and the angle Estimated by one-dimensional spectrum peak search;

[0252] Then, based on the estimated angle and the corresponding Estimated angle β m ,m=1,2,…,M; specifically, β m ,m=1,2,…,M by The eigenvector corresponding to the minimum eigenvalue estimate:

[0253]

[0254] Among them, f m is a non-zero constant, ∠(·) represents the phase angle of the complex number in the brackets; by estimating (α m ,β m ),m=1,2,…,M r get The estimation of the non-circular phase ψ of the non-circular signal is finally obtained. m ,m=1,2,…,M r Estimates:

[0255]

[0256] Among them, g m ≈g m h m for The eigenvector corresponding to the minimum eigenvalue, g m is a non-zero constant. According to the multiple signal classification method, we get the formula

[0257]

[0258] This is true for every circular signal; then use the upper or lower half of the noise subspace to estimate M c The two-dimensional wave direction of arrival of the circular signal is is a non-zero vector, and

[0259]

[0260] For any vector All of them hold, then we have the following formula:

[0261]

[0262] For circular signals, it is true; then α m ,m=M r +1,M r +2,…,M can be estimated by one-dimensional spectrum peak search;

[0263] Angle β m ,m=M r +1,M r +2,…,M can be The eigenvector corresponding to the minimum eigenvalue of Make an estimate:

[0264]

[0265] in, yes The estimate of m is a non-zero constant;

[0266] In summary, the two-dimensional direction of arrival and non-circular phase of the non-circular signal are estimated by formulas (49)-(54), and the two-dimensional direction of arrival of the circular signal is estimated by formulas (55)-(58).

[0267] Actual simulations are carried out to verify the effectiveness of the proposed method, the enhanced signal identifiability and the successful detection probability. The simulation results are as follows: Figure 4-Figure 8 As shown, the specific process is as follows:

[0268] First, set up the first simulation scenario and consider five circular and non-circular coexisting signals, and their two-dimensional wave arrival directions They are (20°, 60°), (50°, 80°), (70°, 30°), (90°, 50°), and (100°, 20°). The first three signals are non-circular signals, and the non-circular phases are -150°, 0°, and 150° respectively. Figure 4 Shown are the corresponding multi-parameter estimation scatter plots with a signal-to-noise ratio of 10 dB and 3000 snapshots. Figure 4 The above sub-figure shows the automatic pairing estimation result of the two-dimensional wave direction of arrival of the mixed signal. Figure 4 The middle sub-figure shows the automatic pairing estimation result of the two-dimensional direction of arrival estimation of the circular signal, and the lower sub-figure shows the non-circular phase estimation result of the non-circular signal. The method of the present invention can effectively distinguish the circular signal from the non-circular signal through the scatter plot.

[0269] For the second simulation scenario, circular and non-circular mixed signals are considered, where there are 15 non-circular signals and 5 circular signals, and the non-circular phases of the non-circular signals are (-168°:-24°:168°) respectively.

[0270] Azimuth of mixed signal The pitch angles are (135°:-15°:60°), (63°:-15°,3°), (169°:-15°,94°), 9°, 24° and 2.5° respectively. They are (19°:10°:69°), (50°:-10°:10°), (32°:10°:82°), 55°, 65° and 35° respectively;

[0271] Figure 5 and Figure 6 Two 2D DOA estimation scatter plots of two signals in this scenario are given, where the signal-to-noise ratio is 30 dB. Figure 5 The corresponding snapshot number is 500,000. Figure 6 The corresponding number of snapshots is 20,000. Therefore, it can be concluded that the method of the present invention can identify more signals than the number of sensors, and the number of identifiable signals meets M r +2M c =25>min{2L x L y -2L x ,2L x L y -2L y Compared with the two-dimensional spectrum scanning method, the number of signals that can be identified by the traditional method is only M r +2M c <min{2L x L y -2L x ,2L x L y -2L y}=24. Therefore, the comparison shows that the method of the present invention can enhance the signal identifiability, not only avoiding high-dimensional spectrum search / scanning, but also outperforming the traditional two-dimensional spectrum scanning method.

[0272] Figure 4 , Figure 5 and Figure 6 The upper subfigure shows the two-dimensional direction of arrival estimation result of the mixed signal. Figure 4 The intermediate subgraph of Figure 5 and Figure 6 The lower sub-figure of shows the two-dimensional DOA estimation result of the circular signal. The estimation result corresponding to the method proposed in the present invention shows that the proposed method can realize multi-dimensional parameter estimation under underdetermined conditions and has a higher degree of freedom. Figure 7 and Figure 8 In order to compare and analyze the probability of successful detection, the comparison method is a two-dimensional spectrum scanning method, in which two non-circular signals {θ m ,φ m} 2 m＝1 The incident angles are (25°, 45°) and (45°, 15°), and their non-circular phases are -150° and 50°, respectively. Figure 7 and Figure 8 The curve showing the relationship between the detection success probability, signal-to-noise ratio and number of snapshots is shown. Figure 7 The corresponding snapshot number is 300. Figure 8 The corresponding signal-to-noise ratio is -2dB. Figure 7 and Figure 8 It can be seen that the method of the present invention has better detection performance than the traditional method. This superiority is mainly attributed to its better utilization of the non-circularity of the signal and the enhanced array freedom obtained by reconstructing the high-dimensional conjugate augmented covariance matrix.

