IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 16, 2017
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Circular Hexagonal Lattice Array and Its Optimization in Aperture Synthesis Radiometers Chol-Hyon Jang, Fei Hu, Feng He, Myong-Hak Jong, and Dong Zhu
Abstract—An innovative array configuration, circular hexagonal lattice array (CHLA), is proposed to overcome nonuniform sampling drawback of a circular array for aperture synthesis radiometers. A differential evolution algorithm is applied to search optimal CHLA configurations taking into account the radiometric sensitivity degradation as well as the sample number. Representative numerical results are illustrated to evaluate performances of the proposed array in comparison to the previously proposed arrays. The performances of obtained optimal CHLAs are reported for 15–32 elements. Index Terms—Aperture synthesis radiometer (ASR), circular array, differential evolution (DE).
I. INTRODUCTION N INTERFEROMETRIC aperture synthesis technique has been widely used in the radio astronomy and remote sensing of the earth as a way to overcome antenna size problems in high spatial resolution applications, particularly at low frequencies and spaceborne instruments [1], [2]. Interferometric aperture synthesis radiometers (ASRs) measure the correlation between antenna pairs with different separations. Every cross correlation is a sample of the so-called visibility function at the spatial frequency defined by the relative coordinates of the corresponding pair of antennas. The spatial frequency domain (Fourier domain) is called the u–v domain or u-v coverage. The brightness temperature map of a scene under observation is then obtained basically by an inverse Fourier transform of the visibility function. Unlike optimization for dense focal plane arrays in push broom radiometers [3], [4], determining an appropriate configuration of an array in the ASRs is essentially an optimal sampling problem. In ASRs, since the spatial frequencies (or visibility function) of the brightness temperature map are directly given by the spacings of the antenna pairs, determining
A
Manuscript received December 16, 2015; accepted August 9, 2016. Date of publication August 11, 2016; date of current version March 27, 2017. This work was supported in part by the National Science Foundation of China under Grant NSFC61172100, the Fundamental Research Funds for the Central Universities under Grant HUST 2015QN093, and the Shanghai Spaceflight Technology Renovation Fund under Grant SAST2015088. C.-H. Jang and D. Zhu are with the School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China. F. Hu and F. He are with the National Key Laboratory of Science and Technology on Multi-spectral Information Processing, School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail:
[email protected]). M.-H. Jong is with the Department of Control Science, Pyongyang University of Science and Technology, Pyongyang 850097, North Korea. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2016.2599706
the optimal configuration of an antenna array is very important in improvement of the quality of the brightness temperature map. Several array configurations such as T-shape, Y-shape, and circular arrays have been investigated [5], [6]. It is known that a Y-shape array configuration provides high angular resolution. The Y-shape array generates hexagonal grid u-v coverage, and hexagonal fast Fourier transform (FFT) that can be applied to the u-v coverage in the image reconstruction was proposed [7]. Circular array configuration permits nonredundant sampling in the u-v domain [6]. However, because of nonuniform distribution of the samples measured by this array, additional processes are required to place these samples in uniform grid for image reconstruction [8]. Circular lattice array (CLA) configuration to generate uniform rectangular grid u-v coverage was proposed [9]. The optimization of a circular array for ASRs generally aims to find the uniform distribution of u-v coverage [6]. The radiometric sensitivity, as well as the uniform sampling, is also a critical consideration to be taken into account in the circular array optimization of ASRs and is defined as the smallest change in the average brightness temperature that can be detected by the instrument [10]. However, the radiometric sensitivity has never been taken into account in optimizing of the circular array for ASRs. This letter proposes an innovative array structure, circular hexagonal lattice array (CHLA), for ASRs. A fitness function to optimize this array both in terms of uniformity of the u-v coverage and the radiometric sensitivity is presented. A differential evolution (DE) algorithm is applied to optimizing the structure of the array. Representative numerical results are illustrated to assess performances of the proposed array in comparison to those of the previously proposed circular arrays and Y-shape array. This letter is organized as follows. The configuration and design method of CHLA are presented in Section II. Section III describes the fitness function to optimize the array placement. Section IV shows numerical results to evaluate the performance of CHLA. Finally, the conclusion is given in Section V. II. CHLAS A. Configuration of CHLA In ASRs, the samples of spatial frequency domain are defined by the relative coordinates of the antenna pairs with different separations. The main reason for nonuniform distribution of samples measured by a uniform circular array (UCA) is that the array elements are placed at irregular interval along x-axis and y-axis. A detailed analysis of the relation between configuration and spatial frequency samples of UCA can be found in [9].
