CN111781573A - Scattering center model parameter estimation method based on improved 3D-ESPRIT algorithm - Google Patents

Scattering center model parameter estimation method based on improved 3D-ESPRIT algorithm Download PDF

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CN111781573A
CN111781573A CN202010666490.4A CN202010666490A CN111781573A CN 111781573 A CN111781573 A CN 111781573A CN 202010666490 A CN202010666490 A CN 202010666490A CN 111781573 A CN111781573 A CN 111781573A
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CN111781573B (en
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郑舒予
张小宽
周剑雄
宗彬锋
赵唯辰
徐嘉华
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Air Force Engineering University of PLA
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Abstract

A scattering center model parameter estimation method based on an improved 3D-ESPRIT algorithm is provided, and comprises the following steps: acquiring target polarized electromagnetic scattering data; establishing a Hankel matrix; square processing; singular value decomposition; constructing a process matrix Fx(ii) a Obtaining signal subspaces corresponding to the y direction and the z direction of a radar coordinate system by using the permutation matrix J in the foregoing; calculating a main eigenvalue vector and corresponding elements thereof; and solving scattering intensity parameters. The method effectively prolongs the length of the available target electromagnetic scattering echo data, and can further improve the parameter estimation performance and noise robustness of the algorithm.

Description

Scattering center model parameter estimation method based on improved 3D-ESPRIT algorithm
Technical Field
The invention relates to a scattering center model parameter estimation and extraction technology based on the geometric diffraction theory (GTD), in particular to a scattering center model parameter estimation method based on an improved 3D-ESPRIT (three-dimensional-relating signal parameter of variable-angle information technology) algorithm.
Background
Summary of the development of scattering centers
Currently, a scattering center model based on geometric diffraction theory (GTD) has become one of the effective expression models for radar target electromagnetic scattering data (POTTER L C, CHIANG D M, CARRIER R.A GTD-based parameter model for radar calibration [ J ]. IEEETransmission on extensions and Propagation, 1995, 43 (10): 1058-1066.). The core mechanism of the GTD scattering center model is: the electromagnetic scattering echo of the radar target in a high-frequency region can be approximately equivalent to coherent superposition synthesis of a limited number of strong scattering centers. GTD scattering center model As a classical model for describing the electromagnetic scattering properties of a target, in the field of radar target identification (1, DING B Y, WEN G J.A regio mapping on 3D scattering center with application to SAR target registration [ J ] IEEE Sensors Journal, 2018, 18 (11): 4623-4632.2, LI T L, DU L. SAR automatic target registration on distribution center model and distributed directional Learning [ J ] IEEE Sensors Journal, 2019, 19 (12): 4598-4611.3, ZHOU J X, SHI Z G, CHENG X, experimental target registration of radar J [ 19 ] SAR extrapolation of strong scattering center spectrum and frequency estimation of radar target frequency, IEEE scattering center model and distributed scattering center model [ 13 ] research center, 3713, scientific extrapolation center of radar frequency, 3719, scientific spectrum, and distributed radar center model, and distributed target tracking center model, and distributed radar frequency estimation of radar target tracking center, 2016.2, Zuijinrong, research on an inversion method of a three-dimensional electromagnetic scattering parameterized model of a target [ D ]. Changsha: the university of defense science and technology, 2016.), radar target three-dimensional reconstruction and other military fields have very wide application prospects (ZHAO Y, ZHANG L, JIU B, et al, three-dimensional reconstruction for space acquisition with multistatic inversion method radar systems [ J ]. EURASIP journal on Advances in Signal Processing, 2019: 2019(40).).
And how to accurately estimate the parameters of the GTD scattering center model through the electromagnetic scattering echo data of the radar target so as to construct an accurate scattering center model, which is very important for characterizing the electromagnetic scattering characteristics of the radar target. For such model parameter estimation, researchers at home and abroad use an improved multiple Signal classification algorithm [ J ] for two-dimensional Signal arrival direction estimation, 2019, 41 (9): 2137-2142.2, Zhengshu, Zhang, Zong, based on the improved scattering center parameter extraction and RCS reconstruction [ J ] for systematic engineering and electronics, 2020, 42 (01): 76-82.3, TAN J, NIE P, polarization modeling and reconstruction, 2020, 42 (01): 1534-9, ESPRIT algorithm (1, LEYMS, KO and simulation for location [ J ] Sensors, 2018, 18 (5): 1534-9), ESPRIT algorithm (1, LEYMS, KO and coding, PIERG. and coding [ J ] for location, S ] and I, S, JN, JJ, RS, Zhang, RS, ZHANG X F, SUN H P, et al.non-cyclic generated-accurate for direction of arrival estimation [ J ]. IET Radar, Sonar & Navigation, 2017, 11 (5): 736-744), etc. to extract the parameters.
GTD scattering center models can be divided into one-dimensional, two-dimensional, and three-dimensional models from a dimensional perspective. With the increase of dimensionality, the GTD scattering center model can be used for describing the electromagnetic scattering characteristics of the target more accurately, but the operation complexity of the algorithm is increased. At present, most researchers mainly perform parameter estimation and extraction on a one-dimensional GTD scattering center model and a two-dimensional GTD scattering center model, and the problem of parameter estimation of the three-dimensional GTD scattering center model is rarely related. According to the method, the parameter estimation extraction is carried out on the three-dimensional GTD scattering center model by using a classical 3D-ESPRIT algorithm.
The classic 3D-ESPRIT algorithm can accurately estimate parameters of the three-dimensional GTD scattering center model, but when the noise of the external environment is high, the parameter estimation performance of the algorithm is remarkably reduced. In order to solve the problem, the invention provides a 3D-ESPRIT (P-FB-3D-ESPRIT) algorithm based on forward and backward smoothing of square, and the improved algorithm can effectively improve the noise robustness and the parameter estimation performance of the algorithm.
