﻿ 基于Chebyshev正交分解的曲线运动轨迹SAR的Chirp Scaling算法
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Chirp Scaling algorithm based on Chebyshev orthogonal decomposition for curve trajectory SAR
MENG Tingting, TAN Gewei, LI Menghui, YANG Jingjing, LI Biao, XU Xiyi
College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
Abstract: Conventional slant range models has difficulty in accurately describing the motion characteristics of the Synthetic Aperture Radar (SAR) with three-dimensional velocity and acceleration, and the curve trajectory increases the range-walk phenomenon and the high-order terms of azimuth time in the slant range equation, further complicating the two-dimensional coupling of the echo signal. Therefore, this paper proposes an improved Chirp Scaling algorithm to solve the imaging problem of the curvilinear trajectory SAR which moves with the three-dimensional velocity and acceleration. The slant range expression for the curvilinear trajectory SAR is firstly established based on the motion equation, followed by the obtainment of the slant range model in the form of equivalent hyperbolic equation based on the Chebyshev approximation of the slant range equation. The range cell migration function with spatial variability and the chirp scaling factor are finally derived, on the basis of which an improved Chirp Scaling algorithm is proposed. Simulation results confirm the effectiveness of the extended equivalent slant range model and the Chirp Scaling algorithm for large synthetic aperture time, and provide the boundary value of 3D acceleration.
Keywords: curve trajectory    Chebyshev orthogonal decomposition    equivalent slant range    spatial variability    range cell migration    modified Chirp Scaling algorithm

1 曲线运动轨迹SAR理论 1.1 Chebyshev近似的曲线运动轨迹SAR的等效斜距模型

 图 1 曲线运动轨迹SAR几何模型 Fig. 1 Geometric mode of curved trajectory SAR

 $R({t_{\rm{v}}}) = \sqrt {{{\left( {{v_x}{t_{\rm{v}}} + \frac{1}{2}{a_x}t_{\rm{v}}^2 - {X_P}} \right)}^2} + {{\left( {{v_y}{t_{\rm{v}}} + \frac{1}{2}{a_y}t_{\rm{v}}^2 - {Y_P}} \right)}^2} + {{\left( {{v_z}{t_{\rm{v}}} + \frac{1}{2}{a_z}t_{\rm{v}}^2 + {H_Q}} \right)}^2}}$ （1）

 $R({t_{\rm{v}}}) = \sqrt {R_P^2 + {A_1}{t_{\rm{v}}} + {A_2}t_{\rm{v}}^2 + {A_3}t_{\rm{v}}^3 + {A_4}t_{\rm{v}}^4}$ （2）

 $\left\{ {\begin{array}{*{20}{l}} {{R_P} = \sqrt {X_P^2 + Y_P^2 + H_Q^2} }\\ {{A_1} = 2({v_z}{H_Q} - {v_x}{X_P} - {v_y}{Y_P})}\\ {{A_2} = v_x^2 + v_y^2 + v_z^2 + {a_z}{H_Q} - {a_x}{X_P} - {a_y}{Y_P}}\\ {{A_3} = {v_x}{a_x} + {v_y}{a_y} + {v_z}{a_z}}\\ {{A_4} = \frac{1}{4}(a_x^2 + a_y^2 + a_z^2)} \end{array}} \right.$

 $R({t_{\rm{v}}}) = \frac{{{c_0}}}{2} + \sum\limits_{i = 1}^n {{c_i}} {T_i}({t_{\rm{v}}})$ （3）

 $R({t_{\rm{v}}}) = {B_0} + {B_1}{t_{\rm{v}}} + {B_2}t_{\rm{v}}^2 + {B_3}t_{\rm{v}}^3 + {B_4}t_{\rm{v}}^4$ （4）

 $\left\{ {\begin{array}{*{20}{l}} {{B_0} = \frac{{{c_0}}}{2} - {c_2} + {c_4}}\\ {{B_1} = ({c_1} - 3{c_3})\frac{2}{{{T_{{\rm{syn}}}}}}}\\ {{B_2} = (2{c_2} - 8{c_4}){{\left( {\frac{2}{{{T_{{\rm{syn}}}}}}} \right)}^2}}\\ {{B_3} = 4{c_3}{{\left( {\frac{2}{{{T_{{\rm{syn}}}}}}} \right)}^3}}\\ {{B_4} = 8{c_4}{{\left( {\frac{2}{{{T_{{\rm{syn}}}}}}} \right)}^4}} \end{array}} \right.$
1.2 误差分析

 参数 数值 载波频率/GHz 10 信号带宽/MHz 100 采样频率/MHz 260 发射信号时宽/μs 5 脉冲重复频率/Hz 1 400 三维速度/(m·s-1) 100, 35, 2 三维加速度/(m·s-2) 0.1, 0.1, -0.1

