互质阵列信号处理研究进展：波达方向估计与自适应波束成形

周成伟; 郑航; 顾宇杰; 王勇; 史治国

doi:10.12000/JR19068

互质阵列信号处理研究进展：波达方向估计与自适应波束成形

DOI: 10.12000/JR19068 CSTR: 32380.14.JR19068

周成伟^1,,
郑航^2,,
顾宇杰^3,,
王勇²,
史治国^2, ,

1.
浙江大学控制科学与工程学院杭州 310027
2.
浙江大学信息与电子工程学院杭州 310027
3.
天普大学电子与计算机工程系费城 19122

基金项目: 中国博士后科学基金(2018M642431, 2019T120515)，国家自然科学基金(61772467)，浙江省杰出青年科学基金(LR16F010002)，中央高校基本科研业务费专项资金(2019FZA5006)

详细信息

作者简介:
周成伟(1990–)，男，浙江临海人，博士，助理研究员。2018年6月在浙江大学信息与电子工程学院获得工学博士学位，现为浙江大学控制科学与工程学院博士后、助理研究员。研究方向为阵列信号处理、波达方向估计、波束成形。E-mail: zhouchw@zju.edu.cn

郑　航(1998–)，男，广东汕头人，浙江大学在读研究生。2019年于同济大学获得工学学士学位，现于浙江大学电子科学与技术专业攻读硕士学位。研究方向为阵列信号处理。E-mail: hangzheng@zju.edu.cn

顾宇杰(1980–)，男，江苏如东人，博士，副研究员。2008年在浙江大学信息与电子工程学系获得博士学位，现为美国天普大学电子与计算机工程系研究员。研究方向为统计与阵列信号处理，压缩感知，波形设计，自适应波束成形等。E-mail: guyujie@hotmail.com

王　勇(1974–)，男，河南郏县人，博士，副教授，2002年3月在浙江大学信息与电子工程学系取得博士学位。现为浙江大学信电学院副教授。研究方向为雷达信号识别技术，超宽带应用技术。E-mail: wangy@zju.edu.cn

史治国(1978–)，男，江苏扬州人，博士，教授。2006年3月在浙江大学信息与电子工程学系获得博士学位，现为浙江大学信息与电子工程学院教授、博士生导师。主要研究方向为信号处理、物联网、群智感知。E-mail: shizg@zju.edu.cn

通讯作者:
史治国　shizg@zju.edu.cn

责任主编：张小飞 Corresponding Editor: ZHANG Xiaofei
中图分类号: TN911.7
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出版历程
- 收稿日期: 2019-07-12
- 修回日期: 2019-10-11
- 网络出版日期: 2019-10-01

Research Progress on Coprime Array Signal Processing: Direction-of-Arrival Estimation and Adaptive Beamforming

ZHOU Chengwei^1
,,
ZHENG Hang^2
,,
GU Yujie^3
,,
WANG Yong²,
SHI Zhiguo^{2
, ,}

1.
College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China
2.
College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China
3.
Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA 19122, USA

Funds: The China Postdoctoral Science Foundation (2018M642431, 2019T120515), The National Natural Science Foundation of China (61772467), Zhejiang Provincial Natural Science Foundation of China (LR16F010002), The Fundamental Research Funds for Central Universities (2019FZA5006)

