The Combined Search of the Different Kind of Solutions to the Optimization Problems of Antenna Synthesis Theory

Abstract

The restriction on the optimized function in the problem of the antenna synthesis leads to the characteristic form of the nonlinear integral equations, which are equivalent to the respective functionals of synthesis problem. The kind of the reduced equation depends on the operator, which is used for the calculation of antenna’s directivity pattern (DP). The two types of restrictions are considered, the first one is that the amplitude of the objective function is prescribed. The second one is that the restrictions on the phase of the objective function are imposed. The problem is actual because it deals with the analytical investigation of integral Hammerstein equations of a new type, on the one hand, and gives a possibility to get new types of solutions to above equations that can be applied at the modelling of antennas of the different appointment, on the other hand. The peculiarity of such equations is the non-uniqueness of their solutions. The accuracy of analytical approach is supported by the numerical data. The method of successive approximations is much effective for solving the reduced equations. The advantage of the approach proposed is a fast convergence of iterative procedures, which are applied for solving the obtained equations.

1. Introduction

Optimization with restrictions is an effective method for solving the different application problems in the various fields of engineering and technology [1, 2, 3, 4]. The problems discussed therein appertain to both the general schemes of algorithm evolution [1, 2] and using the optimization algorithms for solving the specific problems related to engineering and technology applications [3, 4]. The numerous papers cited therein testify that the majority of optimization problems appearing in the engineering design are related to optimization problems with restrictions.

The antenna synthesis theory and technique [5] represent one of the above engineering fields. Usually, while solving the problem of synthesis for the different types of antenna systems, the restrictions on the physical parameters and parameters of radiation are applied. The typical synthesis problem of antenna is reduced to creating the current or field distribution in antenna construction, which produces the prescribed radiation data. In our consideration, the amplitude directivity pattern (DP) is considered as such feature [6]; there various problems of optimization belonging to the different types of antenna systems were examined. The optimization problem is stated in the variational form because the given amplitude DP can not be formed exactly. Such statement provides the ability to create the amplitude DP of antenna in the closest form to what is desired.

The amplitude DP is not the only one given characteristic for optimization in the antenna synthesis problems. The majority of radiation parameters and characteristics are subject to the interest. One is required to take into account the limitations to the antenna construction in many cases [7, 8], the electromagnetic compatibility demands [9, 10, 11], the peculiarities of whole radiation [12, 13, 14], and the level of DP’s sidelobes [15, 16]. All the mentioned problems can be formulated as the problems of multiparameter optimization; therefore it is suitable to apply the variational formulation for their statement and further solving. The above is an additional reason to use a variational approach for solving the optimization problem we considered in this chapter.

The variational formulation the antenna synthesis optimization problems was proposed for the first time in [17]; as a part of this research, the theoretical base, which enables to get a portion of information regarding the solutions of problems, was developed later in [5]. Since not complex DP of the antenna is optimizing function, but only its amplitude, the variational formulation of the synthesis problem by the amplitude DP leads to the need to solve the non-linear integral equations.

The resulting equations are the Hammerstein non-linear integral equations. The non-uniqueness and process of branching their solutions is the peculiar property of such equation [18]. Usually, only one unique solution exists when a characteristic parameter in a kernel of such equations is small; and a solution becomes branching when this parameter grows. In fact, we need to use the numerical methods to solve such equations, because the solution cannot be derived analytically. A series of such methods was developed in [17]. The above method is characterized by that we start with an initial approximation having some property of the phase of DP, and the resulting solution is determined in class of functions, which have the same property of phase DP.

This approach was generalized in [5], the solutions in a wide class of functions with argument (or phase) in form of a finite degree polynomial, was sought for there. Employment of the Newton generalized method, allowing simultaneous search for some branched solution and determination the characteristic parameter, at which the solution becomes branching, is one more excellence of this approach. The proposed approach correlates with the variety of Newton methods developed in [19]; and it can be effectively applied to solve a series of engineering problems for antennas of different kinds. The developed methods were applied in [6] at solving the synthesis problems of microstrip antennas, planar arrays, and antennas with waveguide excitation. The initial requirement while solving the problem of synthesis knows the recipe how the antenna’s DP should be calculated. The method proposed in [20], is applied to derive the closed form of the DP.

The specific of the problems solved in works [5, 6, 17] was either the absence of restrictions on the function to be sought for (the synthesis by the amplitude and phase), or the restrictions are applied to the amplitude of this function (the synthesis by phase), and the restrictions are applied to the phase of this function (the synthesis by amplitude). The problems, which are solved here, belong to the phase and amplitude synthesis in the context of the problems studied in [5, 17, 21].

This chapter is arranged as follows. In Sections 2 and 3, we formulate the problems of phase and amplitude optimization. The non-linear integral equations of the used functionals are reduced here in the general operator form. The specification of such operators for the 1D and 2D cases is given. Some aspects of convergence of the successive approximation method and properties of obtained non-linear equations are discussed as well. The data of numerical modeling presented in Section 4 are related to the different types of antennas and demonstrate the peculiarities of the solutions to respective non-linear equations and rate of convergence of the successive approximation method, which is used.

The concluding remarks finish the considerations and accentuate the engineering importance of the approach proposed.

2. Phase optimization

2.1 The optimization problem outlines

In this Section, we deal with the optimization problem, which is characterized by restrictions on the amplitude of the function is sought for. Such optimization problem is related to solving the phase synthesis of antenna systems corresponding to the linear antennas and arrays. The function to be optimized presents some vector, which describes the phase value of currents $I$ in the elements of the array. The constrains on the amplitudes $\mid I\mid$ of currents are characterized by prescribing some determined function. A non-negative function $F$ is the input data of problem; it is approximated with the amplitude of the created (synthesized) DP $f$. The DP $f$ is determined by some linear bounded operator $A$ using the currents $I$ in the antenna elements.

