A MULTI-BEAM MODEL OF ANTENNA ARRAY PAT- TERN SYNTHESIS BASED ON CONIC TRUST REGION METHOD

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1 Progress In Electromagnetics Research B, Vol. 53, 67 90, 013 A MULTI-BEAM MODEL OF ANTENNA ARRAY PAT- TERN SYNTHESIS BASED ON CONIC TRUST REGION METHOD Tiao-Jun Zeng and Quan-Yuan Feng * School of Information Science and Technology, Southwest Jiaotong University, Sichuan , China Abstract In this paper, we propose a multi-beam model for antenna array pattern synthesis AAPS problem. The model uses a conic trust region algorithm CTRA similarly proposed in this paper to optimize its cost function. Undoubtedly, whole algorithm efficiency ultimately lies on the CTRA, thereof, we propose a method to improve the iterative algorithm s efficiency. Unlike traditional trust region methods that resolve sub-problems, the CTRA efficiently searches a region via solving a inequation, by which it identifies new iteration points when a trial step is rejected. Thus, the proposed algorithm improves computational efficiency. Moreover, the CTRA has strong convergence properties with the local superlinear and quadratic convergence rate under mild conditions, and exhibits high efficiency and robustness. Finally, we apply the combinative algorithm to AAPS. Numerical results show that the method is highly robust, and computer simulations indicate that the algorithm excellently performs AAPS problem. 1. INTRODUCTION Although array pattern synthesis problems have been extensively investigated over the last several decades [1 9], most of them are single-beam methods, and very few focus on multi-beam case [10, 11]. A multi-beam antenna is an antenna which is able to create multibeams simultaneously. Since the size of antennas is fixed by physical considerations, the difficulty of accommodating them on satellites and other facilities with multiple antenna-based functions grows with the number of such functions if a separate antenna Received 15 June 013, Accepted 31 July 013, Scheduled August 013 * Corresponding author: Quan-Yuan Feng fengquanyuan@163.com.

2 68 Zeng and Feng is used for each. In this context, it is desirable for a single antenna to be able to generate multiple radiation patterns, with the excitation corresponding to each being selectable by suitable switching devices [1]. Multibeam antennas can be considered as effective approach to perform independent data streams between a transmitter and a receiver. In this paper, we propose a model to multi-beam AAPS, and this model is the extension of our previous work [7]. Under this model, the AAPS problem can be transformed into an unconstrained optimization problem. In recent years, the global optimization techniques, such as particle swarm optimization and differential evolution, have been used for AAPS problem [6, 13]. However, these algorithms are too time-consuming and limited in their application. Therefore, it is necessary to design a time-saving algorithms. Trust-region method of quadratic model for unconstrained optimization has been studied by many researchers [14 17]. Trust-region methods are robust, can be applied to ill-conditioned problems and have strong global convergence properties. Instead of quadratic approximation to objective function, Davidon [18] proposed conic model to approximate the objective function. Di and Sun [19] first presented a trust region method based on conic model for unconstrained optimization. The conic model has several advantages. First, if the objective function has strong nonquadratic behavior or its curvature changes severely, the quadratic model methods often produce a poor prediction of the minimizer of the function. In this case, conic model approximates the objective function better than a quadratic approximation, because it has more freedom in the model. Second, the quadratic model does not take into account the information concerning the function value in the previous iteration which is useful for algorithms. However, the conic model possesses richer interpolation information and satisfies four interpolation conditions of the function values and the gradient values at the current and the previous points. Using these rich interpolation information may improve the performance of the algorithms. Third, the initial and limited numerical results provided in [19] etc. show that the conic model methods give improvement over the quadratic model ones. Finally, the conic model methods have the similar global and local convergence properties as the quadratic model ones [0]. Recently, Many researchers have recently studied nonmonotone adaptive conic trust region methods NACTRM for unconstrained optimization problems [16, 19, 0]. NACTRM can automatically produce an adaptive trust region radius whenever a trial step is rejected, and decreases functional values after finite iterations. The

