CN104898107A

CN104898107A - Multiple-input multiple-output synthetic aperture ladar signal processing method

Info

Publication number: CN104898107A
Application number: CN201510337542.2A
Authority: CN
Inventors: 唐禹; 秦宝; 汪路锋; 邢孟道
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2015-06-17
Filing date: 2015-06-17
Publication date: 2015-09-09
Anticipated expiration: 2035-06-17
Also published as: CN104898107B

Abstract

The invention provides a multi-transmit and multi-receive synthetic aperture laser radar signal processing method, which can effectively solve the contradiction between high resolution in azimuth and wide survey swath in distance, and realize SAL imaging with high resolution and wide swath. The method comprises the following steps: N array elements respectively receive target echo signals, and each array element performs residual video phase compensation processing and form simplification processing on the target echo signals; Perform azimuth Fourier transform on the echo signal; perform defuzzification processing on the observation vector composed of the aliased target echo Doppler signal of N array elements; each element of the unambiguous target echo Doppler signal vector according to The numerical values of the fuzzy components are arranged, and the complete and unambiguous target echo Doppler signal is obtained by using Doppler spectrum splicing, so as to obtain the synthetic aperture lidar image.

Description

A signal processing method for multi-transmit and multi-receive synthetic aperture lidar

技术领域technical field

本发明属于通信技术领域，具体涉及一种多发多收合成孔径激光雷达信号处理方法。本发明可用于高分辨率宽测绘带合成孔径激光雷达成像。The invention belongs to the technical field of communication, and in particular relates to a multi-transmit and multi-receive synthetic aperture laser radar signal processing method. The invention can be used for high-resolution and wide-swath synthetic aperture lidar imaging.

背景技术Background technique

高分辨率对地观测要求的分辨率较高，观测距离较远，测绘带宽较大，并且朝着多种传感器相互协同工作的方向发展，合成孔径激光雷达(Synthetic Aperture Ladar,SAL)和微波合成孔径雷达(SyntheticAperture Ladar,SAR)有着很好的优势互补特性。对大范围地域可以采用SAR进行普查，然后对于感兴趣的目标设施，可以采用SAL进行更高分辨率的观测，SAL是对目前高分辨率对地观测手段的一个必要的补充手段。High-resolution earth observation requires higher resolution, longer observation distance, and larger mapping bandwidth, and it is developing towards the direction of multiple sensors working together. Synthetic Aperture Lidar (Synthetic Aperture Ladar, SAL) and microwave synthesis Aperture radar (Synthetic Aperture Ladar, SAR) has very good complementary characteristics. SAR can be used for census of large-scale areas, and then for interested target facilities, SAL can be used for higher-resolution observations. SAL is a necessary supplementary method to the current high-resolution earth observation methods.

采用SAL技术可以在远距离实现比目前SAR的分辨率提高至少一个数量级的高分辨率观测。与传统合成孔径雷达相比，合成孔径成像激光雷达的工作波长更短，其可以得到比合成孔径雷达分辨率高得多得图像(分辨率几十微米到几毫米)。The use of SAL technology can achieve high-resolution observations that are at least an order of magnitude higher than the resolution of current SAR at long distances. Compared with the traditional synthetic aperture radar, the operating wavelength of the synthetic aperture imaging lidar is shorter, and it can obtain images with a much higher resolution than the synthetic aperture radar (resolution tens of microns to several millimeters).

SAL技术的研究已经被列入到国家高分辨率对地观测的发展规划当中，西安电子科技大学雷达信号处理国防科技重点实验室在研究中发现单发单收的SAL在高分辨率模式下，其测绘带宽受到极大的限制。国外对SAL技术的研究已经进行了单发单收SAL的机载飞行试验，在一公里作用距离时其测绘带宽只有2米，因此，单发单收SAL窄测绘带宽的性质，严重制约了SAL技术的实用化。The research on SAL technology has been included in the development plan of the national high-resolution earth observation. The Key Laboratory of Radar Signal Processing and National Defense Science and Technology of Xidian University found in the research that the single-transmit and single-receive SAL in the high-resolution mode, Its mapping bandwidth is extremely limited. Foreign research on SAL technology has carried out airborne flight tests of single-engine and single-receive SAL, and its surveying and mapping bandwidth is only 2 meters at a distance of one kilometer. Therefore, the nature of the narrow surveying and mapping bandwidth of single-engine and single-receiving SAL seriously restricts Practicality of technology.

如何解决距离向测绘带和方位向分辨率的矛盾，实现高分辨率宽测绘带SAL成像，是今后SAL研究的核心问题。How to solve the contradiction between the range swath and azimuth resolution and realize high-resolution wide swath SAL imaging is the core issue of SAL research in the future.

发明内容Contents of the invention

针对上述缺点，本发明的目的在于提出一种多发多收合成孔径激光雷达信号处理方法，对多发多收回波多普勒信号作解模糊处理，得到无模糊的多普勒频谱用于后期成像。In view of the above-mentioned shortcomings, the object of the present invention is to propose a multi-input and multi-receiver synthetic aperture laser radar signal processing method, which performs defuzzification processing on the multi-input and multi-recovery wave Doppler signal, and obtains an unambiguous Doppler spectrum for post-imaging.

本发明可以有效的解决方位向高分辨率与距离向宽测绘带的矛盾，实现高分辨率宽测绘带SAL成像。The invention can effectively solve the contradiction between the high resolution in the azimuth direction and the wide surveying zone in the distance, and realize SAL imaging with high resolution and wide surveying zone.

为达到上述目的，本发明采用如下技术方案予以实现。In order to achieve the above object, the present invention adopts the following technical solutions to achieve.

一种多发多收合成孔径激光雷达信号处理方法，所述多发多收合成孔径激光雷达具有N个阵元，所述多发多收合成孔径激光雷达信号处理方法包括以下步骤：A signal processing method for multiple transmission and multiple collection synthetic aperture lidar, the multiple transmission and multiple collection synthetic aperture laser radar has N array elements, and the multiple transmission and multiple collection synthetic aperture laser radar signal processing method comprises the following steps:

步骤1，N个阵元分别接收目标回波信号，每个阵元对所述目标回波信号做剩余视频相位补偿处理和形式简化处理，得到形式简化处理后的目标回波信号；其中，所述阵元为多发多收阵元；Step 1, N array elements respectively receive target echo signals, and each array element performs residual video phase compensation processing and form simplification processing on the target echo signals, and obtains target echo signals after form simplification processing; wherein, The array element is a multi-send and multi-receive array element;

步骤2，所述每个阵元对所述形式简化处理后的目标回波信号做方位向傅立叶变换，得到混迭后的目标回波多普勒信号；Step 2, each array element performs azimuth Fourier transform on the target echo signal after the simplified processing in the form, and obtains the target echo Doppler signal after aliasing;

步骤3，由N个阵元的混迭后的目标回波多普勒信号构成观测向量，对所述观测向量做解模糊处理，得到无模糊的目标回波多普勒信号向量；Step 3, forming an observation vector from the aliased target echo Doppler signals of N array elements, and performing defuzzification processing on the observation vector to obtain an unambiguous target echo Doppler signal vector;

步骤4，对所述无模糊的目标回波多普勒信号向量的各个元素按照模糊分量的数值大小进行排列，采用多普勒频谱拼接得到完整的无模糊的目标回波多普勒信号；Step 4, arranging each element of the unambiguous target echo Doppler signal vector according to the numerical value of the fuzzy component, and using Doppler spectrum splicing to obtain a complete unambiguous target echo Doppler signal;

步骤5，对所述完整的无模糊的目标回波多普勒信号做成像处理，得到合成孔径激光雷达图像。Step 5, performing imaging processing on the complete and unambiguous target echo Doppler signal to obtain a synthetic aperture lidar image.

