Two-dimensional direction-of-arrival estimation method for coprime planar array based on structured coarray tensor processing

Abstract

A two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing, the method includes: deploying a coprime planar array; modeling a tensor of the received signals; deriving the second-order equivalent signals of an augmented virtual array based on cross-correlation tensor transformation; deploying a three-dimensional coarray tensor of the virtual array; deploying a five-dimensional coarray tensor based on a coarray tensor dimension extension strategy; forming a structured coarray tensor including three-dimensional spatial information; and achieving two-dimensional direction-of-arrival estimation through CANDECOMP/PARACFAC decomposition. The present disclosure constructs a processing framework of a structured coarray tensor based on statistical analysis of coprime planar array tensor signals, to achieve multi-source two-dimensional direction-of-arrival estimation in the underdetermined case on the basis of ensuring the performance such as resolution and estimation accuracy, and can be used for multi-target positioning.

Claims

1. A two-dimensional direction-of-arrival (DOA) estimation method for a coprime planar array based on structured coarray tensor signal processing, comprising following steps of: (1) providing a receiver, which is constituted by 4M.sub.xM.sub.y+N.sub.xN.sub.y−1 physical sensors arranged in a coprime planar array; wherein M.sub.x, N.sub.x and M.sub.y, N.sub.y are pairs of coprime integers respectively, and M.sub.x<N.sub.x, M.sub.y<N.sub.y; and the receiver is decomposed into two sparse uniform subarrays, which are respectively a first sparse uniform subarray .sub.1 and a second sparse uniform subarray .sub.2; (2) receiving, by the receiver, signals of K far-field narrowband incoherent sources from directions {(θ.sub.1,φ.sub.1), (θ.sub.2,φ.sub.2), . . . ,(θ.sub.K,φ.sub.K)}, and processing the signals as the following manner: the received signals of the first sparse uniform subarray .sub.1 being expressed by using a three-dimensional tensor .sub.1∈.sup.2M.sup.x.sup.×2M.sup.y.sup.×L (L denotes the number of sampling snapshots) as: $1_{}=\underset{k=1}{\overset{K}{.Math.}}{a}_{Mx}\left({\theta }_k,{\phi }_k\right)\circ a_{My}\left({\theta }_k,{\phi }_k\right)\circ s_k+1_{},$ wherein s.sub.k=[s.sub.k,1, s.sub.k,2, . . . , s.sub.k,L].sup.T denotes a waveform corresponding to the k.sup.th source, [⋅].sup.T denotes a transpose operation, ∘ denotes an exterior product of vectors, .sub.1 denotes an additive Gaussian white noise tensor, and a.sub.Mx(θ.sub.k, φ.sub.k) and a.sub.My(θ.sub.k, φ.sub.k) denote steering vectors of the first sparse uniform subarray .sub.1 along x-axis and y-axis, respectively, a.sub.Mx(θ.sub.k, φ.sub.k) and a.sub.My(θ.sub.k, φ.sub.k) and are defined as: $$\begin{gathered} \begin{array}{cc}{a}_{\mathrm{Mx}}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi u_1^{\left(2\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)},.Math.,e^{-j\pi u_1^{\left(2\mathrm{Mx}\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)}\right]}^T, \\ {a}_{\mathrm{My}}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi v_1^{\left(2\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)},.Math.,e^{-j\pi v_1^{\left(2\mathrm{My}\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)}\right]}^T,& \end{array} \end{gathered}$$ wherein u.sub.1.sup.(i.sup.1.sup.)(i.sub.1=1, 2, . . . , 2M.sub.x) and v.sub.1.sup.(i.sup.2.sup.)(i.sub.2=1, 2, . . . , 2M.sub.y) denote positions of i.sub.1.sup.th and i.sub.2.sup.th physical sensors in the first sparse uniform subarray .sub.1 in the x-axis and y-axis with u.sub.1.sup.(1)=0, v.sub.1.sup.(1)=0, j=√{square root over (−1)}; the received signals of the second sparse uniform subarray .sub.2 being expressed by using another three-dimensional tensor .sub.2∈.sup.N.sup.x.sup.×N.sup.y.sup.×L as: $2_{}=\underset{k=1}{\overset{K}{.Math.}}{a}_{Nx}\left({\theta }_k,{\phi }_k\right)\circ a_{Ny}\left({\theta }_k,{\phi }_k\right)\circ s_k+2_{},$ wherein .sub.2 denotes a noise tensor, and a.sub.Nx(θ.sub.k, φ.sub.k) and a.sub.Ny(θ.sub.k, φ.sub.k) denote steering vectors of the second sparse uniform subarray .sub.2 in the x-axis and y-axis respectively, which are defined as: $$\begin{gathered} \begin{array}{cc}{a}_{Nx}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi u_2^{\left(2\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)},.Math.,e^{-j\pi u_2^{\left(\mathrm{Nx}\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)}\right]}^T, \\ {a}_{Ny}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi v_2^{\left(2\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)},.Math.,e^{-j\pi v_2^{\left(\mathrm{Ny}\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)}\right]}^T,& \end{array} \end{gathered}$$ wherein u.sub.2.sup.(i.sup.3.sup.)(i.sub.3=1, 2, . . . ,N.sub.x) and v.sub.2.sup.(i.sup.4.sup.)(i.sub.4=1, 2, . . . ,N.sub.y) denote positions of i.sub.3.sup.th and i.sub.4.sup.th physical sensors in the second sparse uniform subarray .sub.2 in the x-axis and y-axis with u.sub.2.sup.(1)=0, v.sub.2.sup.(1)=0; calculating a second-order cross-correlation tensor ∈.sup.2M.sup.x.sup.×2M.sup.y.sup.×N.sup.x.sup.×N.sup.y of two three-dimensional tensor signals .sub.1 and .sub.2: $\begin{array}{cc}=\frac{1}{L}\underset{l=1}{\overset{L}{.Math.}}{1}_{}\left(l\right)\circ 2_{*}^{}\left(l\right),& \end{array}$ wherein .sub.1(l) and .sub.2(l) denote an l.sup.th slice of .sub.1 and .sub.2 along a third dimension (i.e., a temporal dimension) respectively, and (⋅)* denotes a conjugate operation; (3) obtaining, by the receiver, an augmented discontinuous virtual planar array based on the cross-correlation tensor , wherein every element in the discontinuous virtual planar array corresponds to a virtual sensor, and the position of each virtual sensor is defined as:
={(−M.sub.xn.sub.xd+N.sub.xm.sub.xd,−M.sub.yn.sub.yd+N.sub.ym.sub.yd)|0≤n.sub.x≤N.sub.x−1,0≤m.sub.x≤2M.sub.x−1,0≤n.sub.y≤N.sub.y−1,0≤m.sub.y≤2M.sub.y−1}, wherein a spacing d is set to half of a signal wavelength) λ, i.e., d=λ/2; comprises a virtual uniform planar array including (M.sub.xN.sub.x+M.sub.x+N.sub.x−1)×(M.sub.yN.sub.y+M.sub.y+N.sub.y−1) virtual sensors with distributing from (−N.sub.x+1)d to (M.sub.xN.sub.x+M.sub.x−1)d and x-axis distributing from (−N.sub.y+1)d to (M.sub.yN.sub.y+M.sub.y−1)d in y-axis, the virtual uniform planar array is defined as:
={(x,y)|x=p.sub.xd,y=p.sub.yd,−N.sub.x+1≤p.sub.x≤M.sub.xN.sub.x+M.sub.x−1, −N.sub.y+1≤p.sub.y≤M.sub.yN.sub.y+M.sub.y−1}, defining dimension sets .sub.1={1, 3} and .sub.2={2, 4}; and reshaping, by the receiver, cross-correlation tensor (noiseless scene) with {.sub.1, .sub.2} to obtain the equivalent second-order signals U∈.sup.2M.sup.x.sup.N.sup.x.sup.×2M.sup.y.sup.N.sup.y of the discontinuous virtual planar array , which is expressed as:
U=Σ.sub.k=1.sup.Kσ.sub.k.sup.2a.sub.x(θ.sub.k,φ.sub.k)○a.sub.y(θ.sub.k,φ.sub.k), wherein a.sub.x(θ.sub.k, φ.sub.k)=a*.sub.Nx(θ.sub.k, φ.sub.k).Math.a.sub.Mx(θ.sub.k, φ.sub.k), a.sub.y(θ.sub.k, φ.sub.k)=a*.sub.Ny(θ.sub.k, φ.sub.k).Math.a.sub.My(θ.sub.k, φ.sub.k) denote steering vectors of the augmented virtual planar array along the x axis and the y axis, σ.sub.k.sup.2 denotes power of a k.sup.th source, and .Math. denotes Kroneker product; an equivalent signal Ũ∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1) of the virtual uniform planar array is obtained by selecting elements in U corresponding to virtual sensor positions in by the receiver, Ũ is modeled as:
