THREE-DIMENSIONAL CO-PRIME CUBIC ARRAY DIRECTION-OF-ARRIVAL ESTIMATION METHOD BASED ON A CROSS-CORRELATION TENSOR

Abstract

The present disclosure discloses a three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor, mainly solving the problems of multi-dimensional signal structured information loss and Nyquist mismatch in existing methods and comprising the following implementing steps: constructing a three-dimensional co-prime cubic array; carrying out tensor modeling on a receiving signal of the three-dimensional co-prime cubic array; calculating six-dimensional second-order cross-correlation tensor statistics; deducing a three-dimensional virtual uniform cubic array equivalent signal tensor based on cross-correlation tensor dimension merging transformation; constructing a four-dimensional virtual domain signal tensor based on mirror image augmentation of the three-dimensional virtual uniform cubic array; constructing a signal and noise subspace in a Kronecker product form through virtual domain signal tensor decomposition; and acquiring a direction-of-arrival estimation result based on three-dimensional spatial spectrum search.

Claims

1. A three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor, comprising the following steps: (1) constructing at a receiving end with +−1 physical antenna array elements in accordance with a structure of a three-dimensional co-prime cubic array, wherein and , and , and and are respectively a pair of co-prime integers, and the three-dimensional co-prime cubic array is decomposable into two sparse and uniform cubic subarrays and ; (2) supposing there are K far-field narrowband non-coherent signal sources from a direction of {(θ.sub.1,φ.sub.1), . . . , (θ.sub.K,φ.sub.K)}, carrying out modeling on a receiving signal of the sparse and uniform cubic subarray of the three-dimensional co-prime cubic array via a four-dimensional tensor ∈×××T (T is a number of sampling snapshots) as follows:
=Σ.sub.k=1.sup.K(μ.sub.k)∘(ν.sub.k)∘(ω.sub.k)∘s.sub.k+, 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 k.sup.th incident signal source, (⋅).sup.T represents a transposition operation, ∘ represents an external product of vectors, ∈×××T is a noise tensor mutually independent from each signal source, (μ.sub.k), (ν.sub.k) and (ω.sub.k) are steering vectors of the three-dimensional sparse and uniform cubic subarray in an x axis, a y axis and a z axis respectively, and a signal source corresponding to a direction-of-arrival of (θ.sub.k,φ.sub.k) is represented as: $a_x^{\left({\mathbb{Q}}_i\right)}\left({\mu }_k\right)={\left[1,e^{-j\pi x_{{\mathbb{Q}}_i}^{\left(2\right)}{\mu }_k},.Math.,e^{-j\pi x_{{\mathbb{Q}}_i}^{\left(M_x^{\left({\mathbb{Q}}_i\right)}\right)}{\mu }_k}\right]}^T,$ $a_y^{\left({\mathbb{Q}}_i\right)}\left(v_k\right)={\left[1,e^{-j\pi y_{{\mathbb{Q}}_i}^{\left(2\right)}{v}_k},.Math.,e^{-j\pi y_{{\mathbb{Q}}_i}^{\left(M_y^{\left({\mathbb{Q}}_i\right)}\right)}{v}_k}\right]}^T,$ $a_z^{\left({\mathbb{Q}}_i\right)}\left({\omega }_k\right)={\left[1,e^{-j\pi z_{{\mathbb{Q}}_i}^{\left(2\right)}{\omega }_k},.Math.,e^{-j\pi z_{{\mathbb{Q}}_i}^{\left(M_z^{\left({\mathbb{Q}}_i\right)}\right)}{\omega }_k}\right]}^T,$ wherein, (i.sub.1=1,2, . . . , )(i.sub.2=1,2, . . . , ) and (i.sub.3=1, 2, . . . , ) respectively represent actual locations of in i.sub.1.sup.th, i.sub.2.sup.th, and i.sub.3.sup.th physical antenna array elements in the x axis, the y axis and the z axis, and ===0, μ.sub.k=sin(φ.sub.k)cos(θ.sub.k), ν.sub.k=sin(φ.sub.k)sin(θ.sub.k), ω.sub.k=cos(φ.sub.k), j=√{square root over (−1)}; (3) based on four-dimensional receiving signal tensors and of the two three-dimensional sparse and uniform cubic subarrays and , solving their cross-correlation statistics to obtain a six-dimensional space information-covered second-order cross-correlation tensor ${ℛ}_{{\mathbb{Q}}_1{\mathbb{Q}}_2}\in {ℂ}^{M_x^{\left({\mathbb{Q}}_1\right)}\times M_y^{\left({\mathbb{Q}}_1\right)}\times M_z^{\left({\mathbb{Q}}_1\right)}\times M_x^{\left({\mathbb{Q}}_2\right)}\times M_y^{\left({\mathbb{Q}}_2\right)}\times M_z^{\left({\mathbb{Q}}_2\right)}}:$ ${ℛ}_{{\mathbb{Q}}_1{\mathbb{Q}}_2}=E\left[{.Math.{𝒳}_{{\mathbb{Q}}_1},{𝒳}_{{\mathbb{Q}}_2}^{*}.Math.}_4\right]=\underset{k=1}{\overset{K}{.Math.}}{\sigma }_k^2{a}_x^{\left({\mathbb{Q}}_1\right)}\left({\mu }_k\right)\circ a_y^{\left({\mathbb{Q}}_1\right)}\left(v_k\right)\circ a_z^{\left({\mathbb{Q}}_1\right)}\left({\omega }_k\right)\circ a_x^{{\left({\mathbb{Q}}_2\right)}^{*}}\left({\mu }_k\right)\circ a_y^{{\left({\mathbb{Q}}_2\right)}^{*}}\left(v_k\right)\circ a_z^{{\left({\mathbb{Q}}_2\right)}^{*}}\left({\omega }_k\right)+{𝒩}_{{\mathbb{Q}}_1{\mathbb{Q}}_2},$ wherein, σ.sub.k.sup.2=E[s.sub.ks.sub.k*] represents a power of a k.sup.th incident signal source, =E[<,>.sub.4] represents a six-dimensional cross-correlation noise tensor, <⋅,⋅>.sub.r represents a tensor contraction operation of two tensors along a r.sup.th dimension, E[⋅] represents an operation of taking a mathematic expectation, and (⋅)* represents a conjugate operation; a six-dimensional tensor merely has an element with a value of σ.sub.n.sup.2 in a (1, 1, 1, 1, 1, 1).sup.th location, σ.sub.n.sup.2 representing a noise power, and with a value of 0 in other locations; (4) as a first dimension and a fourth dimension of the cross-correlation tensor represent space information in a direction of the x axis, a second dimension and a fifth dimension represent space information in a direction of the y axis, and a third dimension and a sixth dimension represent space information in a direction of the z axis, defining dimension sets .sub.1={1,4},.sub.2={2,5} and .sub.3={3,6}, and carrying out tensor transformation of dimension merging on the cross-correlation tensor to obtain a virtual domain second-order equivalent signal tensor ∈××: ${𝒰}_{𝕎}\stackrel{△}{=}{ℛ}_{{\mathbb{Q}}_1{\mathbb{Q}}_{2_{\left\{{𝕁}_1,{𝕁}_2,{𝕁}_3\right\}}}}={.Math.}_{k=1}^K{\sigma }_k^2{b}_x\left({\mu }_k\right)\circ b_y\left(v_k\right)\circ b_z\left({\omega }_k\right),$ wherein, b.sub.x(μ.sub.k)=(μ.sub.k).Math.(μ.sub.k), b.sub.y(ν.sub.k)=(ν.sub.k).Math.(ν.sub.k) and b.sub.z(ω.sub.k)=(ω.sub.k).Math.