本发明的特点和进一步的改进为：Features of the present invention and further improvement are:

(1)步骤1具体包括以下子步骤：(1) Step 1 specifically includes the following sub-steps:

(1a)确定第i个阵元接收到的目标回波信号S_i(t,τ)，(1a) Determine the target echo signal S _i (t,τ) received by the i-th array element,

$\begin{matrix} {S S}_{i i} ((t t,, τ τ)) = = {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i,, k k} exp exp ((- - j j \frac{22 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{22 π π}{c c} {f f}_{c c} (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) exp exp ((j j \frac{44 πγ πγ}{{c c}^{22}} {((\frac{{R R}_{ip ip} + + {R R}_{kp kp}}{22} - - {R R}_{ref ref}))}^{22})) \end{matrix}$

其中，i＝1,2,…,N，N为阵元的数目，t为快时间，τ为慢时间，R_ref为中心作用距离，G_i,k表示第k个阵元发射、第i个阵元接收到的目标回波的增益，R_ip表示第i个阵元到目标的距离，R_kp表示第k个阵元到目标的距离，γ为调频率；Among them, i=1,2,...,N, N is the number of array elements, t is the fast time, τ is the slow time, R _ref is the distance from the center, G _i,k represents the k-th array element emission, the i-th The gain of the target echo received by each array element, R _ip represents the distance from the i-th array element to the target, R _kp represents the distance from the k-th array element to the target, and γ is the modulation frequency;

(1b)对第i个阵元接收到的目标回波信号S_i(t,τ)做剩余视频相位补偿处理，得到剩余视频相位补偿处理后的第i个阵元接收到的目标回波信号S_i′(t,τ)，(1b) Perform residual video phase compensation processing on the target echo signal S _i (t,τ) received by the i-th array element, and obtain the target echo signal received by the i-th array element after residual video phase compensation processing S _i '(t,τ),

$\begin{matrix} {S S}_{i i}^{' '} ((t t,, τ τ)) = = {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i,, k k} exp exp ((- - j j \frac{22 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{22 π π}{c c} {f f}_{c c} (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \end{matrix}$

其中，c为光速，f_c为载波频率；Among them, c is the speed of light, f _c is the carrier frequency;

(1c)对所述剩余视频相位补偿处理后的第i个阵元接收到的目标回波信号S_i′(t,τ)进行形式简化处理，得到形式简化处理后的第i个阵元接收到的目标回波信号S_i″(t,τ)，(1c) Simplify the form of the target echo signal S _i ′(t,τ) received by the i-th array element after the residual video phase compensation processing, and obtain the received i-th array element after the form-simplified process The target echo signal S _i ″(t,τ) arrived,

$\begin{matrix} {S S}_{i i}^{' '' '} ((t t,, τ τ)) \approx \approx {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i k k} exp exp ((- - j j \frac{44 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{\frac{i i + + k k}{22} p p} - - {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{44 π π}{c c} {f f}_{c c} (({R R}_{\frac{i i + + k k}{22},, p p} - - {R R}_{ref ref})))) \end{matrix}$

其中， $R_{\frac{i + k}{2}, p} = \sqrt{{(R_{ref} + r_{p})}^{2} + {(vτ - x_{p} + (\frac{i + k}{2} - (\frac{N + 1}{2})) a)}^{2}},$ v是平台运行的方位向速度，a为相邻两个阵元的间隔，x_p为目标的距离向位置，r_p为中心偏离距离；in, $R_{\frac{i + k}{2}, p} = \sqrt{{(R_{ref} + r_{p})}^{2} + {(vτ - x_{p} + (\frac{i + k}{2} - (\frac{N + 1}{2})) a)}^{2}},$ v is the azimuth velocity of the platform, a is the distance between two adjacent array elements, x _p is the range position of the target, and r _p is the distance from the center;

(1d)依次取i＝1,2,…,N，重复步骤(1b)和(1c)，得到形式简化处理后的N个阵元的目标回波信号S₁″(t,τ)，…,S_N″(t,τ)。(1d) Take i=1, 2,...,N in sequence, repeat steps (1b) and (1c), and obtain the target echo signals S ₁ ″(t,τ),… ,S _N ″(t,τ).

进一步的，further,

在子步骤(1a)中，第i个阵元到目标的距离R_ip具体为：In sub-step (1a), the distance R _ip from the i-th array element to the target is specifically:

${R R}_{ip ip} = = \sqrt{{(({R R}_{ref ref} + + {r r}_{p p}))}^{22} + + ((vτ vτ - - {x x}_{p p} + + ((i i - - ((\frac{N N + + 11}{22})))) a a))}$

其中，i≥1，N为收发阵元数目，v是平台运行的方位向速度。Among them, i≥1, N is the number of transceiver elements, and v is the azimuth speed of the platform.

在子步骤(1a)中，第k个阵元到目标的距离R_kp具体为：In sub-step (1a), the distance R _kp from the kth array element to the target is specifically:

${R R}_{kp kp} = = \sqrt{{(({R R}_{ref ref} + + {r r}_{p p}))}^{22} + + ((vτ vτ - - {x x}_{p p} + + ((k k - - ((\frac{N N + + 11}{22})))) a a))}$

其中，k≤N，N为收发阵元数目，v是平台运行的方位向速度。Among them, k≤N, N is the number of transceiver elements, and v is the azimuth speed of the platform.

(2)步骤2具体包括以下子步骤：(2) Step 2 specifically includes the following sub-steps:

(2a)对形式简化处理后的第i个阵元的目标回波信号S_i″(t,τ)做方位向傅立叶变换，得到第i个阵元的目标回波多普勒信号S_i(t,f_a)，(2a) Perform azimuth Fourier transform on the target echo signal S _i ″(t,τ) of the i-th array element after simplified processing, and obtain the target echo Doppler signal S _i (t , f _a ),

${S S}_{i i} ((t t,, {f f}_{a a})) = = {Σ Σ}_{k k = = 11}^{N N} \begin{matrix} {G G}_{i i,, k k} exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γt γt}{c c}))}^{22} - - {((\frac{{f f}_{a a}}{22 v v}))}^{22}))} - - \frac{22 π π {f f}_{a a}}{v v} {x x}_{p p})) \\ \times \times exp exp ((j j \frac{22 π π {f f}_{a a}}{v v} ((\frac{i i + + k k}{22} - - \frac{N N + + 11}{22})) a a)) \end{matrix}$

其中，i＝1,2,…,N，f_a为多普勒频率，PRF为脉冲重复频率；Wherein, i=1,2,...,N, f _a is the Doppler frequency, PRF is the pulse repetition frequency;

(2b)将所述第i个阵元的目标回波多普勒信号S_i(t,f_a)表示为：(2b) Express the target echo Doppler signal S _i (t, f _a ) of the i-th array element as:

S_i(t,f_a)＝A_i(f_a)S₀(t,f_a)S _i (t,f _a )=A _i (f _a )S ₀ (t,f _a )

其中， $A_{i} (f_{a}) = Σ_{k = 1}^{N} G_{i, k} \exp (j \frac{π f_{a}}{v} (i + k - N - 1) a),$ in, $A_{i} (f_{a}) = Σ_{k = 1}^{N} G_{i, k} \exp (j \frac{π f_{a}}{v} (i + k - N - 1) a),$

$\begin{matrix} {S S}_{00} ((t t,, {f f}_{a a}^{' '})) = = exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γ γ t t}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{{f f}_{a a}^{' '}}{22 v v}))}^{22}))})) \\ \times \times exp exp ((- - j j \frac{22 π π {f f}_{a a}^{' '}}{v v} {x x}_{p p})) \times \times exp exp ((j j \frac{44 π π}{c c} γ γ ((t t + + \frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {R R}_{ref ref})) \end{matrix};;$

(2c)对所述第i个阵元的目标回波多普勒信号S_i(t,f_a)产生混迭，得到第i个阵元混迭后的目标回波多普勒信号S_i′(t,f_a)，(2c) Generate aliasing on the target echo Doppler signal S _i (t, f _a ) of the ith array element, and obtain the target echo Doppler signal S _i '( t, f _a ),

${S S}_{i i}^{' '} ((t t,, {f f}_{a a})) = = {Σ Σ}_{p p = = - - M m}^{M m} {A A}_{i i} ((q q \cdot \cdot PRF PRF + + {f f}_{a a})) {S S}_{00} ((t t,, q q \cdot \cdot PRF PRF + + {f f}_{a a}))$

其中，模糊分量q的取值范围为q＝-M,-M+1,…,M-1,M；Wherein, the value range of the fuzzy component q is q=-M,-M+1,...,M-1,M;

(2d)依次取i＝1,2,…,N，得到N个阵元混迭后的目标回波多普勒信号S₁′(t,f_a)，…，S_N′(t,f_a)。(2d) Take i=1, 2,...,N in sequence to obtain the target echo Doppler signal S ₁ '(t,f _a ),...,S _N ′(t,f _a ).

