Anti-ISRJ Method Based on Intrapulse Frequency-coded Joint Frequency Modulation Slope Agile Radar Waveform

WANG Xiaoge; LI Binbin; CHEN Hui; LIU Weijian; ZHU Yongzhe; NI Mengyu

doi:10.12000/JR24046

Volume 13 Issue 5

Sep. 2024

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Journal of Radars > 2024 > 13(5): 1019-1036

Lin Yuchuan, Zhang Jianyun, Wu Yongjun, Zhou Qingsong. Matrix Inversion Method for Azimuth Reconstruction in Bistatic Spaceborne High-Resolution Wide-Swath SAR System[J]. Journal of Radars, 2017, 6(4): 388-396. doi: 10.12000/JR17060

Citation:

WANG Xiaoge, LI Binbin, CHEN Hui, et al. Anti-ISRJ method based on intrapulse frequency-coded joint frequency modulation slope agile radar waveform[J]. Journal of Radars, 2024, 13(5): 1019–1036. doi: 10.12000/JR24046

Citation:

PDF( 2677 KB)

Anti-ISRJ Method Based on Intrapulse Frequency-coded Joint Frequency Modulation Slope Agile Radar Waveform

DOI: 10.12000/JR24046 CSTR: 32380.14.JR24046

1.
Air Force Early Warning Academy, Wuhan 430019, China
2.
The People’s Liberation Army Unit 93209, Beijing 100094, China

Funds: The National Natural Science Foundation of China (62001510, 62101593)

More Information

Corresponding author: LI Binbin, binbinli_1025@163.com; CHEN Hui, 574667385@qq.com
Received Date: 2024-03-24
Rev Recd Date: 2024-07-08

Available Online: 2024-07-11

Publish Date: 2024-07-24

Abstract

Abstract

Interrupted Sampling Repeater Jamming (ISRJ) is a type of intrapulse coherent jamming that can form multiple realistic false targets that lead or lag behind the actual target, severely affecting radar detection. It is one of the hotspots of current research on electronic counter-countermeasures. To address this problem, an anti-ISRJ method based on an intrapulse frequency-coded joint Frequency Modulation (FM) slope agile waveform is proposed in this paper. In this method, the radar first transmits an intrapulse frequency-coded joint FM slope agile signal to improve the mutual coverability of subpulses by manipulating subpulse center frequency and FM slope agility. Next, the echo signal is divided into several slices according to the subpulse timing of the transmitted signal. Then, the Fuzzy C-Means (FCM) algorithm is used to classify the echo slices. Finally, the interference is suppressed via fractional-domain joint time domain filtering. Simulation results show that the FCM-based method can identify 100% of the interfered echo slices in a jammer synchronous sampling scenario when the Signal-to-Noise Ratio (SNR) is greater than −2.5 dB, and the Jamming-to-Signal Ratio (JSR) is greater than 5 dB. For high JSRs and low SNRs, the proposed method can effectively reduce the target energy loss and suppress the range sidelobes generated via residual interference. Moreover, the target detection probability after interference suppression exceeds 90% when JSR = 50 dB.

FullText(HTML)

1. 引言

双基星载合成孔径雷达(Synthetic Aperture Radar, SAR)利用信号收发平台的分置，能够同时获取不同视角的观测数据，在测绘、干涉测量、地面目标识别、自然灾害监测等领域^[1,2]具有重要的应用价值。以Tandem-L为代表的新一代双基星载SAR系统应用多通道、数字波束形成(Digital Beam Forming, DBF)等技术，实现方位向高分辨率和距离向宽测绘带SAR(High Resolution Wide Swath SAR, HRWS-SAR)成像，系统的成像能力得到显著改善^[3,4]。

HRWS-SAR系统在方位向采用多通道采样降低系统的脉冲重复频率(Pulse Recurrence Frequency, PRF)，在不降低方位分辨率的前提下实现系统的宽测绘带成像。HRWS-SAR系统进行成像处理时，若方位向信号为均匀采样可直接采用传统SAR成像处理方法，而受载星平台轨迹约束及应用场景的限制，方位向的非均匀采样更为普遍^[5]，因而对方位向信号进行重构以获得其均匀采样信号或多普勒谱是HRWS-SAR系统成像处理的一项关键技术。单基星载HRWS-SAR系统的方位向信号重构得到了广泛而深入的研究^[6–17]。重排算法^[6]和插值算法^[7]是两种典型的时域重构算法。重排算法依据接收信号的方位向位置将各通道信号重新排列，只能在特定的PRF得到方位向信号的均匀采样。插值算法则依据文献[7]所提出的周期性非均匀采样信号重构公式通过时域插值得到方位向信号的均匀采样，该算法运算复杂度高且精度依赖于插值核的长度。Krieger等依据广义采样定理提出矩阵求逆算法^[8,9]，该算法通过线性方程求解从混叠的多通道信号中重构出无模糊的多普勒谱。该算法不需要协方差矩阵等先验信息且易于实现，但在重叠采样时该算法无法进行信号重构，接近重叠采样时重构性能也较差。文献[10–13]则采用不同的方法对矩阵求逆算法进行改进，实现重叠采样附近的高性能信号重构。文献[14,15]基于统一的信号模型，选取不同代价函数进行优化，提出了多种自适应波束形成(DBF)算法：正交投影算法、信号最大化算法、最大化信号模糊噪声比算法、最小均方误差算法等。与矩阵求逆算法相比，DBF类算法运算复杂度较高，并假定各通道的噪声为高斯白噪声且相互独立。文献[16]则对方位向信号重构算法的性能进行了仿真对比分析。文献[17]采用NUFFT直接重构方位向非均匀采样信号的多普勒频谱。

上述方位向信号重构方法理论上均能推广到双基星载HRWS-SAR系统。考虑到矩阵求逆算法易于实现且研究较为广泛，本文研究该算法在双基HRWS-SAR系统中的实现。本文首先将方位照射时间内时变的发射接收距离比近似为一个常数，推导了双基星载HRWS-SAR系统与某个单基系统在方位向多信道间传递函数上的等效性，从而构建了双基星载HRWS-SAR系统的方位向信号模型。而后，提出了适用于一般双基构型星载HRWS-SAR系统方位向信号重构的矩阵求逆算法，并给出了信噪比缩放因子及方位模糊比这两个重构性能指标的计算公式。最后，通过对几种典型双基构型的星载HRWS-SAR系统进行方位向信号重构仿真，验证了矩阵求逆算法在一般双基构型星载HRWS-SAR系统中的适用性。

2. 双基星载HRWS-SAR系统的方位向信号模型

为保持双基星载HRWS-SAR系统对地面的持续观测，收发平台应置于同一轨道(顺飞模式)或高度相同的平行轨道(平飞模式)。图1给出了一般构型双基星载HRWS-SAR系统的信号收发几何。发射天线和接收天线的最短距离分别为r_T0, r_R0，接收天线共有M个通道，RX_equ为方位向信号重构后的等效接收通道，通道i到等效接收通道RX_equ的方位向距离为 $\Delta {x_i}$ , t_fd为发射天线和接收天线的零多普勒时间差。