(ω.sub.k) respectively construct augmented virtual arrays in the directions of the x axis, the y axis and the z axis through forming arrays of difference sets on exponential terms, b.sub.x(μ.sub.k) , b.sub.y(ν.sub.k) and b.sub.z(ω.sub.k) are respectively equivalent to steering vectors of the virtual arrays in the x axis, the y axis and the z axis to correspond to signal sources in a direction-of-arrival of (θ.sub.k,φ.sub.k), and .Math. represents a product of Kronecker, so that corresponds to an augmented three-dimensional virtual non-uniform cubic array ; comprises a three-dimensional uniform cubic array with (3−+1)×(3−−1)×−+1) virtual array elements, represented as: ={(x,y,z)|x=p.sub.xd,y=p.sub.yd,z=p.sub.zd,−≤p.sub.x≤−+2,−≤p.sub.y≤−+2,−≤p.sub.z≤−+2}, The equivalent signal tensor ∈−+1)×(3−+1)×(3−+1) of the three-dimensional uniform cubic array is modeled as: =Σ.sub.k=1.sup.Kσ.sub.k.sup.2b.sub.x(μ.sub.k)∘b.sub.y(ν.sub.k)∘b.sub.z(ω.sub.k), wherein, ${\stackrel{\_}{b}}_x\left({\mu }_k\right)={\left[e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k},e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_1\right)}+1\right){\mu }_k},.Math.,e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_2\right)}+2M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k}\right]}^T\in {ℂ}^{3M_x^{\left({\mathbb{Q}}_1\right)}-M_x^{\left({\mathbb{Q}}_2\right)}+1},{\stackrel{\_}{b}}_y\left(v_k\right)={\left[e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k},e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_1\right)}+1\right)v_k},.Math.,e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_2\right)}+2M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k}\right]}^T\in {ℂ}^{3M_y^{\left({\mathbb{Q}}_1\right)}-M_y^{\left({\mathbb{Q}}_2\right)}+1}\mathrm{and}{\stackrel{\_}{b}}_z\left({\omega }_k\right)={\left[e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k},e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_1\right)}+1\right){\omega }_k},.Math.,e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_2\right)}+2M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k}\right]}^T\in {ℂ}^{3M_z^{\left({\mathbb{Q}}_1\right)}-M_z^{\left({\mathbb{Q}}_2\right)}+1}$ represent steering vectors of the three-dimensional virtual uniform cubic array in the x axis, the y axis and the z axis corresponding to signal sources in the direction-of-arrival of (θ.sub.k, φ.sub.k); (5) as a mirror image portion .sub.sym, of the three-dimensional virtual uniform cubic array is represented as:
.sub.sym={(x,y,z)|x={hacek over (p)}.sub.xd,y={hacek over (p)}.sub.yd,z={hacek over (p)}.sub.zd,−2≤{hacek over (p)}.sub.x≤−2≤{hacek over (p)}.sub.y≤,−2≤{hacek over (p)}.sub.z≤}, carrying out transformation by using the equivalent signal tensor of the three-dimensional virtual uniform cubic array to obtain an equivalent signal tensor .sub.sym∈−+1)×(3−+1)×(3−+1) of a three-dimensional mirror image virtual uniform cubic array .sub.sym, specifically comprising: carrying out a conjugate operation on the three-dimensional virtual domain signal tensor to obtain , carrying out position reversal on elements in the along directions of three dimensions successively so as to obtain the equivalent signal tensor .sub.sym corresponding to the .sub.sym ; superposing the equivalent signal tensor of the three-dimensional virtual uniform cubic array and the equivalent signal tensor .sub.sym of the mirror image virtual uniform cubic array .sub.sym in the fourth dimension to obtain a four-dimensional virtual domain signal tensor ∈−+1)×(3−+1)×(3−+1)×2, modeled as: =Σ.sub.k=1.sup.Kσ.sub.k.sup.2b.sub.x(μ.sub.k)∘b.sub.y(ν.sub.k)∘b.sub.z(ω.sub.k)∘c(μ.sub.k,ν.sub.k,ω.sub.k), wherein, $c\left({\mu }_k,v_k,{\omega }_k\right)={\left[1,e^{-j\pi \left(\left(M_x^{\left({\mathbb{Q}}_2\right)}-M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k+\left(M_y^{\left({\mathbb{Q}}_2\right)}-M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k+\left(M_z^{\left({\mathbb{Q}}_2\right)}-M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k\right)}\right]}^T$ is a three-dimensional space mirror image transformation factor vector; (6) carrying out CANDECOMP/PARACFAC decomposition on the four-dimensional virtual domain signal tensor to obtain factor vectors b.sub.x(μ.sub.k),b.sub.y(ν.sub.k), b.sub.z(ω.sub.k) and c(μ.sub.k,ν.sub.k,ω.sub.k), k=1,2, . . . , K, corresponding to four-dimensional space information, and constructing a signal subspace V.sub.s∈.sup.V×K through a form of their Kronecker products: V.sub.s=orth([b.sub.x(μ.sub.1).Math.b.sub.y(ν.sub.1).Math.b.sub.z(ω.sub.1).Math.c(μ.sub.1,ν.sub.1,ω.sub.1), b.sub.x(μ.sub.2).Math.b.sub.y(ν.sub.2).Math.b(ω.sub.2).Math.c(μ.sub.2,ν.sub.2,ω.sub.2), . . . , b.sub.x(μ.sub.K).Math.b.sub.y(ν.sub.K).Math.b.sub.z(ω.sub.K).Math.c(μ.sub.K,ν.sub.K,ω.sub.K)]), wherein, orth(⋅) represents a matrix orthogonalization operation, V=2(3−+1)(3−+1)(3−+1); by using V.sub.n∈.sup.V×(V−K) to represent a noise subspace, V.sub.nV.sub.n.sup.H is obtained by V.sub.s:
V.sub.nV.sub.n.sup.H=I−V.sub.sV.sub.s.sup.H, wherein, I represents a unit matrix; (⋅).sup.H represents a conjugate transposition operation; and (7) traversing a two-dimensional direction-of-arrival of ({tilde over (θ)}, {tilde over (φ)}), calculating corresponding parameters {tilde over (μ)}.sub.k=sin({tilde over (θ)}.sub.k), {tilde over (ν)}.sub.k=sin({tilde over (φ)}.sub.k)sin({tilde over (θ)}.sub.k) and {tilde over (ω)}.sub.k=cos({tilde over (φ)}.sub.k), and constructing a steering vector ({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)∈.sup.V corresponding to the three-dimensional virtual uniform cubic array , represented as: ({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)=b.sub.x({tilde over (μ)}.sub.k).Math.b.sub.y({tilde over (ν)}.sub.k) .Math.b.sub.z({tilde over (ω)}.sub.k).Math.c({tilde over (μ)}.sub.k, {tilde over (ν)}.sub.k, {tilde over (ω)}.sub.k), wherein, {tilde over (θ)}∈[−90°,π°], {tilde over (φ)}∈[0°,180°], A three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) is calculated as follows:
({tilde over (θ)},{tilde over (φ)})=1/(.sup.H({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)(V.sub.nV.sub.n.sup.H)({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)), Spectral peak search is carried out on the three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) to obtain a direction-of-arrival estimation result.