Ũ=Σ.sub.k=1.sup.Kσ.sub.k.sup.2b.sub.x(θ.sub.k,φ.sub.k)○b.sub.y(θ.sub.k,φ.sub.k), wherein, b.sub.x(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.x.sup.+1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), e.sup.−jπ(−N.sup.x.sup.+2)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . , e.sup.−jπ(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)] and b.sub.y(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.y.sup.+1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), e.sup.−jπ(−N.sup.y.sup.+2)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), e.sup.−jπ(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] denote steering vectors of the virtual uniform planar array along the x axis and they axis; (4) expanding, by the receiver, a symmetric part of the virtual uniform planar array , the symmetric being expressed as:
={{hacek over (x)},{hacek over (y)})|{hacek over (x)}={hacek over (p)}.sub.xd,{hacek over (y)}={hacek over (p)}.sub.yd,−M.sub.xN.sub.x−M.sub.x+1≤{hacek over (p)}.sub.x≤N.sub.x−1, −M.sub.yN.sub.y−M.sub.y+1≤{hacek over (p)}.sub.y≤N.sub.y−1}, transforming elements in the equivalent received signal Ũ of the virtual uniform planar array to obtain the equivalent signals Ũ.sub.sym∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1) of a symmetric uniform planar array by the receiver, the equivalent signals Ũ.sub.sym being expressed as:
Ũ.sub.sym=Σ.sub.k=1.sup.Kσ.sub.k.sup.2(b.sub.x(θ.sub.k,φ.sub.k)e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.+N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)).Math.(b.sub.y(θ.sub.k,φ.sub.k)e.sup.(−M.sup.y.sup.N.sup.y.sup.−M.sup.y.sup.+N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)), wherein e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.+N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)) and e.sup.(−M.sup.y.sup.N.sup.y.sup.−M.sup.y.sup.+N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.) are symmetric factors in the x-axis and y-axis, respectively concatenating, by the receiver, the equivalent signals Ũ of the virtual uniform planar array and the equivalent signals Ũ.sub.sym of the symmetric uniform planar array along the third dimension, to obtain a three-dimensional tensor ∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1)×2 for the coprime planar array, the three-dimensional coarray tensor being expressed as: $=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{b}_x\left({\theta }_k,{\phi }_k\right)\circ b_y\left({\theta }_k,{\phi }_k\right)\circ h_k\left({\theta }_k,{\phi }_k\right),$ wherein h.sub.k (θ.sub.k, φ.sub.k)=[1, e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.+N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)+(−M.sub.y.sup.N.sup.y.sup.−M.sup.y+.sup.N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)].sup.T denotes a symmetric factor vector; (5) segmenting, by the receiver, a subarray with a size of P.sub.x×P.sub.y from the virtual uniform planar array , and dividing the virtual uniform planar array into L.sub.x×L.sub.y partially overlapped uniform subarrays which are expressed as .sub.(s.sub.x.sub.,s.sub.y.sub.), s.sub.x=1, 2, . . . , L.sub.x, s.sub.y=1, 2, . . . , L.sub.y, wherein every element in the subarrays .sub.(s.sub.x.sub.,s.sub.y.sub.) corresponds to a virtual sensor, and the position of each virtual sensor in.sub.(s.sub.x.sub.,s.sub.y.sub.) is expressed as:
.sub.(s.sub.x.sub.,s.sub.y.sub.)={(x,y)|x=p.sub.xd,y=p.sub.yd,−N.sub.x+s.sub.x≤p.sub.x≤−N.sub.x+s.sub.x+P.sub.x−1, −N.sub.y+s.sub.y≤p.sub.y≤−N.sub.y+s.sub.y+P.sub.y−1}, obtaining, by the receiver, a sub-coarray tensor .sub.(s.sub.x.sub.,s.sub.y.sub.)∈.sup.P.sup.x.sup.×P.sup.y.sup.×2 in the virtual subarray .sub.(s.sub.x.sub.,s.sub.y.sub.) by selecting elements in the coarray tensors according to position of virtual sensors in the subarray .sub.(s.sub.x.sub.,s.sub.y.sub.):
.sub.(s.sub.x.sub.,s.sub.y.sub.)=Σ.sub.k=1.sup.Kσ.sub.k.sup.2(c.sub.x(θ.sub.k,φ.sub.k)e.sup.(s.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.))○(c.sub.y(θ.sub.k,φ.sub.k)e.sup.(s.sup.y.sup.−1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)○h.sub.k(θ.sub.k,φ.sub.k), where c.sub.x(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.x.sup.+1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), e.sup.−jπ(−N.sup.x.sup.+2)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . , e.sup.−jπ(−N.sup.x.sup.+P.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)] and c.sub.y(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.y.sup.+1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), e.sup.−jπ(−N.sup.y.sup.+2)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), . . . , e.sup.−jπ(−N.sup.y.sup.+P.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] are steering vectors of a virtual subarray .sub.(1,1) along the x axis and the y axis; after the above operations, a total of L.sub.x×L.sub.y three-dimensional sub-coarray tensors .sub.(s.sub.x.sub.,s.sub.y.sub.) with dimensions being all P.sub.x×P.sub.y×2 are obtained by the receiver; the sub-coarray tensors .sub.(s.sub.x.sub.,s.sub.y.sub.) with a same index subscript s.sub.y are concatenated by the receiver along a fourth dimension, to obtain L.sub.y four-dimensional tensors of size P.sub.x×P.sub.y×2×L.sub.x; and the L.sub.y four-dimensional tensors are further concatenated by the receiver along a fifth dimension, to obtain a five-dimensional tensor ∈.sup.P.sup.x.sup.×P.sup.y.sup.×2×L.sup.x.sup.×L.sup.y, the five-dimensional coarray tensor is defined as:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2c.sub.x(θ.sub.k,φ.sub.k)○c.sub.y(θ.sub.k,φ.sub.k)○h.sub.k(θ.sub.k,φ.sub.k)○d.sub.x(θ.sub.k,φ.sub.k)○d.sub.y(θ.sub.k,φ.sub.k), wherein d.sub.x(θ.sub.k, φ.sub.k)=[1, e.sup.−jπ sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . , e.sup.−jπ(L.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)], d.sub.y(θ.sub.k, φ.sub.k)=[1, e.sup.−jπ sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), . . . , e.sup.−jπ(L.sup.y.sup.−1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] are shifting factor vectors along the x-axis and y-axis respectively; (6) reshaping, by the receiver, with {.sub.1, .sub.2, .sub.3}, .sub.1={1,2},.sub.2={3}, .sub.3={4,5}, i.e., combining first and second dimensions and fourth and fifth dimensions of the five-dimensional tensor , and retaining its third dimension, to obtain a three-dimensional structured coarray tensor ∈.sup.P.sup.x.sup.P.sup.y.sup.×2×L.sup.x.sup.L.sup.y as follows:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2g(θ.sub.k,φ.sub.k)○h(θ.sub.k,φ.sub.k)○f(θ.sub.k,φ.sub.k), where g(θ.sub.k, φ.sub.k)=c.sub.y(θ.sub.k, φ.sub.k).Math.c.sub.x(θ.sub.k, φ.sub.k), f(θ.sub.k, φ.sub.k)=d.sub.y(θ.sub.k, φ.sub.k).Math.d.sub.x(θ.sub.k, φ.sub.k) represent the angular information and the shifting information, respectively; (7) performing, by the receiver, CANDECOMP/PARACFAC decomposition on the three-dimensional structured coarray tensor , to obtain a closed-form solution of two-dimensional direction-of-arrivals in the underdetermined case.

