﻿ 基于自适应步长萤火虫-多重信号分类算法的低空目标波达方向估计
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 雷达学报  2015, Vol. 4 Issue (3): 309-316  DOI: 10.12000/JR14142 0

### 引用本文 [复制中英文]

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Zhou Hao, Hu Guo-ping, and Wang Yun. DOA estimation of low altitude target based on adaptive step glowworm swarm optimization-multiple signal classification algorithm.[J] Journal of Radars, 2015, 4(3): 309-316. DOI: 10.12000/JR14142.
[复制英文]

### 文章历史

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DOA Estimation of Low Altitude Target Based on Adaptive Step Glowworm Swarm Optimization-multiple Signal Classification Algorithm
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Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China
Abstract: The traditional MUltiple SIgnal Classification (MUSIC) algorithm requires significant computational effort and can not be employed for the Direction Of Arrival (DOA) estimation of targets in a low-altitude multipath environment. As such, a novel MUSIC approach is proposed on the basis of the algorithm of Adaptive Step Glowworm Swarm Optimization (ASGSO). The virtual spatial smoothing of the matrix formed by each snapshot is used to realize the decorrelation of the multipath signal and the establishment of a full-order correlation matrix. ASGSO optimizes the function and estimates the elevation of the target. The simulation results suggest that the proposed method can overcome the low altitude multipath effect and estimate the DOA of target readily and precisely without radar effective aperture loss.
Key words: MUSIC algorithm    De-correlation    Adaptive Step Glowworm Swarm Optimization (ASGSO)
1 引言

2 低空多径环境下阵列接收信号模型

 图 1 阵列接收信号模型 Fig.1 Model of the received array signal
 \begin{aligned} X(t) = & A(\varphi)S(t) + {n_1}(t)\\ = & {[ {{x_1}(t)} \quad {{x_2}(t)} \ \ \ldots \ \ {{x_N}(t)} ]^{\rm{T}}} \end{aligned} (1)
xi(t)表示第i个阵元接收到的信号[8]

 $A(\varphi) = [ {a({\varphi _{1d}})} \quad {a({\varphi_{1s}})} \ \ \cdots \ \ {a({\varphi _{Pd}})} \quad {a({\varphi _{Ps}})} ]$ (2)
$a({\varphi _{id}}) = {[ 1 \ \; {{{\rm{e}}^{{\rm{j}}\frac { \large{{2π d}} } {\large{\lambda} } \sin {\varphi _{\large{id}}}}}} \ \; \cdots \ \; {{{\rm{e}}^{{\rm{j}}(N - 1)\frac { \large{{2π d}} } {\large{\lambda} } \sin {\varphi _{\large{id}}}}}} ]^{\rm{T}}}$ ,φid为真实目标俯仰角，φis为目标镜像俯仰角，λ代表波长，d代表阵元间距，ais)同理。对于远场目标，若忽略天线阵列高度，近似认为 ${\varphi _{id}} = {\varphi _{is}}$ 。

 $S(t) = {[ {{s_{1d}}(t)} \quad {{s_{1s}}(t)} \quad \cdots \quad {{s_{Pd}}(t)} \quad {{s_{Ps}}(t)} ]^{\rm{T}}}$ (3)
sid(t)=si $\,{{\rm{e}}^{{\rm{j}}{\omega _i}t}}$ 表示第i个目标的直射信号，si为信号幅度，wi为信号频率。sis(t)=Γisid(t) ${{\rm{e}}^{{\rm{j}}{\beta _i}}}$ =ρisid(t) ${{\rm{e}}^{{\rm{j}}{\phi _i}}}$ 表示第i个目标的反射信号，Γii $\,{{\rm{e}}^{{\rm{j}}{\alpha _i}}}$ 代表镜面反射系数，ρi为镜面反射系数幅值，∅iiii为地面反射引起的相移，βi=(2π/λ)ΔRi代表目标直射、反射信号波程差引起的相位差。

n1(t)=[n1(t)  n2(t)  ···  nN(t)]T，假定n1(t),n2(t),···,nN(t)为互不相关的零均值高斯白噪声，且与P个目标信号互不相关。