For our approach, we give $\mid I\mid =C\mid v\mid$, where the value of $\mid v\mid$ is fixed, and $C$ unknown real number to be determined below. The functional, which described an optimization problem, is [5].

$$\sigma \left(\psi ,C\right)={‖C|A(|v|e^{\mathit{i\psi }})|-F‖}^2,$$E1

where $\Vert ·\Vert$ is a norm of vectors, prescribed in form of inner product in the Hilbertian space of functions, to which the DPs belong.

The number $C$ is determined by some relation; i.e. it depends on a phase function $\psi$ in such a way:

$$C\left(\psi \right)=\left(A\left(|v|e^{\mathit{i\psi }}\right),F\right)/{‖A(|v|e^{\mathit{i\psi }})‖}^2$$E2

Using (1) and (2) one can get that $\sigma \left(\psi ,C\right)={‖F‖}^2-\kappa \left(\psi \right)$; the last relation associates $\sigma \left(\psi ,C\right)$ with functional

$$\kappa \left(\psi \right)=\frac{\left(|A\left(|v|e^{\mathit{i\psi }}\right)|,F\right)}{\Vert A\left(|v|e^{\mathit{i\psi }}\right)\Vert }$$E3

One can easy to sure that (3) does not depend on the number $C$. The function-vector $\psi$, which is looking for, maximizes $\kappa \left(\psi \right)$ functional; simultaneously, it gives a minimum for functional (1), when the number $C$ is determined according to (2). Therefore, one can search for a minimum of functional (1) or a maximum of functional (3) when the optimizing function $\psi$ is determined as solution to the phase synthesis problem. I turned out that in contrast to the problem of amplitude-phase synthesis [6], the problem of maximization of functional (3) is more convenient while solving the phase synthesis problem than minimization of functional (1); the above is result of that that one can establish a constructive iterative process, which provides a quick convergence in phase optimization.

2.2 Reducing the non-linear equation

In this subsection, we reduce the Lagrange–Euler equation associated with functional (3). To do this, we add to the function $\psi$ term $\mathit{\delta \psi }$, where $\delta$ is an unknown small parameter (real number), and present the increment in the form

$$v+\mathit{\delta v}=\mid v\mid e^{i\left(\psi +\mathit{\delta \psi }\right)}=\mid v\mid e^{\mathit{i\psi }}\left(1+\mathit{i\delta \psi }\right)$$E4

up to first-order terms.

The above results in

$$\mathit{\delta v}={\mathit{ie}}^{\mathit{i\psi }}\mathit{i\delta \psi }$$E5

and

$$\mathit{\delta f}=\mathit{iA}\left(|v|e^{\mathit{i\psi }}\mathit{i\delta \psi }\right)$$E6

The last formula takes into account the operator $A$ is linear, and $f=A\left(|v|e^{\mathit{i\psi }}\right)$. If to take into account (5) and (6), we reduce the incremental part of functional (3)

$$\mathit{\delta \kappa }\left(\mathit{\delta \psi }\right)=\left(\mathit{\delta \psi },|v|Im\left(e^{-\mathit{i\psi }}{A}^{\ast }\left[{\mathit{Fe}}^{iargf}-\mathit{Cf}\right]\right)\right)$$E7

and $A^{\ast }$ is an adjoint operator to $A$.

2.3 The non-linear Lagrange: Euler equation in the generalized form

To reduce the Lagrange–Euler equation for the unknown function $\psi$, we derive the first variation of functional (3). This variation has form

$$\begin{array}{l}\mathit{\delta \kappa }\left(\psi ,\mathit{\delta \psi }\right)=Im\left(A\left[|v|exp\left(\mathit{i\psi }\right)\mathit{\delta \psi }\right],Fexp\left(iargf\right)-\mathit{Cf}\right)=\\ Im\left(|v|exp\left(\mathit{i\psi }\right)\mathit{\delta \psi },A^{\ast }\left[Fexp\left(iargf\right)-\mathit{Cf}\right]\right)=\\ \left(\mathit{\delta \psi },|v|Im\left(exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargf\right)-\mathit{Cf}\right]\right)\right).\end{array}$$E8

The condition

$$Im\left(e^{-\mathit{i\psi }}{A}^{\ast }\left[{\mathit{Fe}}^{iargf}-\mathit{Cf}\right]\right)=0$$E9

must be met to satisfy the necessary condition of extreme of $\kappa \left(\psi \right)$. Eq. (9) is governing the Lagrange–Euler equation, which allows to search for the unknown distribution $\psi$ of phase.

One can reduce two relating equations using (9); they are:

$$\psi =arg\left(A^{\ast }\left[{\mathit{Fe}}^{iargf}-\mathit{Cf}\right]\right)$$E10

and

$$\psi =arg\left(A^{\ast }\left[\mathit{Cf}-{\mathit{Fe}}^{iargf}\right]\right)$$E11

The last derived equations are equivalent; in this connection, the first one we use for the numerical calculations.

We should emphasize that (10) is non-linear equation with respect to the unknown function $\psi$, the above is result that $\psi$ contained implicitly in the exponent of function $iargf$, and additionally $f=A\left(|v|e^{\mathit{i\psi }}\right)$.

2.4 The simplified equation of Lagrange-Euler

In the process of numerical experiments, more simple functional

$${\kappa }_s\left(\psi \right)=\left(|A\left(|v|e^{\mathit{i\psi }}\right)|,F\right)$$E12

which is the numerator of functional (3), is applied at solving the practical problems. The respective to (12), the Lagrange–Euler equation is

$$Im\left(e^{-\mathit{i\psi }}{A}^{\ast }\left[{\mathit{Fe}}^{iargA\left(|\mathbf{v}|e^{\mathit{i\psi }}\right)}\right]\right)=0$$E13

and (13) provide us with an explicit formula

$$\psi =arg\left(A^{\ast }\left[{\mathit{Fe}}^{iargf}\right]\right)$$E14

for determination of the unknown distributions of the phases of currents. Eqs. (13) and (14) are more suitable for the numerical study than Eqs. (9) and (10), because they are free of an auxiliary number $C$.

