CN111830458B - Parallel linear array single-snapshot two-dimensional direction finding method - Google Patents

Parallel linear array single-snapshot two-dimensional direction finding method Download PDF

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CN111830458B
CN111830458B CN202010672122.0A CN202010672122A CN111830458B CN 111830458 B CN111830458 B CN 111830458B CN 202010672122 A CN202010672122 A CN 202010672122A CN 111830458 B CN111830458 B CN 111830458B
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CN111830458A (en
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李万春
邹炜钦
王丽
周亚文
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University of Electronic Science and Technology of China
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12—Simultaneous equations, e.g. systems of linear equations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/15—Correlation function computation including computation of convolution operations

Abstract

The invention belongs to the technical field of electronic countermeasure, and particularly relates to a parallel linear array single-snapshot two-dimensional direction finding method. The technical scheme adopted by the invention is that a double parallel linear array is divided into three sub-arrays, the three data matrixes are respectively obtained by smoothing each sub-array by utilizing a space smoothing technology, a large matrix is constructed according to autocorrelation and cross-correlation information of the three data matrixes, singular value decomposition is carried out on the matrix to obtain a signal sub-space estimation after dimension reduction, a rotation factor in a two-dimensional direction is calculated by utilizing a rotation invariant sub-space method (ESPRIT), and finally, the characteristic value pairing of the rotation factor is carried out by constructing a pairing matrix to realize two-dimensional direction estimation.

Description

Parallel linear array single-snapshot two-dimensional direction finding method
Technical Field
The invention belongs to the technical field of electronic countermeasure, and particularly relates to a parallel linear array single-snapshot two-dimensional direction finding method.
Background
In electronic countermeasure, it is an important research direction to direct a target signal because the direction information of a target radiation source can help one party to gain the advantage of battlefield information.
The parallel linear array is a common array structure in a direction-finding system, and consists of two parallel uniform linear arrays in space, and direction information of an incident signal is obtained by utilizing autocorrelation information of the linear arrays and cross-correlation information between the two parallel linear arrays. Most importantly, compared with a uniform linear array, the angle measurement in one-dimensional direction can be realized, the angle measurement in two-dimensional direction can be simultaneously carried out by the parallel linear array, more information can be obtained, and the azimuth information of an incoming wave signal can be more accurately obtained, so that the method has more application significance and research value. The ESPRIT (rotation invariant subspace) algorithm is a typical algorithm in spatial spectrum estimation, and by utilizing the rotation invariant characteristic of the signal subspace of each sub-array received data covariance matrix to estimate the direction of a signal, compared with the commonly used MUSIC (multiple signal classification) algorithm, the ESPRIT algorithm has the advantages of small calculation amount, no need of performing spectrum peak search, and capability of directly solving to obtain the incidence angle value of the signal.
Conventional direction finding systems and methods are generally based on receiving a multi-snapshot signal model, with accumulation of correlation information across multiple snapshots (time domain) to suppress the effects of noise. However, in some cases, for example, the duration of a signal is short, the hopping speed is too fast, or in the military field, the signal is difficult to capture, or the requirement on the real-time performance of the system is high, the array received data is difficult to be sufficient, even when the effective data amount is only a single snapshot, the rank of the covariance matrix of the received data is less than the number of the information sources, and at this time, the subspace algorithm cannot be accurately separated, which results in that the estimation of the arrival direction of the incident signal cannot be completed, and therefore, the direction finding method under the multi-snapshot cannot be directly utilized. At present, the public reported research on single-snapshot signal direction finding is very little, and a simple uniform linear array MUSIC-based direction finding method is generally utilized, so that how to apply the single-snapshot direction finding method to a more complex array structure is a direction worthy of research, and the single-snapshot direction finding method can be combined with more subspace algorithms.
Disclosure of Invention
The invention provides a parallel linear array single-snapshot two-dimensional ESPRIT direction finding method, aiming at solving the problem of single-snapshot two-dimensional direction finding of a parallel linear array.