图 1 双基星载HRWS-SAR系统的双基构型

Figure 1. Bistatic configuration of bistatic spaceborne HRWS-SAR

下载: 全尺寸图片幻灯片

不失一般性，假定点目标P在t=0时刻位于接收天线的零多普勒面，容易得到t时刻点P在接收通道i的收发距离和 $R\left( t \right)$ 计算公式为：

$\begin{aligned} R\left( t \right) =& \sqrt {r_{{\rm{T}}0}^2 + {{\left( {vt - v{t_{{\rm{fd}}}}} \right)}^2}} \\ & + \sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt - \Delta {x_i}} \right)}^2}} \end{aligned}$

(1)

忽略天线方向图的影响，接收通道i的冲激响应函数可表示为：

$\begin{aligned} {h_i}\left( t \right) = & \exp \left( { - {\rm j}\frac{{2{{π}} }}{\lambda }R\left( t \right)} \right) \\ = & \exp \Biggr( { - {\rm j}\frac{{2{{π}} }}{\lambda }\biggr( {\sqrt {r_{{\rm{T}}0}^2 + {{\left( {vt - v{t_{{\rm{fd}}}}} \right)}^2}}}} \\ & { {\left.+ {\sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt - \Delta {x_i}} \right)}^2}} } \right)} \Biggr)} \end{aligned}$

(2)

2.1 单基星载HRWS-SAR系统的方位向信号模型

单基星载HRWS-SAR系统可视为r_T0=r_R0=r₀, t_fd=0，发射天线为等效接收通道的特殊情况。由式(2)可得此时接收通道i的冲激响应为：

$\begin{aligned} {\tilde h_i}\left( t \right) = & \exp \Biggr( { - {\rm j}\frac{{2{{π}} }}{\lambda }\biggr( {\sqrt {r_0^2 + {{\left( {vt} \right)}^2}}}} \\ & + {{\sqrt {r_0^2 + {{\left( {vt - \Delta {x_i}} \right)}^2}} } \biggr)} \Biggr) \end{aligned}$

(3)

令此时单基单通道SAR的冲激响应函数为 ${\tilde h_{{\rm{mono}}}}\left( t \right)$ , ${\tilde h_{{\rm{mono}}}}\left( t \right)$ 可由式(4)表示。

${\tilde h_{{\rm{mono}}}}\left( t \right) = \exp \left( { - {\rm j}\frac{{4{{π}} }}{\lambda }\sqrt {r_0^2 + {{\left( {vt} \right)}^2}} } \right)$

(4)

文献[9]对 ${\tilde h_{{\rm{mono}}}}\left( t \right)$ 及 ${\tilde h_i}\left( t \right)$ 的泰勒级数展开进行2阶相位近似处理，可得

$\left\{ \begin{aligned} &{{\tilde h}_{{\rm{mono}}}}\left( t \right) \approx \exp \left( { - {\rm j}\frac{{4{{π}} }}{\lambda }{r_0}} \right) \cdot \exp \left( { - {\rm j}\frac{{2{{π}} {v^2}{t^2}}}{{\lambda {r_0}}}} \right)\\ &{{\tilde h}_i}\left( t \right) \approx \exp \left( { - {\rm j}\frac{{4{{π}} }}{\lambda }{r_0}} \right) \cdot \exp \left( { - {\rm j}\frac{{{{π}} \Delta x_i^2}}{{2\lambda {r_0}}}} \right) \\ & \quad\quad\quad\ \cdot \exp \left( { - {\rm j}\frac{{2{{π}} {v^2}{{\left( {t - \displaystyle\frac{{\Delta {x_i}}}{{2v}}} \right)}^2}}}{{\lambda {r_0}}}} \right) \end{aligned} \right.\quad$

(5)

因而 ${\tilde h_{{\rm{mono}}}}\left( t \right)$ 与 ${\tilde h_i}\left( t \right)$ 间存在式(6)所示的关系，其中 $*$ 为卷积运算。

${\tilde h_i}\left( t \right) \!=\! \exp \left( { \!- {\rm j}\!\frac{{{{π}} \Delta x_i^2}}{{2\lambda {r_0}}}} \right) \!\cdot\! \delta \left( {t \!-\! \frac{{\Delta {x_i}}}{{2v}}} \right)\! *\! {\tilde h_{{\rm{mono}}}}\left( t \right)$

(6)

将式(6)进行Fourier变换，可得单基单通道SAR与接收通道i的系统函数关系：

$\begin{aligned} {\tilde H_i}\left( f \right) = & \exp \left( { - {\rm j}\frac{{{{π}} \Delta x_i^2}}{{2\lambda {r_0}}}} \right) \cdot \exp \left( { - {\rm j}2{{π}} \frac{{\Delta {x_i}}}{{2v}}f} \right) \\ &\cdot {\tilde H_{{\rm{mono}}}}\left( f \right) \end{aligned}$

(7)

单基单通道SAR到接收通道i的传递函数 ${\tilde G_i}\left( f \right)$ 为：

${\tilde G_i}\left( f \right) = \exp \left( { - {\rm j}\frac{{{{π}} \Delta x_i^2}}{{2\lambda {r_0}}}} \right) \cdot \exp \left( { - {\rm j}2{{π}} \frac{{\Delta {x_i}}}{{2v}}f} \right)$

(8)

基于上述分析，单基星载HRWS-SAR系统的方位向信号生成模型可由图2进行描述。方位向各通道信号在时域为单基单通道SAR信号经对应的相位偏移和时间延迟的结果，而在多普勒域为单基单通道SAR信号与对应的常数相位因子和线性相位因子相乘后的结果。PRF小于多普勒带宽时，方位向各通道信号为时域上的欠采样，将导致多普勒域的频谱混叠。

图 2 单基星载HRWS-SAR系统的方位向信号生成模型

Figure 2. Azimuth signal generating model in monostatic spaceborne HRWS-SAR

下载: 全尺寸图片幻灯片

2.2 双基星载HRWS-SAR系统的方位向信号模型

一般双基构型的星载HRWS-SAR系统中r_T0≠r_R0, t_fd≠0，不能通过对h_mono及h_i的泰勒级数展开进行2阶相位近似处理得到与式(6)、式(7)相似的表达式。本节将对式(2)进一步处理，推导一般双基构型星载HRWS-SAR系统中 ${h_{{\rm{mono}}}}\left( t \right)$ 及 ${h_i}\left( t \right)$ 的关系，进而构建其方位向信号生成模型。

引入点目标P的收发距离比函数 $C\left( t \right) =$ $\displaystyle\frac{{\sqrt {r_{{\rm{T}}0}^2 + {{\left( {vt - v{t_{{\rm{fd}}}}} \right)}^2}} }}{{\sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt} \right)}^2}} }}$ ，式(2)等价于：

$\begin{aligned} {h_i}\left( t \right) = & \exp \Biggr( { - {\rm j}\frac{{2{{π}} }}{\lambda }\biggr( {C\left( t \right)\sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt} \right)}^2}}}} \\ & + {{\sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt - \Delta {x_i}} \right)}^2}} } \biggr)} \Biggr) \end{aligned}$