3 Progress In Electromagnetics Research B, Vol. 53, main disadvantage of NACTRM lies in identifying new trial iterations; it requires considerable computational time to repeatedly solve subproblems. Motivated by rectifying this shortcoming, we propose a solving-inequality conic trust region method SICTRM that adopts a different search approach at each iteration. The search direction d k = x k+1 x k is generated by solving the sub-problem of the cost function. If d k is rejected, the sub-problem does not need to be resolved. The search direction is generated by solving an inequality details in Section 3. The rest of the paper is organized as follows. Section presents the Multi-beam model. In Section 3, SICTRM is introduced and then proven as a well-defined algorithm with its convergence properties. Section 4 shows the experimental results and related discussions. Discussion and Conclusions are drawn in Sections 5 and 6, respectively. Notation: T is the transpose, is the complex conjugate, and represents the Frobenius norm.. MULTI-BEAM MODEL The array factor for a linear array with N isotropic elements can be expressed as: N pw, θ = w i e jφiθ = W T Sθ, θ [ 90, 90 ] 1 i=1 where θ is the direction of arrival of signal, j = 1, Sθ = [ 1, e jϕ θ,..., e ] jϕ θ T the steering vector, φi θ = πd i sin θ/λ the phase delay due to propagation, λ the wavelength of the transmitted signal, d i the position of the ith element of the antenna array d 1 = 0, and W = [ w 1, w,..., w N ] T the complexweight vector. Since the optimization approach is only applicable to real variable problems, a complex-to-real transform of the pw, θ is necessary. Firstly, introducing a power function P W, θ of the AF, i.e., P W, θ = pw, θ = pw, θp W, θ, then denote W 1 = [w1 1 w1... wn 1 ]T, W = [w1 w... wn ]T, Sθ = S 1 θ + js θ, S 1 θ = [1 cosφ θ... cosφ N θ ] T, S θ = [0 sinφ θ... sinφ N θ] T, W 1, W, S 1 θ and S θ R N. Thus, we obtain [7] P W, θ = [ W T 1 S 1 + W T 1 S 1 W T S + W T S + W T 1 S W T 1 S W T S 1 + W T S 1 ] R

4 70 Zeng and Feng Transition region Mainlobe region β α 1 Sidelobe region α α 3 α 4 Line 1:Multi-beam pattern Line :Array respond α 5 α θ d 1 θ θ θ θ d θ5θ6 θ7 θ8 d Figure 1. Definition of multi-beam synthesis. In the case where phase constraint is not considered, an optimal weight vector, W opt, is determined so that the pown or amplitude response P W, θ or pw, θ best approximates the desired pattern. The definition of multi-beam synthesis is illustrated in Figure 1, we propose a method for synthesizing multi-beam pattern of antenna array, it is formulated by a discreted angle penalty function based on pown response of antenna array as following: F W, A = F 1 W, A + F W, A 3 d 1 θ [ P m ] θ 1 θ [ P m ] F 1 W, A = α 1 P + + α P + F W, A = θ θ θ 1 θ 4 θ θ 3 θ θ 5 θ d α 1 d 1 [ P m ] f 1 P + + f 1 [ P f P + [ α 4 P + P f α 4 m ] + m ] + θ 3 β P θ [ P α 3 P + d θ θ 4 θ 6 θ θ 7 θ 8 θ [ P m ] + β P + f 4 P + + f 4 θ 6 θ 7 θ θ 5 α 3 α m ] 4 [ P m ] f 3 P + f 3 d 3 θ θ 8 [ α5 P

