迭代稀疏漸近最小方差算法 [ 1] 合成孔徑雷達 [ 2] [ 3]
SAMV算法的公式在DOA估計的背景下作為反問題 給出。假設
M
{\displaystyle M}
均勻線性陣列 （ULA）分別接收從位於
θ
=
{
θ
a
,
…
,
θ
K
}
{\displaystyle \mathbf {\theta } =\{\theta _{a},\ldots ,\theta _{K}\}}
K
{\displaystyle K}
N
{\displaystyle N}
M
×
1
{\displaystyle M\times 1}
y
(
n
)
=
A
x
(
n
)
+
e
(
n
)
,
n
=
1
,
…
,
N
{\displaystyle \mathbf {y} (n)=\mathbf {A} \mathbf {x} (n)+\mathbf {e} (n),n=1,\ldots ,N}
其中
A
=
[
a
(
θ
1
)
,
…
,
a
(
θ
K
)
]
{\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}
轉向矩陣 ,
x
(
n
)
=
[
x
1
(
n
)
,
…
,
x
K
(
n
)
]
T
{\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}
e
(
n
)
{\displaystyle {\bf {e}}(n)}
E
(
e
(
n
)
e
H
(
n
¯
)
)
=
σ
I
M
δ
n
,
n
¯
{\displaystyle \mathbf {E} \left({\bf {e}}(n){\bf {e}}^{H}({\bar {n}})\right)=\sigma {\bf {I}}_{M}\delta _{n,{\bar {n}}}}
δ
n
,
n
¯
{\displaystyle \delta _{n,{\bar {n}}}}
Dirac delta 函數 並且它僅等於1，唯一存在
n
=
n
¯
{\displaystyle n={\bar {n}}}
e
(
n
)
{\displaystyle {\bf {e}}(n)}
x
(
n
)
{\displaystyle {\bf {x}}(n)}
E
(
x
(
n
)
x
H
(
n
¯
)
)
=
P
δ
n
,
n
¯
{\displaystyle \mathbf {E} \left({\bf {x}}(n){\bf {x}}^{H}({\bar {n}})\right)={\bf {P}}\delta _{n,{\bar {n}}}}
P
=
Diag
(
p
1
,
…
,
p
K
)
{\displaystyle {\bf {P}}=\operatorname {Diag} ({p_{1},\ldots ,p_{K}})}
p
{\displaystyle {\bf {p}}}
p
=
[
p
1
,
…
,
p
K
,
σ
]
T
{\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}
y
(
n
)
{\displaystyle {\bf {y}}(n)}
p
{\displaystyle {\boldsymbol {\bf {p}}}}
R
=
A
P
A
H
+
σ
I
.
{\displaystyle {\bf {R}}={\bf {A}}{\bf {P}}{\bf {A}}^{H}+\sigma {\bf {I}}.}
該協方差矩陣可以通過樣本協方差矩陣進行傳統估計
R
N
=
Y
Y
H
/
N
{\displaystyle {\bf {R}}_{N}={\bf {Y}}{\bf {Y}}^{H}/N}
Y
=
[
y
(
1
)
,
…
,
y
(
N
)
]
{\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}
R
{\displaystyle {\bf {R}}}
r
(
p
)
=
vec
(
R
)
{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})}
p
{\displaystyle {\boldsymbol {\bf {p}}}}
r
(
p
)
=
vec
(
R
)
=
S
p
{\displaystyle {\bf {r}}({\boldsymbol {\bf {p}}})=\operatorname {vec} ({\bf {R}})={\bf {S}}{\boldsymbol {\bf {p}}}}
其中
S
=
[
S
1
,
a
¯
K
+
1
]
{\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}
S
1
=
[
a
¯
1
,
…
,
a
¯
K
]
{\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}
a
¯
k
=
a
k
∗
⊗
a
k
{\displaystyle {\bar {\bf {a}}}_{k}={\bf {a}}_{k}^{*}\otimes {\bf {a}}_{k}}
k
=
1
,
…
,
K
{\displaystyle k=1,\ldots ,K}
a
¯
K
+
1
=
vec
(
I
)
{\displaystyle {\bar {\bf {a}}}_{K+1}=\operatorname {vec} ({\bf {I}})}
要從統計的
r
N
{\displaystyle {\bf {r}}_{N}}
p
{\displaystyle {\boldsymbol {\bf {p}}}}
[ 1]
Cov
p
Alg