2. The two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing according to claim 1, wherein the receiver in step (1) is decomposed into two sparse uniform subarrays .sub.1 and .sub.2 on a coordinate system xoy, wherein .sub.1 comprises 2M.sub.x×2M.sub.y sensors, the spacing in the x-axis direction and the spacing in the y-axis direction are N.sub.xd and N.sub.yd, respectively, and sensor coordinates thereof on the xoy plane are {(N.sub.xdm.sub.x, N.sub.ydm.sub.y),m.sub.x=0,1, . . . ,2M.sub.x−1,m.sub.y=0,1, . . . , 2M.sub.y−1}; .sub.2 comprises N.sub.x×N.sub.y sensors, the spacing in the x-axis direction and the spacing in the y-axis direction are M.sub.xd and M.sub.yd, respectively, and the sensor coordinates on the xoy plane are {(M.sub.xdn.sub.x,M.sub.ydn.sub.y), n.sub.x=0, 1, . . . ,N.sub.x−1, n.sub.y=0,1, . . . ,N.sub.y−1}; M.sub.x, N.sub.x and M.sub.y, N.sub.y are a pair of coprime integers respectively, and M.sub.x<N.sub.x, M.sub.y<N.sub.y; overlapping, by the receiver, the subarrays .sub.1 and .sub.2 only at an origin of the coordinate system (0,0), to obtain a coprime planar array comprising 4M.sub.xM.sub.y+N.sub.xN.sub.y−1 physical sensors.

3. The two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing according to claim 1, wherein the cross-correlation tensor in step (3) is ideally modeled (noiseless scene) at the receiver as:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2a.sub.Mx(θ.sub.k,φ.sub.k)○a.sub.My(θ.sub.k,φ.sub.k)○a*.sub.Nx(θ.sub.k,φ.sub.k)○a*.sub.Ny(θ.sub.k,φ.sub.k), a.sub.Mx(θ.sub.k,φ.sub.k)○a*.sub.Nx(θ.sub.k,φ.sub.k) in the cross-correlation tensor derives augmented coarray along the x axis, and a.sub.My(θ.sub.k,φ.sub.k)○a*.sub.Ny(θ.sub.k,φ.sub.k) derives an augmented coarray along the y axis, so as to obtain the augmented discontinuous virtual planar array .

4. The two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing according to claim 1, wherein the equivalent signals of the symmetric of the virtual uniform planar array in step (4) is obtained by transformation of the equivalent signals Ũ of the virtual uniform planar array by the receiver, which specifically comprises: performing, by the receiver, a conjugate operation on Ũ to obtain Ũ*, and flipping elements in Ũ* left and right and then up and down, to obtain the equivalent signals Ũ.sub.sym of the symmetric uniform planar array .

