In this section, we present the multidimensional parameter estimation algorithm in polarimetric MIMO radar using PARAFAC quadrilinear decomposition.
3.1 PARAFAC quadrilinear decomposition model and uniqueness theorem
The definition of PARAFAC quadrilinear decomposition model can be derivatively described from the trilinear decomposition model [19–21].
Definition 1. (Quadrilinear decomposition in tensor format) A quadrilinear decomposition of a four-order tensor is a decomposition of the type , where a
m
,b
m
,c
m
,d
m
are the m th columns of the matrices , and , respectively, and ∘ is the outer product, i.e., (a
m
∘ b
m
∘ c
m
∘ d
m
)
ijkl
= a
im
b
jm
c
km
d
lm
, for all values of the indices i,j,k and l.
Definition 2. (Quadrilinear decomposition in matrix format) Let , and be the four matrix representations of a four-way array. The quadrilinear decomposition of can be written as four equivalent matrices X
(1) = (A ◇ B ◇ C)D
T, X
(2) = (D ◇ A ◇ B)C
T, X
(3) = (C ◇ D ◇A)B
T and X
(4) = (B ◇ C ◇ D)A
T.
Under certain conditions, can be decomposed uniquely into matrices A, B, C and D. These conditions are based on the notion of Kruskal-rank [17, 18].
Definition 3. (Kruskal-rank or k-rank) [18] Consider a matrix . If rank (A) = r, then A contains a collection of r linearly independent columns. Moreover, if every k < M columns of A are linearly independent, but this does not hold for every k+1 columns, then A has k-rank k
A
= k. Note that k
A
< rank (A),∀ A.
Theorem 1. (Uniqueness of quadrilinear decomposition) [25] Consider that a matrix representation of is X = (A ◇ B ◇ C)D
T, where and are the four mode matrices of , and M denotes the common dimension. If
(13)
then A , B, C and D are unique up to permutation and scaling of columns, meaning that any other quadruple , and that gives rise to X is related to A , B , C , D via , , , where Π is a permutation matrix, Λ
1,Λ
2, Λ
3 and Λ
4 are diagonal scaling matrices satisfying Λ
1
Λ
2
Λ
3
Λ
4 = I
M
.
3.2 Parameter matrix estimation using QALS
In this subsection, QALS algorithm is applied to uniquely identify the four parameter matrices A
t
(θ
t
,ϕ
t
), A
r
(θ
r
,ϕ
r
), G(θ
r
,ϕ
r
,γ,η) and B(f
d
) for multidimensional parameter estimation. We write the observed matrix Y in (12) as follows:
(14)
with .
Based on definition 2, it is clear that (14) corresponds to one of the matrix representations of quadrilinear decomposition of a four-way array , with P denoting the common dimension. For clear analysis, we express the 2MN × L matrix Y as Y
(1). Thus, quadrilinear decomposition can obtain an equivalent L MN × 2 matrix representation as the following form:
(15)
an equivalent 2LM × N matrix representation
(16)
and an equivalent 2LN × M matrix representation
(17)
where Y
(1), Y
(2), Y
(3) and Y
(4) denote the slice sets of the four-way array along the snapshot, polarization, reception and transmission ways, respectively.
From definition 3 and theorem I in section 3.1, it is obvious that the k-ranks of A
t
, A
r
, G and B are , k
G
= min(2,P), k
B
= min(L,P), respectively. If multiple targets and enough snapshots are taken into account, i.e. L ≥ P ≥ 2, then the following inequality is always satisfied:
(18)
which means that the four-way array satisfies the Kruskal-rank Theorem. Thus, A
t
, A
r
, G and B can be recovered uniquely up to permutation and scaling ambiguity.
Based on the four matrix representations Y
(1), Y
(2), Y
(3) and Y
(4), the parameter matrices A
t
(θ
t
,ϕ
t
), A
r
(θ
r
,ϕ
r
), G(θ
r
,ϕ
r
,γ,η) and B(f
d
) can be estimated using quadrilinear alternating least square (QALS) to fit PARAFAC model. The basic idea behind QALS is to update one parameter matrix using the least squares algorithm, conditioned on previously obtained estimates for the remaining parameter matrices that define the decomposition. This process is repeated until convergence in the least squares fit. The detailed iterative algorithm is as follows:
-
i)
Initialize the estimation , , and with random matrices, denoting them as , , and . The iteration number is k = 1,2,3,⋯.
-
ii)
Given , and , obtain the least squares solution of the k th iteration of according to
(19)
where ∥∥
F
denote the Frobenius norm of its matrix argument.
-
iii)
Update the k th iteration solution based on , and , such that
(20)
-
iv)
Update the k th iteration solution based on , and , such that
(21)
-
v)
Substitute the estimated , and into
(22)
acquiring the least squares solution of the k th iteration .
-
vi)
Compute the error at the k th iteration. The algorithm has converged when , where ε is an error threshold.
3.3 Multidimensional parameter estimation
Upon obtaining the estimates of , , and , the following problem is to estimate multidimensional parameters of P targets from these parameter matrices. The algorithm is based on available Vandermonde structure [26] and Kronecker product structure in the matrix blocks, also uses the polarization-sensitive array processing technology [27].
