1 Introduction
2 Signal model
3 MSNWF-DOA estimation
3.1 Subarray signal formation
3.2 Recursion algorithm of MSNWF
Forward recursion for i = 1, 2, ⋯, D
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\( {\mathbf{t}}_i={\displaystyle {\sum}_{n=1}^{L-1}{d}_{i-1}^{*}(n){\mathbf{x}}_{i-1}(n)} \)
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t
i
= t
i
/‖t
i
‖
2
| |
\( {d}_i(n)={\mathbf{t}}_i^{\mathrm{H}}{\mathbf{x}}_{i-1}(n),\kern0.24em n=1,\cdots, L-1 \)
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x
i
(n) = x
i− 1(n) − d
i
(n)t
i
, n = 1, ⋯, L − 1 | |
ε
D
(n) = d
D
(n), n = 1, ⋯, L − 1 | |
Backward recursion for i = D − 1, ⋯ , 2, 1 | |
\( {\omega}_{i+1}={\displaystyle {\sum}_{n=1}^{L-1}{d}_i(n){\varepsilon}_{i+11}^{*}\left[n\right]/}{\displaystyle {\sum}_{n=1}^{L-1}{\left|{\varepsilon}_{i+1}(n)\right|}^2} \)
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ε
i
(n) = d
i
(n) − ω
i
+ 1
ε
i+ 1(n), n = 1, ⋯, L − 1 | |
Calculate the Wiener filter coefficient | |
\( {\mathbf{w}}^{(D)}={\displaystyle {\sum}_{i=1}^D{\left(-1\right)}^{i+1}\left\{{\displaystyle {\prod}_{l=1}^i}\left(-{\omega}_l\right)\right\}\times {\mathbf{t}}_i} \)
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3.3 MSNWF-DOA estimation system
Initialization | |
\( {\mathbf{t}}_{A(1)}={r}_k^{*}{\mathbf{y}}_A(n)/{\left\Vert {r}_k^{*}{\mathbf{y}}_A(n)\right\Vert}_2,\kern0.24em n=1,\cdots, \kern0.1em L-1 \)
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Forward recursion for i = 2, 3, ⋯, D
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\( {\mathbf{t}}_{A(i)}={\displaystyle {\sum}_{n=1}^{L-1}{d}_{A\left(i-1\right)}^{*}{\mathbf{x}}_{A\left(i-1\right)}(n)} \)
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t
A(i) = t
A(i)/‖t
A(i)‖
2
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\( {d}_{A(i)}(n)={\mathbf{t}}_{A(i)}^{\mathrm{H}}{\mathbf{x}}_{A\left(i-1\right)}(n),\kern0.24em n=1,\cdots, L-1 \)
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\( \begin{array}{c}\hfill {\mathbf{x}}_{A(i)}(n)={\mathbf{x}}_{A\left(i-1\right)}(n)-{d}_{A(i)}(n){\mathbf{t}}_{A(i)},\kern0.24em \hfill \\ {}\hfill \kern1.44em n=1,\cdots, \kern0.1em L-1\hfill \end{array} \)
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ε
A(D)(n) = d
A(D)(n), n = 1, ⋯, L − 1 | |
Backward recursion for i = D − 1, ⋯, 2 | |
\( {\omega}_{A\left(i+1\right)}={\displaystyle {\sum}_{n=1}^{L-1}{d}_{A(i)}(n){\varepsilon}_{\left(i+1\right)}^{*}(n)/}{\displaystyle {\sum}_{n=1}^{L-1}{\left|{\varepsilon}_{A\left(i+1\right)}(n)\right|}^2} \)
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\( \begin{array}{c}\hfill {\varepsilon}_{A(i)}(n)={d}_{A(i)}(n)-{\omega}_{A\left(i+1\right)}{\varepsilon}_{A\left(i+1\right)}(n),\hfill \\ {}\hfill \kern1.44em n=1,\cdots, \kern0.1em L-1\hfill \end{array} \)
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Calculate the Wiener filter coefficient | |
\( {\mathbf{w}}_k^{A(D)}={\displaystyle {\sum}_{i=1}^D{\left(-1\right)}^{i+1}}\left\{{\displaystyle {\prod}_{l=1}^i}\left(-{\omega}_{A(l)}\right)\right\}\times {\mathbf{t}}_{A(i)} \)
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Initialization | |
\( \begin{array}{c}\hfill {\mathbf{t}}_{B(1)}={\left({e}^{j{\phi}_k}{r}_k\right)}^{*}{\mathbf{y}}_B(n)/{\left\Vert {\left({e}^{j{\phi}_k}{r}_k\right)}^{*}{\mathbf{y}}_B(n)\right\Vert}_2,\kern0.24em \hfill \\ {}\hfill \kern1.44em n=1,\cdots, \kern0.1em L-1\hfill \end{array} \)
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Forward recursion for i = 2, ⋯, D
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\( {\mathbf{t}}_{B(i)}={\displaystyle {\sum}_{i=1}^{L-1}{d}_{B\left(i-1\right)}^{*}{\mathbf{x}}_{B\left(i-1\right)}(n)/}{\displaystyle {\sum}_{i=1}^{L-1}{\left\Vert {d}_{B\left(i-1\right)}^{*}{\mathbf{x}}_{B\left(i-1\right)}(n)\right\Vert}_2} \)
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t
B(i) = t
B(i)/‖t
B(i)‖2
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\( {d}_{B(i)}(n)={\mathbf{t}}_{B(i)}^{\mathrm{H}}{\mathbf{x}}_{B\left(i-1\right)}(n),\kern0.24em n=1,\cdots, L-1 \)
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x
B(i)(n) = x
B(i− 1)(n) − d
B(i)(n)t
B(i), n = 1, ⋯, L − 1 | |
ε
B(D)(n) = d
B(D)(n), n = 1, ⋯, L − 1 | |
Backward recursion for i = D − 1, ⋯, 2 | |
\( {\omega}_{B\left(i+1\right)}={\displaystyle {\sum}_{i=1}^{L-1}{d}_{B(i)}(n){\varepsilon}_{B\left(i+1\right)}^{*}(n)/}{\displaystyle {\sum}_{i=1}^{L-1}{\left|{\varepsilon}_{B\left(i+1\right)}(n)\right|}^2} \)
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ε
B(i)(n) = d
B(i)(n) ‐ ω
B(i+ 1)
ε
B(i+ 1)(n), n = 1, ⋯, L − 1 | |
Calculate the Wiener filter coefficient | |
\( {\mathbf{w}}_k^{B(D)}={\displaystyle {\sum}_{i=1}^D{\left(-1\right)}^{i+1}\left\{{\displaystyle {\prod}_{l=1}^i}\left(-{\omega}_{B(l)}\right)\right\}\times {\mathbf{t}}_{B(i)}} \)
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