3. Optimization problem and antenna array synthesis

A generic planar array of N elements lying on the xy-plane of Figure 4 is assumed. The synthesis function in (4) can be written as follows:

$$F(\phi,\theta) = \frac{r\_0}{r} \frac{f(\phi,\theta)}{f(\phi\_0,\theta\_0)} \sum\_{n=1}^{N} w\_n e^{j\left(k\_x x\_n + k\_y y\_n\right)} \tag{6}$$

point of the coverage area, this always leads to a feasible solution with an acceptable array size. Another criterion for the steering direction choice is to select ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> in

The broadside array is the most considered case for practical usage. Under the hypothesis of symmetric antenna array with respect to the axes origin, the synthesis

> ∑ N=2

wncos kxxn þ kyyn

(10)

(11)

n¼1

In this case, the amplitude excitations wn ∈ R, n∈f g 1; ::; N [18], and, consequently, the function Fð Þ ϕ; θ are real. In this way, the lower bound inequality in (9) can be rewritten as a convex constraint. In fact, since the real function Fð Þ ϕ; θ is close to its maximum value in the bounded area C, it is plausible that within C it is

�<sup>F</sup> <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>≤</sup>10<sup>P</sup>bound=<sup>20</sup>, <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>∈</sup>C that can be included as a convex constraint in

The optimization problem (9) for the broadside direction can now be written as

F ϕ<sup>i</sup> ð Þ ; θ<sup>i</sup> j j≤t, ϕ<sup>i</sup> ð Þ ; θ<sup>i</sup> ∈Σ

F ϕ<sup>k</sup> ð Þ ; θ<sup>k</sup> ≤1, ϕ<sup>k</sup> ð Þ ; θ<sup>k</sup> ∈C �<sup>F</sup> <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>≤</sup> � <sup>10</sup><sup>P</sup>bound

The last constraint has been introduced because in the case of a high number of antennas, the array factor exhibits very large oscillations which might cause the

The optimization problem in (11) can now be written in the form of a linear program as described in [15] with the great advantage of a lower computational

When the mechanical tilt θ<sup>A</sup> cannot be arbitrarily steered to comply with a specific coverage direction, or if it is necessary to synthesize more coverage areas toward different directions, the synthesis function in (4) is not real because

wn ∈ C, n∈ f g 1; ::; N . For this reason, another simplification of the problem is herein

<sup>F</sup> <sup>ϕ</sup><sup>h</sup> j j ð Þ ; <sup>θ</sup><sup>h</sup> <sup>≤</sup> <sup>10</sup><sup>P</sup>bound

<sup>20</sup> , ϕ<sup>j</sup>

; θj <sup>∈</sup><sup>B</sup>

<sup>20</sup> , ϕ<sup>h</sup> ð Þ ; θ<sup>h</sup> ∈ Ω

<sup>20</sup> , ϕ<sup>k</sup> ð Þ ; θ<sup>k</sup> ∈C

order to synthesize the array factor as much symmetrical as possible [15]. The second hypothesis, instead, addresses the most part of practical cases. Based on the choice of the steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> , two main practical cases can be distinguished: the broadside array ð Þ ϕ<sup>0</sup> ¼ 0; θ<sup>0</sup> ¼ 0 and the steered array

ð Þ ϕ<sup>0</sup> 6¼ 0; θ<sup>0</sup> 6¼ 0 .

the optimization.

complexity.

37

3.1.2 Non-broadside array

3.1.1 Broadside array

function in (4) can be written as follows:

Array Pattern Synthesis for ETC Applications DOI: http://dx.doi.org/10.5772/intechopen.80525

Fð Þ¼ ϕ; θ 2

min wn, <sup>n</sup>∈f g <sup>1</sup>;…;<sup>N</sup>

r0 r

also strictly positive; thus, the inequality can be simplified as <sup>F</sup> <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>≥</sup>10<sup>P</sup>bound=<sup>20</sup>, <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>∈</sup>C, and, finally, written in the form

t

s:t: F ϕ<sup>0</sup> ð Þ¼ ; θ<sup>0</sup> 1

F ϕ<sup>j</sup> ; θ<sup>j</sup> <sup>¼</sup> <sup>10</sup><sup>P</sup>bound

function <sup>F</sup>ð Þ <sup>ϕ</sup>; <sup>θ</sup> to be lower than 10<sup>P</sup>bound=<sup>20</sup> within the coverage area.

