Deep Network for SAR Target Recognition Based on Attribute Scattering Center Convolutional Kernel Modulation

LI Yi; DU Lan; ZHOU Ke’er; DU Yuang

doi:10.12000/JR24001

Volume 13 Issue 2

Apr. 2024

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Article Navigation > Journal of Radars > 2024 > 13(2): 443-456

Hu Jingqiu, Liu Falin, Zhou Chongbin, Li Bo, Wang Dongjin. CS-SAR Imaging Method Based on Inverse Omega-K Algorithm[J]. Journal of Radars, 2017, 6(1): 25-33. doi: 10.12000/JR16027

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LI Yi, DU Lan, ZHOU Ke’er, et al. Deep network for SAR target recognition based on attribute scattering center convolutional kernel modulation[J]. Journal of Radars, 2024, 13(2): 443–456. doi: 10.12000/JR24001

Hu Jingqiu, Liu Falin, Zhou Chongbin, Li Bo, Wang Dongjin. CS-SAR Imaging Method Based on Inverse Omega-K Algorithm[J]. Journal of Radars, 2017, 6(1): 25-33. doi: 10.12000/JR16027

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Deep Network for SAR Target Recognition Based on Attribute Scattering Center Convolutional Kernel Modulation

DOI: 10.12000/JR24001

National Key Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

Funds: The National Natural Science Foundation of China (U21B2039)

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Corresponding author: DU Lan, dulan@mail.xidian.edu.cn
Received Date: 2024-01-04
Rev Recd Date: 2024-03-13

Available Online: 2024-03-19

Publish Date: 2024-03-27

Abstract

Abstract

The feature extraction capability of Convolutional Neural Networks (CNNs) is related to the number of their parameters. Generally, using a large number of parameters leads to improved feature extraction capability of CNNs. However, a considerable amount of training data is required to effectively learn these parameters. In practical applications, Synthetic Aperture Radar (SAR) images available for model training are often limited. Reducing the number of parameters in a CNN can decrease the demand for training samples, but the feature expression ability of the CNN is simultaneously diminished, which affects its target recognition performance. To solve this problem, this paper proposes a deep network for SAR target recognition based on Attribute Scattering Center (ASC) convolutional kernel modulation. Given the electromagnetic scattering characteristics of SAR images, the proposed network extracts scattering structures and edge features that are more consistent with the characteristics of SAR targets by modulating a small number of CNN convolutional kernels using predefined ASC kernels with different orientations and lengths. This approach generates additional convolutional kernels, which can reduce the network parameters while ensuring feature extraction capability. In addition, the designed network uses ASC-modulated convolutional kernels at shallow layers to extract scattering structures and edge features that are more consistent with the characteristics of SAR images while utilizing CNN convolutional kernels at deeper layers to extract semantic features of SAR images. The proposed network focuses on the electromagnetic scattering characteristics of SAR targets and shows the feature extraction advantages of CNNs due to the simultaneous use of ASC-modulated and CNN convolutional kernels. Experiments based on the studied SAR images demonstrate that the proposed network can ensure excellent SAR target recognition performance while reducing the demand for training samples.
- Synthetic Aperture Radar (SAR),
- Target recognition,
- Convolutional Neural Network (CNN),
- Attribute Scattering Center (ASC),
- Convolutional kernel modulation

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1. 引言

由于脉冲雷达发射的大功率信号在城市环境下容易干扰电台等其他通讯设备，可以借助通信信号来探测目标。近年来出现的多载波调制(Multi-Carrier Modulation, MCM)技术^[1]引起了人们的关注，由此发展而来的正交频分复用(Orthogonal Frequency Division Multiplex, OFDM)^[2]技术采用多路正交子载波进行信号调制，具有频率分集和波形分集的潜力，在通信、雷达一体化的发展背景下有着重要的研究价值。

相位编码OFDM信号具有多普勒高分辨力，用于雷达动目标检测时可以更精确地估计目标速度。不同的相位编码信号具有不同的自相关特性。霍夫曼编码可以降低信号自相关函数的整体旁瓣水平，但其包络峰均比(Peak-to-Mean Envelope Power Ratio, PMEPR)明显增大^[3]。研究表明，barker码序列具有较好的综合性能^[4]，本文将以13位barker码作为相位编码序列。

