Sparse Planar Array Synthesis Using Matrix Enhancement and Matrix Pencil

Abstract

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.

1. 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 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 𝑁 discrete similarly oriented identical elements arranged arbitrarily in a 3-D space, as shown in Figure 1. Denote the excitation in the 𝑖th element located at (𝑟_𝑖,𝜃_𝑖,𝜙_𝑖) as 𝑅_𝑖.

Figure 1 

Reference coordinates of an arbitrary array.

According to the superposition theorem of electromagnetic waves, we can write the following array factor [12]: 𝐹𝜃,𝜑=^𝑁_𝑖=1𝑅_𝑖exp𝑗𝜙_𝑖+𝑘𝑟_𝑖cos𝛼_𝑖, (1) where 𝑘=2𝜋/𝜆, 𝜆 is the wavelength. Consider, cos𝛼_𝑖=sin𝜃sin𝜃_𝑖cos𝜙−𝜙_𝑖+cos𝜃cos𝜃_𝑖, (2) where 0≤𝜃≤𝜋, and 0≤𝜙≤2𝜋 are the azimuth and elevation, respectively.

For the planar array, (1) can be simplified as 𝐹(𝑢,𝑣)=^𝑁_𝑖=1𝑅_𝑖exp𝑗𝑘𝑑_𝑥𝑢+𝑑_𝑦𝑣, (3) where 𝑑_𝑥 and 𝑑_𝑦 are the abscissa and ordinate of the 𝑖th element, and 𝑢=sin𝜃cos𝜙,𝑣=sin𝜃sin𝜙.

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 {𝑄}𝐹()−minst.Min𝑢,𝑣^𝑄_𝑖=1𝑅^′_𝑖exp𝑗𝑘𝑑^′_𝑥𝑢+𝑑^′_𝑦𝑣≤𝜀, (4) where 𝑅^′_𝑖, 𝑑^′_𝑥, and 𝑑^′_𝑦 are the excitation, abscissa, and ordinate of the new planar array. 𝐿=2 because this paper uses the total least squares criteria.

Firstly, to sample the desired pattern from 𝑢=−1, 𝑣=−1 to 𝑢=1, 𝑣=1 uniformly, then 𝑢_𝑚=𝑚Δ=𝑚/𝑁, 𝑚=−𝑁,…,0,…,𝑁; 𝑣_𝑛=𝑛Δ=𝑛/𝑁, 𝑛=−𝑁,…,0,…,𝑁, which has (2𝑁+1)(2𝑁+1) sampling points.

The value of any sampling point is 𝑓(𝑚,𝑛)=^𝑁_𝑖=1𝑅_𝑖𝑦^𝑚_𝑖𝑧^𝑛_𝑖, (5) where 𝑦_𝑖=𝑒^{𝑗𝑘𝑑𝑥Δ} and 𝑧_𝑖=𝑒^{𝑗𝑘𝑑𝑦Δ}.

Secondly, to construct an enhanced matrix 𝐗_𝑒 using the sampling point data, 𝐗_𝑒=𝐗₀𝐗₁⋯𝐗_{2𝑁−𝐾+1}𝐗₁𝐗₂⋯𝐗_{2𝑁−𝐾+2}𝐗⋮⋮⋮_𝐾−1𝐗_𝐾⋯𝐗_2𝑁, (6) where 𝐗_𝑚=𝑥()𝑥()()𝑥()𝑥()()𝑥()𝑥()()𝑚;0𝑚;1⋯𝑥𝑚;2𝑁−𝐿+1𝑚;1𝑚;2⋯𝑥𝑚;2𝑁−𝐿+2⋮⋮⋮𝑚;𝐿−1𝑚;𝐿⋯𝑥𝑚;2𝑁, (7) where 𝑥(𝑚;𝑛)=𝑓(𝑚−𝑁;𝑛−𝑁). 𝐗_𝑒 is a Hankel block matrix and 𝐗_𝑚 is a Hankel matrix. The choice of parameters {𝐾,𝐿} must satisfy the following conditions: ()𝐾()()𝐾−1𝐿≥𝑁,𝐿−1≥𝑁,2𝑁+2−𝐾)(2𝑁+2−𝐿≥𝑁. (8)