On the other hand, we can consider the additional function

$$w=A^{\ast }\left[{\mathit{Fe}}^{iargf}\right]$$E15

that gives $\psi =argw$, and

$$f=A^{\ast }\left[{\mathit{Fe}}^{iargw}\right]$$E16

Eqs. (15) and (16) are equivalent to Eq. (14). They present non-linear system of equations for two unknown functions $w$ and $f$.

The reduced forms (13), (15), and (16) of non-linear Lagrange–Euler equations results in the subsequent iterative process to determine their solutions

$${\psi }^{\left(p+1\right)}=arg\left(A^{\ast }\left[{\mathit{Fe}}^{iarg{f}^{\left(p\right)}}\right]\right)$$E17

for (14) and

$$f^{\left(p\right)}=A^{\ast }\left[{\mathit{Fe}}^{iarg{w}^{\left(p\right)}}\right]$$E18

$$w^{\left(p+1\right)}=A^{\ast }\left[{\mathit{Fe}}^{iargf\left(p\right)}\right]$$E19

for (15) and (16), respectively. The convergence of the obtained iterative processes was studied firstly in [6]. Eqs. (17)–(19) are non-linear, therefore the peculiarities of their solutions depend on the properties of the $A$ operator, the frequency of radiation, and the geometry of array.

2.5 The operators $A$ and $A^{\ast }$ in the 1D and 2D cases

One can conclude from Eqs. (18)–(19) that the iterative process of its solving needs many actions of the operators $A$ and $A^{\ast }$ on the interim unknown functions. As a rule, the operator $A$ corresponds to calculation of the DP $f$ of array by the currents $I$ on its radiators. The form of operator $A^{\ast }$ is defined by the geometry of array and the used inner products in the spaces of DPs and currents [17].

2.5.1 The operators $A$ and $A^{\ast }$ in the 1D case

If we solve the synthesis problem in the 1D case, the above is related to a linear array, operator $A$ determines the DP of array and acts as

$$f\left(\xi \right)=\mathit{AI}=\sqrt{\frac{c}{2\pi }}\sum _{n=-N}^N{I}_{n}exp\left(\mathit{\text{icnξ}}\right)$$E20

i.e., operator $A$ correspons to the discrete Fourier transform, and it acts from the Euclidean complex $\left(2N+1\right)-$dimensional space $l_2\left(-N,N\right)$ to Hilbertian space $L_2\left(-\pi /c,\pi /c\right)$ of continuous functions. In Eq. (20), parameter $\xi$ is a generalized angular coordinate; it is determined by relation $\xi =sin\theta /sin{\theta }_0$, and value $2{\theta }_0$ is angle, where the function $F$ is not equal to zero; $-\pi /2\le \theta \le \pi /2$. The DP (20) is periodical and its period is equal to $2\pi /c$.

The operator $A$ in formula (20) is isometric [5]. The operator $A^{\ast }$ acts from space $L_2\left(-\pi /c,\pi /c\right)$ into space $l_2\left(-N,N\right)$, it has form

$$A^{\ast }v=\sqrt{\frac{c}{2\pi }}\underset{-\pi /c}{\overset{\pi /c}{\int }}v\left(\xi \right)exp\left(-\mathit{\text{icnξ}}\right)\mathit{d\xi },n=-N,\dots ,N$$E21

We use also the operator ${\mathit{AA}}^{\ast }$: $l_2\left(-N,N\right)\to l_2\left(-N,N\right)$

$$\left[{\mathit{AA}}^{\ast }v\right]\left(\xi \right)=\frac{c}{2\pi }\underset{-\pi /c}{\overset{\pi /c}{\int }}v\left({\xi }^{\text{'}}\right)\frac{sin\left[c\left(N+1/2\right)\left(\xi -{\xi }^{\text{'}}\right)\right]}{sin\left[c\left(\xi -{\xi }^{\text{'}}\right)/2\right]}d{\xi }^{\text{'}}$$E22

which corresponds to kernel of the integral non-linear equation

$$\mathit{Aw}=A\left[A^{\ast }\left({\mathit{Fe}}^{iargf}\right)\right]$$E23

and which is solved to determine the DP $f$. The last equation is applied mainly while the analytical research regarded to the non-uniqueness of solution and the process of branching [5].

2.5.2 The operators $A$ and $A^{\ast }$ in the 2D case

The application of the method developed here is used to a plane equidistant array. We consider an array having the odd number of radiators in the direction of both coordinate axes, and take into account only array factor but not whole DP [11]. Additionally, we use in a far zone the generalized angular coordinate. The above assumptions allow us to consider the DP of array as a product of independent DPs in the $O^{\text{'}}{\xi }_1$ and $O^{\text{'}}{\xi }_2$ planes that differs on use of usual coordinates $\theta$ and $\phi$. We assume also that the current’s phases in the array’s radiators are separated on two independent distribution sets along the axes $\mathit{Ox}$ and $\mathit{Oy}$, and coordinate system ${\xi }_1{O}^{\text{'}}{\xi }_2$ corresponds to a specific set in a far zone. When to take into account the above assumptions, we present the whole DP in form

$$f\left({\xi }_1,{\xi }_2\right)=A\mathbf{I}\equiv \sum _{m=-M_1}^{M_1}\sum _{n=-N_1}^{N_1}{g}_{\mathit{nm}}\left({\xi }_1,{\xi }_2\right)I_{\mathit{nm}}{e}^{i\left(c_{1}n{\xi }_1+c_{2}m{\xi }_2\right)}$$E24

