CN102331577A

CN102331577A - An Improved NCS Imaging Algorithm for Geosynchronous Orbit SAR

Info

Publication number: CN102331577A
Application number: CN201110280669A
Authority: CN
Inventors: 龙腾; 胡程; 刘志鹏; 朱宇
Original assignee: Beijing Institute of Technology BIT
Current assignee: Beijing Institute of Technology BIT
Priority date: 2011-09-21
Filing date: 2011-09-21
Publication date: 2012-01-25
Anticipated expiration: 2031-09-21
Also published as: CN102331577B

Abstract

The invention relates to an improved NCS imaging algorithm suitable for geosynchronous orbit SAR, belonging to the technical field of synthetic aperture radar (SAR) imaging. The improvement of the present invention to the NCS imaging algorithm lies in two parts: one is to establish a curved trajectory signal model to replace the equivalent straight line model in the original NCS imaging algorithm; 2D analytical spectral representation of the NCS imaging algorithm. Compared with the prior art, the present invention has the advantages that a new curved trajectory model suitable for GEO SAR is obtained through the method of high-order Taylor expansion, and the trajectory model can solve the problem that the error of the GEOSAR perigee equivalent straight line model is relatively large. The apogee equivalent straight line model cannot be applied at all; at the same time, based on the equivalent straight line model, a binary spectrum suitable for the NCS algorithm is obtained. Using this spectrum, each compensation function of the NCS algorithm can be obtained, and the GEO SAR is realized. Large scene imaging requirements.

Description

An Improved NCS Imaging Algorithm for Geosynchronous Orbit SAR

技术领域 technical field

本发明涉及一种改进NCS成像算法，特别涉及一种适用于地球同步轨道(GEO)SAR的的改进NCS成像算法，属于合成孔径雷达(SAR)成像技术领域。The invention relates to an improved NCS imaging algorithm, in particular to an improved NCS imaging algorithm suitable for geosynchronous orbit (GEO) SAR, and belongs to the technical field of synthetic aperture radar (SAR) imaging.

背景技术 Background technique

目前的SAR卫星均为低轨卫星，轨道高度不超过1,000km，对特定地区的重复观测周期一般为3到5天，在进行轨道机动时也需要至少1天时间；因此，低轨SAR存在时间分辨率低、应对突发事件滞后时间长的问题。解决此问题的一种有效方法是地球同步轨道合成孔径雷达(GEO SAR)卫星，这是运行在36,000km高度地球同步轨道上的SAR卫星；这种地球同步轨道并非地球静止轨道，它具有一定的倾斜角度，其星下点轨迹为‘8’字形，由此可获得与地面目标的相对运动，实现二维SAR成像。The current SAR satellites are all low-orbit satellites, with an orbital height of no more than 1,000km. The repeated observation period for a specific area is generally 3 to 5 days, and at least 1 day is required for orbital maneuvering; therefore, the existence of low-orbit SAR Low resolution and long lag time to respond to emergencies. An effective method to solve this problem is the Geosynchronous Orbit Synthetic Aperture Radar (GEO SAR) satellite, which is a SAR satellite operating in a geosynchronous orbit at a height of 36,000 km; this geosynchronous orbit is not a geostationary orbit, and it has certain Inclination angle, its sub-satellite point trajectory is in the shape of '8', so that the relative motion with the ground target can be obtained, and two-dimensional SAR imaging can be realized.

目前的SAR成像算法都是基于低轨(LEO)情况建立的，缺陷在于：LEO SAR一般合成孔径时间比较短，卫星飞行的轨迹可以用等效直线模型来近似，但是在GEO SAR中合成孔径时间一般达到上百秒，因而低轨SAR成像算法所依赖的等效直线模型因GEO SAR超长的孔径时间而失效。NCS算法是一种优秀的大场景成像算法，但是它是基于等效直线模型，因此很难直接应用于GEO SAR。针对以上情况，我们提出了一种改进的NCS成像算法去实现GEO SAR大场景成像的要求。The current SAR imaging algorithms are all established based on low-orbit (LEO) conditions. The disadvantages are: LEO SAR generally has a relatively short synthetic aperture time, and the trajectory of satellite flight can be approximated by an equivalent straight line model, but the synthetic aperture time in GEO SAR Generally, it reaches hundreds of seconds, so the equivalent linear model relied on by the low-orbit SAR imaging algorithm is invalid due to the ultra-long aperture time of GEO SAR. The NCS algorithm is an excellent large-scene imaging algorithm, but it is based on the equivalent straight line model, so it is difficult to be directly applied to GEO SAR. In view of the above situation, we propose an improved NCS imaging algorithm to meet the requirements of GEO SAR large scene imaging.

发明内容 Contents of the invention

本发明的目的是为了实现GEO SAR大场景成像，提出了一种适用于地球同步轨道SAR的改进NCS成像算法。The purpose of the present invention is to realize GEO SAR large scene imaging, and propose an improved NCS imaging algorithm suitable for geosynchronous orbit SAR.

本发明是通过以下技术方案实现的。The present invention is achieved through the following technical solutions.

本发明的一种适用于地球同步轨道SAR的改进NCS成像算法，其改进之处在于两个部分：一是建立弯曲轨迹信号模型取代原NCS成像算法中的等效直线模型，二是并在建立弯曲轨迹信号模型的基础上求出适用于NCS成像算法的二维解析频谱表达式，两个部分的具体过程分别为：An improved NCS imaging algorithm suitable for geosynchronous orbit SAR of the present invention, its improvement lies in two parts: one is to establish a curved trajectory signal model to replace the equivalent straight line model in the original NCS imaging algorithm, and the other is to establish On the basis of the curved trajectory signal model, the two-dimensional analytical spectrum expression suitable for the NCS imaging algorithm is obtained. The specific processes of the two parts are as follows:

1)传统的NCS算法是基于等效直线模型，但是等效直线模型在GEO SAR中出现误差比较大甚至失效等问题，因此需要建立一种弯曲轨迹信号模型去近似卫星和目标之间的真实斜距历史，建立弯曲轨迹信号模型的过程为：1) The traditional NCS algorithm is based on the equivalent straight line model, but the equivalent straight line model has problems such as relatively large error or even failure in GEO SAR, so it is necessary to establish a curved trajectory signal model to approximate the real slope between the satellite and the target. From history, the process of building a curved trajectory signal model is:

定义卫星和目标在每个脉冲重复时间(PRT)的坐标分别为

和

卫星和目标之间的真实斜距历史表示为Define the coordinates of the satellite and the target at each pulse repetition time (PRT) as

and

The true slant range history between the satellite and the target is expressed as

${R R}_{n no} = = | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{sn sn} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{gn gn} | | | | - - - - - - ((11))$

对式(1)进行泰勒展开后得到弯曲轨迹模型The curved trajectory model is obtained after Taylor expansion of formula (1)

R_n＝R+k₁·t_a+k₂·t_a ²+k₃·t_a ³+k₄·t_a ⁴+… (2)R _n =R+k ₁ ·t _a +k ₂ ·t _a ² +k ₃ ·t _a ³ +k ₄ ·t _a ⁴ +... (2)

其中t_a为方位向时间，R、k₁、k₂、k₃和k₄为R_n的0到4阶的泰勒展开系数，其中k₁、k₂、k₃和k₄具体表达式分别为：where t _a is the azimuth time, R, k ₁ , k ₂ , k ₃ and k ₄ are the Taylor expansion coefficients of order 0 to 4 of R _n , and the specific expressions of k ₁ , k ₂ , k ₃ and k ₄ are respectively for:

${k k}_{11} = = {k k}_{1010} + + {\overset{. .}{k k}}_{11} \cdot &Center Dot; ((R R - - {R R}_{00})) - - - - - - ((33))$

${k k}_{22} = = {k k}_{2020} + + {\overset{. .}{k k}}_{22} \cdot &Center Dot; ((R R - - {R R}_{00})) - - - - - - ((44))$

${k k}_{33} = = {k k}_{3030} + + {\overset{. .}{k k}}_{33} \cdot \cdot ((R R - - {R R}_{00})) - - - - - - ((55))$

${k k}_{44} = = {k k}_{4040} + + {\overset{. .}{k k}}_{44} \cdot \cdot ((R R - - {R R}_{00})) - - - - - - ((66))$

在式(3)～(6)中，k₁₀～k₄₀，的具体表达式分别为：In formulas (3) to (6), k ₁₀ to k ₄₀ , The specific expressions are:

${k k}_{1010} = = {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} / / | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | | - - - - - - ((77))$

${k k}_{2020} = = \frac{{\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} + + {| | | | {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} | | | |}^{22}}{22 \cdot \cdot | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |} - - \frac{{[[{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}]]}^{22}}{22 \cdot &Center Dot; {| | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |}^{33}} - - - - - - ((88))$

$k_{30} = \frac{{\overset{&RightArrow;}{b}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 3 \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}}{6 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} + \frac{{[{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}]}^{3}}{2 \cdot {| | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |}^{5}}$ (9) $k_{30} = \frac{{\overset{&Right Arrow;}{b}}_{the s 0} &Center Dot; {({\overset{&Right Arrow;}{r}}_{the s 0} - {\overset{&Right Arrow;}{r}}_{g 0})}^{T} + 3 &Center Dot; {\overset{&Right Arrow;}{a}}_{the s 0} \cdot {\overset{&Right Arrow;}{v}}_{the s 0}^{T}}{6 &Center Dot; | | {\overset{&Right Arrow;}{r}}_{the s 0} - {\overset{&Right Arrow;}{r}}_{g 0} | |} + \frac{{[{\overset{&Right Arrow;}{v}}_{the s 0} \cdot {({\overset{&Right Arrow;}{r}}_{the s 0} - {\overset{&Right Arrow;}{r}}_{g 0})}^{T}]}^{3}}{2 \cdot {| | {\overset{&Right Arrow;}{r}}_{the s 0} - {\overset{&Right Arrow;}{r}}_{g 0} | |}^{5}}$ (9)

$- - \frac{{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{22 \cdot &Center Dot; {| | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |}^{33}} - - \frac{{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} \cdot &Center Dot; {| | | | {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} | | | |}^{22}}{22 \cdot &Center Dot; {| | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |}^{33}}$

${k k}_{4040} = = \frac{{\overset{&RightArrow; &Right Arrow;}{d d}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} + + 33 \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{b b}}_{s the s 00} \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00}^{T T}}{24 twenty four \cdot \cdot | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |} + + \frac{{| | | | {\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} | | | |}^{22}}{88 \cdot \cdot | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |} - - \frac{{k k}_{22}^{22} + + 22 \cdot \cdot {k k}_{11} \cdot &Center Dot; {k k}_{33}}{22 \cdot \cdot | | | | {\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00} | | | |} - - - - - - ((1010))$

${\overset{. .}{k k}}_{11} = = \frac{{v v}_{s the s 00 x x}}{{r r}_{s the s 00 x x}} - - \frac{{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{{R R}_{00}^{22}} - - - - - - ((1111))$

${\overset{\cdot \cdot}{k k}}_{22} = = \frac{{a a}_{s the s 00 x x} \cdot &Center Dot; {R R}_{00}^{22} - - {r r}_{s the s 00 x x} \cdot \cdot (({\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} + + {| | | | {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} | | | |}^{22}))}{22 \cdot \cdot {R R}_{00}^{22} \cdot \cdot {r r}_{s the s 00 x x}} - - \frac{{v v}_{s the s 00 x x} \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{{R R}_{00}^{22} \cdot \cdot {r r}_{s the s 00 x x}} + + \frac{33 \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{22 \cdot \cdot {R R}_{00}^{44}} - - - - - - ((1212))$