5. The two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing according to claim 1, wherein said concatenating, by the receiver, the equivalent signals Ũ of the virtual uniform planar array and the equivalent signals Ũ.sub.sym of the symmetric uniform planar array along the third dimension, to obtain a three-dimensional tensor in step (4) comprises: performing, by the receiver, CANDECOMP/PARACFAC decomposition on to achieve two-dimensional direction-of-arrival estimation in the overdetermined case.

6. The two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing according to claim 1, wherein the step (7) comprises: performing, by the receiver, CANDECOMP/PARACFAC decomposition on the three-dimensional structured coarray tensor , to obtain three factor matrixes, G=[g({circumflex over (θ)}.sub.1,{circumflex over (φ)}.sub.1), g({circumflex over (θ)}.sub.2,{circumflex over (φ)}.sub.2), . . . , g({circumflex over (θ)}.sub.K,{circumflex over (φ)}.sub.K)] H=[h({circumflex over (θ)}.sub.1,{circumflex over (φ)}.sub.1), h({circumflex over (θ)}.sub.2,{circumflex over (φ)}.sub.2), . . . , h({circumflex over (θ)}.sub.K,{circumflex over (φ)}.sub.K)], F=[f({circumflex over (θ)}.sub.1,{circumflex over (φ)}.sub.1), f({circumflex over (θ)}.sub.2,{circumflex over (φ)}.sub.2), . . . , f({circumflex over (θ)}.sub.K,{circumflex over (φ)}.sub.K)]; wherein ({circumflex over (θ)}.sub.K,{circumflex over (φ)}.sub.K), k=1, 2, . . . , K is an estimation of (θ.sub.K,φ.sub.K), k=1, 2, . . . , K; dividing, by the receiver, elements in a second row in the factor matrix G by elements in a first row to obtain e.sup.−jπ sin({circumflex over (φ)}.sup.k.sup.)cos({circumflex over (θ)}.sup.k.sup.); dividing, by the receiver, elements in the P.sub.x+1.sup.th row in the factor matrix G by elements in the first row to obtain e.sup.−jπ sin({circumflex over (φ)}.sup.k.sup.)sin({circumflex over (θ)}.sup.k.sup.), after a similar parameter retrieval operation on the factor matrix F, averaging and logarithm processing being performed to parameters extracted from G and F, respectively, to obtain .sub.k=sin({circumflex over (φ)}.sub.k)cos({circumflex over (θ)}.sub.k) and .sub.k=sin({circumflex over (φ)}.sub.k)sin({circumflex over (θ)}.sub.k); and obtaining, by the receive end, closed-form solution of the two-dimensional azimuth and elevation angles ({circumflex over (θ)}.sub.K,{circumflex over (φ)}.sub.K) as follows: $$\begin{gathered} {\hat{\theta }}_k=\arctan \left(\frac{k_{}}{k_{}}\right), \\ {\hat{\phi }}_k=\sqrt{k_2^{}+k_2^{}}. \end{gathered}$$ in the above step, CANDECOMP/PARAFAC decomposition follows a uniqueness condition:
.sub.rank(G)+.sub.rank(H)+.sub.rank(F)≥2K+2, wherein .sub.rank(⋅) denotes a Kruskal rank of a matrix, and .sub.rank(G)=min(P.sub.xP.sub.y,K), .sub.rank(H)=min(L.sub.x,L.sub.y,K), .sub.rank(F)=min(2,K), min(⋅) denotes a minimization operation; optimal P.sub.x and P.sub.y values are obtained by the receiver according to the above inequality, so as to obtain a theoretical maximum value of K, i.e., a theoretical upper bound of the number of distinguishable sources, is obtained by ensuring that the uniqueness condition is satisfied; the value of K exceeds the total number of physical sensors in the coprime planar array 4M.sub.xM.sub.y+N.sub.xN.sub.y−1.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 is an overall flow diagram according to the present disclosure;

(2) FIG. 2 is a schematic structural diagram of a coprime planar array according to the present disclosure;

(3) FIG. 3 is a schematic structural diagram of an augmented virtual planar array derived according to the present disclosure;

(4) FIG. 4 is a schematic diagram of a dimension extension process of a coarray tensor signal of a coprime planar array according to the present disclosure; and

(5) FIG. 5 is an effect diagram of multi-source direction-of-arrival estimation in a method according to the present disclosure.

DESCRIPTION OF EMBODIMENTS

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

(7) In order to solve the problem of loss of degrees-of-freedom in the existing methods, the present disclosure provides a two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing, which establishes an association between a coprime planar array coarray domain and second-order tensor statistics in combination with means such as multi-linear analysis, coarray tensor signal construction, and coarray tensor decomposition, so as to achieve two-dimensional direction-of-arrival estimation in an underdetermined condition. Referring to FIG. 1, the present disclosure is implemented through the following steps:

(8) Step 1: A coprime planar array is deployed. The coprime planar array is deployed with 4 M.sub.xM.sub.y+N.sub.xN.sub.y−1 physical sensors at a receiving end. As shown in FIG. 2, a pair of sparse uniform planar subarrays .sub.1 and .sub.2 are constructed on a coordinate system xoy plane, wherein .sub.1 includes 2M.sub.x×2M.sub.y sensors, spacing in the x-axis direction and spacing in the y-axis direction are N.sub.xd and N.sub.yd respectively, and position coordinates thereof on the xoy are {(N.sub.xdm.sub.x, N.sub.ydm.sub.y), m.sub.x=0, 1, . . . , 2M.sub.x−1, m.sub.y=0, 1, . . . , 2M.sub.y−1}; .sub.2 includes N.sub.x×N.sub.y sensors, spacing in the x-axis direction and spacing in the y-axis direction are M.sub.xd and M.sub.yd respectively, and position coordinates thereof on the xoy are {(M.sub.xdn.sub.x, M.sub.ydn.sub.y), n.sub.x=0, 1, . . . , N.sub.x−1, n.sub.y=0, 1, . . . , N.sub.y−1}, M.sub.x, N.sub.x and M.sub.y, N.sub.y are a pair of coprime integers respectively, and M.sub.x<N.sub.x, M.sub.y<N.sub.y. A spacing d is set to half of an incident narrowband signal wavelength λ, i.e., d=λ/2. Subarray combination is performed on .sub.1 and .sub.2 according to overlap of sensors at a position of a coordinate system (0,0), to obtain a coprime planar array actually including 4M.sub.xM.sub.y+N.sub.xN.sub.y−1 physical sensors.