(9)

等效接收通道RX_equ的冲激响应函数为：

$\begin{aligned} & {h_{{\rm{mono}}}}\left( t \right) \\ & \quad =\exp \Biggr( { - {\rm j}\frac{{2{{π}} }}{\lambda }\left( {\left( {C\left( t \right) + 1} \right)\sqrt {r_{{\rm{R}}0}^2 + {{\left( {vt} \right)}^2}} } \right)} \Biggr)\quad\ \end{aligned}$

(10)

对式(9)、式(10)的泰勒级数展开进行2阶相位近似处理，可得：

$\left\{ \begin{aligned} &{h_i}\left( t \right) \approx \exp \left( { - {\rm j}\frac{{2{{π}} }}{\lambda }\left( {C\left( t \right) + 1} \right){r_{{\rm{R}}0}}} \right) \\ & \quad\quad\quad\ \cdot \exp \left( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \frac{{C\left( t \right)\Delta x_i^2}}{{C\left( t \right) + 1}}} \right) \\ & \quad\quad\quad\ \cdot \exp \Biggr( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \Big( {C^{} \left( t \right) + 1} \Big) }\\ & \quad\quad\quad\ {\cdot {v^2}{{\left( {t - \frac{{\Delta {x_i}}}{{\left( {C\left( t \right) + 1} \right)v}}} \right)}^2}} \Biggr)\\ & {h_{{\rm{mono}}}}\left( t \right) \approx \exp \left( { - {\rm j}\frac{{2{{π}} }}{\lambda }\left( {C\left( t \right) + 1} \right){r_{{\rm{R}}0}}} \right) \\ & \quad\quad\quad\quad\ \cdot \exp \left( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \left( {C\left( t \right) + 1} \right) \cdot {v^2}{t^2}} \right) \end{aligned} \right.$

(11)

在目标照射时间T_a内， $C\left( t \right)$ 的变化极小，用波束中心穿越时刻的取值C₀予以近似。容易得到接收通道i与等效接收通道RX_equ的冲激响应关系为：

$\begin{aligned} {h_i}\left( t \right) = & \exp \left( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \frac{{{C_0}\Delta x_i^2}}{{{C_0} + 1}}} \right) \\ & \ \cdot \delta \left( {t - \frac{{\Delta {x_i}}}{{\left( {{C_0} + 1} \right)v}}} \right) * {h_{{\rm{mono}}}}\left( t \right) \end{aligned}$

(12)

式(12)进行Fourier变换，可得等效接收通道RX_equ与接收通道i的系统函数关系：

$\begin{aligned} {H_i}\left( f \right) = & \exp \left( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \frac{{{C_0}\Delta x_i^2}}{{{C_0} + 1}}} \right) \\ & \cdot \exp \left( { - {\rm j}2{{π}} \frac{{\Delta {x_i}}}{{\left( {{C_0} + 1} \right)v}}f} \right) \cdot {H_{{\rm{mono}}}}\left( f \right) \end{aligned}$

(13)

等效接收通道RX_equ到接收通道i的传递函数为：

$\begin{aligned} {G_i}\left( f \right) = & \exp \left( { - {\rm j}\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \frac{{{C_0}\Delta x_i^2}}{{{C_0} + 1}}} \right) \\ & \cdot \exp \left( { - {\rm j}2{{π}} \frac{{\Delta {x_i}}}{{\left( {{C_0} + 1} \right)v}}f} \right) \end{aligned}$

(14)

考察式(8)、式(14)，传递函数 ${G_i}\left( f \right)$ 与 ${\tilde G_i}\left( f \right)$ 可视为f及 $\Delta {x_i}$ 的二元函数，并存在如下关系：

${G_i}\left( {f,\Delta {x_i}} \right) = {\tilde G_i}\left( {f,\frac{2}{{\left( {{C_0} + 1} \right)}}\Delta {x_i}} \right)$

(15)

式(15)表明双基星载HRWS-SAR系统的方位向信号的生成模型可等效为一单基系统模型，只需将图2中的 $\Delta {x_i}$ 替换为 $2/{{\left( {{C_0} + 1} \right)}}\Delta {x_i}$ 即可予以描述，C₀=1时，双基系统退化为单基系统。

2.3 C(t)近似为C₀的合理性

式(12)、式(13)、式(14)是在 $C\left( t \right)$ 近似为C₀的基础上推导的结果，显然存在近似误差。上述近似误差进行理论上的严格定量分析非常复杂，然而可以从上述表达式中得到简单的定性结果。以式(14)为例，由于在双基星载HRWS-SAR系统中 $\Delta x_i^2 \ll {r_{{\rm{R}}0}}$ ，从而 $\exp \left( { - {\rm j}\displaystyle\frac{{{π}} }{{\lambda {r_{{\rm{R}}0}}}} \cdot \displaystyle\frac{{C\left( t \right)\Delta x_i^2}}{{C\left( t \right) + 1}}} \right)$ 可视为常数1。指数项 $\exp \left( { - {\rm j}2{{π}} \displaystyle\frac{{\Delta {x_i}}}{{\left( {{C_0} + 1} \right)v}}f} \right)$ 的相位近似误差约为 $\Delta \varphi \approx 2{{π}} \left| {\displaystyle\frac{{C\left( t \right) - {C_0}}}{{{C_0} + 1}}} \right| \cdot \left| {\displaystyle\frac{{\Delta {x_i}}}{{\left( {C\left( t \right) + 1} \right)v}}f} \right|$ ，在多普勒带宽B_fd范围内，令接收通道的长度为L_R, $\mathop {\max }\limits_{i = 1,2, \cdot \cdot \cdot ,M} \left| {\Delta {x_i}} \right|\! =\! \alpha {L_{\rm{R}}}$ ，则 $\left| {\displaystyle\frac{{\Delta {x_i}}}{{\left( {C\left( t \right) + 1} \right)v}}f} \right|\!<$ $\left| {\displaystyle\frac{{\alpha {L_a}}}{{\left( {C\left( t \right) + 1} \right)v}}{B_{{\rm{fd}}}}} \right| \approx 0.886\alpha$ 。受天线最小尺寸的限制，L_R不能过小即 $\alpha$ 不会太大，可合理假定 $\alpha$ 在10¹数量级。记 $\varepsilon \left( t \right) \!=\! \left| {\displaystyle\frac{{C\left( t \right) - {C_0}}}{{{C_0} + 1}}} \right|$ ，仿真分析表明 $\mathop {\max }\limits_{t \in \left( { - \frac{{{T_a}}}{2},\frac{{{T_a}}}{2}} \right)} \left( {\varepsilon \left( t \right)} \right)$ 一般不超过10^–3量级，因而 $\Delta \varphi$ 在 $2{{π}} \cdot {10^{ - 2}}$ 量级，为一极小相位。上述分析表明可以忽略式(14)的近似误差，即C₀是 $C\left( t \right)$ 的合理近似。

3. 方位向信号重构的矩阵求逆算法

文献[8]给出了单基HRWS-SAR系统方位向信号重构的矩阵求逆算法，由于双基HRWS-SAR系统在方位向信号模型上与单基系统的等效性，矩阵求逆算法在双基系统同样适用。

