Pattern Optimization of a Reflector Antenna with Planar ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 1, JANUARY 1997

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Pattern Optimization of a Reflector Antenna with Planar-Array Feeds and Cluster Feeds Birsen Saka, Member, IEEE, and Erdem Yazgan, Member, IEEE

Abstract—We study pattern optimization of a parabolic reflector with planar-array feeds and cluster feeds. The optimization is carried out within conjugate gradient method (CGM) in which optimum weight vector of linearly constrained minimum variance (LCMV) beam former is used as the initial condition. The optimization with CGM to determine element weights is found to yield better sidelobe suppression and true settlement of desired scan direction compared to the LCMV beam former and conjugate field matching (CFM) techniques for a continuous beam-scanning case. The proposed method is also applied to the multibeam reflector-antenna design. Index Terms—Reflector antennas, optimization methods.

I. INTRODUCTION

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HERE are many military and civilian applications such as radar, radio astronomy, and microwave communication systems in which high-gain electronically scanned antennas need to scan only some restricted sector of space. The earliest and the most successful limited scan system is an array which is placed in or near the focal region of a reflector or lens antenna to scan its beam. Limited scan techniques have their origin in the development of mechanically scanned reflector and lens geometry using feed tilt and displacement [1]–[4]. Previous studies include single or dual reflector or lens geometry in combination with an array and beam-forming network that produce an electronically scanned beam using phase and amplitude control of array feeds [5]–[23]. Conjugate field matching (CFM) method has been known for quite some time, and is frequently used to obtain the pattern shape and surface distortion compensation to determine the feed-element excitation [8]–[12], [17]. The CFM excitation coefficients for an array feed are proportional to the complex conjugates of the received focal plane fields of each feed element when the reflector is illuminated by a plane wave. Iterative methods can be used to obtain the excitation coefficients of array feeds for pattern shaping [15], [18], [20]. In this paper, we demonstrate that the CFM method is a special case of the optimum weight vector of linearly constrained minimum variance (LCMV) beam former. The LCMV beamformer objective is to find a weight vector that minimizes the contributions in the undesired directions while maintaining a fixed response in specified directions. A difficulty associated Manuscript received April 6, 1995; revised March 8, 1996. This work was supported in part by the Scientific and Technical Research Council of Turkey (TUBITAK). The authors are with the Electrical and Electronics Engineering Department, Hacettepe University, Beytepe, Ankara, 06532 Turkey. Publisher Item Identifier S 0018-926X(97)00445-6.

with LCMV beam former is the requirement of complex valued constraints for the specified directions. In practice, one knows the magnitude but not the complex value of the desired pattern. To circumvent this difficulty, we optimize the power pattern instead of the field pattern using the iterative conjugate gradient method (CGM). We employ Chebyshev polynomials as a template to achieve the desired sidelobe performance. The rate of convergence of iterative methods depends on appropriate initial conditions. For this purpose, we use the LCMV beam-former closed-form solution weight vector for the initial condition of CGM. To illustrate the proposed method, the concave part of the parabolic reflector is fully illuminated by an obliquely incident electromagnetic plane wave. Array feeds are placed at the focus of the reflector for both the planar and cluster feeds. Optimization with CGM is used to determine the performance of the array-feed parabolic reflector with the parameters of directivity, sidelobe level, and steering error in the specified scan range. II. FORMULATION OF SECONDARY FIELDS OF ARRAY ELEMENTS The axially symmetric parabolic reflector with array feeds under consideration is shown in Fig. 1. Feed array geometry may be planar, cluster, or any other shape placed on the focal plane of a paraboloid. An obliquely incident arbitrary electromagnetic (EM) plane wave is incident from the direction . The EM plane wave is scattered by the reflector and produces a secondary field in the focal region. The focal region secondary field is found by transforming the physical optics integral over the reflector surface into a plane wave spectral (PWS) integral. This PWS integral is formed by the Fourier transform integral and is evaluated via the fast Fourier transform (FFT) algorithm [24]. Using the PWS-FFT approach, scattering fields from the reflector to the focal plane