${\overset{\cdot \cdot}{k k}}_{33} = = \frac{{b b}_{s the s 00 x x} \cdot \cdot {R R}_{00}^{22} - - {r r}_{s the s 00 x x} \cdot \cdot [[{\overset{&RightArrow; &Right Arrow;}{b b}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} + + 33 \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00}^{T T}]]}{66 \cdot &Center Dot; {R R}_{00}^{22} \cdot \cdot {r r}_{s the s 00 x x}} + +$

$\frac{33 \cdot \cdot {v v}_{s the s 00 x x} \cdot \cdot {[[{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}]]}^{22} - - 55 \cdot &Center Dot; {r r}_{s the s 00 x x} \cdot &Center Dot; {[[{\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}]]}^{33}}{22 \cdot &Center Dot; {R R}_{00}^{66} \cdot \cdot {r r}_{s the s 00 x x}} - - \frac{{v v}_{s the s 00 x x} \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} + + {a a}_{s the s 00 x x} \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{22 \cdot &Center Dot; {R R}_{00}^{22} \cdot &Center Dot; {r r}_{s the s 00 x x}} - - - - - - ((1313))$

$+ + \frac{33 \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} \cdot &Center Dot; {\overset{&RightArrow; &Right Arrow;}{a a}}_{s the s 00} \cdot &Center Dot; {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T}}{22 \cdot \cdot {R R}_{00}^{44}} - - \frac{{v v}_{s the s 00 x x} \cdot &Center Dot; {| | | | {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} | | | |}^{22}}{22 \cdot \cdot {R R}_{00}^{22} \cdot \cdot {r r}_{s the s 00 x x}} + + \frac{33 \cdot \cdot {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} \cdot \cdot {(({\overset{&RightArrow; &Right Arrow;}{r r}}_{s the s 00} - - {\overset{&RightArrow; &Right Arrow;}{r r}}_{g g 00}))}^{T T} \cdot \cdot {| | | | {\overset{&RightArrow; &Right Arrow;}{v v}}_{s the s 00} | | | |}^{22}}{22 \cdot &Center Dot; {R R}_{00}^{44}}$

${\dot{k}}_{4} = \frac{d_{s 0 x} \cdot {R_{0}}^{2} - r_{s 0 x} \cdot [{\overset{&RightArrow;}{d}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 4 \cdot {\overset{&RightArrow;}{b}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}]}{24 \cdot {R_{0}}^{2} \cdot r_{s 0 x}} - \frac{{| | {\overset{&RightArrow;}{a}}_{s 0} | |}^{2}}{8 \cdot {R_{0}}^{2}}$ (14) ${\dot{k}}_{4} = \frac{d_{the s 0 x} &Center Dot; {R_{0}}^{2} - r_{the s 0 x} &Center Dot; [{\overset{&Right Arrow;}{d}}_{the s 0} \cdot {({\overset{&Right Arrow;}{r}}_{the s 0} - {\overset{&Right Arrow;}{r}}_{g 0})}^{T} + 4 \cdot {\overset{&Right Arrow;}{b}}_{the s 0} &Center Dot; {\overset{&Right Arrow;}{v}}_{the s 0}^{T}]}{twenty four \cdot {R_{0}}^{2} &Center Dot; r_{the s 0 x}} - \frac{{| | {\overset{&Right Arrow;}{a}}_{the s 0} | |}^{2}}{8 &Center Dot; {R_{0}}^{2}}$ (14)

$+ + \frac{{k k}_{2020} \cdot \cdot {\overset{\cdot \cdot}{k k}}_{22} + + {k k}_{3030} \cdot &Center Dot; {\overset{\cdot \cdot}{k k}}_{11} + + {k k}_{1010} \cdot \cdot {\overset{\cdot &Center Dot;}{k k}}_{33}}{{r r}_{s the s 00 x x}} + + \frac{{k k}_{2020}^{22} + + 22 \cdot &Center Dot; {k k}_{1010} \cdot \cdot {k k}_{3030}}{22 \cdot &Center Dot; {R R}_{00}^{22}}$

式(7)～式(14)中，

和表示卫星在孔径中心时刻的位置矢量，

和分别表示卫星在孔径中心时刻的速度矢量、加速度矢量、加加速度矢量和加加加速度矢量，R₀表示卫星和参考点目标在孔径中心时刻的距离，r_s0x、a_s0x、b_s0x、v_s0x和d_s0x分别为

和

在场景坐标系下的距离向分量；In formula (7) ~ formula (14),

and Indicates the position vector of the satellite at the moment of the aperture center,

and respectively represent the velocity vector, acceleration vector, jerk vector and jerk vector of the satellite at the aperture center moment, R ₀ represents the distance between the satellite and the reference point target at the aperture center moment, r _s0x , a _s0x , b _s0x , v _s0x and d _s0x are

and

The distance component in the scene coordinate system;

2)基于弯曲轨迹，求出适用于NCS成像算法的二维解析频谱表达式的过程为：2) Based on the curved trajectory, the process of finding the two-dimensional analytical spectrum expression suitable for the NCS imaging algorithm is as follows:

NCS算法的处理是从二维频域开始的，因此求得弯曲轨迹下的二维解析频谱表达式尤为重要；所提出的弯曲轨迹是高阶泰勒展开，故利用级数反转原理求得驻定相位点后，得到的频谱表达式为The processing of the NCS algorithm starts from the two-dimensional frequency domain, so it is particularly important to obtain the two-dimensional analytical spectrum expression under the curved trajectory; the proposed curved trajectory is a high-order Taylor expansion, so the resident After phasing the points, the obtained spectrum expression is

$S S (({f f}_{r r},, {f f}_{a a})) = = {u u}_{r r} ((\frac{{f f}_{r r}}{{K K}_{r r}})) \cdot &Center Dot; {u u}_{a a} [[{f f}_{a a} + + \frac{22 \cdot &Center Dot; {k k}_{11}}{c c} \cdot &Center Dot; (({f f}_{r r} {+ + f f}_{c c}))]] \cdot &Center Dot; exp exp ((- - j j \cdot \cdot π π \cdot \cdot \frac{{f f}_{r r}^{22}}{{K K}_{r r}}))$

$exp exp {{j j \cdot \cdot 22 \cdot &Center Dot; π π \cdot &Center Dot; [\begin{matrix} - - \frac{22 \cdot \cdot (({f f}_{r r} + + {f f}_{c c}))}{c c} \cdot \cdot R R \\ + + \frac{11}{44 \cdot \cdot {k k}_{22}} \cdot \cdot ((\frac{c c}{22 \cdot \cdot (({f f}_{r r} + + {f f}_{c c}))})) \cdot \cdot {(({f f}_{a a} + + \frac{22 \cdot &Center Dot; {k k}_{11}}{c c} \cdot &Center Dot; (({f f}_{r r} + + {f f}_{c c}))))}^{22} \\ + + \frac{{k k}_{33}}{88 \cdot &Center Dot; {k k}_{22}^{33}} \cdot \cdot {((\frac{c c}{22 \cdot \cdot (({f f}_{r r} + + {f f}_{c c}))}))}^{22} \cdot &Center Dot; {(({f f}_{a a} + + \frac{22 \cdot \cdot {k k}_{11}}{c c} \cdot \cdot (({f f}_{r r} + + {f f}_{c c}))))}^{33} \\ + + \frac{99 \cdot \cdot {k k}_{33}^{22} - - 44 \cdot \cdot {k k}_{22} \cdot &Center Dot; {k k}_{44}}{6464 \cdot &Center Dot; {k k}_{22}^{55}} \cdot &Center Dot; {((\frac{c c}{22 \cdot &Center Dot; (({f f}_{r r} + + {f f}_{c c}))}))}^{33} \cdot &Center Dot; {(({f f}_{a a} + + \frac{22 \cdot &Center Dot; {k k}_{11}}{c c} \cdot &Center Dot; (({f f}_{r r} + + {f f}_{c c}))))}^{44} \end{matrix}]}} - - - - - - ((1515))$

其中，f_r和f_a分别为距离向和方位向频率，u_r(·)和u_a(·)分别为距离向和方位向包络，k_r为距离向调频率，c为光速，f_c为雷达载频；where f _r and f _a are the range and azimuth frequencies respectively, u _r ( ) and u _a ( ) are the range and azimuth envelopes respectively, k _r is the range modulation frequency, c is the speed of light, f _c is the radar carrier frequency;

式(15)不能在NCS算法中直接应用，需要进一步推导，得出Equation (15) cannot be directly applied in the NCS algorithm, and needs to be further deduced to obtain

$\frac{11}{{f f}_{r r} + + {f f}_{c c}} = = \frac{11}{{f f}_{c c}} [[11 - - \frac{{f f}_{r r}}{{f f}_{c c}} + + {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{22} - - {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{33} + + \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;]]$

${((\frac{11}{{f f}_{r r} + + {f f}_{c c}}))}^{22} = = \frac{11}{{f f}_{c c}^{22}} [[11 - - \frac{22 \cdot &Center Dot; {f f}_{r r}}{{f f}_{c c}} + + 33 \cdot &Center Dot; {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{22} - - 44 \cdot &Center Dot; {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{33} + + \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;]] - - - - - - ((1616))$

${((\frac{11}{{f f}_{r r} + + {f f}_{c c}}))}^{33} = = \frac{11}{{f f}_{c c}^{33}} [[11 - - \frac{33 \cdot &Center Dot; {f f}_{r r}}{{f f}_{c c}} + + 66 \cdot &Center Dot; {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{22} - - 1010 \cdot &Center Dot; {((\frac{{f f}_{r r}}{{f f}_{c c}}))}^{33} + + \cdot &Center Dot; \cdot &Center Dot; \cdot \cdot]]$

利用式(16)，经过推导后，得出GEO SAR二维频谱为：Using formula (16), after derivation, the two-dimensional spectrum of GEO SAR is obtained as:

$S (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{az} (f_{a}, R)] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{RP} (R)]$ (17) $S (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) &Center Dot; u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] &Center Dot; \exp [j &Center Dot; 2 &Center Dot; π \cdot φ_{az} (f_{a}, R)] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{RP} (R)]$ (17)

$\cdot &Center Dot; exp exp [[- - j j \cdot &Center Dot; 22 \cdot \cdot π π \cdot &Center Dot; b b (({f f}_{a a},, {f f}_{r r}))]] \cdot \cdot exp exp [[- - j j \cdot \cdot \frac{44 \cdot &Center Dot; π π \cdot \cdot R R}{c c \cdot &Center Dot; M m (({f f}_{a a}))} \cdot &Center Dot; {f f}_{r r}]] \cdot \cdot exp exp [[- - j j \cdot &Center Dot; π π \cdot \cdot \frac{{f f}_{r r}^{22}}{{K K}_{s the s} (({f f}_{a a},, R R))}]] \cdot &Center Dot; exp exp [[j j \cdot &Center Dot; {φ φ}_{33} (({f f}_{a a},, R R)) \cdot &Center Dot; {f f}_{r r}^{33}]]$

对式(17)作进一步的说明如下：A further description of formula (17) is as follows:

2.1式(17)等号右侧的ф_az(f_a，R)为方位向调制函数，具体表达式为2.1 The ф _az (f _a , R) on the right side of the equal sign in formula (17) is the azimuth modulation function, and the specific expression is

${φ φ}_{az az} (({f f}_{a a},, R R)) = = [[\frac{{k k}_{11}}{22 \cdot &Center Dot; {k k}_{22}} + + \frac{33 \cdot &Center Dot; {k k}_{11}^{22} \cdot &Center Dot; {k k}_{33}}{88 \cdot &Center Dot; {k k}_{22}^{33}} + + \frac{{k k}_{11}^{33} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot \cdot {k k}_{44}))}{1616 \cdot &Center Dot; {k k}_{22}^{55}}]] \cdot &Center Dot; {f f}_{a a} + +$

