CN112816936A - Two-dimensional sparse linear array direction of arrival estimation method based on matrix matching - Google Patents

Two-dimensional sparse linear array direction of arrival estimation method based on matrix matching Download PDF

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CN112816936A
CN112816936A CN202011619778.2A CN202011619778A CN112816936A CN 112816936 A CN112816936 A CN 112816936A CN 202011619778 A CN202011619778 A CN 202011619778A CN 112816936 A CN112816936 A CN 112816936A
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CN112816936B (en
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葛启超
郭艺夺
冯为可
胡晓伟
宫健
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Air Force Engineering University of PLA
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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
    • 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
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section

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Abstract

The invention relates to a method for estimating the direction of arrival, in particular to a two-dimensional sparse linear array direction of arrival estimation method based on matrix matching, which comprises the following steps: establishing a basic model of two-dimensional sparse linear array two-dimensional direction of arrival estimation, and defining the incident angle of the ith signal as (theta)l,
Figure DDA0002877933520000017
) Assuming that L far-field narrow-band signals are incident into the array in total, receiving data of two sub-arrays in the array at the time k are obtained, a covariance matrix of the receiving data of the two sub-arrays is obtained according to the received data, virtual receiving data of continuous ULA parts in the virtual array formed by the two sub-arrays on the x axis and the y axis are obtained, a Toeplitz matrix is constructed by utilizing the virtual receiving data, and then estimated values of two-dimensional angles are respectively obtained by constructing a spatial spectrum function
Figure DDA0002877933520000011
And
Figure DDA0002877933520000012
assumptions and estimates
Figure DDA0002877933520000013
In
Figure DDA0002877933520000014
The corresponding other dimensional angle estimate is
Figure DDA0002877933520000015
The least square algorithm and the matrix characteristic decomposition are sequentially carried out to obtain an angle estimation value of
Figure DDA0002877933520000016

Description

Two-dimensional sparse linear array direction of arrival estimation method based on matrix matching
Technical Field
The invention relates to an estimation method of a direction of arrival, in particular to a two-dimensional sparse linear array direction of arrival estimation method based on matrix matching.
Background
Direction of Arrival (DOA) estimation is an important component of radar signal processing, and a linear array structure can only estimate a one-dimensional angle, and cannot simultaneously estimate two-dimensional angle information of a signal, so that a two-dimensional array structure is generally required to be used when pitch and azimuth information of the signal needs to be simultaneously estimated. Due to the fact that the two-dimensional linear array is simple in structure, two-dimensional DOA estimation in the two-dimensional linear array is widely researched, and in order to further expand the degree of freedom of the two-dimensional DOA estimation, the two-dimensional DOA estimation suitable for the two-dimensional sparse linear array is provided. Compared with the application of the traditional two-Dimensional DOA estimation algorithm to a two-Dimensional sparse linear array, the two-Dimensional matching DOA (2Dimensional ordered DOA,2-D PDOA) estimation algorithm is provided in the prior art, the degree of freedom of two-Dimensional DOA estimation is improved by utilizing the sparse characteristic of each Dimensional array of an L-shaped sparse array, after one-Dimensional angles are estimated by utilizing one-Dimensional arrays, the other one-Dimensional angles are estimated by utilizing the cross-correlation matrix of two-Dimensional array received data and the characteristic of mutual matching of the two-Dimensional angles, and the simultaneous estimation and matching of the two-Dimensional DOA are realized.
However, the 2-D PDOA estimation algorithm estimates two-dimensional angles separately, in which one dimension directly performs angle estimation using a sparse array, and the angle estimation for the other dimension is based on the angle estimation of the other dimension, so that the angle estimation error of the other dimension is large.
Disclosure of Invention
The invention aims to provide a two-dimensional sparse linear array direction of arrival estimation method based on matrix matching, and solves the problem that in the current two-dimensional angle estimation of the direction of arrival, the two-dimensional angle estimation process is related, so that the angle estimation error of the other dimension is larger.