Second, GTD scattering center model introduction (basic model for the invention)
The GTD scattering center model can be used as a classical scattering center model to effectively describe the electromagnetic backscattering characteristics of a radar target in a high frequency region, and taking a step frequency radar signal as an example, the three-dimensional GTD scattering center model of the target can be expressed as follows (dawn populus, shich et gao, zhao hong Kong, etc.. a 3D-ESPRIT-based scattering center parameter estimation algorithm [ J ] radar science and technology, 2007, 5 (2): 119-:
Figure BSA0000213161490000031
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000032
a back-scattered echo that represents the target,
Figure BSA0000213161490000033
respectively representing the frequency, azimuth angle and pitch angle of the change,
Figure BSA0000213161490000034
respectively representing the azimuth and elevation angles of the target. I represents the number of scattering centers, { Ai，αi，xi，yi，ziAnd represents the scattering intensity, scattering type, transverse distance, longitudinal distance and vertical distance of the ith scattering center respectively. f. ofm＝f0+mΔf，m＝0，1，...，m1,.., M, wherein f0Is the starting frequency, Δ f is the step frequency, M represents the frequency subscript, and M is the total frequency step number; thetan＝θ0+nΔθ，n＝0，1，...，n1,., N, wherein θ0Is the initial azimuth, delta theta is the stepping azimuth, N is the azimuth subscript, and N is the total azimuth stepping number;
Figure BSA0000213161490000035
wherein
Figure BSA0000213161490000036
In order to start the pitch angle,
Figure BSA0000213161490000037
step pitch angle, K is pitch angle subscript, and K is total pitch angle step number; n.DELTA.theta,
Figure BSA0000213161490000038
C is 3 × 108m/s is the propagation speed of the electromagnetic wave,
Figure BSA0000213161490000039
is complex Gaussian white noise αiThe number of the scattering media is an integral multiple of 0.5, the number of the scattering media is 5, and different α are corresponding to different scattering mediaiValue (Wangbu optical zone radar target scattering center extraction and its application research [ D)]Nanjing: nanjing university of aerospace, 2010), see Table 1.
Table 1 α for a typical scattering structureiValue taking
Figure BSA00002131614900000310
Due to the fact that the selected radar working frequency satisfies delta f/f0< 1, and thus can be approximated as follows:
Figure BSA00002131614900000311
the approximate result in the formula (2) is taken into the formula (1), the obtained formula is transformed to cartesian coordinates and subjected to interpolation normalization processing, and resampling techniques (coordinate transformation, difference normalization processing, and resampling techniques are three basic techniques, which are well known by those skilled in the art and are not described herein again) are utilized, so that the electromagnetic echo data of the target can be represented by the formula (3).
Figure BSA0000213161490000041
Wherein M is 0, M-1, N is 0, N-1, K is 0, K-1
Figure BSA0000213161490000042
Wherein f isx0、fy0、fz0Respectively the initial frequency of the radar signal in the x, y and z directions of a radar coordinate system;
Figure BSA0000213161490000043
Pyi＝exp(-4πjΔfyyi/c) (6)
Pzi＝exp(-4πjΔfzzi/c) (7)
wherein Δ fx、Δfy、ΔfzRespectively representing the stepping frequencies in the x direction, the y direction and the z direction under a radar coordinate system, and the expression formula is shown in formula (8):
Figure BSA0000213161490000044
let fcThe center frequency of the radar and the working bandwidth of the radar are B, then:
Figure BSA0000213161490000045
Figure BSA0000213161490000046
Figure BSA0000213161490000047
FIG. 1 shows a three-dimensional frequency domain data range in which interpolated equally spaced data points are contained within a cube.
Equations (5) - (7) include the type parameter and three types of position parameters of the scattering center, and thus can be solved by equations (12) - (14) below:
αi＝(|Pxi|-1)f0/Δf (12)
Figure BSA0000213161490000051
Figure BSA0000213161490000052
Figure BSA0000213161490000053
wherein Δ fx、Δfy、ΔfzRespectively representing stepping frequencies in x, y and z directions respectively representing a radar coordinate system; angle (.) represents the complex phase angle function found in MATLAB.