More Information

Corresponding author: SHI Zhiguo, shizg@zju.edu.cn

摘要

摘要: 阵列信号处理是雷达领域各类应用的核心技术之一。近年来，互质阵列的提出打破了传统方法受限于奈奎斯特采样速率这一瓶颈，其稀疏布设的阵列结构和互质欠采样的信号处理方式大幅降低了系统所需的软硬件开销，为当前不断提升的实际应用需求提供了理论基础和技术前提。鉴于其在自由度、分辨率及计算复杂度等方面的性能优势，互质阵列信号处理的理论和技术研究受到了国内外学者的广泛关注。该文分别从波达方向估计和自适应波束成形这两个阵列信号处理领域的基本问题出发，介绍了互质阵列信号处理方向的研究进展。在互质阵列波达方向估计方面，该文总结了互质子阵分解方法和虚拟阵列信号处理方法等两类典型技术路线，并以此为基础介绍了压缩感知和无网格化技术在低复杂度和超分辨估计等方面的最新研究工作。在互质阵列波束成形方面，该文剖析了其与互质阵列波达方向估计问题的区别与联系，并介绍了面向互质阵列的高效鲁棒自适应波束成形设计方法。该文旨在通过对互质阵列信号处理研究前沿的分类归纳和总结，探讨各类方法的优势和未来的研究方向，为其在雷达等领域的产业需求和实际应用提供理论和技术参考。
- 阵列信号处理 /
- 互质阵列 /
- 波达方向估计 /
- 自适应波束成形
Abstract: Array signal processing is an essential tool in broad radar applications. The coprime array has recently been proposed to overcome the bottleneck caused by the Nyquist spatial sampling rate. The coprime array, whose sparse structure and undersampling feature drastically decrease necessary computational and hardware cost, provides a theoretical foundation and technical basis for the increasing demands of its practical applications. Considering its superior performance in degrees-of-freedom, spatial resolution, and computational complexity, research on coprime array signal processing has attracted much attention. This paper reviews recent research progress on coprime array signal processing, which has focused on both the Direction-of-Arrival (DOA) estimation and adaptive beamforming. From the perspective of coprime array DOA estimation, this paper summarizes two typical approaches, namely the coprime subarray decomposition-based approach and the virtual array signal processing-based approach. Moreover, recent work on low-complexity and super-resolution DOA estimation via compressive sensing and gridless techniques is also introduced. From the perspective of coprime array adaptive beamforming, the differences and relationships between DOA estimation and beamforming in the framework of coprime array signal processing are discussed, and an efficient, robust, and adaptive beamformer design tailored for the coprime array is introduced. Advantages and the future directions of coprime array signal processing are discussed, along with the theoretical basis and a technical reference for practical radar applications.
- Array signal processing /
- Coprime array /
- Direction-of-Arrival (DOA) estimation /
- Adaptive beamforming

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1. 引言

在传统跟踪方法中，目标假设为点目标，只占据一个分辨单元，但随着传感器分辨率的提高，目标将占据多个分辨单元，称该类目标为扩展目标。检测前跟踪算法是低信杂噪比(Signal To Clutter Noise Ratio, SCNR)下对目标进行处理的一种多帧信号积累方法，它使用传感器原始量测数据，并通过多次扫描量测数据的积累来提高信噪比，而粒子滤波(Particle Filter, PF)方法非常适合处理这类问题。Salmond^[1]在2001年第1次明确提出PF-TBD的概念，并将该方法应用于凝视型光电传感器的单目标检测与跟踪处理中。2005年，Rutten^[2]提出了一种针对观测噪声为Rayleigh情形下的PF-TBD算法。TSOU Haiping^[3]提出了基于极大似然估计的扩展目标跟踪算法，研究了目标发生旋转时的跟踪问题。Gilholm K^[4]利用多假设Kalman滤波来实现扩展目标跟踪，并分析了扩展目标中具有空间分布的散射点。Franken D^[5]提出扩展目标模型为对称正定随机矩阵，并用Bayes方法分析和解决该问题。Ling Fan^[6]建立椭圆模型，并在状态向量中加入目标存在变量和目标形状参数。针对雷达扩展目标，吴兆平^[7]利用传感器获得的数据，确立更接近于实际的线性扩展模型，并推导出了扩展模型的似然比函数，同时，对原始的粒子滤波算法进行了优化。为实现隐身扩展目标的检测与跟踪，于洪波^[8,9]将目标强度和空间长度引入状态向量，利用粒子滤波实现目标的跟踪和目标长度的估计。

上述问题只考虑Gaussian噪声下的微弱目标检测问题，没有考虑有杂波的情况。例如，当雷达工作于下视状态时，经过空时自适应(Space-Time Adaptive Processing, STAP)技术抑制杂波后的量测数据除了受噪声影响外还受剩余杂波的影响，使得传统基于噪声假设的量测模型不准确。同时，随着雷达性能的改善，研究者发现地(海)杂波出现了很长的波动，其杂波的统计特性明显偏离高斯分布。当高分辨雷达处于低视角观测时，通过考察地(海)杂波发现^[10–12]，Weibull分布能较好地刻画实际环境中的杂波统计特性。

针对杂波环境下微弱扩展目标的检测与跟踪问题，本文以杆状物体作为研究对象，对距离和方位进行划分，并给出不同区域上含有Weibull杂波的传感器量测模型。利用扩散函数，在点目标的基础上，推导了扩展目标的似然函数以及粒子权重的计算。在状态向量中加入目标存在状态以及目标形状参数，利用粒子滤波算法分别实现扩展目标状态和形状参数的检测和估计。仿真实验表明，杂波环境下基于粒子滤波的微弱扩展目标检测前跟踪算法具有较好的稳定性。

2. 系统模型

2.1 目标状态模型

考虑x-y平面内运动的杆状目标，目标状态转移具有如下形式：

${x_{k + 1}} = f({x_k}) + {v_k}$

(1)

