2-D DOA Estimation for Coprime Cubic Array: A Cross-correlation Tensor Perspective

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2-D DOA Estimation for Coprime Cubic Array:
A Cross-correlation Tensor Perspective
Hang Zheng†, Chengwei Zhou†, Yong Wang†,B, and Zhiguo Shi†,♯
†College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China
♯Alibaba-Zhejiang University Joint Institute of Frontier Technologies BEmail: wangy@zju.edu.cn
Abstract—In this paper, a cross-correlation coarray tensor-
based 2-D direction-of-arrival (DOA) estimation method for
coprime cubic array (CCA) is proposed. Since the CCA received
signals characterized by two coprime tensors cannot be collec-
tively shaped as a single tensor to operate auto-correlation, the
cross-correlation is introduced to calculate the coarray tensor
statistics of CCA. A Nyquist-matched 4-D coarray tensor is then
obtained for 2-D DOA estimation, where the designed structured
tensorization is applied without spatial smoothing. Simulation
results verify the effectiveness of the proposed method.
Keywords—DOA estimation, coprime cubic array, cross-
correlation, coarray tensor, structured tensorization.
I. INT ROD UC TI ON
Tensor-based direction-of-arrival (DOA) estimation for s-
parse arrays, among which the coprime array [1] is a typical
one, has been a heated topic due to its ability to exploit multi-
dimensional coarray statistics [2]. However, the efforts from
coprime linear array [3] and coprime planar array [4] to 3-D
coprimecubicarray (CCA) are non-trivial, since the coprimal-
ity in the increased dimensions brings hardship in correlating
tensor signals. The existing auto-correlation coarray tensor-
based method [2] is only suitable for linearly combined sparse
arrays. Therefore, performing 2-D DOA estimation for 3-D
CCA with coarray tensor statistics remains challenging.
In this paper, we propose a cross-correlation coarray tensor-
based 2-D DOA estimation method for CCA. The CCA
received signals are naturally represented as two coprime ten-
sors, which cannot be reshaped into a single tensor for auto-
correlation calculation. As such, the cross-correlation tensor
statistics of CCA are calculated, where a structured ten-
sorization technique is then designed for creating a Nyquist-
matched 4-D coarray tensor. The structural coarray tensor
with mild decomposition uniqueness condition allows the 2-D
DOAs to be estiamted without performing spatial smoothing.
II. CRO SS -CO RR EL ATIO N-B AS ED TENSOR STATIS TI CS
The CCA incorporates two sparse UCAs Pi(i= 1,2) with
Mix ×Miy ×Miz sensors as shown in Fig. 1(a), where the
coprime integers (M1x, M2x),(M1y, M2y)and (M1z, M2z)
represent the number of sensors along the x, y, z-axes direc-
tion with inter-element spacings of P1being d1x=M2xd,
d1y=M2ydand d1z=M2zd, respectively. Here, dequals
to a half-wavelength. Accordingly, the inter-element spacings
This work was partially supported by NSFC (No. 61901413, 61772467)
and National K&D Program of China (No. 2018YFE0126300).
The virtual UCA
The sparse UCA
The sparse UCA
(a) (b)
Fig. 1. Array Configurations. (a) Coprime cubic array with M1x=M1y=
M1z= 3,M2x=M2y=M2z= 2; (b) The derived virtual UCA.
of P2are d2x=M1xd,d2y=M1ydand d2z=M1zd,
respectively. Due to the coprimality of the two sparse UCAs,
they only overlap at the origin of the coordinate system.
Suppose that Kuncorrelated narrowband far-field sources
impinge on the CCA from (θk, φk), where θk∈[0◦,180◦]and
φk∈[−90◦,90◦]represent the azimuth and elevation angle
of the k-th source, respectively. The received signals of the
3-D sparse UCA Piat timeslot tcan be denoted as a tensor
Xi(t)∈CMix×Miy ×Miz . To reserve the structural spatial
information, the Tsnapshots are then concatenated along an
additional temporal dimension to formulate a 4-D tensor
Xi=
K
∑
k=1
aix(µk)◦aiy (νk)◦aiz (ωk)◦sk+Ni,(1)
where aix(µk)∈CMix ,aiy (νk)∈CMiy and aiz (ωk)∈CMiz
are the steering vectors with µk= sin φkcos θk,νk=
sin φksin θkand ωk= cos φk,sk∈CTdenotes the waveform,
and Niis the Gaussian noise tensor. Here, ◦denotes the
outer product.