Disclosure of Invention
In order to further improve the utilization of target echo data, the invention provides a scattering center model parameter estimation method based on an improved 3D-ESPRIT (PQ-FB-3D-ESPRIT) algorithm, which comprises the following steps:
the first step is as follows: obtaining target polarized electromagnetic scattering data
Firstly, on the basis of an original three-dimensional GTD scattering center model, the utilization of target polarization information is increased, and the polarization scattering coefficient S is obtainedi，pAnd adding the three-dimensional GTD scattering center model into the fully polarized three-dimensional GTD scattering center model to obtain a fully polarized three-dimensional GTD scattering center model as follows:
Figure BSA0000213161490000054
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000055
a back-scattered echo that represents the target,
Figure BSA0000213161490000056
frequency, azimuth, pitch representing variation, respectively: f. ofm＝f0+mΔf，m＝0，1，...，M，f0Is the starting frequency,. DELTA.f is the step frequency, and m representsFrequency subscript, M is total frequency step number; thetan＝θ0+ N Δ θ, N ═ 0, 1.., N, where θ is0Is the initial azimuth, delta theta is the stepping azimuth, N is the azimuth subscript, and N is the total azimuth stepping number;
Figure BSA0000213161490000061
wherein
Figure BSA0000213161490000062
In order to start the pitch angle,
Figure BSA0000213161490000063
step pitch angle, K is pitch angle subscript, and K is total pitch angle step number; n.DELTA.theta,
Figure BSA0000213161490000064
Respectively a small corner in the azimuth direction and a small corner in the pitching direction; i represents the number of scattering centers; si，pRepresenting the scattering coefficient of the ith scattering center in a p-polarization mode, p ∈ { hh, hv, vh, vv } represents four polarization modes, hh represents horizontal transmission and horizontal reception, hv represents horizontal transmission and vertical reception, vh represents vertical transmission and horizontal reception, vv represents vertical transmission and vertical reception, B represents the scattering coefficient of the ith scattering center in a p-polarization mode, andirepresenting a first transition parameter; pxi、Pyi、PziRespectively representing second, third and fourth transition parameters, the four parameters being used only for the pair
Figure BSA0000213161490000065
Splitting is carried out, so that subsequent parameter estimation is facilitated; { xi，yi，ziRespectively representing the transverse distance, the longitudinal distance and the vertical distance of the ith scattering center;
Figure BSA0000213161490000066
complex white gaussian noise;
table 1 below gives the scattering matrix for some typical targets;
TABLE 1 Scattering matrix for typical targets
Figure BSA0000213161490000067
The second step is that: establishing a Hankel matrix
Firstly, constructing a Hankel matrix based on target backward electromagnetic data;
firstly, smoothing is carried out along the x direction of a radar coordinate system to construct a [ P × Q × L ]]×[(M-P+1)×(N-Q+1)×(K-L+1)]Of (2) an enhancement matrix XxRepresented by the following formula (17); wherein M, N, K is defined by the same formula (1), P is more than or equal to M/2 and less than or equal to 2M/3, Q is more than or equal to N/2 and less than or equal to 2N/3, L is more than or equal to K/2 and less than or equal to 2K/3, and P, Q, L are process variables with values within the range;
Figure BSA0000213161490000071
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000072
Figure BSA0000213161490000073
and performing forward and backward spatial smoothing on the matrix containing the polarization information to obtain a new total covariance matrix R as shown in formula (20):
Figure BSA0000213161490000074
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000075
representative matrix XxThe autocorrelation covariance matrix of (a);
Figure BSA0000213161490000077
representative matrix XxA cross-correlation covariance matrix of sum matrix Y; y ═ JXx；
Figure BSA0000213161490000076
A permutation matrix with one dimension (P × Q × L) × (P × Q × L) and with anti-diagonal elements of 1 and elements of 0 at the remaining positions;
the third step: squaring process
As can be seen from formula (20), the total covariance matrix R is an Hermittan matrix, and therefore satisfies R ═ RHI.e. R1＝RRH＝R2(ii) a The resulting matrix R after squaring1The eigenvalues and eigenvectors of the total covariance matrix R have the following relations:
Figure BSA0000213161490000081
in the formula, λ1And lambda represents the matrix R obtained by squaring1Eigenvalues of the total covariance R, Λ1Λ respectively represent the matrix R obtained after the square1A feature vector associated with the total covariance R;
by the square of the resulting R1The total covariance matrix R is replaced, the difference between the signal characteristic value and the noise characteristic value can be increased, and the original characteristic vector is not changed, so that the signal characteristic value and the noise characteristic value are more easily distinguished when the signal-to-noise ratio is low; from a mathematical relationship, the variance of each parameter is expressed as follows:
Figure BSA0000213161490000082
Figure BSA0000213161490000083
wherein E {. is a variance,
Figure BSA0000213161490000084
omega respectively represents parameters and original parameters estimated by the ith Monte Carlo experiment; sigma2、γiRespectively representing a characteristic value corresponding to the noise and a characteristic value corresponding to the signal; i represents the total number of scattering centers; v. ofiRepresents the ith characteristic value gammaiA corresponding feature matrix; (v)i)HRepresents viThe transposed matrix of (2); v. ofi＝γiE-XxE represents a dimension of [ P × Q × L ]]×[P×Q×L]The identity matrix of (1); gHIs the transposed matrix of G;
G＝[ai，...，aI](23)
Figure BSA0000213161490000085
Figure BSA0000213161490000086
wherein c is 3 × 108m/s is the propagation velocity of electromagnetic waves, αiRepresents the scattering type of the ith scattering center;
then, as shown in equation (22), the difference, variance, between the noise eigenvalue and the signal eigenvalue is increased
Figure BSA0000213161490000091
The variance of the estimation parameters can be reduced; therefore, a final total covariance matrix R obtained by squaring the following equation (26) is constructed1The method is used for replacing the total covariance matrix R, and can equivalently increase the signal-to-noise ratio and effectively improve the estimation precision of parameters;
R1＝RRH＝R2(26)
the fourth step: singular value decomposition
For the enhancement matrix XxSingular value decomposition to give formula (27):
Figure BSA0000213161490000092
in the formula: u shapexS、VxSAll represent signal characteristic value vectors in the X direction of a radar coordinate system, and are respectively represented by XxThe first I main left eigenvectors and the first I main right eigenvectors; wherein U isxN，VxNRepresents XxRespectively by XxNon-dominant left feature vector and non-dominant right feature ofVector composition; dxSA diagonal matrix formed for the signal eigenvalues; dxNA diagonal matrix formed for the noise eigenvalues;
the fifth step: constructing a process matrix Fx
