Method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition

12117545 · 2024-10-15

Assignee

Zhejiang University (Zhejiang, CN)

Inventors

Cpc classification

International classification

Abstract

The disclosure provides a method for estimating a direction of arrival of an L-type coprime array based on coupled tensor decomposition. The method includes: constructing an L-type coprime array with separated sub-arrays and modeling a received signal; deriving a fourth-order covariance tensor of the received signal of the L-type coprime array; deriving a fourth-order virtual domain signal corresponding to an augmented virtual uniform cross array; dividing the virtual uniform cross array by translation; constructing a coupled virtual domain tensor by stacking a translation virtual domain signal; and obtaining a direction of arrival estimation result by coupled virtual domain tensor decomposition. The present invention makes full use of the spatial correlation property of the virtual domain tensor statistics of the constructed L-type coprime array with the separated sub-arrays, and realizes high-precision two-dimensional direction of arrival estimation by coupling the virtual domain tensor processing, which can be used for target positioning.

Claims

1. A method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition, wherein the method comprises the following steps: (1) using 2 custom character ++2+?2 physical antenna array elements by a receiving end to construct the L-type coprime array with separated sub-arrays, wherein the L-type coprime array consists of two coprime linear arrays .sub.i, i=1, 2 located on the x-axis and the y-axis, wherein the first array elements of the two coprime linear arrays custom character .sub.1 and .sub.2 are laid out from positions (1, 0) and (1, 0) in an xoy coordinate system respectively; the coprime linear array .sub.i contains |.sub.i|=2+?1 array elements, wherein, and are a pair of coprime integers, |.Math.| represents the potential of the set; {(, 0)|=[, , . . . , custom character ]d} and {(0, )|=[, , . . . , ]d} are respectively used to represent the positions of each array element in the L-type coprime array on the x-axis and y-axis, wherein, ==1, the unit interval d is taken as a half of the wavelength of incident narrowband signal; assuming that there are K far-field narrow-band incoherent signal sources from {(?.sub.1, ?.sub.1), (?.sub.2, ?.sub.2), . . . , (?.sub.K, ?.sub.K)} directions, modeling a received signal of the coprime linear array custom character .sub.i forming the L-type coprime array as: $X_{?_{i}} = {.Math.}_{k = 1}^{K} a_{?_{i}} (k) ? s_{k} + N_{?_{i}} ? ?^{.Math. ?_{i} .Math. ? T},$ wherein, s.sub.k=[s.sub.k,1, s.sub.k,2, . . . , s.sub.k,T].sup.T is a multi-snapshot sampling signal waveform corresponding to a kth incident signal source, T is the number of sampling snapshots, ? represents an outer product of the vector, custom character is a noise independent of each signal source, (k) is a steering vector of .sub.i, and corresponds to a signal source with an incoming wave direction being (?.sub.k, ?.sub.k), and is expressed as: $a_{?_{i}} (k) = {[e^{- j ? q_{?_{i}}^{(1)} ?_{i} (k)}, e^{- j ? q_{?_{i}}^{(2)} ?_{i} (k)}, .Math., e^{- j ? q_{?_{i}}^{(.Math. ?_{i} .Math.)} ?_{i} (k)}]}^{T},$ wherein ?.sub.1(k)=sin (?.sub.k)cos(?.sub.k), ?.sub.2(k)=sin (?.sub.k)sin(?.sub.k), j=?{square root over (?1)}, [.Math.].sup.T represents a transpose operation; (2) by solving cross-correlation statistics of custom character and , obtaining a second-order cross-correlation matrix ?; $R_{?_{1} ?_{2}} = E {X_{?_{1}} X_{?_{2}}^{H}} = {.Math.}_{k = 1}^{K} ?_{k}^{2} a_{?_{1}} (k) ? a_{?_{2}}^{*} (k),$ wherein, ?.sub.k.sup.2=E{s.sub.k(t)s*.sub.k(t)} represents the power of the kth incident signal source, E{.Math.} represents a mathematical expectation operation, (.Math.).sup.H represents a conjugate transpose operation, (.Math.)* represents a conjugate operation, and on the basis of the second-order cross-correlation matrix, fourth-order statistic of the L-type coprime array with the separated sub-arrays is derived, that is, a fourth-order covariance tensor custom character ? is obtained by calculating the auto-correlation of the second-order cross-correlation matrix : $? = R_{?_{1} ?_{2}} ? R_{?_{1} ?_{2}}^{*} = E {(X_{?_{1}} X_{?_{2}}^{H}) ? {(X_{?_{1}} X_{?_{2}}^{H})}^{*}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} a_{?_{1}} (k) ? a_{?_{2}}^{*} (k) ? a_{?_{1}}^{*} (k) ? a_{?_{2}} (k);$ (3) defining dimension sets custom character .sub.1={1, 3}, .sub.2={2, 4} and obtaining a fourth-order virtual domain signal ? by performing a tensor trans formation of dimension merging on the fourth-order covariance tensor : $V_{?} \overset{?}{=} ?_{{?_{1}, ?_{2}}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} [a_{?_{1}}^{*} (k) .Math. a_{?_{1}} (k)] ? [a_{?_{1}} (k) .Math. a_{?_{2}}^{*} (k)],$ wherein, by forming a difference set array on an exponential term, custom character (k).Math.(k) and (k).Math.