Fixed Beamwidth Electronic Scanning Antenna Array Synthesis and Its

Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 1333

Fixed Beamwidth Electronic Scanning Antenna Array Synthesis and Its Application to Multibeam Pattern Synthesis Amr H. Hussein1 , Haythem H. Abdullah2 , Mohammed Nasr1 , Salah Khamis1 , and Ahmed M. Attiya1 1

Faculty of Engineering, Tanta University, Tanta, Egypt 2 Electronics Research Institute, Dokki, Giza, Egypt

Abstract— In some applications, where, the scanning rate is not a critical factor, mechanically rotating system is sufficient to do fixed beamwidth scanning. The electronic scanning is an optimum solution to the high scanning rate applications, but it suffers from the beamwidth broadeningand gain variations with steering angles. In this paper, a fixed beamwidth electronic scanning algorithm is proposed. The proposed algorithm is based on synthesizing sets of excitation coefficients to direct the main beam at some scanning angles. The synthesis takes into consideration the fixation of the beamwidth at these angles. The synthesis of the excitation coefficients will be done using a scheme based on the moment method due to its accuracy in solving such problems. The optimum spacing between elements will be determined using the genetic algorithm. One of the main advantages of the proposed algorithm is the applicability of synthesizing multibeam antenna array of fixed beamwidths using the superposition principle. 1. INTRODUCTION

The synthesis of fixed beamwidth scanned antenna arrays with minimum number of antenna elements is of main concern in many applications such as Radar systemsand Tracking systems. Some of applications have mechanical steering systems, where a directive antenna array of a specific beam is mounted on a mechanical rotating system that directs the array to any direction keeping its beam unchanged such as rolling radar [1]. But the rate of direction change of the mechanical system is slow. The electronic scanning is used when it is necessary to vary the direction, or rate of change of direction of the array beam faster than is possible by mechanical movement of the aperture [2]. But electronic scanned radars suffer from the beamwidth variations with steering the main beam direction. The variations in the antenna array beamwidth as a function of the steering angle results in corresponding gain variations, and broadening of the main beam beamwidth which may cause signals from different transmitters to interfere with the desired transmitter signal. In this paper, it is introduced a new algorithmbased on a combination between the method of moments (MoM) [3] and the genetic algorithm(GA) [4–7]. The proposed algorithm is used for the synthesis of scannedlinear antenna arrays, and multi-beam antenna arrays with fixed minimum beamwidth at any direction with minimum number of equispaced antenna elements. 2. PROBLEM FORMULATION 2.1. Synthesisof Fixed MinimumBeamwidth Scanned Antenna Arrays

One of the most commonly used scanned antenna arrays is the phase scanned antenna array that is a linear array of uniformly spaced elements, where the direction of the main beam can be steered to any direction by adding progressive phase shift to the array excitation coefficients. But in addition to changing the main beam direction, beam steering changes the beamwidth. The main lobe becomes broader as the beam is steered away from broadside direction. Consequently, for a linear antenna array the end-fire beam is wider than the broadside beam [8]. The array factor of a linear antenna array consisting of isotropic antenna elements positioned symmetrically along the z-axis with uniform element spacing d is given by: AF (θ) =

N X n=1

µ µ ¶ ¶ N +1 an exp j n − kdcos θ 2

(1)

where an is the excitation coefficient of the nth element, d is the element spacing, and k = 2π/λ is the free space wave number [9]. Recently, we presented a synthesis technique based ona combination between the method of moments and the genetic algorithm (MoM/GA) [9]. The MoM/GA is utilized to reconstruct new element locationsand excitations that fulfill the required characteristics

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in the desired pattern. This synthesis method has shown its ability of reducingthe number of elements for linear antenna arrays with pencil-beampatterns and shaped-beam patterns as mentioned in [9]. The MoM/GA algorithm is based on solving a system of linear equations which is written in a matrix form as [Z]M ×M [I]M ×1 = [V ]M ×1 (2) [I]M ×1 is the excitation coefficients vector to be determined, such that [I]M ×1 = [a1 , a2 , a3 , . . . , aM ]T . The elements of the [Z]M ×M matrix take the form Z π zmn = ej(n−m)kd cos(Θ) dΘ (3) 0

To get a radiation pattern with approximately the same minimum beamwidth as the broadside pattern at any steering direction φo the following procedure is followed: A. The excitation coefficients an |φo =90◦ of the broadside pattern AF (Θ)|φo =90◦ are calculated. B. Adding phase shift β = +(90 − φo ) to the angle Θ and replacing an by an |φo =90◦ in Eq. (1). C. As a result, a replica of the broadside pattern is generated and rotated by φo from the array line AF (Θ + 90 − φo ) whose equation is given by Z AF (Θ+90 − φo ) =

N

n=1

(an |φo =90◦ )ej(n−

N +1 2

)kdcosφ(Θ+90−φo )

(4)