(1)

, , and is the scatterwhere ing field characteristic of the parabolic antenna at the constant z plane. The evanescent field contribution is neglected and limits of the integration are chosen as . After the rearrangement of the above integral as a continuous

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Fig. 1. Configuration of an array-feed parabolic reflector illuminated by a plane wave.

we express the output of the array after the beam-forming network as where the superscript implies the conjugate transpose operation and consist of a set of vectors. The objective function for the array-power pattern with a specified shape and direction can be defined as

Fourier transform, we obtain

(2) We may write the above equation as an with windowing as

(5)

point FFT,

(3) and is the where two-dimensional rectangular windowing function. Here and are integers with which the FFT is carried out and , are the sampling rates. Determination of , , and windowing function dimensions depend on interelement spacing of the array elements. III. PATTERN OPTIMIZATION PROBLEM The linearly polarized plane-wave fronts emitted from sources impinge on the reflector surface. These rays scatter from the reflector surface to an array consisting of elements with some known geometry and polarization characteristics. The direction of these sources are represented by . The secondary rays at the output of the th array element are denoted by the vector vector and is expressed as (4) where the vector represents the gain and polarization response of the th array element, and is the secondary field from reflector to the th array element at the . The explicit form of is given in (3). location Letting denote a -dimensional complex vector of adjustable weights (excitation vector when the reflector is transmit mode)

where is a matrix describing the coupling between secondary scattering fields which reach the field elements. is the template function specifying all requirements on the desired pattern. One possibility is which results in minimum average sidelobe level. The optimal weight vector is found from the following constrained minimization problem: min subject to

(6)

where is complex constraint vector and is the number of constraints. This problem is known as LCMV beam former, and the solution of the above constrained problem is formally given by (7) exist. and consist of and provided constraint vectors, respectively. For the special case where the spacing between array elements is large compared to wavelength, coupling between the element receiving patterns is small. Then matrix is nearly diagonal. If our constraint vector is defined only in the main-beam direction , is reduced to the quiescent where response weight vector [25] is some proportionality constant and denotes conjugate operation. In the reflector antenna context, quiescent response weight vector is known as CFM weight vector. LCMV beamformer approach can be applied to the array problem with specified constraints such as sidelobe level and main-beam direction, provided sidelobe peak location and desired field

SAKA AND YAZGAN: REFLECTOR ANTENNA WITH PLANAR-ARRAY AND CLUSTER FEEDS

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amplitudes and phase are specified previously. As another possibility, the desired power pattern instead of the field can be considered. In this case, template function can be chosen as a real function which contains all the constraints in the range of the pattern synthesis. The Chebyshev polynomials are quite suitable for this purpose and defined as [26]–[27] constant (8) . Here, and where denote the desired main-beam direction and equal sidelobe level in decibels. and are some parameters which can be manipulated to describe or approximate a desired pattern ( i.e., main beam width, sidelobe level number in the specified range). Using the template function as a power pattern, objective function minimization can be achieved with an iterative method such as CGM. Rate of convergence of iterative methods depends on the choice of appropriate initial conditions. For this purpose, the optimum weight vector of LCMV beam former with desired main-beam direction constraint is used as an initial condition of the CGM. The optimum weight vector is constructed algorithmically in the CGM as follows: i) calculate , optimum weight vector of LCMV beam former with the main-beam direction constraint, ii) determine the template function with parameters ( , , , ), iii) choose initial value of weight vector as . Let the iteration index be zero, iv) calculate the objective function using (5), v) calculate the gradient of using

where superscript denotes the array element number, vi) find such that is minimum, vii) if let go to step iv), otherwise terminate the iteration. IV. RESULTS In the numerical calculations, the reflector antenna diameter is taken as 540 and array element spacing is chosen as 2. ratio of the reflector is 0.47. The incoming field is polarized in the direction. In (4), array-element polarization is chosen as - directed, and gain response is unity for the all values. Array-element gain response may be chosen as a Huygens source or dipole for realistic calculations. In the cases of interest, reflector diameter is very large with respect to the wavelength and main beam is very narrow, thus, our unity gain response assumption should be reasonable. We consider two array configurations, one of which is a 7 7 planar array and the other one is a 37 element cluster feed. The CGM and LCMV beam forming techniques set out in the

Fig. 2. Directivity of reflector antenna with 37-element cluster feed, D 540 and f=D = 0:479.