Thirdly, to perform the singular value decomposition (SVD) of the enhanced matrix 𝐗_𝑒, 𝐗_𝑒=^min_𝑖=1𝜎_𝑖𝐮_𝑖𝐯^𝐻_𝑖=𝐔_𝑠Σ_𝑠𝐕^𝐻_𝑠+𝐔_𝑛Σ_𝑛𝐕^𝐻_𝑛 (9)min=min{𝐾𝐿,(2𝑁+2−𝐾)(2𝑁+2−𝐿)} which is the smaller dimension of 𝐗_𝑒, and 𝐔_𝑠, Σ_𝑠, and 𝐕_𝑆 contain the 𝑁 principal eigenvalues and eigenvectors. 𝐔_𝑛, Σ_𝑛, and 𝐕_𝑛 contain the remaining eigenvalues and eigenvectors. Specifically,𝐔_𝑠=𝐮₁,𝐮₂,…,𝐮_𝑁,Σ_𝑠=diag𝜎₁,𝜎₁,…,𝜎_𝑁,𝐕_𝑆=𝐯₁,𝐯₂,…,𝐯_𝑁,𝐔_𝑛=𝐮_𝑁+1,𝐮_𝑁+2,…,𝐮_min,Σ_𝑛=diag𝜎_𝑁+1,𝜎_𝑁+2,…,𝜎_min,𝐕_𝑛=𝐯_𝑁+1,𝐯_𝑁+2,…,𝐯_min, (10a)(10b)(10c)(10d)(10e)(10f) where 𝜎₁≥𝜎₂≥⋯≥𝜎_min. The rank of enhanced matrix 𝐗_𝑒 is equal to the number of nonzero singular values. Generally, an array of 𝑁 elements has 𝑁 non-zero singular values; that is, 𝜎_𝑖>0(𝑖=1,…,𝑁), 𝜎_𝑖=0(𝑖>𝑁+1). So, Σ_𝑛 is equal to zero. Equation (9) can be simplified to 𝐗_𝑒=𝐔_𝑠Σ_𝑠𝐕^𝐻_𝑠. (11)

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, 𝐗_𝑄=𝐔_𝑠Σ_𝑄𝐕^𝐻_𝑠, (12) where Σ_𝑄=diag{𝜎₁,𝜎₁,…,𝜎_𝑄,0,…,0}, 𝑄≤𝑁.

In the actual synthesis of array, the minimum value of 𝑄 can be determined by the following formula [11]: 𝑄=min𝑞;∣∑^𝑁_𝑖=𝑞+1𝜎²_𝑖∑^𝑞_𝑖=1𝜎²_𝑖<𝜀, (13) where 𝜀 is a small positive number, and the choice of 𝜀 depends on the approximation between the reconstructed pattern and the original pattern.

3.2. The Solutions of Eigenvalues 𝑦_𝑖 and 𝑧_𝑖

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 𝑦_𝑖 and 𝑧_𝑖.

3.2.1. Extracting 𝑦_𝑖

Once the low rank matrix 𝐗_𝑄 is obtained, to solve the eigenvalue 𝑦_𝑖 is equal to solve the generalized eigenvalue of the following equation: 𝐗_𝑄,𝑓−𝑦𝐗_𝑄,𝑙𝐯=0, (14) where 𝐗_𝑄,𝑓 and 𝐗_𝑄,𝑙 are obtained by deleting the first 𝐿 rows and the last 𝐿 rows of 𝐗_𝑄. To solve (14) is equal to solve the generalized eigenvalue of the following equation: 𝐔₂−𝑦𝐔₁𝐯=0, (15) where 𝐔₂ and 𝐔₁ are obtained by deleting the first 𝐿 rows and the last 𝐿 rows of 𝐔_𝑄 which contains only 𝑄 dominant left-singular vectors of 𝐔_𝑠 in (11).

