METHOD FOR ESTIMATING DIRECTION OF ARRIVAL OF SUB-ARRAY PARTITION TYPE L-SHAPED COPRIME ARRAY BASED ON FOURTH-ORDER SAMPLING COVARIANCE TENSOR DENOISING

Abstract

Disclosed in the present invention is a method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising, which mainly solves problems of a damage to a signal structure and noise term interference to high-order virtual domain statistics in an existing method. The implementation steps are as follows: constructing an L-shaped coprime array partitioned with linear sub-arrays; modeling a receiving signal of the L-shaped coprime array and deriving a second-order cross-correlation matrix thereof, deriving a fourth-order covariance tensor based on the cross-correlation matrix; realizing fourth-order sampling covariance tensor denoising based on kernel tensor thresholding; deriving a fourth-order virtual domain signal based on denoised sampling covariance tensor; constructing a denoised structured virtual domain tensor; obtaining a direction of arrival estimation result by decomposing the structured virtual domain tensor.

Claims

1. A method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising, wherein the method comprises the following steps: (1) constructing a linear sub-array partition type L-shaped coprime array by a receiving end with 2++2+−2 physical antenna array elements, wherein the L-shaped coprime array consists of two coprime linear arrays .sub.i, i=1, 2 located on an x axis and a y axis, and first array elements of the two coprime linear arrays .sub.1 and .sub.2 are laid out from a positions where coordinates are 1 on the x axis and the y axis respectively; the coprime linear array .sub.i contains |.sub.i|=2+−1 array elements, and wherein , and are a pair of coprime integers, <, |⋅| represents a potential of a set; $\left\{\left(x_{{𝕃}_1},0\right)|x_{{𝕃}_1}=\left[c_{{𝕃}_1}^{\left(1\right)},c_{{𝕃}_1}^{\left(2\right)},.Math.,c_{{𝕃}_1}^{\left(.Math.{𝕃}_1.Math.\right)}\right]d\right\}\mathrm{and}\left\{\left(0,y_{{𝕃}_2}\right)|y_{{𝕃}_2}=\left[c_{{𝕃}_2}^{\left(1\right)},c_{{𝕃}_2}^{\left(2\right)},.Math.,c_{{𝕃}_2}^{\left(.Math.{𝕃}_2.Math.\right)}\right]d\right\}$ are respectively used to represent a position of each array element of the L-shaped coprime array on the x axis and y axis, wherein ==1, and a unit interval d is taken as half of a wavelength of an incident narrowband signal; (2) if there are K far-field narrow-band incoherent signal sources from {(θ.sub.1, φ.sub.1), (θ.sub.2, φ.sub.2), . . . , (θ.sub.K, φ.sub.K)} directions, modeling a received signal of the coprime linear array .sub.i forming the L-shaped coprime array as follows: $X_{{𝕃}_i}=\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_i}\left(k\right)\circ s_k+N_{{𝕃}_i}\in {ℂ}^{\left|{𝕃}_i\right|\times T},$ 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, º represents an outer product of a vector, is noise independent of each signal source, (k) is a steering vector of .sub.i, and corresponds to a signal source having an incoming wave direction of (θ.sub.k, φ.sub.k) and is expressed as follows: $a_{{𝕃}_i}\left(k\right)={\left[e^{-j\pi c_{{𝕃}_i}^{\left(1\right)}{\mu }_i\left(k\right)},e^{-j{\mathrm{\pi c}}_{{𝕃}_i}^{\left(2\right)}{\mu }_i\left(k\right)},.Math.,e^{-j\pi c_{{𝕃}_i}^{\left(.Math.{𝕃}_i.Math.\right)}{\mu }_i\left(k\right)}\right]}^T,$ wherein, μ.sub.1(k)=sin(φ.sub.k)cos(θ.sub.k), μ.sub.2(k)=sin(φ.sub.k)sin(θ.sub.k), j=√{square root over (−1)}, [⋅].sup.T represents a transpose operation; a second-order cross-correlation matrix $R_{{𝕃}_1{𝕃}_2}\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_2\right|}$ is obtained by solving cross-correlation statistics of and : $R_{{𝕃}_1{𝕃}_2}=E\left\{X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right\}=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{a}_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right),$ and wherein, σ.sub.k.sup.2=E{s.sub.k(t)s.sub.k*(t)} represents power of the kth incident signal source, E{⋅} represents a mathematical expectation operation, (⋅).sup.H represents a conjugate transpose operation, (⋅)* represents a conjugate operation; (3) calculating an autocorrelation of the second-order cross-correlation matrix to obtain a fourth-order covariance tensor $𝒱\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_2\right|\times \left|{𝕃}_1\right|\times \left|{𝕃}_2\right|}:$ $\begin{array}{c}V=R_{{𝕃}_1{𝕃}_2}\circ R_{{𝕃}_1{𝕃}_2}^{*}=E\left\{\left(X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)\circ {\left(X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)}^{*}\right\}\\ =\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^4{a}_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right)\circ a_{{𝕃}_1}^{*}\left(k\right)\circ a_{{𝕃}_2}\left(k\right);\end{array}$ wherein, the fourth-order covariance tensor is approximated by a fourth-order sampling covariance tensor $\hat{v}\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_2\right|\times \left|{𝕃}_1\right|\times \left|{𝕃}_2\right|},$  that is: $\widehat{𝒱}=\left(\frac{1}{T}{X}_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)\circ {\left(\frac{1}{T}{X}_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)}^{*}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)a_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right)\circ a_{{𝕃}_1}^{*}\left(k\right)\circ a_{{𝕃}_2}\left(k\right)+𝒵,$ $\mathrm{wherein}:$ $\begin{array}{c}𝓏=\left[\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_2}^H\right)+\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_1}^H\right)+\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H\right]\\ {\left[\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_2}^H\right)+\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_1}^H\right)+\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H\right]}^{*}\end{array}$ is a fourth-order sampling noise tensor; the (τ.sub.1, .sub.1, τ.sub.2, .sub.2)th element in is represented as .sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.), τ.sub.1, τ.sub.2=1, 2, . . . , |.sub.2|, .sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.) obeys an approximate complex Gaussian distribution, and an approximate variance thereof σ.sup.2 is expressed as: ${\overline{\sigma }}^2=\frac{1}{T^2}\left[{{\lambda }_1\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2+{\lambda }_2{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2+{\lambda }_3{\sigma }_n^8\right],$ and wherein, λ.sub.1, λ.sub.2 and λ.sub.3 represent a combined weight of three sub-variance terms ${\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2,{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2$  and σ.sub.n.sup.B, σ.sub.n.sup.2 represents a noise power; (4) performing high-order singular value decomposition on the fourth-order sampling covariance tensor :
=×.sub.1Y.sup.(1)×.sub.2Y.sup.(2)×.sub.3Y.sup.(3)×.sub.4Y.sup.(4), wherein, $𝒮\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_2\right|\times \left|{𝕃}_1\right|\times \left|{𝕃}_2\right|}$  represents a kernel tensor, which contains projections from signal and noise components in , $Y^{\left(1\right)}\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_1\right|},Y^{\left(2\right)}\in {ℂ}^{\left|{𝕃}_2\right|\times \left|{𝕃}_2\right|},Y^{\left(3\right)}\in {ℂ}^{\left|{𝕃}_1\right|\times \left|{𝕃}_1\right|}\mathrm{and}$ $Y^{\left(4\right)}\in {ℂ}^{\left|{𝕃}_2\right|\times \left|{𝕃}_2\right|}$  represent singular matrices corresponding to four dimensions of ; a thresholding is performed on , that is, elements in that are less than or equal to a noise threshold ϵ are set to zero, and elements larger than the noise threshold ϵ are reserved, thus obtaining a thresholded kernel tensor .sub.dn, where an element in .sub.dn is expressed as follows: ${𝒮}_{dn\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}=\{\begin{array}{cc}& {𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}|{𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}\text{|>}\in ,\\ 0& .Math.{𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}.Math.\le \in ,\end{array}$ and wherein, S.sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.) represents a (τ.sub.1, .sub.1, τ.sub.2, .sub.2)th element of , the noise threshold ϵ is as follows:
ϵ=σ.sup.2√{square root over (2 log(|.sub.1∥.sub.2∥.sub.1∥.sub.2|))}: the thresholded kernel tensor .sub.dn is multiplied with the four singular matrices Y.sup.(1), Y.sup.(2), Y.sup.(3) and Y.sup.(4) to obtain a denoised sampling covariance tensor .sub.dn, which is expressed as follows:
.sub.dn=.sub.dn×.sub.1Y.sup.(1)×.sub.2Y.sup.(2)×.sub.3Y.sup.(3)×.sub.4Y.sup.(4); (5) defining dimension sets .sub.1={1, 3} and .sub.2={2, 4}, and obtaining a fourth-order virtual domain signal ${\hat{V}}_{𝕎}\in {ℂ}^{|{𝕃}_1{|}^2\times |{𝕃}_2{|}^2}$  by performing tensor transformation of dimension merging on the denoised sampling covariance tensor .sub.dn: ${\hat{V}}_{𝕎}\stackrel{\Delta }{=}{\widehat{𝒱}}_{{\mathrm{dn}}_{\left\{{𝕁}_1,{𝕁}_2\right\}}}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)\left[a_{{𝕃}_1}^{*}\left(k\right).Math.a_{{𝕃}_1}\left(k\right)\right]\circ \left[a_{{𝕃}_2}\left(k\right).Math.a_{{𝕃}_2}^{*}\left(k\right)\right],$ wherein, for (k).Math.(k) and (k).Math.(k), by forming a difference set array on exponent terms respectively, augmented virtual linear arrays on the x axis and on the y axis are constructed, .Math. representing a Kronecker product; corresponds to a two-dimensional non-continuous virtual cross array , contains a virtual uniform cross array =.sub.x∪.sub.y, where .sub.x and .sub.y are respectively virtual uniform linear arrays on the x axis and the y axis; positions of all virtual array elements in .sub.x and .sub.y are respectively expressed as ${𝕍}_x=\left\{\left(x_{𝕍},0\right)|x_{𝕍}=\left[q_{{𝕍}_x}^{\left(1\right)},q_{{𝕍}_x}^{\left(2\right)},.Math.,q_{{𝕍}_x}^{\left(\left|{𝕍}_x\right|\right)}\right]d\right\}\mathrm{and}{𝕍}_y=\left\{\left(0,y_{𝕍}\right)|y_{𝕍}=\left[q_{{𝕍}_y}^{\left(1\right)},q_{{𝕍}_y}^{\left(2\right)},.Math.,q_{{𝕍}_y}^{\left(\left|{𝕍}_y\right|\right)}\right]d\right\},\mathrm{where}{q}_{{𝕍}_x}^{\left(1\right)}=-M_{{𝕃}_1}{N}_{{𝕃}_1}-M_{{𝕃}_1}+2,q_{{𝕍}_x}^{\left(\left|{𝕍}_x\right|\right)}=M_{{𝕃}_1}{N}_{{𝕃}_1}+M_{{𝕃}_1},$ =−−+2, $q_{{𝕍}_y}^{\left(.Math.