(9) Step 2: The tensor signals of the coprime planar array is modeled. Assuming that there are K far-field narrowband incoherent sources from {(θ.sub.1, φ.sub.1), (θ.sub.2, φ.sub.2), . . . , (θ.sub.K, φ.sub.K)} directions, a three-dimensional tensor .sub.1∈.sup.2M.sup.x.sup.×2M.sup.y.sup.×L (L denotes the number of sampling snapshots) may be obtained after sampling snapshots on the sparse uniform subarray .sub.1 of the coprime planar array are superimposed in the third dimension, which is modeled as:

(10) $1_{}=\underset{k=1}{\overset{K}{.Math.}}{a}_{Mx}\left({\theta }_k,{\phi }_k\right)\circ a_{My}\left({\theta }_k,\phi \right)\circ k_{}+1_{},$

(11) wherein s.sub.k=[s.sub.k,1, s.sub.k,2, . . . , s.sub.k,L].sup.T denotes a multi-snapshot signal waveform corresponding to the k.sup.th source, [⋅].sup.T denotes a transpose operation, ∘ denotes an exterior product of vectors, .sub.1 denotes an additive Gaussian white noise tensor, and a.sub.Mx(θ.sub.k, φ.sub.k) and a.sub.My(θ.sub.k, φ.sub.k) denote steering vectors of .sub.1 in x-axis and y-axis directions respectively, corresponding to the source from direction (θ.sub.k, φ.sub.k), and are defined as:

(12) $$\begin{gathered} \begin{array}{cc}{a}_{\mathrm{Mx}}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi u_1^{\left(2\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)},.Math.,e^{-j\pi u_1^{\left(2\mathrm{Mx}\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)}\right]}^T, \\ {a}_{\mathrm{My}}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi v_1^{\left(2\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)},.Math.,e^{-j\pi v_1^{\left(2\mathrm{My}\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)}\right]}^T,& \end{array} \end{gathered}$$

(13) wherein u.sub.1.sup.(i.sup.1.sup.)(i.sub.1=1, 2, . . . , 2M.sub.x) and v.sub.1.sup.(i.sup.2.sup.)(i.sub.2=1, 2, . . . , 2M.sub.y) denote actual positions of i.sub.1.sup.th and i.sub.2.sup.th physical sensors in the sparse subarray .sub.1 in the x-axis and y-axis directions, and u.sub.1.sup.(1)=0, v.sub.1.sup.(1)=0, j=√{square root over (−1)}.

(14) Similarly, a received signals of the sparse uniform subarray .sub.2 may be defined by another three-dimensional tensor .sub.2∈.sup.N.sup.x.sup.×N.sup.y.sup.×L:

(15) 0$2_{}=\underset{k=1}{\overset{K}{.Math.}}{a}_{Nx}\left({\theta }_k,{\phi }_k\right)\circ a_{Ny}\left({\theta }_k,{\phi }_k\right)\circ s_k+2_{},$

(16) wherein .sub.2 denotes a noise tensor, and a.sub.Nx(θ.sub.k, φ.sub.k) and a.sub.Ny(θ.sub.k, φ.sub.k) denote the steering vectors of .sub.2 in the x-axis and y-axis directions respectively, corresponding to a signal source from direction (θ.sub.k, φ.sub.k), and are defined as:

(17) $$\begin{gathered} \begin{array}{cc}{a}_{Nx}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi u_2^{\left(2\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)},.Math.,e^{-j\pi u_2^{\left(\mathrm{Nx}\right)}\sin \left({\phi }_k\right)\cos \left({\theta }_k\right)}\right]}^T, \\ {a}_{Ny}\left({\theta }_k,{\phi }_k\right)={\left[1,e^{-j\pi v_2^{\left(2\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)},.Math.,e^{-j\pi v_2^{\left(\mathrm{Ny}\right)}\sin \left({\phi }_k\right)\sin \left({\theta }_k\right)}\right]}^T,& \end{array} \end{gathered}$$

(18) wherein u.sub.2.sup.(i.sup.3.sup.)(i.sub.3=1, 2, . . . , N.sub.x) and v.sub.2.sup.(i.sup.4.sup.)(i.sub.4=1, 2, . . . , N.sub.y) denote actual positions of i.sub.3.sup.th and i.sub.4.sup.th physical sensors in the sparse subarray .sub.2 in the x-axis and y-axis directions, and u.sub.2.sup.(1)=0, v.sub.2.sup.(1)=0.

(19) Cross-correlation statistics of three-dimensional tensors .sub.1 and .sub.2 sampled by the sparse subarrays .sub.1 and .sub.2 is calculated, to obtain the second-order cross-correlation tensor ∈.sup.2M.sup.x.sup.×2M.sup.y.sup.×N.sup.x.sup.×N.sup.y including four-dimensional spatial information:

(20) $\begin{array}{cc}=\frac{1}{L}\underset{l=1}{\overset{L}{.Math.}}{1}_{}\left(l\right)\circ 2_{*}^{}\left(l\right),& \end{array}$

(21) wherein .sub.1(l) and .sub.2(l) denote the l.sup.th slice of .sub.1 and .sub.2 in the third dimension (i.e., temporal dimension) respectively, and (⋅)* denotes a conjugate operation.

(22) Step 3: A second-order equivalent signals of the virtual array associated with coprime planar array based on cross-correlation tensor statistics is derived. The cross-correlation tensor of the received tensor signals of the two subarrays may be ideally modeled (noiseless scene) as:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2a.sub.Mx(θ.sub.k,φ.sub.k)○a.sub.My(θ.sub.k,φ.sub.k)○a*.sub.Nx(θ.sub.k,φ.sub.k)○a*.sub.Ny(θ.sub.k,φ.sub.k),

(23) wherein σ.sub.k.sup.2 denotes power of the k.sup.th source. In this case, a.sub.Mx(θ.sub.k, φ.sub.k).Math.a*.sub.Nx(θ.sub.k, φ.sub.k) in the cross-correlation tensor is equivalent to an augmented coarray along the x axis, and a.sub.My(θ.sub.k, φ.sub.k)○a*.sub.Ny(θ.sub.k, φ.sub.k) is equivalent to an augmented coarray along the y axis, so as to obtain the augmented discontinuous virtual planar array . As shown in FIG. 3, a position of each virtual sensor is defined as:
={(−M.sub.xn.sub.xd+N.sub.xm.sub.xd,−M.sub.yn.sub.yd+N.sub.ym.sub.yd)|0≤n.sub.x≤N.sub.x−1,0≤m.sub.x≤2M.sub.x−1,0≤n.sub.y≤N.sub.y−1,0≤m.sub.y≤2M.sub.y−1}.

