Simple Matrix Equation (SME) Method for Pattern Synthesis of Conformal Antenna Array with Arbitrary Architecture

. A novel noniterative method for pattern synthesis of conformal antenna array is put forward. First, a new pattern function formula for general conformal antenna array is derived. Ten, according to the new pattern function formula, a new block matrix equation (BME) and a simple matrix equation (SME) are obtained. SME has the same form as the equation of linear array pattern synthesis. Te new method can be applied to pattern synthesis of any conformal array. Moreover, due to SME having the same form as the matrix equation of linear array pattern synthesis, the new simple matrix equation can be dealt with by the approaches for linear array pattern synthesis, which signifcantly expands the existing approaches for conformal antenna array pattern synthesis. Two diferent conformal array confgurations are taken as the examples to demonstrate the advantages of the new method. Results of the simulations show that the new method can fexibly and efectively be applied to synthesize patterns for various conformal array architectures.


Introduction
Design and analysis of conformal array have always been a difcult work since conformal array was presented several decades ago for its complicated architecture.Some special and frequently used conformal array confgurations have been mainly explored by researchers.Te cylindrical conformal antenna arrays were studied in references [1,2].In references [3,4], the conical conformal antenna arrays were investigated.A method of radiation pattern computation of pyramidal conformal antenna array was developed in reference [5].In references [6][7][8][9], the spherical conformal antenna arrays were researched.
Te antenna elements of conformal array are mounted on the surface of the host and the shape of the surface was not allowed to be changed, which can make conformal array meet the requirements of a lot of special individual applications and also make every antenna element axis have diferent orientation.Hence, the conformal array's radiated beam pattern cannot be calculated from array beam pattern multiplication theorem-array element directivity function multiplying array factor.
Pattern synthesis of conformal antenna array is much more complex than linear antenna array, and the methods of conformal array pattern synthesis are much less than those for linear arrays.In references [10,11], Euler's rotation approach was utilized to generate conformal beam pattern.Based on Euler's rotation approach, rectangular microstrip patch antenna on spherical structure was analyzed [6], frequency-invariant pattern of conformal array antenna with low cross-polarisation was synthesized [12], an adaptive wideband beam forming algorithm was presented to form beam for conformal array [13], and optimum pattern synthesis of nonuniform spherical array was discussed in reference [14].Tree smart approaches were presented in the article [15][16][17] to synthesize pattern for conformal array.In reference [18], the constructive analytical phasing (CAP) method based on a new coordinate rotation formula was presented to calculate the pattern of conformal array.In references [19][20][21], the geometric algebra, fnite-diference time-domain (FDTD), and the moment method were developed for conformal array pattern synthesis.In reference [22], a multidimension to onedimension transformation approach was developed for the similar antenna array pattern synthesis.
Most of these methods mentioned above for conformal array pattern synthesis were based on Euler's rotation approach, in which the antenna element pattern was derived through its coordinate rotation operation and the process was complicated.All the solutions of these methods for conformal array were acquired by iterative computation formulas.Tese computation formulas for conformal array were diferent from those for linear array, and their computation processes were much more complicated.Moreover, lots of methods for linear array cannot be applied in the process for conformal array pattern synthesis.Tis situation signifcantly impacts the efectiveness of these methods for conformal array.Moreover, there is convergence problem in smart approaches.Te reported general success rate of smart methods is not more than 60%, and the success case of smart methods for the element number of array more than 50 has not been reported due to the huge complexity and a signifcant drop in the success rate.Te success rate drops very fast while the element number of array increases.
To overcome these problems, this article presents a new simple analytical pattern formula for conformal array pattern synthesis.A novel pattern synthesis block matrix formula for conformal array is put forward frst.Ten, a new method is presented to transform the matrix formula into a form similar to the matrix equation of the linear array, so all the abundant methods for linear array can be utilized to synthesize the beam pattern for conformal array, and in particular, many noniterative computation formulas, such as the least square method and the Fourier transform (FT) method, can be applied for conformal array pattern synthesis, which can signifcantly improve the diversity of the method of pattern synthesis for conformal array and reduce the amount of the computation.In addition, the new method is a general approach that can be applied to any conformal array architecture.
Te rest of the contents of this paper are organized as follows: in Section 2, new mathematical model and analytical formula for pattern synthesis of conformal array are presented; in Section 3, a new block matrix equation for pattern synthesis of conformal array is developed; in Section 4, a new simple matrix equation for pattern synthesis of conformal arrays is presented.It can be easily solved by using methods as similar as those for linear array pattern synthesis; examples are taken to demonstrate the efect of the new method in Section 5; and Section 6 draws conclusions.

