3.1.1 Broadside array

<sup>F</sup>ð Þ¼ <sup>ϕ</sup>; <sup>θ</sup> <sup>r</sup><sup>0</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>

tude excitation, wn ∈ C, n ∈f g 1; ::; N [18], and xn; yn

where kx <sup>¼</sup> <sup>2</sup><sup>π</sup>

Array Pattern Optimization

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>

3.1 Derivation of suboptimal problem

36

r

larger than a certain bound value Pbound (expressed in dB), that is,

it is possible to derive the generic optimization problem as

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

within the area of interest. In particular, the constraint

; θ<sup>j</sup>  t

F ϕj; θ<sup>j</sup> 

min <sup>ϕ</sup>0, <sup>θ</sup>0, wn, <sup>n</sup>∈f g <sup>1</sup>;…;<sup>N</sup>

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

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,

and a coverage area C, where it is desired that the normalized electric field is

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> j j ð Þ ; θ<sup>k</sup> ≤1, ϕ<sup>k</sup> ð Þ ; θ<sup>k</sup> ∈C

 

Some additional constraints are included to better define the function trend

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.

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

∑ N n¼1

wne j kxxnþkyy ð Þ<sup>n</sup> (6)

is the position of the nth

<sup>λ</sup><sup>0</sup> sin ð Þ ϕ sin ð Þθ , wn is the nth element ampli-

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 (7)

<sup>F</sup> <sup>ϕ</sup><sup>k</sup> j j ð Þ ; <sup>θ</sup><sup>k</sup> <sup>≥</sup>10Pbound=<sup>20</sup>, <sup>ϕ</sup><sup>k</sup> ð Þ ; <sup>θ</sup><sup>k</sup> <sup>∈</sup><sup>C</sup> (8)

<sup>¼</sup> <sup>10</sup><sup>P</sup>bound=<sup>20</sup>, <sup>ϕ</sup>j; <sup>θ</sup><sup>j</sup>

∈B fixes the function value on the coverage area

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

<sup>F</sup> <sup>ϕ</sup><sup>k</sup> j j 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><sup>C</sup>

∈B

(9)

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 function in (4) can be written as follows:

$$F(\phi,\theta) = 2\frac{r\_0}{r} \frac{f(\phi,\theta)}{f(\phi\_0,\theta\_0)} \sum\_{n=1}^{N/2} w\_n \cos\left(k\_x \kappa\_n + k\_y \wp\_n\right) \tag{10}$$

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 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 �<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.

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

$$\begin{array}{lcl}\min\_{w\_{u}, u \in \{1, \ldots, N\}} & t\\ \text{s.t.} & F(\phi\_{0}, \theta\_{0}) = 1 \\ & |F(\phi\_{i}, \theta\_{i})| \leq t, (\phi\_{i}, \theta\_{i}) \in \Sigma \\ & F\left(\phi\_{j}, \theta\_{j}\right) = 10^{\frac{\mathsf{P}\_{\text{bound}}}{20}}, \left(\phi\_{j}, \theta\_{j}\right) \in B \\ & |F(\phi\_{h}, \theta\_{h})| \leq 10^{\frac{\mathsf{P}\_{\text{bound}}}{20}}, (\phi\_{h}, \theta\_{h}) \in \Omega \\ & F(\phi\_{k}, \theta\_{k}) \leq 1, (\phi\_{k}, \theta\_{k}) \in C \\ & -F(\phi\_{k}, \theta\_{k}) \leq -10^{\frac{\mathsf{P}\_{\text{bound}}}{20}}, (\phi\_{k}, \theta\_{k}) \in C \end{array} \tag{11}$$

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 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.

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 complexity.

