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International Journal of Antennas and Propagation
Volume 2013 (2013), Article ID 147097, 7 pages
Sparse Planar Array Synthesis Using Matrix Enhancement and Matrix Pencil
1School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
2The Second Research Institute of Civil Aviation of China, Chengdu 610041, China
3School of Astronautics and Aeronautics, University of Electronic Science and Technology of China, Chengdu 611731, China
Received 3 July 2012; Accepted 28 January 2013
Academic Editor: Tat Yeo
Copyright © 2013 Mei-yan Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The matrix enhancement and matrix pencil (MEMP) plays important roles in modern signal processing applications. In this paper, MEMP is applied to attack the problem of two-dimensional sparse array synthesis. Firstly, the desired array radiation pattern, as the original pattern for approximating, is sampled to form an enhanced matrix. After performing the singular value decomposition (SVD) and discarding the insignificant singular values according to the prior approximate error, the minimum number of elements can be obtained. Secondly, in order to obtain the eigenvalues, the generalized eigen-decomposition is employed on the approximate matrix, which is the optimal low-rank approximation of the enhanced matrix corresponding to sparse planar array, and then the ESPRIT algorithm is utilized to pair the eigenvalues related to each dimension of the planar array. Finally, element positions and excitations of the sparse planar array are calculated according to the correct pairing of eigenvalues. Simulation results are presented to illustrate the effectiveness of the proposed approach.
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) , differential evolution algorithm (DEA) , analytical methods [3, 4], matrix pencil method (MPM) , 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 , 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 , azimuth/elevation angel , and two-dimensional DOA . 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 . 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 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.
2. 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 : where , is the wavelength. Consider, where , and are the azimuth and elevation, respectively.
For the planar array, (1) can be simplified as where and are the abscissa and ordinate of the element, and .
3. MEMP to Synthesize Planar Array
3.1. Estimation of the Minimum Number of Elements
Matrix enhancement and matrix pencil is to use as few elements as possible to form a new planar array to approximate the desired pattern. Thus, the optimal mathematical is described as where , , and are the excitation, abscissa, and ordinate of the new planar array. because this paper uses the total least squares criteria.
Firstly, to sample the desired pattern from , to , uniformly, then , ; , , which has sampling points.
The value of any sampling point is where and .
Secondly, to construct an enhanced matrix using the sampling point data, where where . is a Hankel block matrix and 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 , which is the smaller dimension of , and , , and contain the principal eigenvalues and eigenvectors. , , and contain the remaining eigenvalues and eigenvectors. Specifically, where . The rank of enhanced matrix is equal to the number of nonzero singular values. Generally, an array of elements has non-zero singular values; that is, , . So, is equal to zero. Equation (9) can be simplified to
However, some literature results  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 , .
In the actual synthesis of array, the minimum value of can be determined by the following formula : where is a small positive number, and the choice of depends on the approximation between the reconstructed pattern and the original pattern.
3.2. The Solutions of Eigenvalues 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  and pairing algorithm to estimate the correct pairing of and .
Once the low rank matrix is obtained, to solve the eigenvalue is equal to solve the generalized eigenvalue of the following equation: where and are obtained by deleting the first rows and the last rows of . To solve (14) is equal to solve the generalized eigenvalue of the following equation: where and are obtained by deleting the first rows and the last rows of which contains only dominant left-singular vectors of in (11).
Therefore, it can be obviously observed that the matrix pencil can be expressed as where is a diagonal matrix composed of the eigenvalues of .
In order to extract another set of , firstly introduce a permutation matrix as
The element is a vector with one at the position and zero everywhere else. Left multiplying by , 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 and are obtained by deleting the first rows and the last rows of which contains only dominant left-singular vectors of in (18).
Therefore, it can be obviously observed that the matrix pencil can be expressed as where is a diagonal matrix composed of the eigenvalues of .
The eigenvalue decomposition of a linear combination of and is computed to obtain a transformation diagonalizing of these two matrixs as follows: The transformation is applied to and to obtain the matrixs: The values in the diagonal position of and determine the pairing.
3.3. To Solve Elements’ Excitations and Locations
As pointed in , 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: where
The elements’ excitations can be obtained as follows: The diagonal elements of matrix are where where where
4. 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 by the sampling data; is shown as (6).
Step 2. To perform the singular value decomposition (SVD) of enhanced matrix and to calculate the singular values and left singular vector .
Step 3. According to (13), to determine the minimum number of elements .
5. 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.
5.1. Synthesis of Planar Array with Excitation of One
Suppose, there is a rectangular planar array with elements. The elements are arranged in a rectangular grid, , and the elements’ excitations are one. In this example, there are sampling points (by further increasing the sampling points will not improve the approximate accuracy of the pattern). The parameters . According to (13), the required minimum number of elements is when . The desired pattern and the reconstructed pattern are shown in Figure 2. The obtained sidelobe levels are −13.1525 dB 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 unequally spaced distribution of the elements can basically reconstruct the equally spaced distribution of the elements . The example is to save 30% of the array elements. The elements’ locations of the equally and unequally spaced are shown in Figure 4. The equally spaced elements’ locations and the unequally spaced elements’ excitations and locations are shown in Table 1. Because the locations and excitations are symmetrically distributed, only the locations and excitations in the first quadrant are given.
5.2. Synthesis of Chebyshev Planar
Suppose that there is a chebyshev planar array with elements. The elements are arranged in a rectangular grid, , and the required sidelobe level is −20 dB. In this example, there are sampling points, and the parameters . 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 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 −19.3058 dB. 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.
A synthesis method to sparse planar array which is based on MEMP is presented. Compared to stochastic optimization 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.
This work was supported by the National Natural Science Foundation of China (nos. 60736045 and 60702070).
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