Conformal Array Pattern Function Formula
Te confguration of an arbitrary conformal antenna array is considered, as shown in Figure 1.Assuming the element number of the array to be M, the array's m th antenna element location is (x m , y m , z m ), and the current excitation of the m th element is I m .Ten, the beam pattern function of the conformal array can be expressed as follows: wherein S(m, θ, φ) is the m th antenna element's pattern function, φ is the azimuth angle of a far feld point P(r, θ, φ), θ is the elevation angle of point P, r is the distance of point P, and F(θ, φ) is the beam pattern function of the conformal array.
Assuming that all antenna elements of the conformal antenna array are homogeneous but with diferent axis orientations and locations and that the m th antenna element's axis orientation is (β m , c m ), we set up a new local coordinate system for the m th antenna element with z' coordinate along its axis orientation (β m , c m ).In the m th antenna element's new local coordinate system, the space direction coordinate can be written as (θ m ′ , φ m ′ ).Te m th antenna element's pattern function is denoted as f(θ m ′ , φ m ′ ).According to Euler's right-handed rotation method in the xy-z rotation order with the rotation angles α m � 0, β m , and c m , in turn [10], θ m ′ and φ m ′ can be derived as follows: ( In far feld,  International Journal of Antennas and Propagation wherein λ is the wavelength of the radiated signal.Terefore, equation ( 1) can be re-expressed as follows: Let where in equation ( 6), the superscript T denotes transpose operation.Hence, equation ( 1) can be expressed as follows: Equation ( 8) is the pattern function of the conformal array.It can be transformed into a simple matrix equation to solve.
So, equation ( 8) can be re-expressed as follows: Tis is the matrix equation of conformal array pattern synthesis.But this equation is a block matrix equation (BME).

Simple Matrix Equation of Conformal Array Pattern Synthesis
Directly solving equation ( 12) is difcult; we transform this equation into an easily solved new equation and then solve it.Let Similarly, let therefore International Journal of Antennas and Propagation According to equation ( 12), we can get Let b � F(:, 1) Similarly, let Terefore, equation ( 17) can be re-expressed as follows: It is apparent that equation ( 20) is a general simple matrix equation, not a block matrix equation, and b is a column vector.Equation ( 20) has the similar form as the matrix equation of the linear array pattern synthesis.So, equation ( 20) can be solved by using the existing methods for linear array pattern synthesis.On the one hand, equation (20) can be used to calculate the beam pattern of conformal array from A and I. On the other hand, when the expected pattern b is prescribed, the antenna element excitation vector I can be solved from equation (20).Equation ( 20) is named as the simple matrix equation (SME) method for conformal array pattern synthesis.

