US 12,540,996 B2 · Method for estimating direction of arrival of sub-array partition type l-shaped coprime array based on fourth-order sampling covariance tensor denoising

Abstract

Disclosed in the present invention is a method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising. The implementation steps are as follows: constructing an L-shaped coprime array partitioned with linear sub-arrays; modeling a receiving signal of the L-shaped coprime array and deriving a second-order cross-correlation matrix thereof; deriving a fourth-order covariance tensor based on the cross-correlation matrix; realizing fourth-order sampling covariance tensor denoising based on kernel tensor thresholding; deriving a fourth-order virtual domain signal based on denoised sampling covariance tensor; constructing a denoised structured virtual domain tensor; obtaining a direction of arrival estimation result by decomposing the structured virtual domain tensor. The present invention makes full use of the statistical distribution characteristics of the high-order tensor of the constructed sub-array partition type L-shaped coprime array, realizes high-precision two-dimensional direction of arrival estimation through denoised virtual domain tensor signal processing, and can be used for target positioning.

A method for estimating a direction of arrival of a sub-array partition type L-shaped coprime array based on fourth-order sampling covariance tensor denoising, wherein the method comprises the following steps:

(1) constructing a linear sub-array partition type L-shaped coprime array using 2++2+−2 physical antenna array elements, wherein the L-shaped coprime array consists of two coprime linear arrays i, i=1, 2 located on an x axis and a y axis, and first array elements of the two coprime linear arrays 1 and 2 are laid out from positions corresponding to a coordinate of 1 on the x axis and to a coordinate of 1 on the y axis respectively;
the coprime linear array i contains |i|=2+−1 array elements, and wherein and are a pair of coprime integers, <, |⋅| represents a potential of a set;
{
(
x
𝕃
1
,
0
)
|
x
𝕃
1
=
[
c
𝕃
1
(
1
)
,
c
𝕃
1
(
2
)
,
…
,
c
𝕃
1
(
❘
"\[LeftBracketingBar]"
𝕃
1
❘
"\[RightBracketingBar]"
)
]
⁢
d
}
⁢
and
⁢
{
(
0
,
y
𝕃
2
)
|
y
𝕃
2
=
[
c
𝕃
2
(
1
)
,
c
𝕃
2
(
2
)
,
…
,
c
𝕃
2
(
❘
"\[LeftBracketingBar]"
𝕃
2
❘
"\[RightBracketingBar]"
)
]
⁢
d
}
are respectively used to represent a position of each array element of the L-shaped coprime array on the x axis and y axis, wherein
c
𝕃
1
(
1
)
=
c
𝕃
2
(
1
)
=
1
,
and a unit interval d is taken as half of a wavelength of an incident narrowband signal;
(2) modeling, for K far-field narrow-band incoherent signal sources from {(θ1, φ1), (θ2, φ2), . . . , (θK, φK)} directions, where K is an integer greater than or equal to one, a received signal of the coprime linear array i forming the L-shaped coprime array as follows:
X
𝕃
i
=
∑
k
=
