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CN110618403B - Landing aircraft parameter measuring method based on dual-beam radar - Google Patents

Landing aircraft parameter measuring method based on dual-beam radar Download PDF

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CN110618403B
CN110618403B CN201910791681.0A CN201910791681A CN110618403B CN 110618403 B CN110618403 B CN 110618403B CN 201910791681 A CN201910791681 A CN 201910791681A CN 110618403 B CN110618403 B CN 110618403B
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doppler
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CN110618403A (en
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解虎
党红杏
李琪
贺亚鹏
张爱军
李栓劳
高飞
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Xian Institute of Space Radio Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/50—Systems of measurement based on relative movement of target
    • G01S13/58—Velocity or trajectory determination systems; Sense-of-movement determination systems
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/50—Systems of measurement based on relative movement of target
    • G01S13/58—Velocity or trajectory determination systems; Sense-of-movement determination systems
    • G01S13/581—Velocity or trajectory determination systems; Sense-of-movement determination systems using transmission of interrupted pulse modulated waves and based upon the Doppler effect resulting from movement of targets
    • G01S13/582—Velocity or trajectory determination systems; Sense-of-movement determination systems using transmission of interrupted pulse modulated waves and based upon the Doppler effect resulting from movement of targets adapted for simultaneous range and velocity measurements
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/91—Radar or analogous systems specially adapted for specific applications for traffic control
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/28—Details of pulse systems
    • G01S7/285—Receivers
    • G01S7/292—Extracting wanted echo-signals
    • G01S7/2923—Extracting wanted echo-signals based on data belonging to a number of consecutive radar periods
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/415—Identification of targets based on measurements of movement associated with the target

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Abstract

一种基于双波束雷达的着陆飞行器参数测量方法，步骤如下：(1)对装在飞行器上的两个波束相互垂直、且指向地面的雷达回波数据进行处理，构造接收数据矩阵X1和X2；(2)对接收数据矩阵X1和X2进行脉冲压缩、FFT和检测处理，获取两个雷达回波数据的距离‑多普勒信息；(3)将两个雷达回波数据的距离‑多普勒信息进行变换，构造对应的数据矩阵；(4)根据步骤(3)构造的数据矩阵反算出着陆飞行器的三维速度、高度；(5)重复上述步骤，对下一组相关处理时间内的雷达回波数据进行处理，得到飞行器的实时飞行参数。本发明显著降低了现有着陆雷达采用四波束进行着陆器飞行参数测量的系统复杂度，仅采用双波束即可实现飞行器飞行参数的测量，飞行器参数测量精度高。

Figure 201910791681

A method for measuring parameters of a landing aircraft based on a dual-beam radar, the steps are as follows: (1) Process the radar echo data of two beams mounted on the aircraft that are perpendicular to each other and point to the ground, and construct received data matrices X 1 and X 2 ; (2) Pulse compression, FFT and detection processing are performed on the received data matrices X 1 and X 2 to obtain the distance-Doppler information of the two radar echo data; (3) The distance between the two radar echo data is obtained; ‑Doppler information is transformed to construct a corresponding data matrix; (4) inversely calculate the three-dimensional speed and altitude of the landing aircraft according to the data matrix constructed in step (3); The radar echo data inside is processed to obtain the real-time flight parameters of the aircraft. The invention significantly reduces the system complexity of the existing landing radar using four beams to measure the flight parameters of the lander, and can measure the flight parameters of the aircraft only by using two beams, and the measurement accuracy of the parameters of the aircraft is high.

Figure 201910791681

Description

一种基于双波束雷达的着陆飞行器参数测量方法A method for measuring parameters of landing aircraft based on dual-beam radar

技术领域technical field

本发明涉及一种基于双波束雷达的着陆飞行器参数测量方法，可直接应用于多种深空探测着陆平台的飞行参数估计，属于飞行器参数测量领域。The invention relates to a method for measuring parameters of a landing aircraft based on a dual-beam radar, which can be directly applied to the estimation of flight parameters of various deep space exploration landing platforms, and belongs to the field of aircraft parameter measurement.

背景技术Background technique

现有的着陆雷达飞行参数测量主要靠单波束测速、测距，然后根据多个不同指向波束的相对几何关系，利用各自的测量结果联合估计着陆器的飞行参数数据。该体制的飞行参数测量雷达精度取决两方面要素：波束中心估计精度、波束中心多普勒速度估计精度。波束中心估计精度目前主要采用重心法，在大入射角下，波束中心估计精度明显下降，此外多普勒扩展也会影响系统的波束中心估计精度，对波束中心处的多普勒估计精度影响也较大。The existing landing radar flight parameter measurement mainly relies on single-beam speed measurement and ranging, and then uses the respective measurement results to jointly estimate the lander's flight parameter data according to the relative geometric relationship of multiple different pointing beams. The flight parameter measurement radar accuracy of this system depends on two factors: the beam center estimation accuracy and the beam center Doppler velocity estimation accuracy. At present, the estimation accuracy of the beam center mainly adopts the barycentric method. Under the large incident angle, the estimation accuracy of the beam center decreases significantly. In addition, the Doppler expansion will also affect the beam center estimation accuracy of the system, and the Doppler estimation accuracy at the beam center is also affected. larger.

发明内容SUMMARY OF THE INVENTION

本发明的发明内容为：克服现有技术的不足，提供一种基于双波束雷达的着陆飞行器参数测量方法，仅采用两个波束即可估计飞行器飞行参数，估计精度高。The content of the present invention is to overcome the deficiencies of the prior art and provide a method for measuring parameters of a landing aircraft based on a dual-beam radar, which can estimate the flight parameters of the aircraft using only two beams, and has high estimation accuracy.