其中，目标的状态 ${x_k} = {\left[ {\begin{array}{*{20}{c}}{{x_k}}&{{{\dot x}_k}}&{{y_k}}&{{{\dot y}_k}}&{{I_k}}&{{l_k}}\end{array}} \right]^{\rm T}}$ , (x_k, y_k), $\left( {{{\dot x}_k},{{\dot y}_k}} \right)$ 分别表示目标中心的位置、速度，I_k, l_k分别表示扩散目标的强度和空间长度，v_k为过程噪声。用E_k对目标存在状态建模，且描述为2态Markov链，定义目标“新生”概率P_b $\buildrel \Delta \over =$ P $\left\{ {{E_k}\!=\! 1|{E_{k - 1}} \!=\! 0} \right\}$ 和“死亡”概率P_d $\buildrel \Delta \over =$ P{E_k=0|E_k–1=1}，其中，E_k=0表示目标不存在，E_k=1表示目标存在，则Markov转换概率矩阵为：

$\phi =\left[ \begin{matrix} 1-{{P}_{b}} &{{P}_{b}} \\ {{P}_{d}} &1-{{P}_{d}} \\ \end{matrix} \right]$

(2)

2.2 量测模型

考虑经过脉冲压缩处理后的雷达距离-多普勒量测数据。假设距离和方位分别包含N_r和N_a个单元，距离和方位的分辨率分别为D_r和D_a，用 $\varOmega$ $\ \ \buildrel \Delta \over = \ \$ $\left\{ {1,2, \cdots ,{N_{\rm r}}} \right\}$ 和K $\buildrel \Delta \over =$ $\left\{ {1,2, \cdots ,{N_{\rm a}}} \right\}$ 来表示距离和方位上的分辨单元集合。对于距离向^[13]，由于目标的扩展，目标回波所占距离单元表示如下：

$\begin{aligned}{\varOmega _{\rm T}} = & \left\{ {\left\lceil {\frac{{{r_k} - L\left( {{x_k}} \right)/2}}{{{\varDelta _{\rm{r}}}}}} \right\rceil , \cdots ,\left\lceil {\frac{{{r_k}}}{{{\varDelta _{\rm{r}}}}}} \right\rceil , \cdots ,} \right.\\& \left. {\left\lceil {\frac{{{r_k} + L\left( {{x_k}} \right)/2}}{{{\varDelta _{\rm{r}}}}}} \right\rceil } \right\}\end{aligned}$

(3)

其中， $\left\lceil X \right\rceil$ 表示向上求整， ${r_k} = \sqrt {x_k^2 + y_k^2}$ , $L\left( {{x_k}} \right)$ 表示目标距离向长度。

对于方位向，目标回波所占的方位单元表示如下：

$\begin{aligned}{K_T} = & \left\{ {\left\lceil {\frac{{{a_k} - V\left( {{x_k}} \right)/2}}{{{\varDelta _{\rm{r}}}}}} \right\rceil , \cdots ,\left\lceil {\frac{{{a_k}}}{{{\varDelta _{\rm{a}}}}}} \right\rceil , \cdots ,} \right.\\& \left. {\left\lceil {\frac{{{a_k} + V\left( {{x_k}} \right)/2}}{{{\varDelta _{\rm{a}}}}}} \right\rceil } \right\}\end{aligned}$

(4)

其中， ${a_k} = \arctan\left( {{y_k}/{x_k}} \right)$ , V(x_k)表示目标方位向长度。

假设雷达在相同扫描区域已经产生多帧距离-多普勒图像，z_k(i, j)表示k时刻量测数据，具体形式为：

${{z}_{k}}\left( i,j \right)=\left\{ \begin{array}{*{35}{l}} \begin{array}{*{35}{l}} {{v}_{k}}\left( i,j \right)+{{n}_{k}}\left( i,j \right),i\in \Omega ,j\in K, \\ {{E}_{k}}=0 \\ \end{array} \\ \begin{array}{*{35}{l}} {{h}_{k}}\left( i,j \right)+{{v}_{k}}\left( i,j \right)+{{n}_{k}}\left( i,j \right),i\in {{\Omega }_{T}},j\in {{K}_{T}}, \\ {{E}_{k}}=1 \\ \end{array} \\ \begin{array}{*{35}{l}} {{v}_{k}}\left( i,j \right)+{{n}_{k}}\left( i,j \right),i\in \Omega \backslash {{\Omega }_{T}},j\in K\backslash {{K}_{T}}, \\ {{E}_{k}}=1 \\ \end{array} \\ \end{array} \right.$

(6)