3.3.1 2D-DOD estimation
The obtained matrix contains the estimated 2D-DOD. According to (6), the p th column of A
t
(θ
t
,ϕ
t
) for p = 1,⋯,P is the Kronecker product of and . Thus, A
t
(θ
t
,ϕ
t
) can be further expressed as
(23)
with . Equation 23 reveals that A
t
(θ
t
,ϕ
t
) is composed of M
1 blocks according to rows, each block being a M
2 × P matrix with Vandermonde structure. [ξ
t 1,⋯,ξ
tP
] is the ratio between the corresponding elements of the (m
1 + 1)th and M
1th blocks of A
t
(θ
t
,ϕ
t
) for m
1 = 1,⋯,M
1 - 1, and [ζ
t 1,⋯,ζ
tP
] is the ratio between the corresponding elements of the (m
2 + 1)th and M
2th rows in each block of A
t
(θ
t
,ϕ
t
), m
2 = 1,⋯,M
2 - 1. Therefore, calculating the average of M
1(M
2 - 1) ratios respectively yields
(24)
(25)
where [A]
i,j
denotes the (i,j) element of the matrix A, and and are the estimates of ξ
tp
and ζ
tp
, respectively. From (24) and (25), we have , and , where angle(a) denotes the phase of a complex number a and and are the estimates of θ
tp
and ϕ
tp
, respectively. Thus, the estimates of 2D-DOD can be calculated if given and . For and , we have
(26)
(27)
3.3.2 2D-DOA estimation
Similarly, the parameters of 2D-DOA can be attained from the matrix . The estimates of θ
rp
and ϕ
rp
are respectively calculated by
(28)
(29)
for and , where and are the estimates of ξ
rp
and ζ
rp
, respectively, which can be obtained by
(30)
(31)
3.3.3 Polarization parameter estimation
Upon obtaining and , the estimates of two polarization parameters γ
p
and η
p
can be calculated based on the polarization matrix . From (5), the p th column of G(θ
r
,ϕ
r
,γ,η) can be rewritten as
(32)
Define ς
p
as the ratio between the second and first elements of the polarization vector g(θ
rp
,ϕ
rp
,γ
p
,η
p
) in (32)
(33)
The estimate of ς
p
can be calculated by
(34)
where is the estimate of ς
p
. Rewriting ς
p
in (33) yields
(35)
Then we have
(36)
Since tanγ
p
≥ 0 for 0 ≤ γ
p
≤ π/2, and cos θ
rp
> 0 for 0 ≤ θ
rp
< π/2, the estimates of polarization parameters γ
p
and η
p
can be calculated by
(37)
(38)
3.3.4 Doppler frequency estimation
The estimate of Doppler frequency is derived from the matrix . Define , then
(39)
Since [ω
1,⋯,ω
P
] is the ratio between the (l + 1)th and l th rows of the matrix B(f
d
) for l = 1,⋯,L, perform the average of ratios yielding
(40)
where is the estimate of ω
p
. Therefore, the Doppler frequency can be estimated by
(41)
Upon the above analysis, multidimensional parameter θ
t
,ϕ
t
,θ
r
,ϕ
r
,γ,η and f
d
for the p th target can be effectively calculated, p = 1,⋯,P.
3.4 Discussion
Note that the proposed method can obtain unique parametric identification even when multiple targets have close 2D-DODs or close 2D-DOAs while they are diversely polarized. Specifically, we consider the situation that P targets have the same 2D-DOD or the same 2D-DOA, while the other parameters are different. In this case, the inequality in (18) becomes
(42)
From the Kruskal-rank Theorem in section 3.1, the sufficient condition is derived herein to guarantee that the quadrilinear decomposition model is unique. Therefore, with polarization diversity, the 2D-DODs and 2D-DOAs of multiple closely spaced targets in MIMO radar can be uniquely identified, and high-resolution estimation can be achieved. It is mentioned that, based on the signal model described in (14), the trilinear decomposition-based algorithm with TALS can also be applied to identify multidimensional parameters using three matrices A
t
, A
r
◇ G and B as the slice sets of three-way array . However, for trilinear decomposition, the Kruskal-rank condition can not be satisfied in the situation that P targets have the same 2D-DOD, since the inequality is needed to uniquely identify the three parameter matrices.
Besides, robust iterative fitting of multilinear approaches such as linear programming (LP) or weighted median filtering iteration [28] can also be applied here to further yield a better solution for robust estimation in non-Gaussian noise.
The following assumption is made in the article: (i) The antennas are assumed to be ideal, identical isotropic scatterers. In practical MIMO radar systems, the effect of mutual coupling and array self-calibration methods should be considered. (ii) The target amplitude is assumed to remain constant during snapshot collection (Swerling I model). The condition can be relaxed to Swerling II model [29] when Doppler shift is known. (iii) The number of targets is assumed to be known. In practical situation, the detection of the number of targets, such as the minimum description length (MDL) criterion, should be investigated to obtain the rank of the four-way array . (iv) The targets are assumed to be point sources. In practical MIMO radar implementation, distributed target models and methods can be further considered.