﻿ 一种高效的基于对比度的步进频雷达运动补偿算法
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 雷达学报  2016, Vol. 5 Issue (4): 378-388  DOI: 10.12000/JR16068 0

### 引用本文 [复制中英文]

[复制中文]
Hong Yongbin , Zhang Yong , Lu Zhenxing , Huang Wei . An Efficient Contrast-based Motion Compensation Algorithm for Stepped-frequency Radar[J]. Journal of Radars, 2016, 5(4): 378-388. DOI: 10.12000/JR16068.
[复制英文]

### 文章历史

(中国电子科技集团公司第五十四研究所 石家庄 050081)
(航天东方红卫星有限公司 北京 100094)

An Efficient Contrast-based Motion Compensation Algorithm for Stepped-frequency Radar
Hong Yongbin, Zhang Yong, Lu Zhenxing, Huang Wei
(The 54th Research Institute of CETC, Shijiazhuang 050081, China)
(DFH Satellite Co., Ltd, Beijing 100094, China)
Foundation Item: The National Ministries Foundation
Abstract: In stepped-frequency radar, the contrast cost function can be used to estimate the target radial velocity and acceleration, and therefore reduce range profile distortions. However, the contrast cost surface fluctuates sharply in the velocity and acceleration space, which greatly limits the efficiency of the algorithm. In this study, the cause of this fluctuation is analyzed and its elimination is formulated analytically through strict formula derivation. Based on intensive study of the inherent properties of the contrast cost surface, a novel and highly efficient target motion compensation algorithm is presented. Theoretical analysis and simulations confirm the effectiveness and feasibility of this new algorithm.
Key words: Stepped-frequency radar    Synthetic range profile    Contrast    Motion compensation
1 引言

2 步进频信号合成距离像原理 2.1 回波模型及处理方法

 $u\left( t \right)=\sum\limits_{n=0}^{N-1}{p\left( t-n{{T}_{\text{r}}} \right)}\exp \left[ \text{j}2\text{ }\!\!π\!\!\text{ }\left( {{f}_{0}}+n\Delta f \right)t \right]\text{ }$ (1)

 $\begin{array}{*{35}{l}} {{s}_{n}}\left( t \right)=\sum\limits_{i=1}^{K}{{{A}_{i}}}p\left[ t-n{{T}_{\text{r}}}-{{\tau }_{i}}\left( t \right) \right] \\ \qquad \quad \cdot \exp \left[ -\text{j}2\text{ }\!\!π\!\!\text{ }\left( {{f}_{0}}+n\Delta f \right){{\tau }_{i}}\left( t \right) \right] \\ \end{array}$ (2)

 ${R_i}\left( t \right) \cong {R_i}\left( 0 \right) + {v_{\rm r}}t + \frac{1}{2}{a_{\rm r}}{t^2}$ (3)

$t = n{T_{\rm r}} + t'$，其中，nTr称为慢时间，$t'$称为快时间。因为回波复数包络在时间轴上的长度变化通常可以忽略不计，即满足${\tau _i}\left( t \right) \simeq {\tau _i}\left( {n{T_{\text{r}}}} \right)$，所以式(2)可改写为：

 $\begin{array}{l} {s_n}\left( t \right) = {s_{\rm R}}\left( {t',n{T_{\rm r}}} \right) = \sum\limits_{i = 1}^K {{A_i}} p\left[{t' - {\tau _i}\left( {n{T_{\rm r}}} \right)} \right]\\ \quad\quad\quad\quad\!\! \cdot \exp \left[{ - {\text{j}}2\text{π} \left( {{f_0} + n\Delta f} \right){\tau _i}\left( {n{T_{\rm r}} + t'} \right)} \right] \end{array}$ (4)