=

Fig. 3. Directivity of reflector antenna with 49-element planar feed, D = 540 and f=D = 0:479.

previous section were used to optimize the receiving pattern of array-feed paraboloid for continuous scan in the range 0.6 and 0.6 which corresponds to 12 beamwidths. Figs. 2 and 3 show achieved pattern analysis results given as directivity for each scan-angle case. In these figures, results of LCMV beam former and CGM optimization are given by solid lines and crosses, respectively. In the calculations, , , , and are used for the Chebyshev polynomial parameters. Results are given only for positive scan values, since array feed and parabola geometry are symmetric with respect to the focal plane. The directivity or directive gain of the reflector antenna with array feed versus desired scan angles was calculated by

(9)

and are the desired scan direction. In the calwhere culations, is kept at a constant value of . The calculated results in Figs. 2 and 3 (cluster and planar feeds, respectively) show that LCMV beam-former directivity is slightly higher than the CGM optimization for both array-feed types. The maximum achievable directivity [17] is calculated as . In Fig. 3, directivity of the

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Fig. 4. Obtained normalized electric field pattern of reflector antenna with 49-element planar array feeds for 0 scan case.

Fig. 6. Obtained normalized electric-field pattern of reflector antenna with 37-element cluster feeds for 0 scan.

Fig. 5. Obtained normalized electric-field pattern of reflector antenna with 49-element planar array feeds for 0.44 scan.

Fig. 7. Obtained normalized electric-field pattern of reflector antenna with 37-element cluster feeds for 0.62 scan.

LCMV beam former and CGM optimization results for planararray feeds exceed this maximum value and superdirectivity phenomenon occurs. In Fig. 2 (cluster feeds), the LCMV beam former shows superdirectivity for some scan angles, whereas the CGM optimization result always remains below 64.6dBi. Maximum sidelobe level and scan-error performances are shown in Figs. 4–7 to compare the two-pattern analysis methods. The sidelobe level suppression in CGM optimization is above 17 dB in the whole scan range. The sidelobe suppression of LCMV beam former is not stable and falls below 0 dB in certain places. Obtained and desired scan angle results show that CGM optimization settles to the desired scan angle in the range of 0 to 0.68 degrees where LCMV beam former does not. Figs. 4–7 show the achieved pattern relative to maximum field strength for several scanning angles, and we find that CGM optimization gives us more sidelobe suppression and true settlement of desired scan angle. An important result emerging from our calculations is that the reflector antenna with cluster feed improves the maximum scan range as also noted by Lam et al. [17]. Reflector antenna with cluster-array feeds provides steering of main beam up to 0.68 . Using the same reflector size with planar-array feeds only gives steering main beam up to 0.48 . It means that maximum scan angle of the reflector antenna without feed scanning can be increased using good optimization algorithm and array configuration.

Fig. 8. Obtained normalized electric-field pattern of reflector antenna for multibeam purpose s 0:4 .

=6

Fig. 8 illustrates a multibeam reflector-antenna design. Two template functions are derived from (8) steered to and . Two desired main-beam directions are 0.4 degrees. The directive gains are calculated as 45.07 dBi and 51.16 dBi for planar and cluster feeds, respectively. The sidelobe performance is similar for both planar and cluster feeds in Fig. 8. We also considered larger scan angles such as singlebeam case and found similar results. We found that cluster feed causes scan-angle increment for the multibeam design as same value as in single-beam cases.