The technical scheme adopted by the invention is as follows: dividing a double parallel linear array into three sub-arrays, smoothing each sub-array by using a space smoothing technology to respectively obtain three data matrixes, constructing a large matrix according to autocorrelation and cross-correlation information of the three data matrixes, decomposing a singular value of the matrix to obtain a signal sub-space estimation after dimension reduction, further calculating a rotation factor in a two-dimensional direction by using a rotation invariant sub-space method (ESPRIT), and finally constructing a pairing matrix to carry out angle parameter pairing to realize the two-dimensional direction estimation.
A parallel linear array in a space formed by two uniform linear arrays X and Y which are parallel to each other is shown in figure 1, and the distance between the linear array X and the linear array Y along the Y-axis direction is dyThe distance between adjacent array elements on the linear array along the x-axis direction is dx. The linear array X and the linear array Y are respectively provided with M +1 and M array elements, the linear array X is subdivided into two sub-arrays, and the sub-array X is formed by the 1 st array element (the array element at the origin position of the coordinate axis) to the Mth array element of the array X1The 2 nd array element to the M +1 th array element form a subarray X2。
Far-field narrow-band signal S (t) s with N concentric frequencies in space1(t),s2(t),…,sN(t)]TAt different two-dimensional direction angles (alpha)i,βi) I 1, …, N is incident on the parallel linear array, the signal wavelength is λ, where αiE (0, pi) and betaiAnd e (0, pi) respectively represents the included angle between the incident direction of the ith signal and the positive direction of the y axis of the coordinate axis and the included angle between the incident direction of the ith signal and the positive direction of the x axis of the coordinate axis.
The noise output by each array element is independent in statistics and is additive white Gaussian noise with the average value being zero. Using the first array element (the array element at the origin position of the coordinate axis) of the array X as a reference array element to obtain a subarray X1、X2Sum array Y at a certain snapshot time t0The outputs of (a) are:
X1(t0)＝[x1(t0),x2(t0),…,xM(t0)]T (1)
X2(t0)＝[x2(t0),x3(t0),…,xM+1(t0)]T (2)
Y(t0)＝[y1(t0),y2(t0),…,yM(t0)]T (3)
written as a vector matrix in the form of:
X1(t0)＝AMS(t0)+nX1(t0) (4)
X2(t0)＝AMΦXS(t0)+nX2(t0) (5)
Y(t0)＝AMΦYS(t0)+nY(t0) (6)
wherein the M-dimensional vector nX1(t0)、nX2(t0) And nY(t0) Respectively representing sub-arrays X1、X2And the single snapshot noise vector received by array Y. A. theMRepresenting a sub-array X1Array flow pattern of (2), defining ∈x＝2πdx/λ，εy＝2πdy/λ，AMThe mathematical form of (a) is:
Figure BDA0002582684090000031
ΦXand phiYIs a flow pattern transformation matrix, and the mathematical forms are respectively as follows:
Figure BDA0002582684090000032
Figure BDA0002582684090000033
the linear array X can be obtained from the formulas (2) and (3)1Flow pattern A ofMIs transposed into the vandermonde matrix, phiXAnd phiYAre all diagonal matrices, so that subarrays X may be derived2And flow pattern A of array YMΦXAnd AMΦYThe transpose of (a) is also a vandermonde matrix.
When the data received by the array element has only 1 snapshot, the rank of the data covariance matrix is smaller than the number N of the information sources, and the reduction of the rank makes the complete separation of the noise subspace and the signal subspace not realized when the eigenvalue decomposition is carried out on the data covariance matrix, so that the data covariance matrix needs to be reconstructed through space smoothing processing, and the rank of the data covariance matrix is larger than the number N of the incident signals.
