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Reducing the Number of Elements in Multiple-Pattern Linear Arrays by the Extended Matrix Pencil Methods Yanhui Liu, Qing Huo Liu, Fellow, IEEE, and Zaiping Nie, Fellow, IEEE
Abstract—Previously, the matrix pencil method (MPM) and the forward-backward MPM (FBMPM) were used to effectively reduce the number of antenna elements in the single-pattern linear arrays. This work extends the MPM and FBMPM-based synthesis methods to the synthesis of multiple-pattern linear arrays with a smaller number of elements. The extended MPM (resp., the extended FBMPM) method organizes all the multiple pattern data into a composite Hankel (resp., composite Hankel-Toeplitz) matrix from which the minimum number of elements and the common poles corresponding to element positions can be obtained with similar processing used in the original MPM or FBMPM synthesis method. In particular, the extended FBMPM inherits the advantage of the original FBMPM that a useful restriction is put on the distribution of poles, which makes the element positions obtained much more accurate and robust. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed methods. For the tested cases, the element saving is about 20% ~ 25% for reconfigurable shaped patterns, and can be even more for electrically large linear arrays with scanned pencil-beams. Index Terms—Antenna pattern synthesis, extended matrix pencil method (MPM), multiple patterns, nonuniformly spaced arrays, reconfigurable arrays.
I. INTRODUCTION
T
HE reconfigurable array antennas capable of radiating dual or more patterns by varying only element excitations can reduce the number of antennas, the weight and the cost of whole hardware systems. They have been used in many applications, such as air traffic control radars, spacecraft and wireless communications. The synthesis of such reconfigurable arrays has received increasing attention in the past thirty years [1]–[9]. The synthesis methods include the analytical technique [1], the modified Woodward-Lawson technique [2], the projection approaches [3], [4], particle swarm optimization Manuscript received November 04, 2012; revised September 09, 2013; accepted November 08, 2013. Date of publication November 22, 2013; date of current version January 30, 2014. This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant 2012121036, in part by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20120121120027, in part by the National Natural Science Foundation of China (NSFC) under No. 61231001, 61301009 and 41240029, and in part by the NSF of Fujian Province under No. 2013J01252. Y. Liu is with Department of Electronic Science, Xiamen University, Fujian, 361005, China (e-mail:
[email protected]). Q. H. Liu is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA. Z. Nie is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Sichuan, 610054, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2013.2292529
(PSO) [5], improved genetic algorithms (GA) [6]–[8], and other techniques [9]. For most of them, multiple patterns are generated by only controlling the excitation phases of a single array with prefixed or optimized common amplitudes. Such synthesis can simplify the design of feeding network without controlling the amplitudes, but reduces the number of freedoms from the perspective of the pattern synthesis. In addition, most synthesis techniques in the literature deal only with uniformly spaced linear arrays and sometimes require a large number of elements to simultaneously satisfy all the requirements in the multiple-pattern synthesis. Naturally, using nonuniform element spacings has more freedoms and can reduce the total number of elements. For the synthesis of a nonuniformly spaced array with single-pattern, many practical methods have been proposed, such as dynamic programming [10], analytical or deterministic methods [11], [12], Bayesian compressive sampling techniques [13], [14], and stochastic or metaheuristic optimization algorithms [15]–[18]. Despite the success of these synthesis methods, most of them cannot be easily extended to the synthesis of a sparse array with multiple patterns since the best element positions usually change with different patterns. The aim of this paper is to reconstruct a multiple-pattern linear antenna array with as few elements as possible. Different from general pattern synthesis problems where power pattern specifications may be provided with a set of inequalities or masks, this work is focused on the design of a sparse linear array matching user-defined or reference multiple patterns. Previously, we presented a type of sparse linear array reconstruction technique based on the matrix pencil method (MPM) and the forward-backward matrix pencil method (FBMPM) for performance improvement [19], [20]. However, when applied to the case of reconfigurable multiple patterns, the MPM and FBMPM also have the problem that the best element positions may change for different patterns. In this paper, we will show that this problem can be solved by the extended MPM or FBMPM-based synthesis method. Note that the proposed extended MPM is another implementation version of the method presented by Sarkar et al. in [21] and is also the basis of the proposed extended FBMPM. With these methods, the optimal solution of the common element positions can be found for the multiple patterns. In particular, the extended FBMPM maintains a useful constraint of the original FBMPM on the distribution of eigenvalues or poles, which contributes to more accurate and robust estimation of the common element positions. A set of synthesis experiments for different pattern shapes and varying apertures are conducted to validate the accuracy, robustness and efficiency of the proposed methods and to define the choice of their control parameters.