In order to solve the technical problems, the invention adopts the following technical scheme:
a two-dimensional sparse linear array direction of arrival estimation method based on matrix matching comprises the following steps:
s1, establishing a basic model of two-dimensional sparse linear array two-dimensional direction of arrival estimation, wherein the array model comprises M 12d and M2Two subarrays of 3d, where d is a half wavelength;
s2, defining the incident angle of the first signal as
Figure BDA0002877933500000011
Wherein theta isl、
Figure BDA0002877933500000012
Included angles between the signal incidence direction and xoz and yoz surfaces are respectively obtained, and L far-field narrow-band signals are assumed to be incident into the array, so that the received data of two sub-arrays in the array at the time k are obtained;
s3, obtaining a received data covariance matrix of the two sub-arrays according to the received data;
s4, obtaining virtual received data of continuous ULA parts of the two sub-arrays on the x axis and the y axis;
s5, constructing Toeplitz matrix by using the virtual received data, and then respectively obtaining estimated values of two-dimensional angles by constructing a spatial spectrum function
Figure BDA0002877933500000021
And
Figure BDA0002877933500000022
s6, assumption and estimation
Figure BDA0002877933500000023
In
Figure BDA0002877933500000024
The corresponding other dimensional angle estimate is
Figure BDA0002877933500000025
Wherein
Figure BDA0002877933500000026
Is to
Figure BDA0002877933500000027
Is reordered to obtain
Figure BDA0002877933500000028
Where T ∈ {0,1}L×LIs a column transform matrix;
s7, obtaining the result by sequentially carrying out least square algorithm and matrix characteristic decomposition
Figure BDA0002877933500000029
Then, an angle estimation value is obtained as
Figure BDA00028779335000000210
The further technical scheme is that the received data of the two subarrays at the time k are respectively as follows:
Figure BDA00028779335000000211
and
Figure BDA00028779335000000212
a further technical solution is that the covariance matrices of the received data of the two sub-arrays are respectively:
Figure BDA00028779335000000213
and
Figure BDA00028779335000000214
the further technical scheme is that the virtual received data of the continuous ULA parts in the virtual array formed by the two sub-arrays on the x axis and the y axis are respectively as follows:
Figure BDA00028779335000000215
and
Figure BDA00028779335000000216
a further technical solution is that the Toeplitz matrices constructed by using the virtual received data are respectively:
Figure BDA00028779335000000217
and
Figure BDA00028779335000000218
the further technical scheme is that the constructed spatial spectrum function is as follows:
Figure BDA00028779335000000219
compared with the prior art, the invention has the beneficial effects that: compared with the existing 2-D PDOA estimation algorithm, the estimation method respectively estimates two-dimensional angles, avoids the influence of one-dimensional angle estimation errors on the estimation of another-dimensional angle, realizes the matching of two-dimensional direction-of-arrival estimation values, and simultaneously improves the precision of angle estimation.
Drawings
FIG. 1 is a basic model of array receiving signals in two-dimensional sparse linear array two-dimensional direction of arrival estimation in the present invention.
Fig. 2 is a graph of RMSE versus SNR for DOA estimates of the present invention, where K is 200.
