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A New Approach to High-order Range Cell Migration Correction for SAR Ground Moving Targets Based on Phase Tracking
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摘要: 距离徙动校正(RCMC)是合成孔径雷达(SAR)实现运动目标参数估计和聚焦成像的关键环节。当目标或平台运动复杂时，传统低阶RCMC方法将不再适用，而现有基于参数化的高阶RCMC方法易存在模型失配和计算复杂度高的问题、现有非参数化方法在低信噪比下性能也将大幅下降。对此，该文借助扩展卡尔曼滤波(EKF)对造成RCM的相位进行追踪，进而构建RCM补偿函数实现RCMC。所提方法不依赖于RCM的具体模型，追踪得到的相位包含高阶分量，因此可以实现SAR运动目标的高阶RCMC。此外，EKF在进行相位追踪的同时能对信号进行滤波处理，可有效降低所提方法的信噪比(SNR)门限。与传统方法相比，该方法适用性广，计算量适中，且能校正不可忽略的高阶残余距离徙动。该文详细阐释了所提方法的原理及数学模型，并通过多组仿真和实测数据处理验证了所提方法的有效性和优越性。Abstract: Range Cell Migration Correction (RCMC) represents an important advancement in the estimation of moving target parameters and imaging of targets in high-resolution Synthetic Aperture Radar (SAR) systems. When the motion of a target or platform becomes complex, the traditional low-order RCMC method may no longer be suitable. Meanwhile, the existing high-order RCMC method based on parameterization is susceptible to issues such as model mismatch and high computational complexity. Additionally, its performance may decrease significantly under a low Signal-to-Noise Ratio (SNR). This research utilizes Extended Kalman Filter (EKF) to track the phase responsible for RCM and develop a phase compensation function to achieve RCMC. The proposed approach is model-independent and can track high-order components in the phase, thereby enabling high-order RCMC of moving targets in SAR. In addition, EKF can filter signals during phase tracking to effectively lower the SNR threshold of the proposed method. Thus, this method offers broad applicability, moderate computational complexity, and the ability to correct non-negligible high-order residual range cell migrations, thereby distinguishing it from traditional methods. This study thoroughly explains the principles and mathematical model behind the proposed method, demonstrating its effectiveness and superiority through multiple sets of simulations and measured data processing.
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图 9 多频余弦耦合波动下的追踪结果
Figure 9. Phase tracking result under multifrequency cosine-coupled fluctuations
图 10 多频余弦耦合波动下点目标的处理结果
Figure 10. Processing results of point target under multifrequency cosine-coupled fluctuations
图 11 多频余弦耦合波动下多散射点目标的处理结果
Figure 11. Processing results of multi-point target under multifrequency cosine-coupled fluctuations
图 14 所提方法处理结果(人为加入噪声)
Figure 14. Processing results of the proposed method (artifical noise)
1 二元状态空间方程的迭代求解过程
1. Iterative solution of binary state-space equation
输入：目标所在距离单元的方位向采样信号$ s_{{\mathrm{T}}}(k) $ 输出：相位历程$ \phi(k) $的估计值$\tilde \phi(k) $：$\left\{ \begin{aligned} &\tilde \phi(k)=\hat{\boldsymbol{\varphi}}_k(1);\;k \ne N_{\mathrm{a}}-1\\ & \left[ \tilde \phi(N_{\mathrm{a}}-2) \;\;\hat{\boldsymbol{\phi}}(N_{\mathrm{a}}-1)\;\;\hat{\boldsymbol{\phi}}(N_{\mathrm{a}})\right]^{\rm{T}}= \hat{\boldsymbol{\varphi}}_{N_{\mathrm{a}}-1};\;k=N_{\mathrm{a}}-1 \end{aligned}\right. $。 $ O(1) $ 1. 初始化参数：相位状态向量$ {\boldsymbol{\varphi}}_{2} $，对应的协方差矩阵$ \boldsymbol{\varPhi}_{2} $，相位状态方程噪声协方差矩阵$ \boldsymbol{W}_{2} $；幅度状态向量$ {\boldsymbol{p}}_{2} $，对应的协方差矩阵
