CN107958106A - A kind of directional diagram numerical optimization of circle bore planar array antenna - Google Patents
A kind of directional diagram numerical optimization of circle bore planar array antenna Download PDFInfo
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Abstract
The invention discloses a kind of directional diagram numerical optimization of circular bore planar array antenna, the array cell layout of circular aperture distribution is established according to systematic parameters such as satellite transmitted frequencies, array bore shape and size, array element spacing first；The beam position angle specified according to user determines the phase weighting of each unit；Then amplitude weighting optimization vector is set, and initializes amplitude weighting optimization vector matrix, numerical optimization is iterated to amplitude weighting optimization vector matrix using differential evolution algorithm, until the fitness value of optimum optimization vector is less than given threshold.The present invention solves the Numerical Optimization of circular bore rectangular mesh planar array and circular bore triangle gridding planar array well, can rapid Optimum go out required amplitude weighting matrix, the pattern features such as the minor level on array pattern and null level is met index.
Description
Technical field
The present invention relates to the numerical optimization technique field of phased array, especially a kind of side of circular bore planar array antenna
To figure numerical optimization.
Background technology
In the prior art, go to solve using Optimum Theory such as genetic algorithm, particle swarm optimization algorithm and differential evolution algorithms
The certainly Numerical Optimization of array antenna, and the Numerical Optimization of time-modulation array, and achieve lot of research.
But at present in the achievement in research of array Numerical Optimization, being essentially all for one-dimensional even linear array and rectangular aperture square
Shape grid plan battle array, the report studied triangle gridding array and circular bore array are relatively fewer.
When rectangular aperture rectangular grid array is solved the problems, such as, array direction can be decomposed into Z-direction and Y side
To two mutually independent components, so array amplitude weighting matrix is also Z-direction and the one-dimensional amplitude weighting vector of Y-direction two
Multiplication.In Numerical Optimization, which can be greatly reduced operand because it can by optimize vector length by
M × N is reduced to M+N.
Conventional method is when triangle gridding planar array and circular bore planar array is optimized, due to cannot be by array
Directionality carries out the decomposition of Z-direction and Y-direction, can only be included in each cell parameters in optimization vector, will make numerical optimization
Operand sharply increase with the increase of array element number.
The content of the invention
The technical problems to be solved by the invention are, there is provided a kind of directional diagram numerical value of circle bore planar array antenna
Optimization method, the amplitude weighting numerical optimization of circular bore array is carried out using the technology, is reached and is reduced iterations, reduces meter
The purpose of evaluation time and memory consumption.
In order to solve the above technical problems, the present invention provides a kind of directional diagram numerical optimization of circular bore planar array antenna
Method, includes the following steps：
(1) according to systematic parameter input by user, the distribution form of array element, including circular bore rectangular mesh are determined
Array and circular two kinds of bore triangle gridding array, and generation unit Layout matrix F；
(2) the beam main lobe orientation angle (θ set according to user0,φ0), determine the phase weighting of each unit, obtain
Phase weighting matrix
(3) it is as follows to define amplitude weighting optimization vector X：Wherein, LX=MH+NH+ 2, wherein
MH=(MF- 1)/2 and NH=(NF- 1)/2, MFIt is the number of unit of the most middle row of circular aperture array, and largest unit row
Number；NFIt is the number of unit of circular aperture array most middle row, and largest unit columns；
Vector X is optimized based on amplitude weighting, generation amplitude weighting optimization vector matrix X theoretical according to differential evolution algorithmg
It is as follows：Wherein, NpFor the optimization population quantity of user's setting, g represents numerical optimization
Algebraically；
(4) using differential evolution algorithm to optimizing vector matrix XgNumerical optimization is carried out, in Optimized Iterative each time, all
Including making a variation, intersecting and selecting three operations：To optimizing vector matrix XgMutation operation is carried out, obtains variation vector matrix Vg；It is right
XgAnd VgCrossover operation is carried out, obtains experiment vector matrix Ug；In XgAnd UgBetween make choice operation, obtain follow-on optimization
Vector matrix Xg+1。VgAnd UgRepresent as follows：
In the selection operation of differential evolution algorithm, vector X will be optimizedi,gWith experiment vector Ui,gAs target vector, and
The corresponding fitness value of two class target vectors is calculated by object function, using the less target vector of fitness value as the next generation
Optimization vector Xi,g+1, which is a loop iteration process, i.e. i=1,2 ..., NP；
During fitness value calculation, according to target vector Xi,gCalculate the corresponding one-dimensional amplitude weighting vector of Y-direction
AY, and the one-dimensional amplitude weighting vector A of Z-directionZ, further according to AYAnd AZCalculate two dimensional amplitude weighting matrix A, and then computing array
Overall weighting matrix W；According to W computing array directional diagram E, minor level SLL and null electricity are then calculated according to array pattern E
The pattern features such as flat NPL, and by pattern features substitute into object function calculate target vector fitness value；
(5) will optimization vector matrix Xg+1The optimization vector of middle fitness value minimum is denoted as Xbest,g+1If Xbest,g+1Institute
Corresponding fitness value then returns to step (4) and carries out next round iteration optimization still greater than the fitness threshold value of setting；Otherwise terminate
Optimization process；According to optimum optimization vector Xbest,g+1Corresponding overall weighting matrix W is calculated, as the final of array numerical optimization
As a result exported.