For sub-array X of parallel linear array1、X2And the array Y respectively carries out forward space smoothing, the number of the smooth sub-arrays is p, the number of the sub-array elements is M, the relation M is M + p-1, and 2p is more than or equal to N. Subarrays X1、X2And the output of the jth smoothing sub-array of array Y is:
Figure BDA0002582684090000034
Figure BDA0002582684090000035
Figure BDA0002582684090000036
where j is 1,2, …, p,
Figure BDA0002582684090000037
for the single snap noise vector of the jth smooth sub-array on each sub-array, Sj(t0) The mathematical form of (a) is:
Figure BDA0002582684090000038
wherein A ismIs an array X1The array flow pattern of the first subarray above, in mathematical form:
Figure BDA0002582684090000039
define subarrays X1、X2Three data matrixes obtained by carrying out forward spatial smoothing on the sum array Y
Figure BDA0002582684090000041
Yf(t0) Comprises the following steps:
Figure BDA0002582684090000042
Figure BDA0002582684090000043
Figure BDA0002582684090000044
according to three forward smoothed data matrices
Figure BDA0002582684090000045
Yf(t0) An autocorrelation matrix and two cross-correlation matrices can be calculated:
Figure BDA0002582684090000046
Figure BDA0002582684090000047
Figure BDA0002582684090000048
wherein R iss＝S(t0)S(t0)H，
Figure BDA0002582684090000049
Representing the variance of m-dimensional zero mean noise, ImRepresenting m-dimensional identity matrices, three matrices can be obtained from equations (18) to (20)
Figure BDA00025826840900000410
And
Figure BDA00025826840900000411
the mathematical form is as follows:
Figure BDA00025826840900000412
Figure BDA00025826840900000413
Figure BDA00025826840900000414
wherein
Figure BDA00025826840900000415
Sub-array X of parallel linear array by same sub-array division method1、X2And respectively carrying out backward space smoothing on the array Y to obtain a sub-array X1、X2And the output of the jth smoothing sub-array of array Y is:
Figure BDA0002582684090000051
Figure BDA0002582684090000052
Figure BDA0002582684090000053
wherein (·)*Represents three data matrices obtained by backward spatial smoothing of complex conjugate, j-1, 2, …, p
Figure BDA0002582684090000054
Figure BDA0002582684090000055
Yb(t0) Comprises the following steps:
Figure BDA0002582684090000056
Figure BDA0002582684090000057
Figure BDA0002582684090000058
according to three data matrixes after backward smoothing
Figure BDA0002582684090000059
Yb(t0) An autocorrelation matrix and two cross-correlation matrices can be calculated:
Figure BDA00025826840900000510
Figure BDA00025826840900000511
Figure BDA00025826840900000512
a matrix can be obtained from equations (30) to (32)
Figure BDA00025826840900000513
And
Figure BDA00025826840900000514
the mathematical forms are respectively as follows:
Figure BDA00025826840900000515
Figure BDA00025826840900000516
Figure BDA0002582684090000061
wherein
Figure BDA0002582684090000062
A special large matrix C is constructed by equations (21) - (23), (33) - (35), and its mathematical form is:
Figure BDA0002582684090000063
performing singular value decomposition on the C to obtain a signal subspace EsI.e. by
Figure BDA0002582684090000064
Where T is an N-dimensional invertible matrix. The implementation of the ESPRIT algorithm is analyzed from the aspect of least square, and the following fitting idea is established:
Figure BDA0002582684090000065
solving the least square solution of the formula (38) to obtain the subarray X1And X2By the rotation factor Ψ betweenXOf subarray X1Rotation factor Ψ from array YYRespectively as follows:
Figure BDA0002582684090000066
Figure BDA0002582684090000067
to psiXAnd ΨYPerforming characteristic decomposition to obtain phiXAnd phiYIs estimated by
Figure BDA0002582684090000068
And
Figure BDA0002582684090000069
the azimuth and elevation information of the signal is contained in
Figure BDA00025826840900000610
And
Figure BDA00025826840900000611
the diagonal elements of the two matrices.
The azimuth and elevation of the N signals are contained therein
Figure BDA00025826840900000612
And
Figure BDA00025826840900000613
in the diagonal elements of the two matrices, there is a two-dimensional angle matching problem. Theoretical psiXAnd ΨYThe eigenvector matrix obtained by the eigenvalue decomposition is T, but in practice, the two eigenvalue decompositions are performed independently, and the arrangement order of the eigenvector may be different, so that the order of the eigenvalue needs to be adjusted to accurately solve the parameter. T is1And T2Are respectively p- ΨXAnd ΨYAnd (3) constructing a pairing matrix by using a characteristic vector matrix obtained by characteristic value decomposition:
Figure BDA0002582684090000071
because the same signal corresponds to the eigenvector T1And T2Are completely correlated, so it can be obtained from the matrix coordinates of the element having the largest absolute value in each row element in the pairing matrix G
Figure BDA0002582684090000072
And
Figure BDA0002582684090000073
and the one-to-one correspondence of the diagonal elements of the two matrixes is the two-dimensional angle pairing information.