本发明的技术解决方案是：The technical solution of the present invention is:

一种基于双波束雷达的着陆飞行器参数测量方法，包括如下步骤：A method for measuring parameters of a landing aircraft based on a dual-beam radar, comprising the following steps:

(1)对装在飞行器上的两个波束相互垂直、且指向地面的雷达回波数据进行处理，构造两个雷达的接收数据矩阵X1和X2；(1) Process the radar echo data of the two beams installed on the aircraft that are perpendicular to each other and point to the ground, and construct the received data matrices X 1 and X 2 of the two radars;

(2)对接收数据矩阵X1和X2进行脉冲压缩、FFT和检测处理，获取两个雷达回波数据的距离-多普勒信息；(2) Perform pulse compression, FFT and detection processing on the received data matrices X 1 and X 2 to obtain the range-Doppler information of the two radar echo data;

(3)将两个雷达回波数据的距离-多普勒信息进行变换，构造对应的数据矩阵；(3) Transform the range-Doppler information of the two radar echo data to construct a corresponding data matrix;

(4)根据步骤(3)构造的数据矩阵反算出着陆飞行器的三维速度、高度；(4) inversely calculate the three-dimensional speed and altitude of the landing aircraft according to the data matrix constructed in step (3);

(5)重复步骤(1)-(4)，对下一组相关处理时间内的雷达回波数据进行处理，得到飞行器的实时飞行参数。(5) Repeat steps (1)-(4) to process the radar echo data within the next set of relevant processing time to obtain real-time flight parameters of the aircraft.

所述步骤(1)中，

Figure BDA0002179721840000021
Figure BDA0002179721840000022
为L行M列的复矩阵，接收数据矩阵X1中的第l行第m列元素x1(l,m)为第一个雷达的第m个脉冲的第l个基带信号采样数据，接收数据矩阵X2中的第l行第m列元素x2(l,m)为第二个雷达的第m个脉冲的第l个基带信号采样数据。In the step (1),
Figure BDA0002179721840000021
Figure BDA0002179721840000022
is a complex matrix with L rows and M columns, and the element x 1 (l,m) of the l-th row and m-th column in the received data matrix X 1 is the l-th baseband signal sampling data of the m-th pulse of the first radar, receiving The element x 2 (l,m) of the l-th row and the m-th column in the data matrix X 2 is the l-th baseband signal sample data of the m-th pulse of the second radar.

所述步骤(2)中，对接收数据矩阵X1和X2按列进行脉冲压缩，获得距离向高分辨回波数据

Figure BDA0002179721840000023
和
Figure BDA0002179721840000024
获得过程如下：In the step (2), pulse compression is performed on the received data matrices X 1 and X 2 by columns to obtain high-resolution echo data in the range direction.
Figure BDA0002179721840000023
and
Figure BDA0002179721840000024
The obtaining process is as follows:

Figure BDA0002179721840000025
Figure BDA0002179721840000025

Figure BDA0002179721840000026
Figure BDA0002179721840000026

其中X1(:,m)表示接收数据矩阵X1的第m列，X2(:,m)表示接收数据矩阵X2的第m列，

Figure BDA0002179721840000027
表示矩阵
Figure BDA0002179721840000028
的第m列，
Figure BDA0002179721840000029
表示矩阵
Figure BDA00021797218400000210
的第m列，符号
Figure BDA00021797218400000211
表示卷积操作，
Figure BDA00021797218400000212
为雷达发射信号s(t)对应的数字匹配信号，
Figure BDA00021797218400000213
为1行P列的复矩阵，
Figure BDA00021797218400000214
where X 1 (:,m) represents the m-th column of the received data matrix X 1 , X 2 (:, m) represents the m-th column of the received data matrix X 2 ,
Figure BDA0002179721840000027
representation matrix
Figure BDA0002179721840000028
The mth column of ,
Figure BDA0002179721840000029
representation matrix
Figure BDA00021797218400000210
the mth column of , notation
Figure BDA00021797218400000211
represents the convolution operation,
Figure BDA00021797218400000212
is the digital matching signal corresponding to the radar transmit signal s(t),
Figure BDA00021797218400000213
is a complex matrix with 1 row and P columns,
Figure BDA00021797218400000214

所述步骤(2)中，对脉冲压缩后的

Figure BDA00021797218400000215
和
Figure BDA00021797218400000216
按行进行FFT处理，获取两个雷达的回波数据距离多普勒谱
Figure BDA00021797218400000217
和
Figure BDA00021797218400000218
获取过程如下：In the step (2), the pulse-compressed
Figure BDA00021797218400000215
and
Figure BDA00021797218400000216
Perform FFT processing by row to obtain the range Doppler spectrum of the echo data of the two radars
Figure BDA00021797218400000217
and
Figure BDA00021797218400000218
The acquisition process is as follows:

Figure BDA00021797218400000219
Figure BDA00021797218400000219

Figure BDA00021797218400000220
Figure BDA00021797218400000220

其中，

Figure BDA00021797218400000221
in,
Figure BDA00021797218400000221

所述步骤(2)中，对两个雷达的回波数据距离多普勒谱

Figure BDA00021797218400000222
和
Figure BDA00021797218400000223
进行检测，得到两个雷达回波数据的距离-多普勒信息，过程如下：In the step (2), the range Doppler spectrum of the echo data of the two radars is
Figure BDA00021797218400000222
and
Figure BDA00021797218400000223
Perform detection to obtain the distance-Doppler information of the two radar echo data. The process is as follows:

令系统检测门限为ρTH，对矩阵

Figure BDA00021797218400000224
和
Figure BDA00021797218400000225
进行检测，将其中取值大于ρTH的元素对应的回波时间和多普勒值进行记录，记作Let the system detection threshold be ρ TH , for the matrix
Figure BDA00021797218400000224
and
Figure BDA00021797218400000225
Perform detection, record the echo time and Doppler value corresponding to the element whose value is greater than ρ TH , and record it as