5 Progress In Electromagnetics Research B, Vol. 53, P m ] + + α 5 90 d 3 [ α 6 P P m ] + λ α 6 6 lg α i 5 where θ is the angle resolution, P = P W, θ defined by Eq., constant m > 0, β = r α i r is a given value, in general, and r 0 will satisfy the condition: β 100α i, f 1 = β α θ θ 1 θ θ 1 + α, f = β α 3 θ 3 θ 4 θ θ 4+α 3, f 3 = β α 4 θ 6 θ 5 θ θ 4+α 4, f 4 = β α 5 θ 7 θ 8 θ θ 8+α 5. P Similar roles of penalty terms f i, α j and lg α j are mentioned in reference [7]. When F W, θ ε ε is a very small constant, it obtain [ P W, θ m ] θ [ 90, θ 1 +θ 4, θ 5 +θ 8, 90], α i P W, θ + ε ω [θ, θ 3 ] + [θ 6, θ 7 ], β P W, ω ε i.e., P W, θ < α i and P W, ω = β, thereof, P SLL = 10 log P W,ω P W,θ > 10 log β α i > 0 db β > 100α i. Above analysis indicates that F ε is the sufficient condition of weight vector W converging to a optimal solution. In practice, the converging condition is too hard to achieve, because F is a bundle of many sub functions, every sub function has convergence errors generally. Thereof, it is necessary to introduce a more practical F function, here, average cost function ACF, i.e., 180/ θ to index algorithm convergency. The optimal value of the cost function, which is finally obtained by the optimizing method. In this paper, an algorithm based on hybrid trust region method is propped in next section, however, for applying the approach, we must first compute its gradient and Hessian matrices see the details in Appendix A. 3. ALGORITHM BASED ON THE CONIC TRUST REGION ALGORITHM We consider the following unconstrained optimization problem: i=1 min fx, x R n 6 where f : R n R is a continuously differentiable function. The optimization problem has become an important research focus given its wide range of potential applications. Throughout the paper, {x k } is a sequence of points generated by our algorithm. CTRA iteratively solves optimization problems. At each iteration, a trial step d k is α i

6 7 Zeng and Feng generated by solving the sub-problem min ϕ k d = fx k + gt k d 1 αk T d + 1 d T B k d 1 α T k d, α k, d R n 7 s.t. d k, 1 αk T d > 0 where g k = fx k R n, B k R n n is an approximate Hessian matrix of fx k, and k > 0 is a trust region radius. Some criteria are used to decide on accepting a trial step and the manner by which the trust region radius is adjusted. After obtaining a trial step d k, the trust region algorithms compute the ratio ρ k = fx k fx k +d k fx k ϕ k d k. Where ρ k indicates the approximate degree of fx k + d k and ϕ k d k, and ρ k 1 while k 0. In iterative computations, ρ k c 0 a positive constant means that ϕ k d k is approximate to fx k +d k, and trial step d k is accepted, so that k is enlarged to speed up the computations. On the other hand, if d k is not accepted ρ k < c 0, some methods resolve subproblem Eq. 7 by reducing the trust region radius until an acceptable step is found. Therefore, the subproblem may be solved several times during each iteration before an acceptable step is found, and these repetitious processes increase the total computational cost for large-scale problems. In the current paper, we propose a new algorithm to improve the computational efficiency of trust region methods. In the computation, B k is generated using the Broyden-Fletcher- Goldfarb-Shanno BFGS formula and is therefore is positive; d k = β k g k β k 0 is a descent direction of fx k. Meanwhile, fx k +d k ϕ k d k when d k { x x k k }, indicating that d k = β k g k is also the descent direction of ϕ k d under this condition. Thus, we obtain an approximate solution of the subproblem by solving the following inequality: gt k β kg k 1 αk T β kg k 1 β k g k T B k β k g k 1 α T k β k g k fx k fx k + β k g k 8 c 0 Let β = max{β k 8 holds}, then d k = β g k is an approximate solution of the subproblem and k = β g k is the trust region radius of the kth iteration. Algorithm 3.1 conic trust region algorithm for unconstrained optimization Step 0. Given 0.75 < c 0 < 1, c 1 > 1 and an initial symmetric positive definite matrix B 0, choose x 0 and set k = 0. Step 1. Compute g k ; if g k < ε ε is a small positive real number, then stop. Step. calculating solution d k of sub-problem Eq. 7.