{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }}
p
{\displaystyle {\boldsymbol {p}}}
r
N
{\displaystyle {\bf {r}}_{N}}
Cov
p
Alg
≥
[
S
d
H
C
r
−
1
S
d
]
−
1
,
{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }\geq [{\bf {S}}_{d}^{H}{\bf {C}}_{r}^{-1}{\bf {S}}_{d}]^{-1},}
其中
S
d
=
d
r
(
p
)
/
d
p
{\displaystyle {\bf {S}}_{d}={\rm {d}}{\bf {r}}({\boldsymbol {p}})/{\rm {d}}{\boldsymbol {p}}}
p
^
{\displaystyle {\hat {\bf {p}}}}
p
^
=
arg
min
p
f
(
p
)
,
{\displaystyle {\hat {\boldsymbol {p}}}=\arg \min _{\boldsymbol {p}}f({\boldsymbol {p}}),}
其中
f
(
p
)
=
[
r
N
−
r
(
p
)
]
H
C
r
−
1
[
r
N
−
r
(
p
)
]
.
{\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].}
因此，可以迭代地獲
p
{\displaystyle {\boldsymbol {\bf {p}}}}
{
p
^
k
}
k
=
1
K
{\displaystyle \{{\hat {p}}_{k}\}_{k=1}^{K}}
f
(
p
)
{\displaystyle f({\boldsymbol {p}})}
σ
^
{\displaystyle {\hat {\sigma }}}
假設
p
^
k
(
i
)
{\displaystyle {\hat {p}}_{k}^{(i)}}
σ
^
(
i
)
{\displaystyle {\hat {\sigma }}^{(i)}}
i
{\displaystyle i}
(
i
+
1
)
{\displaystyle (i+1)}
p
^
k
(
i
+
1
)
=
a
k
H
R
−
1
(
i
)
R
N
R
−
1
(
i
)
a
k
(
a
k
H
R
−
1
(
i
)
a
k
)
2
+
p
^
k
(
i
)
−
1
a
k
H
R
−
1
(
i
)
a
k
,
k
=
1
,
…
,
K
{\displaystyle {\hat {p}}_{k}^{(i+1)}={\frac {{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {R}}_{N}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}{({\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k})^{2}}}+{\hat {p}}_{k}^{(i)}-{\frac {1}{{\bf {a}}_{k}^{H}{\bf {R}}^{-1{(i)}}{\bf {a}}_{k}}},\quad k=1,\ldots ,K}
σ
^
(
i
+
1
)
=
(
Tr
(
R
−
2
(
i
)
R
N
)
+
σ
^
(
i
)
Tr
(
R
−
2
(
i
)
)
−
Tr
(
R
−
1
(
i
)
)
)
/
Tr
(
R
−
2
(
i
)
)
,
{\displaystyle {\hat {\sigma }}^{(i+1)}=\left(\operatorname {Tr} ({\bf {R}}^{-2^{(i)}}{\bf {R}}_{N})+{\hat {\sigma }}^{(i)}\operatorname {Tr} ({\bf {R}}^{-2^{(i)}})-\operatorname {Tr} ({\bf {R}}^{-1^{(i)}})\right)/{\operatorname {Tr} {({\bf {R}}^{-2^{(i)}})}},}
其中
R
{\displaystyle {\bf {R}}}
i
{\displaystyle i}
R
(
i
)
=
A
P
(
i
)
A
H
+
σ
^
(
i
)
I
{\displaystyle {\bf {R}}^{(i)}={\bf {A}}{\bf {P}}^{(i)}{\bf {A}}^{H}+{\hat {\sigma }}^{(i)}{\bf {I}}}
P
(
i
)
=
Diag
(
p
^
1
(
i
)
,
…
,
p
^
K
(
i
)
)
{\displaystyle {\bf {P}}^{(i)}=\operatorname {Diag} ({\hat {p}}_{1}^{(i)},\ldots ,{\hat {p}}_{K}^{(i)})}
