Direction of arrival estimation method and system for sparse array based on vandermonde decomposition reconstruction
Provided are a direction of arrival (DOA) estimation method and system for a sparse array based on Vandermonde decomposition reconstruction, relating to the technical field of array signal processing. The method includes: constructing a covariance matrix completion optimization model based on sparse array signals and uniform linear array signals, and performing Vandermonde decomposition by using characteristics of a uniform linear array; introducing a nuclear norm to optimize a rank function in the model, and updating the covariance matrix completion optimization model; and introducing an auxiliary variable to transform the model into a solvable optimization problem, and solving the problem by an alternating direction multiplier method to obtain an optimal estimation value. The DOA is estimated using a root multiple signal classification algorithm.
1 . A method for identifying signal sources based on direction of arrival (DOA) estimation for a sparse array based on Vandermonde decomposition reconstruction, which is implemented in a system for identifying signal sources comprising a processor and a memory storing instructions to be executed by the processor to implement the method, wherein, the method comprises:
acquiring sparse array signals by a sparse sensor array, wherein the sparse array signals are composed of a plurality of far-field narrow-band and uncorrelated signals entering the sparse sensor array from any directions;
acquiring uniform linear array signals, wherein the uniform linear array signals are signals received by a presumed uniform linear array, and the presumed uniform linear array is obtained by transforming the sparse sensor array through an interpolation method;
constructing a covariance matrix completion optimization model according to the sparse array signals and the uniform linear array signals, wherein the covariance matrix completion optimization model is a matrix completion model established by using characteristics of Vandermonde decomposition of a covariance matrix of the presumed uniform linear array;
wherein constructing the covariance matrix completion optimization model comprises:
computing a covariance matrix of the sparse array signals and a covariance matrix of the uniform linear array signals;
decomposing the covariance matrix of the presumed uniform linear array into a Vandermonde matrix and a corresponding coefficient vector by using the characteristics of the Vandermonde decomposition; and
constructing the covariance matrix completion optimization model according to the Vandermonde matrix and the coefficient vector;
wherein a formula expression of the covariance matrix completion optimization model is as follows:
min
U
,
u
∑
r
=
1
R
rank
(
ℛ
[
[
U
]
]
)
+
λ
Φ
(
u
)
Φ
H
-
R
^
S
F
2
subject
to
UU
H
=
(
u
)
wherein U= diag([σ 1 , . . . , σ L ]) is a | |×L-dimensional matrix, subject to is a feasible region for limiting an optimization variable, (u) is a covariance matrix and is a Hermitian-Toeplitz matrix, and a first column of the covariance matrix is u, ∥⋅∥ F represents Frobenius norm, is a Hankel matrix transformation operator, λ represents a wavelength of incident signals, R indicates a total number of columns, r is an r -th column, is a column-extraction operator, Φ represents a | |×| |-dimensional compression matrix only comprising 0 and 1, is a sample covariance matrix, H represents conjugate transpose, is a Vandermonde matrix, and σ is a square root of power;
introducing a nuclear norm for the covariance matrix completion optimization model to replace a rank function in the covariance matrix completion optimization model to obtain an updated covariance matrix completion optimization model;
introducing an auxiliary variable for the updated covariance matrix completion optimization model, and transforming the updated covariance matrix completion optimization model into an equivalent form to obtain a solvable optimization problem;
wherein introducing the auxiliary variable for the updated covariance matrix completion optimization model to perform equivalent form transformation on the updated covariance matrix completion optimization model to obtain the solvable optimization problem comprises:
a formula expression of the solvable optimization problem is as follows:
min
U
,
V
,
u
,
B
r
∑
r
=
1
R
B
r
*
+
λ
Φ
(
u
)
Φ
H
-
R
^
S
F
2
;
wherein subject to UV H = (u), U=V, [ [U]]=B r , ℄r∈{1, . . . , R};
solving the solvable optimization problem by an alternating direction multiplier method to obtain an optimal estimation value of the covariance matrix of the sparse array signals;
performing DOA estimation by using a root multiple signal classification algorithm according to the optimal estimation value of the covariance matrix of the sparse array signals; and
identifying signal sources based on the DOA estimation.
2 . The DOA estimation method for the sparse array based on Vandermonde decomposition reconstruction according to claim 1 , wherein a formula expression of the sparse array signals is as follows:
x
𝕊
(
t
)
=
∑
l
=
1
L
a
𝕊
(
θ
l
)
s
l
(
t
)
+
n
𝕊
(
t
)
=
A
𝕊
(
θ
)
s
(
t
)
+
n
𝕊
(
t
)
wherein Σ represents a summation symbol, s(t)=[s 1 (t), s 2 (t), . . . s L (t)] T represents an L-dimensional signal vector, 1∈(1, L); t represents sampling time; (⋅) T represents a transpose operation;
(t) represents a | |-dimensional independent and identically distributed complex-valued additive Gaussian noise vector;
A (θ)=[a (θ 1 ), a (θ 2 ), . . . a (θ L )] represents a | |×L-dimensional array manifold matrix; and a (θ 1 ) is a | |-dimensional steering vector with an angle of θ l .
3 . The DOA estimation method for the sparse array based on Vandermonde decomposition reconstruction according to claim 2 , wherein a formula expression of the uniform linear array signals is as follows:
x
𝕌
(
t
)
=
∑
l
=
1
L
a
𝕌
(
θ
l
)
s
l
(
t
)
+
n
𝕌
(
t
)
=
A
𝕌
(
θ
)
s
(
t
)
+
n
𝕌
(
t
)
wherein is a | -dimensional noise vector; (θ)=[ (θ 1 ), (θ 2 ), . . . , (θ L )] represents a | |×L-dimensional array manifold matrix; and (θ l ) represents a | |-dimensional steering vector with an angle of θ l .
4 . The DOA estimation method for the sparse array based on Vandermonde decomposition reconstruction according to claim 3 , wherein computing the covariance matrix of the sparse array signals comprises:
computing the covariance matrix of the sparse array signals according to a formula
R
𝕊
=
𝔼
{
x
𝕊
(
t
)
x
H
𝕊
(
t
)
}
=
A
𝕊
(
θ
)
PA
H
𝕊
(
θ
)
+
σ
2
n
I
wherein {⋅} represents mathematical expectation,
P
=
diag
(
[
σ
2
1
,
σ
2
2
,
…
,
σ
2
L
]
)
represents a signal covariance matrix, diag(⋅) represents a diagonal matrix generated by taking elements of one vector as diagonal elements;
σ
2
l
represents power of an l -th signal,
σ
2
n
is a noise term, and H represents conjugate transpose.
5 . The DOA estimation method for the sparse array based on Vandermonde decomposition reconstruction according to claim 4 , wherein computing the covariance matrix of the uniform linear array signals comprises:
computing the covariance matrix of the uniform linear array signals according to a formula
R
𝕌
=
𝔼
{
x
𝕌
(
t
)
x
H
𝕌
(
t
)
}
=
A
𝕌
(
θ
)
PA
H
𝕌
(
θ
)
+
σ
2
n
I
=
R
𝕌
s
+
σ
2
n
I
;
wherein represents a noise-free covariance matrix of .
6 . The DOA estimation method for the sparse array based on Vandermonde decomposition reconstruction according to claim 5 , wherein the root multiple signal classification algorithm is as follows:
P
MUSIC
(
θ
)
=
1
a
H
(
θ
)
U
N
U
N
H
a
(
θ
)
,
wherein U N represents a noise sub-space of (u), and a(θ) represents a steering vector.