2. The three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor according to claim 1, wherein the structure of the three-dimensional co-prime cubic array in step (1) is described as: a pair of three-dimensional sparse and uniform cubic subarrays and are constructed in a rectangular coordinate system, wherein comprises ×× antenna array elements, with array element spacings in the directions of the x axis, they axis and the z axis being d, d and d respectively, with locations in the rectangular coordinate system being {(dm.sub.1x,dm.sub.1y,dm.sub.1z), m.sub.1x=0,1, . . . , −1, m.sub.1y=0,1, . . . , −1, m.sub.1z−0,1, . . . , −1}; comprises ×× antenna array elements, with array element spacings in the directions of the x axis, the y axis and the z axis being d, d and d respectively, with locations in the rectangular coordinate system being {(dm.sub.2x,dm.sub.2y,dm.sub.2z), m.sub.2x=0,1, . . . , −1, m.sub.2y=0,1, . . . , −1, m.sub.2z=0,1, . . . , −1}; a unit spacing d has a value half of an incident narrowband signal wavelength) λ, i.e., d=λ/2; subarray combination is carried out on the and in such a way that array elements on the (0, 0, 0) location in the rectangular coordinate system are overlapped so as to obtain a three-dimensional co-prime cubic array actually containing +−1 physical antenna array elements.