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 16, 2017
Fig. 2. Diagram represents details of CHLA configuration. The CHLA configuration is derived from UCA configuration. Circle plots show locations of UCA elements, and triangle plots indicate the set of allowed grids for CHLA elements.
Fig. 1. (a) Fifteen-element CHLA configuration. (b) Its u-v coverage. N sa m = 91.
To overcome nonuniform distribution of UCA samples, we present an innovative circular array, CHLA, which generates hexagonal grid u-v coverage. The presented 15-element array configuration and the corresponding u-v coverage are shown in Fig. 1. The positions of CHLA elements are determined by the closest grid nodes from UCA elements. B. CHLA Design Like the Y-shape array, hexagonally placed array configuration generates hexagonally sampled u-v coverage. As shown in Fig. 2, a hexagonal grid and a UCA configuration are used in CHLA design. Moreover, two circles are employed to point out possible positions for the array elements. The grid intervals and radii of the two circles are given as follows: 4r0
xgrid = fix
8N (N −1) √ 3π
2 ygrid = √ xgrid 3
(1)
(2)
Fig. 3. u-v coverages of (a) staggered-Y array and (b) CHLA for 15-element. The inner hexagons shown in black indicate maximum area not containing the missing (u, v) samples (or holes) for the staggered-Y array. Circle markers show holes inside the inner hexagon.
ygrid 2 r2 = r1 + ygrid r1 = r0 −
(3) (4)
where r0 is radius of the circle placed with UCA elements and N is the element number of UCA. The operator fix [] means √ rounding toward zero, and the constant 8/( 3π) is the areas of the entire u-v coverage and the sampled region with the circle shape. The grid intervals are determined by (1) and (2) so that the sample region consists of approximately N (N − 1) samples. In Fig. 2, the triangles indicate the possible positions for CHLA elements. III. OPTIMIZATION OF CHLA CHLAs and uniform Y-shape arrays produce visibility samples over a hexagonal grid in the spatial frequency domain. The u-v coverages of the CHLA and staggered-Y array [11] are shown in Fig. 3. Unlike uniform Y-shape arrays uniquely determined for given number of antennas, there are a great number of CHLA configurations for any number of antennas. The DE algorithm is applied to find optimal CHLA configurations.
JANG et al.: CIRCULAR HEXAGONAL LATTICE ARRAY AND ITS OPTIMIZATION IN APERTURE SYNTHESIS RADIOMETERS
DE is a simple yet powerful stochastic global optimizer that belongs to the class of evolutionary global optimization techniques. It has been proved to be a very efficient and robust technique for function optimization and has been successfully applied to array synthesis problems and some other engineering problems [12], [13]. The greatest advantage of DE is that it uses the differences of randomly sampled pairs of object vectors to guide the mutation operation instead of using probability distribution functions as other evolutionary algorithms. The DE follows a general procedure of an evolutionary algorithm. The detailed description and advances in the DE algorithm can be found in [13]. As seen in Fig. 3, the CHLA and staggered-Y array have missing (u, v) samples, which are often called holes. The staggeredY array has missing (u, v) samples outside the inner hexagon of the star shown in black in Fig. 3(a), while those of CHLA exist inside as well as outside the inner hexagon. For large Y-shape arrays, visibility power not collected by the missing (u, v) samples is insignificant, and these missing values should be extrapolated in some way [14]. From this fact, we put mid nonhole constraint (i.e., h(a1 ,a2 , . . . ,an ) = 0) on the optimization of CHLA configuration. Here, h(a1 ,a2 , . . . ,an ) is the number of holes displayed in circle marker inside the inner hexagon in Fig. 3(b). On the other hand, for CHLA configurations satisfying the mid nonhole constraint, there are different sample distributions with respect to the same number of samples. The 1/2 sam radiometric sensitivity degradation factor (DF) (ΣN i=0 1/ri ) of each CHLA depends on the concrete sample distribution represented by the redundancy set ri , where ri is the number of times that a relative spacing of ith sample is present in the array [1], [10]. Consequently, taking into account the sample number, Nsam , together with the radiometric sensitivity under the mid nonhole constraint, the fitness function to optimize the CHLA is defined as follows: 1 f = Nsam + Nsam i=1
1 ri
,
i=1
=
+ 1 ri
1 1 + DF nR
TABLE I EFFECTIVE SAMPLE NUMBER COMPARISON AMONG FOUR CIRCULAR ARRAYS n
UCA
Optimal CLA1
Original CHLA2
Optimized CHLA
12 13 14 15 16 17 18 19 20
73 117 87 149 121 207 147 243 179
105 123 129 151 171 205 233 259 271
61 117 95 91 125 213 145 257 193
119 143 155 189 217 235 251 281 329
1 2
Optimal CLAs are those listed in [7, Table III]. Original CHLAs are those derived in the same way as shown in Fig. 1(a).
under h(a1 , a2 , . . . , an ) = 0.