 \begin{align} & {{s}_{\text{R}}}\left( {t}',n{{T}_{\text{r}}} \right)=\sum\limits_{i=1}^{K}{{{A}_{i}}}p\left[ {t}'-{{\tau }_{i}}\left( n{{T}_{\text{r}}} \right) \right] \\ & \quad \quad \quad \quad \cdot \exp \left\{ -\text{j}2π \left( {{\text{f}}_{\text{0}}}\text{+n}\Delta \text{f} \right)\left[ {{\tau }_{i}}\left( n{{T}_{\text{r}}} \right) \right. \right. \\ & \quad \quad \quad \quad \left. \left. +\frac{2{{v}_{\text{r}}}{t}'+2{{a}_{\text{r}}}n{{T}_{\text{r}}}{t}'+{{a}_{\text{r}}}{{{{t}'}}^{2}}}{\text{c}} \right] \right\} \\ \end{align} (5)

 $\begin{array}{l} {s_{\text{R}}}\left( {t',n{T_{\rm r}}} \right) = \sum\limits_{i = 1}^K {{A_i}} p\left[{t' - {\tau _i}\left( {n{T_{\rm r}}} \right)} \right]\exp \left( {{\text{j}}2\text{π} {f_{\text{dn}}}t'} \right)\\ \quad \quad \quad \quad \quad \quad\!\! \cdot\exp \left[{ - {\text{j}}2\text{π} \left( {{f_0} + n\Delta f} \right){\tau _i}\left( {n{T_{\rm r}}} \right)} \right] \end{array}$ (6)

 $\begin{array}{l} {s_{\text{R}}}\left( {t',n{T_{\rm r}}} \right) \!=\! \sum\limits_{i = 1}^K {{A_i}} {\mathop{\rm rect}\nolimits} \left[{t' \!-\! {\tau _i}\left( {n{T_{\rm r}}} \right)} \right]\exp \left( {{\text{j}}2\text{π} {f_{\text{dn}}}t'} \right)\\ \quad \quad \quad \quad \quad \quad\!\!\!\! \cdot\exp \left[{ - {\text{j}}2\text{π} \left( {{f_0} + n\Delta f} \right){\tau _i}\left( {n{T_{\rm r}}} \right)} \right] \end{array}$ (7)

 $h\left( k \right)=\left| \frac{\sin \left[ π \left( \text{k-}{{\text{l}}_{\text{0}}} \right) \right]}{\sin \left[ π \left( \text{k-}{{\text{l}}_{\text{0}}} \right)\text{ /}N \right]} \right|$ (8)

2.2 目标运动对合成距离像的影响

 $\left. \begin{array}{l} {\varphi _{v1}} = - \frac{{4\text{π} }}{\text{c}}{f_0}{v_{\rm r}}{T_{\rm r}}n\\ {\varphi _{v2}} = - \frac{{4\text{π} }}{\text{c}}\Delta f{v_{\rm r}}{T_{\rm r}}{n^2}\\ {\varphi _{a2}} = - \frac{{2\text{π} }}{\text{c}}{f_0}{a_{\rm r}}{T_{\rm r}}^2{n^2}\\ {\varphi _{a3}} = - \frac{{2\text{π} }}{\text{c}}{a_{\rm r}}\Delta f{T_{\rm r}}^2{n^3} \end{array} \right\}$ (9)

 $\begin{array}{*{35}{l}} \Delta {{R}_{v1}}=v\left( {{f}_{0}}\text{ /}\Delta f \right){{T}_{\text{r}}} \\ \Delta {{R}_{v2}}=v\left( N-1 \right){{T}_{\text{r}}} \\ \Delta {{R}_{a2}}={{f}_{0}}a\left( N-1 \right){{T}_{\text{r}}}^{2}/\left( 2\Delta f \right) \\ \Delta {{R}_{a3}}=3a{{\left( N-1 \right)}^{2}}{{T}_{\text{r}}}^{2}/4 \\ \end{array}$ (10)