式中， $i = 1,2, \cdots ,{N_{\rm r}},j = 1,2, \cdots ,{N_{\rm a}}$ , h_k(i, j)为目标在分辨单元(i, j)处的信号，n_k(i, j)表示在(i, j)处的杂波，v_k(i, j)为量测噪声，假设为零均值、方差为 $2{\sigma ^2}$ 的复高斯噪声。h_k(i, j)表示散射点对分辨单元(i, j)的强度影响，一般情况下h_k(i, j)可近似为2维高斯分布形式^[14,15]，则散射点在传感器分辨单元(i, j)的强度影响为^[7]：

$\begin{aligned}{h_k}\left( {i,j} \right) \approx & \frac{{{\varDelta _{\rm r}}{\varDelta _{\rm a}}{I_k}}}{{2 {π} {{{Σ} \;\!}^2}}}\\& \cdot \exp \left\{ { - \frac{{{{\left( {i{\varDelta _{\rm r}} - {x_k}} \right)}^2} + {{\left( {j{\varDelta _{\rm a}} - {y_k}} \right)}^2}}}{{2{{{Σ} \;\!}^2}}}} \right\}\end{aligned}$

(6)

其中 ${Σ}$ 为已知参数，表示传感器模糊斑点数量。k时刻的量测数据可表示为 ${z_k} = \{ {z_k}(i,j),i = 1, ·\!·\!· ,$ ${N_{\rm r}},j = 1, ·\!·\!· ,{N_{\rm a}}\}$ ，直到k时刻的完整量测数据集合表示为 ${Z_k} = \left\{ {{z_i},i = 1, ·\!·\!· ,k} \right\}$ 。

在实际雷达环境中，存在许多地物、海和云雨等各种杂波，经研究表明^[12]，Weibull分布在一定程度内可以与实际杂波吻合。因此，设 $\left| {n_k^{\left( {i,j} \right)}} \right|$ 符合以下Weibull分布，即

$\begin{aligned}p\left( x \right) = & \frac{p}{q}{\left( {\frac{x}{q}} \right)^{p - 1}}\exp \left[ { - {{\left( {\frac{x}{q}} \right)}^p}} \right],\\& x \ge 0,\ p > 0,\ q > 0\end{aligned}$

(7)

其中，p为形状参数，q为尺度参数。

3. 杂波环境下扩展目标的PF-TBD算法

设 ${z'\!\!_k}\left( {i,j} \right) = \left| {{h_k}\left( {i,j} \right) + {v_k}\left( {i,j} \right)} \right|$ 表示信号加噪声幅度，对于分辨单元(i, j)，当E_k=1时， ${z'\!\!_k}\left( {i,j} \right)$ 近似服从非中心的2阶卡方分布，即

$\begin{aligned} & \!\!\!\!\!\!\!\!\!\!\!\!\!\! p\left( {{{z'}\!\!_k}\left( {i,j} \right)|{x_k},{E_k} = 1} \right)\\ = & \frac{1}{{{\sigma ^2}}}\exp \left( { - \frac{{{{z'}\!\!_k}\left( {i,j} \right) + {h_k}\left( {i,j} \right)}}{{{\sigma ^2}}}} \right)\\& \cdot {I_0}\left( {\frac{{2\sqrt {{{z'}\!\!_k}\left( {i,j} \right){h_k}\left( {i,j} \right)} }}{{{\sigma ^2}}}} \right)\end{aligned}$

(8)

当E_k=0时，分辨单元(i, j)内信号的幅度近似服从瑞利分布^[13]，即

$p\left( {z}'{{}_{k}}\left( i,j \right)|{{E}_{k}}=0 \right)=\frac{\left| {z}'{{}_{k}}\left( i,j \right) \right|}{{{\sigma }^{2}}}\exp \left( -\frac{{{\left| {z}'{{}_{k}}\left( i,j \right) \right|}^{2}}}{2{{\sigma }^{2}}} \right)$

(9)

在目标存在时，考虑Weibull杂波因子，量测值z_k(i, j)的幅度服从非中心卡方分布与Weibull的混合分布，借用文献[16]的思想，根据混合概率密度公式

$f\left( y \right) = \rho {f_0}\left( {y,{\lambda _0}} \right) + \left( {1 - \rho } \right){f_1}\left( {y,\theta } \right),\ y > 0$

(10)