(5) Taking into account the radiometric sensitivity DF in preference to the sample number, the fitness function can be defined by 1 f = Nsam
687
Nsam n2 (n − 1) + n
under h(a1 , a2 , . . . , an ) = 0
Fig. 4. Two-dimensional histograms show the distribution of u-v samplings generated by several 21-elements arrays. (a) Staggered-Y array (N sa m = 295). (b) MSCHLA (N sa m = 285). (c) Test CHLA (N sa m = 277). (d) MDCHLA (N sa m = 253).
IV. NUMERICAL RESULTS (6)
(7)
where n is element number of the CHLA and R is redundancy, which is defined as the ratio of theoretic number of samples that included the origin (u,v) = (0, 0) (i.e., n(n−1) + 1) to actual number of samples in u-v domain (Nsam ). The element number of the staggered-Y array for the mid nonholes constraint is determined by 3 × fix[n/3]. Note that for n ≥ M > N , CHLAs can satisfy the mid nonhole constraint. M is a minimum number of elements that produce the mid nonhole u-v coverage, which is determined through a serial of simulations.
To assess performances of the proposed CHLA, two experiments are carried out in this section. One is comparing the sample numbers of maximum sample CHLAs (MSCHLA) to those of the UCAs and the optimal CLAs to evaluate the uniformity. The second is comparing the performances of the different CHLAs optimized using the fitness functions of (5) and (6) to those of the staggered-Y array. The DE algorithm was applied to optimize CHLAs. The adaptation scheme of F and CR has been applied to the DE/best/2/bin strategy. The parameters of DE were set as follows: population size Np = 150, Fm in = 0.4, Fm ax = 0.9, CRm in = 0.5, CRm ax = 0.9, and maximum iteration number itm ax = 4000. In the first experiment, the CHLAs were optimized on purpose to maximize sample number for 12–20 elements. To evaluate the uniformity of u-v coverage sampled by UCA, the u-v coverage
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 16, 2017
TABLE II COMPARISON OF PERFORMANCES BETWEEN DIFFERENT CHLAS AND STAGGERED-Y ARRAY
V. CONCLUSION
Array
n
Nsam
R
DF
Staggered-Y array MSCHLA Test CHLA MDCHLA
21 21 21 21
295 285 277 263
1.427 1.477 1.519 1.601
16.358 15.485 15.177 14.385
TABLE III PERFORMANCE COMPARISON OF CHLAS N
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
M
17 20 20 21 21 24 26 26 28 30 30 30 34 34 36 38 38 40
MSCHLA
MDCHLA
Nsam
DF
Nsam
DF
197 271 271 273 285 391 419 441 503 557 565 573 703 731 809 893 893 961
12.883 15.078 15.078 15.193 15.485 18.177 18.496 19.127 20.408 21.219 21.867 21.605 23.905 24.834 25.606 27.299 27.299 27.254
169 239 239 269 263 347 387 377 445 447 485 519 643 671 719 813 813 797
11.399 13.749 13.749 14.918 14.385 16.634 16.928 16.802 18.581 17.211 18.872 20.051 21.800 22.683 23.464 24.657 24.657 23.378
was divided into quadrangular grids determined by (1) and (2) as shown in Fig. 2. Then, the total number of grids that have at least one sample in a grid was counted. The total number of such grids represents the uniformity of the u-v coverage, and we shall refer to this as effective sample number. For the CHLAs, the effective sample number is equal to sample number, Nsam . The sample number of optimized CHLAs and the effective sample number of some representative circular arrays are given in Table I. Optimized CHLAs outperform UCAs and optimal CLAs in uniformity of u-v coverage. The second comparison study was carried out for 21-element arrays. The CHLAs that were optimized using two fitness functions expressed (5) and (6) are referred to as MSCHLA and minimum degradation factor CHLA (MDCHLA), respectively. Besides MSCHLA and MDCHLA for 21-element, there are many different CHLA configurations with performances comparable to these in terms of Nsam or DF . A CHLA among the many CHLAs was chosen as test CHLA for the comparison. Fig. 4(a)–(d) plots the 2-D histograms of the u-v coverages produced by the staggered-Y array, test CHLA, MSCHLA, and MDCHLA. The performances of these four arrays are compared in Table II. Although the optimized CHLAs are inferior to the staggered-Y array in redundancy, the optimized CHLAs produce the smaller radiometric sensitivity degradations. Optimization of the CHLAs was performed for 15–32 elements and the performance of resulting arrays is summarized in Table III.