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LIU et al.: REDUCING THE NUMBER OF ELEMENTS IN MULTIPLE-PATTERN LINEAR ARRAYS
II. SPARSE ARRAY RECONSTRUCTION METHOD A. Array Reconstruction Problem Consider a reconfigurable linear array of antenna elements which can radiate multiple desired patterns by varying the element excitations. The array factors are given by
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must be satisfied, where . For instance, for the -element array with a uniform spacing of . Hence, we have , where . Then the extended MPM organizes the sampled pattern data into the form of a block-Hankel matrix which is given by
(1) , , , and where is the total number of the desired patterns. is the position of the th element, and denotes the complex excitation of the th element for the th pattern. The problem considered here is to reconstruct a new array that can produce a satisfactory approximation of all the desired patterns with as few elements as possible. For different patterns, the new elements have optimized common positions but with probably different excitations. That is, find the minimum element count ( ), under the following constraint:
.. .
where
(2) , and and ( ) are the where locations and excitations of new elements, respectively. The biggest difficulty in the above problem comes from determining the best common element positions . For the case of single-pattern reconstruction, the optimized element positions can be obtained by the MPM/FBMPM method or many other methods. However, as mentioned previously, the best positions may be varied among the multiple desired patterns. The simplest method is to synthesize the element positions for each single pattern and calculate the average over all the synthesized position results for multiple patterns. Obviously, this is not the best way. In [21], the MPM is extended to estimate the common SEM (singularity expansion method) poles of transient responses without averaging from multiple data observed at different look directions. The idea with this extended MPM can be used here to deal with the estimation of the common element positions. However, the extended MPM does not have a constraint on the distribution of poles just like in the original MPM, which probably results in the presence of nonphysical imaginary parts in the estimated positions [19]. The forward-backward structure in the FBMPM has proven useful for overcoming this problem, but it cannot be compatible with the formulation of [21]. Hence, another version of the extended MPM is presented for the sparse reconstruction of multiple patterns, and the extended FBMPM is straightforwardly obtained by incorporating the forward-backward structure into the proposed extended MPM. B. Extended MPM for Multiple-Pattern Synthesis Similar to the original MPM-based synthesis method [19], the extended MPM uniformly samples all the desired pattern functions in the space of from to , with where . According to the Nyquist sampling theorem [19], the condition that
.. .
.. .
(3)
and . The subscript denotes the transpose of a vector or matrix, and is called the pencil parameter which is chosen such that [23]. It can be seen that consists of Hankel matrices. Then consider the two matrices and , which are obtained from by deleting the first column and the last column, respectively. They can be respectively factorized as (4) and (5), shown at the bottom of the next page, where is an identity matrix and the others are shown in (6)–(11), shown at the bottom of the next page. From (4) and (5), the poles are the generalized eigenvalues of the following problem: where
(12) To reduce the number of elements, the extended MPM performs the singular value decomposition (SVD) of the matrix , as is given by (13)
(13) The optimal lower-rank approximation matrix, say , can be obtained by retaining only (where ) largest singular values. corresponds to an approximation of the multiple desired patterns that can be radiated by fewer elements. Right now, one question is how to determine the value of or what the minimum number of elements required for a satisfactory approximation of the multiple patterns is. In [19] and [20], it was found that a rough estimate of the minimum can be given by
(14)
where is appropriate for the usual case [20]. The true value should be equal or very close to the above estimate. The poles associated with new element positions can be obtained by solving the following generalized eigenvalue problem (15)
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where
where and are obtained from by deleting the first column and the last column, respectively. In addition, a more computationally efficient choice is to solve an ordinary eigenvalue problem given by [19]
and
are shown in (19) and (20)
.. .
.. .
.. .
