Sparse Planar Array Synthesis Using Matrix Enhancement and Matrix Pencil

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Introduction
The question of nonuniform planar array synthesis with fewer number of elements for a desired beam pattern has been a popular concern since early in the 1970s.Reducing the number of elements in planar arrays has long been a problem both practical and theoretical significance being the subject of concern; for example, satellite communication system has limitation on the weight of antenna systems and subsystem accessories which wishes to use the minimum number of elements.Furthermore, fewer elements can lower the cost obviously and simplify the antenna system.
The synthesis of nonuniform array elements' positions, excitations, and phases is a complicated nonlinear problem which contains a number of decision variables.So far, there have been many technologies used to synthesize the nonuniform arrays, including optimization algorithm (such as genetic algorithm (GA) [1], differential evolution algorithm (DEA) [2], analytical methods [3,4], matrix pencil method (MPM) [5], dynamic programming, and particle swarm optimization (PSO)) and other comprehensive technologies.
However, some optimization algorithms such as GA, DEA, and PSO are suitable for solving nonlinear optimization of high dimension complex function, but they are very time consuming for their intrinsic iterative mechanism.Many other comprehensive approaches based on iteration do not guarantee that all variables are global optimal solutions.The new non iterative methods, called MPM [6,7] and MEMP [8], have been successfully used by Yingbo Hua, Tapan K. S. and Odilon Pererira in the parameters estimation of a sum of complex exponentials, such as two-dimensional frequencies [8], azimuth/elevation angel [9], and two-dimensional DOA [10].These years Liu and others find a new application in the synthesis of nonuniform array for MPM [5,11].
MPM is qualified to the synthesis of sparse planar array which is designed to a separable planar array by a product of two orthogonal linear sparse arrays [5].It can guarantee the best features in the two principal plane patterns in this particular case, but it cannot at the same time guarantee that others have the same characteristics.
Further expansion of MPM to the synthesis of inseparable distribution sparse planar array in this paper, just as in signal International Journal of Antennas and Propagation processing applications, the new method similarly called matrix enhancement and matrix pencil (MEMP) in synthesis of sparse arrays.The organization of this paper is listed as follows.In Section 2, the optimal array model is presented.In Section 3, the idea of matrix enhancement and matrix pencil is introduced, and then the method will be applied to estimate the minimum number of elements and to calculate the locations and excitations of the new array.Section 4 introduces the essential steps of MEMP.Some numerical results are presented in Section 5 to show the efficiency of the proposed method.Finally, some conclusions are drawn in Section 6.

The Optimal Array Model
Assume an array of  discrete similarly oriented identical elements arranged arbitrarily in a 3-D space, as shown in Figure 1.Denote the excitation in the th element located at (  ,   ,   ) as   .
According to the superposition theorem of electromagnetic waves, we can write the following array factor [12]: where  = 2/,  is the wavelength.Consider, where 0 ≤  ≤ , and 0 ≤  ≤ 2 are the azimuth and elevation, respectively.For the planar array, (1) can be simplified as where   and   are the abscissa and ordinate of the th element, and  = sin  cos , V = sin  sin .

MEMP to Synthesize Planar Array
where    ,    , and    are the excitation, abscissa, and ordinate of the new planar array. = 2 because this paper uses the total least squares criteria.

𝑥 (𝑚
where (; ) = (−; −).X  is a Hankel block matrix and X  is a Hankel matrix.The choice of parameters {, } must satisfy the following conditions: Thirdly, to perform the singular value decomposition (SVD) of the enhanced matrix X  , min = min{, (2+2−)(2+2−)} which is the smaller dimension of X  , and U  , Σ  , and V  contain the  principal eigenvalues and eigenvectors.U  , Σ  , and V  contain the remaining eigenvalues and eigenvectors.Specifically, where The rank of enhanced matrix X  is equal to the number of nonzero singular values.Generally, an array of  elements has  non-zero singular values; that is,   > 0 ( = 1, . . ., ),   = 0 ( >  + 1).So, Σ  is equal to zero.Equation ( 9) can be simplified to However, some literature results [13] show that the number of important singular values is much smaller than the total number of array elements, that is the unimportant elements' contributions can be replaced by a combination of other elements.Therefore, we can discard the unimportant singular values to get a low-rank approximation of the enhanced matrix, which corresponds to the new array with fewer elements.The usual approach is to set these small singular values to zero.That is, where Σ  = diag{ 1 ,  1 , . . .,   , 0, . . ., 0},  ≤ .
In the actual synthesis of array, the minimum value of  can be determined by the following formula [11]: where  is a small positive number, and the choice of  depends on the approximation between the reconstructed pattern and the original pattern.

The Solutions of Eigenvalues 𝑦
and   .The eigenvalues are obtained by constructing two matrixs and solving the generalized eigenvalue of the matrixes [7,8], then utilizing the ESPRIT algorithm in the literature [14] and pairing algorithm to estimate the correct pairing of   and   .

