CN111143765A - Novel method for designing MxN Nuomon matrix - Google Patents

Novel method for designing MxN Nuomon matrix Download PDF

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CN111143765A
CN111143765A CN201911330918.1A CN201911330918A CN111143765A CN 111143765 A CN111143765 A CN 111143765A CN 201911330918 A CN201911330918 A CN 201911330918A CN 111143765 A CN111143765 A CN 111143765A
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matrix
equation
phase
port
electric field
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CN111143765B (en
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李文廷
孔迪
高式昌
苏海滨
任晓飞
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China Institute of Radio Wave Propagation CETC 22 Research Institute
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China Institute of Radio Wave Propagation CETC 22 Research Institute
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0456—Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0615—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal
    • H04B7/0617—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal for beam forming

Abstract

The invention discloses a new method for designing an MxN Nuomon matrix, wherein the Nuomon matrix comprises a phase shifter and a coupler, and the value of the phase shifter is set as
Figure 125668DEST_PATH_IMAGE002
The scattering matrix of the coupler is [ S ]]From the scattering matrix [ S ]]It can be seen that each input port is isolated from the other input ports, and in order to design a norlon matrix meeting the requirements, it is required to solve the parameters of the phase shifter and the coupler. The norlon matrix or feed network based on the norlon matrix has the advantage of being more convenient and more scalable for any number of antennas (output ports) and radiation beams (input ports). The novel method for designing the MxN norlon matrix disclosed by the invention is simpler and more suitable for programming calculation。