7 Progress In Electromagnetics Research B, Vol. 53, Step 3. We compute ρ k. If ρ k µ, then we proceed to step 4. Otherwise, Solving inequality 8, we obtain d k = β g k and x k+1 = x k + d k. Let We then proceed to step 5. Step 4. We set k+1 = ρ k k and x k+1 = x k + d k k+1 = k, if d k < k k+1 [ k, c 1 k ], if d k = k Step 5. Generate α k+1, modify B k into B k+1 using the BFGS formula as an approximation to fx k+1, then return to step 1. Remarks. The method for generating α k+1 and B k+1 was as previously reported [16 18]. We assume that the matrices B k+1 are uniformly bounded to prove the convergence, and k, σ 0, 1 : α k k σ which ensures that the conic model function ϕ k d is bounded over the trust-region d k. We reiterate that our algorithm is reduced to a quadratic model-based algorithm if α k = 0 for all k. Under the smoothness assumptions made in this paper, note that the objective function is convex quadratic around a local minimizer. Therefore, choosing α k 0 asymptotically is suitable when x k is near the minimizer. The mild following assumptions, commonly used in the convergence analysis of most optimization algorithms [16 18], are proposed to analyze the convergence properties of Algorithm 3.1. Assumption 1. f is twice continuously differentiable and bounded below. Assumption. A strong local minimizer point x exists, such that fx is positive definite. d Set s = 1 α T d, d = s k 1+α T s. From d k, we obtain k s 1+α T s k, s k 1 + αk T s k + k α k s k + σ s, k and s k 1 σ. We consider a sequence {s k = τ k k { 1 σ 1, if g T { } k Bg k 0 τ k = min k g k 3 1 σ gk T Bg, 1, otherwise k s k Bx k, k 1 σ. Lemma 1. In equation fx k ϕ k d = gt k d g T k s 1 st Bs 1 g k min{ k 1 σ, g k B } holds. g k g k }, where. Clearly, s k k 1 σ, 1 α T k d 1 d T B k d = 1 α T k d

8 74 Zeng and Feng Proof: Case 1: g T k Bg k 0 means that τ k = 1. g T k s k 1 st k Bs k = gt k g k 1 g k min{ k 1 σ, g k B }. Case : gk T Bg k > 0. If min{ k g k 3 1 σ gk T Bg, 1} = k k 1 σ gk T s k 1 st k Bs k = g k 4 1 g k min{ k 1 σ, g k If min{ k 1 σ B }. and s k = gt k gk T Bg g k. k Thereof, k 1 σ 1 k 1 σ 1 g T k g T k Bg k k 1 σ g k g k 3 gk T Bg k gk T Bg 1 g k 4 k g k 3 gk T Bg, 1} = 1, then k k 1 σ g T k s k 1 st k Bs k = k 1 σ g k g k, then s k = gt k g k. g T k Bg k g T k Bg k = 1 g k 3 g T k Bg k 1, gk T Bg k g k 4 g T k Bg k 1 g k B g T k Bg k 1 σ k g k 3 1 k 1 σ gt k Bg k g k k 1 σ g k 1 k 1 σ 1 1 σ g k k g k 3 1 k 1 σ g k 1 g k min{ k 1 σ, g k B }. Lemma. Suppose d is a solution of Sub-problem 7; then the following inequality holds: fx k ϕ k d 1 g k min { k 1 σ, g } k B Proof. ϕ k d ϕ k d and Lemma 1, thus, the above inequality holds. Lemma 3. Suppose that Assumption holds, then M exists as a positive constant, such that fx k fx k + d k fx k ϕ k d k M d k, Proof. This proof is similar to that of Lemma 3. [17]. Lemma 4. Suppose that Assumption 1 holds, then Algorithm 3.1 is well defined. Proof. Assuming that the algorithm does not terminate at x k g k 0, then from Lemma and 3, we obtain fx k fx k + d k fx k ϕ k d k fx k ϕ k d k M d k { g k min k k 1 σ, g k B } M k { } g k min k 1 σ, g k B The definition of k, k 0 as β 0 implies that

9 Progress In Electromagnetics Research B, Vol. 53, fx k fx k +d k fx k ϕ k d k fx k ϕ k d k From the definition of ρ k, we obtain 0, i.e., fx k fx k +d k fx k ϕ k d k 1 ρ k = fx k fx k +d k fx k ϕ k d k 1 > c 0 Therefore, we can obtain trial step d k after finite computation, i.e., Algorithm 3.1 is well defined. Lemma 5. Suppose that Assumption 1 and hold, and is the sequence generated by Algorithm 3.1, then lim g k = 0 k Proof. This proof is similar to that of Lemma [7]. Lemma 6. The sequence {x k } is generated by Algorithm 3.1 and converging to x B. Suppose that lim k fx d k k d k = 0, where fx is positive definite. Therefore, the convergence rate is superlinear. Moreover, suppose that fx is Lipschitz-continuous in a neighborhood Nx, δ = { x R N x x < δ }, i.e., Lδ > 0 exists such that fx fy Lδ x y, x, y Nx, δ {x k } then quadratically converges to x. Proof. Given that B lim k fx d k k d k = 0 therefore, B k fx d k = o d k 9 Using Eq. 9 and the Taylor theorem, we obtain fx k fx k + d k fx k ϕ k d k = gt k d k 1 dt k f k d k o d k gk T + d 1 αk T d + 1 d T B k d 1 α T k d Thus, α k 0 as x k x 1 dt k Bk f d k +o d k 1 d T k B k f d k +o d k 1 d T k o d k + o d k = o d k