(3)步骤3具体包括以下子步骤：(3) Step 3 specifically includes the following sub-steps:

(3a)由所述N个阵元的混迭后的目标回波多普勒信号S₁′(t,f_a)，…，S_N′(t,f_a)，构建观测向量X(t_l,f_a)，( _3a ₎ Construct the observation vector _X ( _t _l , f _a ),

$X x (({t t}_{l l},, {f f}_{a a})) \overset{Δ Δ}{= =} {[[{S S}_{11}^{' '} (({t t}_{l l},, {f f}_{a a})),, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; {S S}_{N N}^{' '} (({t t}_{l l},, {f f}_{a a}))]]}^{T T}$

其中，t_l是快时间t中的某一时刻，l＝1,2,…,L，L为快时间离散序列的长度，(·)^T表示矩阵转置操作；Wherein, t _l is a certain moment in the fast time t, l=1,2,...,L, L is the length of the fast time discrete sequence, ( ) ^T represents the matrix transposition operation;

(3b)根据Capon波束形成原理，确定目标回波信号在多普勒频率f_a处的自适应最优权矢量w_opt(f_a(q))，(3b) According to the Capon beamforming principle, determine the adaptive optimal weight vector w _opt (f _a (q)) of the target echo signal at the Doppler frequency f _a ,

${w w}_{opt opt} (({f f}_{a a} ((q q)))) = = \frac{{R R}^{- - 11} (({f f}_{a a})) v v (({f f}_{a a} ((q q))))}{{v v}^{H h} (({f f}_{a a} ((q q)))) {R R}^{- - 11} (({f f}_{a a})) v v (({f f}_{a a} ((q q))))}$

其中，R(f_a)为多普勒频率f_a处的目标回波信号的统计协方差矩阵，v(f_a(q))为多普勒频率f_a处的第q个模糊分量的导向矢量，(·)^-1表示矩阵求逆操作，(·)^H表示矩阵共轭转置操作；Among them, R(f _a ) is the statistical covariance matrix of the target echo signal at the Doppler frequency f _a , v(f _a (q)) is the guidance of the qth fuzzy component at the Doppler frequency f _a Vector, ( ) ^-1 means matrix inversion operation, ( ) ^H means matrix conjugate transpose operation;

(3c)根据自适应最优权矢量w_opt(f_a(q))对观测向量X(t_l,f_a)加权，得到多普勒频率f_a处无模糊的目标回波多普勒信号向量S_DBF(t,f_a)。(3c) Weight the observation vector X(t _l , f _a ) according to the adaptive optimal weight vector w _opt (f _a (q)), and obtain the unambiguous target echo Doppler signal vector at the Doppler frequency f _a S _DBF (t,f _a ).

进一步的，further,

所述多普勒频率f_a处无模糊的目标回波多普勒信号向量S_DBF(t,f_a)为：The target echo Doppler signal vector S _DBF (t, f _a ) without ambiguity at the Doppler frequency f _a is:

S_DBF(t,f_a)＝[S₀(t,-M·PRF+f_a),......S₀(t,-M·PRF+f_a)]^T S _DBF (t,f _a )＝[S ₀ (t,-M·PRF+f _a ),...S ₀ (t,-M·PRF+f _a )] ^T

其中， $\begin{matrix} S_{0} (t, p \cdot PRF + f_{a}^{'}) = \exp (- j 4 π (R_{ref} + r_{p}) \sqrt{({(\frac{f_{c} + γ t}{c} - \frac{2 γ R_{ref}}{c^{2}})}^{2} - {(\frac{p \cdot {PRF + f}_{a}^{'}}{2 v})}^{2})}) \\ \times \exp (- j \frac{2 π (p \cdot PRF + f_{a})}{v} x_{p}) \times \exp (j \frac{4 π}{c} γ (t_{l} + \frac{f_{c}}{γ} - \frac{2 R_{ref}}{c}) R_{ref}) \end{matrix} .$ in, $\begin{matrix} S_{0} (t, p \cdot PRF + f_{a}^{'}) = \exp (- j 4 π (R_{ref} + r_{p}) \sqrt{({(\frac{f_{c} + γ t}{c} - \frac{2 γ R_{ref}}{c^{2}})}^{2} - {(\frac{p \cdot {PRF + f}_{a}^{'}}{2 v})}^{2})}) \\ \times \exp (- j \frac{2 π (p &Center Dot; PRF + f_{a})}{v} x_{p}) \times \exp (j \frac{4 π}{c} γ (t_{l} + \frac{f_{c}}{γ} - \frac{2 R_{ref}}{c}) R_{ref}) \end{matrix} .$

本发明提出的多发多收SAL体制可以有效解决传统单发单收SAL体制中存在的距离向宽测绘带和方位向高分辨率的矛盾问题，实现高分辨率宽测绘带SAL成像，该发明成果将拓展SAL成像的概念和内涵，为未来针对典型应用的高分辨率宽测绘带系统的提出和研制提供理论和方法基础。The multi-transmit and multi-receive SAL system proposed by the present invention can effectively solve the contradictory problems existing in the traditional single-transmit and single-receive SAL system in the wide surveying zone in the distance direction and the high resolution in the azimuth direction, and realize high-resolution wide surveying zone SAL imaging. The concept and connotation of SAL imaging will be expanded, and the theoretical and methodological basis will be provided for the proposal and development of high-resolution wide swath systems for typical applications in the future.

附图说明Description of drawings

为了更清楚地说明本发明实施例或现有技术中的技术方案，下面将对实施例或现有技术描述中所需要使用的附图作简单地介绍，显而易见地，下面描述中的附图仅仅是本发明的一些实施例，对于本领域普通技术人员来讲，在不付出创造性劳动的前提下，还可以根据这些附图获得其他的附图。In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. Those skilled in the art can also obtain other drawings based on these drawings without creative work.

图1为本发明实施例提供的基本流程示意图；Fig. 1 is a schematic flow diagram of the basic flow provided by the embodiment of the present invention;

图2为方位向多发多收合成孔径激光雷达体制示意图；Fig. 2 is a schematic diagram of azimuth multi-transmit and multi-receive synthetic aperture lidar system;

图3为多发多收SAL体制的多普勒频谱示意图，横坐标为多普勒频率，单位为赫兹(Hz)，纵坐标为幅度；Figure 3 is a schematic diagram of the Doppler spectrum of the multi-transmit and multi-receive SAL system, the abscissa is the Doppler frequency, the unit is Hertz (Hz), and the ordinate is the amplitude;

图4为单发单收SAL体制的9个点目标成像的等高线示意图，横坐标为方位单元，纵坐标为距离单元；Figure 4 is a schematic diagram of the contour line of 9-point target imaging in the single-send-single-reception SAL system, the abscissa is the azimuth unit, and the ordinate is the distance unit;

图5为单发单收SAL体制的9个点目标成像的方位脉压剖面示意图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)；Figure 5 is a schematic diagram of the azimuth pulse pressure profile of 9-point target imaging in the single-shot, single-receive SAL system. The abscissa is the azimuth distance in meters (m), and the ordinate is the normalized amplitude in decibels (dB). ;

图6为多发多收SAL体制的9个点目标成像的等高线示意图，横坐标为方位单元，纵坐标为距离单元；Figure 6 is a schematic diagram of the contour line of 9-point target imaging in the multi-transmit and multi-receive SAL system, the abscissa is the azimuth unit, and the ordinate is the distance unit;

图7为方位向多发多收SAL体制的9个点目标成像的方位脉压剖面示意图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)；Figure 7 is a schematic diagram of the azimuth pulse pressure profile of 9-point target imaging in the azimuth multi-transmission and multi-reception SAL system. The abscissa is the azimuth distance in meters (m), and the ordinate is the normalized amplitude in decibels (dB );

图8为图6中中间位置的点目标成像放大后的示意图，横坐标为方位单元，纵坐标为距离单元；Fig. 8 is the enlarged schematic diagram of the point target imaging in the middle position in Fig. 6, the abscissa is the azimuth unit, and the ordinate is the distance unit;

图9为图6中中间位置的点目标方位脉压剖面示意图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)；Fig. 9 is a schematic diagram of the point target azimuth pulse pressure profile at the middle position in Fig. 6, the abscissa is the azimuth distance, the unit is meter (m), and the ordinate is the normalized amplitude, the unit is decibel (dB);

图10为图6中中间位置的点目标距离脉压剖面示意图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)。Fig. 10 is a schematic diagram of the point target distance pulse pressure profile at the middle position in Fig. 6, the abscissa is the azimuth distance, the unit is meter (m), and the ordinate is the normalized amplitude, the unit is decibel (dB).

具体实施方式detailed description

下面将结合本发明实施例中的附图，对本发明实施例中的技术方案进行清楚、完整地描述，显然，所描述的实施例仅仅是本发明一部分实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例，都属于本发明保护的范围。The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

参照图1，本发明的具体实现包括如下步骤：With reference to Fig. 1, concrete realization of the present invention comprises the following steps:

步骤1，N个阵元分别接收目标回波信号，每个阵元对所述目标回波信号做剩余视频相位补偿处理和形式简化处理，得到形式简化处理后的目标回波信号。Step 1: N array elements respectively receive target echo signals, and each array element performs residual video phase compensation processing and form simplification processing on the target echo signals to obtain target echo signals after form simplification processing.

其中，所述阵元为多发多收阵元。Wherein, the array element is a multi-transmit and multi-receive array element.