[0273] Therefore, the present invention adopts a two-step fitting two-dimensional wave direction estimation method based on a planar symmetric sparse array, and proposes a method for estimating the two-dimensional wave direction of a mixed signal and the non-circular phase of a non-circular signal, which can effectively avoid multi-dimensional spectrum search / scanning. The angle estimation order of the proposed method is reversible and the reconstructed conjugate augmented covariance matrix is ​​used to increase the array degree of freedom, thereby being able to identify more signals.

[0274] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array, characterized in that: The following steps are involved: Step 1: Construct a sparse symmetric planar array: Set the plane coordinate system based on the geometric centroid of the sparse symmetric planar array, establish the Cartesian origin at the first antenna, and define the sensor position vector along the x-axis as n x d x , and the position vector of the sensor along the y-axis is defined as n y d y , the arrays along the x and y axes are symmetrical line arrays arranged in a cascade nested array, n x and n y represents a vector consisting of symmetric integers, d x and d y denote the minimum inter-sensor distances along the x and y axes of the sparse symmetric planar array, respectively; The incident signal parameters are expressed as follows: α m , β m are the angles between the arrival direction of the mth signal and the y-axis and x-axis, θ m ,φ m Respectively represent the azimuth and elevation angles of the mth signal; ψ m is the non-circular phase of the mth non-circular signal; Step 2: Signal modeling for sparse symmetric planar array. For sparse symmetric planar array, consider the incident situation of M circular and non-circular mixed signals. The array output formula is as follows: In the above formula, represents the steering vector corresponding to the mth incident signal, where a x,m and a y,m are the steering vectors corresponding to the x-axis subarray and the y-axis subarray of the m-th signal, respectively. represents the Kronecker product, s m (t) is the complex envelope of the mth incident signal, n(t) is zero-mean Gaussian white noise; For a sparse symmetric planar array, the relationship in the two-dimensional direction of arrival is as follows: sinφ m sinθ m =cosα m ,sinφ m cosθ m =cosβ m (2) In the above formula, θ m ∈[0,2π) and φ m ∈[0,π]; finally we get L x Dimensional guide vector a x,m and L y Dimensional guide vector a y,m They are as follows: In the above formula, [·] T represents transpose, λ represents wavelength, Indicates the first x The distance between the sensor and the coordinate origin along the x-axis, Indicates the first y The distance between a sensor and the origin of the coordinate system along the y-axis. The coordinate of the origin of the coordinate system is x1=y1=0; The number of non-circular and circular signals in the incident signal is M respectively. r and M c =MM r , then the array conjugate augmentation output The corresponding conjugate augmented covariance matrix is ​​expressed as: in, and σ 2 Corresponding to the power of the mth signal and noise, respectively, (·) H represents the conjugate transpose operation; Indicated by a m and The block diagonal matrix composed of (·) * represents conjugation, Indicates 2L x L y ×2L x L y -dimensional identity matrix, the corresponding subscripts represent the row and column dimensions of the matrix; Step 3: First step conjugate augmented covariance matrix fitting preprocessing; For formula (5), rewrite the conjugate augmented covariance matrix as follows: In the above formula, is the array output covariance matrix, is the array output conjugate covariance matrix; for After fitting data preprocessing, the final results are as follows: In the above formula, in is the virtual array steering vector corresponding to the mth signal; Step 4: First step is to fit the conjugate augmented covariance matrix data transformation; Pair Matrix Reorganize the data as shown below: in J contains L xv L y Subarray Each sub-matrix J p,q There is only one element equal to 1, and J p,q (q,p)=1, (27) And J p,q All other elements in are 0; the above data reorganization will make the initial matrix Transform to a new matrix as follows: In the above formula, and is the virtual sparse array steering vector corresponding to the mth signal; Step 5: Tail-step conjugate augmented covariance matrix fitting preprocessing; The conjugate augmented covariance matrix data of the first step fitting is processed by the last step fitting to obtain the final conjugate augmented covariance matrix corresponding to the virtual uniform rectangular array as follows: In the above formula, represents the virtual uniform rectangular array steering vector corresponding to the mth signal; Step 6: Sequential decoupling post-processing realizes multi-dimensional parameter estimation of mixed signals and obtains multi-dimensional parameter estimation results of incident signals.

