﻿ 基于修正自适应匹配滤波器的机动目标检测方法
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 雷达学报  0, Vol. 0 Issue (0): 0-0  DOI: 10.12000/JR15105 0

### 引用本文 [复制中英文]

[复制中文]
Li Hai, Liu Xin-long, Zhou Meng, et al. Detection of Maneuvering Target Based on Modified AMF[J]. Journal of Radars, 0, 0(0): 0-0. DOI: 10.12000/JR15105.
[复制英文]

### 文章历史

, , ,
(中国民航大学天津市智能信号与图像处理重点实验室 天津 300300)
(空军预警学院黄陂士官学校 武汉 430019)

Detection of Maneuvering Target Based on Modified AMF
, , ,
(Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China)
(Huangpi NCO School of Air Force Early Warning Academy, Wuhan 430019, China)
Abstract: Owing to the Doppler frequency migration of the return signal of maneuvering targets and finite training samples, it is difficult to detect maneuvering targets by conventional Adaptive Matched Filter (AMF) detectors. To solve this problem, a new method is proposed. First, to minimize sample size impairments, the diagonal loading technique was adopted to decrease the degrees of freedom of the sample space. Second, the Doppler frequency migration was compensated by the estimated acceleration which was estimated by the cubic phase transform, so as to reduce the dimension of matched searching and degrade the heavy calculation load. Finally, accumulation detection was conducted. The simulation results suggest that the proposed method can efficiently detect maneuvering target in finite sample situations with simple computation and constant false alarm rate detection.
Key words: Maneuvering target detection     Adaptive matched filter     Cubic phase transform     Diagonal loading     Doppler frequency migration
1 引言

2 问题描述

 \begin{aligned} {x} = [{x_{11}} ,& {x_{12}} ,\cdots ,{x_{1{N_{\rm{a}}}}},{x_{21}},{x_{22}},\cdots ,{x_{2{N_{\rm{a}}}}},\cdots ,\\ & {x_{{N_{\rm{p}}}1}},{x_{{N_{\rm{p}}}2}},\cdots ,{x_{{N_{\rm{p}}}{N_{\rm{a}}}}}{]^{\rm{T}}} \end{aligned} (1)

 $\left\{ {\begin{array}{*{20}{l}} {{H_0}:x = c + n}\\ {{H_1}:x = \alpha s({\varpi _m},{\vartheta _n}){\rm{ + }}c + n} \end{array}} \right.$ (2)

 ${s}{\rm{(}} {\varpi _m},{\vartheta _n}{\rm{)}} = {b}{\rm{(}} {\varpi _m}{\rm{)}} \otimes {a}{\rm{(}}{\vartheta _n}{\rm{)}}$ (3)

 ${T_{{\rm{AM}}{{\rm{F}}_{mn}}}} = \frac{{{{\left| {{s^{\rm{H}}}({\varpi _m},{\rm{ }}{\vartheta _n}){{\hat R}^{ - 1}}x} \right|}^2}}}{{{s^{\rm{H}}}({\varpi _m},{\rm{ }}{\vartheta _n}){{\hat R}^{ - 1}}s({\varpi _m},{\rm{ }}{\vartheta _n})}}\mathop \gtrless \limits_{{H_0}}^{{H_1}} {\eta _{mn}}$ (4)

 ${f_{\rm{d}}} = \frac{{2v}}{\lambda } + \frac{{2ak}}{{\lambda {f_{\rm{r}}}}},\ k = 1,2,\cdots ,{N_{\rm{p}}}$ (5)

 $\begin{array}{l} b({\varpi _m}) = \left[ {1\;\;{{\rm{e}}^{{\rm{j}}\left( {2\pi \frac{{2v}}{{\lambda {f_{\rm{r}}}}} + \pi \frac{{2a}}{{\lambda f_{\rm{r}}^2}}} \right)\;\;\; \cdots }}} \right.\\ \quad \quad \quad \quad {\left. {{{\rm{e}}^{{\rm{j}}\left( {2\pi (K - 1)\frac{{2v}}{{\lambda {f_{\rm{r}}}}} + \pi {{(K - 1)}^2}\frac{{2a}}{{\lambda f_{\rm{r}}^2}}} \right)}}} \right]^{\rm{T}}} \end{array}$ (6)

 $\hat R{\rm{ = }}\frac{{\rm{1}}}{L}\sum\limits_{l = 1}^L {{x_l}x_l^{\rm{H}}}$ (7)

3 本文算法

3.1 多普勒走动补偿

 $s(k) = \alpha {{\rm{e}}^{{\rm{j}}2\pi \left[ {(k - 1) \cdot \frac{{2v}}{{\lambda {f_{\rm{r}}}}} + {{(k - 1)}^2} \cdot \frac{a}{{\lambda f_{\rm{r}}^2}}} \right]}},\;k = 1,2, \cdots ,{N_{\rm{p}}}$ (8)

