双基地MIMO雷达的未知相关噪声下DOD与DOA联合估计新方法
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本文主要探讨了在未知相关噪声环境下，双基地MIMO（Multiple-Input Multiple-Output）雷达的方向离去角（Direction-Of-Departure, DOD）和方向到达角（Direction-Of-Arrival, DOA）的联合估计问题。研究发表于2015年11月的IEEE Transactions on Vehicular Technology期刊，卷64，第11期。作者包括Hong Jiang（IEEE会员）、Jian-Kang Zhang（IEEE高级会员）和Kon Max Wong（IEEE院士）。
研究背景是考虑到雷达信号模型中，两个独立的子阵分别由M1和M2个传感器发送，而接收端也有两个独立子阵，每个子阵分别有N1和N2个传感器。这些子阵使用一组由M1和M2个正交波形组成的匹配矩阵进行脉冲压缩，以提高信号处理的效率和精度。
在实际应用中，遇到的主要挑战是噪声的未知性和空间相关性。传统的DOA和DOD估计方法可能在面对这类复杂环境时失效。因此，论文提出了一种创新的算法，该算法利用了未知噪声条件下的发射器和接收器子阵，并结合了共轭最小二乘（Canonical Correlation Decomposition, CCD）技术。CCD的优势在于它能够揭示两个变量之间的最大线性相关性，这对于噪声环境中的信号特征提取至关重要。
该算法利用了噪声的特性，尤其是其自相关性，来设计估计策略。通过分析噪声的联合协方差矩阵，研究人员开发出一种能够同时估计DOD和DOA的方法，同时保持了对噪声变化的鲁棒性。这种方法利用了信号的“位移不变性”特性，即信号在不同位置的观测值之间的关系是相同的，这对于提高估计的准确性和稳定性非常关键。
总结来说，这篇论文的核心贡献在于提出了一种有效的双基地MIMO雷达在未知相关噪声环境下的联合DOD和DOA估计方法，该方法依赖于先进的统计分析工具和信号处理技术，旨在提升雷达系统在复杂电磁环境中的性能。这对于现代雷达系统的稳健性和定位精度具有重要意义，特别是在需要高分辨率和抗干扰能力的应用领域。
JIANG et al.: JOINT DOD AND DOA ESTIMATION FOR BISTATIC MIMO RADAR 5115
two receiver subarrays, the received signal at the th snapshot
= 1, 2 ...Lfrom all the P targets is given by
X
()
1
X
()
2
=
A
r1
(φ
1
)
A
r2
(φ
2
)
B
()
A
T
t1
(θ
1
)A
T
t2
(θ
2
)
S
1
S
2
+
Z
()
1
Z
()
2
(2)
where X
()
i
is the N
i
× K received data matrix in
the ith receiver subarray at the th snapshot, B
()
=
diag(β
1
()
...β
P
()
) is the target matrix with β
p
()
being the
zero-mean Gaussian reflection coefficient of the pth target at
the th snapshot, and A
ti
(θ
i
) is the transmitter steering matrix
for the ith transmitter subarray formed by all the P directional
vectors, i.e., A
ti
(θ
i
)=[a
ti
(θ
i1
) a
ti
(θ
i2
) ···a
ti
(θ
iP
)]. Likewise,
A
ri
(φ
i
)=[a
ri
(φ
i1
) a
ri
(φ
i2
) ···a
ri
(φ
iP
)] is the receiver
steering matrix, and Z
i
is the zero-mean complex N
i
× K
(space–time) noise matrix in the ith subarray at the th
snapshot, the covariance of which is such that
E
Z
()
i
Z
()H
j
=
R
z
i
z
i
i = j
0 i = j
,i,j=1, 2(3)
i.e., the two receiver subarrays are sufficiently separated such
that the noise in the two subarrays are uncorrelated. In addition,
R
z
i
z
i
, i = 1, 2 are assumed arbitrary and unknown covariance
matrices.