3 完全解相干MUSIC算法 3.1 算法原理

 \begin{aligned} {y_2}(t) = & A(\varphi ){Λ ^*}S(t) + {n_2}(t)\\ = & {[x_2^ * (t),{x_1}(t),\cdots ,{x_{N - 1}}(t)]^{\rm{T}}} \end{aligned} (4)

 \begin{aligned} Λ = {\rm{diag}}\left( {{{\rm{e}}^{{\rm{j}}\frac {\large{{2π d}}} {\large{\lambda }} \sin ({\varphi _{1d \, }})}}} \quad {{{\rm{e}}^{{\rm{j}}\frac {\large{{2π d}}} {\large{\lambda }} \sin ({\varphi _{1s \, }})}}} \quad \cdots \right.\\ \left. {{{\rm{e}}^{{\rm{j}}\frac {\large{{2π d}}} {\large{\lambda }} \sin ({\varphi _{Pd \, }})}}} \quad {{{\rm{e}}^{{\rm{j}}\frac {\large{{2π d}}} {\large{\lambda }} \sin ({\varphi _{Ps \, }})}}} \right)\hspace{17pt} \end{aligned} (5)
 ${{\bf{n}}_{\rm{2}}}(t) = {[ {n_2^ * (t)} \quad {{n_1}(t)} \quad \cdots \quad {{n_{N - 1}}(t)} ]^{\rm{T}}}$ (6)

 \begin{aligned} {y_N}(t) & = A(\varphi ){({\Lambda ^*})^{N - 1}}S(t) + {n_N}(t)\\ & = {\left[{x_N^ * (t) \quad x_{N - 1}^ * (t) \quad \cdots \quad {x_1}(t)} \right]^{\rm{T}}} \end{aligned} (7)
 ${n_N}(t) = {[n_N^ * (t) \quad n_{N - 1}^ * (t) \quad \cdots \quad {n_1}(t)]^{\rm{T}}}$ (8)

y1(t),y2(t),···,yN(t)组成矩阵Y(t)，将n1(t),n2(t),···,nN(t)组成噪声矩阵N(t)=[n1(t)n2(t)  ···  nN(t)]T，则有：

 \begin{aligned} {Y}(t) = & [\begin{array}{*{20}{c}}\!\! {{{ \, y}_{\rm{1}}}(t)}& {{y_{\rm{2}}}(t)} & \cdots & {{y_N}(t) \, } \end{array}\!\!]\\ = & \left[\!\!\! {\begin{array}{*{20}{c}} {{x_1}(t)} & {x_2^ * (t)} & {x_3^ * (t)} & \cdots & {x_N^ * (t)}\\ {{x_2}(t)} & {{x_1}(t)} & {x_2^ * (t)} & \cdots & {x_{N - 1}^ * (t)}\\ {{x_3}(t)} & {{x_2}(t)} & {{x_1}(t)} & \cdots & {x_{N - 2}^ * (t)}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ { \, {x_N}(t)} & {{x_{N - 1}}(t)} & {{x_{N - 2}}(t)} & \cdots & {{x_1}(t)} \end{array}} \!\!\! \right]\\ = & A({\varphi })[\!\! \begin{array}{*{20}{c}} { \, S(t)} & {{{\Lambda}^{*}}S(t)} & \cdots & {{{({{\Lambda }^*})}^{N - 1}}S(t) \, } \end{array}\!\!] \\ & + N(t) \\ = & A({\varphi}){R_{s}}{A^{\rm{H}}}({\varphi }) + N(t) \end{aligned} (9)
${R_s} = {\rm{diag}}[ {{s_{1d}}(t)} \ {{s_{1s}}(t)} \ \cdots \ {{s_{Pd}}(t)} \ {{s_{Ps}}(t)} ],$ AH(φ)为A(φ)的共轭转置。令
 \begin{aligned} R & = E[x_1^*(t)Y(t)]\\ & = A(\varphi )E[x_1^*(t){R_s}]{A^{\rm{H}}}(\varphi ) + \sigma _{\rm{n}}^2I\\ & = A(\varphi ){R_s^{ \prime }}{A^{\rm{H}}}(\varphi ) + \sigma _{\rm{n}}^2I \end{aligned} (10)