 $\begin{array}{*{20}{l}} {g({k_0},l) = s({k_0} + l)s({k_0} - l)}\\ {\quad \quad \quad = {\alpha ^2} \cdot {{\rm{e}}^{{\rm{j}}2\pi \left[ {({k_0} - 1)\frac{{4v}}{{\lambda {f_{\rm{r}}}}} + {{({k_0} - 1)}^2}\frac{{2a}}{{\lambda f_{\rm{r}}^2}}} \right]}} \cdot {{\rm{e}}^{{\rm{j}}\frac{{4\pi a}}{{\lambda f_{\rm{r}}^2}}{l^2}}}}\\ {\quad \quad \quad = \tilde \alpha {{\rm{e}}^{{\rm{j}}\frac{{4\pi a}}{{\lambda f_{\rm{r}}^2}}{l^2}}}}\\ {\quad \quad \quad = \tilde \alpha {{\rm{e}}^{{\rm{j}}{\omega _0}{l^2}}}} \end{array}$ (9)

 ${\omega _0} = \frac{{4{π} a}}{{\lambda {f_{\rm{r}}^2}}}$ (10)

 \begin{aligned} {\rm{C}}{{\rm{P}}_{\rm{s}}}{\rm{(}}{k_0},\omega ) & = \sum\limits_{l = 0}^{{{{N_{\rm{p}}}} \mathord{\left/ {\vphantom {{{N_{\rm{p}}}} 2}} \right. } 2} - 1} {s({k_0} + l)s({k_0} - l){{\rm e}^{ - {\rm j}\omega {l^2}}}} \\ & = \sum\limits_{l = 0}^{{{{N_{\rm{p}}}} \mathord{\left/ {\vphantom {{{N_{\rm{p}}}} 2}} \right. } 2} - 1} {\tilde \alpha {{\rm e}^{{\rm j}{\omega _0}{l^2}}}{{\rm e}^{ - {\rm j}\omega {l^2}}}} \\ & = \tilde \alpha \sum\limits_{l = 0}^{{{{N_{\rm{p}}}} \mathord{\left/ {\vphantom {{{N_{\rm{p}}}} 2}} \right. } 2} - 1} {{{\rm e}^{{\rm j}({\omega _0} - \omega ){l^2}}}} \end{aligned} (11)

 ${\hat \omega _0} = \arg \mathop {\max }\limits_\omega \left| {{\rm{C}}{{\rm{P}}_{\rm{s}}}({k_0},\omega )} \right|$ (12)

 $\hat a = \frac{{\lambda {f_{\rm{r}}^2}{{\hat \omega }_0}}}{{4{π} }}$ (13)

 $\begin{array}{*{20}{l}} {{s_{\rm{c}}}(k) = s(k){{\rm{e}}^{ - {\rm{j}}2\pi \left[ {{{(k - 1)}^2} \cdot \frac{{\hat a}}{{\lambda f_{\rm{r}}^2}}} \right]}}}\\ {\quad \quad \; = \alpha {{\rm{e}}^{{\rm{j}}2\pi \left[ {(k - 1) \cdot \frac{{2v}}{{\lambda {f_{\rm{r}}}}} + {{(k - 1)}^2} \cdot \frac{{a - \hat a}}{{\lambda f_{\rm{r}}^2}}} \right]}},}\\ {\quad \quad \quad \;k = 1,2, \cdots ,{N_{\rm{p}}}} \end{array}$ (14)

 ${\rm{CAG}} = \frac{1}{{{N_{\rm{p}}}}}\left| {\sum\limits_{n = 0}^{{N_{\rm{p}}} - 1} {{{\rm{e}}^{{\rm{j}}2\pi n\frac{{{a_{\rm{e}}}}}{{\lambda f_{\rm{r}}^{\rm{2}}}}}}} } \right|$ (15)

3.2 DL-AMF检测

 $R = U\Lambda {U^{\rm{H}}} = \sum\limits_{n = 1}^N {{\lambda _n}{u_n}u_n^{\rm{H}}}$ (16)

 $\begin{array}{*{20}{l}} {{w_{{\rm{opt}}}} = \kappa {R^{ - 1}}s}\\ {\quad \quad = \tilde \kappa \left[ {s - \sum\limits_{n = 1}^N {\left( {\frac{{{\lambda _n} - {\lambda _{\min }}}}{{{\lambda _n}}}} \right)\left( {u_n^{\rm{H}}s} \right){u_n}} } \right]} \end{array}$ (17)

 $\mathop \lambda \nolimits_1 \ge \mathop \lambda \nolimits_2 \ge \cdots \mathop \lambda \nolimits_p \gg \mathop \lambda \nolimits_{p + 1} = \cdots \mathop \lambda \nolimits_N = \mathop \sigma \nolimits^2$ (18)

DL[18, 19]可有效改善$\hat{{R}}$小特征值估计不准确而增加NDoF问题。设对角加载量[18]为${{\delta }^{2}}=10{{\sigma }^{2}}$，则加载后的协方差矩阵${{\hat{{R}}}_{\text{d}}}$为：

 ${{\hat{{R}}}_{\text{d}}}=\hat{{R}}+{{\delta }^{2}}{{{I}}_{N}}$ (19)