Constructing a process matrix FxThe following were used:
Figure BSA0000213161490000093
in the formula (I), the compound is shown in the specification,U xS，
Figure BSA0000213161490000094
are respectively a matrix UxSRemoving the back Q × L rows and removing the front Q × L rows to obtain a matrix,
Figure BSA0000213161490000095
representsU xSThe generalized inverse matrix of (2);
and a sixth step: the signal subspaces corresponding to the y direction and the z direction of the radar coordinate system are obtained by utilizing the permutation matrix J in the foregoing
Permutation matrix E under three-dimensional conditionsxy，Eyz，ExzThe following were used:
Figure BSA0000213161490000096
Figure BSA0000213161490000097
Figure BSA0000213161490000098
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000099
representing the Kronecker product,
Figure BSA00002131614900000910
represents an element in the (q, l) position of1, a Q × L matrix with elements 0 elsewhere,
Figure BSA0000213161490000101
represents an L × P matrix with 1 element at the (L, P) position and 0 elements at other positions,
Figure BSA0000213161490000102
represents a P × Q matrix with 1 element at the (P, Q) position and 0 elements at other positions;
according to an amplification matrix E in three directionsxy、Eyz、ExzThe relation between the signal subspaces in different directions of the radar coordinate system is obtained as follows:
UyS＝ExyUxS(32)
UzS＝EyzUyS(33)
UxS＝ExzUzS(34)
in the formula of UySRepresenting a signal characteristic value vector in the y direction of a radar coordinate system; u shapezSRepresenting a signal characteristic vector in the z direction of a radar coordinate system;
therefore, U obtained from equation (27)xSAnd formulae (32) to (33) to give UySAnd UzSFurther, a process matrix F in the y direction and the z direction of the radar coordinate system can be obtainedy、FzThe expressions for both are as follows:
Figure BSA0000213161490000103
Figure BSA0000213161490000104
in the formula (I), the compound is shown in the specification,U yS，
Figure BSA0000213161490000105
are respectively a matrix UySRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;U zS，
Figure BSA0000213161490000106
are respectively a matrix UzSRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;
Figure BSA0000213161490000107
respectively representU ySAndU zSthe generalized inverse matrix of (2);
the seventh step: calculating a principal eigenvalue vector and its corresponding elements
First, a process matrix F is calculated according to the following equations (37) to (39)x、Fy、FzPrincipal eigenvalue vector Ψ of the first I elementsx、Ψy、Ψz；
Figure BSA0000213161490000108
Figure BSA0000213161490000109
Figure BSA00002131614900001010
In the formula, Tx、Ty、TzAre all non-singular matrices, i.e. Tx、Ty、TzAs long as it is a non-singular matrix;
solving for P based on thisxi、Pyi、PziAnd type parameter αiAnd three types of position parameters xi、yi、zi: matrix Ψ obtained by equations (40) to (42)x、Ψy、ΨzThe element on the main diagonal corresponds to Pxi，Pyi，Pzi：
Pxi＝diag(Ψx)，i＝1，...，I (40)
Pyi＝diag(Ψy)，i＝1，...，I (41)
Pzi＝diag(Ψz)，i＝1，...，I (42)
P obtained according to equations (40) to (42)xi，Pyi，PziBringing it back to formulas (12) - (15)
αi＝(|Pxi|-1)f0/Δf (12)
Figure BSA0000213161490000111
Figure BSA0000213161490000112
Figure BSA0000213161490000113
Solution type parameter αiTransverse distance parameter xiLongitudinal distance parameter yiParameter z of distance from verticali；Δfx、Δfy，、ΔfzRespectively representing stepping frequencies in x, y and z directions respectively representing a radar coordinate system; where angle (.) represents the complex phase angle function found in MATLAB;
eighth step: scattering intensity parameter solution
On the basis of obtaining the type parameters and the three types of position parameters by estimation, solving the intensity parameters in the scattering center model by using a least square method, as follows:
Figure BSA0000213161490000114
in the formula (G)HG)-1Representative matrix GHG conjugate transposition; esA process matrix formed for the scattered echo data of the target, EsIs expressed as shown in formula (44),
Figure BSA0000213161490000115
the method firstly adds the polarization information of the target to a three-dimensional GTD scattering center model in a polarization scattering matrix mode, secondly performs forward and backward smoothing processing on a covariance matrix, and finally performs square processing on a total covariance matrix. The method effectively prolongs the length of the available target electromagnetic scattering echo data, and can further improve the parameter estimation performance and noise robustness of the algorithm.
Drawings
FIG. 1 shows a three-dimensional frequency domain data range diagram;
FIG. 2 shows different algorithms, a comparison of the mean-means-square deviations of the various parameters of the GTD scattering center model, wherein FIG. 2(a) shows a comparison of the mean-means-square deviations for the transverse distance x, FIG. 2(b) shows a comparison of the mean-means-square deviations for the longitudinal distance y, FIG. 2(c) shows a comparison of the mean-means-square deviations for the perpendicular distance z, FIG. 2(d) shows a comparison of the mean-square deviations for the scattering type α, and FIG. 2(e) shows a comparison of the mean-square deviations for the intensity A;
fig. 3 shows the SNR of 0dB, comparison of positioning accuracy for different algorithms;
fig. 4 shows the SNR of 10dB, comparison of positioning accuracy for different algorithms.
Detailed Description
The present invention will be further described with reference to the following drawings and examples, which include, but are not limited to, the following examples.
Principles of improved algorithm
The invention provides an improved 3D-ESPRIT algorithm, which effectively utilizes target polarization scattering information so as to improve the parameter estimation performance of the algorithm, and comprises the following steps:
the first step is as follows: target polarized electromagnetic scattering data is acquired.
Firstly, on the basis of an original three-dimensional GTD scattering center model, the utilization of target polarization information is increased, and the polarization scattering coefficient S is obtainedi，pAnd adding the fully polarized three-dimensional GTD scattering center model into a three-dimensional GTD scattering center model to obtain the following fully polarized three-dimensional GTD scattering center model:
Figure BSA0000213161490000121
in the formula, Si，pRepresents the scattering system of the i-th scattering center in the p-polarization modeP ∈ { hh, hv, vh, vv } represents four polarization modes, hh represents horizontal transmission and horizontal reception, hv represents horizontal transmission and vertical reception, vh represents vertical transmission and horizontal reception, vv represents vertical transmission and vertical reception, BiRepresenting a first transition parameter; pxi、Pyi、PziRespectively representing second, third and fourth transition parameters, the four parameters being used only for the pair
Figure BSA0000213161490000131
And splitting is carried out, so that subsequent parameter estimation is facilitated. Table 1 below gives the scattering matrix for some typical targets.
The second step is that: a Hankel (Hankel) matrix is established.