The 4-D coprime tensors X1and X2corresponding to
the coprime pair UCAs P1and P2own different structural
shapes, indicating that they cannot be naturally stacked in the
temporal dimension for auto-correlation calculation. Thus,
cross-correlation, which measures the relevance between two
different shaped tensors, is applied to generate a 6-D cross-
correlation tensor R∈CM1x
×M1y
×M1z
×M2x
×M2y
×M2zas
R=E [<X1,X∗
2>{4}]
=
K
∑
k=1
σ2
ka1x(µk)◦a1y(νk)◦a1z(ωk)
◦a∗
2x(µk)◦a∗
2y(νk)◦a∗
2z(ωk) + N,
(2)
where σ2
k= E [sks∗
k]is the power of the k-th source,
and N= E [<N1,N∗
2>{4}]is the noise term, which is
Proceedings of ISAP2020, Osaka, Japan
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ignored to simplify the analysis. Here, < , >{q}denotes the
tensor contraction operation between two tensors along the
q-th dimension, E[ ·]denotes the statistic expectation, and
[·]∗denotes the conjugate operator. The second-order cross-
correlation tensor statistics of CCA in (2) containing all the
spatial information from the two coprime tensor signals, lay
foundation for the derivation of coarray tensor. In practice,
the cross-correlation tensor Rcan be estimated as
ˆ
R=1
T<X1,X∗
2>{4}.(3)
III. COAR RAY TENSOR MUSIC FOR 2-D
DOA E ST IM ATIO N WI TH OU T SPATI AL S MO OTH IN G
Define the dimensional sets S{1,4},S{2,5}, and S{3,6}
respectively combine the same directional dimensions of
the 6-D cross-correlation tensor R, then the tensorization
on R, i.e., R{1,4},{2,5},{3,6}, yields a 3-D coarray tensor
U∈CM1xM2x
×M1yM2y
×M1zM2zas
U,R{1,4},{2,5},{3,6}=
K
∑
k=1
σ2
kbx(µk)◦by(νk)◦bz(ωk),(4)
where the steering vectors bx(µk) = a∗
2x(µk)⊗a1x(µk),
by(νk) = a∗
2y(νk)⊗a1y(νk), and bz(ωk) = a∗
2z(ωk)⊗
a1z(ωk)generate difference coarrays along the x, y, z-axes
direction, respectively. Here, ⊗denotes the Kronecker product.
The 3-D coarray tensor Ucorresponds to an augmented
discontinuous 3-D virtual array, which contains a 3-D virtual
UCA Wof size (3M2x
−M1x
+1)×(3M2y
−M1y
+1)×(3M2z
−M1z
+1) as shown in Fig. 1(b). By reorganizing the elements
in Uto map the locations of virtual sensors in W, the
corresponding coarray tensor ˜
Uis obtained. Since the cross-
correlation-based virtual UCA Wis non-symmetric in the
coordinate axes, to extend the equivalent coarray aperture, the
symmetry of Wis also constructed, whose corresponding 3-D
coarray tensor ˜
Uscan be obtained by reversing the elements
in each dimension of ˜
U∗. Then, these 3-D tensors ˜
Uand
˜
Usare concatenated in the fourth dimension to create a 4-D
coarray tensor
V=
K
∑
k=1
σ2
k˜
bx(µk)◦˜
by(νk)◦˜
bz(ωk)◦ck,(5)
where ˜
bx(µk)∈C3M2x−M1x+1,˜
by(νk)∈C3M2y−M1y+1 and
˜
bz(ωk)∈C3M2z−M1z+1 serve as the steering vectors of W,
and ck∈C2is the symmetric factor vector. As such, the
structured tensorization technique transforms the original 6-D
cross-correlation tensor to the equivalent structural tensor of
the virtual UCA received signals as in (5). Due to the milder
uniqueness condition for tensor decomposition along with
the non-symmetry property of the constituting factor vectors
in (5), direct CANDECOMP/PARAFAC decomposition on
Vwithout spatial smoothing is automatically unique for
K > 2. The Kronecker product of these factor vectors, i.e.,
vk=˜
bx(µk)⊗˜
by(νk)⊗˜
bz(ωk)⊗ck,k= 1,2,· · · , K,
(a) Tensor-based Coarray MUSIC (b) Matrix-based Coarray MUSIC
Fig. 2. Spatial spectrum comparison (SNR = 0 dB, T= 800).
directly spans the steering function of the 3-D virtual U-
CA W, and thus equally spans the signal subspace Us=
[v1,v2,· · · ,vK].
The noise subspace is then obtained from the orthogonal
complement subspace of the signal subspace Usas
UnUH
n=I−orth(Us)orth(Us)H,(6)
where Iis the identity matrix, orth(·)denotes the or-
thogonalization operation, and (·)Hdenotes the Hermitian
transpose. With the steering function of the virtual UCA
Wfor the scanning bin (¯µ, ¯ν, ¯ω)defined as v(¯µ, ¯ν , ¯ω) =
˜
bx(¯µ)⊗˜
by(¯ν)⊗˜
bz(¯ω)⊗c(¯µ, ¯ν, ¯ω), the coarray tensor MUSIC
spectrum can be calculated as
P(¯µ, ¯ν, ¯ω) = 1
v(¯µ, ¯ν, ¯ω)H(UnUH
n)v(¯µ, ¯ν, ¯ω).(7)
As such, the 2-D DOA estimates (ˆ
θk,ˆφk)can be obtained via
spectrum searching on P(¯µ, ¯ν, ¯ω).
IV. SIM UL ATION
Assume two sources respectively from [20◦,25◦]and [40◦,
45◦]impinge on the CCA as deployed in Fig. 1(a). The
coarray MUSIC spectrum of the proposed method is shown
in Fig. 2(a), and it is clear that two sources corresponding to
two sharp peaks can be successfully identified. Thecompared
coarray MUSIC spectrum with matrix-based processing [5]
fails to recognize these two sources as shown in Fig. 2(b),
demonstrating that the proposed tensor-based method ensures
an improved resolution and accuracy for 2-D DOA estimation.
V. CO NC LU SI ON
Across-correlation coarraytensor-based2-DDOAestimation
method for CCA is proposed. The cross-correlation operation
contributes to the Nyquist-matched coarray tensor of CCA
for accurate 2-D DOA estimation without spatial smoothing.
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