得到似然函数：

$\begin{aligned}& \!\!\!\!\!\!\!\!\!\!\!\! p\left( {\left| {{z_k}\left( {i,j} \right)} \right|\left| {{x_k},{E_k} = 1} \right.} \right)\\ = & \rho \frac{1}{{{\sigma ^2}}}\exp \left( { - \frac{{\left| {{z_k}\left( {i,j} \right)} \right| + {h_k}\left( {i,j} \right)}}{{{\sigma ^2}}}} \right)\\& \cdot {I_0}\left( {\frac{{2\sqrt {\left| {{z_k}\left( {i,j} \right)} \right|{h_k}\left( {i,j} \right)} }}{{{\sigma ^2}}}} \right)\\ & + \left( {1 - \rho } \right)\frac{p}{q}{\left( {\frac{{\left| {{z_k}\left( {i,j} \right)} \right|}}{q}} \right)^{{{p - 1}}}}\\ & \cdot \exp \left[ { - {{\left( {\frac{{\left| {{z_k}\left( {i,j} \right)} \right|}}{q}} \right)}^p}} \right]\end{aligned}$

(11)

同上，当E_k=0时，量测值z_k(i, j)的幅度服从瑞利分布与Weibull分布的混合分布，则相应的似然函数为：

$\begin{aligned}& \!\!\!\!\!\!\!\!\!\!\!\!\! p\left( {\left| {{z_k}\left( {i,j} \right)} \right|\left| {{E_k} = 0} \right.} \right)\\ = & \kappa \frac{{\left| {{z_k}\left( {i,j} \right)} \right|}}{{{\sigma ^2}}}\exp \left( { - \frac{{{{\left| {{z_k}\left( {i,j} \right)} \right|}^2}}}{{2{\sigma ^2}}}} \right)\\ & + \left( {1 - \varepsilon } \right)\frac{p}{q}{\left( {\frac{{\left| {{z_k}\left( {i,j} \right)} \right|}}{q}} \right)^{{{p - 1}}}}\\& \cdot \exp \left[ { - {{\left( {\frac{{\left| {{z_k}\left( {i,j} \right)} \right|}}{q}} \right)}^p}} \right]\end{aligned}$

(12)

式中， $\rho$ 和 $\varepsilon$ 为比例系数( $\rho \in \left[ {0,1} \right],\varepsilon \in \left[ {0,1} \right]$ )。

随着雷达分辨率的提高，目标的物理尺寸大于距离分辨率，这样就会引起目标在距离和方位上的线性扩展，则扩展目标的似然函数为^[17]：

$p\left( {{z_k}|{x_k},{E_k} = 1} \right) = \int {p\left( {{z_k}|\hat x,{E_k} = 1} \right)p\left( {{{\hat x}_k}|{x_k}} \right){\rm{d}}{{\hat x}_k}}$

(13)

其中，x_k为中心点， ${\hat x_k}$ 为扩展点； $p\left( {{{\hat x}_k}|{x_k}} \right)$ 表示从点x_k扩展到 ${\hat x_k}$ 的概率密度函数。

目标在距离和方位角发生2维线性扩展。设线性扩展目标长度为N_z，扩展目标的方向与观察者视线方向的夹角为 $\theta$ ，则目标扩展后的坐标范围^[7]为：

$\begin{aligned}& \left( {\left[ {{l_k} - 0.5{N_z}\tan \theta ,{l_k} + 0.5{N_z}\tan \theta } \right],} \right.\\& \quad\quad\ \left. {\left[ {{d_k} - 0.5{N_z}\cos \theta ,{d_k} + 0.5{N_z}\cos \theta } \right]} \right)\end{aligned}$

(14)

其中，(l_k, d_k)表示扩展目标的中心位置。根据以上条件，则其似然函数为：

$\begin{align} & p\left( {{z}_{k}}|{{x}_{k}},{{E}_{k}}=1 \right)=\int_{-{{n}_{z}}/2}^{{{n}_{z}}/2}{p\left( {{z}_{k}}|{{x}_{k}},{{E}_{k}}=1 \right)p\left( {{{\hat{x}}}_{k}}|{{x}_{k}} \right)\text{d}{{{\hat{x}}}_{k}}} \\ & =\frac{1}{{{n}_{z}}}\int_{-{{n}_{z}}/2}^{{{n}_{z}}/2}{\left( \rho \frac{1}{{{\sigma }^{2}}}\exp \left( -\frac{{{z}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right)+{{h}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right)}{{{\sigma }^{2}}} \right) \right.} \\ & \cdot {{I}_{0}}\left( \frac{2\sqrt{{{z}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right){{h}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right)}}{{{\sigma }^{2}}} \right) \\ & +\left( 1-\rho \right)\frac{p}{q}{{\left( \frac{{{z}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right)}{q} \right)}^{p-1}} \\ & \left. \cdot \exp \left[ -{{\left( \frac{{{z}_{k}}\left( i-u\tan \theta ,j-u\cos \theta \right)}{q} \right)}^{p}} \right] \right)\text{d}u \\ \end{align}$