$[[\frac{λ λ}{88 \cdot &Center Dot; {k k}_{22}} + + \frac{33 \cdot &Center Dot; λ λ \cdot &Center Dot; {k k}_{11} \cdot &Center Dot; {k k}_{33}}{1616 \cdot &Center Dot; {k k}_{22}^{33}} + + \frac{33 \cdot &Center Dot; λ λ \cdot &Center Dot; {k k}_{11}^{22} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot &Center Dot; {k k}_{44}))}{6464 \cdot &Center Dot; {k k}_{22}^{55}}]] \cdot &Center Dot; {f f}_{a a}^{22} + + - - - - - - ((1818))$

$[[\frac{{λ λ}^{22} \cdot &Center Dot; {k k}_{33}}{3232 \cdot &Center Dot; {k k}_{22}^{33}} + + \frac{λ λ \cdot &Center Dot; {k k}_{11} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot &Center Dot; {k k}_{44}))}{6464 \cdot &Center Dot; {k k}_{22}^{55}}]] \cdot \cdot {f f}_{a a}^{33} + + \frac{{λ λ}^{33} \cdot &Center Dot; ((99 \cdot \cdot {k k}_{33}^{22} - - 44 \cdot \cdot {k k}_{22} \cdot &Center Dot; {k k}_{44}))}{512512 \cdot \cdot {k k}_{22}^{55}} \cdot &Center Dot; {f f}_{a a}^{44}$

由于ф_az(f_a，R)只与方位向频率和目标位置有关，与距离向频率无关，因此可以在距离多普勒域进行补偿；Since ф _az (f _a , R) is only related to the azimuth frequency and the target position, and has nothing to do with the range frequency, it can be compensated in the range-Doppler domain;

2.2式(17)等号右侧的ф_RP(R)为精确二维频谱泰勒展开后的剩余相位，表达式为2.2 ф _RP (R) on the right side of the equation (17) is the residual phase after Taylor expansion of the exact two-dimensional spectrum, the expression is

${φ φ}_{RP RP} ((R R)) = = \frac{{k k}_{11}^{22}}{22 \cdot &Center Dot; λ λ \cdot \cdot {k k}_{22}} + + \frac{{k k}_{11}^{33} \cdot \cdot {k k}_{33}}{44 \cdot \cdot λ λ \cdot \cdot {k k}_{22}^{33}} + + \frac{{k k}_{11}^{44} \cdot \cdot ((99 \cdot \cdot {k k}_{33}^{22} - - 44 \cdot \cdot {k k}_{22} \cdot \cdot {k k}_{44}))}{3232 \cdot \cdot λ λ \cdot \cdot {k k}_{22}^{55}} - - \frac{22 \cdot \cdot R R}{λ λ} - - - - - - ((1919))$

该项与方位向频率和距离向频率无关，与目标的距离向位置有关，可以在距离多普勒域予以补偿；This item has nothing to do with the azimuth frequency and the range frequency, but is related to the range position of the target, and can be compensated in the range-Doppler domain;

2.3式(17)等号右侧的b(f_a，f_r)为二维频谱展开时得到的参考点处徙动相位，其表达式为2.3 The b(f _a , f _r ) on the right side of the equation (17) is the migratory phase at the reference point obtained when the two-dimensional spectrum is expanded, and its expression is

$b b (({f f}_{a a},, {f f}_{r r})) = = - - [[\frac{{k k}_{1010}^{22}}{22 \cdot &Center Dot; {k k}_{2020} \cdot \cdot c c} + + \frac{{k k}_{1010}^{33} \cdot \cdot {k k}_{3030}}{44 \cdot \cdot {k k}_{2020}^{33} \cdot &Center Dot; c c} + + \frac{{k k}_{1010} \cdot \cdot ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{3232 \cdot &Center Dot; c c \cdot &Center Dot; {k k}_{2020}^{55}}]] \cdot &Center Dot; {f f}_{r r} + +$

$[\frac{λ}{8 \cdot k_{20} \cdot f_{c}} + \frac{3 \cdot λ \cdot k_{10} \cdot k_{30}}{16 \cdot k_{20}^{3} \cdot f_{c}} + \frac{3 \cdot λ k_{10}^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{64 \cdot f_{c} \cdot k_{20}^{5}}] \cdot f_{a}^{2} \cdot f_{r} +$ (20) $[\frac{λ}{8 \cdot k_{20} &Center Dot; f_{c}} + \frac{3 &Center Dot; λ &Center Dot; k_{10} &Center Dot; k_{30}}{16 \cdot k_{20}^{3} &Center Dot; f_{c}} + \frac{3 &Center Dot; λ k_{10}^{2} &Center Dot; (9 \cdot k_{30}^{2} - 4 &Center Dot; k_{20} \cdot k_{40})}{64 \cdot f_{c} &Center Dot; k_{20}^{5}}] \cdot f_{a}^{2} \cdot f_{r} +$ (20)

$[[\frac{{λ λ}^{22} \cdot &Center Dot; {k k}_{3030}}{1616 \cdot &Center Dot; {k k}_{2020}^{33} \cdot \cdot {f f}_{c c}} + + \frac{{λ λ}^{22} \cdot &Center Dot; {k k}_{1010} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{3232 \cdot &Center Dot; {f f}_{c c} \cdot \cdot {k k}_{2020}^{55}}]] \cdot &Center Dot; {f f}_{a a}^{33} \cdot \cdot {f f}_{r r} + +$

$\frac{33 \cdot \cdot {λ λ}^{22} \cdot \cdot ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot \cdot {k k}_{2020} \cdot \cdot {k k}_{4040}))}{512512 \cdot \cdot {f f}_{c c} \cdot \cdot {k k}_{2020}^{55}} {f f}_{a a}^{44} \cdot \cdot {f f}_{r r} + + ((\frac{22 \cdot \cdot {R R}_{00}}{c c} - - {\overset{\cdot &Center Dot;}{B B}}_{11} \cdot &Center Dot; {R R}_{00})) \cdot \cdot {f f}_{r r}$

其中in

${\overset{\cdot &Center Dot;}{B B}}_{11} = = - - \frac{22 \cdot &Center Dot; {k k}_{1010} \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {\overset{. .}{k k}}_{11} - - {k k}_{1010}^{22} \cdot \cdot {\overset{. .}{k k}}_{22}}{22 \cdot &Center Dot; c c \cdot &Center Dot; {k k}_{2020}^{22}} - - \frac{((33 \cdot \cdot {k k}_{1010}^{22} \cdot \cdot {k k}_{3030} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{11} + + {k k}_{1010}^{33} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{33})) \cdot &Center Dot; {k k}_{2020} - - 33 \cdot &Center Dot; {k k}_{1010}^{33} \cdot \cdot {k k}_{3030} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{22}}{44 \cdot &Center Dot; c c \cdot &Center Dot; {k k}_{2020}^{44}} - - \frac{{A A}_{33} \cdot &Center Dot; {\overset{. .}{k k}}_{11} - - {k k}_{1010} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{A A}}_{33}}{3232 \cdot &Center Dot; c c} + +$

${{- - \frac{λ λ \cdot &Center Dot; {\overset{. .}{k k}}_{22}}{88 \cdot &Center Dot; {f f}_{c c} \cdot &Center Dot; {k k}_{2020}^{22}} + + \frac{33 \cdot &Center Dot; λ λ \cdot &Center Dot; [[(({k k}_{1010} \cdot \cdot {\overset{\cdot &Center Dot;}{k k}}_{33} + + {k k}_{3030} \cdot \cdot {\overset{. .}{k k}}_{11})) \cdot &Center Dot; {k k}_{2020} - - 33 \cdot &Center Dot; {k k}_{1010} \cdot \cdot {k k}_{3030} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{22}]]}{1616 \cdot &Center Dot; {f f}_{c c} \cdot \cdot {k k}_{2020}^{44}} + + \frac{33 \cdot &Center Dot; λ λ \cdot \cdot ((22 \cdot &Center Dot; {k k}_{1010} \cdot &Center Dot; {A A}_{33} \cdot \cdot {\overset{\cdot \cdot}{k k}}_{11} - - {k k}_{1010}^{22} \cdot \cdot {\overset{\cdot &Center Dot;}{A A}}_{33}))}{6464 \cdot &Center Dot; {f f}_{c c}}}} \cdot \cdot {f f}_{a a}^{22} ((21 twenty one))$

$+ + [[\frac{{λ λ}^{22} \cdot &Center Dot; (({k k}_{2020} \cdot \cdot {\overset{\cdot &Center Dot;}{k k}}_{33} - - 33 \cdot \cdot {k k}_{3030} \cdot \cdot {\overset{\cdot &Center Dot;}{k k}}_{22}))}{1616 \cdot &Center Dot; {f f}_{c c} \cdot \cdot {k k}_{2020}^{44}} + + \frac{{λ λ}^{22} \cdot \cdot (({A A}_{33} \cdot &Center Dot; {\overset{\cdot \cdot}{k k}}_{11} - - {k k}_{1010} \cdot &Center Dot; {\overset{\cdot \cdot}{A A}}_{33}))}{3232 \cdot \cdot {f f}_{c c}}]] \cdot &Center Dot; {f f}_{a a}^{33} + + \frac{33 \cdot \cdot {λ λ}^{22} \cdot &Center Dot; {\overset{. .}{A A}}_{33}}{512512 \cdot \cdot {f f}_{c c}} \cdot \cdot {f f}_{a a}^{44} + + \frac{22}{c c}$

${A A}_{33} = = \frac{99 \cdot \cdot {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}}{{k k}_{2020}^{55}} - - - - - - ((22 twenty two))$

${\overset{. .}{A A}}_{33} = = \frac{[[1818 \cdot &Center Dot; {k k}_{3030} \cdot &Center Dot; {\overset{. .}{k k}}_{33} - - 44 \cdot &Center Dot; (({k k}_{2020} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{44} + + {k k}_{4040} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{22}))]] {k k}_{2020} - - 55 \cdot \cdot ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040})) \cdot &Center Dot; {\overset{. .}{k k}}_{22}}{{k k}_{2020}^{66}} - - - - - - ((23 twenty three))$

该相位只是参考点处徙动相位的一部分，在传统的基于等效直线模型的CS、NCS算法中是没有这一项的，而在基于弯曲轨迹的CS算法、NCS算法中这一项是特有的，必须需要补偿；而由于该项不具有空变性，因此可以在二维频谱补偿；This phase is only a part of the migration phase at the reference point, which is not available in the traditional CS and NCS algorithms based on the equivalent straight line model, but is unique to the CS algorithm and NCS algorithm based on curved trajectories , it must be compensated; and since this item does not have space variation, it can be compensated in the two-dimensional spectrum;

2.4在式(17)等号右侧的第四个指数项即

中，

为距离徙动，M(f_a)为徙动因子且2.4 The fourth exponent term on the right side of the equation (17) is

middle,

is the distance migration, M(f _a ) is the migration factor and

$M m (({f f}_{a a})) = = \frac{11}{{\overset{. .}{B B}}_{11} \cdot &Center Dot; c c} - - - - - - ((24 twenty four))$

其中的表达式如(21)所示；in The expression of is shown in (21);

可以发现该项对于不同的距离向位置徙动不一致，在CS算法中主要是在距离多普勒域乘以一个Chirp信号调整这一空变性，在NCS算法中同样需要调整这一空变性；It can be found that this item is inconsistent for different distances to position migration. In the CS algorithm, this spatial variability is mainly multiplied by a Chirp signal in the range Doppler domain. This spatial variability also needs to be adjusted in the NCS algorithm;