 $\begin{array}{*{35}{l}} \left| \Delta {{v}_{1}} \right| & ＜c/\left( 4{{f}_{0}}N{{T}_{\text{r}}} \right) \\ \left| \Delta {{v}_{2}} \right| & ＜c/\left[ 8{{\left( N-1 \right)}^{2}}\Delta f{{T}_{\text{r}}} \right] \\ \left| \Delta {{a}_{2}} \right| & ＜c/\left[ 4{{\left( N-1 \right)}^{2}}{{f}_{0}}{{T}_{\text{r}}}^{2} \right] \\ \left| \Delta {{a}_{3}} \right| & ＜c/\left[ 4{{\left( N-1 \right)}^{3}}\Delta f{{T}_{\text{r}}}^{2} \right] \\ \end{array}$ (11)

3 基于对比度的运动补偿算法

 ${\text{P}} = \left( \begin{array}{l} \;\;\;\;\tilde x\left( {0,0} \right)\;\;\;\;\;\;\;\;\;\tilde x\left( {0,1} \right)\;\;\;\;\;\;\;\; \cdots \;\;\;\;\;\;\tilde x\left( {0,M - 1} \right)\\ \;\;\;\;\tilde x\left( {1,0} \right)\;\;\;\;\;\;\;\;\;\tilde x\left( {1,1} \right)\;\;\;\;\;\;\;\; \cdots \;\;\;\;\;\;\tilde x\left( {1,M - 1} \right)\\ \;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\; \ddots \;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ \tilde x\left( {N - 1,0} \right)\;\;\;\tilde x\left( {N - 1,1} \right)\;\;\; \cdots \;\;\;\tilde x\left( {N - 1,M - 1} \right) \end{array} \right)$ (12)

 ${\text{H}} = \left( \begin{array}{l} \;\;\;\;\;\tilde I\left( {0,0} \right)\;\;\;\;\;\;\;\;\;\;\tilde I\left( {0,1} \right)\;\;\;\;\;\;\;\; \cdots \;\;\;\;\;\;\;\tilde I\left( {0,M - 1} \right)\\ \;\;\;\;\;\tilde I\left( {1,0} \right)\;\;\;\;\;\;\;\;\;\;\tilde I\left( {1,1} \right)\;\;\;\;\;\;\;\; \cdots \;\;\;\;\;\;\;\tilde I\left( {1,M - 1} \right)\\ \;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ \tilde I\left( {{N_{\rm r}} - 1,0} \right)\;\;\;\tilde I\left( {{N_{\rm r}} - 1,1} \right)\;\;\; \cdots \;\;\;\;\tilde I\left( {{N_{\rm r}} - 1,M - 1} \right) \end{array} \right)$ (13)

 \begin{align} & C\left( v,a \right)=\sqrt{\frac{1}{M{{N}_{\text{r}}}}\sum\limits_{m=0}^{M-1}{\sum\limits_{k=0}^{{{N}_{\text{r}}}-1}{{{\left( {{\left| \tilde{I}\left( k,m \right) \right|}^{2}}-\frac{1}{M{{N}_{\text{r}}}}\sum\limits_{m=0}^{M-1}{\sum\limits_{k=0}^{{{N}_{\text{r}}}-1}{{{\left| \tilde{I}\left( k,m \right) \right|}^{2}}}} \right)}^{2}}}}}/ \\ & \left( \frac{1}{M{{N}_{\text{r}}}}\sum\limits_{m=0}^{M-1}{\sum\limits_{k=0}^{{{N}_{\text{r}}}-1}{{{\left| \tilde{I}\left( k,m \right) \right|}^{2}}}} \right) \\ \end{align} (14)

 $\left( {\hat v,\hat a} \right) = {\mathop{\rm argument}\nolimits} \left[{\mathop {\max }\limits_{v,a} C\left( {v,a} \right)} \right]$ (15)

 图 1 Nr=N 时理想单散射点目标的对比度代价面 Fig.1 The contrast cost surface for a single ideal point scatterer target with Nr=N