Finishing diagonal element pairing according to the two-dimensional angle pairing information to obtain the diagonal elements after the sequence is adjusted
Figure BDA0002582684090000074
And
Figure BDA0002582684090000075
the ith diagonal element of (a) is ui(α)、vi(beta), and calculating the two-dimensional direction angle estimation value of each incident signal by the following formula
Figure BDA0002582684090000076
Figure BDA0002582684090000077
Figure BDA0002582684090000078
Where angle (. cndot.) represents the phase angle.
The flow chart of the method provided by the invention is shown in fig. 2, and the specific implementation steps are as follows:
s1, dividing the parallel linear array into sub-arrays X1、X2And Y;
s2, receiving data X for single snapshot signals of three sub-arrays1(t0)、X2(t0) And Y (t)0) Respectively carrying out forward space smoothing to obtain smoothed data matrixes
Figure BDA0002582684090000079
Yf(t0) Three matrices are calculated according to the equations (18) to (23)
Figure BDA00025826840900000710
Figure BDA00025826840900000711
And
Figure BDA00025826840900000712
s3, for X1(t0)、X2(t0) And Y (t)0) Respectively carrying out backward space smoothing to obtain smoothed data matrixes
Figure BDA00025826840900000713
Figure BDA00025826840900000714
Yb(t0) Three matrices are obtained by calculation according to equations (30) to (35)
Figure BDA00025826840900000715
And
Figure BDA00025826840900000716
s4, constructing a large matrix C according to the formula (36), and performing singular value decomposition on the matrix C to obtain a signal subspace Es；
S5, establishing a least square problem of the formula (38), and solving a least square solution according to the formulas (39) and (40) to obtain a sub-array X1And X2By the rotation factor Ψ betweenXOf subarray X1Rotation factor Ψ from array YY；
S6, rotation factor psiX、ΨYRespectively decomposing the characteristic values to obtain characteristic value pairsAngle matrix
Figure BDA0002582684090000081
And a feature vector T1、T2；
S7, constructing a pairing matrix G according to the formula (41), and searching the matrix coordinate of the element with the maximum absolute value in each row element in G to obtain
Figure BDA0002582684090000085
And
Figure BDA0002582684090000086
and the one-to-one correspondence of the diagonal elements of the two matrixes is the two-dimensional angle pairing information.
S8, finishing diagonal element pairing according to the two-dimensional angle pairing information to obtain the diagonal elements after the sequence is adjusted
Figure BDA0002582684090000082
And
Figure BDA0002582684090000083
has a diagonal element of ui(α)、vi(β)。
S9, calculating the two-dimensional direction angle estimation value of each incident signal by the equations (42) and (43)
Figure BDA0002582684090000084
And finishing direction finding.
The invention has the beneficial effects that: a spatial smoothing technology is expanded to a parallel linear array, a data matrix with the rank larger than the number of signals is obtained by carrying out forward and backward spatial smoothing on single snapshot data, and a basis is provided for realizing a subspace type direction finding algorithm. The two-dimensional ESPRIT algorithm is realized by constructing a special matrix, the direction angle is directly solved, the two-dimensional spectrum peak search is avoided, and the calculation amount is greatly reduced. By constructing the pairing matrix, the problem of two-dimensional angle pairing is solved.