Figure BDA00021797218400000226
Figure BDA00021797218400000226

其中t1为矩阵

Figure BDA0002179721840000031
中取值大于ρTH的元素对应的回波时间向量，fd,1为矩阵
Figure BDA0002179721840000032
中取值大于ρTH的元素对应的多普勒值向量，t2为矩阵
Figure BDA0002179721840000033
中取值大于ρTH的元素对应的回波时间向量，fd,2为矩阵
Figure BDA0002179721840000034
中取值大于ρTH的元素对应的多普勒值向量；J,K分别为矩阵
Figure BDA0002179721840000035
中取值大于ρTH的元素个数和矩阵
Figure BDA0002179721840000036
中取值大于ρTH的元素个数；
Figure BDA0002179721840000037
为J行1列的实矩阵，
Figure BDA0002179721840000038
为K行1列的实矩阵；where t 1 is the matrix
Figure BDA0002179721840000031
The echo time vector corresponding to the element whose value is greater than ρ TH , f d,1 is a matrix
Figure BDA0002179721840000032
The Doppler value vector corresponding to the element whose value is greater than ρ TH , t 2 is a matrix
Figure BDA0002179721840000033
The echo time vector corresponding to the element whose value is greater than ρ TH , f d,2 is a matrix
Figure BDA0002179721840000034
The Doppler value vector corresponding to the element whose value is greater than ρ TH ; J and K are matrices respectively
Figure BDA0002179721840000035
The number of elements and the matrix whose value is greater than ρ TH
Figure BDA0002179721840000036
The number of elements whose value is greater than ρ TH ;
Figure BDA0002179721840000037
is a real matrix with J rows and 1 column,
Figure BDA0002179721840000038
is a real matrix with K rows and 1 column;

检测得到两个雷达回波数据的距离-多普勒信息如下：The distance-Doppler information of the two radar echo data is detected as follows:

Figure BDA0002179721840000039
Figure BDA0002179721840000039

Figure BDA00021797218400000310
Figure BDA00021797218400000310

Figure BDA00021797218400000311
Figure BDA00021797218400000311

Figure BDA00021797218400000312
Figure BDA00021797218400000312

其中

Figure BDA00021797218400000313
t1j为t1中的第j个元素，t2k为t2中的第k个元素，j＝1,2,…,J，k＝1,2,…,K，c为光速。in
Figure BDA00021797218400000313
t 1j is the jth element in t 1 , t 2k is the kth element in t 2 , j=1,2,...,J, k=1,2,...,K, c is the speed of light.

所述步骤(3)的实现方式如下：The implementation of the step (3) is as follows:

根据两个雷达回波数据的距离-多普勒信息，构造如下数据矩阵：According to the range-Doppler information of the two radar echo data, the following data matrix is constructed:

a＝[a1 a2 … aJ]a=[a 1 a 2 ... a J ]

b＝[b1 b2 … bJ]b=[b 1 b 2 … b J ]

c＝[c1 c2 … cJ]c=[c 1 c 2 ... c J ]

d＝[d1 d2 … dK]d=[d 1 d 2 ... d K ]

e＝[e1 e2 … eK]e=[e 1 e 2 ... e K ]

f＝[f1 f2 … fK]f=[f 1 f 2 ... f K ]

其中in

Figure BDA0002179721840000041
Figure BDA0002179721840000041

Figure BDA0002179721840000042
Figure BDA0002179721840000042

Figure BDA0002179721840000043
Figure BDA0002179721840000043

Figure BDA0002179721840000044
Figure BDA0002179721840000044

Figure BDA0002179721840000045
Figure BDA0002179721840000045

Figure BDA0002179721840000046
Figure BDA0002179721840000046

设中间变量set intermediate variable

Figure BDA0002179721840000047
Figure BDA0002179721840000047

x2＝vzHx 2 =v z H

Figure BDA0002179721840000048
Figure BDA0002179721840000048

Figure BDA0002179721840000049
Figure BDA0002179721840000049

Figure BDA00021797218400000410
Figure BDA00021797218400000410

其中，vx、vy、vz分别为着陆飞行器的x方向速度、y方向速度、z方向速度，H为着陆飞行器的高度；Among them, v x , v y , v z are the x-direction speed, y-direction speed, and z-direction speed of the landing aircraft, respectively, and H is the height of the landing aircraft;

根据两个雷达波束Doppler以及距离-多普勒信息，构造如下矩阵：According to the two radar beam Doppler and range-Doppler information, the following matrix is constructed:

x＝[x1 x2 x3 x4 x5]T x=[x 1 x 2 x 3 x 4 x 5 ] T

Γ1 Tx+a＝0Γ 1 T x+a=0

Γ2 Tx+d＝0Γ 2 T x+d=0

其中in

Figure BDA00021797218400000411
Figure BDA00021797218400000411

Figure BDA00021797218400000412
Figure BDA00021797218400000412

Figure BDA00021797218400000413
Figure BDA00021797218400000413

Figure BDA00021797218400000414
Figure BDA00021797218400000414

所述步骤(4)的实现方式如下：The implementation of the step (4) is as follows:

(S1)根据步骤(3)构造的数据矩阵，解算中间变量x：(S1) According to the data matrix constructed in step (3), solve the intermediate variable x:

步骤(3)构造的数据矩阵可以写作The data matrix constructed in step (3) can be written as

ΓTx+g＝0Γ T x+g=0

其中in

Figure BDA0002179721840000051
Figure BDA0002179721840000051

Figure BDA0002179721840000052
Figure BDA0002179721840000052

求解可得can be solved

Figure BDA00021797218400000515
Figure BDA00021797218400000515

其中

Figure BDA0002179721840000053
表示矩阵的伪逆；in
Figure BDA0002179721840000053
represents the pseudo-inverse of a matrix;

(S2)根据x，反算得到

Figure BDA0002179721840000054
和
Figure BDA0002179721840000055
(S2) According to x, the inverse calculation is obtained
Figure BDA0002179721840000054
and
Figure BDA0002179721840000055

(S3)利用两个雷达波束之间的几何耦合关系求解

Figure BDA0002179721840000056
(S3) Use the geometric coupling relationship between the two radar beams to solve
Figure BDA0002179721840000056

所述步骤(S2)实现方式如下：The step (S2) is implemented as follows:

Figure BDA0002179721840000057
Figure BDA0002179721840000057

Figure BDA0002179721840000058
Figure BDA0002179721840000058

Figure BDA0002179721840000059
Figure BDA0002179721840000059

Figure BDA00021797218400000510
表示飞行器x方向的估计速度，
Figure BDA00021797218400000511
表示飞行器y方向的估计速度，
Figure BDA00021797218400000512
表示飞行器的估计高度。
Figure BDA00021797218400000510
represents the estimated speed of the aircraft in the x direction,
Figure BDA00021797218400000511
represents the estimated speed of the aircraft in the y direction,
Figure BDA00021797218400000512
Indicates the estimated altitude of the aircraft.