and $g_{\mathit{nm}}\left({\xi }_1,{\xi }_2\right)$ is the DP of single radiator, $\mathbf{I}=\left\{I_{\mathit{nm}}\right\}$ are currents in the radiators of array. The new coordinates are given in form ${\xi }_1=sin\theta cos\phi /sin{\theta }_1$, and ${\xi }_2=sin\theta sin\phi /sin{\theta }_2$, where $k=2\pi /\lambda$ is the wavenumber, $\lambda$ is length of wave; $c_1={\mathit{kd}}_{1}sin{\theta }_1$, $c_2={\mathit{kd}}_{2}sin{\theta }_2$; the values $d_1$, $d_2$ correspond to the distances between separate radiators in the $\mathit{Ox}$ and $\mathit{Oy}$ directions, respectively. We assume also that the prescribed DP $F$ is zero beyond $\left[\pm {\theta }_1,\pm {\theta }_2\right]$.

If the scalar products in the spaces of DPs and currents are determined according to Ref. [6], one can reduce the close form for the corresponding elements of the operator $A^{\ast }$

$$\begin{array}{l}{\left[A^{\ast }f\right]}_{\mathit{nm}}=\frac{c_1{c}_2}{4{\pi }^2}\underset{-1}{\overset{1}{\int }}\underset{-1}{\overset{1}{\int }}f\left({\xi }_1,{\xi }_2\right){g_{\mathit{nm}}}_{\ast }^{}\left({\xi }_1,{\xi }_2\right)\times \\ e^{-i\left(c_{1}n{\xi }_1+c_{2}m{\xi }_2\right)}]d{\xi }_{1}d{\xi }_2.\end{array}$$E25

Eq. (24) is reduced by the assumption that the mutual coupling the adjacent radiators is taken into account by using the DPs $g_{\mathit{nm}}\left({\xi }_1,{\xi }_2\right)$ of separate elements of array. It can be reduced to simpler form if to take this influence while solving the direct (the analysis) problem in the case of plane array [9].

The iterative processes (17) or (18)–(19) can be easy in use, when the function $A^{\ast }f$ is known.

3. Amplitude optimization

3.1 The optimizing functional

The problem of the amplitude optimization reduces to minimization of functional

$$\sigma =\mid {‖A\left[vexp\left(\mathit{i\psi }\right)\right]|-F‖}^2$$E26

where $F$ is real and positive function (this is desired or given amplitude DP), function $\psi$ is phase (argument) of the complex current $u=vexp\left(\mathit{i\psi }\right)$ in antenna; term $\Vert \cdot \Vert$ is used to define the norm of function in the spaces of DPs or currents. Operator $A$ describes the definition of the DP $f$ of antenna by known functions $v$ and $\psi$, i.e. solves a direct electro-dynamical problem. The closed form of operator $A$ is very important when the optimization problems are solved; moreover, this is essential not only in the synthesis problems but also in the problems of determining the material parameters of scattering medium [22]. The additional terms can be included into (26) to impose specific requirements to the given function $F$ [23].

In the case is considered, phase $\psi$ of function $u$ is given, the minimization of (26) is carried out by choosing the function $v$. In functional (26), it is assumed that function $v$ must be real but not necessary positive. Namely, it can have different sign in the points of definition. To do it positive everywhere, we can add jump equal to $\pi$ in function $\psi$, where it is negative. Therefore, the function $v$ is determined in a set of positive functions. The functional (26) can by minimized directly by the gradient methods [24] while the necessity to solve the specific engineering problems; and the reduction to the corresponding Euler equation is one more recipe of minimization. The latter allows to study the properties of the obtained non-linear equations and the convergence of proposed method of the successive approximations.

3.2 The Euler equation for functional

We reduce the first variation with respect to the unknown function $v$ in functional (26) to reduce the corresponding Euler equation. For this end, we substitute function $v$ in formula (26) by $v_0+\mathit{\delta v}$ ($\delta$ is small number), and we deal with the first-order terms with respect to the increment $\mathit{\delta v}$. After this, we have

$$\sigma \left(v_0+\mathit{\delta v}\right)=\sigma \left(v_0\right)+2Re\left(\mathit{\delta f},f\right)-2\left(\delta |f|,F\right),$$E27

where $\left(\cdot ,\cdot \right)$ is a scalar (inner) product of the functions in DPs space, and

$$f=A\left[vexp\left(\mathit{i\psi }\right)\right],$$E28

$$\mathit{\delta f}=A\left[\mathit{\delta v}exp\left(\mathit{i\psi }\right)\right],$$E29

$$\delta \mid f\mid =Re\left[\mathit{\delta f}exp\left(-iargf\right)\right].$$E30

Using two latter equalities, we derive

$$\mathit{\delta v}\left(v,\mathit{\delta v}\right)=2(\mathit{\delta v},Re\{exp\left(-\mathit{i\psi }\right)A^{\ast }\left(f\right)-exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(-iargf\right)\right]\left\}\right),$$E31

where $A^{\ast }$ is operator adjoint to $A$ [17].

The formula (31) allows us to get the Euler equation corresponding to (26)

$$Re\{exp\left(-\mathit{i\psi }\right)A^{\ast }A\left[vexp\left(\mathit{i\psi }\right)\right]-exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[vexp\left(\mathit{i\psi }\right)\right]\right)\right]\}=0.$$E32

In such a way, (32) is the Hammerstein non-linear integral equation, and the iterative process

$$Re\{exp\left(-\mathit{i\psi }\right)A^{\ast }A\left[v^{\left(n+1\right)}exp\left(\mathit{i\psi }\right)\right]-exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[v^{\left(n\right)}exp\left(\mathit{i\psi }\right)\right]\right)\right]\}=0$$E33

is applicable to solve it.