Description

Novel method for designing MxN Nuomon matrix
Technical Field
The invention belongs to the field of matrix equation sets and networks, and particularly relates to a novel method for designing an M multiplied by N norlon matrix in the field.
Background
The search for multi-beam antennas has been in the leading field of research for decades, and the development in this field has mainly been derived from the research of beam forming matrices. A multi-beam antenna is a linear antenna array fed by a multiple-input multiple-output beam forming network (matrix or lens). The beam forming networks provide excitation to the antenna array with appropriate amplitude and phase, with the input ports of each beam forming network spatially corresponding to a discrete radiation beam. Thus, in transmit mode, multiple independent beams are generated from one aperture by sequential excitation of the beamforming input ports; on the other hand, all beams are simultaneously available in the receive mode. Multi-beam antennas have found wide application in the field of satellite communications, and also in radar and electronic warfare systems and point-to-multipoint terrestrial communications systems.
Generally, the multi-beam antenna is mainly classified into two types according to different feeding modes: parallel feeding and serial feeding. The most widely used beamforming matrices are now butler, braus and norlon matrices. The Butler matrix is a parallel multi-beam feeding mode, and the Booth matrix and the Nulon matrix are serial feeding modes; the butler matrix is typically designed for the same number of input and output ports, while the braz and norlon matrices are typically designed for different numbers of input and output ports.
Although the butler matrix has received a great deal of attention, much research work has been done on it, and the butler matrix is also used in a wide range of fields. However, research in recent years has shown that the norlon matrix or feed network based on the norlon matrix has the advantage of being more convenient and more scalable for any number of antennas (output ports) and radiation beams (input ports). However, in the current research on the norlon matrix, some design methods are actually complex, and some designs are not practical during programming calculation, so that it is very important to find a more concise norlon matrix design method which is more suitable for programming calculation.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a new method for designing an M multiplied by N norlon matrix.
The invention adopts the following technical scheme:
in a new method for designing an mxn norlon matrix, the improvement comprising:
the norlon matrix comprises phase shifters and couplers, and the phase shifters are set to have values of
Figure BDA0002329530630000011
The scattering matrix of the coupler is [ S ]]From the scattering matrix [ S ] as shown in equation (1)]It can be seen that each input port is isolated from other input ports, and in order to design a norlon matrix meeting requirements, parameters of a phase shifter and a coupler are required to be solved;
Figure BDA0002329530630000021
input port and output port are respectively designated by a1,a2,…,aMAnd b1,b2,…,bNIs shown as amnAnd fmnRepresenting the value of the electric field, a, at each point in the networkmnRepresenting the value of the electric field, f, for the point connecting the two couplers or for the point of one input portmnRepresenting the value of the electric field at the point of connection of a coupler and a phase shifter or at the point of connection of an output port, then there is FmnAs shown in equation (2), i.e., FmnThe element (b) is the electric field value at the point above the m-th row of phase shifters;
Fmn＝[fm1fm2…fmn]T(2)
order to
Figure BDA0002329530630000022
Representing an input vector of length M, the element of which is the electric field value of each input port;
Figure BDA0002329530630000023
representing an output vector of length N, the element of which is the electric field value of each output port, the input vector
Figure BDA0002329530630000024
And the output vector
Figure BDA0002329530630000025
Is shown in equations (3) and (4):
Figure BDA0002329530630000026
Figure BDA0002329530630000027
reissue to order
Figure BDA0002329530630000028
Representing an excitation vector of length N, the elements of which are present at an input port amExcited by unit power, i.e. am1When the output port is equal to 1, the electric field value of each output port is expressed as formula (5):
Figure BDA0002329530630000029
when all input ports are inputted with vectors
Figure BDA00023295306300000210
Excited, then the vector is output
Figure BDA00023295306300000211
Equation (6) is satisfied due to the mutual isolation between the input ports;
Figure BDA00023295306300000212
then equation (6) can be rewritten as equation (7):
Figure BDA00023295306300000213
for the design of a multi-beam network,
Figure BDA00023295306300000214
it is known how the main solution task is to obtain
Figure BDA0002329530630000031
And thetamnWherein M is more than or equal to 1 and less than or equal to M, N is more than or equal to 1 and less than or equal to N, and the specific design steps are as follows:
step 1: let i equal to 1;
step 2: reissue to order
Figure BDA0002329530630000032
This means that only the input ports of the ith row are excited with unit power, i.e. ai1 When 1, there is a unit output vector
Figure BDA0002329530630000033
According to
Figure BDA0002329530630000034
And FmnCan be defined by
Figure BDA0002329530630000035
Therefore it has the advantages of
Figure BDA0002329530630000036
Figure BDA0002329530630000037
F in formula (9) can be obtained from formula (8)i(N-i+1)，
Figure BDA0002329530630000039
Figure BDA00023295306300000310
Figure BDA00023295306300000311
And step 3: due to Fi(N-i+1)Has been calculated, then the element f thereini1fi2… fi(N-i+1)Can also be obtained from the scattering matrix S of the coupler]It can be seen that port 1 and port 3 are in phase, port 2 and port 4 are in phase, and the phase difference between port 1 and port 2 is 90 °, so equation (10) is as follows:
phase(ai1)＝phase(ai2)＝…＝phase(ai(N-i+1)) (9)
as can be seen from the structure of such a norlon matrix, when N ═ N-i +1, there is formula (11):
Figure BDA00023295306300000312
then, the formula (12) is obtained
Figure BDA00023295306300000313
When N is less than N-i +1, the formula (13)
Figure BDA00023295306300000314
Assume a case as in equation (14):
phase(ain)＝0,(1≤n≤N-i+1) (13)
combining equations (12), (13), and (14), equation (15) can be obtained:
Figure BDA0002329530630000041
in this regard, it is believed that the first,
Figure BDA0002329530630000042
can be calculated;
and 4, step 4: according to the existing calculation method, the coupling coefficient of the i-th row coupler can be calculated as shown in equation (16):
Figure BDA0002329530630000043
considering a in step 2i1Equation (16) can be rewritten as equation (17) for 1,
Figure BDA0002329530630000044
and 5: then, the step 2-4 is repeated by changing i to i +1 until all the values of the coupler and the phase shifter are calculated;
when the above five steps are completed, all the values in the coupler and the phase shifter are calculated, and the mxn norlon matrix is designed.
The invention has the beneficial effects that:
the norlon matrix or feed network based on the norlon matrix has the advantage of being more convenient and more scalable for any number of antennas (output ports) and radiation beams (input ports). The novel method for designing the M x N norlon matrix disclosed by the invention is simpler and more suitable for programming calculation.
Drawings
FIG. 1 is a schematic diagram of the structure of an M N Nuon matrix;
FIG. 2 is a schematic diagram of a typical node of the norlon matrix.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1, this example discloses a new method for designing an M × N norlon matrix, where, under the condition that M excitation vectors of the norlon matrix are orthonormal, a first M columns of a scattering matrix of a coupler are orthonormal, and when only an i-th row input port in the norlon matrix is excited by single-bit power, a formula can be obtained
Figure BDA0002329530630000045
Figure BDA0002329530630000046
Depending on the characteristics of the coupler: the phase relationship between ports 1,2, 3 and 4, the value of the phase shifter can be found
Figure BDA0002329530630000047
Then, calculating a formula according to the coupler coupling value of the ith row:
Figure BDA0002329530630000048
the coupling coefficient theta of the coupler can be obtainedmn. And repeating the steps to obtain the values of all the couplers and the phase shifters in the Nulon matrix, namely the design method of the M multiplied by N Nulon matrix.
As shown in FIGS. 1-2, the norlon matrix includes phase shifters and couplers, with the phase shifters having values of
Figure BDA0002329530630000051
The scattering matrix of the coupler is [ S ]]From the scattering matrix [ S ] as shown in equation (1)]It can be seen that each input port is isolated from other input ports, and in order to design a norlon matrix meeting requirements, parameters of a phase shifter and a coupler are required to be solved;
Figure BDA0002329530630000052
input port and output port are respectively designated by a1,a2,…,aMAnd b1,b2,…,bNIs shown as amnAnd fmnRepresenting the value of the electric field, a, at each point in the networkmnRepresenting the value of the electric field, f, for the point connecting the two couplers or for the point of one input portmnRepresenting the value of the electric field at the point of connection of a coupler and a phase shifter or at the point of connection of an output port, then there is FmnAs shown in equation (2), i.e., FmnThe element (b) is the electric field value at the point above the m-th row of phase shifters;
Fmn＝[fm1fm2… fmn]T(18)
order to
Figure BDA0002329530630000053
Representing an input vector of length M, the element of which is the electric field value of each input port;
Figure BDA0002329530630000054
representing an output vector of length N, the element of which is the electric field value of each output port, the input vector
Figure BDA0002329530630000055
And the output vector
Figure BDA0002329530630000056
Is shown in equations (3) and (4):
Figure BDA0002329530630000057
Figure BDA0002329530630000058
reissue to order
Figure BDA0002329530630000059
Representing an excitation vector of length N, the elements of which are present at an input port amExcited by unit power, i.e. am1When the output port is equal to 1, the electric field value of each output port is expressed as formula (5):
Figure BDA00023295306300000510
when all input ports are inputted with vectors
Figure BDA00023295306300000511
Excited, then the vector is output
Figure BDA00023295306300000512
Equation (6) is satisfied due to the mutual isolation between the input ports;
Figure BDA0002329530630000061
then equation (6) can be rewritten as equation (7):
Figure BDA0002329530630000062
from equation (7), equation (8) can be derived:
Figure BDA0002329530630000063
therefore, the relationship between the input port and the output port is shown in equation (9):
Figure BDA0002329530630000064
the scattering matrix [ SN ] of an MXN Nuomon matrix is shown in equation (10).
Figure BDA0002329530630000065
As can be understood from the formula (8),
Figure BDA0002329530630000066
if each input port is matched and isolated from the other input ports,
Figure BDA0002329530630000067
then equation (10) can be rewritten as equation (11).
Figure BDA0002329530630000068
When we assume that the phase shifters and couplers are lossless, the norlon matrix is also lossless. Therefore, [ SN ] is a unitary matrix, it satisfies equation (12).
[SN]T＝([SN]*)-1(12)
Equation (12) is written as a summation in equation (13):
Figure BDA0002329530630000071
equation (13) indicates that any column of the scattering matrix [ SN ] does a dot product with the conjugate of the same column equal to 1 and does a dot product with the conjugate of a different column equal to 0. This means that the scattering matrix SN is orthonormal.