(24) contains a virtual uniform planar array including (M.sub.xN.sub.x+M.sub.x+N.sub.x−1)×(M.sub.yN.sub.y+M.sub.y+N.sub.y−1) virtual sensors with x-axis distribution from (−N.sub.x+1)d to (M.sub.xN.sub.x+M.sub.x−1)d and y-axis distribution from (−N.sub.y+1)d to (M.sub.yN.sub.y+M.sub.y−1)d, as shown in the dashed box of FIG. 3, which is specifically defined as:
={(x,y)|x=p.sub.xd,y=p.sub.yd,−N.sub.x+1≤p.sub.x≤M.sub.xN.sub.x+M.sub.x−1, −N.sub.y+1≤p.sub.y≤M.sub.yN.sub.y+M.sub.y−1}.

(25) In order to obtain the equivalent signals of the augmented virtual planar array , there is a need to combine the first and third dimensions in the cross-correlation tensor that represent the spatial information in the x-axis direction into one dimension and combine second and fourth dimensions that represent spatial information in the y-axis direction into another dimension. Dimension combination of tensors can be achieved by the tensor reshaping operation. Taking a four-dimensional tensor ∈.sup.I.sup.1.sup.×I.sup.2.sup.×I.sup.3.sup.×I.sup.4=Σ.sub.r=1.sup.Rb.sub.11.Math.b.sub.12○b.sub.21○b.sub.22 as an example, dimension sets .sub.1={1, 2} and .sub.2={3, 4} are defined, and then unfolding of a module {.sub.1, .sub.2} of PARAFAC decomposition of is as follows:

(26) $B\in {I_1{I}_2\times I_3{I}_4}^{}\stackrel{\Delta }{=}{\left\{{𝕋}_1,{𝕋}_2\right\}}_{}=\underset{r=1}{\overset{R}{.Math.}}{b}_1\circ b_2,$

(27) wherein the tensor subscript denotes the tensor reshaping; b.sub.1=b.sub.12 .Math.b.sub.11 and b.sub.2=b.sub.22 .Math.b.sub.21 denote factor vectors of two dimensions after the unfolding respectively. Herein, .Math. denotes Kroneker product. Therefore, dimension sets .sub.1={1, 3} and .sub.2={2, 4} are defined, and a module {.sub.1, .sub.2} of reshaping is performed for an ideal value (noiseless scene) of the cross-correlation tensor , to obtain an equivalent second-order signals U∈.sup.2M.sup.x.sup.N.sup.x.sup.×2M.sup.y.sup.N.sup.y of the augmented virtual planar array :
U=Σ.sub.k=1.sup.Kσ.sub.k.sup.2a.sub.x(θ.sub.k,φ.sub.k)○a.sub.y(θ.sub.k,φ.sub.k),

(28) wherein a.sub.x(θ.sub.k, φ.sub.k)=a*.sub.Nx(θ.sub.k, φ.sub.k).Math.a.sub.Mx(θ.sub.k, φ.sub.k), a.sub.y(θ.sub.k, φ.sub.k)=a*.sub.Ny(θ.sub.k, φ.sub.k).Math.a.sub.My(θ.sub.k, φ.sub.k) denote steering vectors of the augmented virtual planar array corresponding to a (θ.sub.k, φ.sub.k) direction on the x axis and the y axis. Based on the above derivation, the equivalent signals Ũ∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1) of the virtual uniform planar array is obtained by selecting elements in U corresponding to virtual sensor positions in , which is modeled as:

(29) $\tilde{U}=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{b}_x\left({\theta }_k,{\phi }_k\right)\circ b_y\left({\theta }_k,{\phi }_k\right),$

(30) wherein b.sub.x(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.x.sup.+1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), e.sup.−jπ(−N.sup.x.sup.+2)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . ,

(31) e.sup.−jπ(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)] and b.sub.y(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.y.sup.+1)sin(φ.sub.k.sup.)sin(θ.sup.k.sup.),

(32) e.sup.−jπ(−N.sup.y.sup.+2)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), e.sup.−jπ(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] Denote steering vectors of the virtual uniform planar array corresponding to the (θ.sub.k, φ.sub.k) direction on the x axis and the y axis.

(33) Step 4: A three-dimensional tensor signal of the coprime planar array virtual domain is constructed. In order to increase an effective aperture of the virtual planar array and further improve the degree of freedom, a symmetric extension of the virtual uniform planar array is taken into account, which is defined as:
={{hacek over (x)},{hacek over (y)})|{hacek over (x)}={hacek over (p)}.sub.xd,{hacek over (y)}={hacek over (p)}.sub.yd,−M.sub.xN.sub.x−M.sub.x+1≤{hacek over (p)}.sub.x≤N.sub.x−1, −M.sub.yN.sub.y−M.sub.y+1≤{hacek over (p)}.sub.y≤N.sub.y−1}.

(34) In order to obtain the equivalent signals of the symmetric uniform planar array , the equivalent signal Ũ of the virtual uniform planar array may be transformed specifically as follows: performing a conjugate operation on Ũ to obtain Ũ*, and flipping elements in Ũ* left and right and then up and down, to obtain the equivalent signal Ũ.sub.sym∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1) corresponding to the symmetric uniform planar array , which is defined as:
Ũ.sub.sym=Σ.sub.k=1.sup.Kσ.sub.k.sup.2(b.sub.x(θ.sub.k,φ.sub.k)e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.+N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)).Math.(b.sub.y(θ.sub.k,φ.sub.k)e.sup.(−M.sup.y.sup.N.sup.y.sup.−M.sup.y.sup.+N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)),

(35) where e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.) and e.sup.(−M.sup.y.sup.N.sup.y.sup.−M.sup.y.sup.N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)) denote symmetric factors in the x-axis and y-axis directions respectively when mirror transformation is performed on the virtual uniform planar array .