 ${T_{{\rm{DL - AM}}{{\rm{F}}_{mn}}}} = \frac{{{{\left| {{s^{\rm{H}}}({\varpi _m},\;{\vartheta _n})\hat R_{\rm{d}}^{ - 1}x} \right|}^2}}}{{{s^{\rm{H}}}({\varpi _m},\;{\vartheta _n})\hat R_{\rm{d}}^{ - 1}s({\varpi _m},\;{\vartheta _n})}}$ (20)

DL-AMF的精确统计分布比较复杂，得到解析解比较困难，文献[30]推导了DL-AMF的渐进统计分布，记$T_{\text{DL-AM}{{\text{F}}_{mn}}}^{\text{ a}}$为${{T}_{\text{DL-AM}{{\text{F}}_{mn}}}}$的近似表示[30]，则

 $T_{{\rm{DL - AM}}{{\rm{F}}_{mn}}}^{{\rm{ a}}} = \frac{{{{\left| {{s^{\rm{H}}}{\rm{(}}{\varpi _m},\;{\vartheta _n}{\rm{)}}Px} \right|}^2}}}{{{s^{\rm{H}}}{\rm{(}}{\varpi _m},\;{\vartheta _n}{\rm{)}}Ps{\rm{(}}{\varpi _m},\;{\vartheta _n}{\rm{)}}}}$ (21)

 ${{\eta }_{\text{DL-AM}{{\text{F}}_{mn}}}}=-\ln ({{P}_{\text{FA}}})$ (22)

 \begin{aligned} r & = \mathop {\max }\limits_{m,n} \{ {T_{{\rm{DL - AM}}{{\rm{F}}_{mn}}}}\} \\ & = \mathop {\max }\limits_{m,n} \left\{ {\frac{{{{\left| {{{s}^{\rm{H}}}{\rm{(}} {\varpi _m},{\vartheta _n}{\rm{)}}\hat {R}_{\rm{d}}^{ - 1}{x}} \right|}^2}}}{{{{s}^{\rm{H}}}{\rm{(}} {\varpi _m},{\vartheta _n}{\rm{)}}\hat {R}_{\rm{d}}^{ - 1}{s}{\rm{(}} {\varpi _m},{\vartheta _n}{\rm{)}}}}} \right\}\mathop {\gtrless}\limits_{{H_0}}^{{H_1}} {\eta _{\rm{t}}} \end{aligned} (23)

 \begin{aligned} {P_{{\rm{FA}}}} = & \Pr \left\{ {\mathop {\max }\limits_{{\normalsize \scriptstyle1 \le m \le M} \atop \scriptstyle1 \le n \le N} {\Large (}T_{{\rm{DL - AM}}{{\rm{F}}_{mn}}}^{\rm{ \ a}} {\Large )} > {\eta _{\rm{t}}} {\Large |}{H_0}} \right\}\\ = & 1 - \Pr \Bigr\{ {T_{{\rm{DL - AM}}{{\rm{F}}_{11}}}^{\;{\rm{ \ a}}} \le {\eta _{\rm{t}}},T_{{\rm{DL - AM}}{{\rm{F}}_{12}}}^{\;{\rm{ \ a}}} \le {\eta _{\rm{t}}},} \\ & \left. { \cdots ,T_{{\rm{DL - AM}}{{\rm{F}}_{MN}}}^{\rm{\ a}} \le {\eta _{\rm{t}}} {\Large |}{H_0}} \right\}\\ = & 1 - \prod\limits_{n = 1}^N {\prod\limits_{m = 1}^M {\Pr \left\{ {T_{{\rm{DL - AM}}{{\rm{F}}_{mn}}}^{\rm{\ a}} \le {\eta _{\rm{t}}} {\Large |}{H_0}} \right\}} } \\ = & 1 - {(1 - {{\rm e}^{ - {\eta _{\rm{t}}}}})^{MN}} \end{aligned} (24)

 ${{\eta }_{\text{t}}}=-\ln {\Large [}1-{{(1-{{P}_{\text{FA}}})}^{1/MN}}{\Large]}$ (25)

3.3 算法流程图

 图 1 算法流程图 Fig. 1  Flow diagram of algorithm

4 仿真分析

4.1 加速度估计和多普勒走动补偿仿真与分析

 $\text{RMS}{{\text{E}}_{a}}=\sqrt{\frac{1}{M}\sum\limits_{i=1}^{M}{{{(\hat{a}-a)}^{2}}}}$ (26)

 图 2 RMSEa随样本数变化曲线 Fig. 2  RMSEa versus number of training sample

 图 3 CAG随加速度估计误差变化曲线 Fig. 3  CAG versus estimated error of acceleration
4.2 检测性能仿真与分析

 图 4 L=2p和L=2N时检测性能比较 Fig. 4  PD comparison when L=2p and L=2N

 图 5 信杂噪比损失比较 Fig. 5  Comparison of SCNR loss

 图 6 不同样本数时检测性能比较 Fig. 6  PD comparison under different number of training sample

 图 7 检测门限随CNR变化曲线 Fig. 7  Thresholds of the detector versus CNR
4.3 计算量比较

5 结论

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