假设各个(i, j)内的z_k^{(i, j)}是独立的，似然函数可表示为：

$\begin{aligned} & \!\!\!\!\!\!\!\!\!\!\!\! p\left( {\left| {{z_k}} \right||{x_k},{E_k} = 1} \right)\\ \approx & \prod\limits_{i \in {C_i}\left( {{{x}_k}} \right)} {\prod\limits_{j \in {C_{j}}\left( {{x_k}} \right)} p } \left( {\left| {{z_k}\left( {i,j} \right)} \right|{x_k},{E_k} = 1} \right)\\& {\; \cdot \prod\limits_{i \notin {C_{i}}\left( {{{x}_k}} \right)} {\prod\limits_{j \notin {C_{j}}\left( {{x_k}} \right)} p } \left( {\left| {{z_k}\left( {i,j} \right)} \right|{x_k},{E_k} = 0} \right)}\end{aligned}$

(16)

$p\left( {\left| {{z_k}} \right||{E_k} = 0} \right) = \mathop \prod \limits_{i = 1}^{{N_\rm{r}}} \mathop \prod \limits_{j = 1}^{{N_{\rm a}}} p\left( {\left| {{z_k}\left( {i,j} \right)} \right||{E_k} = 0} \right)$

(17)

式中，C_i(x_k)和C_j(x_k)表示(i, j)内受回波影响的集合。

假设该模型中扩展目标每一时刻仅有限个回波(散射点)时^[18]，设散射点个数为M时，则有：

$p\left( {{z_k}|{x_k},{E_k} = 1} \right) \approx \frac{1}{M}\sum\limits_{i = 1}^M {p\left( {{z_k}|x_k^{\left( {i} \right)}} \right)}$

(18)

其中，x_k⁽ⁱ⁾表示在某一时间k相互独立的散射点。

$\begin{aligned}& \!\!\!\!\!\!\!\!\!\!\!\!\! p\left( {{z_k}|{x_k},{E_k} = 1} \right)\\ = & \sum\limits_{r = 0}^{n - 1} {\left( {\rho \frac{1}{{{\sigma ^2}}}\exp \left( { - \frac{{{z_k}\left( {i - {t_{r}}\tan \theta ,j - {t_{r}}\cos \theta } \right) + {h_k}\left( {i - {t_{r}}\tan \theta ,j - {t_r}\cos \theta } \right)}}{{{\sigma ^2}}}} \right)} \right.} \\& \cdot {I_{\rm{0}}}\left( {\frac{{2\sqrt {{z_k}\left( {i - {t_r}\tan \theta ,j - {t_r}\cos \theta } \right){h_k}\left( {i - {t_r}\tan \theta ,j - {t_{\mathop{ r}\nolimits} }\cos \theta } \right)} }}{{{\sigma ^2}}}} \right)\\& + \left( {1 - \rho } \right)\frac{p}{q}{\left( {\frac{{{z_k}\left( {i - {t_{r}}\tan \theta ,j - {t_{r}}\cos \theta } \right)}}{q}} \right)^{p - 1}}\\& \left. { \cdot \exp \left[ { - {{\left( {\frac{{{z_k}\left( {i - {t_{r}}\tan \theta ,j - {t_{r}}\cos \theta } \right)}}{q}} \right)}^p}} \right]} \right)\left( {{x_{{r} + 1}} - {x_{r}}} \right)\end{aligned}$

(19)

定义似然比为：

$L\left( {\left| {{z_k}} \right||x_k^{n},{E_k} = 1} \right) = \frac{{p\left( {\left| {{z_k}} \right||x_k^n,{E_k} = 1} \right)}}{{p\left( {\left| {{z_{k}}} \right||{E_{k}} = 0} \right)}}$

(20)

$\!\!\!\!\!\! L\left( {\left| {{z_k}} \right||{E_k} = 0} \right) = \frac{{p\left( {\left| {{z_k}} \right||{E_k} = 0} \right)}}{{p\left( {\left| {{z_k}} \right||{E_k} = 0} \right)}} = 1$

(21)

因此，对状态为 $x_k^n$ 的目标，将式(8)、式(19)代入式(20)、式(21)中，可以计算出未归一化权重 $\tilde \omega _k^{n}$ 。