3.3 The convergence of method

To prove the convergence of iteration process (33), we study the auxiliary functional [5].

$${\sigma }_{\left(n\right)}\left(v\right)={‖|A\left[vexp\left(\mathit{i\psi }\right)\right]|-Fexp(i{\phi }_{\left(n\right)}‖}^2$$E34

where ${\phi }_{\left(n\right)}=arg\left(f^{\left(n\right)}\right)$, and $f^{\left(n\right)}=A\left[v^{\left(n\right)}exp\left(\mathit{i\psi }\right)\right]$.

The Euler equation for the last functional has form

$$Re\{exp\left(-\mathit{i\psi }\right)A^{\ast }A\left[vexp\left(\mathit{i\psi }\right)\right]-exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(i{\phi }_{\left(n\right)}\right)\right]\}=0,$$E35

and one can prove that its solution coincides with function $v^{\left(n+1\right)}$. This fact testifies that $v^{\left(n+1\right)}$ provides a minimum for (34).

Taking into account the results of [5] (subsection 3.2, page 51), we can write

$${\sigma }_{\left(n\right)}\left(v\right)-\sigma \left(v\right)=2\left(|f|\left[1-cos\left(argf-{\phi }_{\left(n\right)}\right)\right],F\right)\ge 0$$E36

and the last inequality results in the chain let

$$\sigma \left(v^{\left(n+1\right)}\right)\le {\sigma }_{\left(n\right)}\left(v^{\left(n+1\right)}\right)\le {\sigma }_{\left(n\right)}\left(v^{\left(n\right)}\right)=\sigma \left(v^{\left(n\right)}\right)$$E37

which proves the convergence of iterative process (33).

The non-linear Eq. (32) has non-unique solutions, and such solutions branch off when the number $c$ in the kernel $A^{\ast }A$ grows. As a rule, there exists the unique solution only at small $c$. While necessity to find the branching points we should consider the corresponding homogeneous equation and to solve the respective eigenvalue problem.

3.4 Equation to determine the branching points

As in the case of amplitude-phase and phase synthesis problems [17], the non-linear Eq. (32) becomes the non-unique solutions if the characteristic parameter grows in its kernel $A^{\ast }A$ (only a unique solution exists if this parameter is small). The method of perturbations is used to get a homogeneous equation for determination of the points of branching. It is evidently that perturbation of the parameter $c=c_0+\epsilon$ in the kernel leads to perturbation of the operator

$$A=A_0+\epsilon A_1$$E38

and the solution

$$v=v_0+\epsilon v_1$$E39

If we substitute Eqs. (38), (39) into (32), and after some transformations, we get the non-homogeneous linear equation

$$\begin{array}{l}Re[exp\left(-\mathit{i\psi }\right)A_0^{\ast }\left[v_{1}exp\left(\mathit{i\psi }\right)\right]-\mathit{\text{iF}}exp\left(iarg\left(f_0\right)\right)\times \\ Im\frac{A_0(v_{1}exp\left(\mathit{i\psi }\right)}{f_0}]=\\ -Re\left[exp\left(-\mathit{i\psi }\right)A_1^{\ast }{A}_1\left[v_{0}exp\left(\mathit{i\psi }\right)\right]\right]\\ +Re\left[exp\left(-\mathit{i\psi }\right)A_1^{\ast }\left[Fexp\left(iarg{A}_1\left[v_{0}exp\left(\mathit{i\psi }\right)\right]\right)\right]\right]\end{array}$$E40

for the perturbed function $v_1$, and

$$f_0=A_0\left(v_{0}exp\left(\mathit{i\psi }\right)\right)$$E41

Eq. (40) has an unique solution when the homogeneous equation

$$Re[exp\left(-\mathit{i\psi }\right)[A_0^{\ast }\left[v_{1}exp\left(\mathit{i\psi }\right)\right]-\mathit{\text{iF}}exp\left(iarg\left(f_0\right)\right)\times Im\frac{A_0(v_{1}exp\left(\mathit{i\psi }\right)}{f_0}\left]\right]]=0$$E42

does not have the nontrivial solutions. The different solutions to (32) can be exist at the parameter $c$, at which Eq. (42) has a nontrivial solution. Basing on this results, we get the non-linear eigenvalue problem with respect to $c$, and we write it in the form

$$\begin{array}{l}{\lambda }_{n}Re\left[exp\left(-\mathit{i\psi }\right)\left[A_0^{\ast }\left[A_0\left(w_{n}exp\left(\mathit{i\psi }\right)\right)\right]\right]\right]=\\ -Im\left[exp\left(-\mathit{i\psi }\right)A_0^{\ast }\left[Fexp\left(iarg\left(f_0\right)\right)Im\frac{A_0\left(w_{n}exp\left(\mathit{i\psi }\right)\right)}{f_0}\right]\right].\end{array}$$E43

The value $c=c_0$ is the point of branching for solution of Eq. (32) when at least one of its eigenvalues ${\lambda }_n$ in (43) is unity.

3.5 The cases of different operators $A$

One can show that form of Eq. (32) can vary for the different operators $A$. We confine ourselves by operator $A$, which corresponds the DP of a plane continuous antenna (the case of compact operator), and the DP of a linear antenna (the case of isometric operator).

3.5.1 The compact operator $A$

For this case, the method of minimization of (26) is unstable; namely, its solutions become perturbed considerably, if the amplitude DP $F$ or the operator $A$ becomes small perturbation. In order to exclude this drawback, we add tot (26) some stabilization term

$${\sigma }_{\alpha }\left(v\right)=\mid {‖A\left[vexp\left(\mathit{i\psi }\right)\right]|-F‖}^2+\alpha {‖v‖}^2$$E44

where number $\alpha >0$ is the value of regularization parameter.