Considering the definition of the excitation matrix by equation (5), the first M columns of the scattering matrix [ SN ] can be written as shown by equation (14).
Figure BDA0002329530630000072
Due to the scattering matrix SN]The first M columns of (a) are orthonormal,
Figure BDA0002329530630000073
and is also orthonormal. This limitation applies not only to norlon matrix network architectures, but also to other lossless network architectures.
For the design of a multi-beam network,
Figure BDA0002329530630000074
is alreadyKnowing how the main solution task is to obtain
Figure BDA0002329530630000075
And thetamnWherein M is more than or equal to 1 and less than or equal to M, N is more than or equal to 1 and less than or equal to N, and the specific design steps are as follows:
step 1: let i equal to 1;
step 2: reissue to order
Figure BDA0002329530630000076
This means that only the input ports of the ith row are excited with unit power, i.e. ai1When 1, there is a unit output vector
Figure BDA0002329530630000077
According to
Figure BDA0002329530630000078
And FmnCan be defined by
Figure BDA0002329530630000079
Therefore it has the advantages of
Figure BDA00023295306300000710
At this time, we should deduce a very important conclusion, namely equation (15):
Figure BDA00023295306300000711
f can be obtained from the structure of the norlon matrix and formula (1)mn. When N < N-m +1, fmnThe following expression (16):
Figure BDA00023295306300000712
when N is N-m +1, fmnExpression (17) of (a) is:
Figure BDA0002329530630000081
combining equations (16) and (17) yields equation (18):
Figure BDA0002329530630000082
to validate the above derivation, we define some matrices and vectors:
Figure BDA0002329530630000083
Φpq、Apq、
Figure BDA0002329530630000084
and u1q. Wherein E(q-1)Is an identity matrix of (q-1) rows and (q-1) columns. Wherein P is an integer.
Figure BDA0002329530630000085
Figure BDA0002329530630000086
Figure BDA0002329530630000087
Figure BDA0002329530630000088
Apq＝[ap2ap3… apq]T (q-1)×1
Figure BDA0002329530630000089
Figure BDA0002329530630000091
u1q＝[1 0 … 0]T q×1
When N is 1,2, … N-m +1, equation (19) can be derived:
Figure BDA0002329530630000092
by the matrix definition introduced and the pair F in the formula (2)mnEquation (19) can be rewritten as equation (20):
Figure BDA0002329530630000093
known formulae (21), (22), and (23):
Figure BDA0002329530630000094
Figure BDA0002329530630000095
Figure BDA0002329530630000096
substituting the equations (21), (22) and (23) into the equation (20) yields the equation (24):
Figure BDA0002329530630000097
Figure BDA0002329530630000101
based on the structure of the norlon matrix and formula (1), we can also know that am(n+1)The following expression (25):
am(n+1)＝jf(m+1)nsinθmn+amncosθmn,(n＜N-m+1) (25)
it can also be rewritten as formula (26):
-amncosθmn+am(n+1)＝jf(m+1)nsinθmn,(n＜N-m+1) (26)
when N is 1,2, … N-m, equation (27) can be derived:
Figure BDA0002329530630000102
we can define a matrix Ad：
Figure BDA0002329530630000103
Then equation (27) can be written as:
Figure BDA0002329530630000104
from equation (28) it can be derived:
Figure BDA0002329530630000105
when only the input port of the ith row is activated, ap10(p ≠ i). When m +1 is less than or equal to i, and m is not equal to i, am1Equation (29) can be simplified to equation (30) as 0:
Figure BDA0002329530630000106
for N2, … N-m, there is | cos θmnAnd | is less than 1. So matrix AdIs a strictly diagonal dominating matrix, which means that it is invertible, i.e. a matrix
Figure BDA0002329530630000111
Are present. Then for A in equation (30)m(N-m+1)May be represented by equation (31):
Figure BDA0002329530630000112
the formulae (31) and am1Substituting 0 into equation (24) yields equation (32):
Figure BDA0002329530630000113
we can redefine a matrix BmThe expression is formula (33), matrix BmIs an (N-m +1) × (N-m) matrix and is not directly reversible. Formula (34) can be derived by substituting formula (33) into formula (32).
Figure BDA0002329530630000114
Fm(N-m+1)＝Bm·F(m+1)(N-m)(34)
Then, the following matrix can be defined:
Figure BDA0002329530630000115
and Cm. Wherein C ismIs a square matrix.
Figure BDA0002329530630000116
Figure BDA0002329530630000117
Multiplication of formula (34)
Figure BDA0002329530630000118
Equation (37) is obtained:
Figure BDA0002329530630000119
due to equation (38), then equation (37) can be rewritten as equation (39):
Figure BDA00023295306300001110
Fm(N-m)＝Cm·F(m+1)(N-m)(39)
thus, equation (15) proves:
Figure BDA0002329530630000121
f in formula (40) can be obtained from formula (15)i(N-i+1)，
Figure BDA0002329530630000122
Figure BDA0002329530630000123
Figure BDA0002329530630000124
Figure BDA0002329530630000125
And step 3: due to Fi(N-i+1)Has been calculated, then the element f thereini1fi2… fi(N-i+1)Can also be obtained from the scattering matrix S of the coupler]It can be seen that port 1 and port 3 are in phase, port 2 and port 4 are in phase, and the phase difference between port 1 and port 2 is 90 °, so equation (41) is as follows:
phase(ai1)＝phase(ai2)＝…＝phase(ai(N-i+1)) (41)
as can be seen from the structure of the norlon matrix, when N ═ N-i +1, there is formula (42):
Figure BDA0002329530630000126
then, the formula (43) is obtained
Figure BDA0002329530630000127
When N is less than N-i +1, the formula (44)
Figure BDA0002329530630000128
Assume a case as in equation (45):
phase(ain)＝0,(1≤n≤N-i+1) (45)
combining equations (43), (44), and (45), equation (46) can be obtained:
Figure BDA0002329530630000129
in this regard, it is believed that the first,
Figure BDA00023295306300001210
can be calculated;
and 4, step 4: according to the existing calculation method, the coupling coefficient of the i-th row coupler can be calculated, as shown in equation (47):
Figure BDA0002329530630000131
considering a in step 2i1Equation (47) can be rewritten as equation (48) for 1,
Figure BDA0002329530630000132
and 5: then, the step 2-4 is repeated by changing i to i +1 until all the values of the coupler and the phase shifter are calculated;
when the above five steps are completed, all the values in the coupler and the phase shifter are calculated, and the mxn norlon matrix is designed.