4 改进的基于对比度的运动补偿算法 4.1 散射点群位置对对比度影响的消除

 $\tilde x\left( n \right) = x\left( n \right)W_N^{\alpha n},\,\,\ 0 \le n \le N - 1$ (16)

$\tilde I\left( k \right)$$\tilde x\left( n \right)Nr(NrN)点IDFT,{\tilde I_2}\left( k \right) = {\tilde I^*}\left( k \right)\tilde I\left( k \right) = {\left| {\tilde I\left( k \right)} \right|^2}，且{\tilde I_2}\left( k \right)$${\tilde x_2}\left( n \right)$的IDFT，则根据DFT的圆周相关定理和DFT与IDFT的关系，可得${\tilde x_2}\left( n \right)$的表达式为：

 ${\tilde x_2}\left( n \right) = \frac{1}{{{N_{\rm r}}}}\sum\limits_{l = 0}^{{N_{\rm r}} - 1} {{{\tilde x}^*}\left( l \right)\tilde x{{\left( {\left( {l + n} \right)} \right)}_{{N_{\rm r}}}}{R_{{N_{\rm r}}}}\left( n \right)}$ (17)

(1) 当${N_{\rm r}} \ge 2N - 1$时，$\left| {{{\tilde x}_2}\left( n \right)} \right|$可化为：

 $\left| {{{\tilde x}_2}\left( n \right)} \right| = \left\{ \begin{array}{l} \frac{1}{{{N_{\rm r}}}}\left| {\sum\limits_{l = 0}^{N - 1 - n} {{x^*}\left( l \right)x\left( {l + n} \right)} } \right|,\quad \quad \quad \quad \quad \!\!\!\!0 \le n \le N - 1\\ 0,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \!\!N \le n \le {N_{\rm r}} - N\\ \frac{1}{{{N_{\rm r}}}}\left| {\sum\limits_{l = {N_{\rm r}} - n}^{N - 1} {{x^*}\left( l \right)x\left( {l + n - {N_{\rm r}}} \right)} } \right|,\quad \quad {N_{\rm r}} - N + 1 \le n \le {N_{\rm r}} - 1 \end{array} \right.$ (18)

(2) 当$N \le {N_{\rm r}} \le 2N - 2$时，${\tilde x_2}\left( n \right)$可化为：

 \left| {{{\tilde{x}}}_{2}}\left( n \right) \right|=\left\{ \begin{array}{*{35}{l}} \frac{1}{{{N}_{\text{r}}}}\left| \sum\limits_{l=0}^{N-1-n}{{{x}^{*}}\left( l \right)}x\left( l+n \right) \right|,\quad \quad \quad \quad \quad \quad \quad \quad 0\le n\le {{N}_{\text{r}}}-N \\ \frac{1}{{{N}_{\text{r}}}}\left| \begin{align} & \sum\limits_{l=0}^{N-1-n}{{{x}^{*}}\left( l \right)x\left( l+n \right)}+ \\ & \sum\limits_{l={{N}_{\text{r}}}-n}^{N-1}{{{x}^{*}}\left( l \right)x\left( l+n-{{N}_{\text{r}}} \right)W_{N}^{-\alpha {{N}_{\text{r}}}}} \\ \end{align} \right|,{{N}_{\text{r}}}-N+1\le n\le N-1 \\ \frac{1}{{{N}_{\text{r}}}}\left| \sum\limits_{l={{N}_{\text{r}}}-n}^{N-1}{{{x}^{*}}\left( l \right)x\left( l+n-{{N}_{\text{r}}} \right)} \right|,\quad \quad \quad \quad \quad \quad N\le n\le {{N}_{\text{r}}}-1 \\ \end{array} \right. (19)