The operator $A$ in this case becomes form

$$f\left(\phi \right)=\mathit{Au}=\underset{-\pi }{\overset{\pi }{\int }}u\left({\phi }^{\text{'}}\right)exp\left[\mathit{ikr}\left({\phi }^{\text{'}}\right)cos\left(\phi -{\phi }^{\text{'}}\right)\right]{\mathit{dS}}_{\phi \text{'}}$$E45

where angular coordinate $\phi$ and ${\phi }^{\text{'}}$ are points in the antenna and a far zone, the wavenumber $k$ plays role of parameter $c$, and element of arc is ${\mathit{dS}}_{\phi \text{'}}=\sqrt{r^2\left({\phi }^{\text{'}}\right)+{\left(\mathit{dr}\left(\phi \text{'}\right)/\mathit{d\phi }\text{'}\right)}^2}$.

The Euler equation for (44) has form

$$Re\{exp\left(-\mathit{i\psi }\right)A^{\ast }A\left[vexp\left(\mathit{i\psi }\right)\right]-exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[vexp\left(\mathit{i\psi }\right)\right]\right)\right]\}+\mathit{\alpha v}=0,$$E46

and form of the ajoint operator $A^{\ast }$ is determined by form of operator $A$ and the iscalar products in the spaces of the currents and DPs. We use such iterative process to solve it

$$\begin{array}{l}Re\left\{exp\left(-\mathit{i\psi }\right)A^{\ast }A\left[v^{\left(n+1\right)}exp\left(\mathit{i\psi }\right)\right]\right\}+\alpha v^{\left(n+1\right)}\\ =Re\left\{exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[v^{\left(n\right)}exp\left(\mathit{i\psi }\right)\right]\right)\right]\right\}=0.\end{array}$$E47

The corresponding eigenvalue problem looks like similarly to (43), but it is supplemented by term ${\lambda }_n\alpha w_n$ in the first line.

3.5.2 The isometric operator $A$

The equality is true

$$\Vert A\left(vexp\left(\mathit{i\psi }\right)\right)\Vert =\Vert v\Vert$$E48

if operator $A$ is an isometric. It connects the norms of the amplitude $v$ and DP $f=A\left(vexp\left(\mathit{i\psi }\right)\right)$, therefore (26) becomes

$$\sigma \left(v\right)={‖v‖}^2-2\left(|A\left[vexp\left(\mathit{i\psi }\right)\right]|,F\right)+{‖F‖}^2.$$E49

The operator $A$ for determination of the DP is

$$f\left(\xi \right)=\mathit{Au}=\underset{-1}{\overset{1}{\int }}u\left(x\right)exp\left(\mathit{\text{icξx}}\right)\mathit{dx}$$E50

In this case, the Euler equation becomes simpler form

$$v=Re\left\{exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[vexp\left(\mathit{i\psi }\right)\right]\right)\right]\right\}$$E51

and the iterative procedure is

$$v^{\left(n+1\right)}=Re\left\{exp\left(-\mathit{i\psi }\right)A^{\ast }\left[Fexp\left(iargA\left[v^{\left(n\right)}exp\left(\mathit{i\psi }\right)\right]\right)\right]\right\}$$E52

The points of branching of solution to Eq. (51) are that values of the number $c$, at which the corresponding linear homogeneous equation

$${\lambda }_n{w}_n=-Im\left\{Fexp\left(iarg\left(f_0\right)\right)Im\frac{A_0\left[w_{n}exp\left(\mathit{i\psi }\right)\right]}{f_0}\right\}$$E53

has the multiple eigenvalues ${\lambda }_i=1$.

4. Numerical modeling

4.1 Phase optimization

We show in this section the results related to solving the synthesis problems for the 1D and 2D cases, which are related to the engineering implementation for the linear antennas and plane equidistant arrays [14].

In the case if $A$ operator is Fourier transform of some limited function that corresponds to physical nature of a linear antenna, the optimization problem similar to (26) was solved firstly in [6]; the iterative method (the algorithm proposed by Katsenelenbaum and Semenov, [5]) for its solving was proposed and acquitted. In paper [17], it was borrowed to the Fourier transform in the discrete case, and a series of the numerical results was got.

The different solutions supplementing a maximum of (3), which correspond to the non-linear system (18) and (19), are presented here. Such solutions give maximum at the certain values of the characteristic number $c$, which define the frequency of radiation and geometry of antenna. The global maximum is defined by the upper envelope of curves match the different types of solutions.

In our case, system (18), (19) has form

$$w\left(x\right)={\int }_{-1}^{1}F\left(\xi \right)exp\left(iargf\left(\xi \right)\right)exp\left(-\mathit{\text{icxξ}}\right)\mathit{d\xi }$$E54

$$f\left(\xi \right)={\int }_{-1}^{1}U\left(x\right)exp\left(iargw\left(x\right)\right)exp\left(\mathit{\text{icxξ}}\right)\mathit{dx}$$E55

4.1.1 The variety of solutions

In the case if $U$ and $F$ are the even functions of their arguments, the solutions to Eqs. (18) and (19) can be specified within three groups:

1. the functions $w\left(x\right)$, $f\left(\xi \right)$ are real valued for $x\in \left(-1,1\right)$, $\xi \in \left(-1,1\right)$, respectively; either one or both such functions are non-zero;

2. one of function, let $w\left(x\right)$, is real valued and becomes zeros in the interval $\left(-1,1\right)$; the other function $f\left(\xi \right)$ has an odd phase, i.e. $\phi \left(-\xi \right)=-\phi \left(\xi \right)$;

3. the functions $f\left(\xi \right)$, $w\left(x\right)$ have an even phases, namely $\psi \left(-x\right)=\psi \left(x\right)$, $\phi \left(-\xi \right)=\phi \left(\xi \right)$.