 图 2 不同斜距模型的误差比较 Fig. 2 Error comparison of different slant range models

1.3 曲线运动轨迹SAR的双曲等效斜距模型及回波

 $R({t_{\rm{v}}}) = \sqrt {R_{{\rm{eq}}}^2 + \nu _{{\rm{eq}}}^2t_{\rm{v}}^2} + D{t_{\rm{v}}} + Et_{\rm{v}}^3 + Ft_{\rm{v}}^4$ （5）

 $\begin{array}{l} s({t_{\rm{r}}},{t_{\rm{v}}}) = {u_{\rm{r}}}\left( {{t_{\rm{r}}} - \frac{{2R({t_{\rm{v}}})}}{c}} \right){u_{\rm{v}}}({t_{\rm{v}}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( { - {\rm{j}}\frac{{4\pi R({t_{\rm{v}}})}}{\lambda }} \right){\rm{exp}}\left( {{\rm{j}}\pi {K_{\rm{r}}}{{\left( {{t_{\rm{r}}} - \frac{{2R({t_{\rm{v}}})}}{c}} \right)}^2}} \right) \end{array}$ （6）

2 改进的Chirp Scaling成像算法

 $\begin{array}{l} S({f_{\rm{r}}},{t_{\rm{v}}}) = {U_{\rm{r}}}({f_{\rm{r}}}){u_{\rm{v}}}({t_{\rm{v}}}){\rm{exp}}\left( { - {\rm{j}}\pi \frac{{f_{\rm{r}}^2}}{{{K_{\rm{r}}}}}} \right) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( { - {\rm{j}}4\pi \frac{{{f_{\rm{r}}} + {f_{\rm{c}}}}}{c}R({t_{\rm{v}}})} \right) \end{array}$ （7）

 ${H_{{\rm{1rwc}}}}({f_{\rm{r}}},{t_{\rm{v}}}) = {\rm{exp}}\left( {{\rm{j}}4\pi \frac{{{f_{\rm{r}}} + {f_{\rm{c}}}}}{c}{D_{\rm{o}}}{t_{\rm{v}}}} \right)$ （8）

 $\begin{array}{*{20}{c}} {S({f_{\rm{r}}},{t_{\rm{v}}}) = {U_{\rm{r}}}({f_{\rm{r}}}){u_{\rm{v}}}({t_{\rm{v}}}){\rm{exp}}\left( { - {\rm{j}}\pi \frac{{f_{\rm{r}}^2}}{{{K_{\rm{r}}}}}} \right) \cdot }\\ {{\rm{exp}}\left( { - {\rm{j}}4\pi \frac{{{f_{\rm{r}}} + {f_{\rm{c}}}}}{c}(R({t_{\rm{v}}}) - {D_{\rm{o}}}{t_{\rm{v}}})} \right)} \end{array}$ （9）

 $\begin{array}{l} \begin{array}{*{20}{c}} {S({f_{\rm{r}}},{f_{\rm{v}}}) = {U_{\rm{r}}}({f_{\rm{r}}}){U_{\rm{v}}}({f_{\rm{v}}}){\rm{exp}}\left( { - {\rm{j}}\pi \frac{{f_{\rm{r}}^2}}{{{K_{\rm{r}}}}}} \right) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( { - {\rm{j}}\frac{{4\pi {R_{{\rm{eq}}}}}}{\lambda }\sqrt {{{\left( {1 + \frac{{{f_{\rm{r}}}}}{{{f_{\rm{c}}}}}} \right)}^2} - {{\left( {\frac{{\lambda {f_{\rm{v}}}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} - } \right.} \end{array}\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}4\pi \frac{{{f_{\rm{r}}} + {f_{\rm{c}}}}}{c}(Et_{\rm{v}}^{*3} + Ft_{\rm{v}}^{*4})} \right) \end{array}$ （10）

 $\begin{array}{l} \begin{array}{*{20}{l}} {s({t_{\rm{r}}},{f_v}) = {u_{\rm{r}}}({t_{\rm{r}}}){U_{\rm{v}}}({f_{\rm{v}}}) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left\{ {{\rm{j}}\pi {K_{{\rm{eq}}}}{{[{t_{\rm{r}}} - 2{R_{{\rm{eq}}}}/(c\sqrt {1 - \frac{{\lambda f_{\rm{v}}^2}}{{2{\nu _{{\rm{eq}}}}}}} )]}^2} - } \right.} \end{array}\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{{4{R_{{\rm{eq}}}}}}{\lambda }\sqrt {1 - {{\left( {\frac{{\lambda {f_{\rm{v}}}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} - {\rm{j}}\pi \frac{4}{\lambda }(Et_{\rm{v}}^{\# 3} + Ft_{\rm{v}}^{\# 4})} \right\} \end{array}$ （11）