The received signal in (2) can be written in a more compact
form such that
X
()
= A
r
(φ)B
()
A
T
t
(θ)S + Z
()
(4)
where X
()
=[X
()T
1
X
()T
2
]
T
, S =[S
T
1
S
T
2
]
T
, Z
()
=
[Z
()T
1
Z
()T
2
]
T
, A
t
(θ)=[A
t1
(θ
1
)
T
A
t2
(θ
2
)
T
]
T
, and
A
r
(φ)=[A
r1
(φ
1
)
T
A
r2
(φ
2
)
T
]
T
. We note that, since A
r
(φ)
is (N
1
+ N
2
) × P , A
t
(θ) is (M
1
+ M
2
) × P , and S is
(M
1
+ M
2
) × K, then X
()
is of dimension (N
1
+ N
2
) × K,
and so is Z
()
. From (3) above, we can see that the (N
1
+
N
2
) × (N
1
+ N
2
) covariance matrix of Z
()
is given by
R
zz
=E
Z
()
1
Z
()
1
H
Z
()
1
Z
()
2
H
Z
()
2
Z
()
1
H
Z
()
2
Z
()
2
H
=
R
z
1
z
1
0
0R
z
2
z
2
. (5)
Since the M
i
complex waveforms transmitted by the ith trans-
mitter subarray are mutually orthogonal, then we can also see
S
i
S
H
j
=
KI
M
i
i = j
0 i = j
,i,j= 1, 2. (6)
This yields SS
H
= KI
M
.
B. Pulse Compression, Vectorization, and
Noise Characteristics
The purpose of pulse compression is to allow the radar to
employ a longer pulse to have larger radiated energy while
achieving the range resolution of a shorter pulse [14]. This
is accomplished by having the received signal processed by
a matched filter so that the duration of the long pulse is
compressed. Thus, pulse compression can be carried out by
applying the matched filter matrix (1/
√
K)S
H
on the received
signal in (4) so that the data matrix at the th snapshot, after
pulse compression, is given by
Y
()
=
√
KA
r
(φ)B
()
A
T
t
(θ)+
1
√
K
Z
()
S
H
=
Y
()
11
Y
()
12
Y
()
21
Y
()
22
(7)
where Y
()
ij
=
√
KA
ri
(φ
i
)B
()
A
T
tj
(θ
j
)+(1/
√
K)Z
()
i
S
H
j
,
i, j = 1, 2 are the four submatrices of Y
()
.
Denoting the vectorization operator of a matrix [15] by
vec(·), a composite data vector can be constructed by vector-
izing the four submatrices Y
()
ij
,giving
η
()
=
⎡
⎢
⎢
⎢
⎣
vec(Y
()
11
)
vec(Y
()
12
)
vec(Y
()
21
)
vec(Y
()
22
)
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
A
11
(φ
1
, θ
1
)
A
12
(φ
1
, θ
2
)
A
21
(φ
2
, θ
1
)
A
22
(φ
2
, θ
2
)
⎤
⎥
⎥
⎦
b
()
+
⎡
⎢
⎢
⎢
⎣
ζ
()
11
ζ
()
12
ζ
()
21
ζ
()
22
⎤
⎥
⎥
⎥
⎦
(8)
where A
ij
is (N
i
M
j
×P ) such that A
ij
(φ
i
, θ
j
)=[a
tj
(θ
j1
) ⊗
a
ri
(φ
i1
) ···a
tj
(θ
jP
) ⊗ a
ri
(φ
iP
)],b
()
=
√
K[β
()
1
β
()
2
···β
()
P
]
T
,
and ζ
()
=[ζ
()T
11
ζ
()T
12
ζ
()T
21
ζ
()T
22
]
T
, with ζ
()
ij
=(1/
√
K)
vec(Z
()
i
S
H
j
).
Let us examine the statistical characteristics of the noise
vector at the ith receiver array mixed with the jth matched
filter signal. We note that this vectorized subarray noise can be
written as [15]
ζ
()
ij
=
1
√
K
vec
Z
()
i
S
H
j
=
1
√
K
S
∗
j
⊗ I
N
j
vec
Z
()
i
.
(9)
Therefore, using some basic properties of Kronecker products
[15], their autocovariance and cross-covariance matrices are
given by
R
ζ
ij
ζ
mn
=E
ζ
()
ij
ζ
()
H
mn
=
1
K
S
∗
j
⊗ I
N
j
× E
vec
Z
()
i
vec
H
Z
()
m
S
T
n
⊗ I
N
n
=
1
K
S
∗
j
⊗ I
N
j
(I
K
⊗ R
z
i
z
m
)
S
T
n
⊗ I
N
n
=
1
K
S
∗
j
⊗ R
z
i
z
m
S
T
n
⊗ I
N
n
=
1
K
S
∗
j
S
T
n
⊗ R
z
i
z
m
=
I
M
j
⊗ R
z
i
z
i
, if i = m, j = n
0, if i = m or j = n
(10)
where, in the last step, the results of (3) and (6) have been used.
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