Preferably, in step (3), using random device to XgInitialized, g=0 during initialization.
Preferably, described in step (4) according to target vector X, calculate the one-dimensional amplitude weighting vector A of Z-directionZIt is as follows：
Its a length of MF, by AZExpression formula as it can be seen that the amplitude weighting vector of Z-direction is symmetrical.
Preferably, in step (4), according to target vector X, the one-dimensional amplitude weighting vector A of Y-direction is calculatedYIt is as follows：
Its a length of NF, by AYExpression formula as it can be seen that the amplitude weighting vector of Y-direction is symmetrical.
Preferably, described in step (4) according to AYAnd AZTwo dimensional amplitude weighting matrix A is calculated, and then computing array totally adds
Weight matrix W, its step are as follows：
In Z-direction, it is for there are each row of antenna element, defining from the top unit to the length of bottom unitThen it is M by lengthFAZProgress linear interpolation obtains length and is'sAnd willUpper and lower two
End symmetrically mends 0, becomes the vector that a symmetrical length is M；
Similarly, in the Y direction, based on vector AYLinear interpolation and symmetrical zero padding are carried out, obtains corresponding to symmetrical per a line
The amplitude weighting vector of distributionIts length is N；
Then the amplitude weighting matrix A for obtaining array is：
Last array totality weighting matrix W calculates as follows：
Beneficial effects of the present invention are：The present invention forces amplitude when carrying out numerical optimization to amplitude weighting optimization vector
Weighting is symmetrical in Z-direction and Y-direction, significantly reduces operand and iterations；The present invention is by rectangular aperture rectangle net
The Z-direction of lattice planar array and the mutually independent thinking of the directionality of Y-direction, which are incorporated into, solves triangle gridding array and circular port
In the array problem of footpath, the length for optimizing vector is reduced to M+N by M × N, significantly reduces memory consumption and iterations.
Brief description of the drawings
Fig. 1 is the method flow schematic diagram of the present invention.
Fig. 2 (a) is the circular bore rectangular mesh plane Zhen Liao cell layouts schematic diagram of the present invention.
Fig. 2 (b) is circular this column unit schematic layout pattern of bore triangle gridding plane of the present invention.
Fig. 3 (a) is the array phase weighting matrix histogram of the circular bore rectangular mesh planar array of the present invention.
Fig. 3 (b) is the array amplitude weighting matrix histogram of the circular bore rectangular mesh planar array of the present invention.
Fig. 3 (c) is the half space two-dimensional directional figure of the array of the circular bore rectangular mesh planar array of the present invention.
Fig. 3 (d) is the Z-direction one-dimensional square figure of the array of the circular bore rectangular mesh planar array of the present invention.
Fig. 3 (e) is the Y-direction one-dimensional square figure of the array of the circular bore rectangular mesh planar array of the present invention.
Fig. 4 (a) is the array phase weighting matrix histogram of the circular bore triangle gridding planar array of the present invention.
Fig. 4 (b) is the array amplitude weighting matrix histogram of the circular bore triangle gridding planar array of the present invention.
Fig. 4 (c) is the half space two-dimensional directional figure of the array of the circular bore triangle gridding planar array of the present invention.
Fig. 4 (d) is the Z-direction one-dimensional square figure of the array of the circular bore triangle gridding planar array of the present invention.