(16)
(19)
where (resp., ) is obtained by removing the top (resp., bottom) row of which contains only principal right singular vectors of in (13). Note that (16) is slightly different from (47) and (48) in the original extended MPM of [21] which uses principal left singular vectors of to estimate the eigenvalues. Such modification is proposed for combination with the forward-backward matrix pencil. Assume that s are the estimated eigenvalues. The element positions are given by
.. .
.. .
.. .
(20)
and
.. .
.. .
.. .
(21)
(17) The least-squares solution of (18) is given by
Once all the poles and element positions have been computed, the complex excitations can be computed from the following equation [21]:
(22) which is an extension of [19, (15)] to the case of multiple-pattern synthesis.
(18)
(4) (5)
..
.. .
.. .
.
(6)
(7)
.. .
(8)
..
..
.. .
.. .
.
(9)
(10)
.
.. .
(11)
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C. Extended FBMPM for Multiple-Pattern Synthesis
evaluate the pattern reconstruction performance, we define a normalized error as follows:
The extended MPM described in the above can be easily further improved by combining with the idea of FBMPM [20]. The method is called the extended FBMPM in which the multiple pattern data is organized as the following matrix:
.. .
.. .
.. . (23)
.. .
.. .
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.. .
(25)
and ( ) are the original where and reconstructed patterns, respectively. Note that for the case , the above definition is reduced to the one used in of [14]. A. Example 1
where the subscript denotes the complex conjugate. Consider the matrix pencil (24) (resp., ) is obtained from where by deleting the first (resp., the last) column. It can be proven that the above matrix pencil has the same property as that of [20, (5)]. That is, if , is a pair of generalized eigenvalue must be and eigenvector of this matrix pencil, another pair of generalized eigenvalue and eigenvector, where for . This constraint on the s is particularly significant in the current consideration since some of patterns in the synthesis of multiple-pattern may be asymmetric [20]. In the following, the extended FBMPM has exactly the same algorithm steps as those of the extended MPM presented above (different from the extended MPM in [21]). Due to the constraint on the eigenvalue distribution of the extended FBMPM, the element positions calculated by (16) and (17) is usually much more accurate than those obtained from the same equations in the extended MPM, which results in significant improvement in the pattern reconstruction accuracy.
III. NUMERICAL RESULTS In this section, several numerical experiments for the sparse reconstruction of multiple patterns are conducted to verify the performance of the extended MPM and FBMPM. The sampling parameter is chosen depending on the priori maximum array extension, and for the -element array with a spacing of , as previously mentioned. The pencil parameter is chosen such that , and can be roughly estimated by the criterion of (14). The effect on the pattern reconstruction accuracy for different choices of these parameters will be studied in detail with a set of numerical experiments to provide the guideline of applying the proposed methods. To
For the first example, we consider a dual-pattern in [6, Fig. 2]. This dual-pattern, including a pencil beam and a flat-top beam, was synthesized by Mahanti and Chakraborty with the real-coded genetic algorithm (GA) for optimized dynamic range ratio (DRR). They used 20 elements with a uniform spacing of . The two patterns are generated by the common excitation amplitudes but with different phases. We use both the extended MPM and FBMPM-based synthesis methods to reconstruct this dual-pattern with optimized element positions and excitation distributions. Set , and for each , where . At first we fix to study the effect of different combinations of and . Fig. 1(a) shows the normalized error versus for different for the extended FBMPM. It is observed that for different , the accuracy is almost the same with respect to , and the best accuracy is achieved at about except for the case of where we only have . This means that is enough only from point view of sampling condition, while much more points are usually required in terms of the best pencil parameter for the extended FBMPM. In this case, sampling points are appropriate and points are enough. Then the effectiveness is checked for the criterion of (14). Fig. 1(b) shows the estimated value of versus for different ( for , and for ). We can see that the estimated for all maintains exactly the same and all decreases from 17 to 14 as increases. For the case of , the criterion gives an estimate of or 14, depending on the value of . However, the pattern reconstruction in this case is inaccurate and unacceptable. This implies that more reliable estimation of can be achieved by the criterion under the condition that for the -element array with a spacing of . Fig. 2 shows the comparison of the original pattern and the reconstructed pattern by the extended FBMPM with the parameters , , ( ). As can been seen, the reconstructed dual-pattern with only 15 elements agrees well with the original one. Table I shows positions and excitations of the antenna elements reconstructed by the extended FBMPM with (the original array synthesized by [6] has symmetric element positions and excitations, and the reconstructed array also maintains this property). In this case, we
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TABLE I ELEMENT POSITIONS AND EXCITATIONS RECONSTRUCTED BY THE EXTENDED FBMPM WITH 15 ELEMENTS IN FIG. 2 (THE RESULTS ARE SHOWN FOR ONLY 8 ELEMENTS DUE TO SYMMETRIC DISTRIBUTION)
results for the tested parameter values in this example and are not shown here. However, the accuracy performance of the extended MPM does not hold for the case of nonsymmetric patterns, which will be verified by the following example. B. Example 2
Fig. 1. Parameter study of the extended FBMPM for reconstructing the dualfor different pattern shown in [6, Fig. 2]. (a) The normalized error versus at fixed ; (b) the estimated value of versus .