#### 3.1.2 Non-broadside array

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 proposed. In fact, if a particular shape of the weights is chosen, that is, wn ¼ ane �j kx,0xnþky,0<sup>y</sup> ð Þ<sup>n</sup> , where an <sup>∈</sup> <sup>R</sup>, n<sup>∈</sup> f g <sup>1</sup>; ::; <sup>N</sup> , kx,<sup>0</sup> <sup>¼</sup> kx <sup>ϕ</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> ð Þ ; <sup>θ</sup> <sup>¼</sup> <sup>θ</sup><sup>0</sup> , and ky,<sup>0</sup> ¼ ky ϕ ¼ ϕ<sup>0</sup> ð Þ ; θ ¼ θ<sup>0</sup> , the synthesis function (4) reads

$$F(\phi,\theta) = 2\frac{r\_0}{r}\frac{f(\phi,\theta)}{f(\phi\_0,\theta\_0)} \cdot \sum\_{n=1}^{\frac{N}{2}} a\_n \cos\left[ (k\_\text{x} - k\_{\text{x},0})\chi\_n + (k\_\text{y} - k\_{\text{y},0})\chi\_n \right] \tag{12}$$

which is again a real function. As for the broadside array case, the optimization problem (9) in the known direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> can be simplified as (11) and written in the form of a linear program as described in [15].

#### 3.2 Simulation results

In this section, some numerical results which demonstrate the capability of the described method are presented. A circular shape for both the coverage area and the suppression region is considered, with diameter of 3.5 and 6.5 m, respectively. The two regions are centered in 0; y<sup>0</sup> and hA <sup>¼</sup> <sup>5</sup>:5 m. A rectangular array of Nx � Ny elements with uniform interelement distance d ¼ 0:6λ<sup>0</sup> is synthesized, with antenna elements as microstrip patch antennas (theoretical formula of the radiation pattern fð Þ ϕ; θ has been taken as reported in [2, 19]).

Now, the effectiveness of the proposed method is investigated for different

Antenna array normalized electric field with different numbers of antenna elements, with θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, Pbound ¼ �10 dB, and broadside optimization. (a) Normalized electric field in xR axis, within

yR ¼ 3:25 m, and (b) normalized electric field in yR axis, within xR ¼ 0 m.

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

The broadside optimization presented in Section 3.1.1 is firstly analyzed for different numbers of antenna elements. Results are reported in Figure 6 as a function of the coordinates xR and yR, with θ<sup>A</sup> ¼ 60°, y0 ¼ 3:25 m, and

The curve 16 � 12 is the first feasible result which presents a sidelobe level of 35.1 dB within the suppression region. Other curves have been obtained with increased number of antenna elements. Obviously, the sidelobe-level performance improves with the increase of the antenna elements. All the synthesized results

The influence of the mechanical tilt θ<sup>A</sup> choice on the optimization result is herein investigated. Broadside optimization along with the coordinate yR for different mechanical tilt θ<sup>A</sup> is depicted in Figure 7(a). It is worth noting that the coverage

(a) Antenna array normalized electric field with different mechanical tilt θ<sup>A</sup> in xR ¼ 0 m, with

steering directions θ<sup>0</sup> in x<sup>0</sup> ¼ 0 m, with y<sup>0</sup> ¼ 3:25 m, and Pbound ¼ �10 dB.

Pbound ¼ �10 dB, and broadside optimization and (b) antenna array normalized electric field with different

synthesis parameters.

Figure 6.

Pbound ¼ �10 dB.

Figure 7.

39

respect the Pbound constraint.

The linear problem has been solved by the function linprog of the commercial software MATLAB [20]. The optimization has been done with a resolution of 0.25 m on the coverage area plane, with a total of 120,000 points. Both the coverage area and suppression region boundaries have been discretized with four points.

The antenna array normalized electric field when θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, and Pbound ¼ �10 dB is shown in Figure 5(a), achieved with the broadside optimization and an array of size 16 � 12. Figure 5(b) also depicts the contour plot of the synthesized normalized electric field.

Good agreement between the required coverage area and the synthesized one confirms the capability of the proposed method. Furthermore, this result has been obtained in less than 2 minutes with a 2.6 GHz Intel Core i7 processor, which is important to prove the good trade-off between performance and computational complexity are achieved by the described solution.

#### Figure 5.

Antenna array normalized electric field with an array of size 16 � 12 when θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, Pbound ¼ �10 dB, and broadside optimization. (a) Normalized electric field in xR and yR axes and (b) normalized electric field contour plot.

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

#### Figure 6.

proposed. In fact, if a particular shape of the weights is chosen, that is,

ky,<sup>0</sup> ¼ ky ϕ ¼ ϕ<sup>0</sup> ð Þ ; θ ¼ θ<sup>0</sup> , the synthesis function (4) reads

� ∑ N 2 n¼1

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

the form of a linear program as described in [15].

pattern fð Þ ϕ; θ has been taken as reported in [2, 19]).