Fig. 4 (e) is the Y-direction one-dimensional square figure of the array of the circular bore triangle gridding planar array of the present invention.
Embodiment
As shown in Figure 1, circle bore planar array antenna directional diagram numerical optimization proposed by the present invention, first basis
Systematic parameter input by user, determines the distribution form of array element, two choosings in two kinds of layouts of rectangular mesh and triangle gridding
One, generation unit Layout matrix F.Beam position angle (the θ set according to user0,φ0) determine the phase weighting of each unit,
Form phase weighting matrixThe amplitude weighting defined according to this method optimizes vector X, forms amplitude weighting optimization vector matrix
Xg, and initialized using random device.Using differential evolution algorithm to optimizing vector matrix XgNumerical optimization is carried out, is obtained
Follow-on optimization vector matrix Xg+1.Will optimization vector matrix Xg+1The optimization vector of middle fitness value minimum is denoted as Xbest,g+1,
If Xbest,g+1Corresponding fitness value then carries out next round iteration optimization still greater than the fitness threshold value of setting；Otherwise tie
Beam optimization process, exports result of calculation, i.e. optimum optimization vector Xbest,g+1Corresponding array amplitude weighting matrix A and overall weighting
Matrix W.
Comprise the following steps that：
Step 1)：Determine the cell layout of array
According to systematic parameter input by user, such as satellite transmitted frequencies, array caliber size and array element spacing are
System parameter, determines the distribution form of array element, including circular bore rectangular grid array and circular bore triangle gridding array
Two kinds of options, are the planar array in YZ faces as shown in Fig. 2 (a) and Fig. 2 (b).Fig. 2 (a) is that circular bore rectangular mesh is put down
Face array cell layout schemes, and Fig. 2 (b) is circular bore triangle gridding planar array cell layout figure.Circle in figure represents circle
The edge of shape bore array, grey round dot have M rows N row, and some grey round dots are covered by black asterism, are placed with array element
Position.This method can generate cell layout's matrix F of a M rows N row, and corresponding grey dot matrixes, there is the position that unit is placed
(at black asterism) value is 1, and position (at the grey round dot) value placed without unit is 0, for putting for mark array unit
Layout.
Step 2)：Determine the phase weighting of array
According to beam position angle (θ input by user0,φ0) determine the phase weighting of each unit, form a M rows N
The phase weighting matrix of rowIn position, phase weighting existing for no unitThere is position existing for unit
Put, phase weighting is expressed as：
Wherein, dyAnd dzFor Y-direction and the cell spacing of Z-direction, β0For corresponding to transmitting dominant frequency f0Wave number, m=1,
2 ..., M, n=1,2 ..., N.
Step 3)：Amplitude weighting optimizes the initialization of vector matrix
It is defined herein：
MF=(M+1)/2, NF=(N+1)/2
MH=(MF-1)/2,NH=(NF-1)/2
It should be noted that for circular bore array, the number of unit of each row is different, a most middle column unit
Number is most；Unit number per a line is different, and most middle row number of unit is most.Herein, MFIt is the most middle row of array
Number of unit, and largest unit line number；NFIt is the number of unit of array most middle row, and largest unit columns.And MHWith
NHIt is respectively then MFAnd NFRemove the half of number after center cell.
In the method, array far field directionality is decomposed into two mutually independent components of Y-direction and Z-direction, visually
The result being multiplied for two one dimensional linear array directionality of Y-direction and Z-direction.So although the final amplitude weighting matrix A of array is
The two-dimensional matrix of one M rows N row, but the optimization vector X in numerical optimization is a n dimensional vector n.The length for optimizing vector is LX=
MH+NH+ 2, it can be expressed as：
Wherein, M beforeHThe amplitude that+1 element is used for Z-direction optimizes, behind NHThe amplitude that+1 element is used for Y-direction is excellent
Change.
If the population quantity of numerical optimization is Np, then vector matrix X is optimizedgIt can be expressed as：
G herein represents iterations, also referred to as optimizes algebraically, g=0, i.e. the 0th generation during initialization, this method use
The value of random device initialization optimization vector matrix.
Step 4)：The numerical optimization of amplitude weighting
This method is using differential evolution algorithm to optimizing vector matrix XgNumerical optimization is carried out, this is a loop iteration mistake
Journey, iteration is all including three variation, intersection and selection steps each time.
