Patent ID: 11879986
Assignee: ZHEJIANG UNIVERSITY
Field: Measurement (Instruments)
Classification: CPC G | IPC G

Claim 0:
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 {(θ1,φ1), . . . , (θK,φK)}, 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:

=Σk=1K(μk)∘(νk)∘(ωk)∘sk+,

wherein, sk=[sk,1, sk,2, . . . , sk,T]T is a multi-snapshot sampling signal waveform corresponding to a kth incident signal source, (⋅)T represents a transposition operation, ∘ represents an external product of vectors, ∈×××T is a noise tensor mutually independent from each signal source, (μk), (νk) and (ωk) 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 (θk,φk) is represented as:, a
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wherein, (i1=1,2, . . . , )(i2=1,2, . . . , ) and (i3=1, 2, . . . , ) respectively represent actual locations of  in i1th, i2th, and i3th physical antenna array elements in the x axis, the y axis and the z axis, and ===0, μk=sin(φk)cos(θk), νk=sin(φk)sin(θk), ωk=cos(φk), 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
    
     
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wherein, σk2=E[sksk*] represents a power of a kth incident signal source, =E[<,>4] represents a six-dimensional cross-correlation noise tensor, <⋅,⋅>r represents a tensor contraction operation of two tensors along a rth 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 σn2 in a (1, 1, 1, 1, 1, 1)th location, σn2 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 1={1,4},2={2,5} and 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
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wherein, bx(μk)=(μk)⊗(μk), by(νk)=(νk)⊗(νk) and bz(ωk)=(ωk)⊗(ωk) 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, bx(μk) , by(νk) and bz(ωk) 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 (θk,φk), and ⊗ 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=pxd,y=pyd,z=pzd,−≤px≤−+2,−≤py≤−+2,−≤pz≤−+2},
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       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 (θk, φk);
(5) as a mirror image portion sym, of the three-dimensional virtual uniform cubic array  is represented as:

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

carrying out transformation by using the equivalent signal tensor  of the three-dimensional virtual uniform cubic array  to obtain an equivalent signal tensor sym∈−+1)×(3−+1)×(3−+1) of a three-dimensional mirror image virtual uniform cubic array 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 sym corresponding to the sym ; superposing the equivalent signal tensor  of the three-dimensional virtual uniform cubic array  and the equivalent signal tensor sym of the mirror image virtual uniform cubic array sym in the fourth dimension to obtain a four-dimensional virtual domain signal tensor ∈−+1)×(3−+1)×(3−+1)×2, modeled as: =Σk=1Kσk2bx(μk)∘by(νk)∘bz(ωk)∘c(μk,νk,ωk), wherein,, c
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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 bx(μk),by(νk), bz(ωk) and c(μk,νk,ωk), k=1,2, . . . , K, corresponding to four-dimensional space information, and constructing a signal subspace Vs∈V×K through a form of their Kronecker products: Vs=orth([bx(μ1)⊗by(ν1)⊗bz(ω1)⊗c(μ1,ν1,ω1), bx(μ2)⊗by(ν2)⊗b(ω2)⊗c(μ2,ν2,ω2), . . . , bx(μK)⊗by(νK)⊗bz(ωK)⊗c(μK,νK,ωK)]),
wherein, orth(⋅) represents a matrix orthogonalization operation, V=2(3−+1)(3−+1)(3−+1); by using Vn∈V×(V−K) to represent a noise subspace, VnVnH is obtained by Vs:

VnVnH=I−VsVsH,

wherein, I represents a unit matrix; (⋅)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)}k=sin({tilde over (θ)}k), {tilde over (ν)}k=sin({tilde over (φ)}k)sin({tilde over (θ)}k) and {tilde over (ω)}k=cos({tilde over (φ)}k), and constructing a steering vector ({tilde over (μ)}k,{tilde over (ν)}k,{tilde over (ω)}k)∈V corresponding to the three-dimensional virtual uniform cubic array , represented as: ({tilde over (μ)}k,{tilde over (ν)}k,{tilde over (ω)}k)=bx({tilde over (μ)}k)⊗by({tilde over (ν)}k) ⊗bz({tilde over (ω)}k)⊗c({tilde over (μ)}k, {tilde over (ν)}k, {tilde over (ω)}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/(H({tilde over (μ)}k,{tilde over (ν)}k,{tilde over (ω)}k)(VnVnH)({tilde over (μ)}k,{tilde over (ν)}k,{tilde over (ω)}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.