1
K
a
𝕃
i
(
k
)
∘
s
k
+
N
𝕃
i
∈
ℂ
|
𝕃
i
|
×
T
,
wherein, sk=[sk,1>sk,2, . . . , sk,T]T is a multi-snapshot sampling signal waveform corresponding to a kth incident signal source, T is the number of sampling snapshots, ∘ represents an outer product of a vector, is noise independent of each signal source, (k) is a steering vector of i, and corresponds to a signal source having an incoming wave direction of (θk, φk) and is expressed as follows:
a
𝕃
i
(
k
)
=
[
e
-
j
⁢
π
⁢
c
𝕃
i
(
1
)
⁢
μ
i
(
k
)
,
e
-
j
⁢
πc
𝕃
i
(
2
)
⁢
μ
i
(
k
)
,
…
,
e
-
j
⁢
π
⁢
c
𝕃
i
(
❘
"\[LeftBracketingBar]"
𝕃
i
❘
"\[RightBracketingBar]"
)
⁢
μ
i
(
k
)
]
T
,
wherein, μ1(k)=sin(φk)cos(θk), μ2(k)=sin(φk)sin(θk), j=√{square root over (−1)}, [⋅]T represents a transpose operation;
a second-order cross-correlation matrix ∈ obtained by solving cross-correlation statistics of and :
R
𝕃
1
⁢
𝕃
2
=
E
⁢
{
X
𝕃
1
⁢
X
𝕃
2
H
}
=
∑
k
=
1
K
σ
k
2
⁢
a
𝕃
1
(
k
)
∘
a
𝕃
2
*
(
k
)
,
and wherein,
σ
k
2
=
E
⁢
{
s
k
(
t
)
⁢
s
k
*
(
t
)
}
represents power of the kth incident signal source, E{⋅} represents a mathematical expectation operation, (⋅)H represents a conjugate transpose operation, (⋅)* represents a conjugate operation;
(3) calculating an autocorrelation of the second-order cross-correlation matrix to obtain a fourth-order covariance tensor ∈:
V
=
R
𝕃
1
⁢
𝕃
2
∘
R
𝕃
1
⁢
𝕃
2
*
=
E
⁢
{
(
X
𝕃
1
⁢
X
𝕃
2
H
)
∘
(
X
𝕃
1
⁢
X
𝕃
2
H
)
*
}
=
∑
k
=
1
K
σ
k
4
⁢
a
𝕃
1
(
k
)
∘
a
𝕃
2
*
(
k
)
∘
a
𝕃
1
*
(
k
)
∘
a
𝕃
2
(
k
)
;
wherein, the fourth-order covariance tensor is approximated by a fourth-order sampling covariance tensor ∈, that is:
𝒱
^
=
(
1
T
⁢
X
𝕃
1
⁢
X
𝕃
2
H
)
∘
(
1
T
⁢
X
𝕃
1
⁢
X
𝕃
2
H
)
*
=
∑
k
=
1
K
(
1
T
⁢
s
k
T
⁢
s
k
*
)
⁢
a
𝕃
1
(
k
)
∘
a
𝕃
2
*
(
k
)
∘
a
𝕃
1
*
(
k
)
∘
a
𝕃
2
(
k
)
+
𝒵
,
wherein
:
𝓏
=
[
1
T
⁢
∑
k
=
1
K
a
𝕃
1
⁢
(
k
)
∘
(
s
k
T
⁢
N
𝕃
2
H
)
+
1
T
⁢
∑
k
=
1
K
a
𝕃
2
⁢
(
k
)
∘
(
s
k
T
⁢
N
𝕃
1
H
)
+
1
T
⁢
N
𝕃
1
⁢
N
𝕃
2
H
]
[
1
T
⁢
∑
k
=
1
K
a
𝕃
1
⁢
(
k
)
∘
(
s
k
T
⁢
N
𝕃
2
H
)
+
1
T
⁢
∑
k
=
1
K
a
𝕃
2
⁢
(
k
)
∘
(
s
k
T
⁢
N
𝕃
1
H
)
+
1
T
⁢
N
𝕃
1
⁢
N
𝕃
2
H
]
*
is a fourth-order sampling noise tensor;
the (τ1, 1, τ2, 2)th element in is represented as , τ1, τ2−1, 2, . . . , |1|, 1, 2−1, 2, . . . , |2|, obeys an approximate complex Gaussian distribution, and an approximate variance thereof σ2 is expressed as:
σ
¯
2
=
1
T
2
[
λ
1
(
σ
n
2
⁢
∑
k
=
1
K
σ
k
2
)
2
+
λ
2
⁢
σ
n
6
⁢
∑
k
=
1
K
σ
k
2
+
λ
3
⁢
σ
n
8
]
,
and wherein, λ1, λ2 and λ3 represent a combined weight of three sub-variance terms
(
σ
n
2
⁢
∑
k
=
1
K
σ
k
2
)
2
,
σ
n
6
⁢
∑
k
=
1
K
σ
k
2
and
σ
n
8
,
σ
n
2
represents a noise power;
(4) performing high-order singular value decomposition on the fourth-order sampling covariance tensor :
=×1Y(1)×2Y(2)×3Y(3)×4Y(4),
wherein, represents a kernel tensor, which contains projections from signal and noise components in , Y(1)∈, Y(2)∈, Y(3)∈ and Y(4)∈represent singular matrices corresponding to four dimensions of ;