3.1 双基HRWS-SAR系统的矩阵求逆算法

矩阵求逆算法的理论依据是广义采样定理。在双基HRWS-SAR系统中，通过发射到接收通道i的传递函数，能够得到单基单通道SAR的信号 $S\left( f \right) = U\left( f \right) \cdot H\left( f \right)$ 在接收通道i的表达式 ${S_i}\left( f \right) =$ ${G_i}\left( f \right) \cdot S\left( f \right)$ 。对于方位向有M个通道的系统，在无重叠采样的情况下，可以得到 $S\left( f \right)$ 的M种独立表达，且采样率均为PRF。根据广义采样定理，能够恢复带宽最高为 $M \cdot {\rm{PRF}}$ 的信号 $S\left( f \right)$ 。

矩阵求逆算法主要包括3个步骤：

首先，构造系统的传递函数矩阵 ${{G}}\left( f \right)$ 及响应矩阵 ${\tilde{{S}}}\left( f \right)$ ：

${{G}}\left( f \right) = \left( {\begin{array}{*{20}{c}} {{G_1}\left( f \right)} & {{G_1}\left( {f + {\rm{PRF}}} \right)} & ·\!·\!· & {{G_1}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)}\\ {{G_2}\left( f \right)} & {{G_2}\left( {f + {\rm{PRF}}} \right)} & ·\!·\!· & {{G_2}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)}\\ \vdots & \vdots & \ddots & \vdots \\ {{G_M}\left( f \right)} & {{G_M}\left( {f + {\rm{PRF}}} \right)} & ·\!·\!· & {{G_M}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)} \end{array}} \right)$

(16)

${\tilde{{S}}}\left( f \right) = {\Big( {{S_1}\left( f \right),{S_2}\left( f \right), \cdots ,{S_M}\left( f \right)} \Big)^{\rm{T}}}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,$

(17)

然后，对 ${{G}}\left( f \right)$ 进行求逆运算，得到信号重构矩阵 ${{P}}\left( f \right)$ ：

${{P}}\left( f \right) = \left( {\begin{array}{*{20}{c}} {{P_1}\left( f \right)} & {{P_2}\left( f \right)} & ·\!·\!· & {{P_M}\left( f \right)}\\ {{P_1}\left( {f + {\rm{PRF}}} \right)} & {{P_2}\left( {f + {\rm{PRF}}} \right)} & ·\!·\!· & {{P_M}\left( {f + {\rm{PRF}}} \right)}\\ \vdots & \vdots & \ddots & \vdots \\ {{P_1}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)} & {{P_2}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)} & ·\!·\!· & {{P_M}\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)} \end{array}} \right)$

(18)

最后，利用公式 ${{P}}\left( f \right) \cdot {{S}}\left( f \right) = {\big( {S\left( f \right),S\left( {f + } \right.$ ${ {\left. {{\rm{PRF}}} \right), ·\!·\!· ,S\left( {f + \left( {M - 1} \right) \cdot {\rm{PRF}}} \right)} \big)^{\rm{T}}}$ 即可重构出方位向信号的多普勒频谱。矩阵求逆算法进行方位向重构的原理框图及物理实现原理可参看文献[8]。

3.2 矩阵求逆算法的重构性能

星载HRWS-SAR系统的单通道信号带宽为PRF，采用矩阵求逆算法重构后的方位向信号带宽为 $M \cdot {\rm{PRF}}$ ，因而方位向信号重构必然影响到信噪比和方位模糊比这两个重要的SAR系统性能指标。

3.2.1 信噪比缩放因子文献[8]定义信噪比缩放因子表征方位向重构对信噪比的影响，其定义式及计算公式为：

${\varPhi _{{\rm{bf}}}} \!=\! \frac{{{\rm{SN}}{{\rm{R}}_{{\rm{in}}}}/{\rm{SN}}{{\rm{R}}_{{\rm{out}}}}}}{{\left( {{\rm{SN}}{{\rm{R}}_{{\rm{in}}}}/{\rm{SN}}{{\rm{R}}_{{\rm{out}}}}} \right)\left| {_{{\rm{PR}}{{\rm{F}}_{{\rm{uni}}}}}} \right.}} \!=\! \frac{{\displaystyle\sum\limits_{j = 1}^M {{\mathop{\rm E}\nolimits} \left( {{{\left| {{P_j}\left( f \right)} \right|}^2}} \right)} }}{N}$

(19)

其中，SNR_in, SNR_out为方位向信号重构前后的信噪比，PRF_uni表示方位向均匀采样时的PRF, ${P_j}\left( f \right)$ 对应于式(18)中的第j行，为作用于接收通道j的重构函数， ${\mathop{\rm E}\nolimits} \left( \cdot \right)$ 为求期望运算。 ${\varPhi _{{\rm{bf}}}}$ 表征了SNR_in与SNR_out的比值随PRF的变化规律。由式(19)可知，越小的 ${\varPhi _{{\rm{bf}}}}$ 意味着更多的信噪比改善。

${\varPhi _{{\rm{bf}}}}$ 还可以通过式(20)进行计算：

${\varPhi _{{\rm{bf}}}} = \frac{{\displaystyle\sum\limits_{j = 1}^M {{\lambda _j}\left( f \right)} }}{N}$

(20)

其中， ${\lambda _j}\left( f \right)$ 为矩阵 ${{P}}\left( f \right) \cdot {{{P}}^{\rm{H}}}\left( f \right)$ 的特征值， ${{{P}}^{\rm{H}}}\left( f \right)$ 为 $\, {{P}}\left( f \right)$ 的共轭转置。

3.2.2 方位模糊比文献[8]详细推导了单基HRWS-SAR系统的方位模糊比(AASR)计算公式，与之类似可推导出双基HRWS-SAR系统的AASR计算公式，可概括为如下4个计算式：

$\begin{aligned} \\ {e_k}\left( f \right) = & {A_k}\left( f \right)\sum\limits_m^{} {\sum\limits_{j = 1}^M {{G_j}\left( {f + k \cdot {\rm{PRF}}} \right)} } \\ \text{} & \ \cdot {P_j}\left( {f + m \cdot {\rm{PRF}}} \right) \end{aligned}$

(21)

$\begin{aligned} {e_\Sigma }\left( f \right) = & \sum\limits_{k = - \infty ,k \ne 0}^\infty {{e_k}\left( f \right)} \\ = & \sum\limits_{k = - \infty ,k \ne 0}^\infty {\Biggr( {{A_k}\left( f \right)\sum\limits_m^{} {\sum\limits_{j = 1}^M {{G_j}\left( {f + k \cdot {\rm{PRF}}} \right)} } }} \\ \text{ } & {\cdot {P_j}\big( {f + m \cdot {\rm{PRF}}}\big)}\Biggr) \end{aligned}\quad\quad$

(22)

${p_{\rm{s}}} = {\rm{E}}\left( {{{\Big| {A\left( f \right) \cdot {\rm{rect}}\left( {f/{{I}}} \right)} \Big|}^2}} \right)\quad\quad\quad\quad\quad\quad\quad \$

