II. REFLECTOR ANTENNA DESIGN

MFRA consists of a reflector antenna which is fed by a planar array of horn antennas, shown in Fig. 1. The reflector antenna is an offset parabolic reflector antenna with D = 22λ, H = 10λ, and F = 16λ where D is the diameter of the offset parabolic reflector antenna, H is the offset height of the reflector, and F is the focal point of the initial reflector with diameter of Dp (Dp = 2D + 2H = 64λ). Array elements are placed at (x _f–y _f) plane at focal point of the reflector antenna and rotated toward the center point of offset parabolic reflector with angle of ψ _C. One of the advantages of MFRA with parabolic offset reflector is that the feed network illuminates the reflector surface from the offset to prevent the blockage problem. In equations (1–4), the general relationships between the angles and D, H, and F are presented.

(1)$$\displaystyle{{\displaystyle{D \over 2} + H} \over F} = 2\tan \left({\displaystyle{{{\psi _C}} \over 2}} \right)\comma$$

(2)$$\displaystyle{{D + H} \over F} = 2\tan \left({\displaystyle{{{\psi _U}} \over 2}} \right)\comma$$

(3)$$\displaystyle{H \over F} = 2\tan \left({\displaystyle{{{\psi _L}} \over 2}} \right)\comma$$

(4)$${\psi _B} = \displaystyle{{{\psi _U} + {\psi _L}} \over 2}.$$

Fig. 1. Offset parabolic reflector antennas fed by an array of horn antennas [4].

Based on these equations for the feed array network, ψ _U= 90, ψ _L= 35, ψ _C= 67, and ψ _B= 62.5 are achieved.

Each array element has 10 dB gain with waveguide diameter of 0.68λ, flare length of 0.6λ, aperture diameter of 0.9λ, and spacing (center to center) of λ with the next element in both x _f and y _f directions. The feed network consists of 25 horn antennas with rectangular lattice of 5 × 5 at center frequency of 10 GHz. The excitations value of the feed network should be optimized to achieve the desired radiation pattern. The desired radiation pattern is an elliptical-shaped beam in the U–V plane. In equations (5) and (6), the definitions of U and V are given.

(5)$$U = \sin\! \theta \cos \varphi\comma$$

(6)$$V = \sin\! \theta \sin \varphi .$$

III. OPTIMIZATION METHODS

Three conditions, amplitude-only, phase-only, and amplitude–phase optimizations, are considered for pattern synthesis of the desired elliptical-shaped beam. To achieve the desired radiation pattern, the differences between the calculated and desired patterns are minimized. To compare different evolutionary algorithms for the pattern synthesis problem, the error equation or cost value should be defined. Generally, this value is calculated by root-mean-square (RMS) of differences in samples [Reference Foudazi and Mallahzadeh4]. Then, the initial parameters of GA, PSO, and IWO should be defined properly.

Basically, GA is inspired from the biological behavior of genes and their evolution for the next generations. In this paper, the tournament method for mating and the uniform crossover method for providing binary chromosomes for offspring are chosen based on [Reference Haupt and Werner9].

Also, PSO is based on particle swarm to find the source of food. For the PSO, the speed of the swarms are define as equation (7), where w is the inertial weight, P is the personal best, G is the global best, c ₁ is the cognitive rate, c ₂ is the social rate, and V _max is the maximum velocity. Random behavior for the swarm movements are added using random parameters of η ₁ and η ₂ based on [Reference Haupt and Werner9].

(7)$${V_t} = w{V_{t - 1}} + {c_1}{\eta _1}\lpar {P_{t - 1}} - {X_{t - 1}}\rpar + {c_2}{\eta _2}\lpar {G_{t - 1}} - {X_{t - 1}}\rpar .$$

IWO, as a new EA method, is based on the ecology and biology of weeds in nature. In IWO, provided seeds for the next iteration for each flowering planet are spread based on its standard deviation. This parameter is defined based on equation (8) and shows that the provided seeds spread closer to their parents as the number of iteration increased [Reference Foudazi and Mallahzadeh4]. In this equation, n _i is the number of iteration, N is the total iteration, m is the non-linear modulation index, S _min and S _max are the minimum and maximum numbers of produced seeds, and P _max is the total number of produced seed after initial iteration. The standard deviation for the maximum number of 50 iterations with initial and final deviations of σ _ini = 0.1 and σ _f = 0.001 is presented in Fig. 2. Based on equation (8), the standard deviation changes linearly for m = 1. Although m = 2 converged faster than m = 1, the convergence speed is slow and takes long time to find the best result due to wide distribution of seeds at the intermediate stage. Moreover, m = 4 converged very fast and cannot explore appropriately during the convergence. Thus, m = 3 can be a good number for fast and accurate convergence of IWO [Reference Karimkashi and Kishk13].

Fig. 2. Standard deviation of the IWO algorithm for m = 1, 2, 3, and 4 with σ _ini= 0.1, σ _f= 0.001, and N = 50.

(8)$${\sigma _i} = {\left({1 - \displaystyle{{{n_i}} \over N}} \right)^m} \times \lpar {\sigma _{ini}} - {\sigma _f}\rpar + {\sigma _f}.$$

For each number of iteration, the error value is calculated based on the RMS value of the differences between the desired and optimized radiation patterns. The value is calculated based on equation (9).