N个阵元为方位向均匀分布的阵元，N个阵元发射激光线性调频信号，采用解线性调频(dechirping)的方式接收目标回波信号，首先对目标回波信号做剩余视频相位补偿处理，然后再对做剩余视频相位补偿处理后的目标回波信号做形式简化处理，得到形式简化处理后的N个阵元的目标回波信号S₁″(t,τ)，…,S_N″(t,τ)。N array elements are evenly distributed in azimuth, N array elements emit laser chirp signals, and receive target echo signals by dechirping, firstly perform residual video phase compensation processing on target echo signals , and then simplify the form of the target echo signal after the residual video phase compensation processing, and obtain the target echo signal S ₁ ″(t,τ),…,S _N ″ of the N array elements after the simplified form processing (t,τ).

步骤1的具体子步骤为：The specific sub-steps of step 1 are:

(1a)确定第i个阵元接收到的目标回波信号S_i(t,τ)为：(1a) Determine the target echo signal S _i (t,τ) received by the i-th array element as:

其中，c为光速，f_c为载波频率，i＝1,2,…,N，N为阵元数目，t为快时间，τ为慢时间，R_ref为中心作用距离，G_i,k表示第k个发射阵元发射、第i个接收阵元接收回波的增益，R_ip表示第i个接收阵元到点目标的距离，R_kp表示第k个发射单元到点目标的距离，γ为调频率。Among them, c is the speed of light, f _c is the carrier frequency, i=1,2,...,N, N is the number of array elements, t is the fast time, τ is the slow time, R _ref is the center action distance, G _i,k represents The gain of the k-th transmitting element transmitting and the i-th receiving element receiving the echo, R _ip represents the distance from the i-th receiving element to the point target, R _kp represents the distance from the k-th transmitting element to the point target, γ is the tuning frequency.

参照图2，凸透镜的焦距为f，在凸透镜焦平面，有N个阵元沿航迹方向排列(为了阐述简明，图中只画了3个阵元)，相邻两个阵元的间隔为a，将中间的阵元置于坐标(0,0)处。阵元发射激光线性调频信号，调频率为γ，设地面上的点目标位置为p(x_p,R_ref+r_p)，其中，x_p为点目标距离向位置，r_p为中心偏离距离，采用dechirping接收方式接收回波信号，则第i个阵元接收到的目标回波信号S_i(t,τ)为：Referring to Figure 2, the focal length of the convex lens is f. On the focal plane of the convex lens, there are N array elements arranged along the track direction (for the sake of simplicity, only 3 array elements are drawn in the figure), and the distance between two adjacent array elements is a, Place the middle element at coordinate (0,0). The array element emits a laser chirp signal, the modulation frequency is γ, and the point target position on the ground is p(x _p ,R _ref +r _p ), where x _p is the distance position of the point target, and r _p is the distance from the center , using the dechirping receiving method to receive the echo signal, then the target echo signal S _i (t,τ) received by the i-th array element is:

$\begin{matrix} {S S}_{i i} ((t t,, τ τ)) = = {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i,, k k} exp exp ((- - j j \frac{22 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{22 π π}{c c} {f f}_{c c} (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) exp exp ((j j \frac{44 πγ πγ}{{c c}^{22}} {((\frac{{R R}_{ip ip} + + {R R}_{kp kp}}{22} - - {R R}_{ref ref}))}^{22})) \end{matrix} - - - - - - ((11))$

上式(1)中，c为光速，f_c为载波频率，t为快时间，τ为慢时间，R_ref为中心作用距离，G_i,k表示第k个阵元发射、第i个阵元接收到的目标回波的增益，R_ip表示第i个阵元到目标的距离，R_kp表示第k个阵元到目标的距离，In the above formula (1), c is the speed of light, f _c is the carrier frequency, t is the fast time, τ is the slow time, R _ref is the distance from the center, G _i,k represents the emission of the kth array element, the The gain of the target echo received by the element, R _ip represents the distance from the i-th array element to the target, R _kp represents the distance from the k-th array element to the target,

其中，in,

$\begin{matrix} {R R}_{ip ip} = = \sqrt{{(({R R}_{ref ref} + + {r r}_{p p}))}^{22} + + ((vτ vτ - - {x x}_{p p} + + ((i i - - ((\frac{N N + + 11}{22})))) a a))} \\ {R R}_{kp kp} = = \sqrt{{(({R R}_{ref ref} + + {r r}_{p p}))}^{22} + + ((vτ vτ - - {x x}_{p p} + + ((k k - - ((\frac{N N + + 11}{22})))) a a))} \end{matrix} - - - - - - ((22))$

上式(2)中i≥1,k≤N，N为阵元的数目，v是平台运行的方位向速度，a为相邻两个阵元的间隔。In the above formula (2), i≥1, k≤N, N is the number of array elements, v is the azimuth velocity of the platform, and a is the interval between two adjacent array elements.

第i个阵元接收到的目标回波信号S_i(t,τ)中有3个指数项，第一个指数项表示目标回波信号的距离信息，第二个指数项表示目标回波信号的方位多普勒信息，第三个指数项表示目标回波信号经过dechirping方式接收到的剩余视频相位。There are 3 exponential items in the target echo signal S _i (t,τ) received by the i-th array element, the first exponential item Indicates the distance information of the target echo signal, the second index item Represents the azimuth Doppler information of the target echo signal, the third index item Indicates the remaining video phase of the target echo signal received by dechirping.

(1b)对第i个阵元接收到的目标回波信号S_i(t,τ)做剩余视频相位补偿处理，得到剩余视频相位补偿处理后的第i个阵元接收到的目标回波信号S_i′(t,τ)为：(1b) Perform residual video phase compensation processing on the target echo signal S _i (t,τ) received by the i-th array element, and obtain the target echo signal received by the i-th array element after residual video phase compensation processing S _i ′(t,τ) is:

$\begin{matrix} {S S}_{i i}^{' '} ((t t,, τ τ)) = = {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i,, k k} exp exp ((- - j j \frac{22 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{22 π π}{c c} {f f}_{c c} (({R R}_{ip ip} + + {R R}_{kp kp} - - 22 {R R}_{ref ref})))) \end{matrix} - - - - - - ((33))$

(1c)对剩余视频相位补偿处理后的第i个阵元接收到的目标回波信号S_i′(t,τ)进行形式简化处理，得到形式简化处理后的第i个阵元接收到的目标回波信号S_i″(t,τ)为：(1c) Simplify the form of the target echo signal S _i ′(t,τ) received by the i-th array element after the remaining video phase compensation processing, and obtain the The target echo signal S _i ″(t,τ) is:

其中， $R_{\frac{i + k}{2}, p} = \sqrt{{(R_{ref} + r_{p})}^{2} + {(vτ - x_{p} + (\frac{i + k}{2} - (\frac{N + 1}{2})) a)}^{2}},$ v是平台运行的方位向速度，a为相邻两个阵元的间隔，x_p为目标的距离向位置，r_p为中心偏离距离。in, $R_{\frac{i + k}{2}, p} = \sqrt{{(R_{ref} + r_{p})}^{2} + {(vτ - x_{p} + (\frac{i + k}{2} - (\frac{N + 1}{2})) a)}^{2}},$ v is the azimuth velocity of the platform, a is the distance between two adjacent array elements, x _p is the range position of the target, and r _p is the distance from the center.

多发多收SAL的作用距离远远大于收发孔径的长度，即满足：The working distance of multi-transmit and multi-receive SAL is far greater than the length of the transceiver aperture, that is, to meet:

${R R}_{ref ref} > > > > \frac{N N - - 11}{22} a a - - - - - - ((44))$

因此，多发多收SAL的作用距离满足如下关系：Therefore, the working distance of multi-transmit and multi-receive SAL satisfies the following relationship:

${R R}_{ip ip} + + {R R}_{kp kp} \approx \approx 22 {R R}_{\frac{i i + + k k}{22},, p p} - - - - - - ((55))$

上式(5)中：In the above formula (5):

${R R}_{\frac{i i + + k k}{22},, p p} = = \sqrt{{(({R R}_{ref ref} + + {r r}_{p p}))}^{22} + + {((vτ vτ - - {x x}_{p p} + + ((\frac{i i + + k k}{22} - - ((\frac{N N + + 11}{22})))) a a))}^{22}} - - - - - - ((66))$

因此对上式(3)进行形式简化处理，得到形式简化处理后的第i个多发多收阵元接收到的目标回波信号S_i″(t,τ)：Therefore, the above formula (3) is simplified in form to obtain the target echo signal S _i ″(t,τ) received by the i-th multi-transmission and multi-reception array element after the simplified form:

$\begin{matrix} {S S}_{i i}^{' '' '} ((t t,, τ τ)) \approx \approx {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i k k} exp exp ((- - j j \frac{44 π π}{c c} γ γ ((t t - - \frac{22 {R R}_{ref ref}}{c c})) (({R R}_{\frac{i i + + k k}{22} p p} - - {R R}_{ref ref})))) \\ \times \times exp exp ((- - j j \frac{44 π π}{c c} {f f}_{c c} (({R R}_{\frac{i i + + k k}{22},, p p} - - {R R}_{ref ref})))) \end{matrix} - - - - - - ((77))$

(1d)对i进行遍历，重复步骤(1b)和(1c)，得到形式简化处理后的N个多发多收阵元的目标回波信号S₁″(t,τ)，…,S_N″(t,τ)。(1d) Traversing i, repeating steps (1b) and (1c), to obtain target echo signals S ₁ ″(t,τ),…,S _N ″ of N multi-transmission and multi-reception array elements after simplified form (t,τ).