(23)

${\rm{AASR}} = \frac{{{\rm{E}}\left( {{{\left| {{e_\Sigma }\left( f \right)} \right|}^2}} \right)}}{{{p_{\rm{s}}}}}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\$

(24)

$A\left( f \right)$ 为发射天线和接收通道合成的天线方向图，双基系统中t_c≠0时 $A\left( f \right) \ne A\left( { - f} \right)$ ，这是导致双基系统AASR计算公式不同于单基系统的根本原因。 ${A_k}\left( f \right)$ 为区间 $\left[ { - {\rm{PRF}}/{2}{\rm{,}}\ {\rm{PRF}}/{2}} \right] + k \cdot {\rm{PRF}}$ 上的 $A\left( f \right)$ , ${e_k}\left( f \right)$ 为区间 $\left[ { - {\rm{PRF}}/{2}{\rm{,}}\ {\rm{PRF}}/{2}} \right] + k \cdot {\rm{PRF}}$ 内的信号产生的方位模糊信号，I为区间 $\left[ - M \right.$ $\left. \cdot {{\rm{PRF}}/{2}}{\rm{,}}{M \cdot {\rm{PRF}}/{2}} \right]$ , p_s为信号能量。上式中m的取值规则为：k>0时， $\max \left( {M - k + 1,1} \right) \le m \le N$ ; k<0时， $1 \le m \le - k$ 。

4. 仿真分析

本节对几种典型的双基构型星载HRWS-SAR系统进行方位向信号重构进行仿真，验证矩阵求逆算法的正确性，并分析算法的重构性能。表1列出了系统的方位向系统参数，对于单基系统容易计算出多普勒带宽B_fd=5.61 kHz，照射时间T_a=1.05 s。

表 1 双基星载HRWS-SAR系统的方位向系统参数

Table 1. Azimuth parameters in bistatic spaceborne HRWS-SAR

参数	数值
发射天线方位尺寸(m)	2.4
接收通道方位尺寸(m)	2.4
接收通道数目	5
轨道高度(km)	600
接收天线最短距离(km)	700
方位向速度(m/s)	7600
载波波长(cm)	3.1

下载: 导出CSV

| 显示表格

表2列出了7种典型的双基构型。表2中采用发射天线和接收天线的零多普勒时间差t_fd与轨道距离L对双基构型进行表征：t_fd=0时系统工作在平飞模式，L=0时系统工作在顺飞模式，L<0时发射天线位于测绘带近端，反之L>0时接收天线位于测绘带近端，构型Ⅰ中t_fd=0且L=0退化为单基系统。

表 2 双基星载HRWS-SAR系统的7种双基构型

Table 2. Seven configurations for bistatic spaceborne HRWS-SAR

构型编号	t_fd (s)	L (km)
Ⅰ	0	0
Ⅱ	1	0
Ⅲ	10	0
Ⅳ	0	10
Ⅴ	0	100
Ⅵ	0	–10
Ⅶ	0	–100

下载: 导出CSV

| 显示表格

4.1 C₀对C(t)的近似性能

将 $C\left( t \right)$ 近似为C₀是构建双基星载HRWS-SAR系统方位向信号模型及建立矩阵求逆算法的关键步骤，该近似的合理性直接影响到模型及算法的正确性，第3.3小节定性分析了该近似的合理性，本节结合具体的仿真条件予以验证。图3给出了各双基构型下在照射时间范围内的 $C\left( t \right)$ 及 $\varepsilon \!\left( t \right)$ 的变化曲线，图3中对近似误差相对较大的构型Ⅱ、构型Ⅲ采用蓝色线标注，其他构型使用红色线。图3(a)中各双基构型下 $C\left( t \right)$ 曲线近似为直线，直观地的反映出 $C\left( t \right)$ 近似为常数；而图3(b)、图3(c)表明 $\varepsilon \left( t \right)$ 在各双基构型下均小于10^–3，验证了C₀为 $C\left( t \right)$ 的合理近似。图3也表明出目标照射时间T_a及收发天线的零多普勒时间差t_fd是影响 $\varepsilon\! \left( t \right)$ 的主要因素， $\mathop {\max }\limits_t \left( {\varepsilon \left( t \right)} \right)$ 的取值与T_a及t_fd的取值呈正相关性。

图 3 照射时间T_a范围内的

$C\left( t \right)$ 及

$\varepsilon \! \left( t \right)$ 变化曲线

Figure 3.

$C\left( t \right)$ and

$\varepsilon \! \left( t \right)$ variation curve in irradiation time T_a

下载: 全尺寸图片幻灯片

4.2 方位向信号重构结果

各接收通道的信噪比设置为20 dB，取PRF=2.0 kHz，对图1所示的点目标P方位向信号重构过程进行仿真，其结果如图4和图5所示。

图 4 构型Ⅰ方位向信号重构前后的成像结果

Figure 4. Unreconstructed and reconstructed azimuth signal Doppler spectrum for bistatic configuration Ⅰ

下载: 全尺寸图片幻灯片

图 5 方位向信号重构后的成像结果

Figure 5. Imaging for reconstructed azimuth signal Doppler spectrum

下载: 全尺寸图片幻灯片

图4(a)、图4(b)中，由于PRF小于多普勒带宽B_fd，单个接收通道的多普勒频谱有严重的混叠现象，图4(c)则表明采用矩阵求逆算法进行方位向信号重构消除了多普勒频谱中的混叠现象。图4(c)及图5给出了各双基构型下的方位向信号重构后的成像结果，验证了矩阵求逆算法对一般双基构型的适用性，同时也表明重构性能受双基构型的影响，尤其是构型Ⅴ、Ⅶ与其他构型性能差异较为明显。

4.3 方位向信号重构性能

将PRF设置在区间1.4 kHz≤PRF≤2.8 kHz，采用表1的系统参数和表2的双基构型，利用式(20)、式(24)可以得到方位向信号的重构性能曲线，如图6所示。为了便于图6的分析，首先计算各双基构型下的C₀及典型PRF值，计算结果为表3，其中PRF_uni为均匀采样时的PRF值，PRF_rep1及PRF_rep2为PRF范围内出现重叠采样的两个PRF值。构型Ⅰ的PRF_uni, PRF_rep1和PRF_rep2在图6中予以标注。

图 6 方位向信号重构性能曲线

Figure 6. Azimuth signal reconstruction performance curve

下载: 全尺寸图片幻灯片

表 3 C₀及典型PRF值

Table 3. C₀ and typical PRF

构型编号	C₀	PRF_uni (kHz)	PRF_rep1 (kHz)	PRF_rep2 (kHz)
Ⅰ	1.0000	2.533	1.583	2.111
Ⅱ	1.0001	2.533	1.583	2.111
Ⅲ	1.0059	2.540	1.588	2.117
Ⅳ	0.9927	2.524	1.577	2.103
Ⅴ	0.9345	2.450	1.531	2.041
Ⅵ	1.0074	2.542	1.589	2.118
Ⅶ	1.0805	2.635	1.647	2.196