2.5在式(17)等号右侧的第五个指数项即

中

为距离向调制项，其中k_s(f_a，R)为新的距离向调频因子，且2.5 The fifth exponent term on the right side of the equal sign in formula (17) is

middle

is the range modulation item, where k _s (f _a , R) is the new range frequency modulation factor, and

$\frac{1}{K_{s} (f_{a}, R)} = - [\frac{λ}{4 \cdot k_{2} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{1} \cdot k_{3}}{8 \cdot k_{2}^{3} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{1}^{2} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{32 \cdot k_{2}^{5} \cdot f_{c}^{2}}] \cdot f_{a}^{2} -$ (25) $\frac{1}{K_{the s} (f_{a}, R)} = - [\frac{λ}{4 &Center Dot; k_{2} \cdot f_{c}^{2}} + \frac{3 \cdot λ &Center Dot; k_{1} &Center Dot; k_{3}}{8 &Center Dot; k_{2}^{3} &Center Dot; f_{c}^{2}} + \frac{3 &Center Dot; λ \cdot k_{1}^{2} \cdot (9 &Center Dot; k_{3}^{2} - 4 &Center Dot; k_{2} &Center Dot; k_{4})}{32 \cdot k_{2}^{5} &Center Dot; f_{c}^{2}}] \cdot f_{a}^{2} -$ (25)

$[[\frac{33 \cdot &Center Dot; {λ λ}^{22} \cdot &Center Dot; {k k}_{33}}{1616 \cdot &Center Dot; {k k}_{22}^{33} \cdot &Center Dot; {f f}_{c c}^{22}} + + \frac{33 \cdot \cdot {λ λ}^{22} \cdot &Center Dot; {k k}_{11} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot \cdot {k k}_{44}))}{3232 \cdot &Center Dot; {k k}_{22}^{55} \cdot &Center Dot; {f f}_{c c}^{22}}]] \cdot &Center Dot; {f f}_{a a}^{33} - - \frac{33 \cdot &Center Dot; {λ λ}^{33} \cdot \cdot ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot &Center Dot; {k k}_{44}))}{128128 \cdot \cdot {k k}_{22}^{55} \cdot &Center Dot; {f f}_{c c}^{22}} \cdot &Center Dot; {f f}_{a a}^{44} + + \frac{11}{{K K}_{r r}}$

k_s(f_a，R)具有空变性，在CS算法中并不考虑它的空变性，在NCS算法中它是空变性调制的一部分，但是它很难直接应用，为此需要进行近似：k _s (f _a , R) is space-variant, and its space-variation is not considered in the CS algorithm. In the NCS algorithm, it is a part of the space-variation modulation, but it is difficult to apply directly, and an approximation is needed for this:

K_s(f_a，R)＝k_s(f_a，R₀)+Δk_s(f_a)·[τ(f_a，R)-τ(f_a，R₀)] (26)K _s (f _a , R)=k _s (f _a , R ₀ )+Δk _s (f _a )·[τ(f _a ,R)-τ(f _a ,R ₀ )] (26)

其中in

$\frac{1}{K_{s} (f_{a}, R_{0})} = - [\frac{λ}{4 \cdot k_{20} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{10} \cdot k_{30}}{8 \cdot k_{20}^{3} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{10}^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{32 \cdot k_{20}^{5} \cdot f_{c}^{2}}] \cdot f_{a}^{2} -$ (27) $\frac{1}{K_{the s} (f_{a}, R_{0})} = - [\frac{λ}{4 \cdot k_{20} \cdot f_{c}^{2}} + \frac{3 &Center Dot; λ &Center Dot; k_{10} &Center Dot; k_{30}}{8 \cdot k_{20}^{3} &Center Dot; f_{c}^{2}} + \frac{3 \cdot λ &Center Dot; k_{10}^{2} &Center Dot; (9 &Center Dot; k_{30}^{2} - 4 &Center Dot; k_{20} &Center Dot; k_{40})}{32 &Center Dot; k_{20}^{5} &Center Dot; f_{c}^{2}}] &Center Dot; f_{a}^{2} -$ (27)

$[[\frac{33 \cdot &Center Dot; {λ λ}^{22} \cdot \cdot {k k}_{3030}}{1616 \cdot &Center Dot; {k k}_{2020}^{33} \cdot &Center Dot; {f f}_{c c}^{22}} + + \frac{33 \cdot &Center Dot; {λ λ}^{22} \cdot &Center Dot; {k k}_{1010} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{3232 \cdot &Center Dot; {k k}_{2020}^{55} \cdot &Center Dot; {f f}_{c c}^{22}}]] \cdot &Center Dot; {f f}_{a a}^{33} - - \frac{33 \cdot &Center Dot; {λ λ}^{33} \cdot \cdot ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{128128 \cdot &Center Dot; {k k}_{2020}^{55} \cdot &Center Dot; {f f}_{c c}^{22}} \cdot &Center Dot; {f f}_{a a}^{44} + + \frac{11}{{K K}_{r r}}$

${Δk Δk}_{s the s} (({f f}_{a a})) = = \frac{{K K}_{s the s}^{22} (({f f}_{a a},, {R R}_{00})) \cdot &Center Dot; c c \cdot &Center Dot; M m (({f f}_{a a}))}{22} . .$

$\{\begin{matrix} - - \frac{λ λ \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{22} \cdot \cdot {f f}_{a a}^{22}}{44 \cdot \cdot {f f}_{c c}^{22} \cdot \cdot {k k}_{2020}^{22}} + + \frac{33 \cdot \cdot λ λ \cdot \cdot [[(({k k}_{1010} \cdot \cdot {\overset{\cdot \cdot}{k k}}_{33} + + {k k}_{3030} \cdot \cdot {\overset{\cdot \cdot}{k k}}_{11})) \cdot \cdot {k k}_{2020} - - 33 \cdot &Center Dot; {k k}_{1010} \cdot \cdot {k k}_{3030} \cdot &Center Dot; {\overset{\cdot \cdot}{k k}}_{22}]]}{88 \cdot \cdot {f f}_{c c}^{22} \cdot &Center Dot; {k k}_{2020}^{44}} \cdot \cdot {f f}_{a a}^{22} + + \\ \frac{33 \cdot &Center Dot; λ λ \cdot \cdot ((22 \cdot &Center Dot; {k k}_{1010} \cdot \cdot {A A}_{33} \cdot \cdot {\overset{. .}{k k}}_{11} - - {k k}_{1010}^{22} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{A A}}_{33}))}{3232 \cdot \cdot {f f}_{c c}^{22}} \cdot &Center Dot; {f f}_{a a}^{22} + + \frac{33 \cdot &Center Dot; {λ λ}^{22} \cdot &Center Dot; (({k k}_{2020} \cdot \cdot {\overset{\cdot \cdot}{k k}}_{33} - - 33 \cdot &Center Dot; {k k}_{3030} \cdot \cdot {\overset{\cdot &Center Dot;}{k k}}_{22}))}{1616 \cdot \cdot {f f}_{c c}^{22} \cdot \cdot {k k}_{2020}^{44}} \cdot &Center Dot; {f f}_{a a}^{33} \\ + + \frac{33 \cdot &Center Dot; {λ λ}^{22} \cdot \cdot (({A A}_{33} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{k k}}_{11} - - {k k}_{1010} \cdot &Center Dot; {\overset{\cdot &Center Dot;}{A A}}_{33}))}{3232 \cdot &Center Dot; {f f}_{c c}^{22}} \cdot \cdot {f f}_{a a}^{33} + + \frac{33 \cdot &Center Dot; {λ λ}^{33} \cdot &Center Dot; {\overset{\cdot \cdot}{A A}}_{33}}{128128 \cdot &Center Dot; {f f}_{c c}^{22}} {f f}_{a a}^{44} \end{matrix}\} - - - - - - ((2828))$

$τ τ (({f f}_{a a},, R R)) = = \frac{22 \cdot \cdot R R}{c c \cdot &Center Dot; M m (({f f}_{a a}))} - - - - - - ((2929))$

$τ τ (({f f}_{a a},, {R R}_{00})) = = \frac{22 \cdot \cdot {R R}_{00}}{c c \cdot \cdot M m (({f f}_{a a}))} - - - - - - ((3030))$

根据式(26)～(30)，得到在NCS算法中用到的操作因子Y_m(f_a)、q₂和q₃，它们的表达式分别为(31)～(33)，其中Y_m(f_a)一方面要除三次相位的影响，另一方面用于调整由于后续的调频率空变形的调整所引入的残留三次相位误差；q₂和q₃主要用于调整距离徙动的空变性和调频率的空变性。According to formulas (26)-(30), the operation factors Y _m (f _a ), q ₂ and q ₃ used in the NCS algorithm are obtained, and their expressions are (31)-(33), where Y _m (f _a ) On the one hand, it is necessary to remove the influence of the third-order phase, and on the other hand, it is used to adjust the residual third-order phase error introduced by the subsequent adjustment of frequency-space deformation; q ₂ and q ₃ are mainly used to adjust the space of distance migration Spatial variability of denaturation and tuning frequency.

${Y Y}_{m m} (({f f}_{a a})) = = \frac{{Δk Δk}_{s the s} (({f f}_{a a})) \cdot &Center Dot; ((M m (({f f}_{ref ref})) / / M m (({f f}_{a a})) - - 0.5 0.5))}{{K K}_{s the s}^{33} (({f f}_{a a},, {R R}_{00})) \cdot \cdot ((M m (({f f}_{ref ref})) / / M m (({f f}_{a a})) - - 11))} - - - - - - ((3131))$

q₂＝K_s(f_a，R₀)·(M(f_ref)/M(f_a)-1) (32)q ₂ =K _s (f _a , R ₀ )·(M(f _ref )/M(f _a )-1) (32)

${q q}_{33} = = \frac{{Δk Δk}_{s the s} (({f f}_{a a})) \cdot \cdot ((M m (({f f}_{ref ref})) / / M m (({f f}_{a a})) - - 11))}{22} - - - - - - ((3333))$

2.6在式(17)等号右侧的第六个指数项

中的

为在二维频谱解耦时得到的，与距离向频率的三次方有关，且2.6 The sixth exponent term on the right side of the equal sign in formula (17)

middle

is obtained when the two-dimensional spectrum is decoupled, and is related to the cube of the distance to the frequency, and

${φ φ}_{33} (({f f}_{a a},, R R)) = =$

$22 \cdot \cdot π π \cdot \cdot [\begin{matrix} - - \frac{λ λ \cdot \cdot {f f}_{a a}^{22}}{88 \cdot \cdot {k k}_{22} \cdot \cdot {f f}_{c c}^{33}} - - \frac{33 \cdot \cdot λ λ \cdot \cdot {k k}_{11} \cdot \cdot {k k}_{33}}{1616 \cdot &Center Dot; {k k}_{22}^{33} \cdot \cdot {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{22} - - \frac{33 \cdot &Center Dot; λ λ \cdot \cdot {k k}_{11}^{22} \cdot \cdot ((99 \cdot \cdot {k k}_{33}^{22} - - 44 \cdot \cdot {k k}_{22} \cdot \cdot {k k}_{44}))}{6464 \cdot &Center Dot; {k k}_{22}^{55} \cdot &Center Dot; {f f}_{c c}^{33}} \cdot &Center Dot; {f f}_{a a}^{22} \\ - - \frac{{λ λ}^{22} \cdot &Center Dot; {k k}_{33} \cdot &Center Dot; {f f}_{a a}^{33}}{88 \cdot \cdot {k k}_{22}^{33} \cdot &Center Dot; {f f}_{c c}^{33}} - - \frac{{λ λ}^{22} \cdot \cdot {k k}_{11} \cdot &Center Dot; ((99 \cdot \cdot {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot \cdot {k k}_{44}))}{1616 \cdot \cdot {k k}_{22}^{55} \cdot &Center Dot; {f f}_{c c}^{33}} \cdot &Center Dot; {f f}_{a a}^{33} - - \frac{55 \cdot \cdot {λ λ}^{33} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{33}^{22} - - 44 \cdot &Center Dot; {k k}_{22} \cdot &Center Dot; {k k}_{44}))}{256256 \cdot \cdot {k k}_{22}^{55} \cdot \cdot {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{44} \end{matrix}] - - - - - - ((3434))$