(2) 当$N \le {N_{\rm r}} \le 2N - 2$时，$\left| {{{\tilde x}_2}\left( n \right)} \right|$$\alpha有关，从而推出\sum\nolimits_{k = 0}^{{N_{\rm r}} - 1} {{{\left| {\tilde I\left( k \right)} \right|}^4}}$$\alpha$有关，由此进一步推出式(14)中\sum\nolimits_{m = 0}^{M - 1} {\sum\nolimits_{k = 0}^{{N_{\rm r}} - 1} {{{\left| {\tilde I\left( {k,m} \right)} \right|}^4}} }\alpha有关，所以散射点群位置的变化会引起对比度起伏。 图 2给出了利用{N_{\rm r}} = 2N点IFFT合成距离像时理想单散射点目标的对比度代价面，可以看出，图 2完全消除了对比度起伏现象，对比度的代价面在速度-加速度空间内表现为光滑的曲面，对比度沿v = {v_{\rm r}}的切面和沿a = {a_{\rm r}}的切面均表现为凸函数。对比度起伏现象的消除将有助于提高补偿算法的效率，从而增强算法的实用性。  图 2 Nr=2N 时理想单散射点目标的对比度代价面 Fig.2 The contrast cost surface for a single ideal point scatterer target with Nr=2N 4.2 对比度代价面的性质分析 由于脉间一次相位项对对比度无影响，故只需考虑二次相位项和三次相位项。运动补偿后第n个子脉冲的目标回波的二次相位和三次相位之和可表示为：  \begin{align} & \Delta \phi \left( n{{T}_{\text{r}}},v,a \right)=\frac{2π }{\text{c}}{{T}_{\text{r}}}\left[ 2\Delta f\left( v-{{v}_{\text{r}}} \right)+{{f}_{0}}{{T}_{\text{r}}}\left( a-{{a}_{\text{r}}} \right) \right]{{n}^{2}} \\ & \text{ }+\frac{2π }{\text{c}}\Delta f{{T}_{\text{r}}}^{2}\left( a-{{a}_{\text{r}}} \right){{n}^{3}}n \\ \end{align} (20) 式(20)中，va分别表示补偿速度和补偿加速度。\Delta \phi \left( {n{T_{\rm r}},v,a} \right)可以看成连续时间函数\Delta \phi \left( {t,v,a} \right)t = n{T_{\rm r}}时刻的采样值，$\Delta \phi \left( {t,v,a} \right)$的表达式为：

 \begin{align} & \Delta \phi \left( t,v,a \right)=\frac{2π }{\text{c}{{T}_{\text{r}}}}\left[ 2\Delta f\left( v-{{v}_{\text{r}}} \right)+{{f}_{0}}{{T}_{\text{r}}}\left( a-{{a}_{\text{r}}} \right) \right]{{t}^{2}} \\ & \text{ }+\frac{2π }{\text{c}{{T}_{\text{r}}}}\Delta f\left( a-{{a}_{\text{r}}} \right){{t}^{3}},t\in \left[ 0 \right.,\left. \left( N-1 \right){{T}_{\text{r}}} \right]\text{ } \\ \end{align} (21)

$\Delta \phi \left( {t,v,a} \right)$关于t求导，可得t时刻的瞬时频率为：

 ${f_{\Delta \phi }}\left( {t,v,a} \right) = \frac{1}{{2\text{π} }}\frac{{{\rm d}\left[{\Delta \phi \left( {t,v,a} \right)} \right]}}{{{\rm d}t}} = {\beta _1}{t^2} + {\beta _2}t$ (22)

 \begin{align} & {{\beta }_{1}}=\frac{3\Delta f\left( a-{{a}_{\text{r}}} \right)}{\text{c}{{T}_{\text{r}}}}, \\ & {{\beta }_{2}}=\frac{4\Delta f\left( v-{{v}_{\text{r}}} \right)+2{{f}_{0}}{{T}_{\text{r}}}\left( a-{{a}_{\text{r}}} \right)}{\text{c}{{T}_{\text{r}}}} \\ \end{align} (23)