The solutions (c), for which both the functions $f\left(\xi \right)$, $v\left(\xi \right)$ are complex valued, exist in the case of symmetrical and non-symmetrical functions $U\left(x\right)$, $F\left(\xi \right)$. Furthermore, the global extremal points of ${\kappa }_s$ are located between above solutions, except the case if they both are defined by the Eqs. (26) and (27) for the symmetrical data. Taking into account the above, we consider below the solutions that are determined numerically, and the symmetry of data does not take into account.

4.1.2 The numerical solutions

We consider firstly the case, if both the functions $v\left(x\right)$, $F\left(\xi \right)$ are identical. The results for $v\left(x\right)=\text{const}$, $F\left(\xi \right)=\text{const}$ are presented below. The values of $\sigma$ functional for various solutions to Eqs. (18) and (19) are shown in Figure 1. There exists one trivial solution $w_0\left(x\right)=0$, $f_0\left(\xi \right)=0$ up to point $c=\pi$ (this value is denoted by $c_2$). Nevertheless, this solution ceases as optimal earlier, i.e., at the point $c=c_1<c_2.$ This value of $c$ (it is $\approx$2.80) corresponds to the first point, in which the real solution branches off.

Figure 1.

The values of functionals $\sigma =1-{\left|{\kappa }_s\right|}^2$.

The solution $\left\{f_0,w_0\right\}$ branches into four different at the point $c=c_2$. Three of such solutions are real. The first of them (denoted by 0) is symmetrical, and the functions $f_0\left(\xi \right)$, $w_0\left(x\right)$ coincide in the shape for it. The functions ${\phi }_0\left(\xi \right)$ (phase of $f_0\left(\xi \right)$) and ${\psi }_0\left(x\right)$ (phase of $w_0\left(x\right)$) are also the same, of the form ${\phi }_0\left(\xi \right)=\left\{\begin{array}{c}0,f_0\left(\xi \right)\ge 0,\\ \pi ,f_0\left(\xi \right)<0.\end{array}\right.$. Their zeros are shown in Figure 2 as $x_{1,2}^{\left(0\right)},{\xi }_{1,2}^{\left(0\right)}$. Two remaining solutions (marked by $0$ and $0^{\text{'}}$) are characterized by the different $f_{0^{\prime }}\left(\xi \right)$ and $w_{0^{\prime }}\left(x\right)$. They are mirror-symmetrical, i.e. these functions are interchanging in them. One of such functions (let $f_0\left(\xi \right)$) can be calculated by $f_0=\mathit{Av}$; by this, its zeros are shown in Figure 3 as ${\xi }_m,m=1,2,\dots ,5,6$. Another function (namely $f_{0\text{'}}\left(\xi \right)$) has unit sign for the examined parameters of $c$.

Figure 2.

Zero curves of the functions $w\left(x\right)$ and $f\left(\xi \right)$, corresponding to the real and complex symmetrical solutions.

Figure 3.

Zero lines of the real solution $f_0\left(\xi \right)$.

A global maximum for this set of solutions is described by the different solutions at the defined values of $c$ (see Figure 1). One should note that the solution $\left\{f_0,w_0\right)$ is the best immediately after $c=c_2$.

Subsequently, the solution $\left\{f_{0^{\prime }},w_{0\text{'}}\right\}$ at point $c=c_3\approx 6.175$ is better; by this, the curves ${\sigma }_0$ and ${\sigma }_{0^{\prime }}$ cross two times.

The complex optimal solution, marked by index 1, appears at $c>c_1$. It is the symmetrical i.e. the functions $f_1\left(\xi \right)$, $w_1\left(x\right)$ (and ${\phi }_1\left(\xi \right)$, ${\psi }_1\left(x\right)$, respectively) match in shape. The change of their phases is shown in Figure 4 (curve 1).

Figure 4.

Phase variations of the asymmetrical function $f_2\left(\xi \right)$.

Except for the real solutions (with type (a)), and the complex solution (with type(c)), the asymmetrical solutions (with type (b)) is found for the examined data. One of the determined functions (let $w\left(x\right)$) is real asymmetrical; by this, the second function, namely $f\left(\xi \right)$ has an even amplitude and an odd phase in this solutions. For the considered initial data, three mirror-symmetrical pairs of solutions exist. Let us specify the individuals of such pairs denoted by indices ‘2’, ‘3’, and ‘4’, respectively (see Figure 4). The pair $f_2\left(\xi \right)$, $w_2\left(x\right)$ appears at the point $c=c_2$.The function $w_2\left(x\right)$ is real, and its unique zero is marked as $x_1^{\left(2\right)}$ in Figure 2. The function $f_2\left(\xi \right)$ has even modulus, namely $\mid f_2\left(-\xi \right)\mid =\mid f_2\left(\xi \right)\mid$, and odd phase, $\mid {\phi }_2\left(-\xi \right)\mid =-\mid {\phi }_2\left(\xi \right)\mid$. Its phase change $\Delta {\phi }_2$ is shown as curve 2 in Figure 4.

The last two asymmetrical solutions marked by curves ‘3’and ‘4’ are not branched off from any solutions existing before. They appear at the point $c=c_3\approx 5.75$, and they belong to the isolated branches (Figure 1, curves 3 and 4). Both the functions $w_3\left(\xi \right)$ and $w_4\left(\xi \right)$ are also real. Each of the above functions has two zeros $x_1$, $x_2$ shown by the respective curves by the superscripts (3) and (4) in Figure 2, respectively. The changes $\Delta {\phi }_3\left(\xi \right)$, $\Delta {\phi }_4\left(\xi \right)$ of their odds phases are given in Figure 4 (curves 3 and 4). One should note that for the values of $c$ larger than $c_5\approx 7.685$, the solution ($f_3\left(\xi \right)$, $w_3\left(x\right)$) is the best of all non-optimal solutions. It is characterized by the losses in the amount about 30% comparable to the optimal solution at the point $c=10.0$.