{𝕍}_y.Math.\right)}=M_{{𝕃}_2}{N}_{{𝕃}_2}+M_{{𝕃}_2},$ and |.sub.x|=2()−1, |.sub.y|=2(−)−1; elements corresponding to positions of all virtual array elements in the virtual uniform cross array are extracted from the virtual domain signal of a non-contiguous virtual cross array to obtain a fourth-order virtual domain signal ${\hat{U}}_{𝕍}\in {ℂ}^{\left|{𝕍}_x\right|\times \left|{𝕍}_y\right|}$ corresponding to ; (6) respectively extracting sub-arrays ${\mathbb{Q}}_x^{\left(1\right)}=\left\{\left(x_{\mathbb{Q}}^{\left(1\right),},0\right).Math.x_{\mathbb{Q}}^{\left(1\right)}=\left[1,2,.Math.,q_{{\mathbb{Q}}_x}^{\left(\left|{\mathbb{Q}}_x\right|\right)}\right]d\right\},$ ${\mathbb{Q}}_y^{\left(1\right)}=\left\{\left(0,y_{\mathbb{Q}}^{\left(1\right)}\right).Math.y_{\mathbb{Q}}^{\left(1\right)}=\left[1,2,.Math.,q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}\right]d\right\}$  from .sub.x and .sub.y as translation windows; then, respectively translating the translation windows .sub.x.sup.(1) and .sub.y.sup.(1) along a negative semi-axis direction of the axis x and the axis y by a virtual array element interval d, to obtain J.sub.x virtual uniform linear sub-arrays ${\mathbb{Q}}_x^{\left(j_x\right)}=\left\{\left(x_{\mathbb{Q}}^{\left(j_x\right)},0\right).Math.x_{\mathbb{Q}}^{\left(j_x\right)}=\left[2-j_x,3-j_x,.Math.,q_{{\mathbb{Q}}_x}^{\left(\left|{\mathbb{Q}}_x\right|\right)}+1-j_x\right]d\right\}$  and J.sub.y virtual uniform linear sub-arrays ${\mathbb{Q}}_y^{\left(j_y\right)}=\left\{\left(0,y_{\mathbb{Q}}^{\left(j_y\right)}\right).Math.y_{\mathbb{Q}}^{\left(j_y\right)}=\left[2-j_y,3-j_y,.Math.,q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}+1-j_y\right]d\right\},$  j.sub.x=1, 2, . . . , J.sub.x, j.sub.y=1, 2, . . . , J.sub.y, J.sub.x=(|.sub.x|+1)/2, J.sub.y=(|.sub.y|+1)/2, so that a virtual domain signal corresponding to a virtual uniform sub-array .sub.(j.sub.x.sub.,j.sub.y.sub.)=.sub.x.sup.(j.sup.x.sup.)∪.sub.y.sup.(j.sup.y.sup.) be expressed as $U_{{\tilde{\mathbb{Q}}}_{\left(j_x,j_y\right)}}\in {ℂ}^{J_x\times J_y};$  fixing j.sub.y index, superimposing $U_{{\tilde{\mathbb{Q}}}_{\left(:,j_y\right)}}$  in a third dimension to obtain J.sub.y three-dimensional virtual domain tensors, and then, superimposing the J.sub.y three-dimensional virtual domain tensors in a fourth dimension to obtain a four-dimensional denoised structured virtual domain tensor ∈.sup.J.sup.x.sup.×J.sup.y.sup.×J.sup.x.sup.×J.sup.y, which is expressed as follows: $\widetilde{𝒰}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)l_x\left(k\right).Math.l_y\left(k\right).Math.v_x\left(k\right).Math.v_y\left(k\right),$ $\mathrm{wherein}:$ $l_x\left(k\right)={\left[e^{-j{\mathrm{\pi \mu }}_1\left(k\right)},e^{-j{\mathrm{\pi 2\mu }}_1\left(k\right)},.Math.,e^{-j\pi q_{{\mathbb{Q}}_x}^{(|{\mathbb{Q}}_x.Math.)}{\mu }_1\left(k\right)}\right]}^T,$ $l_y\left(k\right)={\left[e^{-j{\mathrm{\pi \mu }}_2\left(k\right)},e^{-j\pi 2{\mu }_2\left(k\right)},.Math.,e^{-j\pi q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}{\mu }_2\left(k\right)}\right]}^T,$ are steering vectors of .sub.x.sup.(1) and .sub.y.sup.(1), respectively, $v_x\left(k\right)={\left[1,e^{-j{\mathrm{\pi \mu }}_1\left(k\right)},.Math.,e^{-j\pi \left(q_{{\mathbb{Q}}_x}^{(|{\mathbb{Q}}_x.Math.)}-1\right){\mu }_1\left(k\right)}\right]}^T,$ $v_y\left(k\right)={\left[1,e^{-j{\mathrm{\pi \mu }}_2\left(k\right)},.Math.,e^{-j\pi \left(q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}-1\right){\mu }_2\left(k\right)}\right]}^T,$ are translation factors along the x axis and the y axis, respectively; and (7) performing tensor decomposition on the denoised structured virtual domain tensor by canonical polyadic decomposition (CPD) to obtain an estimated value of each spatial factor of , that is, {{circumflex over (l)}.sub.x(k), {circumflex over (l)}.sub.y(k), {circumflex over (v)}.sub.x(k), {circumflex over (v)}.sub.y(k)}; extracting parameters {circumflex over (μ)}.sub.1(k) and {circumflex over (μ)}.sub.2(k) from {{circumflex over (l)}.sub.x(k), {circumflex over (l)}.sub.y(k), {circumflex over (v)}.sub.x(k), {circumflex over (v)}.sub.y(k)}, and obtaining a closed-form solution of a two-dimensional direction of arrival estimation ({circumflex over (θ)}.sub.k, {circumflex over (φ)}.sub.k) according to a relationship between {μ.sub.1(k), μ.sub.2(k)} and a two-dimensional direction of arrival (θ.sub.k, φ.sub.k).