 $\begin{array}{*{20}{l}} {{H_{{\rm{cs}}}}({t_{\rm{r}}},{f_{\rm{v}}}) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( {{\rm{j}}\pi {K_{{\rm{eq}}}}{C_{\rm{s}}}{{\left( {{t_{\rm{r}}} - 2\frac{{r({t_{\rm{r}}},{f_{\rm{v}}};{R_{{\rm{ref}}}})}}{c}} \right)}^2}} \right)} \end{array}$ （12）

 $\begin{array}{l} \begin{array}{*{20}{l}} {S({f_{\rm{r}}},{f_{\rm{v}}}) = {U_{\rm{r}}}({f_{\rm{r}}}){U_{\rm{v}}}({f_{\rm{v}}}) \cdot }\\ {\quad {\rm{exp}}\left( { - {\rm{j}}\frac{{4\pi }}{{{c^2}}}{K_{{\rm{eq}}}}{C_{\rm{s}}}(1 + {C_{\rm{s}}}){{({R_{{\rm{eq}}}} - {R_{\rm{ret}}})}^2} - } \right.} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{{4{R_{{\rm{eq}}}}}}{\lambda }\sqrt {1 - {{\left( {\frac{{\lambda {f_{\rm{v}}}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} - {\rm{j}}\pi \frac{4}{\lambda }(Et_{\rm{v}}^{\# 3} + Ft_{\rm{v}}^{\# 4}) - \\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{{f_{\rm{r}}^2}}{{{K_{{\rm{eq}}}}(1 + {C_{\rm{s}}})}} - {\rm{j}}\frac{{4\pi }}{c}({R_{{\rm{eq}}}} + {R_{{\rm{ref}}}}{C_{\rm{s}}}){f_{\rm{r}}}} \right) \end{array}$ （13）

 ${H_{{\rm{rc + src}}}}({f_{\rm{r}}},{f_{\rm{v}}}) = {\rm{exp}}\left( {{\rm{j}}\pi \frac{{f_{\rm{r}}^2}}{{{K_{{\rm{ref}}}}(1 + {C_{\rm{s}}})}}} \right)$ （14）

CS处理解除了距离徙动曲线的空变性。因此，不同距离的距离徙动校正函数都为

 ${H_{{\rm{rcmc}}}}({f_{\rm{r}}},{f_{\rm{v}}}) = {\rm{exp}}\left( {{\rm{j}}\frac{{4\pi }}{c}{R_{{\rm{ref}}}}{C_{\rm{s}}}{f_{\rm{r}}}} \right)$ （15）

 $\begin{array}{l} \begin{array}{*{20}{l}} {S({f_{\rm{r}}},{f_{\rm{v}}}) = {U_{\rm{r}}}({f_{\rm{r}}}){U_{\rm{v}}}({f_{\rm{v}}}) \cdot }\\ {\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left\{ { - {\rm{j}}\frac{{4\pi }}{{{c^2}}}{K_{{\rm{eq}}}}{C_{\rm{s}}}(1 + {C_{\rm{s}}}){{({R_{{\rm{eq}}}} - {R_{{\rm{ret}}}})}^2} - } \right.} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{{4{R_{{\rm{eq}}}}}}{\lambda }\sqrt {1 - {{\left( {\frac{{\lambda {f_{\rm{v}}}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} - \\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{4}{\lambda }(Et_{\rm{v}}^{\# 3} + Ft_{\rm{v}}^{\# 4}) - {\rm{j}}\frac{{4\pi }}{c}{R_{{\rm{eq}}}}{f_{\rm{r}}}} \right\} \end{array}$ （16）

 $\begin{array}{l} s({t_{\rm{r}}},{f_{\rm{v}}}) = {\rm{sinc}} \left( {{t_{\rm{r}}} - \frac{{2{R_{{\rm{eq}}}}}}{c}} \right){U_{\rm{v}}}({f_{\rm{v}}}) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( { - {\rm{j}}\frac{{4\pi }}{{{c^2}}}{K_{{\rm{eq}}}}{C_{\rm{s}}}(1 + {C_{\rm{s}}}){{({R_{{\rm{eq}}}} - {R_{{\rm{ref}}}})}^2} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{j}}\pi \frac{{4{R_{{\rm{eq}}}}}}{\lambda }\sqrt {1 - {{\left( {\frac{{\lambda {f_{\rm{v}}}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} - {\rm{j}}\pi \frac{4}{\lambda }(Et_{\rm{v}}^{\# 3} + Ft_{\rm{v}}^{\# 4})} \right) \end{array}$ （17）