ф₃(f_a，R)具有空变性，但是它随距离向的变化可以忽略，一般用ф₃(f_a，R₀)代替ф₃(f_a，R)，也即在(34)中k₁～k₄要用R₀处的结果k₁₀～k₄₀，因此，ф₃(f_a，R₀)的表达式为ф ₃ (f _a , R) is space-variant, but its variation with distance can be ignored. Generally, ф ₃ (f _a , R ₀ ) is used to replace ф ₃ (f _a , R), that is, in (34) k ₁ ~ k ₄ need to use the result k ₁₀ ~ k ₄₀ at R ₀ , therefore, the expression of ф ₃ (f _a , R ₀ ) is

${φ φ}_{33} (({f f}_{a a},, R R)) = = {φ φ}_{33} (({f f}_{a a},, {R R}_{00})) = =$

$22 \cdot \cdot π π \cdot \cdot [\begin{matrix} - - \frac{λ λ \cdot \cdot {f f}_{a a}^{22}}{88 \cdot \cdot {k k}_{2020} \cdot \cdot {f f}_{c c}^{33}} - - \frac{33 \cdot \cdot λ λ \cdot \cdot {k k}_{1010} \cdot \cdot {k k}_{3030}}{1616 \cdot \cdot {k k}_{2020}^{33} \cdot \cdot {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{22} - - \frac{33 \cdot \cdot λ λ \cdot \cdot {k k}_{1010}^{22} \cdot \cdot ((99 \cdot \cdot {k k}_{3030}^{22} - - 44 \cdot \cdot {k k}_{2020} \cdot \cdot {k k}_{4040}))}{6464 \cdot &Center Dot; {k k}_{2020}^{55} \cdot \cdot {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{22} \\ - - \frac{{λ λ}^{22} \cdot \cdot {k k}_{3030} \cdot \cdot {f f}_{a a}^{33}}{88 \cdot \cdot {k k}_{2020}^{33} \cdot \cdot {f f}_{c c}^{33}} - - \frac{{λ λ}^{22} \cdot &Center Dot; {k k}_{11} \cdot \cdot ((99 \cdot \cdot {k k}_{3030}^{22} - - 44 \cdot \cdot {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{1616 \cdot &Center Dot; {k k}_{2020}^{55} \cdot \cdot {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{33} - - \frac{55 \cdot &Center Dot; {λ λ}^{33} \cdot &Center Dot; ((99 \cdot &Center Dot; {k k}_{3030}^{22} - - 44 \cdot &Center Dot; {k k}_{2020} \cdot &Center Dot; {k k}_{4040}))}{256256 \cdot &Center Dot; {k k}_{2020}^{55} \cdot &Center Dot; {f f}_{c c}^{33}} \cdot \cdot {f f}_{a a}^{44} \end{matrix}] - - - - - - ((3535))$

有益效果Beneficial effect

本发明相对于现有技术相比，其优势在于：通过高阶泰勒展开的方法得到了一种新的适用于GEO SAR的弯曲轨迹模型，该轨迹模型可以解决GEO SAR近地点等效直线模型误差比较大，远地点等效直线模型完全不能应用等缺点；同时基于等效直线模型，得到了一个解析适用于NCS算法的二位频谱，利用此频谱，NCS算法的各个补偿函数都可以求得，实现了GEO SAR大场景成像的要求。Compared with the prior art, the present invention has the advantage that a new curved trajectory model suitable for GEO SAR is obtained through the method of high-order Taylor expansion, and the trajectory model can solve the error comparison of GEO SAR perigee equivalent straight line model. large, the apogee equivalent straight line model cannot be applied at all; at the same time, based on the equivalent straight line model, a binary spectrum suitable for the NCS algorithm is obtained. Using this spectrum, each compensation function of the NCS algorithm can be obtained, and the realization of GEO SAR large scene imaging requirements.

附图说明 Description of drawings

图1为本发明改进后的NCS算法技术方案实施流程图；Fig. 1 is the implementation flowchart of the NCS algorithm technical scheme after the improvement of the present invention;

图2为本发明实施例中的点目标(-30km，-30km)的仿真验证结果图；Fig. 2 is the simulation verification result figure of the point target (-30km,-30km) in the embodiment of the present invention;

图3为本发明实施例中的点目标(30km，-30km)的仿真验证结果图；Fig. 3 is the simulation verification result figure of the point target (30km,-30km) in the embodiment of the present invention;

图4为本发明实施例中的点目标(0km，0km)的仿真验证结果图；Fig. 4 is the simulation verification result figure of the point target (0km, 0km) in the embodiment of the present invention;

图5为本发明实施例中的点目标(-30km，30km)的仿真验证结果图；Fig. 5 is the simulation verification result figure of the point target (-30km, 30km) in the embodiment of the present invention;

图6为本发明实施例中的点目标(30km，30km)的仿真验证结果图；Fig. 6 is the simulation verification result figure of the point target (30km, 30km) in the embodiment of the present invention;

具体实施方式 Detailed ways

下面结合附图对本发明方法的实施方式做详细说明。The implementation of the method of the present invention will be described in detail below in conjunction with the accompanying drawings.

实施例Example

雷达在地球同步轨道上以一定的速度飞行，向地面发射chirp信号，并接受来自地面的回波。对得到的回波进行成像处理，提出一种适用于GEO SAR改进的NCS算法，其具体步骤如图1所示，包括：The radar flies at a certain speed in a geosynchronous orbit, transmits chirp signals to the ground, and receives echoes from the ground. Imaging processing is performed on the obtained echoes, and an improved NCS algorithm suitable for GEO SAR is proposed. The specific steps are shown in Figure 1, including:

1)雷达接收到的目标回波1) The target echo received by the radar

雷达发射一载波频率为f_c的线性调频信号。对接收到的回波经过解调后，可以得到The radar transmits a chirp signal with a carrier frequency _fc . After demodulating the received echo, we can get

$s the s (({t t}_{r r},, {t t}_{a a})) = = {u u}_{r r} (({t t}_{r r} - - \frac{22 \cdot \cdot {R R}_{n no}}{c c})) \cdot &Center Dot; {u u}_{a a} (({t t}_{a a})) \cdot \cdot exp exp [[j j \cdot \cdot π π \cdot &Center Dot; {K K}_{r r} \cdot &Center Dot; {(({t t}_{r r} - - \frac{22 \cdot \cdot {R R}_{n no}}{c c}))}^{22}]] \cdot &Center Dot; exp exp ((- - j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π}{λ λ} \cdot &Center Dot; {R R}_{n no})) - - - - - - ((3636))$

其中u_r(·)和u_a(·)分别为距离向和方位向包络，t_r和t_a分别为距离向和方位向时间，K_r为距离向调频率，c为光速，λ为波长，R_n为目标的斜距历史，具体表达式可以用公式(2)来表示。where u _r (·) and u _a (·) are the range and azimuth envelopes respectively, t _r and t _a are the range and azimuth time respectively, K _r is the range modulation frequency, c is the speed of light, λ is Wavelength, R _n is the slant distance history of the target, and the specific expression can be expressed by formula (2).

2)对回波进行距离徙动初校正和三次相位去除2) Carry out initial range migration correction and triple phase removal on the echo

距离向和方位向FFT后，可以得到SAR回波的二维频谱表达式，即After range and azimuth FFT, the two-dimensional spectrum expression of the SAR echo can be obtained, namely

$S (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{az} (f_{a}, R)] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{RP} (R)]$ (37) $S (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] &Center Dot; \exp [j \cdot 2 &Center Dot; π &Center Dot; φ_{az} (f_{a}, R)] &Center Dot; \exp [j &Center Dot; 2 &Center Dot; π &Center Dot; φ_{RP} (R)]$ (37)

$\cdot \cdot exp exp [[- - j j \cdot \cdot 22 \cdot \cdot π π \cdot \cdot b b (({f f}_{a a},, {f f}_{r r}))]] \cdot \cdot exp exp [[- - j j \cdot \cdot \frac{44 \cdot \cdot π π \cdot \cdot R R}{c c \cdot \cdot M m (({f f}_{a a}))} \cdot \cdot {f f}_{r r}]] exp exp [[- - j j \cdot \cdot π π \cdot \cdot \frac{{f f}_{r r}^{22}}{{K K}_{s the s} (({f f}_{a a},, R R))}]] \cdot \cdot exp exp [[j j \cdot \cdot {φ φ}_{33} (({f f}_{a a},, R R)) \cdot \cdot {f f}_{r r}^{33}]]$

在二维频域需要做两个工作。第一，要去除在二维频谱推导中得到的参考点处的徙动相位b(f_a，f_r)，以方便后续算法的推导，补偿函数的表达式为(38)。需要说明的是这里去除的距离徙动只是参考点距离徙动的一部分；第二，要乘以一个非线性调频函数，表达式为(39)，该函数一方面要去除三次相位的影响，另一方面用于调整由于后续的调频率空变形的调整所引入的残留三次相位误差。In the two-dimensional frequency domain, two tasks need to be done. First, the migratory phase b(f _a , f _r ) at the reference point obtained in the derivation of the two-dimensional spectrum should be removed to facilitate the derivation of the subsequent algorithm. The expression of the compensation function is (38). It should be noted that the distance migration removed here is only a part of the distance migration of the reference point; secondly, it needs to be multiplied by a non-linear frequency modulation function, the expression is (39), on the one hand, this function needs to remove the influence of the third phase, On the other hand, it is used to adjust the residual third-order phase error introduced by the subsequent adjustment of modulation frequency-space deformation.

H₁＝exp[j·2·π·B₁₀(f_a，f_r)] (38)H ₁ = exp[j·2·π·B ₁₀ (f _a , f _r )] (38)

${H h}_{22} = = exp exp [[j j \cdot \cdot \frac{22 \cdot \cdot π π}{33} \cdot &Center Dot; (({f f}_{a a})) \cdot &Center Dot; {f f}_{r r}^{33}]] - - - - - - ((3939))$

其中in

$Y Y (({f f}_{a a})) = = \frac{{Δk Δk}_{s the s} (({f f}_{a a})) \cdot &Center Dot; ((α α - - 0.5 0.5))}{{K K}_{s the s}^{33} (({f f}_{a a},, {R R}_{00})) \cdot \cdot ((α α - - 11))} - - \frac{33}{22 \cdot &Center Dot; π π} \cdot \cdot {φ φ}_{33} (({f f}_{a a},, {R R}_{00})) - - - - - - ((4040))$

在(40)中，α的表达式为In (40), the expression of α is

$α α = = \frac{M m (({f f}_{ref ref}))}{M m (({f f}_{a a}))} - - - - - - ((4141))$

f_ref为方位向参考频率。f _ref is the azimuth reference frequency.