2. The two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array according to claim 1, characterized in that: In step 3, the derivation process of preprocessing is as follows: For formula (6), replace and Decompose into L x ×L x of sub-matrices, the formula is as follows: Then the sub-matrices in the above two equations are expressed as follows: The above two equations can also be expressed as: in, represents the lth y1 The output of the sub-array is represents the lth y2 Output of sub-arrays; Respectively and Vectorization: In the above formula, It is Khatri-Rao. represents the steering matrix corresponding to the x-axis subarray of the sparse symmetric planar array, The angle parameters of the elements in formula (15) and the M signals and power parameters Related, The elements and angle parameters in formula (16) M r Non-circular signal power parameters and noncircular phase parameters Related; When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in and The elements in formula (17) and formula (18) are the same; where, Indicates the lth parallel to the x-axis y1 Virtual output of subarrays, Indicates the lth parallel to the x-axis y2 The virtual output of the sub-array, L xv Indicates the number of virtual array elements of each sub-array; Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L xv +1)-dimensional virtual difference matrix and (2L xv +1)-dimensional virtual and combined arrays, and then derive and As shown below: in, is the virtual sub-array steering vector on the x-axis corresponding to the m-th signal, and the formula is as follows: Reconstructing the covariance matrix and the conjugate covariance matrix The formula is as follows: Then the conjugate augmented covariance matrix of the first-step reconstruction of the virtual sparse plane array is expressed as follows: In the above formula, in represents the first-step reconstructed virtual sparse planar array steering vector corresponding to the m-th signal.

3. The two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array according to claim 2, characterized in that: In the step 5, the specific process of pre-processing is as follows: Transform the data of formula (28), because the sensor arrangement in step 1 is different, then First, Divide into Covariance submatrices: as well as Conjugate covariance submatrices: In the above formula, Indicates the lth parallel to the y-axis xv1 Virtual output of subarrays, It indicates that the lth xv2 Virtual output of sub-arrays; and Also expressed as: The following are and Vectorization: in, The first step is to reconstruct the steering matrix corresponding to the sub-matrix along the y-axis direction of the virtual sparse planar array; Elements and angle parameters Related to the power parameters of the M signals; Elements and angles M r The power of a non-circular signal is related to the non-circular phase parameter; When the sparse symmetric planar array has continuous difference matrix and sum matrix, and The elements in and The elements in are the same; in the above formula, is parallel to the y-axis xv1 sub-array virtual outputs, is parallel to the y-axis xv2 The virtual output of the sub-array, L yv is the number of virtual array elements of each sub-array; Through and The elements in are reorganized to obtain two new vectors and and Corresponding to (2L yv +1)-dimensional virtual difference matrix and (2L yv +1)-dimensional virtual and combined arrays, and then derive and as follows: In the above formula, Represents the y-axis sub-array steering vector of the virtual uniform rectangular array corresponding to the m-th signal: Among them l yv =1,2,…,L yv ; Covariance matrix and the conjugate covariance matrix After reorganization, 2L was finally obtained xv L yv ×2L xv L yv The conjugate augmented covariance matrix of the dimensional virtual uniform rectangular array is shown in formula (46).

4. The two-step fitting two-dimensional direction of arrival estimation method based on planar symmetric sparse array according to claim 3, characterized in that: In step 6, the corresponding parameter estimation process is as follows: By using the multiple signal classification method, according to formula (46), we get the following formula: To M r non-circular signals are established, among which U n Is 2L xv L yv -M r -2M c The noise subspace spanned by the eigenvectors corresponding to the smaller eigenvalues; h m ≠02 and Under these conditions, the following formula is obtained according to the rank deficiency principle: The above formula can be re-expressed as follows: in According to formula (47), (49) and (50) also apply to M c circular signal, then according to the rank deficiency principle, we get the formula: it(F m )=0 (52) The angles corresponding to the M signals are both true; and the angle Estimated by one-dimensional spectrum peak search; Then, based on the estimated angle and the corresponding Estimated angle β m ,m=1,2,…,M; specifically, β m ,m=1,2,…,M by The eigenvector corresponding to the minimum eigenvalue Make an estimate: Among them, f m is a non-zero constant, ∠(·) represents the phase angle of the complex number in the brackets; by estimating (α m ,β m ),m=1,2,…,M r get The estimation of the non-circular phase of the non-circular signal ψ is finally obtained. m ,m=1,2,…,M r Estimates: Among them, g m ≈g m h m for The eigenvector corresponding to the minimum eigenvalue, g m is a non-zero constant. According to the multiple signal classification method, we get the formula This is true for every circular signal; then use the upper or lower half of the noise subspace to estimate M c The two-dimensional wave direction of arrival of the circular signal is is a non-zero vector, and For any vector All of them hold, then we have the following formula: For circular signals, α m ,m=M r +1,M r +2,…,M are estimated by one-dimensional spectrum peak search; Angle β m ,m=M r +1,M r +2,…,M by The eigenvector corresponding to the minimum eigenvalue Make an estimate: in, yes The estimate of m is a non-zero constant; The two-dimensional direction of arrival and non-circular phase of the non-circular signal are estimated by formulas (49)-(54), and the two-dimensional direction of arrival of the circular signal is estimated by formulas (55)-(58).

Images