2. The method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, wherein a structure of the linear sub-array partition type L-shaped coprime array in step (1) is specifically described as follows: the coprime linear array .sub.i forming the L-shaped coprime array is composed of a pair of sparse uniform linear sub-arrays, two sparse uniform linear sub-arrays respectively contain 2M.sub.i and N.sub.i antenna array elements, and array element spacings are respectively d and d; the two sparse linear uniform sub-arrays in .sub.i are combined in a form of overlapping the first array elements to obtain the coprime linear array .sub.i containing |.sub.i|=2+−1 array elements.

3. The method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, wherein, for the fourth-order sampling noise tensor described in step (3), the (τ, )th elements in $\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right).Math.\left(s_k^T{N}_{{𝕃}_2}^H\right),\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right).Math.\left(s_k^T{N}_{{𝕃}_1}^H\right)\mathrm{and}\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H$  are expressed as g.sub.(τ,.sub.), h.sub.(τ,.sub.) and n.sub.(τ,.sub.), τ=1, 2, . . . , |.sub.i|, =1, 2, . . . , |.sub.2| respectively, then the (τ.sub.1, .sub.1, τ.sub.2, .sub.2)th element in is expressed as follows: $\begin{array}{c}{𝒵}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}=\left(g_{\left({\tau }_1,{\varsigma }_1\right)}+h_{\left({\tau }_1,{\varsigma }_1\right)}+n_{\left({\tau }_1,{\varsigma }_1\right)}\right){\left(g_{\left({\tau }_2,{\varsigma }_2\right)}+h_{\left({\tau }_2,{\varsigma }_2\right)}+n_{\left({\tau }_2,{\varsigma }_2\right)}\right)}^{*}\\ =g_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+g_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+g_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*};\end{array}$ g.sub.(τ,.sub.), h.sub.(τ,.sub.) and n.sub.(τ,.sub.) respectively obey the approximate complex Gaussian distribution, that is: $g_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),$ $h_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),$ $n_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^4\right),$ $\mathrm{so}$ $g_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},g_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2\right),$ $g_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},n_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},n_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),$ $n_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\sigma }_n^8\right),$ .sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.) also obeys the approximate complex Gaussian distribution, and an approximate variance thereof σ.sup.2 is expressed as follows: ${\overline{\sigma }}^2=\frac{1}{T^2}\left[{{\lambda }_1\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2+{\lambda }_2{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2+{\lambda }_3{\sigma }_n^8\right].$

4. The method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, wherein, for the fourth-order virtual domain signal derivation described in step (5), the virtual domain signal ${\hat{U}}_{𝕍}\in {ℂ}^{\left|{𝕍}_x\right|\times \left|{𝕍}_y\right|}$  corresponding to the virtual uniform cross array can be expressed as follows: ${\hat{U}}_{𝕍}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)g_x\left(k\right).Math.g_y\left(k\right),$ $\mathrm{wherein}:$ $g_x\left(k\right)={\left[e^{-j\pi q_{{𝕍}_x}^{\left(1\right)}{\mu }_1\left(k\right)},e^{-j\pi q_{{𝕍}_x}^{\left(2\right)}{\mu }_1\left(k\right)},.Math.,e^{-j\pi q_{{𝕍}_x}^{(|{𝕍}_x.Math.)}{\mu }_1\left(k\right)}\right]}^T,$ $g_y\left(k\right)={\left[e^{-j\pi q_{{𝕍}_y}^{\left(1\right)}{\mu }_2\left(k\right)},e^{-j\pi q_{{𝕍}_y}^{\left(2\right)}{\mu }_2\left(k\right)},.Math.,e^{-j\pi q_{{𝕍}_y}^{\left(\left|{𝕍}_y\right|\right)}{\mu }_2\left(k\right)}\right]}^T,$ are steering vectors of .sub.x and .sub.y, respectively.

5. The method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, wherein, for the two-dimensional direction of arrival estimation process described in step (7), parameters {circumflex over (μ)}.sub.1(k) and {circumflex over (μ)}.sub.2 (k) are extracted from {{circumflex over (l)}.sub.x(k), {circumflex over (l)}.sub.y(k), {circumflex over (v)}.sub.x(k), {circumflex over (v)}.sub.y(k)}:
{circumflex over (μ)}.sub.1(k)=∠({circumflex over (l)}.sub.x.sup.T(k){circumflex over (v)}.sub.x(k)/J.sub.x)/π,
{circumflex over (μ)}.sub.2(k)=∠({circumflex over (l)}.sub.y.sup.T(k){circumflex over (v)}.sub.y(k)/J.sub.y)/π, wherein, β(⋅) represents an operation of taking an argument of a complex number; the closed-form solution of the two-dimensional direction of arrival estimation ({circumflex over (θ)}.sub.k, {circumflex over (φ)}.sub.k) is obtained according to the relationship between {μ.sub.1(k), μ.sub.2 (k)} and the two-dimensional direction of arrival (θ.sub.k, φ.sub.k), that is, μ.sub.1(k)=sin(φ.sub.k)cos(θ.sub.k) and μ.sub.2(k)=sin(φ.sub.k)sin(θ.sub.k): ${\hat{\theta }}_k=\mathrm{arc}\tan \left(\frac{{\hat{\mu }}_2\left(k\right)}{{\hat{\mu }}_1\left(k\right)}\right),$ ${\hat{\phi }}_k=\sqrt{{{\hat{\mu }}_1\left(k\right)}^2+{{\hat{\mu }}_2\left(k\right)}^2}.$