The resultant pattern is applied as the desired pattern within the synthesis scheme presented in [9] such that AF d (Θ) = AF (Θ+90 − φo ). In this case, the elements of the vector [V ]M ×1 are given by Z π M +1 Vm = AF d (Θ)e−j(m− 2 )kd cos(Θ) dΘ (5) 0

The excitation coefficients an are determined by solving the linear system of Eq. (2). For multibeam synthesis, the directions of the multibeam need to be assigned first, then by applying the MoM/GA to synthesis each individual beam, the excitation coefficients for each direction will be available. By summing the excitation coefficients at all the desired directions and applying this summation to excite the array, the multibeam pattern will be achieved. 2.2. Estimation of the Optimum Element Spacing

In order to get the optimum element spacing, GA is utilized. The required cost function is written as follows CF = {|SLLd | − |SLLs |}HP BW s =HP BW B (6) SLLd and SLLs are the desired and the synthesized patterns side lobe levels, respectively. HPBW B is the desired pattern half power beamwidth at the broadside direction φo = 90◦ , and HPBW s is the synthesized pattern half power beamwidth at the desired direction φo . In this sequence, it is assumed no error in the HPBW. For a specific number of elements M, the GA searches for the optimum d that provides a zero error in the HPBW and at the same time introduces a minimum errorin the SLL. 2.3. Implementation of Fast and Continuous Range Fixed Beamwidth Scanning

Ordinary digital beam scanning techniques are based on generating a set of excitation coefficients that corresponds to the desired directions, and store them in lock up table (LUT). But these techniques suffer from; the huge memory storage especially for antenna arrays consisting of large number of elements, small scanning angle resolution, and the discrete scanning. For N elements array, N excitation coefficients for each direction are necessary. The proposed scheme solved these problems by generating two polynomial sets, Pnr (φ), and Pni (φ), each of order Q that are used to determine the real and imaginary parts of the excitation coefficients, respectively. The required memory size in this technique is [2 × N × (Q + 1)] to store the 2N polynomials coefficients where (Q+1) is the number of coefficients per polynomial. The polynomials generation follows these steps: Step (1): By solving Eq. (2) for the set of main beam directions


φ0 = 0, 1, 2, 3, . . . , 90, we get 91 different excitation coefficients sets. These excitation coefficients are arranged in a matrix form [C]N X91 that could be expressed as follows   a1 (0) a1 (1) a1 (2) a1 (3) . . . a1 (90)   a2 (0) a2 (1) a2 (2) a2 (3) . . . a2 (90)     a3 (0) a3 (1) a3 (2) a3 (3) . . . a3 (90)    (7) [C]N ×91 =  .. .. .. .. ..   . ... . . . .     .. .. .. .. .. ..   . . . . . . aN (0) aN (1) aN (2) aN (3) . . . aN (90) Each column of the coefficient matrix [C]N ×91 represents the excitation coefficients for a specific tilting angle φ0 . The excitation coefficients are of complex values such that an (φ0 ) = rn (φ0 ) + jbn (φ0 )

n = 1, 2, 3, . . . , N

(8)

The excitation coefficients an (90◦ +φ) equal the conjugate of the excitation coefficients at (90◦ −φ). an (90◦ +φ) = an (90◦ − φ)∗

(9)

Step (2): Substitute each coefficient of Eq. (7) by its real and imaginary parts. By using curve fitting techniques in Matlab (2N ) polynomials of order (Q) are generated. The first set ofpolynomials, Pnr (φ), are used to fit the real parts of the excitation coefficients within each row of the coefficient matrix [rn (0) rn (1) rn (2) rn (3) . . . rn (90)] where Pnr (φ) = pn1 φQ + pn2 φQ−1 + . . . + pn(Q+1) ,

n = 1, 2, 3, . . . , N

(10)

where {pn1 , pn2 , pn3 , pn4 , . . . , pnQ , pn(Q+1) } are the coefficients of the nth polynomial. The second set of polynomials, Pni (φ), are used to fit the imaginary parts of the excitation coefficients within each row of the coefficient matrix [bn (0) bn (1) bn (2) bn (3) . . . bn (90)] where Pni (φ) = cn1 φQ + cn2 φQ−1 + . . . + cn(Q+1) ,

n = 1, 2, 3, . . . , N

(11)

{cn1 , cn2 , cn3 , cn4 , . . . , cnQ , cn(Q+1) } are the coefficients of the nth polynomial. The excitation coefficients at 90◦ < φ0 ≤ 180◦ are obtained from Eq. (9). 3. RESULTS AND DISCUSSIONS 3.1. Synthesis of Multibeam Pattern with Fixed Beamwidths within The Sector (0◦ ≤ φo ≤ 180◦ )