In order to estimate the scattering center model parameters more accurately, the backward electromagnetic scattering data of the target is firstly subjected to spatial smoothing processing to be decorrelated, and the Hankel matrix can replace the spatial smoothing to play a role in decorrelation. Therefore, a Hankel matrix is first constructed based on the target backward electromagnetic data.
Firstly, smoothing is carried out along the x direction of the radar coordinate system in the figure 1 to construct a [ P × Q × L ]]×[(M-P+1)×(N-Q+1)×(K-L+1)]Of (2) an enhancement matrix XxThis is shown in the following formula (17). Wherein M, N, K is defined by the same formula (1), P is more than or equal to M/2 and less than or equal to 2M/3, Q is more than or equal to N/2 and less than or equal to 2N/3, L is more than or equal to K/2 and less than or equal to 2K/3, and P, Q, L are process variables with values within the range.
TABLE 1 Scattering matrix for typical targets
Figure BSA0000213161490000132
Figure BSA0000213161490000141
In the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000142
Figure BSA0000213161490000143
and performing forward and backward spatial smoothing on the matrix containing the polarization information to obtain a new total covariance matrix R as shown in formula (20):
Figure BSA0000213161490000144
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000145
representative matrix XxThe autocorrelation covariance matrix of (a);
Figure BSA0000213161490000148
representative matrix XxA cross-correlation covariance matrix of sum matrix Y; y ═ JXx；
Figure BSA0000213161490000146
Is a permutation matrix with one dimension of (P × Q × L) × (P × Q × L), the anti-diagonal elements of which are 1 and the elements of which are 0 at the rest positions, and XxP, Q, L are as defined above.
The third step: and (6) square processing.
As can be seen from formula (20), the total covariance matrix R is an Hermittan (Hermittan) matrix, and therefore satisfies R ═ RHI.e. R1＝RRH＝R2. The resulting matrix R after squaring1The eigenvalues and eigenvectors of the total covariance matrix R have the following relations:
Figure BSA0000213161490000147
in the formula, λ1And lambda represents the matrix R obtained by squaring1Eigenvalues of the total covariance R, Λ1Λ respectively represent the matrix R obtained after the square1And the feature vector of the total covariance R.
By the square of the resulting R1Instead of the total covariance matrix R, the gap between the signal eigenvalue and the noise eigenvalue can be increased without changingThe original characteristic vector is changed, so that the signal characteristic value and the noise characteristic value can be more easily distinguished when the signal-to-noise ratio is lower. From a mathematical relationship, the variance of each parameter can be expressed as follows:
Figure BSA0000213161490000151
wherein E {. is a variance,
Figure BSA0000213161490000152
respectively representing the parameters and the original parameters estimated by the invention in the ith Monte Carlo experiment; sigma2、γiRespectively representing a characteristic value corresponding to the noise and a characteristic value corresponding to the signal; m, N, K is defined by the same formula (1); i represents the total number of scattering centers; v. ofiRepresents the ith characteristic value gammaiA corresponding feature matrix; (v)i)HRepresents viThe transposed matrix of (2); v. ofi＝γiE-XxE represents a dimension of [ P × Q × L ]]×[P×Q×L]The identity matrix of (1); p, Q, L are as defined above, GHIs the transposed matrix of G.
G＝[a1，...，aI](23)
Figure BSA0000213161490000153
Figure BSA0000213161490000154
Wherein c is 3 × 108m/s is the propagation velocity of electromagnetic waves, αiIndicating the scattering type of the ith scattering center.
Then, as can be seen from equation (22), the difference, variance, between the noise eigenvalue and the signal eigenvalue is increased
Figure BSA0000213161490000155
The variance of the estimated parameters is reduced, i.e. the effect of reducing the variance of the estimated parameters is achieved. Therefore, a final total covariance matrix R obtained by squaring the following equation (26) is constructed1The method is used for replacing the total covariance matrix R, and can equivalently increase the signal-to-noise ratio and effectively improve the estimation precision of the parameters.
R1＝RRH＝R2(26)
The fourth step: and (5) singular value decomposition.
For the enhancement matrix XxSingular value decomposition to give formula (27):
Figure BSA0000213161490000161
in the formula: u shapexS、VxSAll represent signal characteristic value vectors in the X direction of a radar coordinate system, and are respectively represented by XxThe first I main left eigenvectors and the first I main right eigenvectors; wherein U isxN，VxNRepresents XxRespectively by XxThe non-dominant left eigenvector and the non-dominant right eigenvector; dxSA diagonal matrix formed for the signal eigenvalues; dxNA diagonal matrix formed for the noise eigenvalues; i is the total number of scattering centers.
The fifth step: constructing a process matrix Fx。
Constructing a process matrix FxThe following were used:
Figure BSA0000213161490000162
in the formula (I), the compound is shown in the specification,U xS，
Figure BSA0000213161490000163
are respectively a matrix UxSRemoving the back Q × L rows and removing the front Q × L rows to obtain a matrix,
Figure BSA0000213161490000164
representsU xSThe generalized inverse matrix of (2).
And a sixth step: and obtaining signal subspaces corresponding to the y direction and the z direction of the radar coordinate system by using the permutation matrix J in the foregoing.
From the literature (King of cyanine. optical region Radar target ScatteringCenter extraction and application study [ D]Nanjing: nanjing university of aerospace, 2010.) may derive a permutation matrix E under three-dimensional conditionsxy，Eyz，ExzThe following were used:
Figure BSA0000213161490000165
Figure BSA0000213161490000166
Figure BSA0000213161490000167
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000168
representing the Kronecker product,
Figure BSA0000213161490000169
represents a Q × L matrix with elements of 1 at the (Q, L) position and 0 at other positions,
Figure BSA00002131614900001610
represents an L × P matrix with 1 element at the (L, P) position and 0 elements at other positions,
Figure BSA00002131614900001611
represents a P × Q matrix with an element of 1 at the (P, Q) position and an element of 0 at the other positions.