a thresholding is performed on , that is, elements in that are less than or equal to a noise threshold ϵ are set to zero, and elements larger than the noise threshold ϵ are reserved, thus obtaining a thresholded kernel tensor dn, where an element in dn is expressed as follows:
𝒮
d
⁢
n
⁡
(
τ
1
,
ς
1
,
τ
2
,
ς
2
)
=
{
𝒮
(
τ
1
,
ς
1
,
τ
2
,
ς
2
)
|
𝒮
(
τ
1
,
ς
1
,
τ
2
,
ς
2
)
|>
∈
,
0
❘
"\[LeftBracketingBar]"
𝒮
(
τ
1
,
ς
1
,
τ
2
,
ς
2
)
❘
"\[RightBracketingBar]"
≤
∈
,
and wherein, (τ1,1,τ2,2) represents a (τ1, 1, τ2, 2)th element of , the noise threshold ϵ is as follows:
ϵ=σ2√{square root over (2 log(|1∥2∥1∥2|))};
the thresholded kernel tensor dn is multiplied with the four singular matrices Y(1), Y(2), Y(3) and Y(4) to obtain a denoised sampling covariance tensor dn, which is expressed as follows:
dn=dn×1Y(1)×2Y(2)×3Y(3)×4Y(4);
(5) defining dimension sets 1={1,3} and 2={2,4}, and obtaining a fourth-order virtual domain signal ∈ by performing tensor transformation of dimension merging on the denoised sampling covariance tensor dn:
V
^
𝕎
=
Δ
𝒱
ˆ
dn
{
𝕁
1
,
𝕁
2
}
=
∑
k
=
1
K
(
1
T
⁢
s
k
T
⁢
s
k
*
)
[
a
𝕃
1
*
(
k
)
⊗
a
𝕃
1
(
k
)
]
∘
[
a
𝕃
2
(
k
)
⊗
a
𝕃
2
*
(
k
)
]
,
wherein, for
a
𝕃
1
*
(
k
)
⊗
a
𝕃
1
(
k
)
⁢
and
⁢
a
𝕃
2
(
k
)
⊗
a
𝕃
2
*
(
k
)
,
by forming a difference set array on exponent terms respectively, augmented virtual linear arrays on the x axis and on the y axis are constructed, ⊗ representing a Kronecker product;
corresponds to a two-dimensional non-continuous virtual cross array , contains a virtual uniform cross array =x∪y, where x and y are respectively virtual uniform linear arrays on the x axis and the y axis;
positions of all virtual array elements in x and y are respectively expressed as
𝕍
x
=
{
(
x
𝕍
,
0
)
⁢
❘
"\[LeftBracketingBar]"
x
𝕍
=
[
q
𝕍
x
(
1
)
,
q
𝕍
x
(
2
)
,
…
,
q
𝕍
x
(
|
𝕍
x
|
)
]
⁢
d
}
⁢
and
𝕍
y
=
{
(
0
,
y
𝕍
)
⁢
❘
"\[LeftBracketingBar]"
y
𝕍
=
[
q
𝕍
y
(
1
)
,
q
𝕍
y
(
2
)
,
…
,
q
𝕍
y
(
|
𝕍
y
|
)
]
⁢
d
}
,
where
q
𝕍
x
(
1
)
=
-
M
𝕃
1
⁢
N
𝕃
1
-
M
𝕃
1
+
2
,
q
𝕍
x
(
|
𝕍
x
|
)
=
M
𝕃
1
⁢
N
𝕃
1
+
M
𝕃
1
,
q
𝕍
y
(
1
)
=
-
M
𝕃
2
⁢
N
𝕃
2
-
M
𝕃
2
+
2
,
q
𝕍
y
(
❘
"\[LeftBracketingBar]"
𝕍
y
❘
"\[RightBracketingBar]"
)
=
M
𝕃
2
⁢
N
𝕃
2
+
M
𝕃
2
,
and
⁢
❘
"\[LeftBracketingBar]"
𝕍
x
❘
"\[RightBracketingBar]"
=
2
⁢
(
M
𝕃
1
⁢
N
𝕃
1
+
M
𝕃
1
)
-
1
,
❘
"\[LeftBracketingBar]"
𝕍
y
❘
"\[RightBracketingBar]"
=
2
⁢
(
M
𝕃
2
⁢
N
𝕃
2
+
M
𝕃
2
)
-
1
;
elements corresponding to positions of all virtual array elements in the virtual uniform cross array are extracted from the virtual domain signal of a non-contiguous virtual cross array to obtain a fourth-order virtual domain signal ∈ corresponding to ;
(6) respectively extracting sub-arrays
ℚ
x
(
1
)
=
{
(
x
ℚ
(
1
)
,
0
)
⁢
❘
"\[LeftBracketingBar]"
x
ℚ
(
1
)
=
[
1
,
2
,
…
,
q
ℚ
x
(
|
ℚ
x
|
)
]
⁢
d
}
,
ℚ
y
(
1
)
=
{
(
0
,
y
ℚ
(
1
)
)
⁢
❘
"\[LeftBracketingBar]"
y
ℚ
(
1
)
=
[
1
,
2
,
…
,
q
ℚ
y
(
|
ℚ
y
|
)
]
⁢
d
}