(k) each constructs an augmented non-continuous virtual linear array on the x axis and y axis, .Math. represents a Kronecker product, corresponds to a two-dimensional non-continuous virtual cross array , contains a virtual uniform cross array =.sub.x?.sub.y, wherein custom character .sub.x and .sub.y are each a virtual uniform linear array on the x axis and y axis; the positions of all virtual array elements in the .sub.x and .sub.y are expressed as .sub.x={, 0)|=[, , . . . , ]d} and .sub.y={(0, )|=[, , . . . , ]d}, wherein =??+2, =+, =??+2, =+, and |.sub.x|=2(+ custom character )?1, |.sub.y|=2(+)?1; extracting an element corresponding to the position of each virtual array element in the virtual uniform cross array from the virtual domain signal of the non-continuous virtual cross array , and obtaining a virtual domain signal ? corresponding to , which is modeled as: ${\overline{V}}_{?} = {.Math.}_{k = 1}^{K} ?_{k}^{4} b_{x} (k) ? b_{y} (k),$ wherein, $\begin{matrix} b_{x} (k) = {[e^{- j ? q_{?_{x}}^{(1)} ?_{1} (k)}, e^{- j ? q_{?_{x}}^{(2)} ?_{1} (k)}, .Math., e^{- j ? q_{?_{x}}^{(.Math. ?_{x} .Math.)} ?_{1} (k)}]}^{T}, \\ b_{y} (k) = {[e^{- j ? q_{?_{y}}^{(1)} ?_{2} (k)}, e^{- j ? q_{?_{y}}^{(2)} ?_{2} (k)}, .Math., e^{- j ? q_{?_{y}}^{(.Math. ?_{y} .Math.)} ?_{2} (k)}]}^{T}, \end{matrix}$ are steering vectors of custom character .sub.x and .sub.y, respectively; (4) respectively extracting sub-arrays .sub.x.sup.(1)={(, 0)|=[1, 2, . . . , ]d}, .sub.y.sup.(1)={(0, )|=[1, 2, . . . , ]d} from .sub.x and .sub.y as translation windows; translating the translation windows .sub.x.sup.(1) and .sub.y.sup.(1) along negative semi-axis directions of the x axis and the y axis by a virtual array element interval one by one to obtain P.sub.x virtual uniform linear sub-arrays custom character .sub.x.sup.(p.sup.x.sup.)={(, 0)|=[2?p.sub.x, 3?p.sub.x, . . . , +1?p.sub.x]d} and P.sub.y virtual uniform linear sub-arrays .sub.y.sup.(p.sup.y.sup.)={(0, )|=[2?p.sub.y, 3?p.sub.y, . . . , +1?p.sub.y]d}, wherein P.sub.x=(|.sub.x|+1)/2, P.sub.y=(|.sub.y|+1)/2; then the virtual domain signal of the virtual uniform sub-array custom character .sub.(p.sub.x.sub.,p.sub.y.sub.)=.sub.x.sup.(p.sup.x.sup.)?.sub.y.sup.(p.sup.y.sup.) can be expressed as: $U_{{\tilde{?}}_{(p_{x}, p_{y})}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} g_{x}^{(p_{x})} (k) ? g_{y}^{(p_{y})} (k) ? ?^{.Math. ?_{x}^{(p_{x})} .Math. ? ?_{y}^{(p_{y})} .Math.},$ wherein, $\begin{matrix} g_{x}^{(p_{x})} (k) = {[e^{- j ? (2 - p_{x}) ?_{1} (k)}, e^{- j ? (3 - p_{x}) ?_{1} (k)}, .Math., e^{- j ? (q_{?_{x}}^{(.Math. ?_{x} .Math.)} + 1 - p_{x}) ?_{1} (k)}]}^{T}, \\ g_{y}^{(p_{y})} (k) = {[e^{- j ? (2 - p_{y}) ?_{2} (k)}, e^{- j ? (3 - p_{y}) ?_{2} (k)}, .Math., e^{- j ? (q_{?_{y}}^{(.Math. ?_{y} .Math.)} + 1 - p_{y}) ?_{2} (k)}]}^{T}, \end{matrix}$ are steering vectors of custom character .sub.x.sup.(p.sup.x.sup.) and .sub.y.sup.(p.sup.y.sup.), respectively; (5) for P.sub.y virtual uniform sub-arrays .sub.(p.sub.x.sub., :) with the same subscript p.sub.x, superimposing corresponding virtual domain signals of the P.sub.y virtual uniform sub-arrays .sub.(p.sub.x.sub., :) in a third dimension to obtain P.sub.x three-dimensional coupled virtual domain tensors custom character .sub.(p.sub.x.sub.)? $?_{(p_{x})} = {[U_{{\tilde{?}}_{(p_{x}, 1)}}, U_{{\tilde{?}}_{(p_{x}, 2)}}, .Math., U_{{\tilde{?}}_{(p_{x}, p_{y})}}]}_{?_{3}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} g_{x}^{(p_{x})} (k) ? g_{y}^{(1)} (k) ? q_{y} (k) = ? ?_{k}^{4}; G_{x}^{(p_{x})}, G_{y}^{(1)}, Q_{y} ?,$ wherein, g.sub.y.sup.(1)(k) is a guiding vector of the translation window custom character .sub.y.sup.(1), q.sub.y(k)=[1, e.sup.j??.sup.2.sup.(k), . . . , e.sup.j?(P.sup.y.sup.?1)?.sup.2.sup.(k)].sup.T represents a translation factor along an axis direction of y, and G.sub.x.sup.(P.sup.x.sup.)=[g.sub.x.sup.(p.sup.x.sup.)(1), g.sub.x.sup.(p.sup.x.sup.)(2), . . . , g.sub.x.sup.(p.sup.x.sup.)(K)], G.sub.y.sup.(1)=[g.sub.y.sup.(1) (1), g.sub.y.sup.(1) (2), . . . , g.sub.y.sup.(1)(K)] and Q.sub.y=[q.sub.y(1), q.sub.y(2), . . . , q.sub.y(K)] are factor matrices of custom character .sub.(p.sub.x.sub.), [.Math.].sub.a represents a tensor superposition operation on the a.sup.th dimension, and .Math. represents a canonical polyadic model of the tensor; (6) performing a coupled canonical polyadic decomposition on the constructed P.sub.x coupled virtual domain tensors custom character .sub.(p.sub.x.sub.) to obtain an estimated value {?.sub.x.sup.(p.sup.x.sup.), ?.sub.y.sup.(1), {circumflex over (Q)}.sub.y} of the factor matrices {G.sub.x.sup.(p.sup.x.sup.), G.sub.y.sup.(1), Q.sub.y}, which includes an estimated value {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1) (k), {circumflex over (q)}.sub.y(k)} of a spatial factor {g.sub.x.sup.(p.sup.x.sup.)(k), g.sub.y.sup.(1)(k), q.sub.y(k)}; then, extracting a two-dimensional direction of arrival estimate result ({circumflex over (?)}.sub.k, {circumflex over (?)}.sub.k) from the estimated value {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1)(k), {circumflex over (q)}.sub.y(k)} of the spatial factor.

2. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein the structure of the L-type coprime array with the separated sub-arrays in the step (1) is described as: the coprime linear array custom character .sub.i constituting the L-type coprime array is composed of a pair of sparse uniform linear sub-arrays, the two sparse uniform linear sub-arrays respectively contain 2 and antenna elements, and the distances between the array elements are respectively d and d, wherein, and are one pair of coprime integers; a sub-array combination is performed on the two sparse uniform linear sub-arrays in custom character .sub.i by overlapping the first array elements to obtain a coprime linear array .sub.i containing |.sub.i|=2+?1 array elements.

3. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein in the derivation of the fourth-order statistic described in the step (2), the fourth-order covariance tensor custom character ? based on sampling is obtained by calculating fourth-order statistics of the received signals and of the T sampling snapshots: $\hat{?} = (\frac{1}{T} X_{?_{1}} X_{?_{2}}^{H}) ? {(\frac{1}{T} X_{?_{1}} X_{?_{2}}^{H})}^{*} .$

4. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein in the construction of the coupled virtual domain tensors described in step (5), the obtained P.sub.x virtual domain tensors custom character .sub.(p.sub.x.sub.) represent the same spatial information in a second dimension and a third dimension and different spatial information in a first dimension, the P.sub.x virtual domain tensors .sub.(p.sub.x.sub.) have a coupling relationship in the second dimension and the third dimension, the first dimension represents angle information of the virtual uniform linear sub-arrays custom character .sub.x.sup.(p.sup.x.sup.), the second dimension represents angle information of the translation window .sub.y.sup.(1), and the third dimension represents translation information in the y axis direction.

5. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein in the construction process of the coupled virtual domain tensors described in the step (5), the coupled virtual domain tensors are constructed by superimposing translational virtual domain signals in the x axis direction, comprising: for P.sub.x virtual uniform sub-arrays custom character .sub.(:, p.sub.y.sub.) with the same subscript p.sub.y covering the same angle information in the y axis direction and having a spatial translation relationship in the x axis direction, superimposing virtual domain signals corresponding to the P.sub.x virtual uniform sub-arrays .sub.(:, p.sub.y.sub.) in the third dimension, so as to get P.sub.y virtual domain tensors custom character .sub.(p.sub.y.sub.)?: $?_{(p_{y})} = {[U_{{\tilde{?}}_{(1, p_{y})}}, U_{{\tilde{?}}_{(2, p_{y})}}, .Math., U_{{\tilde{?}}_{(p_{x}, p_{y})}}]}_{?_{3}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} g_{x}^{(1)} (k) ? g_{y}^{(p_{y})} (k) ? q_{x} (k) = ? ?_{k}^{4}; G_{x}^{(1)}; G_{y}^{(p_{y})}, Q_{x} ?,$ wherein, g.sub.x.sup.(1)(k) is a steering vector of the translation window custom character .sub.x.sup.(1), q.sub.x(k)=[1, e.sup.j??.sup.1.sup.(k), . . . , e.sup.j?(P.sup.x.sup.?1)?.sup.1.sup.(k)].sup.T represents a translation factor along the x axis direction, G.sub.x.sup.(1)=[g.sub.x.sup.(1)(1), g.sub.x.sup.(1)(2), . . . , g.sub.x.sup.(1)(K)], G.sub.y.sup.(p.sup.y.sup.)=[g.sub.y.sup.(p.sup.y.sup.)(1), g.sub.y.sup.(p.sup.y.sup.)(2), . . . , g.sub.y.sup.(p.sup.y.sup.)(K)] and Q.sub.x=[q.sub.x(1), q.sub.x(2), . . . , q.sub.x(K)] are factor matrices of custom character .sub.(p.sub.y.sub.); the P.sub.y constructed three-dimensional virtual domain tensors .sub.(p.sub.y.sub.) represent the same spatial information in the first and third dimensions and different spatial information in the second dimension, thus the virtual domain tensors .sub.(p.sub.y.sub.) have a coupling relationship in the first and the third dimensions; the constructed P.sub.y three-dimensional virtual domain tensors custom character .sub.(p.sub.y.sub.) are decomposed by coupled canonical polyadic, and the factor matrices {G.sub.x.sup.(1), G.sub.y.sup.(p.sup.y.sup.), Q.sub.x} of the P.sub.y constructed three-dimensional virtual domain tensors .sub.(p.sub.y.sub.) are estimated, wherein the first dimension represents the angle information of the translation window custom character .sub.x.sup.(1), the second dimension represents the angle information of the virtual uniform linear sub-arrays .sub.y.sup.(p.sup.y.sup.), and the third dimension represents the translation information in the x-axis direction.

6. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein in the decomposition of the coupled virtual domain tensors in the step (6), the coupling relationship of the constructed P.sub.x three-dimensional virtual domain tensors custom character .sub.(p.sub.x.sub.) is utilized, the coupled canonical polyadic decomposition is performed on .sub.(p.sub.x.sub.) via a joint least-squares optimization problem: ${{\hat{G}}_{x}^{(p_{x})}, {\hat{G}}_{y}^{(1)}, {\hat{Q}}_{y}} = \min_{{\overset{?}{G}}_{x}^{(p_{x})}, {\overset{?}{G}}_{y}^{(1)}, {\overset{?}{Q}}_{y}} \underset{p_{x}}{.Math.} .Math. ?_{(p_{x})} - {? {\hat{G}}_{x}^{(p_{x})}, {\hat{G}}_{y}^{(1)}, {\hat{Q}}_{y} ?}_{F}^{2},$ wherein, ?.Math.?.sub.F represents the Frobenius norm; solving the joint least squares optimization problem to obtain the estimated value {?.sub.x.sup.(p.sup.x.sup.), ?.sub.y.sup.(1), {circumflex over (Q)}.sub.y} of the factor matrices {G.sub.x.sup.(p.sup.x.sup.), G.sub.y.sup.(1), Q.sub.y}; in the coupled virtual domain tensor decomposition problem, the maximum number of identifiable targets K is | custom character .sub.x.sup.(p.sup.x.sup.)|+P.sub.y?2, which exceeds the actual number of the physical array elements of the constructed L-type coprime array with separated sub-array.

7. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition according to claim 1, wherein in the step (6), for the estimated space factor {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1) (k), {circumflex over (q)}.sub.y(k)}, the parameters {circumflex over (?)}.sub.1(k) and {circumflex over (?)}.sub.2(k) are extracted from the estimated space factor {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1)(k), {circumflex over (q)}.sub.y(k)}: $\begin{matrix} {\overset{?}{?}}_{1} (k) = (\underset{P x}{.Math.} v_{(p_{x})}^{} ? ({\overset{?}{g}}_{x}^{(p_{x})} (k)) / ?) / P_{x}, \\ {\overset{?}{?}}_{2} (k) = (w_{(1)}^{} ? ({\overset{?}{g}}_{y}^{(1)} (k)) / ? + z^{} ? ({\overset{?}{q}}_{y} (k)) / ?) / 2, \end{matrix}$ wherein, v.sub.(p.sub.x.sub.)=[2?p.sub.x, 3?p.sub.x, . . . , custom character +1?p.sub.x].sup.T is a position index of each virtual array element in .sub.x.sup.(p.sup.x.sup.), w.sub.(1)=[1, 2, . . . , ].sup.T is a position index of each virtual array element in .sub.y.sup.(1), z=[0, 1, . . . , P.sub.y?1].sup.T represents a translation step, ?(.Math.) represents a complex argument taking operation, (.Math.).sup. represents a pseudo-inverse operation; finally, according to the relationship of {?.sub.1(k), ?.sub.2(k)} and the two-dimensional direction of arrival (?.sub.k, ?.sub.k), that is, ?.sub.1(k)=sin(?.sub.k)cos(?.sub.k) and ?.sub.2(k)=sin(?.sub.k)sin(?.sub.k), a closed-form solution of the two-dimensional direction of arrival estimation ({circumflex over (?)}.sub.k, {circumflex over (?)}.sub.k) is obtained as: $\begin{matrix} {\overset{?}{?}}_{k} = \arctan (\frac{{\overset{?}{?}}_{2} (k)}{{\overset{?}{?}}_{1} (k)}), \\ {\overset{?}{?}}_{k} = \sqrt{{{\overset{?}{?}}_{1} (k)}^{2} + {{\overset{?}{?}}_{2} (k)}^{2}} . \end{matrix}$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a general flow block diagram of the present invention.

(2) FIG. 2 is a schematic structural diagram of an L-type coprime array with separated sub-arrays proposed by the present invention.

(3) FIG. 3 is a schematic diagram of a virtual uniform cross array and a virtual uniform sub-array constructed by the present invention.

(4) FIG. 4 is a performance comparison diagram of direction of arrival estimation accuracy of the method proposed in the present invention under different signal-to-noise ratio conditions.

(5) FIG. 5 is a performance comparison diagram of direction of arrival estimation accuracy of the method proposed in the present invention under different sampling snapshot amount conditions.

DESCRIPTION OF THE EMBODIMENTS

(6) The technical solutions of the present invention will be described in further detail below with reference to the accompanying drawings.

(7) In order to solve the problems of multi-dimensional signal structure damage and virtual domain signal correlation information loss existing in the existing methods, the present invention proposes a method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition. By deriving a virtual domain signal of the L-type coprime array based on a tensor model, and constructing a coupling idea of the virtual domain tensors, high-precision two-dimensional direction of arrival estimation can be realized by using the correlation information of the virtual domain tensors. Referring to FIG. 1, the implementation steps of the present invention are as follows:

(8) Step 1: constructing an L-type coprime array with separated sub-arrays and modeling a received signal. At a receiving end, using 2 custom character ++2+?2 physical antenna elements to construct the L-type coprime array with the separated sub-arrays, as shown in FIG. 2: constructing coprime linear arrays .sub.i, i=1, 2 on the x-axis and y-axis respectively; .sub.i contains |.sub.i|=2+?1 antenna array elements, wherein, and are a pair of coprime integers, |.Math.| represents a potential of the set; the first array elements of the two coprime linear arrays custom character .sub.1 and .sub.2 are laid out from (1, 0) and (1, 0) positions in the xoy coordinate system respectively, so the two coprime linear arrays .sub.1 and .sub.2 forming the L-type coprime array do not overlap each other; using {(, 0)|=[, , . . . , ]d} and {(0, )|=[, , . . . , ]d} to represent the positions of all array element of the L-type coprime array on the x-axis and y-axis respectively, wherein custom character ==1, and the unit interval d is taken as half of the wavelength of an incident narrowband signal; the coprime linear array .sub.i forming the L-type coprime array is consisted of a pair of sparse uniform linear sub-arrays. The two sparse uniform linear sub-arrays respectively contain 2 custom character and antenna elements, and the spacings of the array elements are respectively d and d; the two sparse uniform linear sub-arrays in .sub.i are combined with sub-arrays in a way that the first array elements overlap to obtain the coprime linear arrays .sub.i containing 2+?1 array elements. assuming that there are K far-field narrow-band incoherent signal sources from {(?.sub.1, ?.sub.1), (?.sub.2, ?.sub.2), . . . , (?.sub.K, ?.sub.K)} directions, modeling a received signals of the two coprime linear arrays custom character .sub.1 and .sub.2 forming the L-type coprime array as:

(9) $X_{?_{i}} = {.Math.}_{k = 1}^{K} a_{?_{i}} (k) ? s_{k} + N_{?_{i}} ? ?^{| ?_{i} | ? T},$ wherein, s.sub.k=[s.sub.k,1, s.sub.k,2, . . . , s.sub.k,T].sup.T is a multi-snapshot sampling signal waveform corresponding to a kth incident signal source, T is the number of sampling snapshots, ? represents an outer product of the vector, custom character is a noise independent of each signal source, (k) is a steering vector of .sub.i, and corresponds to a signal source with an incoming wave direction being (?.sub.k, ?.sub.k), and is expressed as:

(10) $a_{?_{i}} (k) = {[e^{- j ? q_{?_{i}}^{(1)} {?_{i}}^{(k)}}, e^{- j ? q_{?_{i}}^{(2)} u_{i} (k)}, .Math., e^{- j ? q_{?_{i}}^{(.Math. ?_{i} .Math.)} ?_{i} (k)}]}^{T},$ wherein ?.sub.1(k)=sin (?.sub.k)cos(?.sub.k), ?.sub.2(k)=sin (?.sub.k)sin(?.sub.k), j=?{square root over (?1)}, [.Math.].sup.T represents a transpose operation;

(11) Step 2: deriving a fourth-order covariance tensor of the received signal of the L-type coprime array. A second-order cross-correlation matrix custom character ? is obtained by calculating a cross-correlation statistic of the sampled signal and of the coprime linear arrays .sub.1 and .sub.2:

(12) $R_{?_{1} ?_{2}} = E {X_{?_{1}} X_{?_{2}}^{H}} = {.Math.}_{k = 1}^{K} ?_{k}^{2} a_{?_{1}} (k) ? a_{?_{2}}^{*} (k)$

(13) Wherein, ?.sub.k.sup.2=E{s.sub.k(t)s*.sub.k(t)} represents the power of a kth incident signal source. E{.Math.} represents a mathematical expectation operation. (.Math.).sup.H represents a conjugate transpose operation, (.Math.)* represents a conjugate operation; by calculating the cross-correlation matrix of the received signal, the influence of the noise part custom character in the original received signal is effectively eliminated. In order to realize the derivation of augmented virtual array, based on the second-order cross-correlation statistics, the fourth-order statistics of the L-type coprime array are further derived. Calculating the auto-correlation of the second-order cross-correlation matrix custom character to obtain the fourth-order covariance tensor ?:

(14) $? = R_{?_{1} ?_{2}} ? R_{?_{1} ?_{2}}^{*} = E {(X_{?_{1}} X_{?_{2}}^{H}) ? {(X_{?_{1}} X_{?_{2}}^{H})}^{*}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} a_{?_{1}} (k) ? a_{?_{2}}^{*} (k) ? a_{?_{1}}^{*} (k) ? a_{?_{2}} (k) .$

(15) In practice, based on sampled fourth-order covariance tensor custom character ?, by calculating the fourth-order statistic of the received signals and , we can obtain:

(16) 0 $\hat{?} = (\frac{1}{T} X_{?_{1}} X_{?_{2}}^{H}) ? {(\frac{1}{T} X_{?_{1}} X_{?_{2}}^{H})}^{*};$

(17) Step 3: deriving a fourth-order virtual domain signal corresponding to an augmented virtual uniform cross array. By combining the dimensions in the fourth-order covariance tensor custom character that characterize spatial information in the same direction, the conjugate steering vectors {(k), (k)} and {(k), (k)} corresponding to the two coprime linear arrays .sub.1 and .sub.2 can form a difference set array on the exponential term, so that a non-continuous augmented virtual linear array is constructed on the x-axis and y-axis respectively, and a two-dimensional non-continuous virtual cross array custom character is correspondingly obtained. Specifically, the first and third dimensions of the fourth-order covariance tensor represent the spatial information in the x axial direction, and the second and fourth dimensions represent the spatial information in the y axial direction; for this purpose, dimension sets custom character .sub.1={1, 3}, .sub.2={2, 4} are defined, and a fourth-order virtual domain signal ? corresponding to the non-continuous virtual cross array is obtained by performing dimension-merging tensor transformation on the fourth-order covariance tensor :

(18) $V_{?} \overset{?}{=} ?_{{?_{1}, ?_{2}}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} [a_{?_{1}}^{*} (k) .Math. a_{?_{1}} (k)] ? [a_{?_{2}} (k) .Math. a_{?_{2}}^{*} (k)],$ wherein, by forming a difference set array on an exponential term, custom character (k).Math.(k) and (k).Math.(k) each constructs an augmented virtual linear array on the x axis and y axis, .Math. represents a Kronecker product. contains a virtual uniform cross array =.sub.x?.sub.y, as shown in FIG. 3, wherein .sub.x and .sub.y are the virtual uniform linear arrays on the x axis and on the y axis, respectively. Positions in all virtual array elements in custom character .sub.x and .sub.y are respectively denoted as .sub.x={(, 0)|=[, , . . . , ]d} and .sub.y={(0, y)|y=[, , . . . , ]d}, wherein =??+2, =+, =??+2, =+, and |.sub.x|=2(+)?1, |.sub.y|=2(+)?1. extracting an element corresponding to the position of each virtual array element in the virtual uniform cross array G from the virtual domain signal custom character of the non-continuous virtual cross array , and obtaining a virtual domain signal ? corresponding to , which is modeled as:

(19) ${\overset{?}{V}}_{?} = {.Math.}_{k = 1}^{K} ?_{k}^{4} b_{x} (k) ? b_{y} (k),$ wherein,

(20) $b_{x} (k) = {[e^{- j ? q_{?_{x}}^{(1)} ?_{1} (k)}, e^{- j ? q_{?_{x}}^{(2)} ?_{1} (k)}, .Math., e^{- j ? q_{?_{x}}^{(.Math. ?_{x} .Math.)} ?_{1} (k)}]}^{T},$ $b_{y} (k) = {[e^{- j ? q_{?_{y}}^{(1)} ?_{1} (k)}, e^{- j ? q_{?_{y}}^{(2)} ?_{1} (k)}, .Math., e^{- j ? q_{?_{y}}^{(.Math. ?_{y} .Math.)} ?_{1} (k)}]}^{T},$ represent the steering vectors of custom character .sub.x and .sub.y, respectively;

(21) Step 4: dividing the virtual uniform cross array by translation. Considering the two virtual uniform linear arrays custom character .sub.x and .sub.y that make up the virtual uniform cross array are symmetric about the x=1 and y=1 axis, respectively, extracting the sub-arrays .sub.x.sup.(1)={(, 0)|=[1, 2, . . . , ]d} and .sub.y.sup.(1)={(0, )|=[1, 2, . . . , ]d} from .sub.x and .sub.y as the translation windows; then, translating the translation windows custom character .sub.x.sup.(1) and .sub.y.sup.(1) along negative semi-axis directions of the x axis and the y axis by a virtual array element interval one by one to obtain P.sub.x virtual uniform linear sub-arrays .sub.x.sup.(p.sup.x.sup.)={(, 0)|=[2?p.sub.x, 3?p.sub.x, . . . , +1?p.sub.x]d} and P.sub.y virtual uniform linear sub-arrays custom character .sub.y.sup.(p.sup.y.sup.)={(0, )|=[2?p.sub.y, 3?p.sub.y, . . . , +1?p.sub.y]d}, as shown in FIG. 3, wherein P.sub.x=(|.sub.x|+1)/2, P.sub.y=(|.sub.y|+1)/2; then the virtual domain signal of the virtual uniform sub-arrays .sub.(p.sub.x.sub.,p.sub.y.sub.)=.sub.x.sup.(p.sup.x.sup.)?.sub.y.sup.(p.sup.y.sup.) can be expressed as:

(22) $U_{{\tilde{?}}_{(p_{x}, p_{y})}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} g_{x}^{(p_{x})} (k) ? g_{y}^{(p_{y})} (k) = ?^{.Math. ?_{x}^{(p_{x})} .Math. ? .Math. ?_{y}^{(p_{y})} .Math.},$ wherein,

(23) $g_{x}^{(p_{x})} (k) = {[e^{- j ? (2 - p_{x}) ?_{1} (k)}, e^{- j ? (3 - p_{x}) ?_{1} (k)}, .Math., e^{- j ? (q_{?_{x}}^{(.Math. ?_{x} .Math.)} + 1 - p_{x}) ?_{1} (k)}]}^{T},$ $g_{y}^{(p_{y})} (k) = {[e^{- j ? (2 - p_{y}) ?_{2} (k)}, e^{- j ? (3 - p_{y}) ?_{2} (k)}, .Math., e^{- j ? (q_{?_{y}}^{(.Math. ?_{y} .Math.)} + 1 - p_{y}) ?_{1} (k)}]}^{T},$ are the steering vectors of custom character .sub.x.sup.(p.sup.x.sup.) and .sub.y.sup.(p.sup.y.sup.), respectively;

(24) Step 5: constructing coupled virtual domain tensors by superimposing translational virtual domain signals. Since the virtual uniform sub-arrays custom character .sub.(p.sub.x.sub.,p.sub.y.sub.) obtained by translation division have a spatial translation relationship with each other, the virtual domain signals corresponding to these virtual uniform sub-arrays are structurally superimposed to obtain several virtual domain tensors with a coupling relationship. Specifically, for P.sub.y virtual uniform sub-arrays custom character .sub.(p.sub.x.sub.,:) with the same subscript p.sub.x, they cover the same angle information in the x axial direction, and have a spatial translation relationship in the y axial direction. For this reason, their corresponding virtual domain signals are superimposed in the third dimension to obtain P.sub.x three-dimensional coupled virtual domain tensors custom character .sub.(p.sub.x.sub.)?:

(25) $?_{(p_{x})} = {[U_{{\tilde{?}}_{(p_{x}, 1)}}, U_{{\tilde{?}}_{(p_{x}, 2)}}, .Math., U_{{\tilde{?}}_{(p_{x}, p_{y})}}]}_{?_{3}} = {.Math.}_{k = 1}^{K} ?_{k}^{4} g_{x}^{(p_{x})} (k) ? g_{y}^{(1)} (k) ? q_{y} (k) = ? ?_{k}^{4}; G_{x}^{(p_{x})}, G_{y}^{(1)}, Q_{y} ?,$ Wherein, g.sub.y.sup.(1)(k) is the steering vector of the translation window custom character .sub.y.sup.(1), q.sub.y(k)=[1, e.sup.j??.sup.2.sup.(k), . . . , e.sup.j?(P.sup.y.sup.?1)?.sup.2.sup.(k)].sup.T represents a translation factor along the y axis direction, G.sub.x.sup.(P.sup.x.sup.)=[g.sub.x.sup.(p.sup.x.sup.)(1), g.sub.x.sup.(p.sup.x.sup.)(2), . . . , g.sub.x.sup.(p.sup.x.sup.)(K)], G.sub.y.sup.(1)=[g.sub.y.sup.(1) (1), g.sub.y.sup.(1) (2), . . . , g.sub.y.sup.(1)(K)] and Q.sub.y=[q.sub.y(1), q.sub.y(2), . . . , q.sub.y(K)] are factor matrices of custom character .sub.(p.sub.x.sub.), [.Math.].sub.a represents a tensor superposition operation on the ath dimension, and [[.Math.]] represents a canonical polyadic model of the tensors; the constructed P.sub.x three-dimensional virtual domain tensors .sub.(p.sub.x.sub.) represent the same spatial information in the second dimension (the angle information of the translation window custom character .sub.y.sup.(1)) and the third dimension (the translation information in the y axis direction), and different spatial information in the first dimension (the angle information of the virtual uniform linear sub-arrays .sub.x.sup.(p.sup.x.sup.). For this reason, the virtual domain tensors custom character .sub.(p.sub.x.sub.) have a coupling relationship in the second and third dimensions.

(26) Similarly, coupled virtual domain tensors can be constructed by superimposing the translation virtual domain signals in the x-axis direction. Specifically, for P.sub.x virtual uniform sub-arrays custom character .sub.(:, p.sub.y.sub.) with the same subscript p.sub.y, they cover the same angle information in the axial direction, and have a spatial translation relationship in the x axial direction. Their corresponding virtual domain signals may be superimposed in the third dimension to obtain P.sub.y three-dimensional virtual domain tensors custom character .sub.(p.sub.y.sub.)?

(27) $?_{(p_{x})} = {[U_{{\tilde{?}}_{(p_{x}, 1)}}, U_{{\tilde{?}}_{(p_{x}, 2)}}, .Math., U_{{\tilde{?}}_{(p_{x}, p_{y})}}]}_{?_{3}} = \overset{K}{\underset{k = 1}{.Math.}} ?_{k}^{4} g_{x}^{(p_{x})} (k) ? g_{y}^{(1)} (k) ? q_{y} (k) = ? ?_{k}^{4}; G_{x}^{(p_{x})}, G_{y}^{(1)}, Q_{y} ?,$ Wherein, g.sub.x.sup.(1)(k) is a steering vector of the translation window custom character .sub.x.sup.(1), q.sub.x(k)=[1, e.sup.j??.sup.1.sup.(k), . . . , e.sup.j?(P.sup.x.sup.?1)?.sup.1.sup.(k)].sup.T represents a translation factor along the x axis direction, G.sub.x.sup.(1)=[g.sub.x.sup.(1)(1), g.sub.x.sup.(1) (2), . . . , g.sub.x.sup.(1) (K)], G.sub.y.sup.(p.sup.y.sup.)=[g.sub.y.sup.(p.sup.y.sup.)(1), g.sub.y.sup.(p.sup.y.sup.)(2), . . . , g.sub.y.sup.(p.sup.y.sup.)(K)] and Q.sub.x=[q.sub.x(1), q.sub.x(2), . . . , q.sub.x(K)] are factor matrices of custom character .sub.(p.sub.y.sub.); the constructed P.sub.y three-dimensional virtual domain tensors .sub.(p.sub.y.sub.) represent the same spatial information in the first dimension (the angle information of the translation window .sub.x.sup.(1)) and the third dimension (the translation information in the x axis direction), and different spatial information in the second dimension (the angle information of the virtual uniform linear sub-arrays custom character .sub.y.sup.(p.sup.y.sup.)). For this reason, the virtual domain tensors .sub.(p.sub.y.sub.) have a coupling relationship in the first and third dimensions.

(28) Step 6: obtaining a direction of arrival estimation result by decomposition of the coupled virtual domain tensor. The coupling relationship of the constructed P.sub.x virtual domain tensors custom character .sub.(p.sub.x.sub.) is utilized, the coupled canonical polyadic decomposition is performed on .sub.(p.sub.x.sub.) via a joint least-squares optimization problem:

(29) $({\hat{G}}_{x}^{(p_{x})}, {\hat{G}}_{y}^{(1)}, {\hat{Q}}_{y}} = \min_{{\hat{G}}_{x}^{(p_{x})}, {\hat{G}}_{y}^{(1)}, {\hat{Q}}_{y}} \underset{p_{x}}{.Math.} {.Math. ?_{(p_{x})} - ? {\hat{G}}_{x}^{(p_{x})}, {\hat{G}}_{y}^{(1)}, {\hat{Q}}_{y} ? .Math.}_{F}^{2},$ Wherein, {?.sub.x.sup.(p.sup.x.sup.), ?.sub.y.sup.(1), {circumflex over (Q)}.sub.y} represents the estimated value of the factor matrices {G.sub.x.sup.(p.sup.x.sup.), G.sub.y.sup.(1), Q.sub.y}, which is composed of the estimated value {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1) (k), {circumflex over (q)}.sub.y(k)} of a spatial factor {g.sub.x.sup.(p.sup.x.sup.)(k), g.sub.y.sup.(1)(k), q.sub.y(k)}, and ?.Math.?.sub.F represents the Frobenius norm; by solving the joint least squares optimization problem, {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1) (k), {circumflex over (q)}.sub.y(k)} is obtained. In this problem, the maximum number of identifiable targets K is | custom character .sub.x.sup.(p.sup.x.sup.)|+P.sub.y?2, which exceeds the actual number of the physical array elements of the constructed L-type coprime array with separated sub-arrays. Similarly, the constructed P.sub.y three-dimensional virtual domain tensors .sub.(p.sub.y.sub.) can be decomposed by coupled canonical polyadic to estimate its factor matrix {G.sub.x.sup.(1), G.sub.y.sup.(p.sup.y.sup.), Q.sub.x}. Extracting parameters {circumflex over (?)}.sub.1(k) and {circumflex over (?)}.sub.2(k) from estimated values {?.sub.x.sup.(p.sup.x.sup.)(k), ?.sub.y.sup.(1) (k), {circumflex over (q)}.sub.y(k)} of spatial factors:

(30) ${\hat{?}}_{1} (k) = (\underset{p_{x}}{.Math.} v_{(p_{x})}^{} ? ({\hat{g}}_{x}^{(p_{x})} (k)) / ?) / P_{x},$ ${\hat{?}}_{2} (k) = (w_{(1)}^{} ? ({\hat{g}}_{y}^{(1)} (k)) / ? + z^{} ? ({\hat{q}}_{y} (k)) / ?) / 2,$ wherein, v.sub.(p.sub.x.sub.)=[2?p.sub.x, 3?p.sub.x, . . . , custom character +1?p.sub.x].sup.T is a position index of each virtual array element in .sub.x.sup.(p.sup.x.sup.), w.sub.(1)=[1, 2, . . . , ].sup.T represents a position index of each virtual array element in .sub.y.sup.(1), z=[0, 1, . . . , P.sub.y?1].sup.T represents a translation step, ?(.Math.) represents a complex argument taking operation. (.Math.).sup. represents a pseudo-inverse operation. Finally, according to the relationship between {?.sub.1(k), ?.sub.2(k)} and the two-dimensional direction of arrival (?.sub.k, ?.sub.k), namely ?.sub.1(k)=sin (?.sub.k)cos(?.sub.k) and ?.sub.2(k)=sin (?.sub.k)sin(?.sub.k), a closed-form solution of the two-dimensional direction of arrival estimation ({circumflex over (?)}.sub.k, {circumflex over (?)}.sub.k) is obtained as:

(31) 0 ${\overset{?}{?}}_{k} = \arctan (\frac{{\overset{?}{?}}_{2} (k)}{{\overset{?}{?}}_{1} (k)}),$ ${\overset{?}{?}}_{k} = \sqrt{{{\overset{?}{?}}_{1} (k)}^{2} + {{\overset{?}{?}}_{2} (k)}^{2}} .$

(32) The effects of the present invention will be further described below in conjunction with a simulation instance.

(33) The simulation instance: The L-type coprime array is used to receive the incident signal, and its parameters are selected as custom character ==2, ==3, that is, the constructed L-type coprime array contains 2++2+?2=12 antenna elements. Assuming that there are 2 incident narrowband signals, the azimuth and elevation angles of the incident directions are respectively [20.5?, 30.5?] and [45.6?, 40.6?]. The method for estimating direction of arrival of an L-type coprime array based on coupled tensor decomposition and the traditional Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) method based on the vectorized virtual domain signal processing, and the TensorMultipleSignal Classification (Tensor MUSIC) method based on traditional tensor decomposition are compared. FIG. 4 and FIG. 5 respectively compare the performance of the two-dimensional direction of arrival estimation accuracy of the above methods under the conditions of different signal-to-noise ratios and different sampling snapshot amounts.

(34) Under the condition of the number T=300 of sampling snapshots, plotting the performance comparison curve of direction of arrival estimation root-mean-square error as a function of signal-to-noise ratios, as shown in FIG. 4; under the condition of signal-to-noise ratio SNR=0 dB, plotting the performance comparison curve of the root-mean-square error of direction of arrival estimation as a function of the sampling snapshot amounts, as shown in FIG. 5. It can be seen from the comparison results of FIG. 4 and FIG. 5 that the method proposed in the present invention has a performance advantage in the estimation accuracy of direction of arrival no matter in different signal-to-noise ratio scenarios or in different sampling snapshot amount scenarios. Compared with the ESPRIT method based on vectorized virtual domain signal processing, the method proposed in the present invention makes full use of the structural information of the received signal of the L-type coprime array by constructing the virtual domain tensors, thereby having superior direction of arrival estimation accuracy. On the other hand, compared with the TensorMUSIC method based on traditional tensor decomposition, the performance advantage of the method proposed in the present invention comes from making full use of the spatial correlation properties of multi-dimensional virtual domain signals by coupling virtual domain tensor processing, while the traditional tensor decomposition methods only process a single virtual domain tensor after spatial smoothing, resulting in the loss of virtual domain signal correlation information.

(35) To sum up, the present invention constructs the correlation between the multi-dimensional virtual domain of the L-type coprime array and the tensor signal modeling, deduces the sparse tensor signal to the virtual domain tensor model, and deeply excavates the received signal of the L-type coprime array and the multi-dimensional features of the virtual domain; furthermore, the spatial superposition mechanism of the virtual domain signals is established, and the virtual domain tensors with the spatial coupling relationship are constructed without introducing the spatial smoothing; finally, the present invention uses the coupled decomposition of the virtual domain tensors, realizes the accurate estimation of the two-dimensional direction of arrival, and gives its closed-form solution.

(36) The above descriptions are only preferred embodiments of the present invention. Although the present invention has been disclosed above with preferred examples, it is not intended to limit the present invention. Any person skilled in the art, without departing from the scope of the technical solution of the present invention, can make many possible changes and modifications to the technical solution of the present invention by using the methods and technical contents disclosed above, or modify them into equivalent examples of equivalent changes. Therefore, any simple alterations, equivalent changes and modifications made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solutions of the present invention still fall within the protection scope of the technical solutions of the present invention.