6. The method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, wherein, in step (7), according to a uniqueness condition of the CPD, the following condition are met for performing the CPD on :
κ({circumflex over (L)}.sub.x)+κ({circumflex over (L)}.sub.y)+κ({circumflex over (V)}.sub.x)+κ({circumflex over (V)}.sub.y)≥2K+3, wherein, κ(⋅) represents a Kruskal rank of the matrix, {circumflex over (L)}.sub.x= $\begin{array}{cc}\left[{\hat{l}}_x\left(1\right),{\hat{l}}_x\left(2\right),.Math.{\hat{l}}_x\left(K\right)\right]\in {ℂ}^{J_x\times K},& \left(1\right)\end{array}$ ${\hat{L}}_y=\left[{\hat{l}}_y\left(1\right),{\hat{l}}_y\left(2\right),.Math.{\hat{l}}_y\left(K\right)\right]\in {ℂ}^{J_y\times K},$ ${\hat{V}}_x=\left[{\hat{v}}_x\left(1\right),{\hat{v}}_x\left(2\right),.Math.{\hat{v}}_x\left(K\right)\right]\in {ℂ}^{J_x\times K}\mathrm{and}$ ${\hat{V}}_y=\left[{\hat{v}}_y\left(1\right),{\hat{v}}_y\left(2\right),.Math.{\stackrel{}{\hat{v}}}_y\left(K\right)\right]\in {ℂ}^{J_y\times K}$  are factor matrices of ; κ({circumflex over (L)}.sub.x)=min(J.sub.x, K), κ({circumflex over (L)}.sub.y)=min(J.sub.y, K), κ({circumflex over (V)}.sub.x)=min(J.sub.x, K) and κ({circumflex over (V)}.sub.y)=min(J.sub.y, K) are substituted into an uniqueness conditional inequality of the CPD to obtain K≤└(|.sub.x|+|.sub.y|−1)/2┘, where └⋅┘ represents a round-up operation.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] FIG. 1 is a general flow block diagram of the present invention.

[0040] FIG. 2 is a schematic structural diagram of a sub-array partition L-type coprime array proposed by the present invention.

[0041] FIG. 3 is a schematic diagram of virtual uniform cross arrays and virtual uniform sub-arrays thereof constructed by the present invention.

[0042] FIG. 4 is a graph of estimation results of two-dimensional underdetermined direction of arrival of a traditional Tensor MUSIC method.

[0043] FIG. 5 is a graph of estimation results of two-dimensional underdetermined direction of arrival of the method proposed by the present invention.

DESCRIPTION OF THE EMBODIMENTS

[0044] The technical solutions of the present invention will be described in further detail below with reference to the accompanying drawings.

[0045] In order to solve the problems of a damage to a signal structure and noise term interference to high-order virtual domain statistics in an existing method, the present invention proposes a method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising, wherein high-order tensor statistics of the sub-array partition L-shaped coprime array is derived, a denoising technique for the sampling covariance tensor is designed, and a high-precision two-dimensional direction of arrival estimation is realized based on denoised virtual domain tensor signal processing. Refer to FIG. 1, the implementation steps of the present invention are as follows:

[0046] Step 1: constructing a linear sub-array partition type L-shaped coprime array. At a receiving end, using 2++2+−2 physical antenna array elements to construct a linear sub-array partition L-shaped coprime array, as shown in FIG. 2: constructing a coprime linear array .sub.i, i=1, 2 on the x axis and y axis respectively, where .sub.i contains |.sub.i|=2+−1 antenna array elements, wherein, and are a pair of coprime integers, |⋅| represents a potential of the set; the first array elements of the two coprime linear arrays .sub.1 and .sub.2 are laid out from the positions where the coordinates are 1 on the x axis and y axis respectively, so the two coprime linear arrays .sub.1 and .sub.2 that make up the L-shaped coprime array do not overlap with each other; respectively using

[00024]$\left\{\left(x_{{𝕃}_1},0\right).Math.x_{{𝕃}_1}=\left[c_{{𝕃}_1}^{\left(1\right)},c_{{𝕃}_1}^{\left(2\right)},.Math.,c_{{𝕃}_1}^{\left(.Math.{𝕃}_1.Math.\right)}\right]d\right\}$$\mathrm{and}$$\left\{\left(0,y_{{𝕃}_2}\right)|y_{{𝕃}_2}=\left[c_{{𝕃}_2}^{\left(1\right)},c_{{𝕃}_2}^{\left(2\right)},.Math.,c_{{𝕃}_2}^{\left(.Math.{𝕃}_2.Math.\right)}\right]d\right\}$

to represent the positions of all array element of the L-shaped coprime array on the x axis and y axis, where, ==1, and the unit interval d is taken as half of the wavelength of an incident narrowband signal; the two partition coprime linear arrays .sub.i constituting the L-shaped coprime array are respectively composed of a pair of sparse uniform linear sub-arrays, and the two sparse uniform linear sub-arrays respectively contain 2 and antenna array elements, <, and the array element spacings are respectively d and d, and they are combined in a form of overlapping the first array elements to obtain a coprime linear array .sub.i containing 2+−1 array elements.