Examples of the New Method
Two examples are taken to demonstrate the performance of the new method and its efect in this part.
Example 1.A cylinder conformal array confguration is considered in Figure 2. Tere are N � 30 elements equidistantly distributed as the distance λ/2 along a circle centered at the origin point O on the x-o-y plane with radius a � 1 m.Tis circle is the frst layer of the array.Te frst layer is parallelly moved up along + z coordinate axis direction to get other 9 layers of the cylinder array by (k − 1)λ/2 distance with k referring to the k th layer.It is worth mentioning that every other layer is a copy of the frst layer in every aspect aside from having diferent z coordinates.λ is the central wavelength of the signal transmitted by the array.φ is the azimuth angle of a far feld point P(r, θ, φ), θ is the elevation angle of point P, and r is the distance of point P from the origin point O.In the frst layer, it is easy to get the spherical coordinate location of the m th antenna element, that is, (a, π/2, 2π(m − 1)/N).Assuming that all antenna elements are homogeneous half wavelength patch dipole antennae but with diferent axis orientations and locations, let the m th antenna element of the frst layer axis orientation be (β m , c m ).We set up a new local coordinate system x'-y'-z' for the m th antenna element of the frst layer with z' coordinate along its axis orientation (β m , c m ).In the frst layer m th antenna element's new local coordinate system, the space direction coordinate can be written as (θ m ′ , φ m ′ ).According to Euler's right-handed rotation method in the x-y-z rotation order with the rotation angles α m � 0, β m � π/6, and c m � 2π(m − 1)/N in turn [10], θ m ′ and φ m ′ can be derived from equations ( 2) and (3).Te frst layer m th antenna element pattern function f(θ m ′ , φ m ′ ) can be expressed as follows: Because every other layer is a copy of the frst layer in every aspect aside from having diferent z coordinates, for m th antenna element of every layer in every layer's local coordinate system, the pattern function of the element is the same as equation (21).
Here, the target pattern b is generated by using MAT-LAB; the matrix A is obtained according to the new method mentioned above.After I has been obtained from equation (22), the pattern formed by I can be obtained from equation (20).
MATLAB software is applied to simulate this example.Te three-dimension (3D) fgures of the target pattern and the synthesized pattern using the new method are shown in Figures 3(a) and 3(b), respectively.
From Figure 3, it can be seen that the new method well accomplishes the beam pattern synthesis mission of the cylinder conformal array.Te location and the attenuation of the synthesized beam well accord with the target beam.
Te two-dimension (2D) fgures of simulation results are shown in Figures 4(a) and 4(b), where SME denotes the result of the simple matrix equation method, TARG refers to the target beam pattern, and CAP refers to the result of constructive analytical phasing (CAP) method in reference [18] as a comparison object.In order to further verify the new method, ANSYS HFSS software is applied to simulate the example.HFSS simulation results are also shown in Figures 4(a   ANSYS HFSS software is used to simulate this example with the signal wavelength λ � 4sin(π/N) � 7.3 mm.Te rogers5880 material substrate with thickness d 1 � 0.508 mm is used to bear the half wavelength dipole microwave patch antenna with the copper thickness d 2 � 35 μm.Te dielectric permittivity ε r of rogers5880 material is 2.2, and its loss tangent tanδ is 0.001.
From Figure 4, it can be seen that the result of the new method marked by SME has the higher attenuation than CAP and HFSS simulation results.Te diference between SME and HFSS should be due to the near feld mutual coupling among the antenna elements of the array.Te near feld mutual coupling will not be discussed here in order to focus on the new method.
Example 2. A semispherical conformal array confguration is considered and shown in Figure 5. Te semisphere is centered at the origin point O with radius a � 1 m.Let φ be the azimuth angle and θ be the elevation angle of a space direction.As shown in Figure 5, the benchmark line is the arc on the semisphere when θ from 0 to 90 °and φ � 0 °.Tere are N � 11 elements equidistantly distributed as the distance λ/2 along the benchmark line shown as 1 th to N th in Figure 5. Te equator circle is the circle on the semisphere and x-o-y plane.It is easy to see that the N th antenna element is on the equator circle.Te m th latitude circle on the semisphere parallels to the equator circle and goes through the m th antenna element on the benchmark line with m from 1 to N − 1.On the equator circle and all the latitude circles, there are antenna elements equidistantly distributed as the distance λ/2 along each circle.λ is the central wavelength of the signal transmitted by the array.Assuming all antenna elements are homogeneous half wavelength patch dipole antennae but with diferent axis orientations and locations, let the k th antenna element's spherical coordinates be (a, θ k , and φ k ).Assuming its axis orientation being (β k , c k ) and β k � θ k , c k � φ k + π/2.According to Euler's right-handed rotation method in the x-y-z rotation order with the rotation angles α k � 0, β k , and c k in turn, the array k th antenna element pattern function can be obtained from equation (22).
Similar to Example 1, the target pattern b is generated by using MATLAB; the matrix A is obtained according to the new method mentioned above.After I has been obtained from equation (22), the pattern formed by I can be obtained from equation (20).Te 3D fgures of the target pattern and the synthesized pattern using the new method are shown in Figures 6(a) and 6(b), respectively.From Figure 6, it can be learned that the new method well accomplishes the beam pattern synthesis mission of the semisphere conformal array.Te location and the attenuation of the synthesized beam well accord with the target beam.
Te 2D fgures of simulation results are shown in Figures 7(a) and 7(b), where SME denotes the result of the simple matrix equation method, TARG refers to the target beam pattern, and CAP refers to the result of constructive analytical phasing (CAP) method in reference [18] as a comparison object.ANSYS HFSS software simulation results are also shown in Figures 7(a   ANSYS HFSS software is still used to simulate this example with the signal wavelength λ � 4sin(π/4/N) � 5.4 mm.Te rogers5880 material substrate with thickness d 1 � 0.508 mm is used to bear the half wavelength dipole microwave patch antenna with the copper thickness d 2 � 35 μm.
From Figure 7, it can be seen that the results of the new method marked by SME have higher attenuation than CAP and HFSS simulation results.Te diference between SME and HFSS should be due to the near-feld mutual coupling    International Journal of Antennas and Propagation among the antenna elements of the array.Te near feld mutual coupling will not be discussed here in order to focus on the new method.Te SME method is a noniterative method.Te results of SME can be obtained through equation ( 22) straightly.Terefore, the new method has much less computation complexity than published conformal array pattern synthesis approaches that all are iterative methods.Both of the abovementioned examples are simulated on MATLAB software platform on a laptop with Intel i7-7700HQ and 16G memory.It takes no more than one minute to complete the simulation process in each example.Te computing process is swift.
Te results of the examples show that the new method has excellent efect for conformal array pattern synthesis with little amount of computation.In addition, the adjacent   International Journal of Antennas and Propagation spacing between the discrete values of θ and φ can be arbitrarily set wider to further reduce the amount of computation, and satisfactory results can also be achieved in the light of our experiments.