10 76 Zeng and Feng fx k fx k + d k fx k ϕ k d k o d k gt k d 1 α T k d + 1 d T B k d 1 α T k d o d k d k 0 as k. Therefore, 1 = fx k fx k + d k fx k ϕ k d k fx k ϕ k d k o d k 1 dt k B k, g kd k k 0, α k 0 fx k fx k +d k fx k ϕ k d k µ x k+1 = x k + d k holds for all sufficiently large k, and Algorithm 3.1 becomes the Newton method or quasi-newton method that superlinearly converges. Therefore, the sequence {x k } converges to x superlinearly. x k x, k > 0, k > k, x k Nx, δ. Set δ k = x k x. Using the mean value theorem, we obtain Thus, δ k+1 = x k+1 x = x k x + d k = δ k β k g k = δ k β k g k g β k g g = gx 1 ] = β k [ fx k δ k fx + tδ k δ k dt β k g δ k+1 = 0 [ 1 [ = β k fx k fx + tδ k ] ] δ k dt β k g 0 [ 1 β k β k 0 1 [ fx k fx + tδ k ] ] δ k dt β k g 0 fx k fx + tδ k δ k dt + β k g 1 β k Lδ δ k tdt + fx k 1 g 0 δ k+1 lim k δ k 1 Lδ β k lim gx k = g = 0 k implying that {x k } converges to x quadratically. Finally, the general algorithm is derived by minimizing the cost function, obtaining optimal weight vector W o, and outputing amplitude pattern pw o, θ. The algorithm is described as follows: Step 1. The cost function F W, A is constructed as Eq. 3. Step. Let X = W 1 W A then, computing X F and X F

11 Progress In Electromagnetics Research B, Vol. 53, Eqs. A1 and A43 respectively see the details in Appendix A. The two formulas are essential to optimizing algorithm. Finally, Algorithm 3.1 is used to minimize the cost function F X. Step 3. Base on the previous analysis, vector W converges a optimal solution, if 0, then array amplitude pattern is outputed. F 180/ θ 4. SIMULATION RESULTS In this section, several simulations are performed to test the multibeam synthesizing patterns validity Simulation 1: Comparison with NACTRM [16] Consider a synthesis problem using a 1-element half-wave-length spacing linear array with the following configurations: θ 1 = 6.0, θ = 4.5, θ 3 = 15.5, θ 4 = 14.0, θ 5 = 14.0, θ 6 = 15.5, θ 7 = 4.5 Table 1. Comparison of the SICTRM and NACTRM algorithms. No. Algorithm r m θ ACF DPMLL db PSLL db Time 1 SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM SICTRM NACTRM s

12 78 Zeng and Feng and θ 8 = 6.0. The simulations based on the SICTRM and NACTRM algorithms were executed 10 times independently, the results are listed in Table 1. The two iterative algorithms optimize same cost functions in every stimulation, but the results show their distinct performance. Under NACTRM, average wasted time, average DPMLL the difference of peak mainlobe level and average PSLL the peak sidelobe level are 50.0 second, db and db, under SICTRM, they are 4.07 second, db and 1.99 db respectively. Undoubtedly, compared with NACTRM, the SICTRM algorithm exhibits stronger Normalized AF db θ degree a Log F Iteration Figure. a Radiation pattern and b cost function variety of 1- element line antenna array under SICTRM algorithm. Table. Optimal solution of 1-element line antenna array under SICTRM algorithm. Element W 1 W Element W α 1, α 3, W Element W 1 W α 5, b