[0047] Step 2: modeling a received signal of the L-shaped coprime array and deriving a second-order cross-correlation matrix thereof. assuming that there are K far-field narrow-band incoherent signal sources from {(θ.sub.1, φ.sub.1), (θ.sub.2, φ.sub.2), . . . , (θ.sub.K, φ.sub.K)} directions, a received signal of the two coprime linear arrays .sub.1 and .sub.2 forming the L-shaped coprime array is modeled as follows:

[00025]$X_{{𝕃}_i}=\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_i}\left(k\right)\circ s_k+N_{{𝕃}_i}\in {ℂ}^{\left|{𝕃}_i\right|\times T},$

[0048] 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, º represents the outer product of the vector, is noise independent of each signal source, (k) is a steering vector of .sub.i, and corresponds to a signal source having an incoming wave direction of (θ.sub.k, φ.sub.k) and is expressed as follows:

[00026]$a_{{𝕃}_i}\left(k\right)={\left[e^{-j\pi c_{{𝕃}_i}^{\left(1\right)}{\mu }_i\left(k\right)},e^{-j\pi c_{{𝕃}_i}^{\left(2\right)}{\mu }_i\left(k\right)},.Math.,e^{-j\pi c_{{𝕃}_i}^{(|{𝕃}_i.Math.)}{\mu }_i\left(k\right)}\right]}^T,$

[0049] wherein, μ.sub.1(k)=sin(μ.sub.k)cos(θ.sub.k), μ.sub.2(k)=sin(μ.sub.k)sin(θ.sub.k), j=√{square root over (−1)}, [⋅].sup.T represents a transpose operation; a second-order cross-correlation matrix ∈ is obtained by solving cross-correlation statistics of sampling signals and of coprime linear arrays .sub.1 and .sub.2:

[00027]$R_{{𝕃}_1{𝕃}_2}=E\left\{X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right\}=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{a}_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right),$

[0050] wherein, σ.sub.k.sup.2=E{s.sub.k(t)s.sub.k*(t)} represents power of a kth incident signal source, E{⋅} represents a mathematical expectation operation, (⋅).sup.H represents a conjugate transpose operation, (⋅)* represents a conjugate operation; by performing the cross-correlation calculation on the received signals, the noise power term introduced by the autocorrelation calculation of the noise is eliminated, that is, E{}=σ.sub.n.sup.2I, where σ.sub.n.sup.2 an represents the noise power and I represents the identity matrix.

[0051] Step 3: deriving a fourth-order covariance tensor based on the cross-correlation matrix. In order to realize the derivation of an augmented virtual array, based on the second-order cross-correlation statistics, fourth-order statistics of L-type coprime arrays are further derived. Specifically, calculating the autocorrelation of the second-order cross-correlation matrix to obtain a fourth-order covariance tensor ∈:

[00028]$\begin{array}{c}V=R_{{𝕃}_1{𝕃}_2}\circ R_{{𝕃}_1{𝕃}_2}^{*}=E\left\{\left(X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)\circ {\left(X_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)}^{*}\right\}\\ =\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^4{a}_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right)\circ a_{{𝕃}_1}^{*}\left(k\right)\circ a_{{𝕃}_2}\left(k\right).\end{array}$

[0052] In practice, it can be obtained by estimating the fourth-order statistic of the received signals and , that is, the fourth-order sampling covariance tensor ∈:

[00029]$\widehat{𝒱}=\left(\frac{1}{T}{X}_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)\circ {\left(\frac{1}{T}{X}_{{𝕃}_1}{X}_{{𝕃}_2}^H\right)}^{*}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)a_{{𝕃}_1}\left(k\right)\circ a_{{𝕃}_2}^{*}\left(k\right)\circ a_{{𝕃}_1}^{*}\left(k\right)\circ a_{{𝕃}_2}\left(k\right)+𝒵,$$\mathrm{wherein}:$$\begin{array}{c}𝒵=\left[\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_2}^H\right)+\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_1}^H\right)+\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H\right]\\ {\left[\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_2}^H\right)+\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_1}^H\right)+\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H\right]}^{*}\end{array}$

[0053] is the fourth-order sampling noise tensor. The (τ, )th elements in

[00030]$\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_1}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_2}^H\right)\frac{1}{T}\underset{k=1}{\overset{K}{.Math.}}{a}_{{𝕃}_2}\left(k\right)\circ \left(s_k^T{N}_{{𝕃}_1}^H\right)\mathrm{and}\frac{1}{T}{N}_{{𝕃}_1}{N}_{{𝕃}_2}^H$

are expressed as g.sub.(τ,.sub.), h.sub.(τ,.sub.) and n(τ,), τ=1, 2, . . . , |.sub.1|, =1, 2, . . . , |.sub.2| respectively, then the (τ.sub.1, .sub.1, τ.sub.2, .sub.2)th element in may be expressed as follows:

[00031]${𝒵}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}=\left(g_{\left({\tau }_1,{\varsigma }_1\right)}+h_{\left({\tau }_1{\varsigma }_1\right)}+n_{\left({\tau }_1,{\varsigma }_1\right)}\right){\left(g_{\left({\tau }_2,{\varsigma }_2\right)}+h_{\left({\tau }_2,{\varsigma }_2\right)}+n_{\left({\tau }_2,{\varsigma }_2\right)}\right)}^{*}=g_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+g_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+g_{\left({\tau }_1{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2.{\varsigma }_2\right)}^{*}+h_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}+n_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},$