(36) The equivalent signals Ũ of the virtual uniform planar array and the equivalent signal Ũ.sub.sym of the symmetric uniform planar array are superimposed in the third dimension, to obtain a three-dimensional coarray tensor ∈.sup.(M.sup.x.sup.N.sup.x.sup.+M.sup.x.sup.+N.sup.x.sup.−1)×(M.sup.y.sup.N.sup.y.sup.+M.sup.y.sup.+N.sup.y.sup.−1)×2 for the coprime planar array, the structure thereof is as shown in FIG. 4, and the three-dimensional coarray tensor is defined as:

(37) $=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{b}_x\left({\theta }_k,{\phi }_k\right)\circ b_y({\theta }_k,{\phi }_k\circ h_k\left({\theta }_k,{\phi }_k\right),$

(38) wherein h.sub.k (θ.sub.k, φ.sub.k)=[1, e.sup.(−M.sup.x.sup.N.sup.x.sup.−M.sup.x.sup.+N.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)+(−M.sup.y.sup.N.sup.y.sup.−M.sup.y.sup.+N.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)].sup.T denotes a symmetric transformation factor vector.

(39) Step 5: A five-dimensional coarray tensor is constructed based on a coarray tensor dimension extension strategy. As shown in FIG. 4, a subarray with a size of P.sub.x×P.sub.y is taken, from the virtual uniform planar array , every other sensor along the x-axis and y-axis directions respectively, and then the virtual uniform planar array may be divided into L.sub.x×L.sub.y uniform subarrays partially overlapping each other. L.sub.x, L.sub.y, P.sub.x, P.sub.y satisfy the following relations:
P.sub.x+L.sub.x−1=M.sub.xN.sub.x+M.sub.x+N.sub.x−1,
P.sub.y+L.sub.y−1=M.sub.yN.sub.y+M.sub.y+N.sub.y−1.

(40) The subarray is defined as .sub.(s.sub.x.sub.,s.sub.y.sub.), s.sub.x=1, 2, . . . , L.sub.x, s.sub.y=1, 2, . . . , L.sub.y, and a position of an virtual sensor in .sub.(s.sub.x.sub.,s.sub.y.sub.) is defined as:
.sub.(s.sub.x.sub.,s.sub.y.sub.)={(x,y)|x=p.sub.xd,y=p.sub.yd,−N.sub.x+s.sub.x≤p.sub.x≤−N.sub.x+s.sub.x+P.sub.x−1, −N.sub.y+s.sub.y≤p.sub.y≤−N.sub.y+s.sub.y+P.sub.y−1}.

(41) A tensor signal .sub.(s.sub.x.sub.,s.sub.y.sub.)∈.sup.P.sup.x.sup.×P.sup.y.sup.×2 in the virtual subarray .sub.(s.sub.x.sub.,s.sub.y.sub.) is obtained according to corresponding position elements in a coarray tensor signal corresponding to the subarray .sub.(s.sub.x.sub.,s.sub.y.sub.).
.sub.(s.sub.x.sub.,s.sub.y.sub.)=Σ.sub.k=1.sup.Kσ.sub.k.sup.2(c.sub.x(θ.sub.k,φ.sub.k)e.sup.(s.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)).Math.(c.sub.y(θ.sub.k,φ.sub.k)e.sup.(s.sup.y.sup.−1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.).Math.h.sub.k(θ.sub.k,φ.sub.k),

(42) where c.sub.x(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.x.sup.+1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), e.sup.−jπ(−N.sup.x.sup.+2)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . ,

(43) e.sup.−jπ(−N.sup.x.sup.+P.sup.x.sup.)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)] and c.sub.y(θ.sub.k, φ.sub.k)=[e.sup.−jπ(−N.sup.y.sup.+1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.),

(44) e.sup.−jπ(−N.sup.y.sup.+2)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), . . . , e.sup.−jπ(−N.sup.y.sup.+P.sup.y.sup.)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] denote steering vectors of a virtual subarray .sub.1,1) corresponding to the (θ.sub.k, φ.sub.k) direction on the x axis and they axis. After the above operations, a total of L.sub.x×L.sub.y three-dimensional tensors .sub.(s.sub.x.sub., s.sub.y.sub.) whose dimensions are all P.sub.x×P.sub.y×2 are obtained. In order to extend the dimension of the coarray tensor, firstly, tensors in the three-dimensional sub-coarray tensors .sub.(s.sub.x.sub., s.sub.y.sub.) with the same index subscript s.sub.y are concatenated in the fourth dimension, to obtain L.sub.y four-dimensional tensors with size of P.sub.x×P.sub.y×2×L.sub.x; and further, the L.sub.y four-dimensional tensors are concatenated in the fifth dimension, to obtain a five-dimensional coarray tensor ∈.sup.P.sup.x.sup.×P.sup.y.sup.×2×L.sup.x.sup.×L.sup.y which is defined as:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2c.sub.x(θ.sub.k,φ.sub.k)○c.sub.y(θ.sub.k,φ.sub.k)○h.sub.k(θ.sub.k,φ.sub.k)○d.sub.x(θ.sub.k,φ.sub.k)○d.sub.y(θ.sub.k,φ.sub.k),

(45) where d.sub.x(θ.sub.k, φ.sub.k)=[1, e.sup.−jπ sin(φ.sup.k.sup.)cos(θ.sup.k.sup.), . . . , e.sup.−jπ(L.sup.x.sup.−1)sin(φ.sup.k.sup.)cos(θ.sup.k.sup.)], d.sub.y(θ.sub.k, φ.sub.k)=[1, e.sup.−jπ sin(φ.sup.k.sup.)sin(θ.sup.k.sup.), . . . , e.sup.−jπ(L.sup.y.sup.−1)sin(φ.sup.k.sup.)sin(θ.sup.k.sup.)] denote the shifting factor vectors corresponding to the x-axis and y-axis directions respectively during coarray tensor dimension extension and construction.

(46) Step 6: A structured coarray tensor including three-dimensional spatial information is formed. In order to obtain the structured coarray tensor, the five-dimensional coarray tensor after dimension extension is combined along first and second dimensions representing angular information and is also combined along fourth and fifth dimensions representing shifting information, and the third dimension representing symmetric transformation information is retained, which includes the following specific operations: defining dimension sets .sub.1={1, 2}, .sub.2={3}, .sub.3={4, 5}, and unfolding a module {.sub.1, .sub.2, .sub.3} of reshaping of , to obtain a three-dimensional structured coarray tensor ∈.sup.P.sup.x.sup.P.sup.y.sup.×2×L.sup.x.sup.L.sup.y:
=Σ.sub.k=1.sup.Kσ.sub.k.sup.2g(θ.sub.k,φ.sub.k)○h(θ.sub.k,φ.sub.k)○f(θ.sub.k,φ.sub.k),

(47) where g(θ.sub.k, φ.sub.k)=c.sub.y(θ.sub.k, φ.sub.k).Math.c.sub.x(θ.sub.k, φ.sub.k), f(θ.sub.k, φ.sub.k)=d.sub.y(θ.sub.k, φ.sub.k).Math.d.sub.x(θ.sub.k, φ.sub.k). Three dimensions of the structured coarray tensor represent angular information, symmetric transformation information, and shifting information respectively.