Therefore, it can be obviously observed that the matrix pencil 𝐔₂−𝑦𝐔₁ can be expressed as 𝐔₂−𝑦𝐔₁=𝐄₁𝐘_𝑑−𝑦𝐈𝐓, (16) where 𝐘_𝑑 is a diagonal matrix composed of the eigenvalues of {𝑦_𝑖,𝑖=1,…,𝑄}.

3.2.2. Extracting 𝑧_𝑖

In order to extract another set of {𝑧_𝑖,𝑖=1,…,𝑄}, firstly introduce a permutation matrix 𝐏 as 𝐏=𝐩^𝑇(1),𝐩^𝑇(1+𝐿),…,𝐩^𝑇(()𝐿),𝐩1+𝐾−1_𝑇(2),𝐩^𝑇()𝐩2+𝐿,…,_𝑇(1+(𝐾−1)𝐿),…,…,𝐩^𝑇(𝐿),𝐩^𝑇(𝐿+𝐿),…,𝐩^𝑇(1+(𝐾−1)𝐿)]^𝑇. (17)

The element 𝐩(𝑖) is a 𝐾𝐿×1 vector with one at the 𝑖th position and zero everywhere else. Left multiplying 𝐔_𝑠 by 𝐏, we can get 𝐔_𝑠𝑃=𝐏𝐔_𝑠. (18)

According to (15), we can conclude that to solve the eigenvalues 𝑧_𝑖 is equal to solve the generalized eigenvalues of the following equation: 𝐔_2𝑃−𝑧𝐔_1𝑃𝐯=0, (19) where 𝐔_2𝑃 and 𝐔_1𝑃 are obtained by deleting the first 𝐾 rows and the last 𝐾 rows of 𝐔_𝑄𝑃 which contains only 𝑄 dominant left-singular vectors of 𝐔_𝑠𝑃 in (18).

Therefore, it can be obviously observed that the matrix pencil 𝐔_2𝑃−𝑧𝐔_1𝑃 can be expressed as 𝐔_2𝑃−𝑧𝐔_1𝑃=𝐄_1𝑃𝐙_𝑑−𝑧𝐈𝐓, (20) where 𝐙_𝑑 is a diagonal matrix composed of the eigenvalues of {𝑧_𝑖,𝑖=1,…,𝑄}.

3.2.3. Pairing

According to (16) and (20), we can deduce that 𝐅₁=𝐔⁺₁𝐔₂=𝐓⁻¹𝐘_𝑑𝐅𝐓,₂=𝐔⁺_1𝑃𝐔_2𝑃=𝐓⁻¹𝐙_𝑑𝐓. (21) 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 𝐅₁ and 𝐅₂ is computed to obtain a transformation diagonalizing of these two matrixs as follows: 𝛽𝐅₁+1−𝛽𝐅₁=𝐓⁻¹𝐃𝐓. (22) The transformation 𝐓 is applied to 𝐅₁ and 𝐅₂ to obtain the matrixs: 𝐘_𝑑=𝐓𝐅₁𝐓⁻¹,𝐙_𝑑=𝐓𝐅₂𝐓⁻¹. (23) The values in the diagonal position of 𝐘_𝑑 and 𝐙_𝑑 determine the pairing.

3.3. To Solve Elements’ Excitations and Locations

Once 𝑦_𝑖 and 𝑧_𝑖 are obtained, the locations can be solved according to (14) in the literature [13]; that is, 𝑑^′_𝑥=1𝑦𝑗𝑘Δln_𝑖,𝑑^′_𝑦=1𝑧𝑗𝑘Δln_𝑖. (24)

As pointed in [9], a variety of synthetical examples show that, for the symmetrical pencil beam pattern, the imaginary parts of 𝑑^′_𝑖 are much smaller than the real parts. For example, the imaginary parts of 𝑑^′_𝑖 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 𝑑^′_𝑖 directly. The elements’ locations are calculated as follows: 𝑑^′_𝑥=1𝑦𝑗𝑘Δln_𝑖,𝑑^′_𝑦=1𝑧𝑗𝑘Δln_𝑖, (25) where 𝑦_𝑖=𝑦_𝑖𝑦_𝑖,𝑧_𝑖=𝑧_𝑖𝑧_𝑖. (26)