[0054] wherein, τ.sub.1, τ.sub.2=1, 2, . . . , |.sub.1|, .sub.1, .sub.2=1, 2, . . . , |.sub.2|.Math.g.sub.(τ, .sub.), h.sub.(τ, .sub.) and n(τ, ) respectively obey the approximate complex Gaussian distribution, that is:

[00032]$g_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),$$h_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),$$n_{\left(\tau ,\varsigma \right)}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T}{\sigma }_n^4\right),$$\mathrm{so}$$\begin{array}{c}{g}_{\left({\tau }_1,{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},g_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),\\ g_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},h_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},n_{\left({\tau }_1{\varsigma }_1\right)}{g}_{\left({\tau }_2,{\varsigma }_2\right)}^{*},n_{\left({\tau }_1,{\varsigma }_1\right)}{h}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right),\end{array}$$n_{\left({\tau }_1,{\varsigma }_1\right)}{n}_{\left({\tau }_2,{\varsigma }_2\right)}^{*}\sim \mathrm{As}\mathrm{𝒞𝒩}\left(0,\frac{1}{T^2}{\sigma }_n^8\right),$

[0055] .sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.) also obeys the approximate complex Gaussian distribution, and an approximate variance thereof σ.sup.2 is expressed as follows:

[00033]${\overline{\sigma }}^2=\frac{1}{T^2}\left[{{\lambda }_1\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2+{\lambda }_2{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2+{\lambda }_3{\sigma }_n^8\right],$

[0056] wherein, λ.sub.1, λ.sub.2 and λ.sub.3 represent a combined weight of three sub-variance terms

[00034]${\left({\sigma }_n^2\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\right)}^2,{\sigma }_n^6\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2\mathrm{and}{\sigma }_n^8.$

[0057] Step 4: implementing fourth-order sampling covariance tensor denoising based on kernel tensor thresholding. Performing high-order singular value decomposition on the fourth-order sampling covariance tensor :

=×.sub.1Y.sup.(1)×.sub.2Y.sup.(2)×.sub.3Y.sup.(3)×.sub.4Y.sup.(4),

[0058] wherein, ∈ represents a kernel tensor, which contains projections from signal and noise components in , Y.sup.(1)∈, Y.sup.(2)∈, Y.sup.(3)∈and Y.sup.(4)∈ represent singular matrices corresponding to four dimensions of ; the thresholding is performed on , that is, elements in that are less than or equal to a noise threshold ϵ are set to zero, and elements larger than the noise threshold ϵ are reserved, thus obtaining a thresholded kernel tensor .sub.dn, where an element in .sub.dn is expressed as follows:

[00035]${𝒮}_{dn\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}=\{\begin{array}{cc}& {𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}.Math.{𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)}.Math.>\in ,\\ 0& .Math.{𝒮}_{\left({\tau }_1,{\varsigma }_1,{\tau }_2,{\varsigma }_2\right)})\le \in ,\end{array}$

[0059] and wherein, .sub.(τ.sub.1.sub.,.sub.1.sub.,τ.sub.2.sub.,.sub.2.sub.) represents a (τ.sub.1, .sub.1, τ.sub.2, .sub.2)th element of , the noise threshold ϵ is as follows:

ϵ=σ.sup.2√{square root over (2 log(|.sub.1∥.sub.2∥.sub.1∥.sub.2|))}.

[0060] Further, the thresholded kernel tensor .sub.dn is multiplied with the four singular matrices Y.sup.(1), Y.sup.(2), Y.sup.(3) and Y.sup.(4) to obtain a denoised sampling covariance tensor .sub.dn, which is expressed as follows:

.sub.dn=.sub.dn×.sub.1Y.sup.(1)×.sub.2Y.sup.(2)×.sub.3Y.sup.(3)×.sub.4Y.sup.(4).

[0061] Step 5: deriving a fourth-order virtual domain signal based on the denoised sampling covariance tensor. By merging dimensions representing spatial information in the same direction in the denoised sampling covariance tensor .sub.dn, 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 augmented virtual linear arrays are respectively constructed on the x axis and the y axis, corresponding to a two-dimensional non-continuous virtual cross array . Specifically, the first and third dimensions of the denoised sampling covariance tensor .sub.dn represent the spatial information in the x axis direction, and the second and fourth dimensions represent the spatial information in the y axis direction; to this end, the dimension sets .sub.1={1, 3} and .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 the tensor transformation of dimension merging on the denoised sampling covariance tensor .sub.dn:

[00036]${\hat{V}}_{𝕎}\stackrel{\Delta }{=}{\widehat{𝒱}}_{{\mathrm{dn}}_{\left\{{𝕁}_1,{𝕁}_2\right\}}}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)\left[a_{{𝕃}_1}^{*}\left(k\right).Math.a_{{𝕃}_1}\left(k\right)\right]\circ \left[a_{{𝕃}_2}\left(k\right).Math.a_{{𝕃}_2}^{*}\left(k\right)\right],$

[0062] wherein, by forming difference set arrays on the exponential term, respectively, (k).Math.(k) and (k).Math.(k) construct the augmented virtual linear arrays on the x axis and y axis, and .Math. represents the Kronecker product. contains a virtual uniform cross array =.sub.x∪.sub.y, the structure of is shown in FIG. 3, where .sub.x and .sub.y are virtual uniform linear arrays corresponding to the x axis and y axis, respectively. The positions of all virtual array elements in .sub.x and .sub.y are respectively