According to an amplification matrix E in three directionsxy、Eyz、ExzThe relation between the signal subspaces in different directions of the radar coordinate system can be obtained as follows:
UyS＝ExyUxS(32)
UzS＝EyzUyS(33)
UxS＝ExzUzS(34)
in the formula of UySRepresentational mineA vector of signal eigenvalues up to the y-direction of the coordinate system; u shapezSRepresenting the signal feature vector in the z-direction of the radar coordinate system.
Therefore, U obtained from equation (27)xSAnd formulae (32) to (33) to give UysAnd UzsFurther, a process matrix F in the y direction and the z direction of the radar coordinate system can be obtainedy、FzThe expressions for both are as follows:
Figure BSA0000213161490000171
Figure BSA0000213161490000172
in the formula (I), the compound is shown in the specification,U yS，
Figure BSA0000213161490000173
are respectively a matrix UySRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;U zS，
Figure BSA0000213161490000174
are respectively a matrix UzSRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;
Figure BSA0000213161490000175
respectively representU ySAndU zSthe generalized inverse matrix of (2); q, L are as defined above.
The seventh step: the principal eigenvalue vector and its corresponding elements are calculated.
First, a process matrix F is calculated according to the following equations (37) to (39)x、Fy、FzPrincipal eigenvalue vector Ψ of the first I elementsx、Ψy、Ψz。
Figure BSA0000213161490000176
Figure BSA0000213161490000177
Figure BSA0000213161490000178
In the formula, Tx、Ty、TzAre all non-singular matrices, i.e. Tx、Ty、TzAs long as it is a non-singular matrix.
Solving for P based on thisxi、Pyi、PziAnd type parameter αiAnd three types of position parameters xi、yi、zi: matrix Ψ obtained by equations (40) to (42)x、Ψy、ΨzThe element on the main diagonal corresponds to Pxi，Pyi，PziNamely:
Pxi＝diag(Ψx)，i＝1，...，I (40)
Pyi＝diag(Ψy)，i＝1，...，I (41)
Pzi＝diag(Ψz)，i＝1，...，I (42)
p obtained according to equations (40) to (42)xi，Pyi，PziAnd brought back to equations (12) - (15), the type parameter, lateral distance parameter, longitudinal distance parameter, and vertical distance parameter can be solved.
Eighth step: and solving scattering intensity parameters.
On the basis of obtaining type parameters and three types of position parameters by estimation, the intensity parameters in the scattering center model can be solved by using a least square method (Zhangxiada. modern signal processing [ M ]. Beijing: Qinghua university Press, 2002.) as follows:
Figure BSA0000213161490000181
wherein G is as defined for formula (23); gHRepresents the transpose of G (G)HG)-1Representative matrix GHG conjugate transposition; esConstructed for scattered echo data of the targetThe process matrix, subscript s stands for the first letter of the English letter Scattering, EsIs expressed as shown in formula (44),
Figure BSA0000213161490000182
wherein P, Q, L is as defined above.
Second, simulation experiment analysis
Simulation experiment 1: and (5) comparing the mean square deviations. Due to space limitation of the article, 200 Monte Carlo experiments are respectively carried out under the signal-to-noise ratios of-10 dB to 20dB, different algorithm estimations are compared to obtain various parameters, and mean-root-mean-errors (MRMSE) of the parameters are compared, so that the content of the method is more refined. The simulation results are shown in fig. 2. Wherein, the mean square error MRMSE is defined as follows:
Figure BSA0000213161490000183
in the formula (I), the compound is shown in the specification,
Figure BSA0000213161490000184
and D represents the number of Monte Carlo experiments corresponding to each signal-to-noise ratio, and I is the total number of scattering centers.
As can be seen from (a) - (e) in FIG. 2, compared with the classical 3D-ESPRIT algorithm, the mean square error of the Q-FB-3D-ESPRIT algorithm is slightly lower, the parameter estimation precision is higher, the mean square error of each scattering center model parameter of the improved algorithm is smaller than that of the classical 3D-ESPRIT algorithm and that of the Q-FB-3D-ESPRIT algorithm, and the advantages are more obvious under the simulation condition of low signal-to-noise ratio of-10 dB-0 dB; and with the increase of the signal-to-noise ratio, the parameter estimation precision of the three algorithms is improved and tends to be consistent. Simulation experiments verify the effectiveness and the advancement of the improved algorithm, namely the length of electromagnetic scattering data can be prolonged by utilizing the target polarization information, and the noise robustness and the parameter estimation performance of the algorithm are effectively improved.
Simulation experiment 2: in order to further verify the effectiveness and the advancement of the algorithm, under the simulation conditions that the signal-to-noise ratio is 0dB and 10dB, each signal-to-noise ratio corresponds to 200 Monte Carlo experiments, and the positions of four scattering centers are positioned and compared by respectively utilizing a classical 3D-ESPRIT algorithm, an improved Q-FB-3D-ESPRIT algorithm and the improved PQ-FB-3D-ESPRIT algorithm.
As can be seen from FIG. 3, under the condition that the signal-to-noise ratio is 0dB, the classical 3D-ESPRIT algorithm cannot accurately locate all four scattering centers; the Q-FB-3D-ESPRIT algorithm can accurately position three scattering centers, and the positioning of the other scattering center has slight deviation; the improved PQ-FB-3D-ESPRIT algorithm provided by the invention can accurately position the positions of four scattering centers. As can be seen from FIG. 4, under the condition that the SNR is 10dB, the classical 3D-ESPRIT algorithm cannot accurately locate the positions of the four scattering centers; and the Q-FB-3D-ESPRIT algorithm and the improved algorithm of the invention can accurately position the positions of the four scattering centers. From the two simulation experiments, the parameter estimation performance and the noise robustness of the classic 3D-ESPRIT algorithm are the worst, and the parameter estimation performance and the noise robustness of the proposed improved algorithm are the best, so that the effectiveness and the superiority of the proposed improved algorithm are verified. The invention provides an improved 3D-ESPRIT algorithm, which effectively improves the estimation precision of GTD scattering center model parameters, and the superiority of the improved algorithm in the invention is mainly shown in the following two aspects:
1. the improved algorithm increases the utilization of the target polarization scattering information, effectively prolongs the length information of available data, and greatly improves the utilization rate of the target electromagnetic scattering data;
2. the improved algorithm reduces the influence of noise on the estimation performance of the algorithm parameters by squaring and bidirectional smoothing the covariance matrix, and effectively improves the noise robustness of the algorithm.