经过处理后的回波表达式为The processed echo expression is

$S_{1} (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{az} (f_{a}, R)] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{RP} (R)]$ (42) $S_{1} (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{{2 k}_{1}}{c} (f_{r} + f_{c})] &Center Dot; \exp [j &Center Dot; 2 &Center Dot; π &Center Dot; φ_{az} (f_{a}, R)] &Center Dot; \exp [j \cdot 2 \cdot π &Center Dot; φ_{RP} (R)]$ (42)

$\cdot &Center Dot; exp exp [[- - j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π \cdot \cdot R R}{c c \cdot \cdot M m (({f f}_{a a}))} \cdot \cdot {f f}_{r r}]] \cdot &Center Dot; exp exp [[- - j j \cdot &Center Dot; π π \cdot &Center Dot; \frac{{f f}_{r r}^{22}}{{K K}_{s the s} (({f f}_{a a},, R R))}]] \cdot &Center Dot; exp exp [[j j \cdot &Center Dot; \frac{22 \cdot &Center Dot; π π}{33} \cdot &Center Dot; {Y Y}_{m m} (({f f}_{a a})) \cdot &Center Dot; {f f}_{r r}^{33}]]$

其中in

${Y Y}_{m m} (({f f}_{a a})) = = \frac{{Δk Δk}_{s the s} (({f f}_{a a})) \cdot &Center Dot; ((α α - - 0.5 0.5))}{{K K}_{s the s}^{33} (({f f}_{a a},, {R R}_{00})) \cdot &Center Dot; ((α α - - 11))} - - - - - - ((4343))$

3)对回波数据进行距离徙动的空变性和调频率的空变性的调整距离向IFFT，得到回波的距离多普勒域的表达式为3) Adjust the range-wise IFFT of the space-variation of the range migration and the space-variation of the modulation frequency on the echo data, and obtain the expression of the range-Doppler domain of the echo as

${S S}_{11} (({t t}_{r r},, {f f}_{a a})) = = {u u}_{r r} {{\frac{{K K}_{s the s} (({f f}_{a a},, R R))}{{k k}_{r r}} [[{t t}_{r r} - - \frac{22 \cdot \cdot R R}{c c \cdot \cdot M m (({f f}_{a a}))}]]}} \cdot \cdot {u u}_{a a} (({f f}_{a a})) \cdot \cdot exp exp [[j j \cdot \cdot 22 \cdot \cdot π π \cdot \cdot {φ φ}_{az az} (({f f}_{a a},, R R))]] \cdot \cdot$

$exp exp [[j j \cdot \cdot 22 \cdot \cdot π π \cdot \cdot {φ φ}_{RP RP} ((R R))]] \cdot &Center Dot; exp exp {{j j \cdot \cdot π π \cdot \cdot {K K}_{s the s} (({f f}_{a a},, R R)) {[[{t t}_{r r} - - \frac{22 \cdot \cdot R R}{c c \cdot &Center Dot; M m (({f f}_{a a}))}]]}^{22}}} \cdot &Center Dot; - - - - - - ((4444))$

$exp exp {{j j \cdot &Center Dot; \frac{22 \cdot \cdot π π}{33} \cdot &Center Dot; {Y Y}_{m m} (({f f}_{a a})) \cdot &Center Dot; {K K}_{s the s} (({f f}_{a a},, R R)) {[[{t t}_{r r} - - \frac{22 \cdot &Center Dot; R R}{c c \cdot &Center Dot; M m (({f f}_{a a}))}]]}^{33}}}$

在距离多普勒域进行非线性CS操作，主要是调整距离徙动的空变性和调频率的空变性。经过此步的操作后，场景内的空变性去除，实现距离徙动可以在二维频域进行统一处理。非线性CS操作函数为The non-linear CS operation in the range-Doppler domain is mainly to adjust the spatial variation of the range migration and the spatial variation of the modulation frequency. After this step, the spatial variability in the scene is removed, and distance migration can be processed uniformly in the two-dimensional frequency domain. The nonlinear CS operation function is

${H h}_{33} = = exp exp {{j j \cdot \cdot π π \cdot &Center Dot; {q q}_{22} \cdot \cdot {[[{t t}_{r r} - - τ τ (({f f}_{a a},, {R R}_{00}))]]}^{22}}} \cdot &Center Dot; exp exp {{j j \cdot &Center Dot; \frac{22 \cdot &Center Dot; π π}{33} \cdot &Center Dot; {q q}_{33} \cdot &Center Dot; {[[{t t}_{r r} - - τ τ (({f f}_{a a},, {R R}_{00}))]]}^{33}}} - - - - - - ((4545))$

其中in

q₂＝q₂＝k_s(f_a，R₀)·(α-1) (46)q ₂ =q ₂ =k _s (f _a ,R ₀ )·(α-1) (46)

${q q}_{33} = = {q q}_{33} = = \frac{{Δk Δk}_{s the s} (({f f}_{a a})) \cdot &Center Dot; ((α α - - 11))}{22} - - - - - - ((4747))$

4)对回波进行距离向压缩和距离徙动校正4) Perform range compression and range migration correction on the echo

(45)和(44)相乘后，再进行距离向FFT。此时的回波数据在二维频域，此时回波表达式为(45) and (44) are multiplied, and then the range FFT is performed. The echo data at this time is in the two-dimensional frequency domain, and the echo expression at this time is

${S S}_{22} (({f f}_{r r},, {f f}_{a a})) = = {u u}_{r r} [[\frac{{f f}_{r r}}{{K K}_{s the s} (({f f}_{a a},, {R R}_{00})) \cdot &Center Dot; α α}]] \cdot &Center Dot; {u u}_{a a} (({f f}_{a a})) \cdot &Center Dot; exp exp [[j j \cdot &Center Dot; 22 \cdot &Center Dot; π π \cdot &Center Dot; {φ φ}_{az az} (({f f}_{a a},, R R))]] \cdot &Center Dot; exp exp [[j j \cdot \cdot 22 \cdot \cdot {φ φ}_{RP RP} ((R R))]] \cdot &Center Dot;$

$exp exp [[- - j j \cdot \cdot π π \cdot \cdot \frac{{f f}_{r r}^{22}}{α α \cdot \cdot {K K}_{s the s} (({f f}_{a a},, {R R}_{00}))}]] \cdot \cdot exp exp {{j j \cdot \cdot \frac{22 \cdot \cdot π π}{33} \cdot \cdot \frac{[[{Y Y}_{m m} (({f f}_{a a})) \cdot \cdot {K K}_{s the s}^{33} (({f f}_{a a},, {R R}_{00})) + + {q q}_{33}]]}{{[[α α \cdot \cdot {K K}_{s the s} (({f f}_{a a},, {R R}_{00}))]]}^{33}} \cdot \cdot {f f}_{r r}^{33}}} \cdot \cdot - - - - - - ((4848))$

$exp exp {{- - j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π \cdot &Center Dot; {R R}_{00}}{c c} \cdot &Center Dot; [[\frac{11}{M m (({f f}_{a a}))} - - \frac{11}{M m (({f f}_{ref ref}))}]] \cdot &Center Dot; {f f}_{r r}}} exp exp [[- - j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π \cdot &Center Dot; R R}{c c \cdot \cdot M m (({f f}_{ref ref}))} \cdot &Center Dot; {f f}_{r r}]] \cdot \cdot exp exp ((j j \cdot &Center Dot; π π \cdot &Center Dot; {C C}_{00}))$

在(48)中，最后一个指数项是关于C₀，它是在求解Y_m(f_a)、q₂和q₃过程中残留下的，具体表达式为In (48), the last exponential term is about C ₀ , which is left over from the process of solving Y _m (f _a ), q ₂ and q ₃ , and the specific expression is

$C_{0} = K_{s} (f_{a}, R) \cdot {Δτ}^{2} \cdot {(\frac{1}{α} - 1)}^{2} + \frac{2}{3} \cdot Y_{m} (f_{a}) \cdot K_{s}^{3} (f_{a}, R) \cdot {Δτ}^{3} \cdot {(\frac{1}{α} - 1)}^{3}$ (49) $C_{0} = K_{the s} (f_{a}, R) &Center Dot; {Δτ}^{2} &Center Dot; {(\frac{1}{α} - 1)}^{2} + \frac{2}{3} \cdot Y_{m} (f_{a}) \cdot K_{the s}^{3} (f_{a}, R) &Center Dot; {Δτ}^{3} &Center Dot; {(\frac{1}{α} - 1)}^{3}$ (49)

$+ + {q q}_{22} \cdot &Center Dot; {((\frac{Δτ Δτ}{α α}))}^{22} + + \frac{22}{33} \cdot &Center Dot; {q q}_{33} \cdot &Center Dot; {((\frac{Δτ Δτ}{α α}))}^{33}$

其中in

Δτ＝τ(f_a，R)-τ(f_a，R₀) (50)Δτ=τ(f _a ,R)-τ(f _a ,R ₀ ) (50)

C₀不仅与方位向频率有关，而且是沿着距离向在变化，因此(48)中最后一个指数项的补偿只能在距离多普勒域进行。C ₀ is not only related to the azimuth frequency, but also changes along the range direction, so the compensation of the last exponential term in (48) can only be performed in the range-Doppler domain.

由于经过上一步的非线性CS操作，距离徙动校正可以在方位频域进行统一处理，而且在二维频域要进行距离向压缩和二次距离压缩等。距离向压缩函数为Due to the nonlinear CS operation in the previous step, range migration correction can be processed uniformly in the azimuth frequency domain, and range compression and secondary range compression must be performed in the two-dimensional frequency domain. The distance compression function is

${H h}_{44} = = exp exp [[j j \cdot &Center Dot; π π \cdot &Center Dot; \frac{{f f}_{r r}^{22}}{α α \cdot &Center Dot; {K K}_{s the s} (({f f}_{a a},, {R R}_{00}))}]] - - - - - - ((5151))$

二次距离压缩函数为The quadratic distance compression function is

${H h}_{55} = = exp exp {{- - j j \cdot &Center Dot; \frac{22 \cdot &Center Dot; π π}{33} \cdot &Center Dot; \frac{[[{Y Y}_{m m} (({f f}_{a a})) \cdot &Center Dot; {K K}_{s the s}^{33} (({f f}_{a a},, {R R}_{00})) + + {q q}_{33}]]}{{[[α α \cdot &Center Dot; {K K}_{s the s} (({f f}_{a a},, {R R}_{00}))]]}^{33}} \cdot &Center Dot; {f f}_{r r}^{33}}} - - - - - - ((5252))$

距离徙动校正函数为The distance migration correction function is

${H h}_{66} = = exp exp {{j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π \cdot &Center Dot; {R R}_{00}}{c c} \cdot &Center Dot; [[\frac{11}{M m (({f f}_{a a}))} - - \frac{11}{M m (({f f}_{ref ref}))}]] \cdot &Center Dot; {f f}_{r r}}} - - - - - - ((5353))$

距离向压缩、二次距离压缩和距离徙动校正后的回波表达式为The echo expression after range compression, secondary range compression and range migration correction is

$S_{3} (f_{r}, f_{a}) = u_{r} {\frac{f_{r}}{K_{s} (f_{a}, R_{0}) \cdot α}} \cdot u_{a} (f_{a}) \cdot \exp [j \cdot 2 \cdot π \cdot φ_{az} (f_{a}, R)] \cdot$ (54) $S_{3} (f_{r}, f_{a}) = u_{r} {\frac{f_{r}}{K_{the s} (f_{a}, R_{0}) &Center Dot; α}} &Center Dot; u_{a} (f_{a}) &Center Dot; \exp [j \cdot 2 &Center Dot; π &Center Dot; φ_{az} (f_{a}, R)] &Center Dot;$ (54)

$exp exp [[j j \cdot &Center Dot; 22 \cdot &Center Dot; π π \cdot &Center Dot; {φ φ}_{RP RP} ((R R))]] \cdot &Center Dot; exp exp [[- - j j \cdot &Center Dot; \frac{44 \cdot &Center Dot; π π \cdot &Center Dot; R R}{c c \cdot &Center Dot; M m (({f f}_{ref ref}))} \cdot &Center Dot; {f f}_{r r}]] \cdot &Center Dot; exp exp ((j j \cdot &Center Dot; π π \cdot &Center Dot; {C C}_{00}))$