Variation link, to optimize vector matrix XgIn each optimization vector Xi,g, produce a corresponding variation arrow
Measure Vi,g.Since i is to change to N from 1p, so it is as follows just to form a variation vector matrix：
Intersecting link, to each group of optimization vector Xi,gWith variation vector Vi,g, produce a corresponding experiment vector
Ui,g.Since i is to change to N from 1p, so it is as follows just to form an experiment vector matrix：
In selection link, vector U will be testedi,gWith optimization vector Xi,gIt is compared, there is more excellent fitness value in the two
Individual enter of future generation by selected, become Xi,g+1.Fitness value is calculated by object function.Pursued in this method
The minimum of object function, so its mathematic(al) representation is：
Wherein, f () represents object function, to weigh optimization vector Xi,gWith experiment vector Ui,gPerformance, optimize at this time
Vector Xi,gWith experiment vector Ui,gIt is used as target vector.
First by Xi,gAs target vector, the object function in this method is defined as：
F (X)=cSLL*abs[SLL(X)-SLVL]+cNPL*abs[NPL(X)-NPVL]
Wherein, SLL represents the corresponding minor levels of target vector X, and NPL represents the corresponding null level of target vector X.
SLVL represents the minor level index of user's setting, and NPVL represents the null level index of user's setting, and abs () represents to ask exhausted
To the function of value.cSLLAnd cNPLRespectively correspond to the proportionality coefficient of SLL and NPL, and have cSLLAdd cNPLEqual to 1.0.In we
In method, object function describes current minor level and null level and sets the gap between index, so it is worth smaller get over
It is good.
As it can be seen that calculating minor level and null level are then the core missions of object function, its step is as follows：
According to target vector X, the amplitude weighting vector that can obtain Z-direction is：
Its a length of MF.By AZExpression formula as it can be seen that the amplitude weighting vector of Z-direction is symmetrical.
According to target vector X, the amplitude weighting vector that can obtain Y-direction is：
Its a length of NF.By AYExpression formula as it can be seen that the amplitude weighting vector of Y-direction is symmetrical.
Check that cell layout's matrix F is understood, whether circular bore rectangular mesh face battle array, or circular bore triangle gridding
Face battle array, the number of unit per a line is all different, and the number of unit of each row is also different.This method uses linear interpolation
Technology, to produce the amplitude weighting matrix of circular bore grid distribution face battle array.
By taking Z-direction as an example, for there are each row of antenna element, defining from the top unit to the unit of bottom
Length (with grey round dot count) beThen it is M by lengthFAZLinear interpolation is into length'sAnd willUpper and lower ends symmetrically mend 0, become the vector that a symmetrical length is M.
Similarly, in the Y direction, based on vector AYLinear interpolation and symmetrical zero padding are carried out, obtains the symmetrical of every a line
Amplitude weighting vectorIts length is N.
Then the amplitude weighting matrix A for obtaining array is：
The overall weighting matrix for finally obtaining array is：
, can be with computing array directional diagram, to obtain the feature such as minor level and null level according to array weight W.Then
The fitness value of target vector is calculated by object function.
So far, optimization vector X has been obtainedi,gCorresponding fitness value.Again by Ui,gAs target vector, with same
Step calculates Ui,gCorresponding fitness value.The less target vector of fitness will become follow-on optimization vector Xi,g+1。
When i changes to N from 1P, selection opertor picks out each optimization vector Xi,g+1, that is, obtain entirely optimizing vector moment
Battle array Xg+1.By Xg+1The optimization vector of middle fitness value minimum is denoted as the optimum optimization vector X in g+1 generationsbest,g+1。
Step 5)：Iteration control
If optimum optimization vector Xbest,g+1Fitness value be more than setting fitness index, then return to step 4) progress
The iteration optimization of next round；If optimum optimization vector Xbest,g+1Fitness value met setting fitness index, then
Terminate optimization process.Then optimum results, i.e. optimum optimization vector X are exportedbest,g+1Corresponding array amplitude weighting matrix A, it
Minor level, null level isotropy feature can be made to be satisfied by requiring.