Extracting 𝑦 𝑖 .
Once the low rank matrix X  is obtained, to solve the eigenvalue   is equal to solve the generalized eigenvalue of the following equation: where X , and X , are obtained by deleting the first  rows and the last  rows of X  .To solve ( 14) is equal to solve the generalized eigenvalue of the following equation: where U 2 and U 1 are obtained by deleting the first  rows and the last  rows of U  which contains only  dominant leftsingular vectors of U  in (11).Therefore, it can be obviously observed that the matrix pencil U 2 − U 1 can be expressed as where Y  is a diagonal matrix composed of the eigenvalues of {  ,  = 1, . . ., }.

Extracting
The element p() is a  × 1 vector with one at the th position and zero everywhere else.Left multiplying U  by P, we can get According to (15), we can conclude that to solve the eigenvalues   is equal to solve the generalized eigenvalues of the following equation: where U 2 and U 1 are obtained by deleting the first  rows and the last  rows of U  which contains only  dominant left-singular vectors of U  in (18).Therefore, it can be obviously observed that the matrix pencil U 2 − U 1 can be expressed as where Z  is a diagonal matrix composed of the eigenvalues of {  ,  = 1, . . ., }.

Pairing.
According to ( 16) and (20), we can deduce that It can be seen from ( 21) that the pairing information of the eigenvalues is included in T. The eigenvalue decomposition of a linear combination of F 1 and F 2 is computed to obtain a transformation diagonalizing of these two matrixs as follows: The transformation T is applied to F 1 and F 2 to obtain the matrixs: The values in the diagonal position of Y  and Z  determine the pairing.

To Solve Elements' Excitations and Locations.
Once   and   are obtained, the locations can be solved according to (14) in the literature [13]; that is, As pointed in [9], a variety of synthetical examples show that, for the symmetrical pencil beam pattern, the imaginary parts of    are much smaller than the real parts.For example, the imaginary parts of    are ten orders smaller than the real parts for the synthesis of traditional Chebyshev planar pattern.Under this case, we can use the real parts of    directly.The elements' locations are calculated as follows: The elements' excitations can be obtained as follows: The diagonal elements of matrix R are   ( = 1, . . ., ) where where ] , where

The Steps of MEMP
The essential steps of MEMP are summarized as follows.
Step 1.To sample the desired pattern uniformly and to construct an enhanced matrix X  by the sampling data; X  is shown as (6).
Step 2. To perform the singular value decomposition (SVD) of enhanced matrix X  and to calculate the singular values and left singular vector U  .
Step 3.According to (13), to determine the minimum number of elements .
Step 4. According to ( 16) and (20), to extract the eigenvalues   and   , then to pair them as (22).

Simulation Results
In order to illustrate the effectiveness of MEMP, this paper gives two examples to reduce the number of elements and let the reconstructed array keep the characteristics of the original array.

Synthesis of Planar Array with Excitation of One.
Suppose, there is a rectangular planar array with 6 × 6 elements.The elements are arranged in a rectangular grid,   =   = /2, and the elements' excitations are one.In this example, there are (2×36+1)(2×36+1) sampling points (by further increasing the sampling points will not improve the approximate accuracy of the pattern).The parameters  =  = 37.According to (13), the required minimum number of elements is  = 25 when  = 10 −5 .The desired pattern and the reconstructed pattern are shown in Figure 2. The obtained sidelobe levels are −13.1525dB and −13.1522 dB, respectively.Figure 3 compares the cross-section of the reconstructed pattern and the desired pattern.The figure shows that the array with  = 25 unequally spaced distribution of the elements can basically   =  = 37.Chebyshev planar array has the limitation of the number of elements in every row which is equal to the number of elements in every column which so, we use 5 × 5 elements to approximate the desired Chebyshev planar array.The desired pattern and the reconstructed pattern are shown in Figure 5.The obtained sidelobe level is −19.3058dB. Figure 6 compares the cross section of the reconstructed pattern with the desired pattern.Although there is a slight elevation (0.694 dB) of the sidelobe level, we can save 30% elements.The elements' locations of the equally and unequally spaced are shown in Figure 7.The elements' excitations and locations of equally and unequally spaced arrays are shown in Table 2.

Conclusion
A synthesis method to sparse planar array which is based on MEMP is presented.Compared to stochastic optimization International Journal of Antennas and Propagation 7 algorithm, MEMP is a non-iterative algorithm, which is suitable for designing of sparse array with the requirements of narrow beam, low sidelobe level.In addition, compared to MPM, MEMP can guarantee the best features of either plane.The synthesis of planar array in the paper does not consider the mutual coupling [15,16] between elements.But there exists mutual coupling between actual elements.Therefore, the synthesis results gotten under the condition of unconsidering mutual coupling can be solved by adjusting the elements' excitations or locations.The discussion of this paper enriches the application of MEMP to the synthesis of planar array and provides a valuable reference in engineering applications dealing with complexity of problems.

Figure 1 :
Figure 1: Reference coordinates of an arbitrary array.

Table 1 :
The locations (excitations) of uniformly and nonuniformly spaced arrays.

Table 2 :
The locations (excitations) of Chebyshev and nonuniformly spaced array.