(48) Step 7: Two-dimensional direction-of-arrival estimation is obtained through CANDECOMP/PARACFAC decomposition of the structured coarray tensor. CANDECOMP/PARACFAC decomposition is performed on the three-dimensional structured coarray tensor , to obtain three factor matrixes, G=[g({circumflex over (θ)}.sub.1, {circumflex over (φ)}.sub.1), g({circumflex over (θ)}.sub.2, {circumflex over (φ)}.sub.2), . . . , g({circumflex over (θ)}.sub.K, {circumflex over (φ)}.sub.K)], H=[h({circumflex over (θ)}.sub.1, {circumflex over (φ)}.sub.1), h({circumflex over (θ)}.sub.2, {circumflex over (φ)}.sub.2), . . . , h({circumflex over (θ)}.sub.K, {circumflex over (φ)}.sub.K)], F=[f({circumflex over (θ)}.sub.1, {circumflex over (φ)}.sub.1), f({circumflex over (θ)}.sub.2, {circumflex over (φ)}.sub.2), . . . , f({circumflex over (θ)}.sub.K, {circumflex over (φ)}.sub.K)]; where ({circumflex over (θ)}.sub.k, {circumflex over (φ)}.sub.k), k=1, 2, . . . , K is the estimated value of each incident angle (θ.sub.k, φ.sub.k), k=1, 2, . . . , K; elements in the second row in the factor matrix G are divided by elements in the first row to obtain e.sup.−jπ sin({circumflex over (φ)}.sup.k.sup.)cos({circumflex over (θ)}.sup.k.sup.), and elements in the P.sub.x+1.sup.th row in the factor matrix G are divided by elements in the first row to obtain e.sup.−jπ sin({circumflex over (φ)}.sup.k.sup.)sin({circumflex over (θ)}.sup.k.sup.); after a similar parameter retrieval operation is also performed on the factor matrix F, averaging and logarithm processing are performed on parameters extracted from G and F respectively, to obtain .sub.k=sin({circumflex over (φ)}.sub.k)cos({circumflex over (θ)}.sub.k) and .sub.k=sin({circumflex over (φ)}.sub.k)sin({circumflex over (θ)}.sub.k), and then the closed-form solution of the two-dimensional direction-of-arrival estimation ({circumflex over (θ)}.sub.k, {circumflex over (φ)}.sub.k) is:

(49) $$\begin{gathered} {\hat{\theta }}_k=\arctan \left(\frac{k_{}}{k_{}}\right), \\ {\hat{\phi }}_k=\sqrt{k_2^{}+k_2^{}}. \end{gathered}$$

(50) In the above step, CANDECOMP/PARAFAC decomposition follows the following uniqueness condition:
.sub.rank(G)+.sub.rank(H)+.sub.rank(F)≥2K+2,

(51) wherein .sub.rank(⋅) denotes a Kruskal's rank of a matrix, and .sub.rank(G)=min(P.sub.xP.sub.y,K), .sub.rank(H)=min(L.sub.x,L.sub.y,K), .sub.rank(F)=min(2,K), min(⋅) denotes a minimization operation.

(52) Optimal P.sub.x and P.sub.y values are obtained according to the above inequality, so as to obtain a theoretical maximum value of K, i.e., a theoretical upper bound of the distinguishable sources, is obtained by ensuring that the uniqueness condition is satisfied. Herein, the value of K exceeds the total number 4M.sub.xM.sub.y+N.sub.xN.sub.y−1 of actual physical sensors of the coprime planar array due to construction and processing of the structured coarray tensor, which indicates that the degrees-of-freedom of direction-of-arrival estimation is improved.

(53) The effect of the present disclosure is further described below with reference to a simulation example.

(54) Simulation example: a coprime planar array is used to receive incident signals, parameters thereof are selected as M.sub.x=2, M.sub.y=3, N.sub.x=3, N.sub.y=4, that is, the coprime planar array includes a total of 4M.sub.xM.sub.y+N.sub.xN.sub.y 1=35 physical sensors. Assuming that the number of incident narrowband sources is 50 and azimuth angles in an incident direction are evenly distributed over [−65°, 5°]∪[5°, 65° ], elevation angles are evenly distributed within a space angle domain range of [5°, 65° ]. 500 noiseless sampling snapshots are used for a simulation experiment.

(55) Estimation results of the two-dimensional direction-of-arrival estimation method for a coprime planar array based on structured coarray tensor processing provided in the present disclosure are as shown in FIG. 5, among which x and y axes represent elevation and azimuth angles of incident signal sources respectively. It can be seen that the method provided in the present disclosure can effectively distinguish the 50 incident sources. For the traditional direction-of-arrival estimation method using a uniform planar array, 35 physical sensors can be used to distinguish only at most 34 incident signals. The above results indicate that the method provided in the present disclosure achieves the increase of the degree of freedom.

(56) Based on the above, the present disclosure fully considers an association between a two-dimensional coarray of a coprime planar array and the tensorial space, derives the coarray equivalent signals through second-order statistic analysis of the tensor signal, and retains structural information of the multi-dimensional received signal and the coarray. Moreover, coarray tensor dimension extension and structured coarray tensor construction mechanisms are established, which lays a theoretical foundation for maximizing the number of degrees-of-freedom. Finally, the present disclosure performs multidimensional feature extraction on the structured coarray tensor to form a closed-form solution of two-dimensional direction-of-arrival estimation, and achieves a breakthrough in the degree of freedom performance.

(57) The above are only preferred implementations of the present disclosure. Although the present disclosure has been disclosed above with preferred embodiments, the preferred embodiments are not intended to limit the present disclosure. Any person skilled in the art can make, without departing from the scope of the technical solution of the present disclosure, many possible variations and modifications to the technical solution of the present disclosure or modify the technical solution as equivalent embodiments of equivalent changes by using the method and technical contents disclosed above. Therefore, any simple alteration, equivalent change, or modification made to the above embodiments according to the technical essence of the present disclosure without departing from the contents of the technical solution of the present disclosure still fall within the protection scope of the technical solution of the present disclosure.