步骤2，所述每个阵元对所述形式简化处理后的目标回波信号做方位向傅立叶变换，得到混迭后的目标回波多普勒信号。In step 2, each array element performs azimuth Fourier transform on the target echo signal after the simplified processing to obtain an aliased target echo Doppler signal.

步骤2具体包括如下子步骤：Step 2 specifically includes the following sub-steps:

(2a)对形式简化处理后的第i个阵元的目标回波信号做方位向傅立叶变换，得到第i个阵元的目标回波多普勒信号。(2a) Perform azimuth Fourier transform on the target echo signal of the i-th array element after simplified processing to obtain the target echo Doppler signal of the i-th array element.

利用驻定相位定理对形式简化处理后的第i个阵元的目标回波信号S_i″(t,τ)做方位向傅立叶变换，得到第i个阵元的目标回波多普勒信号S_i(t,f_a)为：Using the stationary phase theorem, perform azimuth Fourier transform on the target echo signal S _i ″(t,τ) of the i-th array element after simplified processing, and obtain the target echo Doppler signal S _i of the i-th array element (t, f _a ) is:

${S S}_{i i} ((t t,, {f f}_{a a})) = = {Σ Σ}_{k k = = 11}^{N N} \begin{matrix} {G G}_{i i,, k k} exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γt γt}{c c}))}^{22} - - {((\frac{{f f}_{a a}}{22 v v}))}^{22}))} - - \frac{22 π π {f f}_{a a}}{v v} {x x}_{p p})) \\ \times \times exp exp ((j j \frac{22 π π {f f}_{a a}}{v v} ((\frac{i i + + k k}{22} - - \frac{N N + + 11}{22})) a a)) \end{matrix} - - - - - - ((88))$

其中，i＝1,2,…,N，f_a为多普勒频率，PRF为脉冲重复频率。Wherein, i=1,2,...,N, f _a is the Doppler frequency, PRF is the pulse repetition frequency.

(2b)将第i个阵元的目标回波多普勒信号S_i(t,f_a)表示为：(2b) Express the target echo Doppler signal S _i (t,f _a ) of the i-th array element as:

S_i(t,f_a)＝A_i(f_a)S₀(t,f_a) (9)S _i (t,f _a )=A _i (f _a )S ₀ (t,f _a ) (9)

其中，in,

${A A}_{i i} (({f f}_{a a})) = = {Σ Σ}_{k k = = 11}^{N N} {G G}_{i i,, k k} exp exp ((j j \frac{π π {f f}_{a a}}{v v} ((i i + + k k - - N N - - 11)) a a)) - - - - - - ((1010))$

$\begin{matrix} {S S}_{00} ((t t,, {f f}_{a a}^{' '})) = = exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γ γ t t}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{{f f}_{a a}^{' '}}{22 v v}))}^{22}))})) \\ \times \times exp exp ((- - j j \frac{22 π π {f f}_{a a}^{' '}}{v v} {x x}_{p p})) \times \times exp exp ((j j \frac{44 π π}{c c} γ γ ((t t + + \frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {R R}_{ref ref})) \end{matrix} - - - - - - ((1111))$

(2c)对第i个阵元的目标回波多普勒信号S_i(t,f_a)产生混迭，得到第i个阵元混迭后的目标回波多普勒信号S_i′(t,f_a)为：(2c) Generate aliasing for the target echo Doppler signal S _i (t, f a ) of the i-th array element, and obtain the target echo Doppler signal S _i ′(t, f _a ) of the i-th array element after aliasing f _a ) is:

其中，模糊分量q的取值范围q＝-M,-M+1,…,M-1,M，模糊分量的数目为2×M+1。Wherein, the value range of the fuzzy component q is q=-M, -M+1, . . . , M-1, M, and the number of fuzzy components is 2×M+1.

参照图3给出的多发多收SAL的多普勒频谱示意图。根据图2可以知道，多发多收阵元不是严格工作在正侧视模式，其中，中间位置的多发多收阵元工作在正侧视模式，前面的多发多收阵元工作在后斜视模式，后面的多发多收阵元工作在前斜视模式，因此多发多收SAL的回波多普勒频带要比单发单收系统的多普勒频带宽。我们使用较小的脉冲重复频率保证获得较大的测绘带宽，因此，对于整个多普勒带宽来说是欠采样的，而欠采样处理会使整个多普勒频带混迭。第i个阵元的目标回波多普勒信号S_i(t,f_a)产生混迭后，得到第i个阵元混迭后的目标回波多普勒信号为：Refer to the schematic diagram of the Doppler spectrum of the MFMA SAL shown in FIG. 3 . According to Figure 2, it can be known that the multi-transmission and multi-reception array elements do not strictly work in the front and side view mode. Among them, the multi-transmission and multi-reception array elements in the middle work in the front and side view mode, and the front multi-transmission and multi-reception array elements work in the rear squint mode. The rear multi-transmission and multi-reception array elements work in the forward squint mode, so the echo Doppler frequency band of the multi-transmission and multi-reception SAL is wider than that of the single-transmission and single-reception system. We use a small pulse repetition frequency to guarantee a large mapping bandwidth, and therefore undersample for the entire Doppler bandwidth, and the undersampling process aliases the entire Doppler band. After the target echo Doppler signal S _i (t, f _a ) of the i-th array element is aliased, the target echo Doppler signal of the i-th array element after aliasing is obtained as:

${S S}_{i i}^{' '} ((t t,, {f f}_{a a})) = = {Σ Σ}_{p p = = - - M m}^{M m} {A A}_{i i} ((q q \cdot &Center Dot; PRF PRF + + {f f}_{a a})) {S S}_{00} ((t t,, q q \cdot \cdot PRF PRF + + {f f}_{a a})) - - - - - - ((1212))$

(2d)对i遍历，得到N个阵元混迭后的目标回波多普勒信号S₁′(t,f_a)，…，S_N′(t,f_a)。(2d) Traverse i to obtain target echo Doppler signals S ₁ ′(t,f _a ), . . . , S _N ′(t,f _a ) after N array elements are mixed.

步骤3，由各个阵元的混迭后的目标回波多普勒信号构成观测向量，对所述观测向量做解模糊处理，得到无模糊的目标回波多普勒信号向量。In step 3, an observation vector is formed from the aliased target echo Doppler signals of each array element, and a defuzzification process is performed on the observation vector to obtain an unambiguous target echo Doppler signal vector.

步骤3的具体子步骤为：The specific sub-steps of step 3 are:

(3a)对N个阵元混迭后的目标回波多普勒信号S₁′(t,f_a)，…，S_N′(t,f_a)，构建观测向量X(t_l,f_a)：(3a) Construct the observation vector X(t _l ,f _a ) for the target echo Doppler signal S ₁ ′(t,f _a ),…,S _N ′(t,f _a ) mixed by N array elements ):

$X x (({t t}_{l l},, {f f}_{a a})) \overset{Δ Δ}{= =} {[[{S S}_{11}^{' '} (({t t}_{l l},, {f f}_{a a})),, \cdot \cdot \cdot &Center Dot; \cdot \cdot \cdot \cdot \cdot \cdot \cdot &Center Dot; {S S}_{N N}^{' '} (({t t}_{l l},, {f f}_{a a}))]]}^{T T}$

根据步骤2得到所有阵元混迭后的目标回波多普勒信号S₁′(t,f_a)，…，S_N′(t,f_a)，构建出观测向量X(t_l,f_a)：Obtain the target echo Doppler signal S ₁ ′(t,f _a ),…,S _N ′(t,f _a ) after all array elements are aliased according to step 2, and construct the observation vector X(t _l ,f _a ):

$X x (({t t}_{l l},, {f f}_{a a})) \overset{Δ Δ}{= =} {[[{S S}_{11}^{' '} (({t t}_{l l},, {f f}_{a a})),, \cdot \cdot \cdot \cdot \cdot &Center Dot; \cdot \cdot \cdot \cdot \cdot &Center Dot; {S S}_{N N}^{' '} (({t t}_{l l},, {f f}_{a a}))]]}^{T T} - - - - - - ((1313))$

其中，t_l是快时间t中的某一时刻，l＝1,2,…,L，L为快时间离散序列的长度。(·)^T表示矩阵转置操作。Wherein, t _l is a certain moment in the fast time t, l=1, 2,..., L, and L is the length of the fast time discrete sequence. (·) ^T represents the matrix transpose operation.