FIG. 3 is a performance analysis diagram of a two-dimensional sparse linear array DOA estimation algorithm based on matrix transformation in the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1:
the two-dimensional sparse linear array direction of arrival estimation method based on matrix matching in the embodiment specifically comprises the following steps:
s1, establishing a basic model of two-dimensional sparse linear array two-dimensional direction of arrival estimation, wherein the array model comprises M 12d and M2Two subarrays of 3d, where d is a half wavelength;
s2, defining the incident angle of the first signal as
Figure BDA0002877933500000031
Wherein theta isl、
Figure BDA0002877933500000032
Included angles between the signal incidence direction and xoz and yoz surfaces are respectively obtained, and L far-field narrow-band signals are assumed to be incident into the array, so that the received data of two sub-arrays in the array at the time k are obtained;
s3, obtaining a received data covariance matrix of the two sub-arrays according to the received data;
s4, obtaining virtual received data of continuous ULA parts of the two sub-arrays on the x axis and the y axis;
s5, constructing Toeplitz matrix by using virtual received data, and then constructing spaceSpectral functions, respectively obtaining two-dimensional angle estimates
Figure BDA0002877933500000033
And
Figure BDA0002877933500000034
s6, assumption and estimation
Figure BDA0002877933500000035
In
Figure BDA0002877933500000036
The corresponding other dimensional angle estimate is
Figure BDA0002877933500000037
Wherein
Figure BDA0002877933500000038
Is to
Figure BDA0002877933500000039
Is reordered to obtain
Figure BDA00028779335000000310
Where T ∈ {0,1}L×LIs a column transform matrix;
s7, obtaining the result by sequentially carrying out least square algorithm and matrix characteristic decomposition
Figure BDA00028779335000000311
Then, an angle estimation value is obtained as
Figure BDA00028779335000000312
The received data of the two subarrays at the time k are respectively:
Figure BDA00028779335000000313
and
Figure BDA00028779335000000314
wherein, the covariance matrixes of the received data of the two sub-arrays are respectively:
Figure BDA0002877933500000041
and
Figure BDA0002877933500000042
the two sub-arrays form virtual receiving data of continuous ULA parts in the virtual array on an x axis and a y axis, and the virtual receiving data are respectively as follows:
Figure BDA0002877933500000043
and
Figure BDA0002877933500000044
wherein, the Toeplitz matrix constructed by using the virtual received data is respectively:
Figure BDA0002877933500000045
and
Figure BDA0002877933500000046
wherein the constructed spatial spectrum function is as follows:
Figure BDA0002877933500000047
and
Figure BDA0002877933500000048
example 2:
as shown in FIG. 1, a basic model for establishing two-dimensional DOA estimation of a two-dimensional sparse linear array by taking an L-type Coprime array as an example is a two-array structure phaseThe same Coprime array forms an L-shaped Coprime array, and the two array models comprise M 12d and M2An array of 3d where d is half wavelength and then the angle of incidence of the first signal is defined as
Figure BDA0002877933500000049
Wherein theta isl、
Figure BDA00028779335000000410
The included angles of the signal incidence direction and the xoz and yoz surfaces are respectively, and for the convenience of identification, two Coprime arrays on the x axis and the y axis are respectively called as SxAnd SyWherein, in the step (A),
Figure BDA00028779335000000411
for the set of positions of Coprime array elements on the x-axis, MxThe total number of array elements of the Coprime array on the x axis is;
Figure BDA00028779335000000412
for the set of positions of Coprime array elements on the y-axis, MyThe total number of array elements of the Coprime array on the y axis. Assuming that a total of L far-field narrow-band signals enter the array, the received data of two sub-arrays in the array at the time k are respectively:
Figure BDA00028779335000000413
and
Figure BDA00028779335000000414
wherein the content of the first and second substances,
Figure BDA00028779335000000415
denotes SxOn the upper part
Figure BDA00028779335000000416
A steering vector of direction;
Figure BDA00028779335000000417
denotes SxAn array manifold matrix of (a);
Figure BDA0002877933500000051
denotes SyUpper thetalA steering vector of direction;
Figure BDA0002877933500000052
denotes SyAn array manifold matrix of (a);
Figure BDA0002877933500000053
representing an incident signal waveform;
Figure BDA0002877933500000054
and
Figure BDA0002877933500000055
respectively represent SxAnd SyAnd the received noise signal.
The covariance matrices of the received data of the two Coprime arrays are respectively:
Figure BDA0002877933500000056
and
Figure BDA0002877933500000057
wherein the content of the first and second substances,
Figure BDA0002877933500000058
an autocorrelation matrix representing the incident signal;
Figure BDA0002877933500000059
and
Figure BDA00028779335000000510
respectively, identity matrices of corresponding dimensions.