Drawings
FIG. 1 is a schematic diagram of parallel linear arrays in space
FIG. 2 is a flow of a parallel linear array single-snapshot two-dimensional ESPRIT direction-finding method
FIG. 3 is a distribution of direction finding results
FIG. 4 shows the direction-finding distribution of the incident signal at (70, 110) degree
FIG. 5 shows the direction-finding distribution of incident signals at (90, 90) degree
FIG. 6 shows the distribution of the incident signal direction at (110, 70) degree
FIG. 7 is a plot of direction-finding RMSE versus SNR for angles alpha and beta
FIG. 8 is a plot of direction-finding RMSE versus SNR for multiple signal incident angles alpha
FIG. 9 is a curve of direction-finding RMSE of angles alpha and beta along with the change of the number of linear array elements
Detailed Description
The invention utilizes matlab software to verify the direction-finding algorithm scheme of the multi-common-frequency information source phase interferometer, and for the sake of simplification, the following assumptions are made for the algorithm model:
1. all array element channels in the parallel linear array have consistency and no array element channel radiation phase error;
2. the incident signal is a far-field narrow-band signal and is propagated to the parallel linear arrays in the space in a plane wave mode;
3. the phase difference between adjacent array elements is not fuzzy, namely the distance between the array elements and the distance between the two parallel linear arrays are both smaller than the half wavelength of an incident signal.
The method comprises the following steps of validity verification:
considering the double parallel uniform linear arrays X, Y, the distance between the linear array X and the linear array Y is d, and the adjacent array elements on the linear arrays are d. The number of parallel array elements X array elements is 34, the number of the first 33 array elements is divided into a sub-array X1, the number of the last 33 array elements is divided into X2, the number of array elements Y is 33, and all array element channels have consistency. When the space smoothing is carried out, the number of the subarrays is 17, and the number of the subarray elements is 17. Three far-field narrow-band signals are incident on the double parallel uniform linear arrays, and the direction angle coordinates (alpha, beta) are respectively as follows: the signal amplitudes of (70, 110) degrees, (90, 90) degrees and (110, 70) degrees are all 1, the signal-to-noise ratio is 20dB, the array element receiving data is single snapshot, and 1000 Monte Carlo tests are carried out to obtain three-beam signal direction-finding distribution (figure 3), (70, 110) degree direction incident signal direction-finding distribution (figure 4), (90, 90) degree direction incident signal direction-finding distribution (figure 5) and (110, 70) degree direction incident signal direction-finding distribution (figure 6).
As can be seen from fig. 3, through multiple monte carlo experiments, the direction-finding distribution results are distributed in three clusters, and the two-dimensional angles corresponding to the horizontal and vertical coordinates of the central positions of the three clusters exactly correspond to the real two-dimensional direction angles of incidence of the three beams of signals set in the experiments, which indicates that the method provided by the invention can simultaneously realize accurate estimation of the directions of the multiple beams of single-snapshot incoming wave signals under the condition of a certain signal-to-noise ratio. Further observing the direction-finding distribution of the three-beam signals with direction angle coordinates (alpha, beta) of (70, 110) degrees, (90, 90) degrees and (110, 70) degrees corresponding to fig. 4, 5 and 6, the direction-finding results are distributed around the coordinate corresponding to the target real incidence direction, the absolute value of the angle deviation on the angles alpha and beta does not exceed 2 degrees, and the surface method has better direction-finding precision.
Performance of the method at different signal-to-noise ratios:
considering the double parallel uniform linear arrays X, Y, the distance between the linear array X and the linear array Y is d, and the adjacent array elements on the linear arrays are d. The number of the array elements X is 34, the number of the first 33 array elements is divided into a sub-array X1, the number of the last 33 array elements is divided into X2, the number of the array elements Y is 33, and all array element channels have consistency. When the space smoothing is carried out, the number of the subarrays is 17, and the number of the subarray elements is 17. Far-field narrow-band signals are incident on the double parallel uniform linear arrays, the signal amplitude is 1, and the array element receiving data is single snapshot. And carrying out 1000 Monte Carlo tests under different signal-to-noise ratios, and reflecting the direction-finding precision of the algorithm by calculating the mean square error of the direction-finding angle.