$\delta f\left( {v,a} \right)$表示${f_{\Delta \phi }}\left( {t,v,a} \right)$在一帧步进频脉冲串内的频率变化量，那么

 $\delta f\left( {v,a} \right) = \mathop {\max }\limits_t \left[{{f_{\Delta \phi }}\left( {t,v,a} \right)} \right] - \mathop {\min }\limits_t \left[{{f_{\Delta \phi }}\left( {t,v,a} \right)} \right]$ (24)

 $\left. \begin{array}{l} - \frac{{{\beta _2}}}{{2{\beta _1}}} = \frac{{\left( {N - 1} \right){T_{\rm r}}}}{2},\ a \ne {a_{\rm r}}\\ v = {v_{\rm r}},\,\,a = {a_{\rm r}} \end{array} \right\}$ (25)

 $v = - \frac{{\left[{2{f_0} + 3\left( {N - 1} \right)\Delta f} \right]{T_{\rm r}}}}{{4\Delta f}}\left( {a - {a_{\rm r}}} \right) + {v_{\rm r}}$ (26)

 $\begin{array}{*{35}{l}} \delta {{f}_{\min }}\left( a \right)=\underset{v}{\mathop{\min }}\,\left[ \delta f\left( v,a \right) \right] \\ \quad \quad \quad \quad =\left| {{f}_{\Delta \phi }}\left( \left( N-1 \right){{T}_{\text{r}}}/2,v,a \right)-{{f}_{\Delta \phi }}\left( 0,v,a \right) \right| \\ \end{array}$ (27)

 $\delta {f_{\min }}\left( a \right) = \left| {\frac{{3{{\left( {N - 1} \right)}^2}\Delta f{T_{\rm r}}\left( {a - {a_{\rm r}}} \right)}}{{4{\rm c}}}} \right|$ (28)

 图 3 4种场景下的对比度代价面投影 Fig.3 The contrast cost surface projections in four different scenes

 $\left. \begin{array}{l} {L_{{\rm m}1}}:v = - 1.597926a - 1079.896285\\ {L_{{\rm m}2}}:v = - 1.600474a - 1079.989844\\ {L_{{\rm m}3}}:v = - 1.601744a - 1080.449445\\ {L_{{\rm m}4}}:v = - 1.594944a - 1079.073461 \end{array} \right\}$ (29)

 ${L_{\rm m}}:v = - 1.597925a - 1079.896250$ (30)

(1) 在速度-加速度投影平面上存在一个条带，在条带内对比度取值较大且变化不明显，在条带外对比度取值较小。多散射点目标可能会引起条带的展宽，但条带的形状和斜率对目标特性并不敏感。

(2) 对比度峰值确定的直线包含在性质(1)所述的条带内。对于不同特性的目标，对比度峰值确定的直线斜率均可用式(26)给出的直线斜率近似表示。

4.3 一种高效的搜索对比度最大值位置的方法

 图 4 一种高效的搜索对比度最大值位置的方法示意图 Fig.4 The diagram of an efficient method of searching for the location corresponding to the max contrast

4.4 算法运算量分析

5 仿真分析

 图 5 运动参数估计误差与信噪比的关系 Fig.5 The relationship between estimation errors of motion parameters and SNR

 图 6 利用不同补偿方法得到的1维距离像 Fig.6 One-dimension range profiles obtained by different compensation methods

(1) 对于不同特性的目标，均能够抑制距离像的发散，基本保持目标距离像的固有形状，并且在低信噪比下依然适用。

(2) 实际工程中不能实现对速度和加速度的精确估计。其根本原因为运动目标回波的脉间相位中，速度产生的二次相位项和加速度产生的二次相位项存在耦合，且加速度产生的三次相位项通常很小，所以速度估计值和加速度估计值对目标特性和信噪比非常敏感，无法实现精确估计。

6 结论