(9)$${\rm Cost} = \sqrt {\displaystyle{1 \over L}\sum {{{\lpar {G_i}\lpar \theta\comma \; \phi \rpar - {G_d}\lpar \theta\comma \; \phi \rpar \rpar }^2}} }\comma$$

(10)$$L = {L_\theta } \times {L_\phi }\comma$$

which G _d(θ, φ) is the desired radiation pattern and G _i(θ, φ) is the optimized radiation pattern in iteration of n _i. Also, L is described in equation (10), which is based on the number of total samples in the θ and φ planes to calculate the differences between desired and optimized radiation patterns. In order to compare these three methods, the maximum number of iterations for GA, PSO, and IWO are the same and fixed at N = 50. The maximum number of iteration is the number that the process of optimization is stopped irrespective to the RMS value of error.

IV. RESULTS AND DISCUSSION

A) Amplitude-only

To initiate the optimization algorithms, the initial population is 40 with the total number of 50 iteration (N = 50). The initial search areas for these three optimization methods are considered to be the same to be able to compare them based on their performance. While the same number of populations is chosen, the comparison is based on the speed of convergence with the same initial space and dimensions. In [Reference Foudazi and Mallahzadeh4, Reference Foudazi, Mallahzadeh and Samadi-Taheri5] and [Reference Chandrasekar and Ramana15], the comparison between different evolutionary methods were presented based on the same initial population size. Initial populations were chosen randomly with the same size and dimension for each method. Also, it is considered that the population size is 40 after the initial iteration. The dimension of the optimizer program for the feed network with 5 × 5 elements is 25 (only 25 amplitudes). Thus, 40 different points in a 25-dimensional space is created for each iterations and changes in [0–1] to find appropriate amplitude excitation. For PSO algorithm, the main parameters are c ₁= 2, c ₂= 2, w = 0.9, η ₁= 0.99, η ₂= 0.99, and V _max= 0.1. The main parameters of IWO algorithm are m = 3, S _min= 0, S _max= 5, σ _ini= 0.1, and σ _f = 0.001. In Figs 3 and 4, the optimization results are presented. As shown in Fig. 3, the final RSM error values based on equation (9) for L = 133 samples (L = L _θ × L _φ) are around 0.28 for PSO and IWO and 0.24 for GA. The PSO, IWO, and GA converged after 8, 12, and 17 iterations, respectively. As depicted in Fig. 4, initial (i = 1) and final (i = 50) iteration results for GA, PSO, and IWO are presented. The radiation patterns of GA, PSO, and IWO are acceptable at the final iteration. For the amplitude-only optimization, good radiation pattern for synthesis of simple elliptical-shaped beam of MFRAs is achieved for IWO. Although PSO and GA have been used widely as accurate optimization processes, IWO shows promise to be an applicable and accurate method for pattern synthesis problems with simple algorithm and few tuning parameters.

Fig. 3. Cost diagram of MFRA with GA, PSO, and IWO for the amplitude-only optimization.

Fig. 4. Optimized radiation pattern of MFRA with GA, PSO, and IWO for the amplitude-only optimization (G _d(θ, φ) is dashed line and G _i(θ, φ) is solid line).

B) Phase-only

In this section, phase of 25 fed antennas is considered to be optimized to achieve the desired simple elliptical-shaped beam in the U–V plane. To initiate the optimization algorithms, the initial parameters for GA, PSO, and IWO are the same as the previous section, while the phase changes in [0–360] for optimizations process. In Fig. 5, the convergence diagram of phase-only optimization for MFRA with GA, PSO, and IWO is presented. It is shown that the RSM value of error between desired and calculated radiation patterns at L = 133 samples becomes converged around 0.4 after about half of iterations (around 25 iterations). The GA and PSO become converged after 40 iterations. Also, it is obvious that IWO converged after 34 iterations. As presented in Fig. 6, the radiation patterns of GA, PSO, and IWO are acceptable at the final iteration. For phase-only optimization, it is obvious that IWO converged at 0.36 and provides good radiation pattern for synthesis of simple elliptical-shaped beam for MFRA. Also, in comparison with amplitude-only optimization, it is clear that the results of amplitude-only converged at smaller error than phase-only optimization. In other words, it means that better results can be achieved using amplitude optimization than phase optimization. The capability of the IWO method shows the promise of having robust, simple, and accurate method for phase optimization of array antennas.

Fig. 5. Cost diagram of MFRA with GA, PSO, and IWO for the phase-only optimization.

Fig. 6. Optimized radiation pattern of MFRA with GA, PSO, and IWO for the phase-only optimization (G _d(θ, φ) is dashed line and G _i(θ, φ) is solid line).

C) Amplitude–phase

In this section, the amplitude and phase of 25 fed antennas is considered to be optimized to achieve the desired radiation pattern. To initiate the optimization algorithms, the initial parameters for GA, PSO, and IWO are the same as the first section. The dimension of the optimization space is 50 in this case, whereas in the two previous conditions it was 25. In Fig. 7, the convergence diagram of amplitude–phase optimization for MFRA with GA, PSO, and IWO is presented. It is shown that the RSM value of error between desired and calculated radiation patterns at L = 133 samples becomes converged around 0.25 before reaching half of iterations (around 20 iterations). Although PSO converged sooner than IWO, but IWO error is less than PSO. In addition, GA is converged later than IWO with worse error values. The PSO and GA become converged after 18 and 27 iterations, respectively. Also, it is obvious that IWO converged after 22 iterations. As presented in Fig. 8, the radiation patterns of GA, PSO, and IWO are acceptable at the final iteration. For amplitude–phase-only optimization, it is obvious that IWO converged at 0.36 and provides good radiation pattern for synthesis of simple elliptical-shaped beam of MFRAs.

Fig. 7. Cost diagram of MFRA with GA, PSO, and IWO for the amplitude–phase optimization.

Fig. 8. Optimized radiation pattern of MFRA with GA, PSO, and IWO for the amplitude–phase optimization (G _d(θ, φ) is dashed line and G _i(θ, φ) is solid line).