[00037]${𝕍}_x=\left\{\left(x_{𝕍},0\right).Math.x_{𝕍}=\left[q_{{𝕍}_x}^{\left(1\right)},q_{{𝕍}_x}^{\left(2\right)},.Math.,q_{{𝕍}_x}^{\left(\left|{𝕍}_x\right|\right)}\right]d\right\}\mathrm{and}{𝕍}_y=\left\{\left(0,y_{𝕍}\right).Math.y_{𝕍}=\left[q_{{𝕍}_y}^{\left(1\right),},q_{{𝕍}_y}^{\left(2\right),},.Math.,q_{{𝕍}_y}^{\left(\left|{𝕍}_y\right|\right)}\right]d\right\},$

where =−−+2,

[00038]$q_{{𝕍}_x}^{\left(\left|{𝕍}_x\right|\right)}=M_{{𝕃}_1}{N}_{{𝕃}_1}+M_{{𝕃}_1},$

=−−+2,

[00039]$q_{{𝕍}_y}^{\left(.Math.{𝕍}_y.Math.\right)}=M_{{𝕃}_2}{N}_{{𝕃}_2}+M_{{𝕃}_2},$

and |.sub.x|=2(+)−1, |.sub.y|=2(+)−1.

[0063] The elements corresponding to the positions of all virtual array elements in the virtual uniform cross array are extracted from the virtual domain signal of the non-continuous virtual cross array to obtain the virtual domain signal ∈ corresponding to , which is modeled as follows:

[00040]${\hat{U}}_{𝕍}=\underset{k=1}{\overset{K}{.Math.}}\left(\frac{1}{T}{s}_k^T{s}_k^{*}\right)g_x\left(k\right)\circ g_y\left(k\right),$$\mathrm{wherein}:$$\begin{array}{c}{g}_x\left(k\right)={\left[e^{-j\pi q_{{𝕍}_x}^{\left(1\right)}{\mu }_1\left(k\right)},e^{-j\pi q_{{𝕍}_x}^{\left(2\right)}{\mu }_1\left(k\right)},.Math.,e^{-j\pi q_{{𝕍}_x}^{(|{𝕍}_x.Math.)}{\mu }_1\left(k\right)}\right]}^T,\\ g_y\left(k\right)={\left[e^{-j\pi q_{{𝕍}_y}^{\left(1\right)}{\mu }_2\left(k\right)},e^{-j\pi q_{{𝕍}_y}^{\left(2\right)}{\mu }_2\left(k\right)},.Math.,e^{-j\pi q_{{𝕍}_y}^{(|{𝕍}_y.Math.)}{\mu }_2\left(k\right)}\right]}^T,\end{array}$

[0064] are steering vectors of .sub.x and .sub.y, respectively,

[0065] Step 6: constructing a denoised structured virtual domain tensor. Considering the two virtual uniform linear arrays .sub.x and .sub.y that make up the virtual uniform cross array are respectively symmetric about the x=1 axis and y=1 axis, respectively extracting sub-arrays

[00041]${\mathbb{Q}}_x^{\left(1\right)}=\left\{\left(x_{\mathbb{Q}}^{\left(1\right)},0\right).Math.x_{\mathbb{Q}}^{\left(1\right)}=\left[1,2,.Math.,q_{{\mathbb{Q}}_x}^{\left(\left|{\mathbb{Q}}_x\right|\right)}\right]d\right\},{\mathbb{Q}}_y^{\left(1\right)}=\left\{\left(0,y_{\mathbb{Q}}^{\left(1\right)}\right).Math.y_{\mathbb{Q}}^{\left(1\right)}=\left[1,2,.Math.,q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}\right]d\right\}$

from .sub.x and .sub.y as translation windows; then, respectively translating the translation windows .sub.x.sup.(1) and .sub.y.sup.(1) along a negative semi-axis direction of the x axis and the y axis by a virtual array element interval d, to obtain J.sub.x virtual uniform linear sub-arrays ={(, 0)|=[2−j.sub.x, 3−j.sub.x, . . . , +1−j.sub.x]d} and J.sub.y virtual uniform linear sub-arrays

[00042]${\mathbb{Q}}_y^{\left(j_y\right)}=\left\{\left(0,y_{\mathbb{Q}}^{\left(j_y\right)}\right).Math.y_{\mathbb{Q}}^{\left(j_y\right)}=\left[2-j_y,3-j_y,.Math.,q_{{\mathbb{Q}}_y}^{\left(\left|{\mathbb{Q}}_y\right|\right)}+1-j_y\right]d\right\},$

as shown in FIG. 3. Here, j.sub.x=1, 2, . . . , J.sub.x, j.sub.y=1, 2, . . . , J.sub.y, J.sub.x=(|.sub.x|+1)/2, J.sub.y=(||+1)/2, and the virtual domain signal corresponding to the virtual uniform sub-array .sub.(j.sub.x.sub.,j.sub.y.sub.)=.sub.x.sup.(j.sup.x.sup.)∪.sub.y.sup.(j.sup.y.sup.) can be expressed as

[00043]$U_{{\tilde{\mathbb{Q}}}_{\left(j_x,j_y\right)}}\mathrm{and}{U}_{{\tilde{\mathbb{Q}}}_{\left(j_x,j_y+1\right)}}$

There is a one-step translation relationship in the y axial direction between the virtual domain signals

[00044]$U_{{\tilde{\mathbb{Q}}}_{\left(j_x,j_y\right)}}\mathrm{and}{U}_{{\tilde{\mathbb{Q}}}_{\left(j_x,j_y+1\right)}}$