�j kx,0xnþky,0<sup>y</sup> ð Þ<sup>n</sup> , where an <sup>∈</sup> <sup>R</sup>, n<sup>∈</sup> f g <sup>1</sup>; ::; <sup>N</sup> , kx,<sup>0</sup> <sup>¼</sup> kx <sup>ϕ</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> ð Þ ; <sup>θ</sup> <sup>¼</sup> <sup>θ</sup><sup>0</sup> , and

which is again a real function. As for the broadside array case, the optimization problem (9) in the known direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> can be simplified as (11) and written in

In this section, some numerical results which demonstrate the capability of the described method are presented. A circular shape for both the coverage area and the suppression region is considered, with diameter of 3.5 and 6.5 m, respectively. The

The linear problem has been solved by the function linprog of the commercial

Good agreement between the required coverage area and the synthesized one confirms the capability of the proposed method. Furthermore, this result has been obtained in less than 2 minutes with a 2.6 GHz Intel Core i7 processor, which is important to prove the good trade-off between performance and com-

elements with uniform interelement distance d ¼ 0:6λ<sup>0</sup> is synthesized, with antenna elements as microstrip patch antennas (theoretical formula of the radiation

software MATLAB [20]. The optimization has been done with a resolution of 0.25 m on the coverage area plane, with a total of 120,000 points. Both the coverage area and suppression region boundaries have been discretized with four points. The antenna array normalized electric field when θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, and Pbound ¼ �10 dB is shown in Figure 5(a), achieved with the broadside optimization and an array of size 16 � 12. Figure 5(b) also depicts the contour plot of the

putational complexity are achieved by the described solution.

Antenna array normalized electric field with an array of size 16 � 12 when θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, Pbound ¼ �10 dB, and broadside optimization. (a) Normalized electric field in xR and yR axes and (b)

ancos ð Þ kx � kx,<sup>0</sup> xn þ ky � ky,<sup>0</sup>

and hA <sup>¼</sup> <sup>5</sup>:5 m. A rectangular array of Nx � Ny

 yn (12)

wn ¼ ane

Fð Þ¼ ϕ; θ 2

Array Pattern Optimization

3.2 Simulation results

two regions are centered in 0; y<sup>0</sup>

synthesized normalized electric field.

Figure 5.

38

normalized electric field contour plot.

r0 r

Antenna array normalized electric field with different numbers of antenna elements, with θ<sup>A</sup> ¼ 60°, y<sup>0</sup> ¼ 3:25 m, Pbound ¼ �10 dB, and broadside optimization. (a) Normalized electric field in xR axis, within yR ¼ 3:25 m, and (b) normalized electric field in yR axis, within xR ¼ 0 m.

Now, the effectiveness of the proposed method is investigated for different synthesis parameters.

The broadside optimization presented in Section 3.1.1 is firstly analyzed for different numbers of antenna elements. Results are reported in Figure 6 as a function of the coordinates xR and yR, with θ<sup>A</sup> ¼ 60°, y0 ¼ 3:25 m, and Pbound ¼ �10 dB.

The curve 16 � 12 is the first feasible result which presents a sidelobe level of 35.1 dB within the suppression region. Other curves have been obtained with increased number of antenna elements. Obviously, the sidelobe-level performance improves with the increase of the antenna elements. All the synthesized results respect the Pbound constraint.

The influence of the mechanical tilt θ<sup>A</sup> choice on the optimization result is herein investigated. Broadside optimization along with the coordinate yR for different mechanical tilt θ<sup>A</sup> is depicted in Figure 7(a). It is worth noting that the coverage

#### Figure 7.

(a) Antenna array normalized electric field with different mechanical tilt θ<sup>A</sup> in xR ¼ 0 m, with Pbound ¼ �10 dB, and broadside optimization and (b) antenna array normalized electric field with different steering directions θ<sup>0</sup> in x<sup>0</sup> ¼ 0 m, with y<sup>0</sup> ¼ 3:25 m, and Pbound ¼ �10 dB.