最近10年，国内外在OFDM信号特性与波形设计上取得了较好的研究成果^[5–7]，因此OFDM雷达的信号处理问题受到了研究者的关注。张卫等人^[8]利用Keystone变换在信号子载波域、快时间域和慢时间域进行联合解耦合处理，解决了目标的跨距离-多普勒单元走动问题，进而可估计出匀速运动下的多目标参数信息，但计算量较大；Lellouch等人^[9]利用回波与发射信号的载频相位信息得到了点目标距离及径向速度估计，但要求目标在一个距离门内运动，即不发生越距离单元走动现象。除此以外，还需进一步研究OFDM雷达回波处理中面临的一些特殊问题，如速度补偿和多普勒解模糊。

在信号处理领域，最大似然估计是一种渐进有效估计量，但是对于多测量的非线性模型而言其计算量较大，不利于实际应用。为了提高计算效率，本文借鉴文献[10]中MIMO雷达信号处理的思路，结合OFDM信号多载波正交结构的特点，对信号进行通道分离，形成多通道信号。通过相关处理得到不同子载波上的距离像；利用Keystone变换进行速度补偿并解多普勒模糊，对同一载波的相同距离单元进行脉冲多普勒处理，得到每个子载波对应的多普勒频谱；进一步在子载波域作相参积累，得到距离-多普勒2维谱。通过谱峰搜索和CLEAN技术^[11]的运用，从中提取出峰值位置对应的时延和多普勒参数。将其作为初值，结合观测数据的似然函数，利用牛顿迭代法获得更精确的参数估计，本文称之为近似最大似然估计。近似最大似然估计量可构成复合假设检验的重要环节，提升检测器的目标检测性能。论文组织结构如下：第2节给出了相位编码OFDM信号的回波模型；第3节给出了目标距离和速度参数的最大似然估计模型；为了提高目标运动参数估计的运算效率，第4节提出一种基于通道分离的近似最大似然估计算法；第5节利用仿真实验验证了算法的性能；第6节是总结。

2. 相位编码OFDM信号模型

设雷达发射相位编码OFDM信号为：

${s_{\rm T}\, }(t) = {{\rm{e}}^{{\,\rm{j2{{π}} }}{f_{\!\rm c}}t}}\sum\limits_{n = 0}^{N - 1} {u(t - n{T_{\rm r}}} ){\rm{rect}}\left( \frac{{t - n{T_{\rm r}}}}{{{T_{\rm r}}}}\right)$

(1)

其中， ${\rm{rect}}(t) = \left\{ $\begin{array}{l}1,\;\; 0 \le t \le 1\\0,\;\; {\rm{else}}\end{array}$ \right.$ 表示窗函数， ${f\!_{\rm c}}$ 为雷达信号的工作频率，N是脉冲个数， ${T_{\rm r}}$ 是脉冲重复周期， $u(t)$ 为相位编码OFDM信号的复包络：

$u(t) = \sum\limits_{k = 0}^{K - 1} {\sum\limits_{m = 0}^{M - 1} {{a_{k,m}}} } {{\rm{e}}^{{\,\rm{j2{{π}} }}k\Delta ft}} \;{\rm{rect}}\left(\frac{{t - m{t_{\rm c}}}}{{{t_{\rm c}}}}\right)$

(2)

其中，K是载波个数，第k个子载波上的相位编码序列 ${a_{k,m}}$ 由M个码元组成， ${t_{\rm c}}$ 为码元宽度，B是信号总带宽，为了满足子载波间的正交性，子载波间隔 $\Delta f = B/K = 1/{t_{\rm c}}$ 。

对目标散射的回波进行下变频处理，可获得相应的OFDM基带信号：

${s_{\rm r}}(t) = {A_0}{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}{f\!_{\rm c}}t}}{s_{\rm T}\,}(t - \tau )$

(3)

其中， ${A_0}$ 为目标散射强度， $ \tau = \displaystyle\frac{{2(R - vt)}}{c} = $ $ {\tau _0} \!-\! \displaystyle\frac{{2vt}}{c}$ 是匀速运动目标对应于t 时刻的时延。 ${\tau _0}{\rm{ = }}\displaystyle\frac{{2R}}{c}$ 表示 $t = 0$ 时刻的目标距离时延， $v$ 是目标的径向速度。若满足条件 $ \displaystyle\frac{{2vt}}{c} \ll {t_{\rm c}}$ ，则式(3)可以简化为：