(3b)根据Capon波束形成原理，确定目标回波信号在多普勒频率f_a处的自适应最优权矢量w_opt(f_a(q))为：(3b) According to the principle of Capon beamforming, determine the adaptive optimal weight vector w _opt (f _a (q)) of the target echo signal at the Doppler frequency f _a as:

根据Capon波束形成原理，多普勒频率f_a处的自适应最优权矢量w(f_a)满足如下条件：According to the principle of Capon beamforming, the adaptive optimal weight vector w(f _a ) at the Doppler frequency f _a satisfies the following conditions:

$\begin{matrix} Minmize Minmize : : {w w}^{H h} (({f f}_{a a})) R R (({f f}_{a a})) w w (({f f}_{a a})) \\ Subject to Subject to : : {w w}^{H h} (({f f}_{a a})) v v (({f f}_{a a} ((q q)))) = = 11 \end{matrix} - - - - - - ((1414))$

上式(14)中v(f_a(q))为多普勒频率f_a处的第q个模糊分量的导向矢量。In the above formula (14), v(f _a (q)) is the steering vector of the qth fuzzy component at the Doppler frequency f _a .

$\begin{matrix} v v (({f f}_{a a} ((q q)))) = = {[[{A A}_{11} ((q q)),, {A A}_{22} ((q q)),, \cdot &Center Dot; \cdot \cdot \cdot &Center Dot;,, {A A}_{N N} ((q q))]]}^{T T} \\ = = [\begin{matrix} {Σ Σ}_{k k = = 11}^{N N} {G G}_{11,, k k} exp exp ((j j \frac{π π ((q q \cdot &Center Dot; PRF PRF + + {f f}_{a a}))}{v v} ((11 + + k k - - N N - - 11)) a a)) \\ {Σ Σ}_{k k = = 11}^{N N} {G G}_{22,, k k} exp exp ((j j \frac{π π ((q q \cdot &Center Dot; PRE PRE + + {f f}_{a a}))}{v v} ((22 + + k k - - N N - - 11)) a a)) \\ \cdot \cdot \\ \cdot \cdot \\ \cdot &Center Dot; \\ {Σ Σ}_{k k = = 11}^{N N} {G G}_{N N,, k k} exp exp ((j j \frac{π π ((q q \cdot \cdot PRF PRF + + {f f}_{a a}))}{v v} ((N N + + k k - - N N + + 11)) a a)) \end{matrix}] \end{matrix} - - - - - - ((1515))$

R(f_a)为多普勒频率f_a处的回波信号的统计协方差矩阵，由于噪声不可知，统计协方差矩阵可以利用样本协方差矩阵估计，R(f _a ) is the statistical covariance matrix of the echo signal at the Doppler frequency f _a , since the noise is unknown, the statistical covariance matrix can be estimated by using the sample covariance matrix,

$\begin{matrix} R R (({f f}_{a a})) = = \frac{11}{L L} {Σ Σ}_{l l = = 11}^{L L} {X x}_{l l} {X x}_{l l}^{H h} \\ = = \frac{11}{L L} {Σ Σ}_{l l = = 11}^{L L} [\begin{matrix} {S S}_{11}^{' '} (({t t}_{l l},, {f f}_{a a})) \\ {S S}_{22}^{' '} (({t t}_{l l},, {f f}_{a a})) \\ \cdot \cdot \\ \cdot \cdot \\ \cdot &Center Dot; \\ {S S}_{N N}^{' '} (({t t}_{l l},, {f f}_{a a})) \end{matrix}] [[{(({S S}_{11}^{' '} (({t t}_{l l},, {f f}_{a a}))))}^{* *},, {(({S S}_{22}^{' '} (({t t}_{l l},, {f f}_{a a}))))}^{* *},, \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot,, {(({S S}_{N N}^{' '} (({t t}_{l l},, {f f}_{a a}))))}^{* *}]] \\ = = \frac{11}{L L} [[{X x}_{11},, {X x}_{22},, \cdot \cdot \cdot \cdot \cdot \cdot,, {X x}_{L L}]] {[[{X x}_{11},, {X x}_{22},, \cdot \cdot \cdot \cdot \cdot \cdot,, {X x}_{L L}]]}^{H h} \end{matrix} - - - - - - ((1616))$

上式中(·)^H表示矩阵共轭转置操作，(·)^*表示信号取共轭操作。In the above formula, (·) ^H represents the matrix conjugate transpose operation, and (·) ^* represents the signal conjugate operation.

根据式(15)得到多普勒频率f_a处的自适应最优权矢量w_opt(f_a(q))为：According to formula (15), the adaptive optimal weight vector w _opt (f _a (q)) at the Doppler frequency f _a is obtained as:

${w w}_{opt opt} (({f f}_{a a} ((q q)))) = = \frac{{R R}^{- - 11} (({f f}_{a a})) v v (({f f}_{a a} ((q q))))}{{v v}^{H h} (({f f}_{a a} ((q q)))) {R R}^{- - 11} (({f f}_{a a})) v v (({f f}_{a a} ((q q))))} - - - - - - ((1717))$

上式中(·)^-1表示矩阵求逆操作。In the above formula (·) ^-1 represents the matrix inversion operation.

(3c)根据自适应最优权矢量w_opt(f_a(q))对观测向量X(t_l,f_a)加权，得到多普勒频率f_a处无模糊的目标回波信号向量S_DBF(t,f_a)为：(3c) Weight the observation vector X(t _l ,f _a ) according to the adaptive optimal weight vector w _opt (f _a (q)), and obtain the unambiguous target echo signal vector S _DBF at the Doppler frequency f _a (t, f _a ) is:

利用步骤(3b)得到的自适应最优权矢量w_opt(f_a(q))对观测向量X(t_l,f_a)加权，得到多普勒频率f_a处的无模糊信号向量S_DBF(t,f_a)：Use the adaptive optimal weight vector w _opt (f _a (q)) obtained in step (3b) to weight the observation vector X(t _l , f _a ), and obtain the unambiguous signal vector S _DBF at the Doppler frequency f _a (t, f _a ):

S_DBF(t,f_a)＝[S₀(t,-M·PRF+f_a),......S₀(t,-M·PRF+f_a)]^T (18)S _DBF (t,f _a )＝[S ₀ (t,-M·PRF+f _a ),...S ₀ (t,-M·PRF+f _a )] ^T (18)

上式中，In the above formula,

$\begin{matrix} {S S}_{00} ((t t,, p p \cdot &Center Dot; PRF PRF + + {f f}_{a a}^{' '})) = = exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γ γ t t}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{p p \cdot \cdot {PRF PRF + + f f}_{a a}^{' '}}{22 v v}))}^{22}))})) \\ \times \times exp exp ((- - j j \frac{22 π π ((p p \cdot &Center Dot; PRF PRF + + {f f}_{a a}))}{v v} {x x}_{p p})) \times \times exp exp ((j j \frac{44 π π}{c c} γ γ (({t t}_{l l} + + \frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {R R}_{ref ref})) \end{matrix} - - - - - - ((1919))$

步骤4，对无模糊的目标回波多普勒信号向量的各个元素按照模糊分量的数值大小进行排列，采用多普勒频谱拼接得到完整的无模糊的目标回波多普勒信号Step 4, arrange each element of the unambiguous target echo Doppler signal vector according to the numerical value of the fuzzy component, and use Doppler spectrum splicing to obtain a complete unambiguous target echo Doppler signal

对无模糊的目标回波多普勒信号向量S_DBF(t,f_a)中各个元素S₀(t,p·PRF+f_a)按照模糊分量q的数值大小进行排列，并做多普勒频谱拼接得到完整的无模糊的目标回波多普勒信号S_UnAmb(t,f_a′)。Arrange each element S ₀ (t,p·PRF+f _a ) in the unambiguous target echo Doppler signal vector S _DBF (t,f _a ) according to the value of the fuzzy component q, and make a Doppler spectrum The complete unambiguous target echo Doppler signal S _UnAmb (t, f _a ′) is obtained by splicing.