area center position y<sup>0</sup> has been progressively increased to be the points on the coverage area plane which corresponds to the broadside direction, that is, <sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup>:25 m with <sup>θ</sup><sup>A</sup> <sup>¼</sup> <sup>60</sup>° , <sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>5</sup>:5 m with <sup>θ</sup><sup>A</sup> <sup>¼</sup> 45° , and <sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>9</sup>:5 m with <sup>θ</sup><sup>A</sup> <sup>¼</sup> 30° , and that the array is assumed to be of minimum size.

according to the free space propagation model, the communication area is defined

λ0 4πr � � <sup>≥</sup>Ptag,th

j k½ � <sup>x</sup>ð Þ <sup>n</sup>�<sup>1</sup> dxþkyð Þ <sup>m</sup>�<sup>1</sup> dy (14)

þ 10 log <sup>10</sup>M ≥Preader,th

(13)

λ0 4πr � �

as the set of coordinates xR and yR in the reference plane zR ¼ htag in which

� � <sup>¼</sup> Pt <sup>þ</sup> Gtð Þþ <sup>ϕ</sup>; <sup>θ</sup> Gr <sup>þ</sup> 20 log <sup>10</sup>

� � <sup>þ</sup> Gtð Þþ <sup>ϕ</sup>; <sup>θ</sup> Gr <sup>þ</sup> 20 log <sup>10</sup>

where Pt is the transmitted power, Gtð Þ¼ ϕ; θ Gt,max þ 20 log <sup>10</sup>Fð Þ ϕ; θ represents the antenna array gain pattern (which includes the normalized synthesis function), Gr is the tag gain, and M is the modulation factor (for a passive tag, M ¼ 0:25 [22]).

Let us consider the normalized synthesis function in (6) for a rectangular planar array of Nx � Ny elements with uniform interelement distances dx and dy which can

Pforward xR; yR

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

4.2 Antenna array synthesis with iterative method

r

complex coefficients wn,m are taken as in (12), that is,

The synthesis problem is basically the definition of:

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

∑ Nx n¼1 ∑ Ny

m¼1

A simple iterative method to synthesize the coverage area with the objective of

Then, the synthesis process can be performed according to the following steps:

Rx ¼ Ry ¼ 10 dB, and the parameters dx and dy according to the antenna design requirements, for example, directivity, mutual coupling, size constraints, etc.

2. Starting from a minimum size Nx ¼ 2 and Ny ¼ 2, increase the antenna array dimension Ny to cover a little bit more than the required transverse width.

3. Adjust the sidelobe level Ry according to the required horizontal power margin

4.Increase the antenna array dimension Nx in accordance with the antenna gain

5. Adjust the sidelobe level Rx to obtain the required vertical power margin.

stretching its length toward the travel direction is described [16]. In this case,

wn,m <sup>¼</sup> an,me j k ½ � ð Þ <sup>x</sup>�kx,<sup>0</sup> ð Þ <sup>n</sup>�<sup>1</sup> dxþð Þ ky�ky,<sup>0</sup> ð Þ <sup>m</sup>�<sup>1</sup> dy , with an,m based on Tschebyscheff

coefficients and Rx and Ry the Tschebyscheff design sidelobe level [2].

1. Initialize the steering direction toward broadside ð Þ ϕ<sup>0</sup> ¼ 0; θ<sup>0</sup> ¼ 0 ,

wn,me

<sup>F</sup>ð Þ¼ <sup>ϕ</sup>; <sup>θ</sup> <sup>r</sup><sup>0</sup>

• Number of antenna elements Nx and Ny

• Interelement distances dx and dy

• Complex excitations wn,m

requirements.

requirements.

41

� � <sup>¼</sup> <sup>P</sup>forward xR; yR

Pback xR; yR

be rewritten as

8 >>><

>>>:

It is of interest to observe that a decrease in mechanical tilt leads to a decrease in the required beamwidth and, consequently, an increase in the required array size. The achieved sidelobe levels are larger than 20 dB.

The non-broadside case is also considered. In Figure 7(b) the performance of the broadside optimization and the non-broadside optimization is compared in order to confirm the steering direction choice discussed in Section 3.1.

It is clear from Figure 7(b) that the choice of the steering direction affects the sidelobe level outside the coverage area. In fact, a 1° decrease in the steering direction with respect to the broadside, that is, the steering direction which corresponds to the coverage area center point, yields a sidelobe-level improvement of 22.5 dB. On the other hand, an increase of 1° leads to a sidelobe deterioration.