所述步骤(S3)实现方式如下：The step (S3) is implemented as follows:

Figure BDA00021797218400000513
Figure BDA00021797218400000513

其中mean(·)表示对向量取平均值，α,β为where mean( ) means taking the average value of the vector, α, β are

α＝[α1 α2 … αJ]α=[α 1 α 2 ... α J ]

β＝[β1 β2 … βK]β=[β 1 β 2 … β K ]

其中in

Figure BDA00021797218400000514
Figure BDA00021797218400000514

Figure BDA0002179721840000061
Figure BDA0002179721840000061

本发明与现有技术相比的优点在于：The advantages of the present invention compared with the prior art are:

(1)本发明仅采用两个波束即可估计平台飞行参数，且采用双波束联合处理的方式，处理精度高。(1) The present invention can estimate the flight parameters of the platform by only using two beams, and adopts the joint processing method of the dual beams, and the processing accuracy is high.

(2)本发明双波束雷达采用两个长条状平面天线，保证天线在方位向波束较窄，距离向波束较宽，从而提高单个距离环的多普勒分辨率，增加距离环数目，最终实现平台参数的准确估计，安装简单，减轻了重量。(2) The dual-beam radar of the present invention adopts two long strip planar antennas to ensure that the antenna has a narrow beam in the azimuth direction and a wide beam in the distance direction, thereby improving the Doppler resolution of a single range loop, increasing the number of range loops, and finally Accurate estimation of platform parameters is achieved, installation is simple, and weight is reduced.

(3)本发明利用平台运动导致地面回波的距离-多普勒依赖特性反算出平台的运动参数，估计精度高。(3) The present invention inversely calculates the motion parameters of the platform by using the distance-Doppler dependence characteristic of the ground echo caused by the motion of the platform, and the estimation accuracy is high.

附图说明Description of drawings

图1为本发明流程图；Fig. 1 is the flow chart of the present invention;

图2为飞行器飞行参数与回波的距离-多普勒耦合关系示意图；Figure 2 is a schematic diagram of the distance-Doppler coupling relationship between the flight parameters of the aircraft and the echo;

图3为不同速度下的等距离环和等多普勒图；Figure 3 is the equidistant loop and equi-Doppler map at different speeds;

图4为不同波束Doppler估计偏差下速度估计误差；Figure 4 shows the velocity estimation error under different beam Doppler estimation deviations;

图5为不同波束Doppler估计偏差下高度估计误差。Figure 5 shows the height estimation errors under different beam Doppler estimation deviations.

具体实施方式Detailed ways

本发明的基本原理如下：1)利用平台运动导致地面回波的距离-多普勒依赖特性反算出平台的运动参数；2)双波束雷达采用两个长条状平面天线，保证天线在方位向波束较窄，距离向波束较宽，从而提高单个距离环的多普勒分辨率，增加距离环数目，最终实现平台参数的准确估计。The basic principle of the present invention is as follows: 1) the motion parameters of the platform are inversely calculated by the distance-Doppler dependence characteristic of the ground echo caused by the motion of the platform; 2) the dual-beam radar adopts two long-strip planar antennas to ensure that the antennas are in the azimuth direction The beam is narrower and the range beam is wider, thereby improving the Doppler resolution of a single range loop, increasing the number of range loops, and finally realizing accurate estimation of platform parameters.

如图1所示，本发明包括如下步骤：As shown in Figure 1, the present invention comprises the following steps:

(1)对装在飞行器上的两个波束相互垂直、且指向地面的雷达回波数据进行处理，构造两个雷达的接收数据矩阵X1和X2。(1) Process the radar echo data of the two beams installed on the aircraft that are perpendicular to each other and point to the ground, and construct the received data matrices X 1 and X 2 of the two radars.

Figure BDA0002179721840000062
Figure BDA0002179721840000063
为L行M列的复矩阵，接收数据矩阵X1中的第l行第m列元素x1(l,m)为第一个雷达的第m个脉冲的第l个基带信号采样数据，接收数据矩阵X2中的第l行第m列元素x2(l,m)为第二个雷达的第m个脉冲的第l个基带信号采样数据。
Figure BDA0002179721840000062
Figure BDA0002179721840000063
is a complex matrix with L rows and M columns, and the element x 1 (l,m) of the l-th row and m-th column in the received data matrix X 1 is the l-th baseband signal sampling data of the m-th pulse of the first radar, receiving The element x 2 (l,m) of the l-th row and the m-th column in the data matrix X 2 is the l-th baseband signal sample data of the m-th pulse of the second radar.

(2)对接收数据矩阵进行脉冲压缩、FFT、检测处理，获取两个雷达回波数据的距离-多普勒信息。(2) Perform pulse compression, FFT, and detection processing on the received data matrix to obtain the range-Doppler information of the two radar echo data.