Fig. 2. Dual-pattern synthesized by [6] with 20 elements and the pattern re, constructed by the extended FBMPM-based synthesis method with , ( ).
save 25% antenna elements, although the additional amplitude control is required in the reconstructed array. Note that due to the symmetry of the original pattern, the extended MPM-based synthesis method gets almost the same
To compare the performance of the extended MPM and FBMPM, we consider a more complicated multiple-pattern in [2, Fig. 2] which consists of a pencil-beam, a flat-top beam, and a cosec-squared pattern that is asymmetric. This pattern was synthesized by the modified Woodward-Lawson method , where with 20 equispaced elements. Set . Fig. 3(a) shows the normalized error versus for different at fixed for both the extended MPM and FBMPM. It is seen that for or , the minimum is achieved at about for the extended MPM for the extended FBMPM. and about Compared with the extended MPM, the extended FBMPM . The criteobtains much better accuracy for all the tested rion of (14) is tested again, and Fig. 3(b) shows the estimated value of versus for the both proposed for , and for methods ( ). For the case of and , the estimated decreases from 16 to 15 for the extended MPM and from 17 to 15 for the extended FBMPM. This estimate is reliable in terms of pattern reconstruction accuracy from Fig. 3(a). Fig. 4 shows the original pattern and the reconstructed patterns by the extended MPM and FBMPM, both with , , ( ). As can been seen, the extended FBMPM gives an acceptable reconstruction of the original pattern with 16 elements, while the extended MPM gives inaccurate reconstruction result due to the asymmetric distribution of the cosec-squared pattern. The comparative performance can be further verified by observing their pole distributions shown in Fig. 5. All the poles of the extended FBMPM fall exactly on the unit circle. However, some of the poles obtained from the extended MPM lie off the unit circle. This explains the poor accuracy of the extended MPM. Table II shows positions and excitations of the elements reconstructed by the extended FBMPM. In this case, we save 20% antenna elements.
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Fig. 4. Multiple-pattern synthesized by [2] with 20 elements and the patterns reconstructed by (a) the extended FBMPM and (b) the extended MPM, both , , ( ). with
Fig. 3. Parameter study of the extended MPM and FBMPM for reconstructing for the dual-pattern shown in [2, Fig. 2]. (a) The normalized error versus ; (b) the estimated value of versus ( different at fixed for , and for ).