13 Progress In Electromagnetics Research B, Vol. 53, search ability in these simulations. The reported minimum PSLL for synthesis of 1-element multi-beam array is 4.76 db in [11], our results is better than their result. We show the optimal normalized AF amplitude pattern, ACF variety and optimal solution in Figures a, b and Table respectively. 4.. Simulation : Comparison with DA Proposed in [7] and Dynamic Differential Evolution DDE [13] The DA method, a gradient-based optimizing algorithm, adopts a computing inequality to define step length for improving computation efficiency, the method has exhibited strong convergence properties. DDE is a new version of differential evolution algorithm, which used only one array to search the optimization. DDE significantly outperforms the differential evolution strategy in efficiency, robustness, and memory requirement [1]. Consider a synthesis problem using a 17-element half-wave-length spacing linear array with the following configurations: θ 1 = 4.0, θ = 3.0, θ 3 = 17.0, θ 4 = 16.0, θ 5 = 6.0, θ 6 = 7.0, θ 7 = 33.0 and θ 8 = Executing the three algorithm 10 times independently and listing results in Table 3 as following. Under DA, average iterative time, minimum PSLL, average DPMLL and PSLL are second, 8.75 db, db, and.98 db, respectively. Similarly, under DDE, they are second, 6.71 db, db, and 3.96 db, respectively. However, under SICTRM, they are 5.79 second, db, db, and 5.88 db, respectively. Obviously, the result indicates that the proposed algorithm has stronger ability to suppress sidelobe level. Finally, we show the optimal radiation pattern, ACF variety and optimal solution in Figures 3a, b, and Table 4, respectively Simulation 3: SICTRM Performance with 17 Mutual Coupling Elements In this simulation, we test the SICTRM performance in the presence of antenna elements coupled to each other. Taking the mutual coupling into account, the SICTRM is closer to the actual situation. The coupling coefficient Between any two antenna elements can be calculated []. In the case of an uniform linear array, these coefficients can be represented by a symmetric Toeplitz matrix [3, 4]. The mutual coupling between two elements is inversely related to their distance, and thus it is assumed that when the distance between two array elements is more than k inter-element spacing, the mutual

14 80 Zeng and Feng Table 3. Comparison of the SICTRM, DA and DDE algorithms. No. Algorithm r m θ ACF DPMLL PSLL Time db db s 1 SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE SICTRM DA DDE coupling coefficients are assumed to be zero. Therefor, the mutual coupling matrix can be sufficiently modeled as follows: M = T opelitz 1, c 1, c,..., c k 1, 0 1 N k 10

15 Progress In Electromagnetics Research B, Vol. 53, Normalized AF db θ degree a LogF Iteration Figure 3. a Radiation pattern and b cost function variety of 17- element line antenna array under SICTRM. b Table 4. SICTRM. Optimal solution of 17-element line antenna array under Element W 1 W Element W α 1, α 3, W Element W 1 W α 5, Under this model, the true steering vector should be rewritten as S t = MSθ 11 Make minor modifications to the original program accordingly and execute 10 times. The results are shown in Table 5. In the simulation, the mutual coupling matrix

16 8 Zeng and Feng Table 5. SICTRM performance with 17 mutual coupling elements. No. Algorithm r m θ ACF DPMLL PSLL Time db db s 1 SICTRM SICTRM SICTRM SICTRM SICTRM SICTRM SICTRM SICTRM SICTRM SICTRM M = T opelitz 1, j, j, j, Other parameters are same as those of simulation. Under SICTRM, average iterative time, minimum PSLL, average DPMLL and PSLL are 7.96 second, 5.47 db, db and 1.57 db, respectively. The results show that our algorithm is valid even in the presence of larger array element coupling. Finally, we show the comparing radiation patterns in isolated and coupled cases, ACF variety and optimal solution in Figures 4a, b and Table 6 respectively. These two patterns are well matched. Normalized AF db Isolated coupled θ degree a Log F Iteration Figure 4. a Comparing radiation patterns and b cost function variety of 17-element line antenna array under SICTRM. b