3. The three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor according to claim 1, wherein the second-order cross-correlation tensor statistics of the three-dimensional co-prime cubic array in step (3) are estimated by calculating cross-correlation statistics of T sampling snapshots of the receiving signal tensors (t) and (t) in reality: ${\widehat{ℛ}}_{{\mathbb{Q}}_1{\mathbb{Q}}_2}=\frac{1}{T}{.Math.}_{t=1}^T{𝒳}_{{\mathbb{Q}}_1}\left(t\right)\circ {𝒳}_{{\mathbb{Q}}_2}^{*}\left(t\right).$

4. The three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor according to claim 1, wherein the equivalent signal tensor of the three-dimensional virtual uniform cubic array in step (4) can be obtained by selecting elements in the equivalent signal tensor of the three-dimensional virtual nonuniform cubic array corresponding to locations of virtual array elements in the .

5. The three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor according to claim 1, wherein in step (6), CANDECOMP/PARAFAC decomposition is carried out on the four-dimensional virtual domain signal tensor to obtain factor matrixes B.sub.x=[b.sub.x(μ.sub.1), b.sub.x(μ.sub.2), . . . b.sub.x(μ.sub.K)], B.sub.y=[b.sub.y(ν.sub.1), b.sub.y(ν.sub.z), . . . , b.sub.y(ν.sub.K)], B.sub.z=[b.sub.z(ω.sub.1), b.sub.z(ω.sub.2), . . . , b.sub.z(ω.sub.K)] and C=[c(μ.sub.1,ν.sub.1,ω.sub.1), c(μ.sub.2,ν.sub.2,ω.sub.2), . . . , c(μ.sub.K,ν.sub.K,ω.sub.K)], wherein, CANDECOMP/PARAFAC decomposition of the four-dimensional virtual domain signal tensor follows a uniqueness condition as follows: .sub.rank(B.sub.x)+.sub.rank(B.sub.y)+.sub.rank(B.sub.z)+.sub.rank(C)≥2K+3, wherein, .sub.rank(⋅) represents a Kruskal rank of a matrix, and .sub.rank(B.sub.x)=min (3−+1, K), .sub.rank(B.sub.y)=min(3−M−1, K), .sub.rank(B.sub.z)=min(3−+1, K), .sub.rank(C)=min(2, K), min (⋅) represents an operation of taking a minimum value; when spatial smoothing is not introduced to process the deduced four-dimensional virtual domain signal tensor , a uniqueness inequation of the above CANDECOMP/PARACFAC decomposition is established.

6. The three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor according to claim 1, wherein in step (7), a process of obtaining a direction-of-arrival estimation result by three-dimensional spatial spectrum search specifically comprises: fixing a value of {tilde over (φ)} at 0°, gradually increasing {tilde over (θ)} to 90° from −90° at an interval of 0.1°, increasing the {tilde over (θ)} to 0.1° from 0°, increasing the θ to 90° from −90° at an interval of 0.1° once again, and repeating this process until the {tilde over (φ)} is increased to 180°, calculating a corresponding ({tilde over (θ)},{tilde over (φ)}) in each two-dimensional direction-of-arrival of ({tilde over (θ)},{tilde over (φ)}) so as to construct a three-dimensional spatial spectrum on a two-dimensional direction-of-arrival plane; searching peak values of the three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) in the two-dimensional direction-of-arrival plane, permutating response values corresponding to these peak values in a descending order, and taking two-dimensional angle values corresponding to first K spectral peaks as the direction-of-arrival estimation result of a corresponding signal source.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] FIG. 1 is an overall flow block diagram of the present disclosure.

[0025] FIG. 2 is a schematically structural diagram of a three-dimensional co-prime cubic array of the present disclosure.

[0026] FIG. 3 is a schematically structural diagram of a deduced three-dimensional virtual uniform cubic array of the present disclosure.

[0027] FIG. 4 is an effect diagram of direction-of-arrival estimation of a traditional method based on vectorized signal processing.

[0028] FIG. 5 is an effect diagram of direction-of-arrival estimation of a method put forward by the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0029] The technical solution of the present disclosure will be further explained in detail by referring to the appended drawings.

[0030] In order to solve the problems of multi-dimensional signal structured information loss and Nyquist mismatch in existing methods, the present disclosure puts forwards a three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor. In combination with cross-correlation tensor statistic analysis, multi-dimensional virtual domain tensor space extension, cross-correlation virtual domain signal tensor decomposition and other means, relevance between three-dimensional co-prime cubic array cross-correlation tensor statistics and a virtual domain is established to realize Nyquist matched two-dimensional direction-of-arrival estimation. As shown in FIG. 1, the present disclosure comprises the following implementing steps:

[0031] step 1: constructing a three-dimensional co-prime cubic array. The three-dimensional co-prime cubic array is constructed at a receiving end with +−1 physical antenna array elements; as shown in FIG. 2, a pair of three-dimensional sparse and uniform cubic subarrays and are constructed in a rectangular coordinate system, wherein comprises ×× antenna array elements, with array element spacings in the directions of the x axis, the y axis and the z axis being d, d and d respectively, with locations in the rectangular coordinate system being {(dm.sub.1x,dm.sub.1y,dm.sub.1z), m.sub.1x=0,1, . . . , −1, m.sub.1y=0,1, . . . , −1, m.sub.1z=0,1, . . . , −1}; comprises ×× antenna array elements, with array element spacings in the directions of the x axis, the y axis and the z axis being d, d and d respectively, with locations in the rectangular coordinate system being {(dm.sub.2x,dm.sub.2y,dm.sub.2z), m.sub.2x=0,1, . . . , −1, m.sub.2y=0,1, . . . , −1, 3.sub.2z=0,1, . . . , −1}; wherein and , and , and and are respectively a pair of co-prime integers; a unit spacing d has a value half of an incident narrowband signal wavelength λ, i.e., d=λ/2; subarray combination is carried out on the and in such a way that array elements on the (0,0,0) location in the rectangular coordinate system are overlapped so as to obtain a three-dimensional co-prime cubic array actually containing +−1 physical antenna array elements.