 $\begin{array}{l} {H_{\rm{a}}}({t_{\rm{r}}},{f_{\rm{v}}}) = {\rm{exp}}\left( {{\rm{j}}2\pi \left[ {\frac{2}{{{c^2}}}{K_{{\rm{eq}}}}{C_{\rm{s}}}(1 + {C_{\rm{s}}}) \cdot } \right.} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {({R_{{\rm{eq}}}} - {R_{{\rm{ref}}}})^2} + \frac{{2{R_{{\rm{eq}}}}}}{\lambda }\sqrt {1 - {{\left( {\frac{{\rm{v}}}{{2{\nu _{{\rm{eq}}}}}}} \right)}^2}} + \\ \left. {\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{2}{\lambda }(Et_{\rm{v}}^{\# 3} + Ft_{\rm{v}}^{\# 4})} \right]} \right) \end{array}$ （18）

 图 3 改进的CS算法流程图 Fig. 3 Improved CS algorithm flow chart
3 实验仿真分析

 图 4 5×5点阵分布图 Fig. 4 Distribution map of 5×5 point targets

 图 5 改进的CS算法对点阵的成像效果图 Fig. 5 Imaging results of improved CS algorithm on point targets

 图 6 场景中心点P0的等高线图 Fig. 6 Contour map of scene center point P0

 图 7 场景中心点P0的冲激响应比较 Fig. 7 Comparison of impulse responses of scene center point P0

 图 8 场景边缘点P2的等高线图 Fig. 8 Contour map of scene edge point P2
 图 9 场景边缘点P2的冲击响应比较 Fig. 9 Comparison of impulse responses of scene edge point P2

 点目标 算法 距离向 方位向 PSLR/dB ISLR/dB IRW/m PSLR/dB ISLR/dB IRW/m P0 CS算法 -13.473 1 -10.623 8 1.427 6 -13.037 2 -10.568 1 1.796 6 文献[18] -12.580 2 -10.531 2 1.418 9 -10.191 4 -8.466 8 1.893 6 文献[19] -11.857 5 -10.506 7 1.433 3 -13.009 9 -10.554 3 1.796 1 P1 CS算法 -13.246 6 -10.595 7 1.680 3 -13.045 3 -10.566 8 1.794 5 文献[18] -12.052 9 -10.470 0 1.458 5 -10.291 4 -8.464 3 1.933 8 文献[19] -11.644 8 -10.449 4 1.433 0 -13.007 9 -10.554 3 1.795 9 P2 CS算法 -13.146 2 -10.691 6 1.423 1 -13.072 1 -10.583 9 1.795 0 文献[18] -12.754 3 -10.608 9 1.423 8 -10.301 8 -8.469 9 1.935 8 文献[19] -12.980 2 -10.622 2 1.426 6 -13.015 4 -10.555 9 1.795 7

 图 10 不同模型的斜距及多普勒相位误差比较 Fig. 10 Comparison of slant range and doppler phase error of different models

 $\begin{array}{*{20}{l}} {\Delta R \le }\\ {\quad \left( {\frac{{12{A_4}}}{{{R_p}}} - \frac{{3(A_2^2 + 2{A_1}{A_3})}}{{R_p^3}} + \frac{{9A_1^2{A_2}}}{{2R_p^5}} - \frac{{15A_1^4}}{{16R_p^7}}} \right)t_{\rm{v}}^4} \end{array}$ （19）

 $\begin{array}{l} \Delta {\varPhi _{\rm{m}}} = \frac{{4\pi }}{\lambda }\Delta {R_{\rm{m}}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\frac{{12{A_4}}}{{{R_p}}} - \frac{{3(A_2^2 + 2{A_1}{A_3})}}{{R_p^3}} + \frac{{9A_1^2{A_2}}}{{2R_p^5}} - \frac{{15A_1^4}}{{16R_p^7}}} \right) \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\pi T_{{\rm{syn}}}^4}}{{4\lambda }} \end{array}$ （20）

SAR成像要求ΔΦm < $\frac{\pi}{4}$，由此可推导系统参数的相互制约条件。忽略Rp的高次项，可推算出本算法适用的三维加速度满足的方程为

 $a_{x}^{2}+a_{y}^{2}+a_{z}^{2}<\frac{\lambda R_{P}}{3 T_{\mathrm{syn}}^{4}}$ （21）

 图 11 不同三维加速度下中心点P0的等高线图 Fig. 11 Contour map of scene center point P0 under different three-dimensional acceleration

 图 12 不同合成孔径时间下的等高线图 Fig. 12 Contour map of different synthetic aperture time
4 结论

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http://dx.doi.org/10.7527/S1000-6893.2020.23741

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#### 文章信息

MENG Tingting, TAN Gewei, LI Menghui, YANG Jingjing, LI Biao, XU Xiyi

Chirp Scaling algorithm based on Chebyshev orthogonal decomposition for curve trajectory SAR

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 323741.
http://dx.doi.org/10.7527/S1000-6893.2020.23741