$\begin{align} {s_{\rm r}}(t) \approx & \sum\limits_{n = 0}^{N - 1} \sum\limits_{k = 0}^{K - 1} \sum\limits_{m = 0}^{M - 1} {{A_0}{a_{k,m}}{\rm{rect}}\left(\frac{{t - n{T_{\rm r}} - {\tau _0}}}{{{T_{\rm r}}}}\right)}\\ & \cdot {\rm{rect}}\left( \frac{{t - n{T_{\rm r}} - {\tau _0} - m{t_{\rm c}}}}{{{t_{\rm c}}}}\right){{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({f_{\rm c}} + k\Delta f){\tau _0}}} \\ &\cdot {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta ft}}\,{{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f_{\rm d}} + {f_{{\rm d}k}})t}}\,{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}k\Delta fn{T_{\rm r}}}} \end{align}$

(4)

其中，子载波间的多普勒频差 ${f_{{\rm d}k}} = \displaystyle\frac{{2vk}}{c}\Delta f$ 。记全时间 $t = n{T_{\rm r}} + \tilde t$ ，其中快时间 $\tilde t \in [0,{T_{\rm r}})$ ，则式(4)表示为：

$\begin{align} {s_{\rm r}}(\tilde t,n) =& \sum\limits_{k = 0}^{K - 1} \sum\limits_{m = 0}^{M - 1} {A_0}{a_{k,m}} {\rm{rect}}\left(\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}}}}\right)\\ & \cdot{\rm{rect}}\left( \frac{{\tilde t - {\tau _0} - m{t_{\rm c}}}}{{{t_{\rm c}}}}\right){{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({f_{\rm c}} + k\Delta f){\tau _0}}}\\ & \cdot {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta f\tilde t} \;{{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f_{\rm d}} + {f_{{\rm d}k}}) (\tilde t + n{T_{\rm r}})}} \end{align}$

(5)

其中，第2个指数项表示不同子载波的回波信号具有不同的频率偏移；由第3个指数项可以看出，子载波分别与慢时间和快时间相耦合。

在加性高斯白噪声背景下，观测信号可以表示为：

$y(\tilde t,n) = {s_{\rm r}}(\tilde t,n) + w(\tilde t,n)$

(6)

其中， ${\rm{E}}\left( {w(\tilde t,n)} \right) = 0$ , ${\rm{Var}}\left( {w(\tilde t,n)} \right) = {\sigma ^2}$ 。

3. 最大似然估计

针对式(6)在快时间域采样，采样时刻为 $\tilde t = m{t_{\rm c}} + p{T_{\rm s}}$ ，其中， ${T_{\rm s}} = \displaystyle\frac{1}{{K\Delta f}}$ , $p \in [0,K - 1]$ , $m \in [0,M - 1]$ 。由于 ${f_{{\rm d}k}}\tilde t \ll 1$ ，式(5)进一步化简

$\begin{align} {s_{\rm r}}(p,m,n) =& \sum\limits_{k = 0}^{K - 1} {A_0}{a_{k,m}}{\rm{rect}}\left(\frac{{m{t_{\rm c}} + p{T_{\rm s}} - {\tau _0}}}{{{T_{\rm r}}}}\right)\\ & \cdot {\rm{rect}}\left(\frac{{p{T_{\rm s}} - {\tau _0}}}{{{t_{\rm c}}}}\right){{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({f_{\rm c}} + k\Delta f){\tau _0}}} \\ & \cdot {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f\!_{\rm d}} + k\Delta f) (m{t_{\rm c}} + p{T_{\rm s}})}} {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f\!_{\rm d}} + {f\!_{{\rm d}k}})n{T_{\rm r}}}} \end{align}$

(7)

令 $A = {A_0}{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}{f_{\rm c}}{\tau _0}}}$ ，则采样信号为：

${{Y}} = A{{S}} + {{W}}$

(8)