参照图3，将无模糊的目标回波多普勒信号按照方位频率顺序排列，拼接成一个完整的无模糊信号S_UnAmb(t,f_a′)为：Referring to Fig. 3, the unambiguous target echo Doppler signals are arranged in order of azimuth frequency, and spliced into a complete unambiguous signal S _UnAmb (t, f _a ′) as follows:

$\begin{matrix} {S S}_{UnAmb UnAmb} (({t t}_{l l},, {f f}_{a a}^{' '})) = = exp exp ((- - j j 44 π π (({R R}_{ref ref} + + {r r}_{p p})) \sqrt{(({((\frac{{f f}_{c c} + + γ γ {t t}_{l l}}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{{f f}_{a a}^{' '}}{22 v v}))}^{22}))})) \\ \times \times exp exp ((- - j j \frac{22 π π {f f}_{a a}^{' '}}{v v} {x x}_{p p})) \times \times exp exp ((j j \frac{44 π π}{c c} γ γ (({t t}_{l l} + + \frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {R R}_{ref ref})) \end{matrix} - - - - - - ((2020))$

其中， ${f_{a}}^{'} &Element; [- \frac{(2 M + 1) \cdot PRF}{2}, \frac{(2 M + 1) \cdot PRF}{2}] .$ in, ${f_{a}}^{'} &Element; [- \frac{(2 m + 1) \cdot PRF}{2}, \frac{(2 m + 1) &Center Dot; PRF}{2}] .$

步骤5，对完整的无模糊的目标回波多普勒信号做成像处理，得到合成孔径激光雷达SAL图像。Step 5, performing imaging processing on the complete and unambiguous target echo Doppler signal to obtain a synthetic aperture lidar SAL image.

其具体子步骤为：Its specific sub-steps are:

(5a)引入方位向匹配函数：(5a) Introduce the azimuth matching function:

${H h}_{11} = = exp exp ((j j 44 π π {R R}_{ref ref} \sqrt{(({((\frac{{f f}_{c c} + + γt γt}{c c} - - \frac{22 r r {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{{f f}_{a a}^{' '}}{22 v v}))}^{22}))})) - - - - - - ((21 twenty one))$

${H h}_{22} = = exp exp ((- - j j \frac{44 π π}{c c} γ γ ((t t + + \frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {R R}_{ref ref})) - - - - - - ((22 twenty two))$

方位向匹配后的目标回波多普勒信号S_match(t_l,f_a′)为：The target echo Doppler signal S _match (t _l , f _a ′) after azimuth matching is:

${S S}_{match match} (({t t}_{l l},, {f f}_{a a}^{' '})) = = = = exp exp ((- - j j 44 π π {r r}_{p p} \sqrt{(({((\frac{{f f}_{c c} + + γt γt}{c c} - - \frac{22 r r {R R}_{ref ref}}{{c c}^{22}}))}^{22} - - {((\frac{{f f}_{a a}^{' '}}{22 v v}))}^{22}))})) \times \times exp exp ((- - j j \frac{22 π π {f f}_{a a}^{' '}}{v v} {x x}_{p p})) - - - - - - ((23 twenty three))$

(5b)令A(f_a)在f_a′＝0处的泰勒级数展开近似为：(5b) order The Taylor series expansion of A(f _a ) at f _a ′=0 is approximated as:

$A A (({f f}_{a a}^{' '})) = = A A ((00)) + + {A A}^{' '} ((00)) {f f}_{a a}^{' '} = = \frac{{f f}_{c c} + + γt γt}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}} - - - - - - ((24 twenty four))$

因此式(23)可以近似为：So formula (23) can be approximated as:

${S S}_{match match}^{' '} (({t t}_{l l},, {f f}_{a a}^{' '})) \approx \approx exp exp ((- - j j 44 π π {r r}_{p p} ((\frac{{f f}_{c c} + + γt γt}{c c} - - \frac{22 γ γ {R R}_{ref ref}}{{c c}^{22}})))) \times \times exp exp ((- - j j \frac{22 π π {f f}_{a a}^{' '}}{v v} {x x}_{p p})) - - - - - - ((2525))$

(5c)将方位向匹配后的近似目标回波多普勒信号S_match′(t_l,f_a′)作方位向逆傅里叶变换和距离向傅立叶变换，得到SAL图像S_SAL(f_r,τ)为：(5c) Perform azimuth inverse Fourier transform and range Fourier transform on the approximate target echo Doppler signal S _match ′( _t _l ,f _a ′) after azimuth matching, and obtain the SAL image S _SAL (fr , τ) is:

${S S}_{SAL SAL} (({f f}_{r r},, τ τ)) = = A A sin sin c c {{π π {T T}_{p p} (({f f}_{r r} + + \frac{22 γ γ {r r}_{p p}}{c c}))}} sin sin c c {{{B B}_{a a}^{' '} ((τ τ - - \frac{{x x}_{p p}}{v v}))}} \times \times exp exp ((j j \frac{44 π π}{c c} γ γ ((\frac{{f f}_{c c}}{γ γ} - - \frac{22 {R R}_{ref ref}}{c c})) {r r}_{p p})) - - - - - - ((2626))$

上式中，A为包络，f_r为距离向频率，f_s为采样频率，T_p为脉宽，B_a为多普勒带宽。In the above formula, A is the envelope, f _r is the range frequency, f _s is the sampling frequency, T _p is the pulse width, and _Ba is the Doppler bandwidth.

至此，本发明的多发多收合成孔径激光雷达成像完成。So far, the multi-input and multi-intake synthetic aperture lidar imaging of the present invention is completed.

以下通过仿真进一步说明本发明实现合成孔径激光雷达成像的有效性。The effectiveness of the present invention in realizing synthetic aperture lidar imaging is further illustrated through simulation below.

1、仿真条件1. Simulation conditions

为了方便起见，我们采用了三发三收的模式，仿真参数如表1所示。For convenience, we adopt the mode of three transmissions and three receptions, and the simulation parameters are shown in Table 1.

表1三发三收SAL系统仿真参数Table 1 Simulation parameters of three transmitters and three receivers SAL system

如果要实现方位向1.5mm分辨率的SAL图像,要求最小的脉冲重复频率为：If a SAL image with a resolution of 1.5mm in azimuth is to be realized, the minimum pulse repetition frequency is required to be:

$\frac{v v}{{ρ ρ}_{a a}} = = 23.2 23.2 kHz kHz$

而实际发射信号的PRF为10kHz，我们采用将三个收发单元的数据相干合成一个全分辨率SAL图像。The PRF of the actual transmitted signal is 10kHz, and we coherently synthesize the data of the three transceiver units into a full-resolution SAL image.

2、仿真内容2. Simulation content

采用三个收发阵元进行三发三收，三个阵元的位置为：(-a,0)，(0,0)，(a,0)，a为收发阵元间隔，对地面上的9个点目标仿真。图4为单发单收SAL体制的9个点目标成像的等高线图，横坐标为方位单元，纵坐标为距离单元。图5为单发单收SAL体制的9个点目标成像的方位脉压剖面图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)。图6为多发多收SAL体制的9个点目标成像的等高线图，横坐标为方位单元，纵坐标为距离单元。图7为方位向SAL体制的9个点目标成像的方位脉压剖面图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)。图8为图6中中间位置的点目标成像放大图，横坐标为方位单元，纵坐标为距离单元。图9为图6中中间位置的点目标方位脉压剖面图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)。图10为图6中中间位置的点目标距离脉压剖面图，横坐标为方位向距离，单位为米(m)，纵坐标为归一化幅度，单位为分贝(dB)。Three transmit and receive array elements are used for three transmit and three receive. The positions of the three array elements are: (-a,0), (0,0), (a,0), and a is the interval between transmit and receive array elements. 9 point target simulations. Figure 4 is the contour map of 9-point target imaging in the single-send-single-reception SAL system, the abscissa is the azimuth unit, and the ordinate is the distance unit. Figure 5 is the azimuth pulse pressure profile of 9-point target imaging in the single-shot, single-receive SAL system. The abscissa is the azimuth distance in meters (m), and the ordinate is the normalized amplitude in decibels (dB). . Figure 6 is the contour map of 9-point target imaging in the multi-transmit and multi-receive SAL system, the abscissa is the azimuth unit, and the ordinate is the distance unit. Figure 7 is the azimuth pulse pressure profile of 9-point target imaging in the azimuth SAL system. The abscissa is the azimuth distance in meters (m), and the ordinate is the normalized amplitude in decibels (dB). Fig. 8 is an enlarged image of a point target in the middle position in Fig. 6, the abscissa is the azimuth unit, and the ordinate is the distance unit. Fig. 9 is a point target azimuth pulse pressure profile at the middle position in Fig. 6, the abscissa is the azimuth distance, the unit is meter (m), and the ordinate is the normalized amplitude, the unit is decibel (dB). Fig. 10 is a point target distance pulse pressure profile at the middle position in Fig. 6, the abscissa is the azimuth distance, the unit is meter (m), and the ordinate is the normalized amplitude, the unit is decibel (dB).

