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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) [

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 [

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

Assume an array of

Reference coordinates of an arbitrary array.

According to the superposition theorem of electromagnetic waves, we can write the following array factor [

For the planar array, (

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

Firstly, to sample the desired pattern from

The value of any sampling point is

Secondly, to construct an enhanced matrix

Thirdly, to perform the singular value decomposition (SVD) of the enhanced matrix

However, some literature results [

In the actual synthesis of array, the minimum value of

The eigenvalues are obtained by constructing two matrixs and solving the generalized eigenvalue of the matrixes [

Once the low rank matrix

Therefore, it can be obviously observed that the matrix pencil

In order to extract another set of

The element

According to (

Therefore, it can be obviously observed that the matrix pencil

According to (

The eigenvalue decomposition of a linear combination of

Once

As pointed in [

The elements’ excitations can be obtained as follows:

The essential steps of MEMP are summarized as follows.

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.

Suppose, there is a rectangular planar array with

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

The locations of uniformly spaced array | The locations (excitations) of nonuniformly spaced array | ||||
---|---|---|---|---|---|

0.25, 1.25 | 0.75, 1.25 | 1.25, 1.25 | 0, 1.2308 (1.4579) | 0.6423, 1.2308 (1.3936) | 1.2308, 1.2308 (1.2226) |

0.25, 0.75 | 0.75, 0.75 | 1.25, 0.75 | 0, 0.6423 (1.6619) | 0.6423, 0.6423 (1.5886) | 1.2308, 0.6423 (1.3936) |

0.25, 0.25 | 0.75, 0.25 | 1.25, 0.25 | 0, 0 (1.7386) | 0.6423, 0 (1.6619) | 1.2308, 0 (1.4579) |

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

The cross-section of the pattern.

The elements’ location.

Suppose that there is a chebyshev planar array with

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

The locations (excitations) of Chebyshev planar array |
||||

| ||||

0.25, 1.25 (0.3654) | 0.75, 1.25 (0.1822) | 1.25, 1.25 (0.0364) | ||

0.25, 0.75 (0.2640) | 0.75, 0.75 (0.3917) | 1.25, 0.75 (0.1822) | ||

0.25, 0.25 (0.4503) | 0.75, 0.25 (0.2640) | 1.25, 0.25 (0.3645) | ||

| ||||

The locations (excitations) of nonuniform planar array | ||||

| ||||

−1.2319, −1.2463 (0.0538) | −0.6505, −1.2237 (0.3215) | −0.0914, −1.1991 (0.7001) | 0.3336, −1.1513 (0.8382) | 0.9598, −0.9605 (0.5520) |

−1.2237, −0.7295 (0.2870) | −0.6472, −0.6505 (0.6144) | 0, −0.6238 (0.6207) | 0.6472, −0.5680 (0.9560) | 1.2237, −0.3341 (0.5179) |

−1.1513, −0.2376 (0.6953) | −0.5680, 0 (0.9462) | 0, 0 (0.7766) | 0.5680, 0 (0.8475) | 1.1513, 0.2376 (0.3763) |

−1.1981, 0.3341 (0.6218) | −0.5971, 0.6505 (0.6486) | 0, 0.5680 (0.8748) | 0.5971, 0.7295 (0.8196) | 1.1981, 0.6238 (0.6182) |

−0.9598, 0.9605 (0.5523) | −0.3336, 1.2237 (0.6272) | 0.0914, 1.1991 (0.7050) | 0.6505, 1.1513 (0.4298) | 1.2319, 1.2463 (0.0526) |

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

The cross section of the pattern.

The elements’ locations.

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 [

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