Claims (1)

1. A new method for designing an mxn norlon matrix, characterized in that:
the norlon matrix comprises phase shifters and couplers, and the phase shifters are set to have values of
Figure FDA0002329530620000011
Scattering matrix of couplerIs [ S ]]From the scattering matrix [ S ] as shown in equation (1)]It can be seen that each input port is isolated from other input ports, and in order to design a norlon matrix meeting requirements, parameters of a phase shifter and a coupler are required to be solved;
Figure FDA0002329530620000012
where i represents the number of rows in which the phase shifter or coupler is located, j represents the number of columns in which the phase shifter or coupler is located, and θmnCharacteristic values of the coupler in the mth row and nth column are represented by a for the input port and the output port respectively1,a2,…,aMAnd b1,b2,…,bNIs shown as amnAnd fmnRepresenting the value of the electric field, a, at each point in the networkmnRepresenting the value of the electric field, f, for the point connecting the two couplers or for the point of one input portmnRepresenting the value of the electric field at the point of connection of a coupler and a phase shifter or at the point of connection of an output port, then there is FmnAs shown in equation (2), i.e., FmnThe element in (b) is the electric field value at the upper point of the m-th row phase shifter, FmnA vector consisting of electric field values representing points above the m-th row of phaseshifters;
Fmn＝[fm1fm2…fmn]T(2)
order to
Figure FDA0002329530620000013
Representing an input vector of length M, the element of which is the electric field value of each input port;
Figure FDA0002329530620000014
representing an output vector of length N, the element of which is the electric field value of each output port, the input vector
Figure FDA0002329530620000015
And the output vector
Figure FDA0002329530620000016
Is shown in equations (3) and (4):
Figure FDA0002329530620000017
Figure FDA0002329530620000018
reissue to order
Figure FDA0002329530620000019
Representing an excitation vector of length N, the elements of which are present at an input port amExcited by unit power, i.e. am1When the output port is equal to 1, the electric field value of each output port is expressed as formula (5):
Figure FDA00023295306200000110
when all input ports are inputted with vectors
Figure FDA00023295306200000111
Excited, then the vector is output
Figure FDA00023295306200000112
Equation (6) is satisfied due to the mutual isolation between the input ports;
Figure FDA0002329530620000021
then equation (6) can be rewritten as equation (7):
Figure FDA0002329530620000022
for the design of a multi-beam network,
Figure FDA0002329530620000023
it is known how the main solution task is to obtain
Figure FDA0002329530620000024
And thetamnWherein M is more than or equal to 1 and less than or equal to M, N is more than or equal to 1 and less than or equal to N, and the specific design steps are as follows:
step 1: let i equal to 1;
step 2: reissue to order
Figure FDA0002329530620000025
This means that only the input ports of the ith row are excited with unit power, i.e. ai1When 1, there is a unit output vector
Figure FDA0002329530620000026
According to
Figure FDA0002329530620000027
And FmnCan be defined by
Figure FDA0002329530620000028
Therefore it has the advantages of
Figure FDA0002329530620000029
Figure FDA00023295306200000210
Figure FDA00023295306200000211
The matrix is the relation between the electric field value of the point above the m-th row of phase shifters and the electric field value of the point above the m + 1-th row of phase shifters;
f in formula (9) can be obtained from formula (8)i(N-i+1)，
Figure FDA00023295306200000212
Figure FDA00023295306200000213
Figure FDA00023295306200000214
Figure FDA00023295306200000215
And step 3: due to Fi(N-i+1)Has been calculated, then the element f thereini1fi2…fi(N-i+1)Can also be obtained from the scattering matrix S of the coupler]It can be seen that port 1 and port 3 are in phase, port 2 and port 4 are in phase, and the phase difference between port 1 and port 2 is 90 °, so equation (10) is as follows:
phase(ai1)＝phase(ai2)＝…＝phase(ai(N-i+1)) (9)
phase is a function of Phase;
as can be seen from the structure of such a norlon matrix, when N ═ N-i +1, there is formula (11):
Figure FDA00023295306200000216
finand ainRepresenting the electric field value at each node in the network, and in representing the position of the node;
then, the formula (12) is obtained
Figure FDA0002329530620000031
When N is less than N-i +1, the formula (13)
Figure FDA0002329530620000032
Assume a case as in equation (14):
phase(ain)＝0,(1≤n≤N-i+1) (13)
combining equations (12), (13), and (14), equation (15) can be obtained:
Figure FDA0002329530620000033
angle is a function of the phase;
in this regard, it is believed that the first,
Figure FDA0002329530620000034
can be calculated;
and 4, step 4: according to the existing calculation method, the coupling coefficient of the i-th row coupler can be calculated as shown in equation (16):
Figure FDA0002329530620000035
fiprepresenting the electric field value at a point above the ith row and the pth column phase shifters;
considering a in step 2i1Equation (16) can be rewritten as equation (17) for 1,
Figure FDA0002329530620000036
and 5: then, the step 2-4 is repeated by changing i to i +1 until all the values of the coupler and the phase shifter are calculated;
when the above five steps are completed, all the values in the coupler and the phase shifter are calculated, and the mxn norlon matrix is designed.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113095021A (en) * 2021-03-15 2021-07-09 南京理工大学 Matrix factorization based non-2nDesign method of Butler matrix
CN113659354A (en) * 2021-08-10 2021-11-16 大连海事大学 3X 3 norlon matrix based on crossing directional coupling lines