Definition of SxAnd SyThe successive ULA portions in forming the virtual array are each
Figure BDA00028779335000000511
And
Figure BDA00028779335000000512
it can be known that
Figure BDA00028779335000000513
And
Figure BDA00028779335000000514
the virtual reception data of (a) can be respectively expressed as:
Figure BDA00028779335000000515
and
Figure BDA00028779335000000516
wherein the content of the first and second substances,
Figure BDA00028779335000000517
to represent
Figure BDA00028779335000000518
The element located at array element u. (for example, when the distribution of the positions of the elements S ═ 1,3,5, the received data is zS＝[0.2,0.5,0.4]TWhen there is<zS>1＝0.2、<zS>3＝0.5、<zS>5＝0.4。)
Wherein, Ux(U) and Uy(u) are defined as:
Figure BDA00028779335000000519
and
Figure BDA00028779335000000520
wherein the content of the first and second substances,
Figure BDA00028779335000000521
to be located at position m1And m2And the cross correlation value between the received data of the real array elements is defined as:
Figure BDA00028779335000000522
constructing a Toeplitz matrix by using the virtual received data to obtain:
Figure BDA0002877933500000061
and
Figure BDA0002877933500000062
wherein the content of the first and second substances,
Figure BDA0002877933500000063
by constructing the following spatial spectral functions, the respective estimates can be obtained
Figure BDA0002877933500000064
And
Figure BDA0002877933500000065
Figure BDA0002877933500000066
and
Figure BDA0002877933500000067
the first two formulas are subjected to spectral peak search to respectively obtain estimated values of two-dimensional angles
Figure BDA0002877933500000068
And
Figure BDA0002877933500000069
but since it is not known
Figure BDA00028779335000000610
And
Figure BDA00028779335000000611
thus, two angles need to be matched.
The matching method is as follows:
suppose that the estimated angles are respectively
Figure BDA00028779335000000612
And
Figure BDA00028779335000000613
and are connected with
Figure BDA00028779335000000614
In
Figure BDA00028779335000000615
The arrangement sequence of the angles in the other dimension corresponding to one dimension is
Figure BDA00028779335000000616
Can obviously be considered as
Figure BDA00028779335000000617
Is to
Figure BDA00028779335000000618
Reordering is performed, i.e. the following can be written:
Figure BDA00028779335000000619
wherein T is ∈ {0,1}L×LThe matrix is a column transformation matrix, namely, only one element in each column of each row is 1, and the rest elements are 0.
Suppose that
Figure BDA00028779335000000620
The array manifold matrix obtained by calculation is
Figure BDA00028779335000000621
The formula can be seen as follows:
Figure BDA00028779335000000622
thus, a matrix can be considered
Figure BDA00028779335000000623
Sum matrix
Figure BDA00028779335000000624
Are matched, i.e. there are:
Figure BDA00028779335000000625
will be provided with
Figure BDA00028779335000000626
Brought into
Figure BDA00028779335000000627
In (1), can obtain
Figure BDA00028779335000000628
The matching problem of the angle is converted into a solving problem of a transformation matrix, and the matrix T can be solved through a minimum optimization model as follows:
Figure BDA00028779335000000629
in particular, when L.ltoreq.M is satisfiedT,xThe first equation is a least squares problem, matrix
Figure BDA0002877933500000071
With explicit expressions:
Figure BDA0002877933500000072
for matrix RSxThe characteristic decomposition is carried out to obtain:
Figure BDA0002877933500000073
wherein the content of the first and second substances,
Figure BDA0002877933500000074
respectively represent matrices
Figure BDA0002877933500000075
The signal subspace and the noise subspace of (1);
Figure BDA0002877933500000076
then it is by the corresponding Ux,s、Ux,nThe eigenvalues of (c) form a diagonal matrix.
Due to the fact that
Figure BDA0002877933500000077
Can likewise be spanned into subspaces, so that the matrix RsIt can be estimated that:
Figure BDA0002877933500000078
will be provided with
Figure BDA0002877933500000079
Bringing in
Figure BDA00028779335000000710
The following can be obtained:
Figure BDA00028779335000000711
for the estimated matrix
Figure BDA00028779335000000712
There is no guarantee that there are only 1 element per row and column and 1, so for the estimated matrix
Figure BDA00028779335000000713
Further processing is required in the following way:
setting the maximum element of each row and each column to be 1 and setting the rest elements to be 0 to satisfy
Figure BDA00028779335000000714
Medium, and ultimately results in:
Figure BDA00028779335000000715
the final angle estimate obtained is
Figure BDA00028779335000000716
As shown in fig. 2 and fig. 3, compared with the existing estimation algorithm, the results of the one-dimensional estimation algorithm are completely overlapped and are consistent with the existing algorithm, and the deviation of the other bit is smaller than that of the existing algorithm, i.e. compared with the existing algorithm and the 2-D PDOA estimation algorithm, the two-dimensional angles are respectively estimated, so that the influence of one-dimensional angle estimation error on the estimation of the other-dimensional angle is avoided, the two-dimensional DOA estimation value matching is realized, and the angle estimation precision is improved.