C. Example 3 For comparison with other state-of-the-art sparse synthesis techniques, we consider two shaped patterns as shown in Fig. 6: one is the pattern synthesized by Cid in [24] and the other is proposed by Akdagli in [25], both with 20 elements. Each of them was used as a reference pattern for single-pattern sparse array synthesis by the Bayesian compressive sampling (BCS) and the multitask BCS (MT-BCS) technique [13], [14]. Fig. 7 shows the results of the error versus the element count for the BCS, MT-BCS and the extended FBMPM. Consider the sparse synthesis with only 15 elements. The BCS has the error ( at ) of larger than 1.0 for the both patterns [14], and the MT-BCS achieves much better accuracy with for the pattern of [24] and 1.7 for the pattern of [25], respectively. The extended FBMPM can synthesize the two patterns produced by only one array with different excitations. For 15 elements, the dual-pattern synthesis error is (with and ). The extended FBMPM can be also used to do the single-pattern synthesis, with the error for the pattern of [24]
and 2.9 for the pattern of [25], respectively. In that case, it is reduced to the original FBMPM. It is obvious that if we consider the reconfigurable case where two or more patterns are required, the extended FBMPM can save much more elements than the MT-BCS and the original FBMPM. For example, only 16 elements are required for the accuracy of for this dual-pattern synthesis. The reconstructed two patterns are shown in Fig. 6 for comparison with the original patterns. If this accuracy is acceptable, the element saving is 48.4% with respect to the MT-BCS that requires two different arrays of 15 plus 16 elements for producing the two patterns. Note that in the above we assumed the two arrays synthesized by the MT-BCS do not have the common elements. D. Example 4 At last, we check the effectiveness of the extended FBMPM for dealing with the case of multiple scanned beams. A set of simulations are carried out with the scanned Chebyshev patterns (with ) for different array’s apertures ( ) and different scanning ranges (the maximum scanning angle is from the array’s broadside). For this example, we set , and is determined by (14) with fixed . The minimum error is obtained by searching for the best for each case with a certain array’s aperture. Table III shows the results of , and for the tested cases. As can be seen, the extended FBMPM is robust
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TABLE II ELEMENT POSITIONS AND EXCITATIONS RECONSTRUCTED BY THE EXTENDED FBMPM WITH 16 ELEMENTS IN FIG. 4(A)
Fig. 5. Distributions of 16 poles for Fig. 4.
Fig. 7. Normalized error versus element count for single and dual-pattern reconstructions (the reference patterns are given in [24] and [25], respectively).
Fig. 6. Reference patterns ([24] and [25]), each of them produced by 20 uniformly spaced elements, and the reconstructed patterns by the extended FBMPM with 16 elements.
and effective even for an electrically large array with multiple scanned beams. For example, for the case of , the element count required for an accurate pattern reconstruction is 148, 164 and 180, for the scanning range of , , , respectively. The element saving is 42.2%, 35.9% and 29.7% for these cases. The efficiency of the extended FBMPM is also tested, and the CPU time versus the array’s aperture is shown in Table III (on
Thinkpad T430s with I5-3210M CPU @2.5 GHz). It is observed that the time cost is negligible for electrically small apertures, but increases rapidly as the aperture gets large. The computational complexity is close to the order of for large . This is reasonable due to the SVD operation and the solution to eigenvalue problems required by the extended FBMPM. Nevertheless, this method is still much more efficient than many other algorithms for the sparse array reconstruction application. For example, for the case of the 256-element linear array with 7 patterns, we take only 152.38 seconds to find the optimized positions, excitation amplitudes and phases for 180 new elements, which may be very hard for some stochastic optimization algorithms. Note that for all the tested cases, the memory occupation by the synthesis procedure is less than 900 Megabytes (MB). IV. CONCLUSION We have presented the extended MPM and FBMPM-based synthesis methods for reconstructing multiple-pattern linear ar-
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TABLE III RECONSTRUCTION OF MULTIPLE SCANNED CHEBYSHEV PATTERNS WITH DIFFERENT ARRAY’S APERTURES BY THE EXTENDED FBMPM WITH