$ {\boldsymbol{P}}_{2} $，幅度状态方程噪声协方差矩阵$ {\boldsymbol{V}}_{2} $；观测方程噪声协方差矩阵$ \boldsymbol{R}_{2} $。2. EKF循环迭代开始，令 $ k={3} $ 3. 提取观测数据采样值：$ {\boldsymbol{s}}_{k}=\left[s_{{\mathrm{T}}}(k-1) \;\;s_{{\mathrm{T}}}(k) \;\;s_{{\mathrm{T}}}(k+1)\right]^{\mathrm{T}} $，并通过式(18)计算得$ {\boldsymbol{y}}_{k} $； $ O(1) $ 4. 状态向量预测值更新：$ \hat{{\boldsymbol{\varphi}}}_{k|k-1}={\boldsymbol{B}} \hat{{\boldsymbol{\varphi}}}_{k-1} ;\,\hat{{\boldsymbol{p}}}_{k| k-1}=\boldsymbol{B} {\hat{\boldsymbol{p}}_{k-1}} $； $ O\left(M^{2}\right) $ 5. 状态向量协方差矩阵计算：$ {\boldsymbol{\varPhi}}_{k|k-1}={\boldsymbol{B \varPhi}}_{k-1} {\boldsymbol{B}}^{{\mathrm{T}}}+{\boldsymbol{W}}_{k-1} ; {\boldsymbol{P}}_{k|k-1}={\boldsymbol{B P}}_{k|k-1} {\boldsymbol{B}}^{{\mathrm{T}}}+{\boldsymbol{V}}_{k-1} $； $ O\left(M^{3}+M^{2}\right) $ 6. 雅可比矩阵计算：$ {\boldsymbol{H}}_{1 k}=\partial f\left({\boldsymbol{p}}_{k}, {\boldsymbol{\varphi}}_{k}\right) / \partial {\boldsymbol{\varphi}}_{k };\, \boldsymbol{H}_{2k}=\partial f\left({\boldsymbol{p}}_{k}, {\boldsymbol{\varphi}}_{k}\right) / \partial {\boldsymbol{p}}_{k} $； $ O\left(M^{2}\right) $ 7. 卡尔曼增益计算：
${\boldsymbol{K}}_{1k}={\boldsymbol{\varPhi}}_{k|k-1} {\boldsymbol{H}}^{\mathrm{T}}_{1k}({\boldsymbol{H}}_{1k}{\boldsymbol{\varPhi}}_{k|k-1}{\boldsymbol{H}}^{\mathrm{T}}_{1k}+{\boldsymbol{R}}_{k-1})^{-1};\;{\boldsymbol{K}}_{2k}={\boldsymbol{P}}_{k|k-1}{\boldsymbol{H}}^{\mathrm{T}}_{2k}({\boldsymbol{H}}_{2k}{\boldsymbol{P}}_{k|k-1}{\boldsymbol{H}}^{\mathrm{T}}_{2k}+{\boldsymbol{R}}_{k-1})^{-1} $；$ O\left(M^{3}+M^{2}\right) $ 8. 根据ADMM策略，令幅度状态值为$ \hat{{\boldsymbol{p}}}_{k} \approx \hat{{\boldsymbol{p}}}_{k|k-1} $； $ O(1) $ 9. 状态向量更新：$ \hat{\boldsymbol{\varphi}}_k=\hat{\boldsymbol{\varphi}}_{k|k-1}+{\boldsymbol{K}}_{1k}\left[ {{{\boldsymbol{y}}_{{k}}} - \hat {\boldsymbol{A}}'_k \odot {{\boldsymbol{h}}_2}({{\hat {\boldsymbol{\varphi}} }_{k|k - 1}})} \right] ;{\hat {\boldsymbol{p}}_k} = {\hat {\boldsymbol{p}}_{k|{{k - 1}}}} + {\boldsymbol{K}}_{2k}\left[ {{{\boldsymbol{y}}_{{k}}} - \hat {\boldsymbol{A}}'_k \odot {{\boldsymbol{h}}_2}({{\hat {\boldsymbol{\varphi}} }_{k|k - 1}})} \right] $
其中，$ \hat{{\boldsymbol{A}}}_{k}^{\prime}=\left[{\boldsymbol{p}}_{k}\; {\boldsymbol{p}}_{k}\right]^{{\mathrm{T}}} $。$ O\left(M^{2}+M\right) $ 10. 状态向量协方差矩阵更新：${\boldsymbol{\varPhi}}_k=({\boldsymbol{I}}_{3 \times 3}-{\boldsymbol{K}}_{1k}{\boldsymbol{H}}_{1k}){\boldsymbol{\varPhi}}_{k|k-1};\;{\boldsymbol{P}}_k=({\boldsymbol{I}}_{3 \times 3 }-{\boldsymbol{K}}_{2k}{\boldsymbol{H}}_{2k}){\boldsymbol{P}}_{k|k-1} $； $ O\left(M^{3}+M^{2}\right) $ 11. 根据文献[36,37]，对噪声协方差矩阵进行自适应更新：