Conclusions
An analytical function formula is developed for conformal antenna array pattern synthesis by using Euler's rotation approach.By a new way, the function is discretized to a block matrix equation (BME).Based on the BME, a simple matrix equation method is developed for conformal array pattern synthesis.Due to the similar form with the matrix equation of the linear array pattern synthesis, SME can be solved by many approaches for linear array pattern synthesis.Te result of SME can be straightly obtained through a noniterative matrix equation.Terefore, the new method has much less computation complexity than published conformal array pattern synthesis approaches that all are iterative methods.Two examples are taken to demonstrate the efectiveness and fexibility of the new method.Te results of the simulations show that the new method is highly efcient with less amount of computation.

Figure 2 :
Figure 2: Architecture of the cylinder conformal array.
) and 4(b) with the data marked by HFSS.

Figure 3 :
Figure 3: 3D beam patterns of simulation results of cylinder array are as follows: (a) the target pattern and (b) the synthesized pattern of the new method.
) and 7(b) with the data marked by HFSS.

Figure 4 :
Figure 4: 2D beam patterns of simulation results of cylinder array and the comparison with CAP and HFSS simulation results are as follows: (a) φ � 60 °and (b) θ � 60 °.

Figure 6 :
Figure 6: 3D beam patterns of simulation results of semisphere array are as follows: (a) the target pattern and (b) the synthesized pattern of the new method.

Figure 7 :
Figure 7: 2D beam patterns of simulation results of semisphere array and the comparison with CAP and HFSS simulation results as follows: (a) φ � 60 °and (b) θ � 60 °.