对接收数据矩阵X1和X2按列进行脉冲压缩，获得距离向高分辨回波数据

Figure BDA0002179721840000071
和
Figure BDA0002179721840000072
获得过程如下：Perform pulse compression on the received data matrices X 1 and X 2 by column to obtain high-resolution echo data in the range direction
Figure BDA0002179721840000071
and
Figure BDA0002179721840000072
The obtaining process is as follows:

Figure BDA0002179721840000073
Figure BDA0002179721840000073

Figure BDA0002179721840000074
Figure BDA0002179721840000074

其中X1(:,m)表示接收数据矩阵X1的第m列，X2(:,m)表示接收数据矩阵X2的第m列，

Figure BDA0002179721840000075
表示矩阵
Figure BDA0002179721840000076
的第m列，
Figure BDA0002179721840000077
表示矩阵
Figure BDA0002179721840000078
的第m列，符号
Figure BDA0002179721840000079
表示卷积操作，
Figure BDA00021797218400000710
为雷达发射信号s(t)对应的数字匹配信号，
Figure BDA00021797218400000711
为1行P列的复矩阵，
Figure BDA00021797218400000712
where X 1 (:,m) represents the m-th column of the received data matrix X 1 , X 2 (:, m) represents the m-th column of the received data matrix X 2 ,
Figure BDA0002179721840000075
representation matrix
Figure BDA0002179721840000076
The mth column of ,
Figure BDA0002179721840000077
representation matrix
Figure BDA0002179721840000078
the mth column of , notation
Figure BDA0002179721840000079
represents the convolution operation,
Figure BDA00021797218400000710
is the digital matching signal corresponding to the radar transmit signal s(t),
Figure BDA00021797218400000711
is a complex matrix with 1 row and P columns,
Figure BDA00021797218400000712

对脉冲压缩后的

Figure BDA00021797218400000728
和
Figure BDA00021797218400000714
按行进行FFT处理，获取两个雷达的回波数据距离多普勒谱
Figure BDA00021797218400000715
和
Figure BDA00021797218400000716
获取过程如下：After pulse compression
Figure BDA00021797218400000728
and
Figure BDA00021797218400000714
Perform FFT processing by row to obtain the range Doppler spectrum of the echo data of the two radars
Figure BDA00021797218400000715
and
Figure BDA00021797218400000716
The acquisition process is as follows:

Figure BDA00021797218400000717
Figure BDA00021797218400000717

Figure BDA00021797218400000718
Figure BDA00021797218400000718

其中，

Figure BDA00021797218400000719
in,
Figure BDA00021797218400000719

对两个雷达的回波数据距离多普勒谱

Figure BDA00021797218400000720
和
Figure BDA00021797218400000721
进行检测，得到两个雷达回波数据的距离-多普勒信息，过程如下：Range Doppler spectrum of echo data for two radars
Figure BDA00021797218400000720
and
Figure BDA00021797218400000721
Perform detection to obtain the distance-Doppler information of the two radar echo data. The process is as follows:

令系统检测门限为ρTH，对矩阵

Figure BDA00021797218400000722
和
Figure BDA00021797218400000723
进行检测，将其中取值大于ρTH的元素对应的回波时间和多普勒值进行记录，记作Let the system detection threshold be ρ TH , for the matrix
Figure BDA00021797218400000722
and
Figure BDA00021797218400000723
Perform detection, record the echo time and Doppler value corresponding to the element whose value is greater than ρ TH , and record it as

Figure BDA00021797218400000724
Figure BDA00021797218400000724

其中t1为矩阵

Figure BDA00021797218400000725
中取值大于ρTH的元素对应的回波时间向量，fd,1为矩阵
Figure BDA00021797218400000726
中取值大于ρTH的元素对应的多普勒值向量，t2为矩阵
Figure BDA00021797218400000727
中取值大于ρTH的元素对应的回波时间向量，fd,2为矩阵
Figure BDA0002179721840000081
中取值大于ρTH的元素对应的多普勒值向量；J,K分别为矩阵
Figure BDA0002179721840000082
中取值大于ρTH的元素个数和矩阵
Figure BDA0002179721840000083
中取值大于ρTH的元素个数；
Figure BDA0002179721840000084
表示J行1列的实矩阵，
Figure BDA0002179721840000085
表示K行1列的实矩阵；where t 1 is the matrix
Figure BDA00021797218400000725
The echo time vector corresponding to the element whose value is greater than ρ TH , f d,1 is a matrix
Figure BDA00021797218400000726
The Doppler value vector corresponding to the element whose value is greater than ρ TH , t 2 is a matrix
Figure BDA00021797218400000727
The echo time vector corresponding to the element whose value is greater than ρ TH , f d,2 is a matrix
Figure BDA0002179721840000081
The Doppler value vector corresponding to the element whose value is greater than ρ TH ; J and K are matrices respectively
Figure BDA0002179721840000082
The number of elements and the matrix whose value is greater than ρ TH
Figure BDA0002179721840000083
The number of elements whose value is greater than ρ TH ;
Figure BDA0002179721840000084
represents a real matrix with J rows and 1 column,
Figure BDA0002179721840000085
Represents a real matrix with K rows and 1 column;

检测得到两个雷达回波数据的距离-多普勒信息如下：The distance-Doppler information of the two radar echo data is detected as follows:

Figure BDA0002179721840000086
Figure BDA0002179721840000086

Figure BDA0002179721840000087
Figure BDA0002179721840000087

Figure BDA0002179721840000088
Figure BDA0002179721840000088

Figure BDA0002179721840000089
Figure BDA0002179721840000089

其中

Figure BDA00021797218400000810
t1j为t1中的第j个元素，t2k为t2中的第k个元素，j＝1,2,…,J，k＝1,2,…,K，c为光速。in
Figure BDA00021797218400000810
t 1j is the jth element in t 1 , t 2k is the kth element in t 2 , j=1,2,...,J, k=1,2,...,K, c is the speed of light.

(3)将两个雷达回波数据的距离-多普勒信息进行变换，构造对应的数据矩阵。(3) Transform the range-Doppler information of the two radar echo data to construct the corresponding data matrix.