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CN103594801A (en) * 2013-11-21 2014-02-19 天津中兴智联科技有限公司 Butler matrix structure
EP3244488A1 (en) * 2016-05-13 2017-11-15 Nxp B.V. Multi beam former
CN109613512A (en) * 2018-12-06 2019-04-12 上海交通大学 N × M based on promise human relations matrix integrates multi-beam laser radar emission system

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CN103594801A (en) * 2013-11-21 2014-02-19 天津中兴智联科技有限公司 Butler matrix structure
EP3244488A1 (en) * 2016-05-13 2017-11-15 Nxp B.V. Multi beam former
CN109613512A (en) * 2018-12-06 2019-04-12 上海交通大学 N × M based on promise human relations matrix integrates multi-beam laser radar emission system

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113095021A (en) * 2021-03-15 2021-07-09 南京理工大学 Matrix factorization based non-2nDesign method of Butler matrix
CN113095021B (en) * 2021-03-15 2023-06-06 南京理工大学 Non-2 based on matrix decomposition n Butler matrix design method
CN113659354A (en) * 2021-08-10 2021-11-16 大连海事大学 3X 3 norlon matrix based on crossing directional coupling lines
CN113659354B (en) * 2021-08-10 2022-11-18 大连海事大学 3X 3 norlon matrix based on crossing directional coupling lines

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