IN阶单位阵，相关矩阵为：

 \begin{aligned} R_s^\prime & = {\rm{diag}}\left[{{{s_{1d}^\prime }}(t) \quad {{s_{1s}^\prime }}(t) \quad \cdots \quad {{s_{Pd}^\prime }}(t) \quad {{s_{Ps}^\prime }}(t)} \right]\\ & = {\left[\!\! {\begin{array}{*{20}{c}} {{{s_{1d}^\prime }}(t)} & 0 & 0 & 0 & 0 \\ 0 & {{{s_{1s}^\prime }}(t)} & 0 & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & {{{s_{Pd}^\prime }}(t)} & 0 \\ 0 & 0 & 0 & 0 & {{{s_{Ps}^\prime }}(t)} \end{array}} \!\! \right]_{2P \times 2P}} \end{aligned} (11)

 \begin{aligned} {{s_{id}^\prime }}(t) & = {\rm{E}}[{s_{id}}(t)x_1^ * (t)]\\ & = {\rm{E}}\left\{ {{s_{id}}(t) \left [\sum\limits_{j = 1}^P {s_{jd}^ * } (t) + \sum\limits_{j = 1}^P {s_{js}^ * } (t) + n_1^*(t) \right]} \right\}\\ & = {\rm{E}}[{s_{id}}(t)s_{id}^ * (t) + {s_{id}}(t)s_{is}^ * (t)]\\ & = \left( {1 + {\rho _i}{{\rm{e}}^{ - {\rm{j}}{\phi _i}}}} \right)\sigma _{{\rm{s}}\_i}^2 \end{aligned} (12)
 \begin{aligned} {{s_{is}^prime }}(t) & = {\rm{E}}[{s_{is}}(t)x_1^ * (t)]\\ & = {\rm{E}}\left\{ {{s_{is}}(t)\left[{\sum\limits_{j = 1}^P {s_{jd}^ * } (t) + \sum\limits_{j = 1}^P {s_{js}^ * } (t) + n_1^*(t)} \right]} \right\}\\ & = {\rm{E}}[{s_{is}}(t)s_{id}^ * (t) + {s_{is}}(t)s_{is}^ * (t)]\\ & = \left( {{\rho _i} + {{\rm{e}}^{{\rm{j}}{\phi _i}}}} \right){\rho _i}\sigma _{{\rm{s}}\_i}^2 \end{aligned} (13)

 \begin{aligned} {{R_{s}^\prime }} & \!\! = \!\! {\rm{diag}}\left( {{{s_{1d}^\prime }}(t) \quad {{s_{1s}^\prime }}(t) \quad \cdots \quad {{s_{Pd}^\prime }}(t) \quad {{s_{Ps}^\prime }}(t)} \right)\\ & \!\! = \!\! {\left[\!\! {\begin{array}{*{20}{c}} {(1 \! + \! {\rho _1}{{\rm{e}}^{ \! - \! {\rm{j}}{\phi _1}}})\sigma _{{\rm{s}}\_1}^2} & 0 & \cdots & 0 & 0 \\ 0 & {(\rho _1^2 \! + \! {\rho _1}{{\rm{e}}^{{\rm{j}}{\phi _1}}})\sigma _{{\rm{s}}\_1}^2} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & {(1 \! + \! {\rho \! \; \! _P} \,{{\rm{e}}^{ \! - \! {\rm{j}}{\phi _P}}})\sigma _{{\rm{s}}\_P}^2} & 0 \\ 0 & 0 & \cdots & 0 & {(\rho _P^2 \! + \! {{\rho} \! \; \! _{P}} \,{{\rm{e}}^{{\rm{j}}{\phi _P}}})\sigma _{{\rm{s}}\_P}^2} \end{array}} \!\! \right] \!\! } \end{aligned} (14)
3.2 算法适用性分析

4 自适应步长萤火虫算法

4.1 基本萤火虫算法

 图 2 GSO算法流程 Fig.2 Flow chart of GSO algorithm
4.2 自适应步长萤火虫算法

 $\bar d_i(t) = \frac{{\sum\limits_{j = 1}^m {{d_{ij}}(t)} }}{m} = \frac{{\sum\limits_{j = 1}^m {\left\| {{x_j}(t) - {x_i}(t)} \right\|} }}{m}$ (15)

4.3 参数选择问题

5 算法仿真

 图 3 解相干效果对比图 Fig.3 Comparison of de-correlation performance

 图 4 不同个体数下估计偏差 Fig.4 Estimate deviation under different individual numbers

 图 5 不同个体数下估计方差 Fig.5 Estimate variance under different individual numbers

 图 6 不同信噪比下的估计偏差 Fig.6 Estimate deviation under different SNRs

 图 7 不同信噪比下的估计方差 Fig.7 Estimate variance under different SNRs

MUSIC algorithm (s)

6 结束语

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