Embodiment 1：This example carries out the amplitude weighting matrix numerical optimization of circular bore rectangular mesh planar array, parameter setting
It is as follows：Launch dominant frequency 15GHz, 0.2 meter of array bore diameter, cell spacing is half wavelength, number of unit 317；Wave beam refers to
It is arranged to (80 °, 10 °) to angle；Null angle 1 is (110 °, 10 °), and null width is 1；Null angle 2 is (80 °, 35 °), zero
It is 5 to fall into width；Optimize 2 times that population quantity is optimization vector length, mutagenic factor 0.6, crossover probability 0.9.Optimization refers to
It is designated as minor level and reaches -40dB, null level reaches -60dB.
Shown in optimum results such as Fig. 3 (a)-(e).Fig. 3 (a) is the linear phase needed for main beam position (80 °, 10 °)；Figure
The amplitude weighting matrix that 3 (b) comes for optimization, as seen from the figure, no matter being all in symmetrical in Z-direction or in the Y direction amplitude weighting
Distribution；Fig. 3 (c) is the two-dimensional directional figure that optimization obtains；Fig. 3 (d) and (e) are the Z-direction and Y-direction by (80 °, 10 °)
One-dimensional square figure, as seen from the figure, the minor level of directional diagram can reach -40dB, can be reached in the null level of two null angles
To -60dB, meet setting index.
Embodiment 2：This example carries out the amplitude weighting matrix numerical optimization of circular bore triangle gridding planar array, parameter setting
It is as follows：Launch dominant frequency 15GHz, 0.2 meter of array bore diameter, cell spacing is half wavelength, number of unit 367；Wave beam refers to
It is arranged to (100 °, -10 °) to angle；Null angle 1 is (130 °, -10 °), and null width is 1；Null angle 2 for (100 °,
20 °), null width is 5；Optimize 2 times that population quantity is optimization vector length, mutagenic factor 0.6, crossover probability 0.9.
Optimizing index reaches -40dB for minor level, and null level reaches -60dB.
Shown in optimum results such as Fig. 4 (a)-(e).Fig. 4 (a) is the linear phase needed for main beam position (100 °, -10 °)；
The amplitude weighting matrix that Fig. 4 (b) comes for optimization, as seen from the figure, no matter being all in pair in Z-direction or in the Y direction amplitude weighting
Claim distribution；Fig. 4 (c) is the two-dimensional directional figure that optimization obtains；Fig. 4 (d) and (e) are Z-direction and the Y side by (100 °, -10 °)
To one-dimensional square figure, as seen from the figure, the minor level of directional diagram can reach -40dB, in the null level of two null angles
- 60dB can be reached, meet setting index.
Although the present invention is illustrated and has been described with regard to preferred embodiment, it is understood by those skilled in the art that
Without departing from scope defined by the claims of the present invention, variations and modifications can be carried out to the present invention.
Claims (5)
1. a kind of directional diagram numerical optimization of circle bore planar array antenna, it is characterised in that include the following steps：
(1) according to systematic parameter input by user, the distribution form of array element, including circular bore rectangular grid array are determined
With circular two kinds of bore triangle gridding array, and generation unit Layout matrix F；
(2) the beam main lobe orientation angle (θ set according to user0,φ0), determine the phase weighting of each unit, obtain phase
Weighting matrix
(3) it is as follows to define amplitude weighting optimization vector X：Wherein, LX=MH+NH+ 2, wherein MH=
(MF- 1)/2 and NH=(NF- 1)/2, MFIt is the number of unit of the most middle row of circular aperture array, and largest unit line number；NF
It is the number of unit of circular aperture array most middle row, and largest unit columns；
Vector X is optimized based on amplitude weighting, generation amplitude weighting optimization vector matrix X theoretical according to differential evolution algorithmgIt is as follows：Wherein, NpFor the optimization population quantity of user's setting, g represents the generation of numerical optimization
Number；
(4) using differential evolution algorithm to optimizing vector matrix XgNumerical optimization is carried out, in Optimized Iterative each time, is all included
Variation, intersect and select three operations：To optimizing vector matrix XgMutation operation is carried out, obtains variation vector matrix Vg；To XgWith
VgCrossover operation is carried out, obtains experiment vector matrix Ug；In XgAnd UgBetween make choice operation, obtain it is follow-on optimization arrow
Moment matrix Xg+1。VgAnd UgRepresent as follows：
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In the selection operation of differential evolution algorithm, vector X will be optimizedi,gWith experiment vector Ui,gAs target vector, and pass through
Object function calculates the corresponding fitness value of two class target vectors, using the less target vector of fitness value as follow-on excellent
Change vector Xi,g+1, which is a loop iteration process, i.e. i=1,2 ..., NP；
During fitness value calculation, according to target vector Xi,gCalculate the corresponding one-dimensional amplitude weighting vector A of Y-directionY, and
The one-dimensional amplitude weighting vector A of Z-directionZ, further according to AYAnd AZTwo dimensional amplitude weighting matrix A is calculated, and then computing array totally weights
Matrix W；According to W computing array directional diagram E, minor level SLL and null level NPL etc. are then calculated according to array pattern E
Pattern features, and by pattern features substitute into object function calculate target vector fitness value；
(5) will optimization vector matrix Xg+1The optimization vector of middle fitness value minimum is denoted as Xbest,g+1If Xbest,g+1Corresponding
Fitness value then returns to step (4) and carries out next round iteration optimization still greater than the fitness threshold value of setting；Otherwise terminate to optimize
Journey；According to optimum optimization vector Xbest,g+1Calculate corresponding overall weighting matrix W, as array numerical optimization final result into
Row output.