The elements’ excitations can be obtained as follows: 𝐑=𝐄⁺_𝐿𝐗_𝑒𝐄⁺_𝑅. (27) The diagonal elements of matrix 𝐑 are 𝑅_𝑖(𝑖=1,…,𝑄) where 𝐄⁺_𝐿=𝐄^𝐻_𝐿𝐄_𝐿⁻¹𝐄^𝐻_𝐿,𝐄⁺_𝑅=𝐄^𝐻_𝑅𝐄_𝑅𝐄^𝐻_𝑅⁻¹, (28) where 𝐄_𝐿=𝐙_𝐿𝐘_𝑑^−𝑁𝐙_𝐿𝐘_𝑑^−𝑁+1⋮𝐙_𝐿𝐘_𝑑^{𝐾−𝑁−1},𝐄_𝑅=𝐙_𝑅,𝐘_𝑑𝐙_𝑅,…,𝐘_𝑑^{2𝑁−𝐾+1}𝐙_𝑅, (29) where 𝐙_𝐿=𝑧₁^−𝑁𝑧₂^−𝑁⋯𝑧_𝑄^−𝑁𝑧₁^−𝑁+1𝑧₂^−𝑁+1⋯𝑧_𝑄^−𝑁+1𝑧⋮⋮⋮_{1𝐿−1−𝑁}(𝑧₂)^{𝐿−1−𝑁}⋯𝑧_𝑄^{𝐿−1−𝑁}𝐲_𝑑,=diag𝑦₁,𝑦₂,…,𝑦_𝑄𝐙_𝑅,=1𝑧₁⋯𝑧₁^{2𝑁−𝐿+1}1𝑧₂⋯𝑧₂^{2𝑁−𝐿+1}1𝑧⋮⋮⋮_𝑄⋯𝑧_𝑄^{2𝑁−𝐿+1}. (30)(31)(32)

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 𝐗_𝑒 by the sampling data; 𝐗_𝑒 is shown as (6).

Step 2. To perform the singular value decomposition (SVD) of enhanced matrix 𝐗_𝑒 and to calculate the singular values and left singular vector 𝐔_𝑠.

Step 3. According to (13), to determine the minimum number of elements 𝑄.

Step 4. According to (16) and (20), to extract the eigenvalues 𝑦_𝑖 and 𝑧_𝑖, then to pair them as (22).

Step 5. According to (25) and (27), to solve the elements’ excitations and locations.

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 6×6 elements. The elements are arranged in a rectangular grid, 𝑑_𝑥=𝑑_𝑦=𝜆/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 𝐾=𝐿=37. According to (13), the required minimum number of elements is 𝑄=25 when 𝜀=10⁻⁵. 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 𝑄=25 unequally spaced distribution of the elements can basically reconstruct the equally spaced distribution of the elements 𝑁=36. 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.

Table 1 

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

(a)

(b)

Figure 2 

(a) The desired radiation pattern. (b) The reconstructed radiation pattern.

Figure 3 

The cross-section of the pattern.

Figure 4 

The elements’ location.

5.2. Synthesis of Chebyshev Planar

Suppose that there is a chebyshev planar array with 6×6 elements. The elements are arranged in a rectangular grid, 𝑑_𝑥=𝑑_𝑦=𝜆/2, and the required sidelobe level is −20 dB. In this example, there are (2×36+1)(2×36+1) sampling points, and the parameters 𝐾=𝐿=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 −19.3058 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.

Table 2 

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

(a)

(b)

Figure 5 

(a) The Chebyshev radiation pattern. (b) The reconstructed radiation pattern.

Figure 6 

The cross section of the pattern.

Figure 7 

The elements’ locations.

6. Conclusion

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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (nos. 60736045 and 60702070).