4.2 Amplitude optimization

The data are given for the linear antenna (the operator $A$ is isometric). The given amplitude DP is

$$F\left(\xi \right)=cos\left(\mathit{\pi \xi }/2\right),$$E56

and the modified angular coordinate $\xi$ is related the angular coordinate $\phi$ by $\xi =sin\phi /sin\alpha$; $\alpha$ is angle where the DP $F\left(\xi \right)$ is non zero. Use of the modified angular coordinate $\xi$ has some advantage, because this reduces the calculations when the adjoint operator $A^{\ast }$ is calculated. In the case, if we use together the normalized coordinate $x$ in the linear antenna and the modified angular coordinate $\xi$ in a far zone, the explicit form of $A^{\ast }$ becomes as integration over the interval $\left[-1,1\right]$ with the complex conjugated exponent as integrand

$$u\left(x\right)=A^{\ast }g=\underset{-1}{\overset{1}{\int }}g\left(\xi \right)exp\left(-\mathit{\text{icxξ}}\right)\mathit{d\xi }$$E57

i.e. the operators $A$ and $A^{\ast }$ are pairs of two Fourier transforms having the complex conjugated exponents.

The iterative process (52) initiates with the approximation $u_0\equiv 1.0$. The characteristic of convergence, namely value of $max\mid u_{n+1}-u_m\mid$ is shown in Figure 5. One can conclude that only several iteration we need to attain the accuracy, which is equal to $2\times {10}^{-14}$. In spite of that the differences in the first iteration are the values 0.86, 0.69, 0.84, and 0.27 for $c=2.0$, $c=5.0$, $c=7.0$, and $c=10.0$, respectively, the high exactness $2\times {10}^{-14}$ is characteristic in the sixth iteration at all values of $c$, which are considered.

Figure 5.

The character of convergence versus number of iterations.

The optimal calculated amplitudes of $v$ for the studied parameters $c$ are shown in Figure 6. The amplitude $v$ is like constant for $c=2.0$, and it oscillates if $c$ grows. The above corresponds to the physical nature of such antennas that demonstrate the growth of the number of waves per length of antenna when frequency increases.

Figure 6.

The optimal amplitudes at the different values of $c$.

One can sure that the iterative process (47) converges quickly, and the optimal amplitudes can be easily realized for the antennas at the studied parameters of problem. The small value of the mean-square deviation $\sigma$ confirms the effectiveness of the developed iterative method. This value is given for some values of $c$ in Figure 7. One can see that $\sigma$ does not decrease if $c$ grows as it expected at the first view. One can conclude from data in Figure 7 that $\sigma$ grows when $c$ increases. The above is the result that $c$ must correspond to the length of antenna equal to integer numbers of wave with one half of wave. Such condition is not valid for the given $c$. Therefore, choosing the optimal value of $c$ is additional way to get the synthesis of results more corresponding to the practice.

Figure 7.

The values of $\sigma$ versus parameter $c$.

In Figure 8, the optimal created amplitude DPs $\mid f\mid$ are marked by the color curves; the given amplitude DP $F$ is depicted by the thick black curve. The oscillations in DPs $\mid f\mid$ are missing at smaller $c$, and function $\mid f\mid$ oscillates at $c=10.0$; this is characteristic for the higher frequency radiation. The amplitudes $\mid f\mid$ are given after additional improvement related to decrease the level of radiation. Such procedure improves the approximation of amplitude $\mid f\mid$ to given function $F$ greatly, i.e. the received values of $\sigma$ functional become 0.0512, 0.0132, 0.947, and 0.1604 for $c=5.0$, $c=7.0$, $c=10.0$, and $c=12.0$, respectively. The last values of $\sigma$ are considerably smaller comparably to those that are given in Figure 7.

Figure 8.

The synthesized amplitude DPs at the different $c$.

The numerical data testify that the quality of approximation to the function $F$ depends not only on the parameter $c$; the form of function $F$ is important as well. The results in Figure 9 show the optimal amplitude DPs $\mid f\mid$, which corresponds to function.

Figure 9.

The synthesized amplitude DPs for the case of narrow $F$.

$F\left(\xi \right)={\left(cos\left(\mathit{\pi \xi }/2\right)\right)}^{16}$ (black thick curve). The quality of approximation to $F\left(\xi \right)$ improves if $c$ increases. The values of (26) for this $F\left(\xi \right)$ are equal to 0.9103, 0.2014, 0.0601, and 0.0087 for the parameters $c=2.0$, $c=5.0$, $c=7.0$, and $c=10.0$, respectively.

5. Conclusions

The variational methods are applied for the formulation of the synthesis problems of antennas; and the problems the reduced to numerical solving the corresponding non-linear integral equations. The restrictions applied to the amplitude or phase of function to be optimized leads to the specific form of such equations, which in turn are solved using the method of successive approximations. The obtained results show the effectiveness of approach proposed in [17], which was used for solving the similar integral equations. The approach proposed in [5] and developed here to two types of operator $A$ allowed to determine a set of branching solutions as well as the points of their branching. The try to consider more complex functions $F\left(\xi \right)$ demonstrates the significant computational severities that can be overcome by the modification of the proposed method of successive approximations. This will open the perspective to study of a new class of the Hammerstein non-linear integral equations and provide ability to solve the engineering synthesis problems for the different types of antennas.

Conflict of interest

The authors declare no conflict of interest.