fð Þ ϕ; θ f ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup>

where kx <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>λ</sup><sup>0</sup> cosð Þ <sup>ϕ</sup> sin ð Þ<sup>θ</sup> , ky <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>λ</sup><sup>0</sup> sin ð Þ ϕ sin ð Þθ , wn is the nth element amplitude excitation, wn ∈ C, n ∈f g 1; ::; N [18], and xn; yn is the position of the nth antenna element on the xy-plane. Let us now consider the case of a synthesis on the plane zR ¼ 0 as illustrated in Figure 4. Defining a suppression region Σ, where the maximum sidelobe level t ¼ max <sup>ϕ</sup><sup>i</sup> ð Þ ;θ<sup>i</sup> <sup>∈</sup><sup>Σ</sup> F ϕ<sup>i</sup> ð Þ ; θ<sup>i</sup> j j has to be minimized, that is,

$$|F(\phi\_i, \theta\_i)| \le t, \ (\phi\_i, \theta\_i) \in \Sigma \tag{7}$$

and a coverage area C, where it is desired that the normalized electric field is larger than a certain bound value Pbound (expressed in dB), that is,

$$|F(\phi\_k, \theta\_k)| \ge \mathbf{1} \mathbf{0}^{p\_{\text{bound}}/20}, (\phi\_k, \theta\_k) \in \mathbf{C} \tag{8}$$

it is possible to derive the generic optimization problem as

$$\begin{array}{lcl} \min\_{\boldsymbol{\phi}\_{0}, \boldsymbol{\theta}\_{0}, w\_{u}, \boldsymbol{n} \in \{1, \ldots, N\}} & t \\ \text{s.t.} & F(\boldsymbol{\phi}\_{0}, \boldsymbol{\theta}\_{0}) = \mathbf{1} \\ & |F(\boldsymbol{\phi}\_{i}, \boldsymbol{\theta}\_{i})| \leq t, (\boldsymbol{\phi}\_{i}, \boldsymbol{\theta}\_{i}) \in \boldsymbol{\Sigma} \\ & \left| F\left(\boldsymbol{\phi}\_{i}, \boldsymbol{\theta}\_{i}\right) \right| = \mathbf{1} \mathbf{0}^{\mathsf{P}\_{\text{bound}}/20}, \left(\boldsymbol{\phi}\_{i}, \boldsymbol{\theta}\_{i}\right) \in \boldsymbol{B} \\ & |F(\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k})| \leq \mathbf{1} \mathbf{0}^{\mathsf{P}\_{\text{bound}}/20}, (\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k}) \in \boldsymbol{\Omega} \\ & |F(\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k})| \leq \mathbf{1}, (\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k}) \in \mathbf{C} \\ & |F(\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k})| \geq \mathbf{1} \mathbf{0}^{\mathsf{P}\_{\text{bound}}/20}, (\boldsymbol{\phi}\_{k}, \boldsymbol{\theta}\_{k}) \in \mathbf{C} \end{array} \tag{9}$$

Some additional constraints are included to better define the function trend within the area of interest. In particular, the constraint F ϕj; θ<sup>j</sup> <sup>¼</sup> <sup>10</sup><sup>P</sup>bound=<sup>20</sup>, <sup>ϕ</sup><sup>j</sup> ; θ<sup>j</sup> <sup>∈</sup><sup>B</sup> fixes the function value on the coverage area bound <sup>B</sup>, while <sup>F</sup> <sup>ϕ</sup><sup>h</sup> j j ð Þ ; <sup>θ</sup><sup>h</sup> <sup>≤</sup> <sup>10</sup><sup>P</sup>bound=<sup>20</sup>, <sup>ϕ</sup><sup>h</sup> ð Þ ; <sup>θ</sup><sup>h</sup> <sup>∈</sup> <sup>Ω</sup> defines a criterion within <sup>Ω</sup> that is the space between the coverage area and the suppression region. It is worth noting that ϕ<sup>i</sup> ð Þ ; θ<sup>i</sup> are related to the coordinates xR and yR according to (5).

The mechanical tilt θ<sup>A</sup> has not been included in the optimization problem because its choice is usually not arbitrary. It could be preliminarily selected to radiate toward a specific direction, and its choice is left to common sense.

### 3.1 Derivation of suboptimal problem

The steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> and the last inequality in (9) lead to a nonlinear optimization problem with a non-convex constraint, and according to [5], the global optimality cannot be guaranteed, with computation time extremely large.

Two hypotheses have been considered for reducing the problem complexity. In particular, a known steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> and symmetric antenna array with respect to the axes origin are assumed. Since there is no way to know a priori the optimum steering direction, the first hypothesis will lead to a suboptimal solution based on a common sense selection of the steering direction. Furthermore, it has been observed experimentally that if the array pattern is steered toward the center Array Pattern Synthesis for ETC Applications DOI: http://dx.doi.org/10.5772/intechopen.80525

point of the coverage area, this always leads to a feasible solution with an acceptable array size. Another criterion for the steering direction choice is to select ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> in order to synthesize the array factor as much symmetrical as possible [15].

The second hypothesis, instead, addresses the most part of practical cases.

Based on the choice of the steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> , two main practical cases can be distinguished: the broadside array ð Þ ϕ<sup>0</sup> ¼ 0; θ<sup>0</sup> ¼ 0 and the steered array ð Þ ϕ<sup>0</sup> 6¼ 0; θ<sup>0</sup> 6¼ 0 .