下载: 导出CSV

| 显示表格

对比图6和表3，可得到如下结论：

(1) 均匀采样时PRF=PRF_uni，方位向信号的重构性能达到局部最优；重叠采样时PRF=PRF_rep，信噪比缩放因子趋向于无穷大，矩阵求逆算法将不能实现信号重构；PRF位于PRF_rep附近时，矩阵求逆算法的性能急剧下降。从矩阵理论的观点分析，PRF=PRF_rep时，传递函数 ${{G}}\left( f \right)$ 为非满秩矩阵，因而 ${{G}}\left( f \right)$ 及重构矩阵 ${{P}}\left( f \right)$ 的条件数为无穷大，此时信噪比缩放因子 ${\varPhi _{{\rm{bf}}}}$ 也为无穷大，从而无法完成方位向信号的重构。PRF趋于PRF_rep时，重构矩阵 ${{P}}\left( f \right)$ 的条件数急剧增大，方位向信号的重构性能也急剧下降。

(2) C₀值的差异直接导致同一PRF时方位向信号重构性能差异。与其他构型相比构型Ⅴ、Ⅶ的C₀值差异较大因而构性能差异也大，与图5的结论相一致。根本原因在于与双基HRWS-SAR系统等效的单基系统，通道间的方位向距离为 $2/{{\left( {{C_0} + 1} \right)}}$ $\cdot \Delta {x_i}$ , C₀的取值直接影响到方位向采样的均匀性，进而影响到矩阵求逆算法的重构性能。

5. 结束语

本文通过分析单基与双基星载HRWS-SAR系统的方位向信号模型，给出了适用于一般双基构型星载HRWS-SAR系统方位向信号重构的矩阵求逆算法，并使用信噪比缩放因子及方位模糊比两个指标分析了该算法的重构性能。本文的分析方法对其他方位向信号重构算法推广到双基星载HRWS-SAR系统具有借鉴意义。双基星载HRWS-SAR系统方位向信号重构的工程应用中，需改进矩阵求逆算法以改善重叠采样附近的重构性能。

References(32)