5)对回波进行方位向压缩5) Azimuth compression of the echo

经过距离向压缩和距离徙动校正后，此时的回波在距离向上已经聚焦好，需要进行的是方位向压缩，方位向压缩函数要沿着不同的距离门进行更新。首先将在二维频域经过距离压缩和徙动校正处理后的回波数据进行距离向IFFT，得到距离多普勒域的回波表达式After range compression and range migration correction, the echo at this time has been focused in the range direction, and what needs to be performed is azimuth compression, and the azimuth compression function needs to be updated along different range gates. Firstly, the range-directed IFFT is performed on the echo data processed by range compression and migration correction in the two-dimensional frequency domain, and the echo expression in the range-Doppler domain is obtained

${S S}_{33} (({t t}_{r r},, {f f}_{a a})) = = sin sin c c [[{t t}_{r r} - - \frac{22 \cdot &Center Dot; R R}{c c \cdot &Center Dot; M m (({f f}_{ref ref}))}]] \cdot &Center Dot; {u u}_{a a} (({f f}_{a a})) \cdot &Center Dot; - - - - - - ((5555))$

$exp exp [[j j \cdot &Center Dot; 22 \cdot &Center Dot; π π \cdot &Center Dot; {φ φ}_{az az} (({f f}_{a a},, R R))]] \cdot &Center Dot; exp exp [[j j \cdot \cdot 22 \cdot &Center Dot; π π \cdot &Center Dot; {φ φ}_{RP RP} ((R R))]] \cdot &Center Dot; exp exp ((j j \cdot &Center Dot; π π \cdot \cdot {C C}_{00}))$

进行方位向压缩，压缩函数为Perform azimuth compression, the compression function is

H₇＝exp[-j·2·π·ф_az(f_a，R)] (56)H ₇ ＝exp[-j·2·π·ф _az (f _a , R)] (56)

然后再进行残留相位的去除，表达式为Then remove the residual phase, the expression is

H₈＝exp[-j·2·π·ф_RP(R)-j·π·C₀] (57)H ₈ ＝exp[-j·2·π·ф _RP (R)-j·π·C ₀ ] (57)

6)最后一步是方位向IFFT，可以将回波变到二维时间域，得到聚焦好的SAR图像。6) The last step is the azimuth IFFT, which can transform the echo into the two-dimensional time domain and obtain a well-focused SAR image.

下面进行仿真验证。这里利用如下参数进行仿真验证：距离向带宽18MHz，采样频率20MHz，PRF为200Hz，脉冲宽度20us，合成孔径时间为100s。得到的结果如图2～图6所示，其中图2、3、5、6为场景边缘点的仿真结果，图4为场景中心点的仿真结果。可以发现这5个点的二维旁瓣清晰可见，没有耦合现象的发生。The simulation verification is carried out below. The following parameters are used here for simulation verification: the range bandwidth is 18MHz, the sampling frequency is 20MHz, the PRF is 200Hz, the pulse width is 20us, and the synthetic aperture time is 100s. The obtained results are shown in Figures 2 to 6, in which Figures 2, 3, 5, and 6 are the simulation results of the edge points of the scene, and Figure 4 is the simulation result of the center point of the scene. It can be found that the two-dimensional side lobes of these five points are clearly visible, and no coupling phenomenon occurs.

以上所述为本发明的较佳实施例而已，本发明不应该局限于该实施例和附图所公开的内容。凡是不脱离本发明所公开的精神下完成的等效或修改，都落入本发明保护的范围。The above description is only a preferred embodiment of the present invention, and the present invention should not be limited to the content disclosed in this embodiment and the accompanying drawings. All equivalents or modifications accomplished without departing from the disclosed spirit of the present invention fall within the protection scope of the present invention.

Claims

1. improvement NCS imaging algorithm that is applicable to geostationary orbit SAR; It is characterized in that its improvements are two parts: the one, set up the serpentine track signal model and replace the equivalent straight line model in the former NCS imaging algorithm; The 2nd, and on the basis of setting up the serpentine track signal model, obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm, the detailed process of two parts is respectively:

1) process of setting up the serpentine track signal model is:

Definition satellite and the target in each pulse repetition time (PRT) coordinates were

and

satellite and target slant range between the real history is expressed as

R_{n} = | | {\overset{&RightArrow;}{r}}_{sn} - {\overset{&RightArrow;}{r}}_{gn} | | - - - (1)

Formula (1) is carried out obtaining the serpentine track model after the Taylor expansion

R _n＝R+k ₁·t _a+k ₂·t _a ²+k ₃·t _a ³+k ₄·t _a ⁴+… (2)

T wherein _aFor the orientation to the time, R, k ₁, k ₂, k ₃And k ₄Be R _nThe Taylor expansion coefficient on 0 to 4 rank, k wherein ₁, k ₂, k ₃And k ₄The formula of embodying is respectively:

k_{1} = k_{10} + {\dot{k}}_{1} \cdot (R - R_{0}) - - - (3)

k_{2} = k_{20} + {\dot{k}}_{2} \cdot (R - R_{0}) - - - (4)

k_{3} = k_{30} + {\dot{k}}_{3} \cdot (R - R_{0}) - - - (5)

k_{4} = k_{40} + {\dot{k}}_{4} \cdot (R - R_{0}) - - - (6)

In formula (3)～(6), k ₁₀～k ₄₀,

The formula that embodies be respectively:

k_{10} = {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} / | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | | - - - (7)

k_{20} = \frac{{\overset{&RightArrow;}{a}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + {| | {\overset{&RightArrow;}{v}}_{s 0} | |}^{2}}{2 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} - \frac{{[{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}]}^{2}}{2 \cdot {| | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |}^{3}} - - - (8)

k_{30} = \frac{{\overset{&RightArrow;}{b}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 3 \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}}{6 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} + \frac{{[{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}]}^{3}}{2 \cdot {| | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |}^{5}}

(9)

- \frac{{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{2 \cdot {| | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |}^{3}} - \frac{{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} \cdot {| | {\overset{&RightArrow;}{v}}_{s 0} | |}^{2}}{2 \cdot {| | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |}^{3}}

k_{40} = \frac{{\overset{&RightArrow;}{d}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 3 \cdot {\overset{&RightArrow;}{b}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}}{24 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} + \frac{{| | {\overset{&RightArrow;}{a}}_{s 0} | |}^{2}}{8 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} - \frac{k_{2}^{2} + 2 \cdot k_{1} \cdot k_{3}}{2 \cdot | | {\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0} | |} - - - (10)

{\dot{k}}_{1} = \frac{v_{s 0 x}}{r_{s 0 x}} - \frac{{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{{R_{0}}^{2}} - - - (11)

{\overset{\cdot}{k}}_{2} = \frac{a_{s 0 x} \cdot {R_{0}}^{2} - r_{s 0 x} \cdot ({\overset{&RightArrow;}{a}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + {| | {\overset{&RightArrow;}{v}}_{s 0} | |}^{2})}{2 \cdot {R_{0}}^{2} \cdot r_{s 0 x}} - \frac{v_{s 0 x} \cdot {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{{R_{0}}^{2} \cdot r_{s 0 x}} + \frac{3 \cdot {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{2 \cdot {R_{0}}^{4}}

(12)

{\overset{\cdot}{k}}_{3} = \frac{b_{s 0 x} \cdot {R_{0}}^{2} - r_{s 0 x} \cdot [{\overset{&RightArrow;}{b}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 3 \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}]}{6 \cdot {R_{0}}^{2} \cdot r_{s 0 x}} +

\frac{3 \cdot v_{s 0 x} \cdot {[{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}]}^{2} - 5 \cdot r_{s 0 x} \cdot {[{\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}]}^{3}}{2 \cdot {R_{0}}^{6} \cdot r_{s 0 x}} - - - (13)

- \frac{v_{s 0 x} \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + a_{s 0 x} \cdot {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{2 \cdot {R_{0}}^{2} \cdot r_{s 0 x}}

+ \frac{3 \cdot {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} \cdot {\overset{&RightArrow;}{a}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T}}{2 \cdot {R_{0}}^{4}} - \frac{v_{s 0 x} \cdot {| | {\overset{&RightArrow;}{v}}_{s 0} | |}^{2}}{2 \cdot {R_{0}}^{2} \cdot r_{s 0 x}} + \frac{3 \cdot {\overset{&RightArrow;}{v}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} \cdot {| | {\overset{&RightArrow;}{v}}_{s 0} | |}^{2}}{2 \cdot {R_{0}}^{4}}

{\dot{k}}_{4} = \frac{d_{s 0 x} \cdot {R_{0}}^{2} - r_{s 0 x} \cdot [{\overset{&RightArrow;}{d}}_{s 0} \cdot {({\overset{&RightArrow;}{r}}_{s 0} - {\overset{&RightArrow;}{r}}_{g 0})}^{T} + 4 \cdot {\overset{&RightArrow;}{b}}_{s 0} \cdot {\overset{&RightArrow;}{v}}_{s 0}^{T}]}{24 \cdot {R_{0}}^{2} \cdot r_{s 0 x}} - \frac{{| | {\overset{&RightArrow;}{a}}_{s 0} | |}^{2}}{8 \cdot {R_{0}}^{2}}

(14)

+ \frac{k_{20} \cdot {\overset{\cdot}{k}}_{2} + k_{30} \cdot {\dot{k}}_{1} + k_{10} \cdot {\dot{k}}_{3}}{r_{s 0 x}} + \frac{k_{20}^{2} + 2 \cdot k_{10} \cdot k_{30}}{2 \cdot {R_{0}}^{2}}

In formula (7)～formula (14),

With

Represent satellite at aperture center position vector constantly, With

Represent respectively satellite aperture center constantly velocity, acceleration, acceleration vector and add acceleration vector, R ₀Expression satellite and reference point target are at aperture center distance constantly, r _S0x, a _S0x, b _S0x, v _S0xAnd d _S0xBe respectively

With

Distance under the scene coordinate system is to component;

2), obtain the two-dimensional analysis frequency spectrum table 2-d spectrum that is applicable to the NCS imaging algorithm and do based on serpentine track

S (f_{r}, f_{a}) = u_{r} (\frac{f_{r}}{K_{r}}) \cdot u_{a} [f_{a} + \frac{2 \cdot k_{1}}{c} \cdot (f_{r} {+ f}_{c})] \cdot

\exp [j \cdot 2 \cdot π \cdot φ_{az} (f_{a}, R)] \cdot \exp [j \cdot 2 \cdot π \cdot φ_{RP} (R)]

\cdot \exp [- j \cdot 2 \cdot π \cdot b (f_{a}, f_{r})] \cdot \exp [- j \cdot \frac{4 \cdot π \cdot R}{c \cdot M (f_{a})} \cdot f_{r}] \cdot - - - (15)

\exp [- j \cdot π \cdot \frac{f_{r}^{2}}{K_{s} (f_{a}, R)}] \cdot \exp [j \cdot φ_{3} (f_{a}, R) \cdot f_{r}^{3}]

Formula (15) is further described as follows:

2.1 the ф on formula (15) equal sign right side _Az(f _a, R) be the orientation to modulating function, the formula of embodying does

φ_{az} (f_{a}, R) = [\frac{k_{1}}{2 \cdot k_{2}} + \frac{3 \cdot k_{1}^{2} \cdot k_{3}}{8 \cdot k_{2}^{3}} + \frac{k_{1}^{3} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{16 \cdot k_{2}^{5}}] \cdot f_{a} +