根据两个雷达回波数据的距离-多普勒信息，构造如下数据矩阵：According to the range-Doppler information of the two radar echo data, the following data matrix is constructed:

a＝[a1 a2 … aJ]a=[a 1 a 2 ... a J ]

b＝[b1 b2 … bJ]b=[b 1 b 2 … b J ]

c＝[c1 c2 … cJ]c=[c 1 c 2 ... c J ]

d＝[d1 d2 … dK]d=[d 1 d 2 ... d K ]

e＝[e1 e2 … eK]e=[e 1 e 2 ... e K ]

f＝[f1 f2 … fK] (5)f=[f 1 f 2 … f K ] (5)

其中in

Figure BDA00021797218400000811
Figure BDA00021797218400000811

Figure BDA00021797218400000812
Figure BDA00021797218400000812

Figure BDA00021797218400000813
Figure BDA00021797218400000813

Figure BDA00021797218400000814
Figure BDA00021797218400000814

Figure BDA00021797218400000815
Figure BDA00021797218400000815

Figure BDA00021797218400000816
Figure BDA00021797218400000816

以及中间变量and intermediate variables

Figure BDA0002179721840000091
Figure BDA0002179721840000091

其中，vx、vy、vz分别为着陆飞行器的x方向速度、y方向速度、z方向速度，H为着陆飞行器的高度；Among them, v x , v y , v z are the x-direction speed, y-direction speed, and z-direction speed of the landing aircraft, respectively, and H is the height of the landing aircraft;

通过上式构造如下矩阵：The following matrix is constructed by the above formula:

x＝[x1 x2 x3 x4 x5]T (8)x=[x 1 x 2 x 3 x 4 x 5 ] T (8)

假设波束指向平面为XZ平面，由图2可以看出其距离-Doppler耦合关系仅与vx,H,有关，其关系如下：Assuming that the beam pointing plane is the XZ plane, it can be seen from Figure 2 that the distance-Doppler coupling relationship is only related to v x , H, and the relationship is as follows:

Figure BDA0002179721840000092
Figure BDA0002179721840000092

fd,Rs表示距离Rs处的多普勒频率，Rs表示斜距；图3所示为不同飞行速度下底面雷达回波的多普勒分布图，根据上式及图3可以看出，飞行速度高度与地面雷达回波的多普勒分布存在一一对应的关系，利用雷达回波的距离多普勒信息，可以对飞行器飞行参数进行有效估计。f d, Rs represents the Doppler frequency at the distance Rs, and R s represents the slant range; Figure 3 shows the Doppler distribution of the bottom radar echo at different flight speeds. According to the above formula and Figure 3, it can be seen that, There is a one-to-one correspondence between the flight speed and altitude and the Doppler distribution of ground radar echoes. Using the range Doppler information of radar echoes, the flight parameters of the aircraft can be effectively estimated.

显然单波束仅能测量两维速度和平台高度，引入与其垂直的波束可以解决vy的估计。为区分两个波束，记XZ平面波束为1，YZ平面波束为2，其对应的Doppler和距离分别为fd1、fd2和RS1、RS2，则对于波束1第j个距离环的距离

Figure BDA0002179721840000093
和其对应的
Figure BDA0002179721840000094
存在以下耦合关系Obviously, a single beam can only measure two-dimensional velocity and platform height, and the introduction of a beam perpendicular to it can solve the estimation of v y . In order to distinguish the two beams, denote the XZ plane beam as 1 and the YZ plane beam as 2, and the corresponding Doppler and distance are f d1 , f d2 and R S1 , R S2 respectively, then for beam 1, the distance of the jth distance ring
Figure BDA0002179721840000093
and its corresponding
Figure BDA0002179721840000094
There are the following coupling relationships

Figure BDA0002179721840000095
Figure BDA0002179721840000095

同理波束2的第k个距离环的距离

Figure BDA0002179721840000096
和其对应的
Figure BDA0002179721840000097
存在以下耦合关系Similarly, the distance of the kth distance ring of beam 2
Figure BDA0002179721840000096
and its corresponding
Figure BDA0002179721840000097
There are the following coupling relationships

Figure BDA0002179721840000101
Figure BDA0002179721840000101

考虑波束1，对上式进行变化得到：Considering beam 1, the above equation is changed to get:

Figure BDA0002179721840000102
Figure BDA0002179721840000102

两边分别取平方得到Take the square of both sides to get

Figure BDA0002179721840000103
Figure BDA0002179721840000103

同理对于波束2，也有Similarly for beam 2, there is also

Figure BDA0002179721840000104
Figure BDA0002179721840000104

上面两式简化为The above two equations are simplified to

aj+x1+b x2+cjx3＝0a j +x 1 +bx 2 +c j x 3 =0

dk+x4+ekx2+fkx5＝0 (15)d k +x 4 +e k x 2 +f k x 5 =0 (15)

其中in

Figure BDA0002179721840000105
Figure BDA0002179721840000105

Figure BDA0002179721840000106
Figure BDA0002179721840000106

Figure BDA0002179721840000107
Figure BDA0002179721840000107

Figure BDA0002179721840000108
Figure BDA0002179721840000108

Figure BDA0002179721840000109
Figure BDA0002179721840000109

Figure BDA00021797218400001010
Figure BDA00021797218400001010

以及中间变量and intermediate variables

Figure BDA00021797218400001011
Figure BDA00021797218400001011

据此构造如下关系矩阵According to this, the following relationship matrix is constructed

Γ1,j Tx+aj＝0Γ 1,j T x+a j =0

Γ2,k Tx+dk＝0 (18)Γ 2,k T x+d k =0 (18)

其中in

Figure BDA0002179721840000111
Figure BDA0002179721840000111

波束1得到J个距离-Doppler数据，波束2得到K个距离-Doppler数据进行联立x得到:Beam 1 obtains J distance-Doppler data, and beam 2 obtains K distance-Doppler data for simultaneous x to obtain:

Γ1 Tx+a＝0Γ 1 T x+a=0

Γ2 Tx+d＝0 (20)Γ 2 T x+d=0 (20)

其中in

Figure BDA0002179721840000112
Figure BDA0002179721840000112

上式联合可以写作The combination of the above formula can be written

ΓTx+g＝0 (22)Γ T x+g=0 (22)

其中in

Figure BDA0002179721840000113
Figure BDA0002179721840000113

Figure BDA0002179721840000114
Figure BDA0002179721840000114

求解可得can be solved

Figure BDA0002179721840000115
Figure BDA0002179721840000115

其中

Figure BDA0002179721840000116
表示矩阵的伪逆。通过式可以给出系统参量估计的显式解，可以降低系统的计算量。in
Figure BDA0002179721840000116
Represents the pseudo-inverse of a matrix. The explicit solution of the parameter estimation of the system can be given by the formula, which can reduce the calculation amount of the system.