17 Progress In Electromagnetics Research B, Vol. 53, Table 6. Optimal solution of 17-coupled-element line antenna array under SICTRM. Element W 1 W Element W α 1, α 3, W Element W 1 W α 5, DISCUSSION We have designed three simulations to validate our algorithm. Simulation results indicate that the proposed algorithm is valid to multi-beam antenna array pattern synthesis in both isolated and coupled cases. Of course, the performance of the algorithms relates to some pre-assumed parameters. In these parameters, the parameter r has impact on the convergence of the algorithm greatly. If r is too small r < 0, antenna sidelobe level will not be suppressed effectively. However, if r is too is too large, the algorithm is difficult to converge. The simulations results show that our assumptions 0 r 4 are reasonable. Generally, parameter m > 0. In the three simulations, m is more than 0.1 and less than 0.4. Similarly, if it is too big, the algorithm is difficult to converge. In theory, when the angle resolution θ is smaller, the antenna sidelobe level will be suppressed more effectively. But that will increase the computational burden. In fact, θ = 0.1 is a good choice. The parameter ACF is a convergence indicator and is assumed less than 1.

18 84 Zeng and Feng 6. CONCLUSION In this paper, we have presented a multi-beam model for antenna array pattern synthesis problem. Under the model, an antenna is able to create multibeams simultaneously. In order to quickly calculate the objective function to obtain the optimal solution, an iterative optimization algorithm named SICTRM is proposed. The SICTRM algorithm, unlike traditional trust region methods that resolve subproblems, can improve computational efficiency as indicated by the theoretical analysis and stimulation results. Computer simulations illustrate the general algorithm s good performance as it is applied to the AAPS problem. APPENDIX A. THE DERIVATION OF GRADIENT AND HESSIAN MATRIX OF COST FUNCTION The similar derivation can be seen in [7]. Let t 1 = S T 1 W 1, t = S T W, t 3 = S T W 1, and t 4 = S T 1 W, we obtain W1 P = t 1 + t S 1 + t 3 + t 4 S A1 For convenience, let F 11 = α 1 P + P m, α1 F1 = α P + m, m, F13 = f 1 P + F14 = β P, F 15 = f P + P α P f P f1 m, and F16 = α 3 P + P m. α3 Thereof, W1 F 11 = P α 1 W1 P + m α 1 P α 1 m 1 W1 P A Similarly, W1 F 1 = P α W1 P + m α P α m 1 W1 P A3 W1 F 13 = P f 1 W1 P + m f 1 P f 1 m 1 W1 P W1 F 14 = P β W1 P W1 F 15 = P f W1 P + m f P f m 1 W1 P A4 A5 A6 W1 F 16 = P α 3 W1 P + m α 3 P α 3 m 1 W1 P A7

19 Progress In Electromagnetics Research B, Vol. 53, From Eqs. A1 A7, we obtain, W1 F 1 W, A = 1 W1 F 11 + W1 F W1 F W1 F W1 F W1 F 16 A8 Similarly, W1 F W, A = 7 W1 F W1 F + 9 W1 F W1 F W1 F W1 F 6 A9 where W1 F 1 = P α 4 W1 P + m α 4 P α 4 m 1 W1 P A10 W1 F = P f 3 W1 P + m f 3 P f 3 m 1 W1 P W1 F 3 = P β W1 P W1 F 4 = P f 4 W1 P + m f 4 P f 4 m 1 W1 P A11 A1 A13 W1 F 5 = P α 5 W1 P + m α 5 P α 5 m 1 W1 P A14 W1 F 6 = P α 6 W1 P + m α 6 P α 6 m 1 W1 P A15 So, Similarly, W1 F W, A = W1 F 1 W, A + W1 F W, A W P = t 4 t 3 S 1 + t 1 + t S A16 A17 Thereof, Eq. A17 is substituted W1 P above equations correspondingly and obtains the formulations of W F 1 W, A and W F W, A. Finally, we get W F W, A= W F 1 W, A + W F W, A F 11 α 1 =α 1 P mp m α m 1 1, A18 F 13 α 1 =f 1 P f 1 α 1 mp m f m 1 1 f 1 α 1, F 14 α 1 =rβ P,