Consider a twenty elements λ/2 Tschebyscheff array with SLL = −30 dB [11] where the broadside beam pattern is chosen as the desired pattern at all directions. The radiation pattern at the broadside direction has the minimum half power beamwidth (HP BW min = 6.1879◦ ), while the end-fire direction has the maximum half power beamwidth (HP BW max = 38.674◦ ) in case of normal phase scanning. The change in the HPBW from end-fire to broadside direction equals ∆HP BW = HP BW max − HP BW min = 38.674◦ − 6.1879◦ = 32.4861◦ . For the fixed beamwidth synthesis within the sector (0◦ ≤ φo ≤ 180◦ ), the end-fire main beam at φo = 0◦ is required to be close to that of the broadside main beam. This can be achieved using numbr of antenna elements M = Mmin = 2N = 40 elements with optimum element spacing d = 0.33λ. There is a slight change in the synthesized end-fire pattern beamwidth compared to the broadside beamwidth of ∆HP BW = 6.961◦ − 6.1879◦ = 0.7731◦ . An acceptable side lobe level SLL = −25.7368 dB is noticed.To get the same HP BW = 6.961◦ , and SLL = −25.7368 dB as the synthesized array using the ordinary λ/2 Tschebyscheff array it requires at least N = 480 antenna elements. Using this array structure,one can synthesize multibeam patterns with relatively fixed beamwidths at all directions. The synthesized multibeam pattern at angles of 0◦ , 40◦ , and 120◦ is shown in Fig. 1. 3.2. Implementation of Fixed Beamwidth Scanning Using The Coefficient Fitting Procedure

Consider a 20 elements λ/2 Tschebyscheff pattern. Applying the proposed coefficient fitting procedureand by using the fitting polynomials with Mmin = 2N = 40, it is found that the optimum element spacing using the GA is d = 0.33λ and the best fitting ploynomials are of order Q = 24.

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Figure 1: Synthesized multibeam pattern of fixed beamwidth beams directed at φ1 = 0◦ , φ2 = 40◦ and φ3 = 120◦ compared to the ordinary multibeam λ/2 Tschebyscheff pattern.

Figure 2: Synthesized pattern of the minimum beamwidth HP BWmin = 6.1879◦ directed at φo = 40.345◦ .

Figure 3: Synthesized pattern of the minimum beamwidth HP BWmin = 6.1879◦ directed at φo = 161.077◦ .

So the memory locations required to store the polynomial coefficients is 2 × 40 × (24 + 1) = 2000. Fixed beamwidth patterns at φo = 40.345◦ and φo = 161.077◦ are synthesized using the polynomials fitting functionsthat gives the same HP BW = 6.1879◦ at the broadside direction as shown in Fig. 2 and Fig. 3. 4. CONCLUSION

An efficient use of the MoM/GA algorithm to synthesis fixed beam width radiation pattern at different angles is presented. The MoM is used to estimate the excitation coefficients of the antenna array while the GA is used to estimate the optimum element spacing. The array synthesis with fixed beamwidth is performed with the minimum number of antenna elements which is much lower than the required number of elements in case of the ordinary phased scanningarrays. The proposed algorithm has efficiently overcome the beamwidth broadeningand gain variations with steering angles in the phase scanned arrays. In addition, a high rate and continuous fixed beamwidth scanning technique is presented. The proposed technique requires much lower memory storage than the traditional digital beam scanning techniques which are based on the look-up tables. Also the proposed technique can be used when it is necessary to vary the direction, or rate of change of direction of the array beam faster than that is possible by the mechanical movement of the stationary beam arrays. REFERENCES

1. Tietjen, B. W., “The rolling radar,” 2005 IEEE International Radar Conference, 16–21, May 9– 12, 2005. 2. Radford, M. F., “Electronically scanned antenna systems,” Proceedings of the Institution of Electrical Engineers, Vol. 125, No. 11, 1100–1112, Nov. 1978. 3. Harrington, R. F., Field Computation by Moment Methods, IEEE Press, 1993. 4. Rahmat-Samii, Y. and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms, John Wiley & Sons, Inc., 1999.


5. Houck, C. R., J. A. Joines, and M. G. Kay, “A genetic algorithm for function optimization: A matlab implementation,” http://www.ie.ncsu.edu:80/mirage/GAToolBox/gaot/. 6. Ting, C.-K., “On the mean convergence time of multi-parent genetic algorithms without selection,” Advances in Artificial Life, 403–412, 2005, ISBN 978-3-540-28848-0. 7. Haupt, R. L. and D. H. Werner, Genetic Algorithms in Electromagnetics, IEEE Press WileyInterscience, 2007. 8. Hansen, R. C., Phased Array Antennas, Wiley & Sons, 1998. 9. Hussein, A. H., H. H. Abdullah, A. M. Salem, S. Khamis, and M. Nasr, “Optimum design of linear antenna arrays using a hybrid MoM/GA algorithms,” IEEE Antennas Wireless Propag. Lett., Vol. 10, 2011. 10. Balanis, C. A., Antenna Theory: Analysis and Design, 3rd Edition, Wiley, New York, 2005. 11. Orfanidis, S. J., Electromagnetic Waves and Antenna, 2004, Available: http://www.ece.rutgers.edu/∼orfanidi/ewa/.