$ {\boldsymbol{W}}_{k}=\left(1-d_{k}\right) {\boldsymbol{W}}_{k-1}+d_{k} {\boldsymbol{K}}_{1 k}\left({\boldsymbol{\varepsilon}}_{k} {\boldsymbol{\varepsilon}}_{k}^{{\mathrm{T}}}\right) {\boldsymbol{K}}_{1k}^{{\mathrm{T}}};\; {\boldsymbol{V}}_{k}=\left(1-d_{k}\right) {\boldsymbol{V}}_{k-1}+d_{k} {\boldsymbol{K}}_{2k}\left({\boldsymbol{\varepsilon}}_{k} {\boldsymbol{\varepsilon}}_{k}^{{\mathrm{T}}}\right) {\boldsymbol{K}}_{2k}^{{\mathrm{T}}} $,
$ \boldsymbol{R}_{k}=\left(1-d_{k}\right) {\boldsymbol{R}}_{k-1}+d_{k}\left({\boldsymbol{\varepsilon}}_{k} {\boldsymbol{\varepsilon}}_{k}^{\mathrm{T}}-\boldsymbol{H}_{1k} {\boldsymbol{\varPhi}}_{k|k-1} \boldsymbol{H}_{1 k}^{\mathrm{T}}\right) $
其中，$ {\boldsymbol{\varepsilon}}_{k}={\boldsymbol{y}}_{k}-\hat{{\boldsymbol{A}}}'_{k} \odot {\boldsymbol{h}}_{2}\left(\hat{{\boldsymbol{\varphi}}}_{k|k-1}\right) $，且$ d_{k}=(1-b)/\left(1-b^{k}\right),(0<b<1) $，$ b $表示遗忘因子。$ O\left(M^{3}+M^{2}\right) $ 12. 更新EKF循环索引值：$ k=k+1 $； $ O(1) $ 13. 重复上述步骤3—步骤12，直至$ k=N_{\rm a}-1 $，结束循环迭代。 
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表 1 计算复杂度
Table 1. Computation complexity
方法 计算复杂度 GRFT $ O\left(N_{\rm r} N_{\rm a} N_{0}^{L}\right) $ FAR $ O\left(N_{\rm a} N_{\rm r} \log N_{\rm r}\right) $ ACCF $ O\left(N_{\rm a} N_{\rm r} \log N_{\rm r}\right) $ ACCM $ O\left(N_{\rm a} N_{\rm r}\right) $ BFGS-WOA $ O\left(N_{I} N_{{\mathrm{r}}} N_{\rm a}\right)+O\left(N_{I} N_{\rm a}^{3}\right) $ 所提方法 $ O\left(N_{\rm r} N_{\rm a}^{2}\right)+O\left(M^{3} N_{\rm a}\right) $
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表 2 雷达仿真参数
Table 2. Radar simulation parameters
参数 数值 中心频率(GHz) 10 信号带宽(MHz) 300 距离向采样率(MHz) 360 信号脉宽(μs) 10 脉冲重复频率(Hz) 2000 中心斜距(km) 10 距离向采样点数 4096 方位向采样点数 8192
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表 3 平台与目标运动参数
Table 3. Platform and target motion parameters
参数 数值 平台速度$ V $(m/s) 100 目标距离向速度$ v_{\rm r} $(m/s) –10 目标距离向加速度$ a_{\rm r} $(m/s2) –2 目标方位向速度$ v_{\rm a} $(m/s) 10 目标方位向加速度$ a_{\rm a} $(m/s2) 2
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表 4 不同SNR下各方法的图像对比度
Table 4. Image contrast of each method under different SNR
方法 SNR (dB) 5 2 0 –2 –4 –5 –6 5阶GRFT 210.76 136.79 115.13 86.72 30.70 28.74 1.08 FAR 10.05 8.32 7.56 7.43 5.31 3.48 1.36 ACCF 12.31 7.24 6.37 4.86 4.52 1.38 1.04 ACCM 10.61 6.15 2.61 1.22 1.08 1.05 1.04 BFGS-WOA 206.30 134.21 106.30 76.33 24.49 10.16 1.01 所提方法 491.58 243.74 157.03 112.70 110.76 100.62 100.22
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表 5 各方法的计算时间
Table 5. Calculation time under different methods
方法 平均耗时(s) 5阶GRFT 8870.32 FAR 13.48 ACCF 12.86 ACCM 10.04 BFGS-WOA 36.25 所提方法 19.64
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表 6 MMW雷达参数
Table 6. MMW radar parameters
参数 数值 中心频率(GHz) 77 信号带宽(GHz) 2.56 距离向采样率(MHz) 10 脉冲重复频率(Hz) 100 中心斜距(m) 10 雷达速度(cm/s) 2.13
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表 7 Ka波段雷达参数
Table 7. Ka-band radar parameters
参数 数值 中心频率(GHz) Ka波段 距离分辨率(m) 0.15 观测时间(s) 70 飞行半径(m) 6000 飞行高度(km) 3 雷达速度(m/s) 80
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