下面给出PF-TBD算法^[19]。首先，引入混合状态向量 ${y_k} = {\left[ {x_k^{\rm T} {E_k}} \right]^{\rm{T}}}$ 。设在k–1时刻的联合PDF为 $p\left( {{y_{k - 1}}|{Z_{k - 1}}} \right)$ 可由粒子集 $\left\{ {y_{k - 1}^n,\omega _{k - 1}^n} \right\}_{n = 1}^N$ 来描述，N为粒子数量。则该算法运算步骤如下所示：

$\left[ {\left\{ {y_k^n} \right\}_{n\rm{ = 1}}^N} \right] = {\rm PF} - {\rm TBD}\left[ {\left\{ {y_{k\rm{ - 1}}^n} \right\}_{n\rm{ = 1}}^N,{z_k}} \right]$

(22)

步骤1 计算目标的存在变量 $\left[ {\left\{ {E_k^n} \right\}_{n\rm{ = 1}}^N} \right]$ 目标存在变量转移 $\left[ {\left\{ {E_{{k - 1}}^{n}} \right\}_{n\rm{ = 1}}^N,\varPhi } \right]$ ；

步骤2 从X_kⁿ中抽取M个散射点：

$\left\{ {\chi _{k}^{{n,m}}} \right\}_{m\rm{ = 1}}^M \sim \psi \left( {{\chi _{k}}|X_k^n} \right)$

(23)

步骤3 FOR n=1:N

(1) 对“新生”(即 $E_{{k - 1}}^n = 0,E_k^n = 1$ )粒子采样 $x_k^n \sim {p_\rm{b}}\left( {{x_k}|{z_k}} \right)$ ；

(2) 对“存活”(即 $E_{k\rm{ - 1}}^n = 1,E_k^n = 1$ )粒子采样 $x_k^n \sim p\left( {{x_k}|x_{k\rm{ - 1}}^n} \right)$ ；

(3) 计算权重 $\tilde \omega _k^n$ ；

步骤4 用 $x_k^n \sim {p_\rm{b}}\left( {{x_k}{|}{z_k}} \right)$ 替换权重比较低的 $x_k^n \sim p\left( {{x_k}|x_{k\rm{ - 1}}^n} \right)$ ，并计算相应的权重；

步骤5 归一化粒子权重， $\left\{ \omega _{k}^{n}=\tilde{\omega }_{k}^{n}/\sum\limits_{n=1}^{N}{\tilde{\omega }_{k}^{n}} \right\}_{n=1}^{N}\text{ }$ ；

步骤6 粒子重采样， $\left[ {\left\{ {y_k^n,1/N} \right\}_{n\rm{ = 1}}^N} \right]$ =重采样 $\left[ {\left\{ {y_k^n,\omega _k^n} \right\}_{n\rm{ = 1}}^N} \right]$ 。

目标在k时刻后验PDF为P_k $\buildrel \wedge \over =$ P{E_k=1|Z_k}，因此

${\hat P_k} = \sum\limits_{n = 1}^N {{{E_k^n} / {N, \ \ 0 \le }}} \hat P \le 1$

(24)

估计目标状态为：

${\hat x_k} = \sum\limits_{n = 1}^N {\left( {x_k^nE_k^n} \right) \biggr/ \sum\limits_{n = 1}^N {E_k^n} }$

(25)

4. 仿真实验

本节将通过仿真实验来说明在杂波环境下基于粒子滤波的雷达扩展目标TBD算法的有效性。假设目标的运动方程为：

${{x}_k} = {F}{{x}_{k - 1}} + {{w}}_{k - 1}$

(26)

其中，

${F} = \left[ {\begin{array}{*{20}{c}}1&1&0&0&0&0\\0&1&0&0&0&0\\0&0&1&1&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\end{array}} \right]$

(27)

目标在x-y平面内做匀速直线运动，w_k–1为零均值Gaussian噪声，其协方差Q为：

${Q} = \left[ {\begin{array}{*{20}{c}}{\frac{{{q_1}}}{3}{T^\rm{3}}}&{\frac{{{q_1}}}{2}{T^2}}&0&0&0&0\\{\frac{{{q_1}}}{2}{T^2}}&{{q_1}T}&0&0&0&0\\0&0&{\frac{{{q_1}}}{3}{T^3}}&{\frac{{{q_1}}}{2}{T^2}}&0&0\\0&0&{\frac{{{q_1}}}{2}{T^2}}&{{q_1}T}&0&0\\0&0&0&0&{{q_2}T}&0\\0&0&0&0&0&{{q_3}T}\end{array}} \right]$

(28)