[\frac{λ}{8 \cdot k_{2}} + \frac{3 \cdot λ \cdot k_{1} \cdot k_{3}}{16 \cdot k_{2}^{3}} + \frac{3 \cdot λ \cdot k_{1}^{2} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{64 \cdot k_{2}^{5}}] \cdot f_{a}^{2} + - - - (16)

[\frac{λ^{2} \cdot k_{3}}{32 \cdot k_{2}^{3}} + \frac{λ \cdot k_{1} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{64 \cdot k_{2}^{5}}] \cdot f_{a}^{3} + \frac{λ^{3} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{512 \cdot k_{2}^{5}} \cdot f_{a}^{4}

2.2 the ф on formula (15) equal sign right side _RP(R) be excess phase after the accurate 2-d spectrum Taylor expansion, expression formula does

φ_{RP} (R) = \frac{k_{1}^{2}}{2 \cdot λ \cdot k_{2}} + \frac{k_{1}^{3} \cdot k_{3}}{4 \cdot λ \cdot k_{2}^{3}} + \frac{k_{1}^{4} \cdot (9 \cdot k_{3}^{2} - 4 \cdot k_{2} \cdot k_{4})}{32 \cdot λ \cdot k_{2}^{5}} - \frac{2 \cdot R}{λ} - - - (17)

2.3 the b (f on formula (15) equal sign right side _a, f _r) the RP place migration phase place that obtains when launching for 2-d spectrum, its expression formula does

b (f_{a}, f_{r}) = - [\frac{k_{10}^{2}}{2 \cdot k_{20} \cdot c} + \frac{k_{10}^{3} \cdot k_{30}}{4 \cdot k_{20}^{3} \cdot c} + \frac{k_{10} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{32 \cdot c \cdot k_{20}^{5}}] \cdot f_{r} +

[\frac{λ}{8 \cdot k_{20} \cdot f_{c}} + \frac{3 \cdot λ \cdot k_{10} \cdot k_{30}}{16 \cdot k_{20}^{3} \cdot f_{c}} + \frac{3 \cdot λ k_{10}^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{64 \cdot f_{c} \cdot k_{20}^{5}}] \cdot f_{a}^{2} \cdot f_{r} +

(18)

[\frac{λ^{2} \cdot k_{30}}{16 \cdot k_{20}^{3} \cdot f_{c}} + \frac{λ^{2} \cdot k_{10} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{32 \cdot f_{c} \cdot k_{20}^{5}}] \cdot f_{a}^{3} \cdot f_{r} +

\frac{3 \cdot λ^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{512 \cdot f_{c} \cdot k_{20}^{5}} f_{a}^{4} \cdot f_{r} + (\frac{2 \cdot R_{0}}{c} - {\overset{\cdot}{B}}_{1} \cdot R_{0}) \cdot f_{r}

Wherein

{\overset{\cdot}{B}}_{1} = - \frac{2 \cdot k_{10} \cdot k_{20} \cdot {\dot{k}}_{1} - k_{10}^{2} \cdot {\dot{k}}_{2}}{2 \cdot c \cdot k_{20}^{2}} - \frac{(3 \cdot k_{10}^{2} \cdot k_{30} \cdot {\overset{\cdot}{k}}_{1} + k_{10}^{3} \cdot {\overset{\cdot}{k}}_{3}) \cdot k_{20} - 3 \cdot k_{10}^{3} \cdot k_{30} \cdot {\overset{\cdot}{k}}_{2}}{4 \cdot c \cdot k_{20}^{4}} - \frac{A_{3} \cdot {\dot{k}}_{1} - k_{10} \cdot {\overset{\cdot}{A}}_{3}}{32 \cdot c} +

\{\begin{matrix} - \frac{λ \cdot {\overset{\cdot}{k}}_{2}}{8 \cdot f_{c} \cdot k_{20}^{2}} + \frac{3 \cdot λ \cdot [(k_{10} \cdot {\overset{\cdot}{k}}_{3} + k_{30} \cdot {\overset{\cdot}{k}}_{1}) \cdot k_{20} - 3 \cdot k_{10} \cdot k_{30} \cdot {\overset{\cdot}{k}}_{2}]}{16 \cdot f_{c} \cdot k_{20}^{4}} \\ + \frac{3 \cdot λ \cdot (2 \cdot k_{10} \cdot A_{3} \cdot {\dot{k}}_{1} - k_{10}^{2} \cdot {\overset{\cdot}{A}}_{3})}{64 \cdot f_{c}} \end{matrix}\} \cdot f_{a}^{2}

+ [\frac{λ^{2} \cdot (k_{20} \cdot {\overset{\cdot}{k}}_{3} - 3 \cdot k_{30} \cdot {\overset{\cdot}{k}}_{2})}{16 \cdot f_{c} \cdot k_{20}^{4}} + \frac{λ^{2} \cdot (A_{3} \cdot {\overset{\cdot}{k}}_{1} - k_{10} \cdot {\overset{\cdot}{A}}_{3})}{32 \cdot f_{c}}] \cdot f_{a}^{3} + \frac{3 \cdot λ^{2} \cdot {\dot{A}}_{3}}{512 \cdot f_{c}} \cdot f_{a}^{4} + \frac{2}{c}

(19)

A_{3} = \frac{9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40}}{k_{20}^{5}} - - - (20)

{\dot{A}}_{3} = \frac{[18 \cdot k_{30} \cdot {\dot{k}}_{3} - 4 \cdot (k_{20} \cdot {\overset{\cdot}{k}}_{4} + k_{40} \cdot {\overset{\cdot}{k}}_{2})] k_{20} - 5 \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40}) \cdot {\dot{k}}_{2}}{k_{20}^{6}} - - - (21)

2.4 the 4th exponential term on formula (15) equal sign right side is

In,

Be range migration, M (f _a) be the migration factor and

M (f_{a}) = \frac{1}{{\dot{B}}_{1} \cdot c} - - - (22)

2.5 the 5th exponential term on formula (15) equal sign right side is

In

For distance to modulation item, K wherein _s(f _a, R) be new distance to frequency modulation factor, and

K _s(f _a，R)＝K _s(f _a，R ₀)+Δk _s(f _a)·[τ(f _a，R)-τ(f _a，R ₀)] (23)

Wherein

\frac{1}{K_{s} (f_{a}, R_{0})} = - [\frac{λ}{4 \cdot k_{20} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{10} \cdot k_{30}}{8 \cdot k_{20}^{3} \cdot f_{c}^{2}} + \frac{3 \cdot λ \cdot k_{10}^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{32 \cdot k_{20}^{5} \cdot f_{c}^{2}}] \cdot f_{a}^{2} -

[\frac{3 \cdot λ^{2} \cdot k_{30}}{16 \cdot k_{20}^{3} \cdot f_{c}^{2}} + \frac{3 \cdot λ^{2} \cdot k_{10} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{32 \cdot k_{20}^{5} \cdot f_{c}^{2}}] \cdot f_{a}^{3} - - - (24)

- \frac{3 \cdot λ^{3} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{128 \cdot k_{20}^{5} \cdot f_{c}^{2}} \cdot f_{a}^{4} + \frac{1}{K_{r}}

{Δk}_{s} (f_{a}) = \frac{K_{s}^{2} (f_{a}, R_{0}) \cdot c \cdot M (f_{a})}{2} .

\{\begin{matrix} - \frac{λ \cdot {\overset{\cdot}{k}}_{2} \cdot f_{a}^{2}}{4 \cdot f_{c}^{2} \cdot k_{20}^{2}} + \frac{3 \cdot λ \cdot [(k_{10} \cdot {\overset{\cdot}{k}}_{3} + k_{30} \cdot {\overset{\cdot}{k}}_{1}) \cdot k_{20} - 3 \cdot k_{10} \cdot k_{30} \cdot {\overset{\cdot}{k}}_{2}]}{8 \cdot f_{c}^{2} \cdot k_{20}^{4}} \cdot f_{a}^{2} + \\ \frac{3 \cdot λ \cdot (2 \cdot k_{10} \cdot A_{3} \cdot {\dot{k}}_{1} - k_{10}^{2} \cdot {\overset{\cdot}{A}}_{3})}{32 \cdot f_{c}^{2}} \cdot f_{a}^{2} + \frac{3 \cdot λ^{2} \cdot (k_{20} \cdot {\overset{\cdot}{k}}_{3} - 3 \cdot k_{30} \cdot {\overset{\cdot}{k}}_{2})}{16 \cdot f_{c}^{2} \cdot k_{20}^{4}} \cdot f_{a}^{3} \\ + \frac{3 \cdot λ^{2} \cdot (A_{3} \cdot {\overset{\cdot}{k}}_{1} - k_{10} \cdot {\overset{\cdot}{A}}_{3})}{32 \cdot f_{c}^{2}} \cdot f_{a}^{3} + \frac{3 \cdot λ^{3} \cdot {\overset{\cdot}{A}}_{3}}{128 \cdot f_{c}^{2}} f_{a}^{4} \end{matrix}\} - - - (25)

τ (f_{a}, R) = \frac{2 \cdot R}{c \cdot M (f_{a})} - - - (26)

τ (f_{a}, R_{0}) = \frac{2 \cdot R_{0}}{c \cdot M (f_{a})} - - - (27)

According to formula (23)～(27), obtain the operations factor Y that in the NCS algorithm, uses _m(f _a), q ₂And q ₃, its expression formula is respectively (28)～(30), wherein Y _m(f _a) be used to eliminate the influence of three phase places and be used to adjust residual three phase errors of introducing owing to the adjustment of follow-up frequency modulation rate space-variant shape, q ₂And q ₃Be used to adjust the space-variant property of range migration and the space-variant property of frequency modulation rate.

Y_{m} (f_{a}) = \frac{{Δk}_{s} (f_{a}) \cdot (M (f_{ref}) / M (f_{a}) - 0.5)}{K_{s}^{3} (f_{a}, R_{0}) \cdot (M (f_{ref}) / M (f_{a}) - 1)} - - - (28)

q ₂＝K _s(f _a，R ₀)·(M(f _ref)/M(f _a)-1) (29)

q_{3} = \frac{{Δk}_{s} (f_{a}) \cdot (M (f_{ref}) / M (f_{a}) - 1)}{2} - - - (30)

2.6 the 6th exponential term on formula (15) equal sign right side is In For what obtain when the 2-d spectrum decoupling zero, relevant with distance to the cube of frequency, adopt RP R here ₀The value at place, promptly

φ_{3} (f_{a}, R) = φ_{3} (f_{a}, R_{0}) =

2 \cdot π \cdot [\begin{matrix} - \frac{λ \cdot f_{a}^{2}}{8 \cdot k_{20} \cdot f_{c}^{3}} - \frac{3 \cdot λ \cdot k_{10} \cdot k_{30}}{16 \cdot k_{20}^{3} \cdot f_{c}^{3}} \cdot f_{a}^{2} - \frac{3 \cdot λ \cdot k_{10}^{2} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{64 \cdot k_{20}^{5} \cdot f_{c}^{3}} \cdot f_{a}^{2} \\ - \frac{λ^{2} \cdot k_{30} \cdot f_{a}^{3}}{8 \cdot k_{20}^{3} \cdot f_{c}^{3}} - \frac{λ^{2} \cdot k_{1} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{16 \cdot k_{20}^{5} \cdot f_{c}^{3}} \cdot f_{a}^{3} - \frac{5 \cdot λ^{3} \cdot (9 \cdot k_{30}^{2} - 4 \cdot k_{20} \cdot k_{40})}{256 \cdot k_{20}^{5} \cdot f_{c}^{3}} \cdot f_{a}^{4} \end{matrix}]

(31)