2. the directional diagram numerical optimization of circle bore planar array antenna as claimed in claim 1, it is characterised in that step
Suddenly in (3), using random device to XgInitialized, g=0 during initialization.
3. the directional diagram numerical optimization of circle bore planar array antenna as claimed in claim 1, it is characterised in that step
Suddenly described in (4) according to target vector X, calculate the one-dimensional amplitude weighting vector A of Z-directionZIt is as follows：
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Its a length of MF, by AZExpression formula as it can be seen that the amplitude weighting vector of Z-direction is symmetrical.
4. the directional diagram numerical optimization of circle bore planar array antenna as claimed in claim 1, it is characterised in that step
Suddenly in (4), according to target vector X, the one-dimensional amplitude weighting vector A of Y-direction is calculatedYIt is as follows：
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Its a length of NF, by AYExpression formula as it can be seen that the amplitude weighting vector of Y-direction is symmetrical.
5. the directional diagram numerical optimization of circle bore planar array antenna as claimed in claim 1, it is characterised in that step
Suddenly described in (4) according to AYAnd AZTwo dimensional amplitude weighting matrix A, and then computing array totality weighting matrix W are calculated, its step is such as
Under：
In Z-direction, it is for there are each row of antenna element, defining from the top unit to the length of bottom unitThen it is M by lengthFAZProgress linear interpolation obtains length and is'sAnd willUpper and lower two
End symmetrically mends 0, becomes the vector that a symmetrical length is M；
Similarly, in the Y direction, based on vector AYLinear interpolation and symmetrical zero padding are carried out, obtains corresponding to symmetrical per a line
Amplitude weighting vectorIts length is N；
Then the amplitude weighting matrix A for obtaining array is：
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</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Last array totality weighting matrix W calculates as follows：
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| CN111525285A (en) * | 2020-05-20 | 2020-08-11 | 西安黄河机电有限公司 | Thin-array antenna and design method thereof |
| CN111817766B (en) * | 2020-06-29 | 2023-05-26 | 西南电子技术研究所（中国电子科技集团公司第十研究所） | Beam forming method of combined array antenna unit directional diagram |
| CN111817766A (en) * | 2020-06-29 | 2020-10-23 | 西南电子技术研究所（中国电子科技集团公司第十研究所） | Beam forming method of combined array antenna unit directional diagram |
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| CN112016662B (en) * | 2020-08-21 | 2023-05-05 | 西安电子科技大学 | Array directional diagram synthesis method based on hybrid differential evolution algorithm and weighted total least square method |
| CN112542684B (en) * | 2020-12-15 | 2022-11-04 | 西安黄河机电有限公司 | Annular thin cloth antenna array design method and annular thin cloth antenna |
| CN112542684A (en) * | 2020-12-15 | 2021-03-23 | 西安黄河机电有限公司 | Annular thin cloth antenna array design method and annular thin cloth antenna |
| CN115470671A (en) * | 2022-09-01 | 2022-12-13 | 电子科技大学 | Optimization design method for enhancing directivity of any planar array end-fire wave beam |
| CN115470671B (en) * | 2022-09-01 | 2023-11-24 | 电子科技大学 | Optimal design method for enhancing directivity of end-fire beam of arbitrary planar array |
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