from x and y as translation windows;
then, respectively translating the translation windows
ℚ
x
(
1
)
⁢
and
⁢
ℚ
y
(
1
)
along a negative semi-axis direction of the axis x and the axis y by a virtual array element interval d, to obtain Jx virtual uniform linear sub-arrays
ℚ
x
(
j
x
)
=
{
(
x
ℚ
(
j
x
)
,
0
)
⁢
❘
"\[LeftBracketingBar]"
x
ℚ
(
j
x
)
=
[
2
-
j
x
,
3
-
j
x
,
…
,
q
ℚ
x
(
|
ℚ
x
|
)
+
1
-
j
x
]
⁢
d
}
and Jy virtual uniform linear sub-arrays
ℚ
y
(
j
y
)
=
{
(
0
,
y
ℚ
(
j
y
)
)
⁢
❘
"\[LeftBracketingBar]"
y
ℚ
(
j
y
)
=
[
2
-
j
y
,
3
-
j
y
,
…
,
q
ℚ
y
(
|
ℚ
y
|
)
+
1
-
j
y
]
⁢
d
}
,
jx=1, 2, . . . , Jx, jy=1, 2, . . . , Jy, Jx=(|x|+1)/2, Jy=(|y|+1)/2, so that a virtual domain signal corresponding to a virtual uniform sub-array
ℚ
~
(
j
x
,
j
y
)
=
ℚ
x
(
j
x
)
⋃
ℚ
y
(
j
y
)
be expressed as
U
ℚ
~
(
j
x
,
j
y
)
∈
ℂ
J
x
×
J
y
;
fixing jy index, superimposing
U
ℚ
~
(
:
,
j
y
)
in a third dimension to obtain Jy three-dimensional virtual domain tensors, and then, superimposing the Jy three-dimensional virtual domain tensors in a fourth dimension to obtain a four-dimensional denoised structured virtual domain tensor ∈Jx×Jy×Jx×Jy, which is expressed as follows:
𝒰
~
=
∑
k
=
1
K
(
1
T
⁢
s
k
T
⁢
s
k
*
)
⁢
l
x
(
k
)
·
l
y
(
k
)
·
v
x
(
k
)
·
v
y
(
k
)
,
wherein
:
l
x
(
k
)
=
[
e
-
j
⁢
πμ
1
(
k
)
,
e
-
j
⁢
π2μ
1
(
k
)
,
…
,
e
-
j
⁢
π
⁢
q
ℚ
x
(
|
ℚ
x
❘
"\[RightBracketingBar]"
)
⁢
μ
1
(
k
)
]
T
,
l
y
(
k
)
=
[
e
-
j
⁢
πμ
2
(
k
)
,
e
-
j
⁢
π
⁢
2
⁢
μ
2
(
k
)
,
…
,
e
-
j
⁢
π
⁢
q
ℚ
y
(
|
ℚ
y
|
)
⁢
μ
2
(
k
)
]
T
,
are steering vectors of
ℚ
x
(
1
)
⁢
and
⁢
ℚ
y
(
1
)
,
respectively,
v
x
⁢
(
k
)
=
[
1
,
e
-
j
⁢
πμ
1
(
k
)
,
…
,
e
-
j
⁢
π
⁢
(
q
ℚ
x
(
|
ℚ
x
❘
"\[RightBracketingBar]"
)
-
1
)
⁢
μ
1
(
k
)
]
T
,
v
y
(
k
)
=
[
1
,
e
-
j
⁢
πμ
2
(
k
)
,
…
,
e
-
j
⁢
π
⁡
(
q
ℚ
y
(
|
ℚ
y
|
)
-
1
)
⁢
μ
2
(
k
)
]
T
,
are translation factors along the x axis and the y axis, respectively;
and
(7) performing tensor decomposition on the denoised structured virtual domain tensor by canonical polyadic decomposition (CPD) to obtain an estimated value of each spatial factor of , that is, {Îx(k),Îy(k),{circumflex over (v)}x(k),{circumflex over (v)}y(k)};
extracting parameters {circumflex over (μ)}1(k) and {circumflex over (μ)}2(k) from {Îx(k),Îy(k),{circumflex over (v)}x(k),{circumflex over (v)}y(k)}, and obtaining a closed-form solution of a two-dimensional direction of arrival estimation ({circumflex over (θ)}k, {circumflex over (φ)}k) according to a relationship between {μ1(k), μ2(k)} and a two-dimensional direction of arrival (θk, φk).

1,664 more elements

Method for estimating direction of arrival of sub-array partition type l-shaped coprime array based on fourth-order sampling covariance tensor denoising

Life of this patent

Method for estimating direction of arrival of sub-array partition type l-shaped coprime array based on fourth-order sampling covariance tensor denoising

Life of this patent

Abstract

Claims6