(4)根据构造的数据矩阵计算出中间变量值，然后反算出平台的三维速度、高度。(4) Calculate the intermediate variable value according to the constructed data matrix, and then inversely calculate the three-dimensional speed and height of the platform.

利用上一步求解得到x，反算得到Use the previous step to solve to get x, and inversely calculate to get

Figure BDA0002179721840000121
Figure BDA0002179721840000121

Figure BDA0002179721840000122
Figure BDA0002179721840000122

Figure BDA0002179721840000123
Figure BDA0002179721840000123

但是vz与x3 x3 x3均存在耦合关系，为精确求解vz，需要利用波束1和2之间的几何耦合关系。vx、vy、vz分别为着陆飞行器的x方向速度、y方向速度、z方向速度，H为着陆飞行器的高度。However, there is a coupling relationship between v z and x 3 x 3 x 3. In order to accurately solve v z , the geometric coupling relationship between beams 1 and 2 needs to be used. v x , v y , v z are the speed in the x direction, the speed in the y direction, and the speed in the z direction of the landing aircraft, respectively, and H is the height of the landing aircraft.

将式(25)的估计结果带入(10)和(11)得到仅关于vz的方程组Bring the estimation result of equation (25) into (10) and (11) to get a system of equations only about v z

Figure BDA0002179721840000124
Figure BDA0002179721840000124

Figure BDA0002179721840000125
Figure BDA0002179721840000125

其中in

Figure BDA0002179721840000126
Figure BDA0002179721840000126

Figure BDA0002179721840000127
Figure BDA0002179721840000127

那么

Figure BDA0002179721840000128
的估计值为So
Figure BDA0002179721840000128
is estimated to be

Figure BDA0002179721840000129
Figure BDA0002179721840000129

其中mean(·)表示对向量取平均值，α,β为where mean( ) represents the mean value of the vector, and α and β are

α＝[α1 α2 … αJ]α=[α 1 α 2 … α J ]

β＝[β1 β2 … βK] (30)β=[β 1 β 2 … β K ] (30)

(5)重复上述步骤对对下一组雷达回波数据进行处理，得到飞行器的实时飞行参数。(5) Repeat the above steps to process the next group of radar echo data to obtain the real-time flight parameters of the aircraft.

仿真实验结果：Simulation results:

平台参数：飞行高度H＝200m；X方向速度：vx＝600m/s；Y方向速度vy＝-400m/s；vz＝10m/s。Platform parameters: flight height H=200m; X-direction speed: vx =600m/s; Y-direction speed vy = -400m/s; vz=10m/s.

图4、图5所示为考虑多普勒估计偏差对系统测量精度的影响分析，可以看出本发明方法对多普勒的估计精度具有较好的抑制度，可以获得较好的速度与高度估计精度。Figures 4 and 5 show the analysis of the influence of Doppler estimation deviation on the measurement accuracy of the system. It can be seen that the method of the present invention has a good degree of suppression on the Doppler estimation accuracy, and can obtain better speed and altitude. Estimation accuracy.

本发明针对现有着陆雷达飞行参数测量的问题，提出了一种基于双波束雷达的着陆器飞行参数(高度和三维速度)测量方法，显著降低了现有着陆雷达采用四波束进行着陆器飞行参数测量的系统复杂度，仅采用双波束即可实现飞行器飞行参数的测量。Aiming at the problem of measuring the flight parameters of the existing landing radar, the present invention proposes a method for measuring the flight parameters (altitude and three-dimensional velocity) of the lander based on the dual-beam radar, which significantly reduces the use of four beams for the existing landing radar to measure the flight parameters of the lander. The complexity of the measurement system, the measurement of the flight parameters of the aircraft can be achieved only by using dual beams.

本发明说明书中未作详细描述的内容属于本领域专业技术人员的公知技术。Contents that are not described in detail in the specification of the present invention belong to the well-known technology of those skilled in the art.

Claims (9)