SAKA AND YAZGAN: REFLECTOR ANTENNA WITH PLANAR-ARRAY AND CLUSTER FEEDS

V. CONCLUSION In this paper, we studied two methods, LCMV beam former and CGM optimization for the determination of the weight vectors or the excitation coefficients of planar-array feeds and cluster feeds of a reflector antenna. The CGM optimization with optimum weight vector of LCMV beam former as initial weight vector for determining element weight was found to yield better sidelobe level suppression and scan-error performances than the LCMV or CFM techniques. Our results also show that the continuous scanning up to 12 beamwidths for cluster feeds, and 8.5 beamwidths for planar feed case are obtained using the CGM optimization. Cluster feed with CGM optimization method combine to yield a wide scanning range and no superdirectivity problem occurs. A brief example is also have been given for the multibeam reflector-antenna design. REFERENCES [1] J. Ruze, “Laterally-feed displacement in a paraboloid,” IEEE Trans. Antennas Propagat., vol. AP-8, pp-368–379, 1960. [2] A. W. Rudge, “Offset-reflector antennas with offset feeds,” Electron. Lett., vol. 9, no. 16, pp. 611–613, Dec. 1973. [3] A. W. Rudge, “Multiple-beam antennas: offset reflectors with offset feeds,” IEEE Trans. Antennas Propagat., vol. AP-23, pp. 317–322, 1975. [4] C. M. Rappaport, “An offset bifocal reflector antenna design for wideangle beam scanning,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 1196–1204, Nov. 1984. [5] A. W. Mrstik and P. G. Smith, “Scanning capability of large parabolic cylinder reflector antennas with phased-array feeds,” IEEE Trans. Antennas Propagat. , vol. AP-29, pp. 455–462, May 1981. [6] S. W. Lee and Y. Rahmat-Samii, “Simple formulas for designing an offset multibeam parabolic reflector,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 472–478, May 1981. [7] Y. Rahmat-Samii and S. W. Lee, “Directivity of planar array feeds for satellite reflector applications,” IEEE Trans. Antennas Propagat., vol. AP-31, pp. 462–470, May 1983. [8] P. C. Loux and R. W. Martin, “Efficient aberration correction with a transverse focal plane array technique,” IEEE Conv. Record, vol. 12, pt. 2, pp. 125–131, 1964. [9] R. N. Assaly and L. J. Ricardi, “A theoretical study of a multi-element scanning feed system for a parabolic cylinder,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 601–605, Sept. 1966. [10] A. W. Rudge and M. J. Withers, “New technique for beam steering with fixed parabolic reflector,” Proc. IEE, vol. 118, pp. 857–863, July 1971. [11] T. S. Bird, J. L. Boomars, and P. J. B. Clarricoats, “Multiple-beam dualoffset reflector antenna with an array feed,” Electron. Lett., vol. 14, pp. 439–441, July 1978. [12] C. C. Hung and R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propagat., vol. AP-31, pp. 756–763, Sept. 1983. [13] S. J. Blank and W. A. Imbrella, “Array feed synthesis for correction of reflector distortion and Vernier beamsteering,”IEEE Trans. Antennas Propagat. , vol. AP-36, pp. 1351–1358, Oct. 1988. [14] A. R. Cherrette, R. J. Acosta, P. T. Lam, and S. W. Lee, “Compensation of reflector antenna surface distortion using an array feed,” IEEE Trans. Antennas Propagat., vol. 37, pp. 966–978, Aug. 1989. [15] H. Serizawa and K. Hongo, “Synthesis of offset parabola with open ended waveguides feed,” IEEE Trans. Antennas Propagat., vol. 39, pp. 535–542, Apr. 1991. [16] R. M. Davis, C. C. Cha, S. G. Kamak, and A. Sadigh, “A scanning reflector using an off-axis space-fed phased-array feed,” IEEE Trans. Antennas Propagat., vol. 39, pp. 391–399, Mar. 1991. [17] P. T. Lam, S. W. Lee, D. C. D. Chang, and K. C. Lang, “Directivity optimization of a reflector antenna with cluster feeds: Closed-form solution,” IEEE Trans. Antennas Propagat., vol. AP-33, pp. 1163–1174, Nov. 1985.