其中， ${{S}} = {[{{{s}}_0},·\!·\!·\!,{{{s}}_{N - 1}}]^{\rm T} }$ , ${{Y}} = {[{{{y}}_0},·\!·\!·\!,{{{y}}_{N - 1}}]^{\rm T}}$ ${{{y}}_n} = {\left[y({\tilde t_0},n),·\!·\!·\!,\,y({\tilde t_{MK - 1}},n)\right]^{\rm T}\ }$ ${{{s}}_n} = \displaystyle\sum\limits_{k = 0}^{K - 1} {a_{k,m}} $ ${{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} (}}{f_{\rm d}} + {f_{{\rm d}k}})n{T_{\rm r}}}}$ 　　 $ \cdot{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}k\Delta f{\tau _0}}}\left[{{\rm{e}}^{{\rm{j}}2{\rm{{{π}} }}({f_{\rm d}} + k\Delta f){{\tilde t}_0}}}\right.,·\!·\!·\!, $ $\left.{{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f_{\rm d}} + k\Delta f){{\tilde t}_{MK - 1}}}}\right]^{\rm T}$

记待估参数向量 $ {u} = {[A,v,R]^{\rm T}}$ ，噪声的协方差矩阵R=σ²I，则观测数据Y对应的似然函数为：

$p({{Y}}\left| {{u}} \right.) = \prod\limits_{n = 0}^{N - 1} {\frac{1}{{{{\rm{{{π}} }}^{MK}}}}} {\left| {{R}} \right|^{ - 1}} {{\rm{e}}^{{\rm{ - }}{{({{{y}}_n} - A{{{s}}_n})}^{\rm H}}{{{R}}^{ - 1}}({{{y}}_n} - A{{{s}}_n})}}$

(9)

式(8)是关于A的条件线性模型，其最大似然估计为：

$\hat A = \sum\limits_{n = 0}^{N - 1} {{{s}}_n^{\rm H}{{{R}}^{ - 1}}{{{y}}_n}} {\Biggr/} \sum\limits_{n = 0}^{N - 1} {{{s}}_n^{\rm H}{{{R}}^{ - 1}}{{{s}}_n}}$

(10)

将其代入式(9)，则距离和速度参数的最大似然估计为：

$\begin{align} [\hat R,\hat v] =& \mathop {\arg \min }\limits_{[R,v]} \sum\limits_{n = 0}^{N - 1} {{{\left({{{y}}_n} - \hat A{{{s}}_n}\right)}^{\rm H}}} {{{R}}^{ - 1}}\left({{{y}}_n} - \hat A{{{s}}_n}\right)\\ =& \mathop {\arg \max }\limits_{[R,v]} {{{{\left| {\sum\limits_{n = 0}^{N - 1} {{{s}}_n^{\rm H}{{{R}}^{ - 1}}{{{y}}_n}} } \right|}^2}} \mathord{\left/ {\vphantom {{{{\left| {\sum\limits_{n = 0}^{N - 1} {{{s}}_n^H{{{R}}^{ - 1}}{{{y}}_n}} } \right|}^2}} {\sum\limits_{n = 0}^{N - 1} {{{s}}_n^{\rm H}{{{R}}^{ - 1}}{{{s}}_n}} }}} \right. } {\sum\limits_{n = 0}^{N - 1} {{{s}}_n^{\rm H}{{{R}}^{ - 1}}{{{s}}_n}} }}\\ = &\! {({\sigma ^2}KN)^{ - 1}}\! \mathop {\arg \max }\limits_{[R,v]} \Biggr| \!{\sum\limits_{n = 0}^{N - 1} {\sum\limits_{p = 0}^{K - 1} {\sum\limits_{m = 0}^{M - 1} {\sum\limits_{k = 0}^{K - 1}\! {y(p,m,n)} } } } } \\ & \cdot a_{k,m}^*{{\rm{e}}^{{\,\rm{j}}2{{π}}k\Delta f{\tau _0}}} {{\rm{e}}^{{\rm{ - j}}2{{π}}({f_{\rm d}} + k\Delta f) (m{t_{\rm c}} + p{T_{\rm s}})}} \\ & \cdot {{\rm{e}}^{{\rm{ - j}}2{{π}}({f_{\rm d}} + {f_{{\rm d}k}})n{T_{\rm r}}}} \Biggr|^{2} \end{align}$

(11)

针对该模型，通过由粗到精的网格搜索可以得到参数的估计值。记速度搜索范围为 $[ - {v_{\max }},{v_{\max }}]$ ，距离搜索范围为 $[0,{R_{\max }}]$ ，则搜索次数对应为 $U = {\rm{ceil}}\left( {2{v_{\max }}/\Delta v} \right)+ 1$ 和 $V = {\rm{ceil}}\left( {{R_{\max }}/\Delta R} \right)+ 1$ 。其中， ${\rm{ceil}}(x)$ 表示不小于x 的最小整数， $\Delta v$ 和 $\Delta R$ 表示搜索步长。算法计算复杂度为 $O\left( {UVKN{{\log }_2}(KN)} \right)$ 。