Claims (1)

1. The scattering center model parameter estimation method based on the improved 3D-ESPRIT algorithm is characterized by comprising the following steps of:
the first step is as follows: obtaining target polarized electromagnetic scattering data
Firstly, on the basis of an original three-dimensional GTD scattering center model, the utilization of target polarization information is increased, and the polarization scattering coefficient S is obtainedi，pAnd adding the three-dimensional GTD scattering center model into the fully polarized three-dimensional GTD scattering center model to obtain a fully polarized three-dimensional GTD scattering center model as follows:
Figure FSA0000213161480000011
in the formula (I), the compound is shown in the specification,
Figure FSA0000213161480000012
a back-scattered echo that represents the target,
Figure FSA0000213161480000013
frequency, azimuth, pitch representing variation, respectively: f. ofm＝f0+mΔf，m＝0，1，...，M，f0Is the starting frequency, Δ f is the step frequency, M represents the frequency subscript, and M is the total frequency step number; thetan＝θ0+ N Δ θ, N ═ 0, 1.., N, where θ is0Is the initial azimuth, delta theta is the stepping azimuth, N is the azimuth subscript, and N is the total azimuth stepping number;
Figure FSA0000213161480000014
wherein
Figure FSA0000213161480000015
In order to start the pitch angle,
Figure FSA0000213161480000016
step pitch angle, K is pitch angle subscript, and K is total pitch angle step number; n.DELTA.theta,
Figure FSA0000213161480000017
Respectively a small corner in the azimuth direction and a small corner in the pitching direction; i represents the number of scattering centers; si，pIndicating the scattering coefficient of the ith scattering center in the p-polarized mode, p ∈ hh, hv,vh, vv } represents four polarization modes: hh represents horizontal transmission, horizontal reception; hv represents horizontal transmission, vertical reception; vh represents vertical transmission, horizontal reception; vv represents vertical transmission, vertical reception; b isiRepresenting a first transition parameter; pxi、Pyi、PziRespectively representing second, third and fourth transition parameters, the four parameters being used only for the pair
Figure FSA0000213161480000018
Splitting is carried out, so that subsequent parameter estimation is facilitated; { xi，yi，ziRespectively representing the transverse distance, the longitudinal distance and the vertical distance of the ith scattering center;
Figure FSA0000213161480000019
complex white gaussian noise;
table 1 below gives the scattering matrix for some typical targets;
TABLE 1 Scattering matrix for typical targets
Figure 1
Figure FSA0000213161480000021
The second step is that: establishing a Hankel matrix
Firstly, constructing a Hankel matrix based on target backward electromagnetic data;
firstly, smoothing is carried out along the x direction of a radar coordinate system to construct a [ P × Q × l ]]×[(M-P+1)×(N-Q+1)×(K-L+1)]Of (2) an enhancement matrix XxRepresented by the following formula (17); wherein M, N, K is defined by the same formula (1), P is more than or equal to M/2 and less than or equal to 2M/3, Q is more than or equal to N/2 and less than or equal to 2N/3, L is more than or equal to K/2 and less than or equal to 2K/3, and P, Q, L are process variables with values within the range;
Figure FSA0000213161480000022
in the formula (I), the compound is shown in the specification,
Figure 2
Figure FSA0000213161480000031
and performing forward and backward spatial smoothing on the matrix containing the polarization information to obtain a new total covariance matrix R as shown in formula (20):
Figure FSA0000213161480000032
in the formula (I), the compound is shown in the specification,
Figure FSA0000213161480000033
representative matrix XxThe autocorrelation covariance matrix of (a);
Figure FSA0000213161480000034
representative matrix XxA cross-correlation covariance matrix of sum matrix Y;
Figure FSA0000213161480000035
a permutation matrix with one dimension (P × Q × L) × (P × Q × L) and with anti-diagonal elements of 1 and elements of 0 at the remaining positions;
the third step: squaring process
As can be seen from formula (20), the total covariance matrix R is an Hermittan matrix, and therefore satisfies R ═ RHI.e. R1＝RRH＝R2(ii) a The resulting matrix R after squaring1The eigenvalues and eigenvectors of the total covariance matrix R have the following relations:
Figure FSA0000213161480000036
in the formula, λ1And lambda represents the matrix R obtained by squaring1Eigenvalues of the total covariance R, Λ1Λ respectively represent the matrix R obtained after the square1A feature vector associated with the total covariance R;
by the square of the resulting R1The total covariance matrix R is replaced, the difference between the signal characteristic value and the noise characteristic value can be increased, and the original characteristic vector is not changed, so that the signal characteristic value and the noise characteristic value are more easily distinguished when the signal-to-noise ratio is low; from a mathematical relationship, the variance of each parameter is expressed as follows:
Figure FSA0000213161480000037
Figure FSA0000213161480000041
wherein E {. is a variance,
Figure FSA0000213161480000042
omega respectively represents parameters and original parameters estimated by the ith Monte Carlo experiment; sigma2、γiRespectively representing a characteristic value corresponding to the noise and a characteristic value corresponding to the signal; i represents the total number of scattering centers; v. ofiRepresents the ith characteristic value gammaiA corresponding feature matrix; (v)i)HRepresents viThe transposed matrix of (2); v. ofi＝γiE-XxE represents a dimension of [ P × Q × L ]]×[P×Q×L]The identity matrix of (1); gHIs the transposed matrix of G;