20 86 Zeng and Feng F 15 α 1 =f P f α 1 mp m f m 1 f α 1, F α 1 =f 3 P f 3 α 1 mp m f m 1 3 f 3 α 1, F 3 α 1 =rβ P, F 4 α 1 =f 4 P f 4 α 1 mp m f m 1 4 f 4 α 1, lg α 1 =1/α 1 F α 1 = 1 F 11 α F 13 α F 14 α F 15 α F α F 3 α F 4 α 1 λ/α 1 A19 Similarly, it is not difficult to compute F α i i =, 3,..., 6, finally, we can obtain A F = F α 1 F α... F α 6 T A0 Using Eqs. A16, A18 and A0, we obtain the formulation X F W, A as following: let X F = W1 F W F A F A1 Next, we calculate the Hessian matrices of the cost function. First, h 11 =t 1 +t S 1 +t 3 +t 4 S R N 1, h 1 =h 11 h T 11+h T 11 h 11 R N N h 1 =S 1 S1 T + S S T R N N, h = h 1 + h T 1 R N N h 31 =t 4 t 3 S 1 +t 1 +t S R N 1, h 3 =h 31 h T 31+h T 31 h 31 R N N then W 1 F 11 = +mm 1P m α1 h1 + P α 1 +mp m 1 α1 h A W 1 F 1 = +mm 1P m α h1 + P α +mp m 1 α h A3 W 1 F 13 = +mm 1P m f1 h1 + P f 1 +mp m 1 f1 h A4 W 1 F 14 =4h 1 + P βh A5 W 1 F 15 = +mm 1P m f m W 1 F 16 = +mm 1P m α3 m W 1 F 1 = +mm 1P m α4 m h1 + P f +mp m 1 f h A6 h1 + P α 3 +mp m 1 α h A7 h1 + P α 4 +mp m 1 α h A8 3 4

21 Progress In Electromagnetics Research B, Vol. 53, W 1 F = +mm 1P m f3 h1 + P f 3 +mp m 1 f3 h A9 W 1 F 3 = 4h 1 + P βh A30 W 1 F 4 = +mm 1P m f4 h1 + P f 4 +mp m 1 f4 h A31 W 1 F 5 = +mm 1P m α5 h1 + P α 5 +mp m 1 α5 h A3 W 1 F 6 = +mm 1P m α6 h1 + P α 6 +mp m 1 α6 h A33 W 1 F = W 1 F 1i + W 1 F i i = 1,,..., 6 A34 i i Similarly, h 3 is substituted h 1 above Eqs. A A33 correspondingly and obtains the formulations of W F 1i and W F i. Finally, we get W F = i W F 1i + i W F i i = 1,,... 6 A35 Based on the definition of Hessian matrix, we obtain A F and W 1 W F as following: F F α 1 α 1 α 1 α... F α 1 α 6 AF = α F F F α 1 α... α 6 = F F α α 1 α... F α α 6.. A F F α 6 α 6 α 1 α 6 α... F α 6 W 1 W F = = w 1 1 w 1. wn 1 F F w1 1 w 1 F w 1 w1. F wn 1 w1 w 1 F w F w1 1 w F w 1 w w 1 N. F... wn F w... F w w w 1 N w N F w N F w N

22 88 Zeng and Feng = W1 W F 1 W1 W F... W1 W F N N N F F F T W F =... A37 Similarly, w 1 w w N W 1 AF = W1 A F 1 W1 A F... W1 A F 6 N 6 A38 W W 1 F = W1 W F T A39 N N W AF = W A F 1 W A F... W A F 6 N 6 A40 AW 1 F = W1 AF T A41 6 N AW F = W AF T A4 6 N Using Eqs. A36 A41, we finally obtain W 1 F W 1 W F W 1 A F XF = W1 W F T W F W A F W1 A F T W A F T A F ACKNOWLEDGMENT N+6 N+6 A43 This work was supported by the National Natural Science Foundation of China NNSF under Grants , and , the National 863 Project of China under Grant 01AA01305, Sichuan Provincial Science and technology support Project under Grant 01GZ0101, and Chengdu Science and technology support Project under Grant 1DXYB347JH-00. REFERENCES 1. Li, X., W.-T. Li, X.-W. Shi, J. Yang, and J.-F. Yu, Modified differential evolution algorithm for pattern synthesis of antenna arrays, Progress In Electromagnetics Research, Vol. 137, , Lin, C., A.-Y. Qing, and Q.-Y. Feng, Synthesis of unequally spaced antenna arrays by using differential evolution, IEEE Transactions on Antennas and Propagation, Vol. 58, , Li, R., L. Xu, X.-W. Shi, N. Zhang, and Z.-Q. Lv, Improved differential evolution strategy for antenna array pattern synthesis

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