[0032] step 2: carrying out tensor modeling on a receiving signal of the three-dimensional co-prime cubic array. Supposing there are K far-field narrowband non-coherent signal sources from a direction of {(θ.sub.1,φ.sub.1), (θ.sub.2,φ.sub.2), . . . , (θ.sub.K,φ.sub.K)}, a sampling snapshot signal at the time t of the sparse and uniform cubic subarray of the three-dimensional co-prime cubic array is represented by a three-dimensional space information-covered tensor (t)∈××, receiving signal tensors (t) of T sampling snapshots are superposed in a fourth dimension (i.e., time dimension) to obtain a four-dimensional receiving signal tensor ∈×××T corresponding to the sparse and uniform cubic subarray , modeled as:

=Σ.sub.k=1.sup.K(μ.sub.k)∘(ν.sub.k)∘(ω.sub.k)∘s.sub.k+,

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 k.sup.th incident signal source, (⋅).sup.T represents a transposition operation, ∘ represents an external product of vectors, ×××T is a noise tensor mutually independent from each signal source, (μ.sub.k), (ν.sub.k) and (ω.sub.k) are steering vectors of the three-dimensional sparse and uniform cubic subarray in the directions of the x axis, the y axis and the z axis respectively, and a signal source corresponding to a direction-of-arrival of (θ.sub.k,φ.sub.k) is represented as:

[00007]$a_x^{\left({\mathbb{Q}}_i\right)}\left({\mu }_k\right)={\left[1,e^{-j\pi x_{{\mathbb{Q}}_i}^{\left(2\right)}{\mu }_k},.Math.,e^{-j\pi x_{{\mathbb{Q}}_i}^{\left(M_x^{\left({\mathbb{Q}}_i\right)}\right)}{\mu }_k}\right]}^T,$$a_y^{\left({\mathbb{Q}}_i\right)}\left(v_k\right)={\left[1,e^{-j\pi y_{{\mathbb{Q}}_i}^{\left(2\right)}{v}_k},.Math.,e^{-j\pi y_{{\mathbb{Q}}_i}^{\left(M_y^{\left({\mathbb{Q}}_i\right)}\right)}{v}_k}\right]}^T,$$a_z^{\left({\mathbb{Q}}_i\right)}\left({\omega }_k\right)={\left[1,e^{-j\pi z_{{\mathbb{Q}}_i}^{\left(2\right)}{\omega }_k},.Math.,e^{-j\pi z_{{\mathbb{Q}}_i}^{\left(M_z^{\left({\mathbb{Q}}_i\right)}\right)}}{\omega }_k\right]}^T,$

wherein, (i.sub.1=1,2, . . . , ) , (i.sub.2=1,2, . . . , ) and (i.sub.3=1,2, . . . , ) respectively represent actual locations of in i.sub.1.sup.th, i.sub.2.sup.th and i.sub.3.sup.th physical antenna array elements in the directions of the x axis, the y axis and the z axis, and ===0, μ.sub.k=sin(φ.sub.k)cos(θ.sub.k), ν.sub.k=sin(φ.sub.k)sin(θ.sub.k), ω.sub.k=cos(φ.sub.k), j=√{square root over (−1)};

[0033] step 3: calculating six-dimensional second-order cross-correlation tensor statistics. As receiving signal tensors and of the two subarrays and meet characteristics of co-prime numbers in structure size, it is unable to superpose and into a tensor signal and then to calculate its second-order autocorrelation statistics. Therefore, their cross-correlation statistics are solved to obtain a six-dimensional space information-covered second-order cross-correlation tensor ∈×××××, calculated as:

[00008]${ℛ}_{{\mathbb{Q}}_1{\mathbb{Q}}_2}=E\left[<{𝒳}_{{\mathbb{Q}}_1},{𝒳}_{{\mathbb{Q}}_2}^{*}{>}_4\right]$

wherein, σ.sub.k.sup.2=E[s.sub.ks.sub.k*] represents a power of a k.sup.th incident signal source, =E[<, >.sub.4] represents a six-dimensional cross-correlation noise tensor, <⋅,⋅>.sub.r represents a tensor contraction operation of two tensors along a r.sup.th dimension, E[⋅] represents an operation of taking a mathematic expectation, and (⋅)* represents a conjugate operation. A six-dimensional tensor merely has an element with a value of σ.sub.n.sup.2 on a (1,1,1,1,1,1).sup.th location, σ.sub.n.sup.2 representing a noise power, and with a value of 0 on other locations. In fact, their cross-correlation statistics of receiving signal tensors (t) and (t) of T sampling snapshots are solved to obtain a second-order sampling cross-correlation tensor ∈×××××:

[00009]${\widehat{ℛ}}_{{\mathbb{Q}}_1{\mathbb{Q}}_2}=\frac{1}{T}{.Math.}_{t=1}^T{𝒳}_{{\mathbb{Q}}_1}\left(t\right)\circ {𝒳}_{{\mathbb{Q}}_2}^{*}\left(t\right);$

[0034] step 4: deducing a three-dimensional virtual uniform cubic array equivalent signal tensor based on cross-correlation tensor dimension merging transformation. As a cross-correlation tensor contains three-dimensional space information respectively corresponding to the two sparse and uniform cubic subarrays and , and by merging dimensions representing space information of a same direction in the , arrays of difference sets are formed on exponential terms of the steering vectors corresponding to the two co-prime subarrays so as to construct an augmented virtual array in a three-dimensional space. Specifically, as a first dimension and a fourth dimension (represented by steering vectors (μ.sub.k) and (μ.sub.k)) of the cross-correlation tensor represent space information in the direction of the x axis, a second dimension and a fifth dimension (represented by steering vectors (ν.sub.k) and (ν.sub.k)) represent space information in the direction of the y axis, and a third dimension and a sixth dimension (represented by steering vectors (ω.sub.k) and (ω.sub.k)) represent space information in the direction of the z axis, dimension sets ={1,4}, ={2,5} and ={3,6} are defined, and tensor transformation of dimension merging is carried out on the cross-correlation tensor to obtain a virtual domain second-order equivalent signal tensor ∈××:

[00010]${𝒰}_{𝕎}\stackrel{\Delta }{=}{ℛ}_{{\mathbb{Q}}_1{\mathbb{Q}}_{2[{𝕁}_1,{𝕁}_2,{𝕁}_3\}}}={.Math.}_{k=1}^K{\sigma }_k^2{b}_x\left({\mu }_k\right)\circ b_y\left(v_k\right)\circ b_z\left({\omega }_k\right),$

wherein, b.sub.x(μ.sub.k)=(μ.sub.k).Math.(μ.sub.k), b.sub.y(ν.sub.k)=(ν.sub.k).Math.(ν.sub.k) and b.sub.z(ω.sub.k)=(ω.sub.k).Math.(ω.sub.k) respectively construct augmented virtual arrays in the directions of the x axis, the y axis and the z axis through forming arrays of difference sets on exponential terms, b.sub.x(μ.sub.k), b.sub.y(ν.sub.k) and b.sub.z(ω.sub.k) are respectively equivalent to steering vectors of the virtual arrays in the x axis, the y axis and the z axis to correspond to signal sources in a direction-of-arrival of (θ.sub.k,φ.sub.k) and .Math. represents a product of Kronecker. Therefore, corresponds to an augmented three-dimensional virtual non-uniform cubic array . To simplify a deduction process, a six-dimensional noise tensor is omitted in a theoretical modeling step about . However, in fact, by replacing theoretical cross-correlation tensor statistics with sampling cross-correlation tensor statistics , is naturally covered in a cross-correlation tensor signal statistic processing process;

[0035] comprises a three-dimensional uniform cubic array with (3−+1)×(3−+1)×(3−+1) virtual array elements, represented as:

={(x,y,z)|x=p.sub.xd,y=p.sub.yd,z=p.sub.zd,−≤p.sub.x≤−+2, −≤p.sub.y≤−+2, −≤p.sub.z≤−+2}.

[0036] The equivalent signal tensor ∈−+1)×(3−+1)×(3−+1) of the three-dimensional virtual uniform cubic array can be obtained by selecting elements in that correspond to locations of virtual array elements in the , modeled as:

=Σ.sub.k=1.sup.Kσ.sub.k.sup.2b.sub.x(μ.sub.k)∘b.sub.y(ν.sub.k)∘b.sub.z(ω.sub.k),

[0037] wherein,

[00011]${\stackrel{\_}{b}}_x\left({\mu }_k\right)={\left[e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k},e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_1\right)}+1\right){\mu }_k},.Math.,e^{-j\pi \left(-M_x^{\left({\mathbb{Q}}_2\right)}+2M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k}\right]}^T\in {ℂ}^{3M_x^{\left({\mathbb{Q}}_1\right)}-M_x^{\left({\mathbb{Q}}_2\right)}+1},{\stackrel{\_}{b}}_y\left(v_k\right)={\left[e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k},e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_1\right)}+1\right)v_k},.Math.,e^{-j\pi \left(-M_y^{\left({\mathbb{Q}}_2\right)}+2M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k}\right]}^T\in {ℂ}^{3M_y^{\left({\mathbb{Q}}_1\right)}-M_y^{\left({\mathbb{Q}}_2\right)}+1}\mathrm{and}{\stackrel{\_}{b}}_z\left({\omega }_k\right)={\left[e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k},e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_1\right)}+1\right){\omega }_k},.Math.,e^{-j\pi \left(-M_z^{\left({\mathbb{Q}}_2\right)}+2M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k}\right]}^T\in {ℂ}^{3M_z^{\left({\mathbb{Q}}_1\right)}-M_z^{\left({\mathbb{Q}}_2\right)}+1}$

respectively represent steering vectors of the three-dimensional virtual uniform cubic array in the x axis, the y axis and the z axis corresponding to signal sources in the direction-of-arrival of (θ.sub.k,φ.sub.k);

[0038] step 5: constructing a four-dimensional virtual domain signal tensor based on mirror image augmentation of the three-dimensional virtual uniform cubic array. As the three-dimensional virtual uniform cubic array obtained based on cross-correlation tensor dimension merging transformation is not symmetric about a coordinate axis, in order to increase an effective aperture of a virtual array, considering that a mirror image portion .sub.sym of the three-dimensional virtual uniform cubic array is represented as:

.sub.sym={(x,y,z)|x={hacek over (p)}.sub.xd,y={hacek over (p)}.sub.yd,z={hacek over (p)}.sub.zd,−2≤{hacek over (p)}.sub.x≤, −2≤{hacek over (p)}.sub.y≤,−2≤{hacek over (p)}.sub.z≤}.