令 $\displaystyle\sum\nolimits_{k = 0}^{K - 1} {|{a_{k,m}}} {|^2} = 1$ ，由文献[12]可知，待估参数的Cramer-Rao下限为：

$\begin{align} {\rm{CRLB}}(v) =& {{{c^{\,2}}{\sigma ^2}N} \mathord{\left/ {\vphantom {{{c^2}{\sigma ^2}N} {\left( {32{{({{{π} }}{f_{\rm c}}{T_{\rm r}}{A_0})}^2}} \right.}}} \right.} {\left( {32{{({{{π} }}{f_{\rm c}}{T_{\rm r}}{A_0})}^2}} \right.}}\cdot \Bigr[{N^2}(N \!-\! 1)\\ & \left.\left.(2N \!-\! 1)/6 \!-\! {N^2}{{(N \!-\! 1)}^2}/4\right] \right)\quad\quad\quad\quad\ \end{align}$

(12)

${\rm{CRLB}}(R) = 3{c^2}{\sigma ^2}\Bigr/\left( {8{{({\rm{{{π}} }}\Delta f{A_0})}^2} N\left({K^2} - 1\right)} \right)$

(13)

4. 基于通道分离的近似最大似然估计

直接利用式(11)求最大似然估计时计算量较大，在此考虑提高运算效率的参数估计方法。式(11)中相位项 ${{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}{f_{{\rm d}k}}n{T_{\rm r}}}}$ 和 ${{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta f(m{t_{\rm c}} + p{T_{\rm s}})}}$ 的子载波k分别与慢时间和快时间相耦合。针对不同的子载波，多脉冲联合处理时目标将在距离单元和多普勒单元上走动。鉴于此，文献[8]利用Keystone变换在信号子载波域、快时间域和慢时间域进行联合解耦合处理，解决了目标的跨距离多普勒单元走动问题，计算量约为 $3{K^3}N{\log _2}(KN)$ 。事实上，式(11)中的相位项 ${{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta fp{T_{\rm s}}}}$ 体现了不同子载波的回波信号具有不同的频率偏移，因此可以将多载波正交结构的OFDM雷达信号进行分离，通过多通道接收的方式增大距离分辨单元，避免目标的跨距离单元走动和3维Keystone变换。当 $\left| {{f_{\rm d}} + {f_{{\rm d}k}}} \right| < {{\Delta f}}/{2}$ 时，对式(5)进行通道分离，即用参考信号 ${{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta f\tilde t}}$ 与回波混频并经过低通滤波处理，得到各子载波通道上的信号

$\begin{align} x(\tilde t,n,k) =& \sum\limits_{m = 0}^{M - 1} {A_0}{a_{k,m}}{\rm{rect}}\left(\frac{{\tilde t - {\tau _0}}}{{{T_r}}}\right)\\ & \cdot {\rm{rect}}\left( \frac{{\tilde t - {\tau _0} - m{t_c}}}{{{t_{\rm c}}}}\right) {{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({f_{\rm c}} + k\Delta f){\tau _0}}}\\ & \cdot {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}{f_{\rm d}}\tilde t}} {{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}({f_{\rm d}} + {f_{{\rm d}k}})n{T_{\rm r}}}} \end{align}$

(14)

其中，快时间域的采样时刻 $\tilde t = m{t_{\rm c}} + p{T_{\rm s}}$ 。

用相位编码信号作为参考对式(14)作相关处理，得到第i个距离单元上的信号

$\begin{align} {z_k}(\tilde t,n,i) = & {\sum\limits_{m,l = 0}^{M - 1} {{A_0}{a_{k,m}}} a_{k,l}^*{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}} - \left| {{\tau _0} - i{t_{\rm c}}} \right|}}} \right)}\\ & \cdot {{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{\tilde t - \left| {{\tau _0} - \left( {i + l - m} \right){t_{\rm c}}} \right|}}} \right)}\\ {} & { \cdot {{\rm{e}}^{{\rm{ - j}}2{{π}} ({f_{\rm c}} + k\Delta f){\tau _0}}}{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}\tilde t}}{{\rm{e}}^{{\,\rm{j}}2{{π}} ({f_{\rm d}} + {f_{{\rm d}k}})n{T_{\rm r}}}}} \end{align}$