3、仿真结果分析3. Simulation result analysis

单发单收SAL体制的脉冲重复频率PRF约为多普勒带宽的三分之一，方位向多普勒重叠三次，从图4和图5可以看出多普勒模糊在二维成像图中表现为将一个目标点分散成三个点，除了在正确的方位位置上出现目标，还会在这个方位位置的左右两侧各出现一个虚假目标。方位向多发多收SAL体制利用方位向上的3个收发单元的回波数据合成大带宽无模糊数据成像，可以消除多普勒模糊，通过图6和图7可以清楚的看出频带合成后的图像，只在正确的方位位置出现目标点，虚假目标已经被消除。The pulse repetition frequency PRF of the single-shot and single-receive SAL system is about one-third of the Doppler bandwidth, and the azimuth Doppler overlaps three times. From Figure 4 and Figure 5, it can be seen that the Doppler ambiguity in the two-dimensional imaging map The performance is that one target point is dispersed into three points. In addition to the target appearing at the correct azimuth position, a false target will appear on the left and right sides of this azimuth position. The azimuth multi-transmission and multi-reception SAL system uses the echo data of the three transceiver units in the azimuth direction to synthesize large-bandwidth and unambiguous data imaging, which can eliminate Doppler ambiguity. The image after frequency band synthesis can be clearly seen from Figure 6 and Figure 7 , the target point appears only at the correct azimuth position, and the false target has been eliminated.

选取9点中的中间点目标分析单个目标图像方位分辨率和压缩效果。根据图8可以看出单个目标成像得到较好的十字成像。三发三收SAL体制利用方位向的三个收发单元将方位向波束宽度扩大了三倍，通过频带合成获得大带宽无模糊数据，方位向分辨率提高约三倍。通过图9和图10可以看到方位脉压和距离脉压的峰值旁瓣比分别为-13.24dB和-13.27dB，距离压缩和方位压缩效果良好。单发单收SAL系统的方位向分辨率为经过三发三收SAL体制解模糊和频带合成后得到的方位向分辨率约为ρ_am＝0.0012m，比单发单收SAL系统提高了约3倍。Select the middle point target among the 9 points to analyze the azimuth resolution and compression effect of a single target image. According to Fig. 8, it can be seen that a single target imaging can obtain better cross imaging. The three-transmit and three-receive SAL system uses three transceiver units in azimuth to triple the beamwidth in azimuth, obtain large bandwidth and unambiguous data through frequency band synthesis, and increase the resolution in azimuth by about three times. It can be seen from Figure 9 and Figure 10 that the peak side lobe ratios of the azimuth pulse pressure and distance pulse pressure are -13.24dB and -13.27dB respectively, and the distance compression and azimuth compression effects are good. The azimuth resolution of the single-send-single-receive SAL system is The azimuth resolution obtained after the three-transmission and three-reception SAL system is defuzzified and the frequency band is synthesized is about ρ _am =0.0012m, which is about 3 times higher than that of the single-transmission and single-reception SAL system.

以上所述，仅为本发明的具体实施方式，但本发明的保护范围并不局限于此，任何熟悉本技术领域的技术人员在本发明揭露的技术范围内，可轻易想到变化或替换，都应涵盖在本发明的保护范围之内。因此，本发明的保护范围应以所述权利要求的保护范围为准。The above is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Anyone skilled in the art can easily think of changes or substitutions within the technical scope disclosed in the present invention. Should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the protection scope of the claims.

Claims

1. A method for processing multiple-transmit multiple-receive synthetic aperture laser radar signals, the multiple-transmit multiple-receive synthetic aperture laser radar having N array elements, the method comprising:

step 1, N array elements respectively receive target echo signals, and each array element performs residual video phase compensation processing and form simplification processing on the target echo signals to obtain the target echo signals after the form simplification processing; wherein, the array element is a multiple-sending multiple-receiving array element;

step 2, each array element performs azimuth Fourier transform on the target echo signal subjected to the form simplification processing to obtain a target echo Doppler signal subjected to aliasing;

step 3, forming an observation vector by the target echo Doppler signals of N array elements after mixing and overlapping, and performing ambiguity resolution processing on the observation vector to obtain an unambiguous target echo Doppler signal vector;

step 4, arranging all elements of the target echo Doppler signal vector without ambiguity according to the numerical value of the ambiguity component, and splicing by adopting Doppler frequency spectrum to obtain a complete target echo Doppler signal without ambiguity;

and 5, imaging the complete target echo Doppler signal without blurring to obtain a synthetic aperture laser radar image.

2. The multiple-shot multiple-receiver synthetic aperture lidar signal processing method of claim 1, wherein step 1 comprises the following sub-steps:

(1a) determining a target echo signal S received by the ith array element_i(t，τ)，

Where i is 1, 2., N is the number of array elements, t is the fast time, τ is the slow time, R is the number of array elements_refAs a central action distance, G_i，kRepresents the gain of the target echo transmitted by the kth array element and received by the ith array element, R_ipIndicating the distance from the ith array element to the target, R_kpThe distance between the kth array element and a target is shown, and gamma is the modulation frequency;

(1b) for the target echo signal S received by the ith array element_i(t, tau) performing residual video phase compensation processing to obtain a target echo signal S received by the ith array element after residual video phase compensation processing_i′(t，τ)，

Where c is the speed of light, f_cIs the carrier frequency;

(1c) the target echo signal S received by the ith array element after the phase compensation processing of the residual video_i' (t, tau) performing simplified form processing to obtain a target echo signal S received by the ith array element after the simplified form processing_i″(t，τ)，

Wherein,

v is the azimuth velocity of the platform operation, a is the interval between two adjacent array elements, x_pIs the range position of the target, r_pA center offset distance;

(1d) sequentially taking 1,2, 1, N, repeating the steps (1b) and (1c) to obtain target echo signals S of N array elements with simplified forms₁″(t，τ)，...，S_N″(t，τ)。

3. The multiple-shot multiple-receive synthetic aperture lidar signal processing method of claim 2, wherein in sub-step (1a), the distance R from the ith array element to the target_ipThe method specifically comprises the following steps:

wherein i is more than or equal to 1, N is the number of the receiving and transmitting array elements, and v is the azimuth speed of the platform operation.

4. The multiple-shot multiple-receive synthetic aperture lidar signal processing method of claim 2, wherein in sub-step (1a), the kth array element is a distance R to the target_kpThe method specifically comprises the following steps:

and k is less than or equal to N, N is the number of the transmitting and receiving array elements, and v is the azimuth speed of the platform in operation.

5. The multiple-shot multiple-receiver synthetic aperture lidar signal processing method of claim 1, wherein step 2 comprises the following sub-steps:

(2a) the target echo signal S of the ith array element after form simplification processing_iPerforming azimuth Fourier transform to obtain target echo Doppler signal S of ith array element_i(t，f_a)，

Wherein, i is 1,2_aIs the frequency of the doppler frequency and is,PRF is pulse repetition frequency;

(2b) the target echo Doppler signal S of the ith array element_i(t，f_a) Expressed as:

S_i(t，f_a)＝A_i(f_a)S₀(t，f_a)

wherein,

(2c) for the target echo Doppler signal S of the ith array element_i(t，f_a) Generating aliasing to obtain the target echo Doppler signal S after the aliasing of the ith array element_i′(t，f_a)，

Wherein, the value range of the fuzzy component q is q ═ M, -M +1,.., M-1, M;

(2d) sequentially taking 1,2, 1, N to obtain a target echo Doppler signal S after N array elements are mixed and overlapped₁′(t，f_a)，...，S_N′(t，f_a)。

6. The multiple-shot multiple-receiver synthetic aperture lidar signal processing method of claim 1, wherein step 3 comprises the following sub-steps:

(3a) the target echo Doppler signal S after the aliasing of the N array elements₁′(t，f_a)，...，S_N′(t，f_a) Constructing an observation vector X (t)_l，f_a)，

Wherein, t_lIs a certain time in the fast time t, L ═ 1, 2., L is the length of the fast time discrete sequence, (·)^TRepresenting a matrix transpose operation;

(3b) determining the Doppler frequency f of the target echo signal according to the Capon beam forming principle_aAn adaptive optimal weight vector w of_opt(f_a(q))，

w_{opt} (f_{a} (q)) = \frac{R^{- 1} (f_{a}) v (f_{a} (q))}{v^{H} (f_{a} (q)) R^{- 1} (f_{a}) v (f_{a} (q))}

Wherein R (f)_a) Is the Doppler frequency f_aA statistical covariance matrix of the target echo signals of (f), v (f)_a(q)) is the Doppler frequency f_aThe guide vector of the q-th blur component of (1)^-1Representation matrix inversion operation, (-)^HRepresenting a matrix conjugate transpose operation;

(3c) according to the self-adaptive optimal weight vector w_opt(f_a(q)) to the observation vector X (t)_l，f_a) Weighting to obtain Doppler frequency f_aTarget echo Doppler signal vector S without ambiguity_DBF(t，f_a)。

7. The method of claim 6, wherein the Doppler frequency f is_aTarget echo Doppler signal vector S without ambiguity_DBF(t，f_a) Comprises the following steps:

S_DBF(t，f_a)＝[S₀(t，-M·PRF+f_a)，......S₀(t，-M·PRF+f_a)]^T

wherein,