In this letter, CHLA to produce hexagonal grid u-v coverage is proposed for ASRs and is optimized in terms of new optimization objectives. To overcome nonuniform sampling of the existing circular arrays, CHLA consisting of circularly placed antennas over hexagonal grid is proposed. The hexagonal samplings provided by CHLA can be inversed by standard rectangular FFT in ASRs, avoiding the need of interpolations [7]. Furthermore, two fitness functions to optimize the CHLA are presented taking into account the sample number together with the radiometric sensitivity under the mid nonhole constraint. The DE algorithm with self-adapting control parameters is applied to find the optimal CHLA configuration using sample number fitness function and those proposed. Unlike the uniform Y-shape array uniquely determined for a given number of antennas, there are many different CHLA configurations comparable to the Yshape array in spatial resolution and radiometric sensitivity for several numbers of antennas. Therefore, CHLA design is quite flexible for ASRs with high spatial resolution and satisfactory sensitivity. The capabilities of optimal CHLAs are summarized for 15–32 elements. REFERENCES [1] C. S. Ruf, C. T. Swift, A. B. Tanner, and D. M. Le Vine, “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. Geosci. Remote Sens., vol. 26, no. 5, pp. 597–611, Sep. 1988. [2] M. Martin Neira, Y. Menard, J. M. Goutoule, and U. Kraft, “MIRAS, a two-dimensional aperture synthesis radiometer,” in Proc. Int. Geosci. Remote Sens. Symp., 1994, pp. 1323–1325. [3] O. A. Iupikov et al., “Dense focal plane arrays for pushbroom satellite radiometers,” in Proc. 8th Eur. Conf. Antennas Propag., The Hague, The Netherlands, Apr. 6–11, 2014, pp. 3536–3540. [4] O. A. Iupikov et al., “An optimal beamforming algorithm for phased-array antennas used in multi-beam spaceborne radiometers,” in Proc. 9th Eur. Conf. Antennas Propag., Lisbon, Portugal, May 13–17, 2015, pp. 1–5. [5] U. R. Kraft, “Two-dimensional aperture synthesis radiometers in a low earth orbit mission and instrument analysis,” in Proc. Int. Geosci. Remote Sens. Symp., vol. 2, 1996, pp. 866–868. [6] T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propag., vol. 36, no. 8, pp. 1165–1167, Aug. 1988. [7] A. Camps, J. Bara, I. Corbella, and F. Torres, “The processing of hexagonally sampled signals with standard rectangular techniques: Application to 2-D large aperture synthesis interferometric radiometers,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 1, pp. 183–190, Jan. 1997. [8] J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Selection of a convolution function for Fourier inversion using gridding,” IEEE Trans. Med. Imag., vol. 10, no. 3, pp. 473–478, Sep. 1991. [9] C. H. Jang, F. Hu, F. He, and L. Wu, “A novel circular array structure and particle swarm optimization in aperture synthesis radiometers,” IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 1758–1761, 2015. [10] J. Dong, Q. Li, R. Shi, L. Gui, and W. Guo, “The placement of antenna elements in aperture synthesis microwave radiometers for optimum radiometric sensitivity,” IEEE Trans. Antennas Propag., vol. 59, no. 11, pp. 4103–4114, Nov. 2011. [11] F. Torres, A. B. Tanner, S. T. Brown, and B. H. Lambrigsten, “Robust array configuration for a microwave interferometric radiometer: Application to the GeoSTAR project,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 1, pp. 97–101, Jan. 2007. [12] P. Rocca, G. Oliveri, and A. Massa, “Differential evolution as applied to electromagnetics,” IEEE Antennas Propag. Mag., vol. 53, no. 1, pp. 38–49, Feb. 2011. [13] A. Qing, Differential Evolution: Fundamentals and Applications in Electrical Engineering. Singapore: Wiley, 2009. [14] J. Bara, A. Camps, I. Corbella, and F. Torres, “Bidimensional discrete formulation for aperture synthesis radiometers,” CNN2 to Work Order no 10 to ESTEC Contract no 9777/92/NL/PB.