(15)

此时，信号的距离分辨单元 $\Delta R =c/(2\Delta f)$ 相对较大，目标不易产生跨距离单元走动。当最大多普勒频差小于多普勒分辨单元，即 $(({2v})/{c})(K - 1)\Delta f < $ $ {1}/(N{T_{\rm r}})$ 时，子载波和慢时间之间的耦合可以忽略，本文称之为低速运动，否则，称之为高速运动。此时，可利用Keystone变换消除子载波k与慢时间n之间的耦合，即 $({f_{\rm d}} + {f_{{\rm d}k}})n = {f_{\rm d}}h$ ，实现多普勒维对齐。解耦合后的信号

$\begin{align} {{g_k}(\tilde t,h,i) = }&{{z_k}\left(\tilde t,\frac{{{f_{\rm c}}}}{{{f_{\rm c}} + k\Delta f}}h,i\right)}\\ \approx &{\sum\limits_{m,l = 0}^{M - 1} {{A_0}{a_{k,m}}} a_{k,l}^*{\,\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}} - \left| {{\tau _0} - i{t_{\rm c}}} \right|}}} \right)}\\ {}&{ \cdot {\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{\tilde t - \left| {{\tau _0} - \left( {i + l - m} \right){t_{\rm c}}} \right|}}} \right)}\\ {}&{ \cdot {{\rm{e}}^{{\rm{ - j}}2{{π}} ({f_{\rm c}} + k\Delta f){\tau _0}}}{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}\tilde t}}{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}h{T_{\rm r}}}}} \end{align}$

(16)

经Keystone变换后的信号在慢时间域可能出现多普勒速度模糊现象( ${f_{\rm d}} = {\tilde f_{\rm d}} + r/{T_{\rm r}}$ ，其中 $\left| {{{\tilde f}_{\rm d}}} \right| < 1/(2{T_{\rm r}})$ ，折叠因子 $r \in \{ - 8, - 7, ·\!·\!·\! ,7,8\} $ )，则式(16)改为：

$\begin{align} {{g_k}\left( {\tilde t,h,i} \right) \approx }&{\sum\limits_{m,l = 0}^{M - 1} {{A_0}{a_{k,m}}} a_{k,l}^*{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}} - \left| {{\tau _0} - i{t_{\rm c}}} \right|}}} \right)}\\ {}&{ \cdot {\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{\tilde t - \left| {{\tau _0} - \left( {i + l - m} \right){t_{\rm c}}} \right|}}} \right)}\\ {}&{ \cdot {{\rm{e}}^{{\rm{ - j}}2{{π}} ({f_{\rm c}} + k\Delta f){\tau _0}}}{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}\tilde t}}{{\rm{e}}^{{\rm{j}}2{{π}} {{\tilde f}_{\rm d}}h{T_{\rm r}}}}}\\ {}& \cdot {{\rm{e}}^{{\,\rm{j}}2{{π}} r \displaystyle\scriptsize\frac{{{f_{\rm c}}}}{{{f_{\rm c}} + k\Delta f}}h}} \end{align}$

(17)

经过折叠因子补偿后的信号

${f_k}(\tilde t,h,i) = {g_k}(\tilde t,h,i) {{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}r \displaystyle\scriptsize\frac{{{f_{\rm c}}}}{{{f_{\rm c}} + k\Delta f}}h}}$

(18)

对式(18)进行脉冲多普勒处理，即关于慢时间作相参积累，可得第k个子载波上的多普勒频谱

$\begin{align} {{F_k}(\tilde t,u,i) = }&{\sum\limits_{m,l = 0}^{M - 1} {{A_0}{a_{k,m}}} a_{k,l}^*{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}} - \left| {{\tau _0} - i{t_{\rm c}}} \right|}}} \right)}\\ &{ \cdot {\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{\tilde t - \left| {{\tau _0} - \left( {i + l - m} \right){t_{\rm c}}} \right|}}} \right)}\\ & \cdot {\frac{{{\rm{sin}}({{π}} s)}}{{{\rm{sin}}({{π}} s/N)}}{{\rm{e}}^{{\,\rm{j}}{{π}} \displaystyle\scriptsize\frac{{N - 1}}{N}s}}{{\rm{e}}^{{\rm{ - j}}2{{π}} ({f_{\rm c}} + k\Delta f){\tau _0}}}}\\ & \cdot{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}\tilde t}} \end{align}$