When a signal is incident at an angle of directivity (90, 90 degrees), as can be seen in FIG. 7, the direction-finding RMSE at angles alpha and beta decreases as the signal-to-noise ratio (SNR) increases. When the signal-to-noise ratio is 0dB, the RMSE is still less than 4 degrees, which shows that the method has certain angle resolution capability in a severe noise environment. When the signal-to-noise ratio is higher than 20dB, the RSME is less than 0.5, which shows that the method has better direction-finding precision under the condition of better signal-to-noise ratio. When the three beams of signals are incident on the parallel linear arrays in space at (40, 90) degrees, (60, 90) degrees and (80, 90) degrees, respectively, it is understood from fig. 8 that the direction-finding RMSE at the angle alpha when the multiple signals are incident decreases as the SNR increases, and the corresponding direction-finding accuracy increases.
By combining the results of fig. 7 and fig. 8, the direction-finding precision of the parallel linear array single-snapshot two-dimensional ESPRIT algorithm provided by the invention is improved along with the rise of the signal-to-noise ratio, has certain requirements on the signal-to-noise ratio, and has better direction-finding performance when the signal-to-noise ratio is higher; the two curves of the direction-finding RMSE on the direction angle alpha and the direction-finding RMSE on the direction angle beta change along with the signal-to-noise ratio are basically overlapped, and the expression that the direction-finding performance of the angle alpha and the direction-finding performance of the angle beta are approximately consistent when the algorithm carries out two-dimensional angle estimation.
The method has the following performances under different array element numbers:
considering the double parallel uniform linear arrays X, Y, the distance between the linear array X and the linear array Y is d, and the adjacent array elements on the linear arrays are d. The array element X array element number M +1, the former M array element numbers are divided into sub-arrays X1, the latter M array elements are divided into X2, the array element Y number is M, and each array element channel has consistency. The number of sub-arrays when performing spatial smoothing is
Figure BDA0002582684090000101
The number of subarray elements is
Figure BDA0002582684090000102
Wherein
Figure BDA0002582684090000103
Meaning that the rounding is done down,
Figure BDA0002582684090000104
indicating a rounding down. A beam of far-field narrow-band signals with the signal amplitude of 1 is incident on the double parallel uniform linear arrays at the direction angles (90, 90 degrees), and the data received by the array elements are single-shot. The array element number M is changed, 500 Monte Carlo tests are carried out under different array element numbers, the direction-finding precision of the algorithm is reflected by calculating the mean square error of the direction-finding angle, and the curve that the direction-finding RMSE of the angle alpha and the angle beta changes along with the array element number M is obtained and is shown in figure 9.
As can be seen from fig. 9, the direction-finding RMSE of the angles alpha and beta both decrease with the increase of the number of linear array elements, and when the signal-to-noise ratio is 20dB, even when the sparse array with the number of linear array elements M of 10 is used for receiving, the RMSE is less than 0.7 degrees, which indicates that the method has better direction-finding performance with enough array elements.

Claims (1)

1. A single-snapshot two-dimensional direction finding method for parallel linear arrays is provided, wherein the parallel linear arrays are composed of two mutually parallel uniform linear arrays X and Y, and the distance between the linear arrays X and Y along the Y-axis direction is dyThe distance between adjacent array elements on the linear array along the x-axis direction is dxThe linear array X and the linear array Y respectively have M +1 array elements and M array elements, and N far-field narrow-band signals S (t) ([ s ]) with the same central frequency are arranged in the space1(t),s2(t),…,sN(t)]TAt different two-dimensional direction angles (alpha)i,βi) I 1, …, N is incident on the parallel linear array, the signal wavelength is λ, where αiE (0, pi) and betaiE (0, pi) respectively represents an included angle between the incident direction of the ith signal and the positive direction of the y axis of the coordinate axis and an included angle between the incident direction of the ith signal and the positive direction of the x axis of the coordinate axis, and the direction finding method is characterized by comprising the following steps of:
s1, dividing the linear array X into sub-arrays X1、X2Wherein the 1 st array element to the Mth array element of X form a subarray X1The 2 nd array element to the M +1 th array element form a subarray X2The parallel linear array is defined as three sub-arrays, X respectively1、X2And Y;
s2, single snapshot signals of three sub-arrays at time t0Received data X of1(t0)、X2(t0) And Y (t)0) Forward spatial smoothing is performed separately, wherein:
X1(t0)＝[x1(t0),x2(t0),…,xM(t0)]T
X2(t0)＝[x2(t0),x3(t0),…,xM+1(t0)]T
Y(t0)＝[y1(t0),y2(t0),…,yM(t0)]T
obtaining a smoothed data matrix
Figure FDA0002582684080000011
Yf(t0) Comprises the following steps:
Figure FDA0002582684080000012
Figure FDA0002582684080000013
Yf(t0)＝[yf(1)(t0),yf(2)(t0),…,yf(p)(t0)]T
wherein p is the number of smooth subarrays, the number of array elements of each smooth subarray is M, and the relation M is M + p-1, and 2p is more than or equal to N; obtaining an autocorrelation matrix and two cross-correlation matrices according to the three forward smoothed data matrices:
Figure FDA0002582684080000014
Figure FDA0002582684080000021
Figure FDA0002582684080000022
wherein R iss＝S(t0)S(t0)H，
Figure FDA0002582684080000023
Representing the variance of m-dimensional zero mean noise, ImRepresenting an m-dimensional identity matrix;
Amis an array X1Array flow pattern of the first subarray above:
Figure FDA0002582684080000024
Figure FDA0002582684080000025
wherein epsilonx＝2πdx/λ，εy＝2πdyλ, λ being the signal wavelength; phiXAnd phiYIs a flow pattern transformation matrix:
Figure FDA0002582684080000026
Figure FDA0002582684080000027
thereby obtaining three covariance matrixes
Figure FDA0002582684080000028
And
Figure FDA0002582684080000029
Figure FDA00025826840800000210
Figure FDA00025826840800000211
Figure FDA00025826840800000212
wherein
Figure FDA00025826840800000213
S3, for X1(t0)、X2(t0) And Y (t)0) Respectively carrying out backward space smoothing to obtain smoothed data matrixes
Figure FDA00025826840800000214
Figure FDA00025826840800000215
Yb(t0)：
Figure FDA0002582684080000031
Figure FDA0002582684080000032
Yb(t0)＝[yb(1)(t0),yb(2)(t0),…,yb(p)(t0)]T
Obtaining an autocorrelation matrix and two cross-correlation matrices according to the three data matrices after backward smoothing:
Figure FDA0002582684080000033
Figure FDA0002582684080000034
Figure FDA0002582684080000035
three covariances can be obtained as wellMatrix array
Figure FDA0002582684080000036
And
Figure FDA0002582684080000037
Figure FDA0002582684080000038
Figure FDA0002582684080000039
Figure FDA00025826840800000310
wherein
Figure FDA00025826840800000311
S4, constructing a large matrix C according to the three matrixes obtained in the step S2 and the three matrixes obtained in the step S3:
Figure FDA00025826840800000312
and performing singular value decomposition on the matrix C to obtain a signal subspace Es：
Figure FDA0002582684080000041
Wherein T is an N-dimensional reversible matrix, and a model is established based on a rotation invariant subspace algorithm:
Figure FDA0002582684080000042
s5, solving the least square solution of the model established in the step S4 to obtain a subarray X1And X2By the rotation factor Ψ betweenXOf subarray X1Rotation factor Ψ from array YY：
Figure FDA0002582684080000043
Figure FDA0002582684080000044
S6, rotation factor psiX、ΨYRespectively decomposing the characteristic values to obtain an angular matrix of the characteristic values
Figure FDA0002582684080000045
And a feature vector T1、T2；
S7, constructing a pairing matrix G:
Figure FDA0002582684080000046
finding the matrix coordinate of the element with the maximum absolute value in each row element in G to obtain
Figure FDA0002582684080000047
And
Figure FDA0002582684080000048
the one-to-one correspondence relationship of the diagonal elements of the two matrices, namely two-dimensional angle pairing information;
s8, finishing diagonal element pairing according to the two-dimensional angle pairing information to obtain the diagonal elements after the sequence is adjusted
Figure FDA0002582684080000049
And
Figure FDA00025826840800000410
has a diagonal element of ui(α)、vi(β)；
S9, calculating two-dimensional direction angle estimated values of all incident signals, and finishing direction finding:
Figure FDA00025826840800000411
Figure FDA00025826840800000412
where angle (. cndot.) represents the phase angle.
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