rays with fewer elements. In particular, the extended FBMPMbased synthesis method inherits a useful constraint of the original FBMPM on the distribution of the poles, and consequently achieves more accurate and robust synthesis performance. By using the proposed methods, we can perform a complete optimization of the common element positions, individual amplitudes and phases for a multiple-pattern array, and finally reduce the number of elements considerably. The performance of the proposed methods is summarized as follows: • The sampling parameter should be chosen according to the Nyquist sampling theorem. Larger is required for obtaining the best pencil parameter to achieve the minimum pattern reconstruction error. For the -element linear array with a spacing of , or is suggested. • The parameter affects the accuracy of the proposed methods, and the best is within 0.5 ~ 0.8 from many tested cases. • The element count can be estimated by the proposed criterion with the parameter within . • By using appropriate parameters, the extended FBMPM is robust and effective for different pattern shapes and different array’s apertures including electrically large beamscanning linear arrays. • From the tested cases, the element saving with the extended FBMPM is about 20% ~ 25% for the reconfigurable shaped patterns, and can be even more for electrically large arrays with scanned pencil-beams. • The proposed methods are very computationally efficient, although the computational complexity is close to (please refer to the time costs shown in Table III). It should be emphasized that the proposed methods deal with the sparse array reconstruction problem, and need to start from the reference field patterns or some original source distributions. This is similar to the other field synthesis methods such as the original MPM/FBMPM, and the BCS/MT-BCS methods. It would be very useful to integrate with the existing power pattern synthesis algorithms where pattern specifications are given in the form of inequality constraints (an example is given for the original FBMPM in [20]). Another limitation of the proposed methods is its inability of specifying some constraints, such as the minimum/maximum inter-element spacing and the dynamic range ratio of the synthesized amplitudes. Nevertheless, from all the tested cases, we found that by searching for the best pencil parameters, the extended FBMPM can always find
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FOR
,
a pretty good synthesis result, e.g., enlarged inter-element spacings while maintaining almost the same array’s aperture and realizable dynamic range ratio, compared with the original array. Finally, we note that the proposed extended MPM and FBMPM can be also applicable to many other problems related to the mathematical model of a sum of damped/undamped complex exponentials. REFERENCES [1] A. Chakraborty, B. N. Das, and G. S. Sanyal, “Beam shaping using nonlinear phase distribution in a uniformly spaced array,” IEEE Trans. Antennas Propag., vol. AP-30, no. 5, pp. 1031–1034, May 1982. [2] M. Durr, A. Trastoy, and F. Ares, “Multiple-pattern linear antenna arrays with single prefixed amplitude distributions: Modified WoodwardLawson synthesis,” Electron. Lett., vol. 36, no. 16, pp. 1345–1346, 2000. [3] O. M. Bucci, G. Mazzarella, and G. Panariello, “Reconfigurable arrays by phase-only control,” IEEE Trans. Antennas Propag., vol. 39, no. 7, pp. 919–925, Jul. 1991. [4] R. Vescovo, “Reconfigurability and beam scanning with phase-only control for antenna arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1555–1565, Jun. 2008. [5] D. Gies and Y. Rahmat-samii, “Particle swarm optimization for reconfigurable phase differentiated array design,” Microw. Opt. Technol. Lett., vol. 38, pp. 168–175, 2003. [6] G. K. Mahanti, S. Das, and A. Chakraborty, “Design of phase-differentiated reconfigurable array antennas with minimum dynamic range ratio,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 262–264, 2006. [7] G. K. Mahanti and A. Chakraborty, “Phase-only and amplitude-phase synthesis of dual-pattern linear antenna arrays using floating-point genetic algorithms,” Prog. Electromag. Res., vol. PIER 68, pp. 247–259, 2007. [8] S. M. Vaitheeswaran, “Dual beam synthesis using element position perturbations and the G3-GA algorithm,” Prog. Electromag. Res., vol. PIER 87, pp. 43–61, 2008. [9] A. F. Morabito, A. Massa, P. Rocca, and T. Isernia, “An effective approach to the synthesis of phase-only reconfigurable linear arrays,” IEEE Trans. Antennas Propag., vol. 60, no. 8, pp. 3622–3631, Aug. 2012. [10] M. I. Skolnik, G. Nemhauser, and J. W. Sherman, III, “Dynamic programming applied to unequally spaced arrays,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 35–43, Jan. 1964. [11] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 511–523, Mar. 1999. [12] O. M. Bucci, M. D. Urso, T. Isernia, P. Angeletti, and G. Toso, “Deterministic synthesis of uniform amplitude sparse arrays via new flexible density taper techniques,” IEEE Trans. Antennas Propag., vol. 58, no. 6, pp. 1949–1958, Jun. 2010. [13] G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 467–481, Feb. 2011. [14] G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag., vol. 60, no. 5, pp. 2309–2326, May 2012.