(19)

其中， $s = u - {\tilde f_{\rm d}}N{T_{\rm r}}$ , $u \in [0,N - 1]$ 。

进一步，关于k作相参积累可得距离-多普勒谱：

$\begin{align} {F(\tilde t,u,i) = }&{\sum\limits_{m,l = 0}^{M - 1} {{A_0}{a_{k,m}}} a_{k,l}^*{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{{T_{\rm r}} - \left| {{\tau _0} - i{t_{\rm c}}} \right|}}} \right)}\\ {}& \cdot {{\rm{rect}}\left( {\frac{{\tilde t - {\tau _0}}}{{\tilde t - \left| {{\tau _0} - \left( {i + l - m} \right){t_{\rm c}}} \right|}}} \right)}\\ &\cdot {\frac{{{\rm{sin}}({{π}} s)}}{{{\rm{sin}}({{π}} s/N)}}\frac{{{\rm{sin}}({{π}} p)}}{{{\rm{sin}}({{π}} p/K)}}}\\ &{ \cdot {{\rm{e}}^{{\,\rm{j}}{{π}} \displaystyle\scriptsize\frac{{N - 1}}{N}s}}{{\rm{e}}^{{\,\rm{j}}{{π}} \displaystyle\scriptsize\frac{{K - 1}}{K}p}}{{\rm{e}}^{{\rm{ - j}}2{{π}} {f_{\rm c}}{\tau _0}}}{{\rm{e}}^{{\,\rm{j}}2{{π}} {f_{\rm d}}\tilde t}}} \end{align}$

(20)

其中， $p = q + K\Delta f{\tau _0}$ , $q \in [0,K - 1]$ 。

上述基于通道分离和Keystone相结合的方法是对式(11)模型的简化处理。为了进一步提高算法的估计精度，将得到的参数值作为初值，根据式(11)进行牛顿迭代，得到近似最大似然估计量。算法计算量约为 $KN{\log _2}(KN) \!+\! N{\log _2}N \!+\! 2K{\log _2}K \!+\! {K^2}N$ 。

上述模型均是以单目标为例进行讨论的。在多目标情况，可以利用CFAR检测器在距离-多普勒谱域进行谱峰搜索，根据最大峰值对应的参数估计值重构相应的子信号

$b(\tilde t,n) = \sum\limits_{k = 0}^{K - 1} {{{\rm{e}}^{{\,\rm{j}}2{\rm{{{π}} }}k\Delta f{{\hat \tau }_0}}}{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({{\hat f}_{\rm d}} + k\Delta f)\tilde t}}{{\rm{e}}^{{\rm{ - j}}2{\rm{{{π}} }}({{\hat f}_{\rm d}} + {{\hat f}_{{\rm d}k}})n{T_{\rm r}}}}}$

(21)

利用CLEAN技术从原信号中减去重构的子信号，然后对剩余信号继续上述的操作，直到CFAR检测不出峰值为止。算法处理流程如图1所示。

图 1 算法处理流程图

Figure 1. Algorithm processing flow chart

下载: 全尺寸图片幻灯片

5. 仿真实验

仿真参数设置如表1所示。根据表中数据并结合前文的分析可知，目标产生速度模糊的阈值是 ${v_{\rm una}}{\rm{ = }}15\;{\rm{m/s}}$ ，出现跨距离单元走动的速度临界值是 ${v_{\rm t}} \approx {\rm{4}}{\rm{.68}}\,{\rm{m/s}}$ ，速度分辨率 $\Delta {v_{\rm r}}=0.06\,{\rm{m/s}}$ ，且满足 $({{2v{T_{\rm r}}}})/{c} \ll {t_{\rm c}}$ 。由于 $\left| {{f_{\rm d}} + {f_{{\rm d}k}}} \right| \ll {{\Delta f}}/{2}$ ，多普勒频偏导致的子载频间串扰可忽略不计。在高斯白噪声背景下，2个目标距离为 ${R_1} = 10.0\;{\rm{km}} $ , $ {R_2} = 1.8\ {\rm{km}}$ , ${R_{\rm{3}}} \!\!=\!\! 17.0\;{\rm{km}}$ ，径向速度 ${v_1} \!\!=\!\! 20\;{\rm{m/s}} $ , ${v_2} \!=\! {\rm{24}}\;{\rm{m/s}} $ , ${v_{\rm{3}}} \!= $ 34 m/s，对应的目标散射强度分别为 ${A_{01}} = 5 $ , $ {A_{02}} = 2$ , ${A_{0{\rm{3}}}} = {\rm{1}}$ 。虚警率 ${P_{\rm fa}} \!=\! {10^{ - 3}}$ ，蒙特卡洛仿真100次。