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[18] W. T. Smith and W. L. Stutzman, “A pattern synthesis technique for array feeds to improve radiation performance of large distorted reflector antennas,” IEEE Trans. Antennas Propagat., vol. 40, pp. 57–62, Jan. 1992. [19] B. Saka and E. Yazgan, “Reflector antenna pattern improvement using optimized array feed,” Can. J. Phys., vol. 73, pp. 407–411, July/Aug. 1995. [20] B. Saka and E. Yazgan, “Adaptive beam forming for array feed parabolic reflector,” in Int. Symp. Antennas—JINA, Nice, France, Nov. 1994, pp. 540–543. [21] R. A. Shore, “Adaptive nulling in hybrid reflector antennas,” Electromagn., vol. 15, pp. 93–121, Jan./Feb. 1995. [22] T. S. Bird, “Contoured-beam synthesis for array-fed reflector antennas by field correlation,” IEE Proc., vol. 129, pp. 293–298, Dec. 1982. [23] M. C. Bailey, “Determination of array feed excitation to improve performance of distorted or scanned reflector antennas,” in AP-S Symp. Dig., June 1991, pp. 175–178. [24] A. Nagamune and P. H. Pathak, “An efficient plane wave spectral analysis to predict the focal region fields of parabolic reflector antennas for small and wide angle scanning,” IEEE Trans. Antennas Propagat., vol. 38, pp. 1746–1756, Nov. 1990. [25] C. T. Tseng and L. J. Griffiths, “A unified approach to the design of linear constraint in minimum variance adaptive beamformers,” IEEE Trans. Antennas Propagat., vol. 40, pp. 1533–1542, Nov. 1992. [26] A. T. Villeneuve, “Taylor patterns for discrete arrays,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 1089–1093, Oct. 1984. [27] B. P. Ng, M. H. Er, and C. Cot, “A flexible array synthesis method using quadratic programming,” IEEE Trans. Antennas Propagat., vol. 41, pp. 1541–1550, Nov. 1993.

Birsen Saka (S’90–M’96) received the B.S., M.S., and Ph.D. degrees in electrical engineering from Hacettepe University, Ankara, Turkey, in 1986, 1989, and 1996, respectively. She worked at the Military Electronics Corporation (ASELAN), Ankara, Turkey, as a Research Engineer from 1986–1989. She is currently an Assistant Professor in the Electrical Engineering Department at Hacettepe University, Ankara, Turkey. Her research interests include reflector antennas, array processing, asymptotic techniques, and optimization problems.

Erdem Yazgan (M’91) received the B.S. and M.S. degrees in electrical engineering from the Middle East Technical University (METU), Ankara, Turkey, in 1971 and 1973, respectively, and the Ph.D. degree from Hacettepe University, Ankara, Turkey, in 1980. In 1984, she obtained an Associate Professor position in Hacettepe University, Ankara, Turkey. Since 1990, she has been a Full Professor in the same university. In 1994 she was a Fulbright Visiting Professor at the ElectroScience Laboratory, Ohio State University, Columbus, OH. In 1989 she was a Visiting Professor in Essex University, U.K. From 1989 to 1993 she worked as an Advisor in the World Bank Project of Technical Schools, Ankara, Turkey. Currently, she is the head of the Electromagnetic Fields and Microwave Theory Science Division, Hacettepe University, and the Chairman of the Radio and TV Special Standardization Committee of Turkish Standardization Institute, Ankara. Her research interests are high-frequency asymptotic solutions, calculation of radiation patterns and the effect of surface loading in reflector antennas and the effect of ground topography in low-altitude radar, Gaussian-beam solutions, and the application of electromagnetic waves in medical, high-frequency propagation, microstrip resonator, and microwave-integrated circuit design.