References

[1]	HANBALI S B S and KASTANTIN R. A review of self-protection deceptive jamming against chirp radars[J]. International Journal of Microwave and Wireless Technologies, 2017, 9(9): 1853–1861. doi: 10.1017/S1759078717000708.
[2]	SPARROW M J and CIKALO J. ECM techniques to counter pulse compression radar[P]. US, 7081846, 2006.
[3]	RIABUKHA V P, SEMENIAKA A V, KATIUSHYN Y A, et al. Pulse DRFM jamming formation and its mathematical simulation[C]. 2022 IEEE 2nd Ukrainian Microwave Week, Ukraine, 2022: 654–659. doi: 10.1109/UkrMW58013.2022.10037145.
[4]	WANG Xuesong, LIU Jiancheng, ZHANG Wenming, et al. Mathematic principles of interrupted-sampling repeater jamming (ISRJ)[J]. Science in China Series F: Information Sciences, 2007, 50(1): 113–123. doi: 10.1007/s11432-007-2017-y.
[5]	LAN Lan, MARINO A, AUBRY A, et al. GLRT-based adaptive target detection in FDA-MIMO radar[J]. IEEE Transactions on Aerospace and Electronic Systems, 2021, 57(1): 597–613. doi: 10.1109/TAES.2020.3028485.
[6]	LAN Lan, XU Jingwei, LIAO Guisheng, et al. Suppression of mainbeam deceptive jammer with FDA-MIMO radar[J]. IEEE Transactions on Vehicular Technology, 2020, 69(10): 11584–11598. doi: 10.1109/TVT.2020.3014689.
[7]	全英汇, 方文, 沙明辉, 等. 频率捷变雷达波形对抗技术现状与展望[J]. 系统工程与电子技术, 2021, 43(11): 3126–3136. doi: 10.12305/j.issn.1001-506X.2021.11.11. QUAN Yinghui, FANG Wen, SHA Minghui, et al. Present situation and prospects of frequency agility radar waveform countermeasures[J]. Systems Engineering and Electronics, 2021, 43(11): 3126–3136. doi: 10.12305/j.issn.1001-506X.2021.11.11.
[8]	王晓戈, 陈辉, 倪萌钰, 等. 基于相位调制的雷达抗假目标干扰方法[J]. 系统工程与电子技术, 2021, 43(9): 2476–2483. doi: 10.12305/j.issn.1001-506X.2021.09.14. WANG Xiaoge, CHEN Hui, NI Mengyu, et al. Radar anti-false target jamming method based on phase modulation[J] Systems Engineering and Electronics, 2021, 43(9): 2476–2483. doi: 10.12305/j.issn.1001-506X.2021.09.14.
[9]	YAN Yifei, CHEN Hao, and SU Junhai. Overview on anti-jamming technology in main lobe of radar[C]. 2021 IEEE 4th International Conference on Automation, Electronics and Electrical Engineering, Shenyang, China, 2021: 67–71. doi: 10.1109/AUTEEE52864.2021.9668666.
[10]	张建中, 穆贺强, 文树梁, 等. 基于LFM分段脉冲压缩的抗间歇采样转发干扰方法[J]. 电子与信息学报, 2019, 41(7): 1712–1720. doi: 10.11999/JEIT180851. ZHANG Jianzhong, MU Heqiang, WEN Shuliang, et al. Anti-intermittent sampling repeater jamming method based on LFM segmented pulse compression[J]. Journal of Electronics & Information Technology, 2019, 41(7): 1712–1720. doi: 10.11999/JEIT180851.
[11]	万鹏程, 白渭雄, 付孝龙. 基于FrFT的LFM间歇采样转发干扰对抗方法[J]. 火力与指挥控制, 2018, 43(10): 35–39. doi: 10.3969/j.issn.1002-0640.2018.10.007. WAN Pengcheng, BAI Weixiong, and FU Xiaolong. Fractional fourier transform-based LFM radars for countering interrupted-sampling repeater jamming[J]. Fire Control & Command Control, 2018, 43(10): 35–39. doi: 10.3969/j.issn.1002-0640.2018.10.007.
[12]	CHEN Jian, WU Wenzhen, XU Shiyou, et al. Band pass filter design against interrupted-sampling repeater jamming based on time-frequency analysis[J]. IET Radar, Sonar & Navigation, 2019, 13(10): 1646–1654. doi: 10.1049/iet-rsn.2018.5658.
[13]	YUAN Hui, WANG Chunyang, LI Xin, et al. A method against interrupted-sampling repeater jamming based on energy function detection and band-pass filtering[J]. International Journal of Antennas and Propagation, 2017, 2017: 6759169. doi: 10.1155/2017/6759169.
[14]	周超, 刘泉华, 胡程. 间歇采样转发式干扰的时频域辨识与抑制[J]. 雷达学报, 2019, 8(1): 100–106. doi: 10.12000/JR18080. ZHOU Chao, LIU Quanhua, and HU Cheng. Time-frequency analysis techniques for recognition and suppression of interrupted sampling repeater jamming[J]. Journal of Radars, 2019, 8(1): 100–106. doi: 10.12000/JR18080.
[15]	盖季妤, 姜维, 张凯翔, 等. 基于差分特征的间歇采样转发干扰辨识与抑制方法[J]. 雷达学报, 2023, 12(1): 186–196. doi: 10.12000/JR22058. GAI Jiyu, JIANG Wei, ZHANG Kaixiang, et al. A method for interrupted-sampling repeater jamming identification and suppression based on differential features[J]. Journal of Radars, 2023, 12(1): 186–196. doi: 10.12000/JR22058.
[16]	ZHOU Chao, LIU Quanhua, and CHEN Xinliang. Parameter estimation and suppression for DRFM-based interrupted sampling repeater jammer[J]. IET Radar, Sonar & Navigation, 2018, 12(1): 56–63. doi: 10.1049/iet-rsn.2017.0114.
[17]	刘孟斐, 陈吉源, 潘小义, 等. 基于信息熵的分段脉压间歇采样干扰抑制[J]. 雷达科学与技术, 2023, 21(3): 264–272, 281. doi: 10.3969/j.issn.1672-2337.2023.03.004. LIU Mengfei, CHEN Jiyuan, PAN Xiaoyi, et al. Interrupted sampling jamming suppression based on piecewise pulse compression and shannon entropy[J]. Radar Science and Technology, 2023, 21(3): 264–272, 281. doi: 10.3969/j.issn.1672-2337.2023.03.004.
[18]	LU Lu and GAO Meiguo. An improved sliding matched filter method for interrupted sampling repeater jamming suppression based on jamming reconstruction[J]. IEEE Sensors Journal, 2022, 22(10): 9675–9684. doi: 10.1109/JSEN.2022.3159561.
[19]	张建中, 穆贺强, 文树梁, 等. 基于脉内步进LFM波形的抗间歇采样转发干扰方法[J]. 系统工程与电子技术, 2019, 41(5): 1013–1020. doi: 10.3969/j.issn.1001-506X.2019.05.12. ZHANG Jianzhong, MU Heqiang, WEN Shuliang, et al. Anti interrupted-sampling repeater jamming method based on stepped LFM waveform[J]. Systems Engineering and Electronics, 2019, 41(5): 1013–1020. doi: 10.3969/j.issn.1001-506X.2019.05.12.
[20]	ZHOU Kai, LI Dexin, SU Yi, et al. Joint design of transmit waveform and mismatch filter in the presence of interrupted sampling repeater jamming[J]. IEEE Signal Processing Letters, 2020, 27: 1610–1614. doi: 10.1109/LSP.2020.3021667.
[21]	周凯, 何峰, 粟毅. 一种快速抗间歇采样转发干扰波形和滤波器联合设计算法[J]. 雷达学报, 2022, 11(2): 264–277. doi: 10.12000/JR22015. ZHOU Kai, HE Feng, and SU Yi. Fast algorithm for joint waveform and filter design against interrupted sampling repeater jamming[J]. Journal of Radars, 2022, 11(2): 264–277. doi: 10.12000/JR22015.
[22]	WANG Fulai, LI Nanjun, PANG Chen, et al. Complementary sequences and receiving filters design for suppressing interrupted sampling repeater jamming[J]. IEEE Geoscience and Remote Sensing Letters, 2022, 19: 4022305. doi: 10.1109/LGRS.2022.3156164.
[23]	董淑仙, 吴耀君, 方文, 等. 频率捷变雷达联合模糊C均值抗间歇采样干扰[J]. 雷达学报, 2022, 11(2): 289–300. doi: 10.12000/JR21205. DONG Shuxian, WU Yaojun, FANG Wen, et al. Anti-interrupted sampling repeater jamming method based on frequency-agile radar joint fuzzy C-means[J]. Journal of Radars, 2022, 11(2): 289–300. doi: 10.12000/JR21205.
[24]	刘智星, 杜思予, 吴耀君, 等. 脉间-脉内捷变频雷达抗间歇采样干扰方法[J]. 雷达学报, 2022, 11(2): 301–312. doi: 10.12000/JR22001. LIU Zhixing, DU Siyu, WU Yaojun, et al. Anti-interrupted sampling repeater jamming method for interpulse and intrapulse frequency-agile radar[J]. Journal of Radars, 2022, 11(2): 301–312. doi: 10.12000/JR22001.
[25]	杜思予, 刘智星, 吴耀君, 等. 频率捷变波形联合时频滤波器抗间歇采样转发干扰[J]. 系统工程与电子技术, 2023, 45(12): 3819–3827. doi: 10.12305/j.issn.1001-506X.2023.12.11. DU Siyu, LIU Zhixing, WU Yaojun, et al. Frequency agility waveform combined with time-frequency filter to suppress interrupted-sampling repeater jamming[J]. Systems Engineering and Electronics, 2023, 45(12): 3819–3827. doi: 10.12305/j.issn.1001-506X.2023.12.11.
[26]	牛闯, 林强, 段敏, 等. 脉内频率-时延捷变雷达抗间歇采样转发干扰方法[J]. 系统工程与电子技术, 2024, 46(5): 1583–1598. doi: 10.12305/j.issn.1001-506X.2024.05.13. NIU Chuang, LIN Qiang, DUAN Min, et al. Anti-interrupted sampling and repeater jamming method for intra-pulse frequency and time delay agile radar[J]. Systems Engineering and Electronics, 2024, 46(5): 1583–1598. doi: 10.12305/j.issn.1001-506X.2024.05.13.
[27]	张亮, 王国宏, 张翔宇, 等. 基于分数阶字典的间歇采样转发干扰自适应抑制算法[J]. 系统工程与电子技术, 2020, 42(7): 1439–1448. doi: 10.3969/j.issn.1001-506X.2020.07.02. ZHANG Liang, WANG Guohong, ZHANG Xiangyu, et al. Interrupted-sampling repeater jamming adaptive suppression algorithm based on fractional dictionary[J]. Systems Engineering and Electronics, 2020, 42(7): 1439–1448. doi: 10.3969/j.issn.1001-506X.2020.07.02.
[28]	BAHER S and HANBALI S. Countering self-protection smeared spectrum jamming against chirp radars[J]. IET Radar, Sonar & Navigation, 2021, 15(4): 382–389. doi: 10.1049/rsn2.12046.
[29]	BEZDEK J C, EHRLICH R, and FULL W. FCM: The fuzzy c-means clustering algorithm[J]. Computers & Geosciences, 1984, 10(2/3): 191–203. doi: 10.1016/0098-3004(84)90020-7.
[30]	WANG Cong, ZHOU Mengchu, PEDRYCZ W, et al. Comparative study on noise-estimation-based fuzzy C-means clustering for image segmentation[J]. IEEE Transactions on Cybernetics, 2024, 54(1): 241–253. doi: 10.1109/TCYB.2022.3217897.3.
[31]	ALMEIDA L B. The fractional Fourier transform and time-frequency representations[J]. IEEE Transactions on Signal Processing, 1994, 42(11): 3084–3091. doi: 10.1109/78.330368.
[32]	WANG Xiaoge, CHEN Hui, LIU Weijian, et al. Echo preprocessing-based smeared spectrum interference suppression[J]. Electronics, 2023, 12(17): 3690. doi: 10.3390/electronics12173690.

