The optimization formulation

The optimization problem of ULAAs for the interference suppression is defined as: maximize the main-beam gain and minimize the PSLL along with the constraints of null depth in the desired directions through optimizing the electrical parameters of each array element. The optimization problem can be formulated as:

(1)$$\left\{{\matrix{ {{\rm find}\colon {\rm \;\;}{\boldsymbol \chi }} \hfill \cr {{\rm min}\colon {\rm \;\;}{\rm \Phi }_0 = {-}G_m + {\rm max}( {{\boldsymbol G}_{{\rm SLL}}} ) } \hfill \cr {{\rm s}.{\rm t}.\colon \;\;{\boldsymbol ZJ} = {\boldsymbol V}( {\boldsymbol \chi } ) } \hfill \cr {{\rm max}( {{\boldsymbol G}_{{\rm NULL}}} ) \le G_0} \hfill \cr {{\boldsymbol \chi }_{{\rm min}} \le {\boldsymbol \chi } \le {\boldsymbol \chi }_{{\rm max}}} \hfill \cr } } \right.$$

where G _m is the maximum of the gain in the main-beam direction. ${\boldsymbol G}_{{\rm SLL}}$ and ${\boldsymbol G}_{{\rm NULL}}$ are the vectors of the gains within the sidelobe region and in the desired null directions, respectively. The design objective Φ₀ is an explicit function that can describe a specific functional requirement, and evaluate the high-quality optimization solution. The constraint is defined to control the null depth by constraining the maximum gain value of the null depth ${\boldsymbol G}_{{\rm NULL}}$ in all specified directions less than a given value G ₀. Φ₁ is defined as a constraint function to characterize the difference between the null depth constraint and G ₀ in the optimization process. The constraint of null depth and Φ₁ in the ULAAs optimization problem are expressed as:

(2a)$${\rm max}( {{\boldsymbol G}_{{\rm NULL}}} ) = \ln \left({\mathop \sum \limits_{m = 1}^{M_0} e^{{-}qG_{{\rm NULL}, {\rm m}}{\rm \;}}} \right)^{1/q} \le G_0$$

(2b)$${\rm \Phi }_1 = {\rm max}( {{\boldsymbol G}_{{\rm NULL}}} ) -G_0$$

where q is the operator that calculates the approximate maximum value, which is determined to be a sufficiently large positive integer. M ₀ is the total number of nulls. G _NULL,m represents the gain value of m-th null.

The optimization procedure

The GOA takes the advantage of the analytical properties of the optimization problem to improve the searching efficiency. In order to carry out the GOA to achieve the optimization design of coupled antenna arrays, it is necessary to establish the relationship among Φ₀ and Φ₁ (collectively referred to as Φ_k, k = 0 or 1) and ${\boldsymbol \chi }$. This relationship requires that the change in Φ_k caused by the iteratively changed arrangements of array parameters can be accurately calculated. According to the relationships, the sensitivities of Φ_k and ${\boldsymbol \chi }$ can be derived analytically. Then, the sensitivity information is introduced into the optimization procedure to speed up the convergence.

The computational steps for solving the optimization problem by the GOA are summarized as:

1. (1) The array parameter to be optimized is set as the design variable ${\boldsymbol \chi }$ and the upper and lower bounds are limited to the range of $[ {{\boldsymbol \chi }_{{\rm min}}, \;{\boldsymbol \chi }_{{\rm max}}} ]$. Here, ${\boldsymbol \chi }$ represents the vector of the excitation amplitude or phase of each array element. In the design of ULAAs, the 2N symmetric geometric elements are assumed to be placed along the x-axis, as shown in Fig. 1. At the beginning of the optimization, a parameterized ULAAs model is generated from ${\boldsymbol \chi }$ and an initial arrangement of ${\boldsymbol \chi }_{{\rm ini}}$ is set up.

2. (2) Define Φ_k of the optimization problem. Φ_k can be calculated through an appropriate analysis method which requires the radiation pattern performances. The full-wave MoM is a common electromagnetic analysis method, and the radiation performance of the ULAAs can be accurately calculated by the full-wave MoM simulation. The following equation is formulated in matrix form as:

   (3)$$\eqalign{& \left[{\matrix{ {\matrix{ {{\boldsymbol Z}_{11}} & \ldots & {{\boldsymbol Z}_{1j}} \cr \vdots & \ddots & \vdots \cr {{\boldsymbol Z}_{i1}} & \ldots & {{\boldsymbol Z}_{ij}} \cr } } & {\matrix{ \ldots & {{\boldsymbol Z}_{1N}} \cr \ddots & \vdots \cr \ldots & {{\boldsymbol Z}_{iN}} \cr } } \cr {\matrix{ \vdots & \ddots & \vdots \cr {{\boldsymbol Z}_{N1}} & \ldots & {{\boldsymbol Z}_{Nj}} \cr } } & {\matrix{ \ddots & \vdots \cr \ldots & {{\boldsymbol Z}_{NN}} \cr } } \cr } } \right]\left\{{\matrix{ {\matrix{ {{\boldsymbol J}_1} \cr \vdots \cr {{\boldsymbol J}_i} \cr } } \cr {\matrix{ \vdots \cr {{\boldsymbol J}_N} \cr } } \cr } } \right\} \cr & \quad = \left\{{\matrix{ {\matrix{ {{\boldsymbol V}_1( {\chi_1} ) } \cr \vdots \cr {{\boldsymbol V}_i( {\chi_i} ) } \cr } } \cr {\matrix{ \vdots \cr {{\boldsymbol V}_N( {\chi_N} ) } \cr } } \cr } } \right\}}$$
   where ${\boldsymbol Z}_{ij}$ represents the mutual impedance matrix between the i-th and j-th array element. ${\boldsymbol J}_i$ and ${\boldsymbol V}_i$ represent the unknown surface current and the incident field at the i-th element, respectively. The unknown currents ${\boldsymbol J}$ can be solved by assembling the ${\boldsymbol Z}$ and ${\boldsymbol V}$, and the gain of the radiation pattern G can be calculated by the equivalent dipole model [Reference Wang28]. The desired Φ_k can be obtained by the G _m, ${\boldsymbol G}_{{\rm SLL}}$, and ${\boldsymbol G}_{{\rm NULL}}$, subsequently.

3. (3) Calculate the sensitivities of Φ_k with respect to the ${\boldsymbol \chi }$. Based on the parametric governing equation, the sensitivity of Φ_k with respect to ${\boldsymbol \chi }$ is derived as:

   (4)$$\displaystyle{{d{\rm \Phi }_k} \over {d{\boldsymbol \chi }}} = \displaystyle{{df( G ) } \over {dG}}\displaystyle{{dG} \over {d{\boldsymbol \chi }}}.$$
   The derivation of ∂f(G)/∂G depends on the specific Φ₀ or Φ₁. According to the chain rule, the sensitivity of G with respect to ${\boldsymbol \chi }$ is derived as:
   (5)$$\displaystyle{{dG} \over {d{\boldsymbol \chi }}} = \displaystyle{{\partial G} \over {\partial {\boldsymbol \chi }}} + \displaystyle{{\partial G} \over {\partial {\boldsymbol J}}}\displaystyle{{\partial {\boldsymbol J}} \over {\partial {\boldsymbol \chi }}}, \;$$
   where only ${\boldsymbol J}$ is an implicit function of ${\boldsymbol \chi }$, and it is difficult to manifest the function expression. Therefore, a complex adjoint vector $\;{\boldsymbol \lambda }$ is determined to eliminate the unknown $\partial {\boldsymbol J}/\partial {\boldsymbol \chi }$ by the adjoint method [Reference Tortorelli26]. The following derivation is given by:
   (6)$$\matrix{ {\displaystyle{{dG} \over {d{\boldsymbol \chi }}} = \displaystyle{{\partial G} \over {\partial {\boldsymbol \chi }}} + \displaystyle{{\partial G} \over {\partial {\boldsymbol J}}}\displaystyle{{\partial {\boldsymbol J}} \over {\partial {\boldsymbol \chi }}} + {\boldsymbol \lambda }^{\dagger} \displaystyle{{\partial [ {{\boldsymbol ZJ}-{\boldsymbol V}( {\boldsymbol \chi } ) } ] } \over {\partial {\boldsymbol \chi }}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cr { = \displaystyle{{\partial G} \over {\partial {\boldsymbol \chi }}} + \left({\displaystyle{{\partial G} \over {\partial {\boldsymbol J}}} + {\boldsymbol \lambda }^{\dagger} {\boldsymbol Z}} \right)\displaystyle{{\partial {\boldsymbol J}} \over {\partial {\boldsymbol \chi }}} + {\rm \Re }\left({{\boldsymbol \lambda }^{\dagger} \displaystyle{{\partial {\boldsymbol Z}} \over {\partial {\boldsymbol \chi }}}{\boldsymbol J}-{\boldsymbol \lambda }^{\dagger} \displaystyle{{\partial {\boldsymbol V}( {\boldsymbol \chi } ) } \over {\partial {\boldsymbol \chi }}}} \right), \;} \cr } $$
   where ${\rm \Re }$ is the real operator and ${\dagger}$ is the conjugated transpose operator. To complete the sensitivity analysis, the ${\boldsymbol \lambda }$ should satisfy the following equation:
   (7)$${\boldsymbol Z}^{\dagger} {\boldsymbol \lambda } = {-}\left({\displaystyle{{\partial G} \over {\partial {\boldsymbol J}}}} \right)^{\dagger} .$$
   Finally, the sensitivity analysis is completed and the formula (6) can be reduced to
   (8)$$\displaystyle{{dG} \over {d{\boldsymbol \chi }}} = {-}{\rm \Re }\left({{\boldsymbol \lambda }^{\dagger} \displaystyle{{\partial {\boldsymbol V}( {\boldsymbol \chi } ) } \over {\partial {\boldsymbol \chi }}}} \right).$$

4. (4) Update the next generation of the ${\boldsymbol \chi }$. It is noted that ${\boldsymbol V}( {\boldsymbol \chi } )$ should be automatically updated with ${\boldsymbol \chi }$ during the iteration process. Then, update Φ_k of the optimization problem and calculate their sensitivities. The optimization procedure needs to be evaluated based on the iteration of Φ_k and ${\boldsymbol \chi }$. If the convergence criterion or the maximum number of iterations is satisfied, the optimization will stop, otherwise, the optimization will go to (2).