^ 1.0 1.1 1.2 1.3 Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim. Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing (PDF) . IEEE Transactions on Signal Processing. 2013, 61 (4): 933–944 [2018-09-14 ] . Bibcode:2013ITSP...61..933A ISSN 1053-587X arXiv:1802.03070 doi:10.1109/tsp.2012.2231676 存檔 (PDF) 於2021-02-24）. ^ Glentis, George-Othon; Zhao, Kexin; Jakobsson, Andreas; Abeida, Habti; Li, Jian. SAR imaging via efficient implementations of sparse ML approaches. Signal Processing. 2014, 95 : 15–26. doi:10.1016/j.sigpro.2013.08.003 ^ Yang, Xuemin; Li, Guangjun; Zheng, Zhi. DOA Estimation of Noncircular Signal Based on Sparse Representation. Wireless Personal Communications. 2015-02-03, 82 (4): 2363–2375. doi:10.1007/s11277-015-2352-z ^ Malioutov, D.; Cetin, M.; Willsky, A.S. A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Transactions on Signal Processing. 2005, 53 (8): 3010–3022. Bibcode:2005ITSP...53.3010M doi:10.1109/tsp.2005.850882 ^ 5.0 5.1 Yardibi, Tarik; Li, Jian; Stoica, Petre; Xue, Ming; Baggeroer, Arthur B. Source Localization and Sensing: A Nonparametric Iterative Adaptive Approach Based on Weighted Least Squares. IEEE Transactions on Aerospace and Electronic Systems. 2010, 46 (1): 425–443. Bibcode:2010ITAES..46..425Y doi:10.1109/taes.2010.5417172
![{\displaystyle \mathbf {A} =[\mathbf {a} (\theta _{1}),\ldots ,\mathbf {a} (\theta _{K})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fcbe78e2e522636e76b2f6c803b2e7d124990d)
![{\displaystyle {\bf {x}}(n)=[{\bf {x}}_{1}(n),\ldots ,{\bf {x}}_{K}(n)]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/643ebb9ed7c697b14fe3bc42d854740365f34ba0)
![{\displaystyle {\bf {p}}=[p_{1},\ldots ,p_{K},\sigma ]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f58e2b0d49c71656f0aeade621a9c868df1066)
![{\displaystyle {\bf {Y}}=[{\bf {y}}(1),\ldots ,{\bf {y}}(N)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88683322a32ac4a60ee717316b02ddc9f019c391)
![{\displaystyle {\bf {S}}=[{\bf {S}}_{1},{\bar {\bf {a}}}_{K+1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8700568a80a1b2e7834b6da983e743219d0bd91)
![{\displaystyle {\bf {S}}_{1}=[{\bar {\bf {a}}}_{1},\ldots ,{\bar {\bf {a}}}_{K}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/223d2224ef830e8dbc6fb307a39cb9f1a4c1e31f)
![{\displaystyle \operatorname {Cov} _{\boldsymbol {p}}^{\operatorname {Alg} }\geq [{\bf {S}}_{d}^{H}{\bf {C}}_{r}^{-1}{\bf {S}}_{d}]^{-1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95412087ea6bb7532ba838759115b5a5e1b0e45)
![{\displaystyle f({\boldsymbol {p}})=[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})]^{H}{\bf {C}}_{r}^{-1}[{\bf {r}}_{N}-{\bf {r}}({\boldsymbol {p}})].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15df0f7c10b4df3f894bdf0b7733ad03987e86ee)
. doi:10.1109/tsp.2012.2231676. （原始內容存檔 (PDF)於2021-02-24）.