The method utilizes the characteristic construction optimization problem calculation transformation matrix of the incoming wave direction two-dimensional angle matching to realize the process of first estimation and then matching, and avoids the phenomenon that the error of one-dimensional angle estimation is increased by the estimation error of another one-dimensional angle.
Although the invention has been described herein with reference to a number of illustrative embodiments thereof, it should be understood that numerous other modifications and embodiments can be devised by those skilled in the art that will fall within the spirit and scope of the principles of this disclosure. More specifically, various variations and modifications are possible in the component parts and/or arrangements of the subject combination arrangement within the scope of the disclosure, the drawings and the appended claims. In addition to variations and modifications in the component parts and/or arrangements, other uses will also be apparent to those skilled in the art.

Claims (6)

1. A two-dimensional sparse linear array direction of arrival estimation method based on matrix matching is characterized by comprising the following steps:
s1, establishing a basic model of two-dimensional sparse linear array two-dimensional direction of arrival estimation, wherein the array model comprises M12d and M2Two subarrays of 3d, where d is a half wavelength;
s2, defining the incident angle of the first signal as
Figure FDA0002877933490000011
Wherein theta isl、
Figure FDA0002877933490000012
Included angles between the signal incidence direction and xoz and yoz surfaces are respectively obtained, and L far-field narrow-band signals are assumed to be incident into the array, so that the received data of two sub-arrays in the array at the time k are obtained;
s3, obtaining a received data covariance matrix of the two sub-arrays according to the received data;
s4, obtaining virtual received data of continuous ULA parts of the two sub-arrays on the x axis and the y axis;
s5, constructing Toeplitz matrix by using the virtual received data, and then respectively obtaining estimated values of two-dimensional angles by constructing a spatial spectrum function
Figure FDA0002877933490000013
And
Figure FDA0002877933490000014
s6, assumption and estimation
Figure FDA0002877933490000015
In
Figure FDA0002877933490000016
The corresponding other dimensional angle estimate is
Figure FDA0002877933490000017
Wherein
Figure FDA0002877933490000018
Is to
Figure FDA0002877933490000019
Is reordered to obtain
Figure FDA00028779334900000110
Where T ∈ {0,1}L×LIs a column transform matrix;
s7, obtaining the result by sequentially carrying out least square algorithm and matrix characteristic decomposition
Figure FDA00028779334900000111
Then, an angle estimation value is obtained as
Figure FDA00028779334900000112
2. The matrix matching-based two-dimensional sparse linear array direction of arrival estimation method of claim 1, wherein: the received data of the two subarrays at the time k are respectively:
Figure FDA00028779334900000113
and
Figure FDA00028779334900000114
3. the matrix matching-based two-dimensional sparse linear array direction of arrival estimation method of claim 1, wherein: received data covariance of two subarraysThe matrices are respectively:
Figure FDA00028779334900000115
and
Figure FDA00028779334900000116
4. the matrix matching-based two-dimensional sparse linear array direction of arrival estimation method of claim 1, wherein: the virtual received data of the two sub-arrays forming successive ULA parts in the virtual array on the x-axis and the y-axis are respectively:
Figure FDA00028779334900000117
and
Figure FDA00028779334900000118
5. the matrix matching-based two-dimensional sparse linear array direction of arrival estimation method of claim 1, wherein: the Toeplitz matrices constructed by using the virtual received data are respectively:
Figure FDA0002877933490000021
and
Figure FDA0002877933490000022
6. the matrix matching-based two-dimensional sparse linear array direction of arrival estimation method of claim 1, wherein:
the spatial spectrum function is constructed as follows:
Figure FDA0002877933490000023
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