Relative Articles

[1]	LAN Lan, ZHANG Xiang, XU Jingwei, LIAO Guisheng. Main-lobe Deceptive Jammers with Array Radars Using Space-time Multidimensional Coding[J]. Journal of Radars, 2025, 14(2): 439-455. doi: 10.12000/JR24229
[2]	ZHANG Jiaxiang, ZHANG Kaixiang, LIANG Zhennan, CHEN Xinliang, LIU Quanhua. An Intelligent Frequency Decision Method for a Frequency Agile Radar Based on Deep Reinforcement Learning[J]. Journal of Radars, 2024, 13(1): 227-239. doi: 10.12000/JR23197
[3]	GAO Yuhang, ZHANG Kaixiang, FAN Huayu, LIU Quanhua, LIU Zihao, WANG Chaoxu. Range-Doppler Two-dimensional Jamming Reconstruction Algorithm Based on Interpulse Code Agile Waveform[J]. Journal of Radars, 2024, 13(1): 187-199. doi: 10.12000/JR23196
[4]	XU Heng, XU Hong, QUAN Yinghui, PAN Qin, SHA Minghui, CHEN Hui, CHENG Qiang, ZHOU Xiaoyang. A Radar Jamming Method Based on Time Domain Coding Metasurface Intrapulse and Interpulse Coding Optimization[J]. Journal of Radars, 2024, 13(1): 215-226. doi: 10.12000/JR23186
[5]	WANG Yingfu, YIN Jiapeng, LU Zhonghao, PANG Chen, HU Weidong. Analysis of the Influence of Distributed Interrupted-sampling Repeating Signals on Airborne Interferometer Parameter Measurements[J]. Journal of Radars, 2024, 13(5): 1037-1048. doi: 10.12000/JR24090
[6]	SU Hanning, PAN Jiameng, BAO Qinglong, GUO Fucheng, HU Weidong. Anti-interrupted Sampling Repeater Jamming Method in the Waveform Domain before Matched Filtering[J]. Journal of Radars, 2024, 13(1): 240-252. doi: 10.12000/JR23149
[7]	WANG Rongqing, XIE Jingyang, TIAN Biao, XU Shiyou, CHEN Zengping. Integrated Jamming Perception and Parameter Estimation Method for Anti-interrupted Sampling Repeater Jamming[J]. Journal of Radars, 2024, 13(6): 1337-1354. doi: 10.12000/JR24153
[8]	DU Siyu, LIU Zhixing, WU Yaojun, SHA Minghui, QUAN Yinghui. Dense-repeated Jamming Suppression Algorithm Based on the Support Vector Machine for Frequency Agility Radar[J]. Journal of Radars, 2023, 12(1): 173-185. doi: 10.12000/JR22065
[9]	GAI Jiyu, JIANG Wei, ZHANG Kaixiang, LIANG Zhennan, CHEN Xinliang, LIU Quanhua. A Method for Interrupted-Sampling Repeater Jamming Identification and Suppression Based on Differential Features[J]. Journal of Radars, 2023, 12(1): 186-196. doi: 10.12000/JR22058
[10]	DONG Shuxian, WU Yaojun, FANG Wen, QUAN Yinghui. Anti-interrupted Sampling Repeater Jamming Method Based on Frequency-agile Radar Joint Fuzzy C-means[J]. Journal of Radars, 2022, 11(2): 289-300. doi: 10.12000/JR21205
[11]	LIU Zhixing, DU Siyu, WU Yaojun, SHA Minghui, XING Mengdao, QUAN Yinghui. Anti-interrupted Sampling Repeater Jamming Method for Interpulse and Intrapulse Frequency-agile Radar[J]. Journal of Radars, 2022, 11(2): 301-312. doi: 10.12000/JR22001
[12]	WANG Fulai, PANG Chen, YIN Jiapeng, LI Nanjun, LI Yongzhen, WANG Xuesong. Joint Design of Doppler-tolerant Complementary Sequences and Receiving Filters Against Interrupted Sampling Repeater Jamming[J]. Journal of Radars, 2022, 11(2): 278-288. doi: 10.12000/JR22020
[13]	ZHOU Kai, HE Feng, SU Yi. Fast Algorithm for Joint Waveform and Filter Design against Interrupted Sampling Repeater Jamming[J]. Journal of Radars, 2022, 11(2): 264-277. doi: 10.12000/JR22015
[14]	FANG Zuqi, CHENG Qiang, CUI Tiejun. Nonlinear Quasi-Bessel Beam Generation Based on the Time-domain Digital-Coding Metasurface[J]. Journal of Radars, 2021, 10(2): 267-273. doi: 10.12000/JR21043
[15]	PEI Jiazheng, HUANG Yong, CHEN Baoxin, GUAN Jian, CAI Mi, CHEN Xiaolong. Long Time Coherent Integration Method Based on Combining Pulse Compression and Radon-Fourier Transform[J]. Journal of Radars, 2021, 10(6): 956-969. doi: 10.12000/JR21068
[16]	YANG Huiting, ZHOU Yu, GU Yabin, ZHANG Linrang. Design of Integrated Radar and Communication Signal Based on Multicarrier Parameter Modulation Signal[J]. Journal of Radars, 2019, 8(1): 54-63. doi: 10.12000/JR18001
[17]	ZHOU Chao, LIU Quanhua, HU Cheng. Time-frequency Analysis Techniques for Recognition and Suppression of Interrupted Sampling Repeater Jamming[J]. Journal of Radars, 2019, 8(1): 100-106. doi: 10.12000/JR18080
[18]	Chen Fangxiang, Yi Wei, Zhou Tao, Kong Lingjiang. Passive Direct Location Determination for Multiple Sources Based on FRFT[J]. Journal of Radars, 2018, 7(4): 523-530. doi: 10.12000/JR18027
[19]	Zhou Yang, Bi Daping, Shen Aiguo, Fang Mingxing. Intermittent Sampling Repeater Shading Jamming Method Based on Motion Modulation for SAR-GMTI[J]. Journal of Radars, 2017, 6(4): 359-367. doi: 10.12000/JR16075
[20]	Zhe Xiao-qiang, Chou Xiao-lan, Han Bing, Lei Bin. An Improved Doppler Rate Estimation Approach for Sliding Spotlight SAR Data Based on the Transposition Domain[J]. Journal of Radars, 2014, 3(4): 419-427. doi: 10.3724/SP.J.1300.2014.14008

Supplements(0)

Cited By

Cited by
Periodical cited type(1)
1. 叶恺，禹卫东，王伟. 基于矩阵束方法的星载MEB SAR俯仰向DBF处理方法. 电子与信息学报. 2018(11): 2659-2666 .
Other cited types(3)

Proportional views

Proportional views

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Figures(19) / Tables(3)

Get Citation

PDF

XML

Article views(533) PDF downloads(203)

Anti-ISRJ Method Based on Intrapulse Frequency-coded Joint Frequency Modulation Slope Agile Radar Waveform

DOI: 10.12000/JR24046 CSTR: 32380.14.JR24046