其中，q₁和q₂分别表示运动和强度的过程噪声方差。设置q₁=0.001, q₂=0.01, q₃=0.01设置T为1 s。其他相关参数设置为D_x=D_y=1, ${\Large\varSigma} = 0.9$ , ${N_\rm{r}} = {N_\rm{a}} = 20$ , ${\alpha _{m}} = 0.3$ ，观测噪声方差为 ${\sigma ^2} = 1$ 。取 ${{x}_\rm{1}} = {\left[ {\begin{array}{*{20}{c}}0&1&2&0&I\end{array}} \right]^\rm{T}}$ ，在整个跟踪过程中，共有32帧。其中，第7到25帧时E_k=1，共19帧，而杂波和噪声从第1帧到第32帧一直存在。信杂噪比计算公式为：

${\rm SCNR} = \frac{S}{{V + C}}$

(29)

其中，S为信号功率，C为杂波功率，V为噪声功率。

图1模拟了散射点模型。根据散射点状态可知，从状态点到扩散点满足均匀分布。在图中，为了清楚地体现扩散效果，围绕着中心点取了8个扩散点。

图 1 目标散射点

Figure 1. Target scatter point

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图2给出了扩展目标在x-y平面上的运动轨迹，其中x轴、y轴只是相对值。在整个过程中，目标做匀速直线运动。目标的初始位置为(10, 2)，设设杆长l为2，即占据2个分辨率单元。

图 2 扩展目标真实航迹图

Figure 2. Real track of extended target

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滤波参数设置为：目标的p_b和p_d都取为0.1，初始存在概率 ${\mu _1}$ =0.05，信号强度门限 $\gamma$ =0.5, v_max=1, I_min=5, I_max=50, p=2。

图3是不同SCNR下扩展目标的平均存在概率图，设置门限为0.5，即目标的平均概率超过0.5，认为能够检测到目标，否则相反。从整体上来看，SCNR=3 dB, 6 dB, 9 dB和12 dB均能够检测到目标存在。SCNR=12 dB能够准确检测出目标出现时刻和消失时刻。SCNR=9 dB在目标出现时有2个时刻延迟，但能够准确检测到目标消失时刻。SCNR=3 dB, 6 dB时在目标出现时有2个时刻延迟，而目标消失时刻则有1个延时。

图 3 目标的平均存在概率

Figure 3. Average existence probability of target

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从图4可以看出，在目标开始出现时，不同SCNR下，误差都比较大，但经过几个时刻后，误差趋于稳定，但仍有一些波动。比较这4种不同情况，SCNR=3 dB的误差最大，这是比较符合实际情况。

图 4 目标位置的RMSE

Figure 4. RMSE of target position

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图5是目标长度的随时间变化的RMSE，从图中可以看出，目标刚出现时，误差比较大，随着时间的积累，误差逐渐减少。比较不同SCNR下的误差值，可以得出，SCNR=12 dB时，误差最小，也比较平缓，SCNR=3 dB估计值不太理想，误差一直居高不小。SCNR=6 dB和SCNR=9 dB时，误差均可以接受。

图 5 目标长度的RMSE

Figure 5. RMSE of target length

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5. 总结

针对杂波环境下的雷达扩展目标检测前跟踪问题，采用了含有Weibull杂波分布的杆状量测模型，运用散射点函数，推导出扩展目标的似然函数和粒子权重。基于粒子滤波，抽样得到散射点集合。实验结果表明，Weibull杂波环境下PF-TBD算法具有较好的稳定性。

图 1 互质阵列结构示意图

Figure 1. Illustration of the coprime array structure

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图 2 互质子阵列的MUSIC空间谱相位模糊示意图

Figure 2. Phase ambiguity of the pair of coprime subarray in the MUSIC spatial spectrum

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图 3 波达方向估计性能对比

Figure 3. Comparison of DOA estimation performance

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图 4 互质阵列及其对应的各种虚拟域阵列结构示意图

Figure 4. Illustration of the coprime array and its corresponding virtual array structures

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图 5 单个随机信号源情况下的波达方向估计性能对比

Figure 5. Comparison of DOA estimation performance under the single random source scenario

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图 6 算法复杂度性能对比

Figure 6. Comparison of algorithm complexity

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图 7 观测方向存在随机误差情况下的输出性能对比

Figure 7. Comparison of output SINR performance under the random signal look direction mismatch scenario

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1. 引言
2. 系统模型
2.1 目标状态模型
2.2 量测模型
3. 杂波环境下扩展目标的PF-TBD算法
4. 仿真实验
5. 总结

互质阵列信号处理研究进展：波达方向估计与自适应波束成形

DOI: 10.12000/JR19068 CSTR: 32380.14.JR19068

通讯作者:
史治国　shizg@zju.edu.cn

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