Transformation is carried out by using the equivalent signal tensor of the three-dimensional virtual uniform cubic array to obtain an equivalent signal tensor .sub.sym∈−+1)×(3−+1)×(3−+1) of a mirror image virtual uniform cubic array .sub.sym. It specifically comprises: a conjugate operation is carried out on the three-dimensional virtual domain signal tensor to obtain , position reversal is carried out on elements in the along directions of three dimensions successively so as to obtain the equivalent signal tensor .sub.sym corresponding to the .sub.sym;

[0039] the equivalent signal tensor of the three-dimensional virtual uniform cubic array and the equivalent signal tensor .sub.sym of the mirror image virtual uniform cubic array .sub.sym are superposed in the fourth dimension (i.e., a dimension representing mirror image transformation information) to obtain a four-dimensional virtual domain signal tensor ∈−+1)×(3−+1)×(3−+1)×2, modeled as:

=Σ.sub.k=1.sup.Kσ.sub.k.sup.2b.sub.x(μ.sub.k)∘b.sub.y(ν.sub.k)∘b.sub.z(ω.sub.k)∘c(μ.sub.k,ν.sub.k,ω.sub.k),

wherein,

[00012]$c\left({\mu }_k,v_k,{\omega }_k\right)={\left[1,e^{-j\pi \left(\left(M_x^{\left({\mathbb{Q}}_2\right)}-M_x^{\left({\mathbb{Q}}_1\right)}\right){\mu }_k+\left(M_y^{\left({\mathbb{Q}}_2\right)}-M_y^{\left({\mathbb{Q}}_1\right)}\right)v_k+\left(M_z^{\left({\mathbb{Q}}_2\right)}-M_z^{\left({\mathbb{Q}}_1\right)}\right){\omega }_k\right)}\right]}^T$

is a three-dimensional space mirror image transformation factor vector;

[0040] step 6: constructing a signal and noise subspace in the form of a Kronecker product by virtual domain signal tensor decomposition. CANDECOMP/PARACFAC decomposition is carried out on the four-dimensional virtual domain signal tensor to obtain factor vectors b.sub.x(μ.sub.k), b.sub.y(ν.sub.k), b.sub.z(ω.sub.k) and c(μ.sub.k,ν.sub.k,ω.sub.k), k=1,2, . . . , K, corresponding to four-dimensional space information, and B.sub.x=[b.sub.x(μ.sub.1) b.sub.x(μ.sub.2), . . . , b.sub.x(μ.sub.K)], B.sub.y=[b.sub.y(ν.sub.1), b.sub.y(ν.sub.2), . . . , b.sub.y(ν.sub.K)],B.sub.z=[b.sub.z(ω.sub.1), b.sub.z(ω.sub.2), . . . , b.sub.z(ω.sub.K)] and C=[c(μ.sub.1,ν.sub.1,ω.sub.1), c(μ.sub.2,ν.sub.2ω.sub.2), . . . , c(μ.sub.K, ν.sub.K,ω.sub.K)] are used to represent factor matrixes. At this point, CANDECOMP/PARAFAC decomposition of the four-dimensional virtual domain signal tensor follows a uniqueness condition as follows:

.sub.rank(B.sub.x)+.sub.rank(B.sub.y)+.sub.rank(B.sub.z)+.sub.rank(C)≥2K+3,

wherein, .sub.rank(⋅) represents a Kruskal rank of a matrix, and .sub.rank(B.sub.x)=min (3−+1, K), .sub.rank(B.sub.y)=min (3−+1, K), .sub.rank(B.sub.z)=min(3−+1, K), .sub.rank(C)=min(2, K), min (⋅) represents an operation of taking a minimum value; when spatial smoothing is not introduced to process the deduced four-dimensional virtual domain signal tensor , a uniqueness inequation of the above CANDECOMP/PARACFAC decomposition is established, indicating that angle information of a signal source can be effectively extracted by the method in the present disclosure in no need of a spatial smoothing step. Further, factor vectors b.sub.x(μ.sub.k), b.sub.y(ν.sub.k),b.sub.z(ω.sub.k) and c(μ.sub.k,ν.sub.k,ω.sub.k) are obtained by tensor decomposition, and a signal subspace V.sub.s∈.sup.V×K is constructed through a form of their Kronecker products:

V.sub.s=orth([b.sub.x(μ.sub.1).Math.b.sub.y(ν.sub.1).Math.b.sub.z(ω.sub.1).Math.c(μ.sub.1,ν.sub.1,ω.sub.1), b.sub.x(μ.sub.2).Math.b.sub.y(ν.sub.2).Math.b.sub.z(ω.sub.2).Math.c(μ.sub.2,ν.sub.2,ω.sub.2), . . . , b.sub.x(μ.sub.K).Math.b.sub.y(ν.sub.K).Math.b.sub.z(ω.sub.K).Math.c(μ.sub.K,ν.sub.K,ω.sub.K)]),

wherein, orth(⋅) represents a matrix orthogonalization operation, V=2(3−+1)(3−+1)(3−+1); by using V.sub.n∈.sup.V×(V−K) to represent a noise subspace, V.sub.nV.sub.n.sup.H is obtained by V.sub.s:

V.sub.nV.sub.n.sup.H=I−V.sub.sV.sub.s.sup.H,

wherein, I represents a unit matrix; (⋅).sup.H represents a conjugate transposition operation;

[0041] step 7: obtaining a direction-of-arrival estimation result based on three-dimensional spatial spectrum search. A two-dimensional direction-of-arrival of ({tilde over (θ)},{tilde over (φ)}) is traversed, corresponding parameters {tilde over (μ)}.sub.k=sin({tilde over (φ)}.sub.k)cos({tilde over (θ)}.sub.k), {tilde over (ν)}.sub.k=sin({tilde over (φ)}.sub.k)sin({tilde over (θ)}.sub.k) and {tilde over (ω)}.sub.k=cos({tilde over (φ)}.sub.k) are calculated, and a steering vector ({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)∈.sup.V corresponding to the three-dimensional virtual uniform cubic array W is constructed, represented as:

({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)=b.sub.x(μ.sub.k).Math.b.sub.y({tilde over (ν)}.sub.k).Math.{tilde over (b)}.sub.z({tilde over (ω)}.sub.k).Math.c({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k),

wherein, {tilde over (θ)}∈[−90°,90°], {tilde over (φ)}∈[0°,180°]. A three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) is calculated as follows:

({tilde over (θ)},{tilde over (φ)})=1/(.sup.H({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)(V.sub.nV.sub.n.sup.H)({tilde over (μ)}.sub.k,{tilde over (ν)}.sub.k,{tilde over (ω)}.sub.k)).

Spectral peak search is carried out on the three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) to obtain a direction-of-arrival estimation result. It specially comprises: a value of {tilde over (φ)} is fixed at 0°, {tilde over (θ)} is gradually increased to 90° from −90° at an interval of 0.1°, then the {tilde over (φ)} is increased to 0.1° from 0°, the {tilde over (θ)} is increased to 90° from −90° at an interval of 0.1° once again, and this process is repeated until the {tilde over (φ)} is increased to 180°, a corresponding ({tilde over (θ)},{tilde over (φ)}) is calculated in each two-dimensional direction-of-arrival of ({tilde over (θ)},{tilde over (φ)}) so as to construct a three-dimensional spatial spectrum in a two-dimensional direction-of-arrival plane; peak values of the three-dimensional spatial spectrum ({tilde over (θ)},{tilde over (φ)}) are searched in the two-dimensional direction-of-arrival plane, response values corresponding to these peak values are permutated in a descending order, and two-dimensional angle values corresponding to first K spectral peaks are taken as the direction-of-arrival estimation result of a corresponding signal source.

[0042] The effects of the present disclosure will be further described with conjunction of a simulation example.

[0043] In the simulation example, a three-dimensional co-prime cubic array is used to receive an incident signal, with selected parameters being ===2, ===3, i.e., the constructed three-dimensional co-prime cubic array comprises +−1=34 physical array elements in total. Supposing there are two incident narrowband signals, azimuths and pitch angles in the incident direction are respectively [25°,20°] and [45°,40°], and when SNR=0 dB, 800 sampling snapshots are adopted for a simulation experiment.

[0044] An estimation result of a traditional three-dimensional co-prime cubic array direction-of-arrival estimation method based on vectorized signal processing is as shown in FIG. 4, wherein the x axis and the y axis represent a pitch angle and an azimuth of an incident signal source respectively. An estimation result of a three-dimensional co-prime cubic array direction-of-arrival estimation method based on a cross-correlation tensor provided by the present disclosure is as shown in FIG. 5. It is seen upon comparison that, the method of the present disclosure can preciously estimate the two incident signal sources, while the traditional method based on vectorized signal processing is unable to effectively identify the two incident signal sources, showing the advantages of the method of the present disclosure in direction-of-arrival estimation on resolution and performance.

[0045] In conclusion, in the present disclosure, relevance between a three-dimensional co-prime cubic array multi-dimensional virtual domain and cross-correlation tensor statistics is established, a virtual domain signal tensor is deduced by transformation of cross-correlation tensor statistics, and cross-correlation virtual domain tensor representation with a multi-dimensional space information structure retained is constructed; identifiability of a signal source is ensured by constructing a signal source identification mechanism for a virtual domain signal tensor without introducing a spatial smoothing step; and the Nyquist matched direction-of-arrival estimation is realized by means of virtual domain signal tensor decomposition.

[0046] The above merely shows preferred embodiments of the present disclosure. Although the present disclosure has been disclosed as above in the preferred embodiments, it is not used to limit thereto. Without departing from the scope of the technical solution of the present disclosure, any person skilled in the art can make many possible variations and modifications to the technical solution of the present disclosure by using the methods and technical contents disclosed above, or modify it into equivalent embodiments with equivalent changes. Therefore, any simple alteration, equivalent variation and modification of the above embodiments according to the technical essence of the present disclosure without departing from the technical solution of the invention still fall in the protection scope of the technical solution of the present disclosure.