G＝[a1，...，aI](23)
Figure FSA0000213161480000043
Figure FSA0000213161480000044
wherein c is 3 × 108m/s is the propagation velocity of electromagnetic waves, αiScattering class representing the ith scattering centerMolding;
then, as shown in equation (22), the difference, variance, between the noise eigenvalue and the signal eigenvalue is increased
Figure FSA0000213161480000046
The variance of the estimation parameters can be reduced; therefore, a final total covariance matrix R obtained by squaring the following equation (26) is constructed1The method is used for replacing the total covariance matrix R, and can equivalently increase the signal-to-noise ratio and effectively improve the estimation precision of parameters;
R1＝RRH＝R2(26)
the fourth step: singular value decomposition
For the enhancement matrix XxSingular value decomposition to give formula (27):
Figure FSA0000213161480000045
in the formula: u shapexS、VxSAll represent signal characteristic value vectors in the X direction of a radar coordinate system, and are respectively represented by XxThe first I main left eigenvectors and the first I main right eigenvectors; wherein U isxN，VxNRepresents XxRespectively by XxThe non-dominant left eigenvector and the non-dominant right eigenvector; dxSA diagonal matrix formed for the signal eigenvalues; dxNA diagonal matrix formed for the noise eigenvalues;
the fifth step: constructing a process matrix Fx
Constructing a process matrix FxThe following were used:
Figure FSA0000213161480000051
in the formula (I), the compound is shown in the specification,U xS，
Figure FSA0000213161480000052
are respectively a matrix UxSMatrix obtained by removing rear Q × L rows and removing front Q × L rows，
Figure FSA0000213161480000053
RepresentsU xSThe generalized inverse matrix of (2);
and a sixth step: the signal subspaces corresponding to the y direction and the z direction of the radar coordinate system are obtained by utilizing the permutation matrix J in the foregoing
Permutation matrix E under three-dimensional conditionsxy，Eyz，ExzThe following were used:
Figure FSA0000213161480000054
Figure FSA0000213161480000055
Figure FSA0000213161480000056
in the formula (I), the compound is shown in the specification,
Figure FSA0000213161480000057
representing the Kronecker product,
Figure FSA0000213161480000058
represents a Q × L matrix with elements of 1 at the (Q, L) position and 0 at other positions,
Figure FSA0000213161480000059
represents an L × P matrix with 1 element at the (L, P) position and 0 elements at other positions,
Figure FSA00002131614800000510
represents a P × Q matrix with 1 element at the (P, Q) position and 0 elements at other positions;
according to the three directions to expand the matrix Exy、Eyz、ExzThe relation between the two signals obtains the signals of the radar coordinate system in different directionsThe relationship between the number spaces is as follows:
UyS＝ExyUxS(32)
UzS＝EyzUyS(33)
UxS＝ExzUzS(34)
in the formula of UySRepresenting a signal characteristic value vector in the y direction of a radar coordinate system; u shapezSRepresenting a signal characteristic vector in the z direction of a radar coordinate system;
therefore, U obtained from equation (27)xSAnd formulae (32) to (33) to give UySAnd UzSFurther, a process matrix F in the y direction and the z direction of the radar coordinate system can be obtainedy、FzThe expressions for both are as follows:
Figure FSA0000213161480000061
Figure FSA0000213161480000062
in the formula (I), the compound is shown in the specification,U yS，
Figure FSA0000213161480000063
are respectively a matrix UySRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;U zS，
Figure FSA0000213161480000064
are respectively a matrix UzSRemoving the rear Q × L rows and removing the front Q × L rows to obtain a matrix;
Figure FSA0000213161480000065
respectively representU ySAndU zSthe generalized inverse matrix of (2);
the seventh step: calculating a principal eigenvalue vector and its corresponding elements
First, a process matrix F is calculated according to the following equations (37) to (39)x、Fy、FzPrincipal eigenvalue vector Ψ of the first I elementsx、Ψy、Ψz；
Ψx＝TxFxTx -1(37)
Ψy＝TyFyTy -1(38)
Ψz＝TzFzTz -1(39)
In the formula, Tx、Ty、TzAre all non-singular matrices, i.e. Tx、Ty、TzAs long as it is a non-singular matrix;
solving for P based on thisxi、Pyi、PziAnd type parameter αiAnd three types of position parameters xi、yi、zi: matrix Ψ obtained by equations (40) to (42)x、Ψy、ΨzThe element on the main diagonal corresponds to Pxi，Pyi，Pzi：
Pxi＝diag(Ψx)，i＝1，...，I (40)
Pyi＝diag(Ψy)，i＝1，...，I (41)
Pzi＝diag(Ψz)，i＝1，...，I (42)
P obtained according to equations (40) to (42)xi，Pyi，PziBringing it back to formulas (12) - (15)
αi＝(|Pxi|-1)f0/Δf (12)
Figure FSA0000213161480000066
Figure 3
Figure FSA0000213161480000071
Solution type parameter αiTransverse distance parameter xiLongitudinal distance parameter yiParameter z of distance from verticali；Δfx、Δfy、ΔfzRespectively representing stepping frequencies in x, y and z directions respectively representing a radar coordinate system; where angle (.) represents the complex phase angle function found in MATLAB;
eighth step: scattering intensity parameter solution
On the basis of obtaining the type parameters and the three types of position parameters by estimation, solving the intensity parameters in the scattering center model by using a least square method, as follows:
Figure FSA0000213161480000072
in the formula (G)HG)-1Representative matrix GHG conjugate transposition; esA process matrix formed for the scattered echo data of the target, EsIs expressed as shown in formula (44),
Figure FSA0000213161480000073
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