1. A landing aircraft parameter measurement method based on dual-beam radar is characterized by comprising the following steps:
(1) processing radar echo data which are arranged on an aircraft and have two mutually vertical wave beams and point to the ground to construct a received data matrix X of two radars1And X2；
(2) To the received data matrix X1And X2Performing pulse compression, FFT and detection processing to obtain distance-Doppler information of two radar echo data;
(3) transforming the distance-Doppler information of the two radar echo data to construct a corresponding data matrix;
(4) inversely calculating the three-dimensional speed and the height of the landing aircraft according to the data matrix constructed in the step (3);
(5) and (5) repeating the steps (1) to (4), and processing the radar echo data in the next group of related processing time to obtain the real-time flight parameters of the aircraft.
2. The method of claim 1, wherein in step (1),
Figure FDA0002179721830000011
Figure FDA0002179721830000012
receiving data for a complex matrix of L rows and M columnsMatrix X1Row i and column m element x in (1)1(l, m) is the data sampled from the l base band signal of the m pulse of the first radar, and the data matrix X is received2Row i and column m element x in (1)2(l, m) is the l baseband signal sample data for the m pulse of the second radar.
3. The method of claim 1, wherein in step (2), the received data matrix X is a matrix of X1And X2Pulse compression is carried out according to columns to obtain range direction high-resolution echo data
Figure FDA0002179721830000013
And
Figure FDA0002179721830000014
the obtaining process is as follows:
Figure FDA0002179721830000015
Figure FDA0002179721830000016
wherein X1(m) denotes a received data matrix X1Column m of (1), X2(m) denotes a received data matrix X2The (c) th column (c) of (c),
Figure FDA0002179721830000017
representation matrix
Figure FDA0002179721830000018
The (c) th column (c) of (c),
Figure FDA0002179721830000019
representation matrix
Figure FDA00021797218300000110
M column of (2), symbol
Figure FDA00021797218300000111
Which represents a convolution operation, the operation of the convolution,
Figure FDA0002179721830000021
for the corresponding digital matching signal of the radar transmission signal s (t),
Figure FDA0002179721830000022
is a complex matrix of 1 row and P columns,
Figure FDA0002179721830000023
4. the method of claim 3, wherein in step (2), the compressed pulses are measured
Figure FDA0002179721830000024
And
Figure FDA0002179721830000025
FFT processing is carried out according to rows to obtain the echo data range Doppler spectrums of two radars
Figure FDA0002179721830000026
And
Figure FDA0002179721830000027
the acquisition process is as follows:
Figure FDA0002179721830000028
Figure FDA0002179721830000029
wherein,
Figure FDA00021797218300000210
5. the method of claim 4, wherein in step (2), the range-Doppler spectra of the echo data of two radars are measured
Figure FDA00021797218300000211
And
Figure FDA00021797218300000212
and detecting to obtain the range-Doppler information of two radar echo data, wherein the process is as follows:
let the system detection threshold be rhoTHTo matrix
Figure FDA00021797218300000213
And
Figure FDA00021797218300000214
detecting to obtain a value greater than rhoTHThe echo time and Doppler value corresponding to the element of (1) are recorded and recorded
Figure FDA00021797218300000215
Wherein t is1Is a matrix
Figure FDA00021797218300000216
Median value greater than rhoTHThe echo time vector corresponding to the element of (f)d,1Is a matrix
Figure FDA00021797218300000217
Median value greater than rhoTHThe element of (a) corresponds to the Doppler value vector, t2Is a matrix
Figure FDA00021797218300000218
Median value greater than rhoTHThe echo time vector corresponding to the element of (f)d,2Is a matrix
Figure FDA00021797218300000219
Median value greater than rhoTHThe corresponding doppler value vector of the element of (a); j and K are respectively a matrix
Figure FDA00021797218300000220
Median value greater than rhoTHNumber of elements and matrix
Figure FDA00021797218300000221
Median value greater than rhoTHThe number of elements (c);
Figure FDA00021797218300000222
is a real matrix of J rows and 1 columns,
Figure FDA00021797218300000223
a real matrix of K rows and 1 column;
the range-doppler information of two radar echo data is detected as follows:
Figure FDA00021797218300000224
Figure FDA00021797218300000225
Figure FDA00021797218300000226
Figure FDA00021797218300000227
wherein
Figure FDA0002179721830000031
t1jIs t1The jth element of (1), t2kIs t2J is 1,2, …, J, K is 1,2, …, K, c is the speed of light.
6. A method for measuring landing aircraft parameters based on dual beam radar as claimed in claim 5, characterized in that said step (3) is implemented as follows:
according to the range-Doppler information of two radar echo data, the following data matrix is constructed:
a＝[a1 a2 … aJ]
b＝[b1 b2 … bJ]
c＝[c1 c2 … cJ]
d＝[d1 d2 … dK]
e＝[e1 e2 … eK]
f＝[f1 f2 … fK]
wherein
Figure FDA0002179721830000032
Figure FDA0002179721830000033
Figure FDA0002179721830000034
Figure FDA0002179721830000035
Figure FDA0002179721830000036
Figure FDA0002179721830000037
Setting intermediate variables
Figure FDA0002179721830000038
x2＝vzH
Figure FDA0002179721830000039
Figure FDA00021797218300000310
Figure FDA00021797218300000311
Wherein v isx、vy、vzThe speed in the x direction, the speed in the y direction and the speed in the z direction of the landing aircraft are respectively, and H is the height of the landing aircraft;
from the two radar beam Doppler and range-Doppler information, the following matrix is constructed:
x＝[x1 x2 x3 x4 x5]T
Γ1 Tx+a＝0
Γ2 Tx+d＝0
wherein
Figure FDA0002179721830000041
Figure FDA0002179721830000042
Figure FDA0002179721830000043
Figure FDA0002179721830000044
7. A method for measuring landing aircraft parameters based on dual beam radar as claimed in claim 6, characterized in that said step (4) is implemented as follows:
(S1) solving an intermediate variable x according to the data matrix constructed in the step (3):
the data matrix constructed in the step (3) can be written
ΓTx+g＝0
Wherein
Figure FDA0002179721830000045
Figure FDA0002179721830000046
Solved to obtain
Figure FDA0002179721830000047
Wherein
Figure FDA0002179721830000048
Representing a pseudo-inverse of the matrix;
(S2) according to x, the inverse calculation is obtained
Figure FDA0002179721830000049
And
Figure FDA00021797218300000410
(S3) solving for a geometric coupling relationship between two radar beams
Figure FDA00021797218300000411
8. A method for measuring landing aircraft parameters based on dual beam radar as claimed in claim 7, characterized in that said step (S2) is implemented as follows:
Figure FDA00021797218300000412
Figure FDA00021797218300000413
Figure FDA00021797218300000414
Figure FDA0002179721830000051
represents the estimated velocity of the aircraft in the x-direction,
Figure FDA0002179721830000052
represents the estimated velocity of the aircraft in the y-direction,
Figure FDA0002179721830000053
representing the estimated altitude of the aircraft.
9. A method for measuring landing aircraft parameters based on dual beam radar as claimed in claim 7, characterized in that said step (S3) is implemented as follows:
Figure FDA0002179721830000054
where mean (-) represents the mean value of the vector, α, β is
α＝[α1 α2 … αJ]
β＝[β1 β2 … βK]
Wherein
Figure FDA0002179721830000055
Figure FDA0002179721830000056
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