5. (5) Obtain the optimal solution ${\boldsymbol \chi }_{{\rm opt}}$ and synthesize the desired radiation pattern.

The flow chart of the GOA is illustrated in Fig. 2, where ζ is each iteration step in the optimization procedure and M is the maximum number of iterations. Tol is the calculation tolerance given based on the convergence criterion. According to the above optimization procedure, the optimization problem of the ULAAs can be solved efficiently.

Fig. 1.

The geometry of 2N-elements symmetric ULAAs.

Fig. 2.

The optimization flow chart of the GOA.

Excitation amplitudes optimization

The first example is to optimize the excitation amplitudes of the 24-element ULAA with the double nulls at ${\pm} 30^\circ$, ${\pm} 40^\circ$. Each dipole element is excited by a 1χ V voltage gap generator with 50 Ω characteristic impedance, where χ represents the excitation amplitude of each array element. ${\boldsymbol \chi }_{{\rm min}}$ and ${\boldsymbol \chi }_{{\rm max}}$ are set to 0 and 1, respectively. The initial arrangement ${\boldsymbol \chi }_{{\rm ini}}$ is set to 0.5. The optimization information and the pattern properties obtained based on the three algorithms are shown in Table 1, which includes the maximum gain in the main-beam direction, the PSLL, the null positions, the iterations of optimization, and the CPU running time. Table 2 describes the design results of the excitation amplitudes optimization. The comparisons of the radiation pattern performances are calculated by the full-wave MoM analysis, which is illustrated in Fig. 3.

Fig. 3.

The comparisons of the radiation pattern performances in the excitation amplitudes optimization.

Table 1.

The information and pattern properties of the excitation amplitudes optimization

Table 2.

The design result of the excitation amplitudes optimization

It is seen that GA and CS algorithms can provide the G _m of 16.66 and 16.74 dB with a PSLL minimization of −2.79 and −3.59 dB, respectively. The CPU time of optimization is 210.1 and 315.8 min, respectively. The CS algorithm uses the MoM to analyze the radiation performances twice in each iteration for obtaining a good optimization result. The design results by the GOA can provide the G _m of 16.74 dB with a PSLL minimization of −3.25 dB, and the CPU time of optimization is 10.6 min. It is illustrated that the proposed method reduces the CPU time of optimization compared with the GA and CS. The iterations of optimization based on the three algorithms may increase due to the null-depth constraint of multiple positions. All the design results maintain the nulls at the specific directions $( { \pm 30^\circ , \;\pm 40^\circ } )$ by the full-wave MoM analysis and satisfy the constraint of the nulls. By comparing the optimization information and the pattern properties, the amplitude-only optimization by the GOA can greatly shorten the computational time, and obtain nearly the same radiation pattern performance as the other two algorithms.

Excitation phases optimization

The second example is the 32-element ULAA with the null positions at ${\pm} 9^\circ$ and each dipole element is excited by a 1e ^jχ V voltage gap generator, where χ represents the excitation phase of each array element. It is determined to match with a 50 Ω transmission line. ${\boldsymbol \chi }_{{\rm min}}$ and ${\boldsymbol \chi }_{{\rm max}}$ are set to 0 and 2π, respectively, where ${\boldsymbol \chi }_{{\rm ini}}$ is set to the middle value in the range of [0, 2π]. Table 3 shows the optimization information and the pattern properties based on the three algorithms, and Table 4 shows the design results of the excitation phases optimization. Figure 4 describes the comparisons of the radiation pattern performances. In the design optimization of the excitation phases, the constraints of null-depth are satisfied less than −45 dB. The design results are calculated by the full-wave MoM analysis for controlling the nulls at the specific directions $( { \pm 9^\circ } )$ precisely and synthesizing the radiation patterns accurately. Compared with GA and CS algorithm, the results of the GOA provide the lowest PSLL, and the G _m is higher than GA. The iterations and the computation time of optimization by the GOA are 232 and 9.9 min, respectively. This validates that solving the optimization problem of the ULAAs through the GOA can enhance the searching efficiency of the optimization efficiently and improve the radiation performances.

Fig. 4.

The comparisons of the radiation pattern performances in the excitation phases optimization.

Table 3.

The information and pattern properties of the excitation phases optimization

Table 4.

The design result of the excitation phases optimization