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[15] K. K. Yan and Y. Lu, “Sidelobe reduction in array-pattern synthesis using genetic algorithm,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1117–1122, Jul. 1997. [16] L. Cen, W. Ser, Z. Yu, S. Rahardja, and W. Cen, “Linear sparse array synthesis with minimum number of sensors,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 720–726, Mar. 2010. [17] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2210–2217, Sep. 2003. [18] S. L. Ho and S. Yang, “The cross-entropy method and its application to inverse problems,” IEEE Trans. Magn., vol. 46, no. 8, pp. 3401–3404, Aug. 2010. [19] Y. Liu, Z. Nie, and Q. H. Liu, “Reducing the number of elements in a linear antenna array by the matrix pencil method,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2955–2962, Sep. 2008. [20] Y. Liu, Q. H. Liu, and Z. Nie, “Reducing the number of elements in the synthesis of shaped-beam patterns by the forward-backward matrix pencil method,” IEEE Trans. Antennas Propag., vol. 58, no. 2, pp. 604–608, Feb. 2010. [21] T. K. Sarkar, S. Park, J. Koh, and S. M. Rao, “Application of the matrix pencil method for estimating the SEM (singularity expansion method) poles of source-free transient responses from multiple look directions,” IEEE Trans. Antennas Propag., vol. 48, no. 4, pp. 48–55, Apr. 2000. [22] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters by a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, no. 1, pp. 48–55, Feb. 1995. [23] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 5, pp. 814–824, May 1990. [24] J. M. Cid, J. A. Rodriguez, and F. Ares, “Shaped power patterns produced by equispaced linear arrays: Optimized synthesis using orthogbeams,” J. Electromagn. Waves Appl., vol. 13, onal no. 7, pp. 985–992, 1999. [25] A. Akdagli and K. Guney, “Touring ant colony optimization algorithm for shaped-beam pattern synthesis of linear antenna arrays,” Electromagnetics, vol. 26, no. 6, pp. 615–628, 2006. Yanhui Liu received the B.S. and Ph.D. degrees, both in electrical engineering, from the University of Electronic Science and Technology of China (UESTC), Sichuan, China, in 2004 and 2009, respectively. From September 2007 to June 2009, he was a Visiting Scholar in the Department of Electrical Engineering at Duke University, Durham, NC, USA. Since July 2011, he has been an Associate Professor with the Department of Electronic Science, Xiamen University, Fujian, China. He has authored and coauthored over 30 peer-reviewed journal or conference papers. He received the UESTC Outstanding Graduate Award in 2004 and the Excellent Doctoral Thesis Award of Sichuan Province of China in 2012. His research interests include antenna array design, signal processing in electromagnetics, and microwave imaging methods.
Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, China, in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 1989. He was with the Electromagnetics Laboratory at the University of Illinois at Urbana-Champaign as a Research Assistant from September 1986 to December 1988, and as a Postdoctoral Research Associate from January 1989 to February 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1990 to 1995. From 1996 to May 1999, he was an Associate Professor with New Mexico State University, Las Cruces, NM, USA. Since June 1999, he has been with Duke University, Durham, NC, USA, where he is now a Professor of electrical and computer engineering. He has published over 500 papers in refereed journals and conference proceedings. His research interests include computational electromagnetics and acoustics, inverse problems, geophysical subsurface sensing, biomedical imaging, electronic packaging, and the simulation of photonic and nano devices. Dr. Liu is a Fellow of the Acoustical Society of America, a member of Phi Kappa Phi, Tau Beta Pi, and a full member of U.S. National Committee of URSI Commissions B and F. Currently, he serves as the Deputy Editor-in-Chief of Progress in Electromagnetics Research, an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, and an Editor for Computational Acoustics. He was a Guest Editor in Chief of the PROCEEDINGS OF THE IEEE for a special issue on large-scale computational electromagnetics published in 2013. He received the 1996 Presidential Early Career Award for Scientists and Engineers (PECASE) from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation.
Zaiping Nie (F’12) was born in Xi’an, China, in 1946. He received the B.S. degree in electronic engineering and the M.S. degree in electromagnetic field and microwave technology from the Chengdu Institute of Radio Engineering (now UESTC: University of Electronic Science and Technology of China), Chengdu, China, in 1968 and 1981, respectively. From 1987 to 1989, he was a Visiting Scholar with the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, USA. He was an Associate Professor from 1989 to 1993 and has been a Professor since 1994 with the Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu, China. He has published more than 300 journal papers. His research interests include waves and fields in inhomogeneous media, computational electromagnetics, antenna theory and techniques, electromagnetic scattering, and inverse scattering.