表 1 仿真参数

Table 1. Simulation parameters

参数	数值
工作频率 ${f_{\rm c}}$ (GHz)	5
带宽B (MHz)	64
脉冲重复周期 ${T_{\rm r}}$ (ms)	2
脉冲数N	250
载波数K	64

下载: 导出CSV

| 显示表格

信号经过通道分离和相关处理后，在子载波-多普勒平面的投影如图2所示。由图2可以看出，目标的多普勒频移随子载波的变化而变化，因此不能直接进行子载波域的相参积累。

图 2 原始子载波-多普勒平面投影

Figure 2. The contour of the original subcarrier-Doppler plane

下载: 全尺寸图片幻灯片

经过Keystone变换和CLEAN处理后，相应子载波-多普勒平面的投影如图3所示。可见，多普勒频移与子载波之间的耦合得到校正。

图 3 校正后的子载波-多普勒平面投影

Figure 3. The contour of the corrected subcarrier-Doppler plane

下载: 全尺寸图片幻灯片

对Keystone变换的数据在子载波域进行相参积累，获得信号的距离-多普勒2维谱如图4所示。其中图4(a)是聚焦于第1个目标的结果，图4(b)是剔除前两个目标后聚焦于第3个目标的结果。结果表明，本文所述的补偿方法能在快时间域、慢时间域以及子载波域将目标的回波能量积累起来，有利于目标检测和后续的参数估计。

图 4 回波信号的距离-多普勒2维谱

Figure 4. Echo signal range-Doppler two-dimensional spectrum

下载: 全尺寸图片幻灯片

下面考察近似最大似然估计量的性能。当 $v = 20\;{\rm{m/s}}$ , ${\rm{SN}}{{\rm{R}}_i}$ 在0～20 dB之间变化时，分别利用文献[8]所述的联合Keystone变换法、通道分离与Keystone变换组合法以及本文所提的近似最大似然估计法进行速度估计，估计量的均方根误差(RMSE)曲线如图5(a)所示。图5(b)给出了各种算法对应的计算量随脉冲数的变化情况。

图 5 各算法的参数估计精度及计算量

Figure 5. Parameter estimation accuracy and calculation amount of each algorithm

下载: 全尺寸图片幻灯片

由图5(a)可知，文献[8](点划线)与基于通道分离和Keystone相结合的方法(实线)的速度估计的RMSE曲线变化趋势类似。在相同的估计精度下，与上述两种方法相比，本文基于通道分离的近似最大似然估计法(点线)的输入信噪比 ${\rm{SN}}{{\rm{R}}_i}$ 改善约4 dB。同时，本文算法得到的RMSE非常接近Cramer-Rao下限。由图5(b)可知，在相同脉冲数下，本文算法的计算量较文献[8]的算法大幅降低。由于使用了牛顿迭代法，与基于通道分离和Keystone相结合的方法相比，计算量有所增加，但仍然远小于文献[8]中算法的计算量。因此，本文提出的基于通道分离的近似最大似然估计算法在估计精度和计算复杂度上具有综合优势。

6. 总结

本文将OFDM通信信号应用在雷达动目标探测中，在通信、雷达一体化的发展背景下有着重要的应用前景。本文参考多载频MIMO雷达通道分离得到目标高分辨距离信息的方法，将OFDM信号的多载波正交结构与脉冲多普勒处理相结合，并借助Keystone变换解决了多普勒偏移问题。为了得到更好的估计精度，利用牛顿迭代法对似然函数进行优化，得到了基于通道分离的近似最大似然估计方法。仿真结果验证了算法的综合性能。今后还可针对机动目标相干化处理以及参数估计问题展开研究，扩展OFDM雷达的应用范围。

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