2. Problem statement and reference system description

Let us consider the design of an antenna array. The total electric field radiated by an array of identical antenna elements can be written by using the well-known pattern multiplication property [2] and reads

$$\overline{E}\_{\rm tot}(r,\phi,\theta) = \overline{E}\_0(r,\phi,\theta) \cdot AF(\phi,\theta) \tag{1}$$

where E0ð Þ r; ϕ; θ is the single antenna electric field vector and AFð Þ ϕ; θ is the array factor. By assuming that the single antenna beamwidth is broader than the desired one, only the term AFð Þ ϕ; θ can be considered in the design. Nonetheless, the single antenna radiation pattern can also be included in the synthesis process. In fact, if E0ð Þ r; ϕ; θ can be decomposed as

$$\overline{E}\_0(r,\phi,\theta) = \frac{e^{-jk\_0r}}{r} \left( E\_\phi \cdot \hat{\phi} + E\_\theta \cdot \hat{\theta} \right) \tag{2}$$

where k<sup>0</sup> is the wavenumber, and if the maximum absolute value of the electric field components is E0, then it is possible to define the function:

$$f(\phi, \theta) = \sqrt{\left(\frac{E\_{\phi}}{E\_0}\right)^2 + \left(\frac{E\_{\theta}}{E\_0}\right)^2} \tag{3}$$

and to include it into the synthesis process, that is, the function that has to be synthesized becomes fð Þ� ϕ; θ ARð Þ ϕ; θ . The function fð Þ ϕ; θ is usually called antenna pattern.

It is now clear that the distance r is not included in the synthesis process. For this reason, the synthesized pattern preserves its characteristics uniquely on an r-constant surface, that is, a spherical surface. If an arbitrarily beam-shaped pattern is required to be synthesized (a pattern defined on a nonspherical surface), the rdecay factor of E0ð Þ r; ϕ; θ must be included into the synthesis process. For this reason, the normalized function that has to be optimized becomes

$$F(\phi,\theta) = \frac{r\_0}{r} \frac{f(\phi,\theta)}{f(\phi\_0,\theta\_0)} \cdot AF(\phi,\theta) \tag{4}$$

The antenna array reference system can be obtained by a rototranslation of the coverage area reference system [17]. Particularly, the following relations can be

> xR ¼ y yR ¼ cosð Þ θ<sup>A</sup> z � sinð Þ θ<sup>A</sup> x zR ¼ sinð Þ θ<sup>A</sup> z þ cosð Þ θ<sup>A</sup> x þ hA

x ¼ sinð Þ θ<sup>A</sup> yR þ cosð Þ θ<sup>A</sup> ð Þ zR � hA y ¼ xR z ¼ cosð Þ θ<sup>A</sup> yR � sinð Þ θ<sup>A</sup> ð Þ zR � hA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>R</sup> þ ð Þ zR � hA

xR sinð Þ θ<sup>A</sup> yR þ cosð Þ θ<sup>A</sup> ð Þ zR � hA � �

cosð Þ <sup>θ</sup><sup>A</sup> yR � sinð Þ <sup>θ</sup><sup>A</sup> ð Þ zR � hA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>R</sup> þ ð Þ zR � hA

2

2

9 >= (5)

>;

8 >><

>>:

8 >><

>>:

8

Antenna array and coverage area reference systems.

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

>>>>>>>>>><

>>>>>>>>>>:

ϕ ¼ arctan

θ ¼ arccos

normalized function F rð Þ ; ϕ; θ in (4) becomes F ¼ Fð Þ ϕ; θ .

3. Optimization problem and antenna array synthesis

assumed. The synthesis function in (4) can be written as follows:

r ¼

8 ><

>:

x2 <sup>R</sup> þ y<sup>2</sup>

> x2 <sup>R</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup>

A generic planar array of N elements lying on the xy-plane of Figure 4 is

It should be noted that other synthesis surfaces could be considered with the method herein presented. For the sake of comprehension simplification, and also because it represents a practical situation, the case zR constant is herein described. In this case, it is straightforward to understand that r ¼ rð Þ ϕ; θ , and then also the

q

q

obtained:

35

Figure 4.

where ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> is the electrical steering direction and r<sup>0</sup> is the reference distance from the antenna array to the synthesis surface (included for function normalization).

Let us assume a RSU with an antenna array placed at height hA which can be mechanically tilted by an angle θ<sup>A</sup> (this can be required to better address a specific coverage area requirement on a planar surface). In this case, both the array electrical steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> and the mechanical tilt steer the beam pattern. Figure 3(a) describes this scenario. The coverage area (herein defined as the region where the normalized total electric field on the road surface is larger than a certain threshold value) could be arbitrarily assigned in shape, even if circular or elliptical is a more realistic hypothesis. A coverage area might be required for high-power reception within a high data-rate service spatial area or to guarantee signal reception as it will be described later for the specific case of RFID-based ETC. Furthermore, the synthesized beam sidelobe-level control might also be important to avoid signal interference with other coverage zones illuminated by other RSUs as in Figure 3(b). Finally, other situations could require to limit the coverage area extension toward a specific direction in order to avoid possible overlap with other coverage areas.

Figure 4 depicts the antenna array reference system in spherical coordinates r, ϕ, and θ and in Cartesian coordinates x, y, and z and the coverage area reference system in Cartesian coordinates xR, yR, and zR. Moreover, a RSU is placed on a hA height pole (or a highway gate tolling station), and the coverage area is defined on the road plane, that is, zR ¼ 0. However, the presented methodology can also address the case in which zR ¼ htag . It is worth noting that htag which represents the OBU height (usually installed on the vehicle windshield) depends on the vehicle model and a univocal solution for the coverage area at a fixed height cannot be specified. For this reason, a reference OBU height can be defined for carrying out the synthesis process of the antenna array, and then synthesis results at different heights htag should be verified.

Figure 3.

Example of RSU displacement and coverage areas. (a) RSU with a coverage area defined on a road surface and (b) RSUs with close coverage areas.

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

It is now clear that the distance r is not included in the synthesis process. For this reason, the synthesized pattern preserves its characteristics uniquely on an r-constant surface, that is, a spherical surface. If an arbitrarily beam-shaped pattern is required to be synthesized (a pattern defined on a nonspherical surface), the rdecay factor of E0ð Þ r; ϕ; θ must be included into the synthesis process. For this

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

where ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> is the electrical steering direction and r<sup>0</sup> is the reference distance from the antenna array to the synthesis surface (included for function normaliza-

Let us assume a RSU with an antenna array placed at height hA which can be mechanically tilted by an angle θ<sup>A</sup> (this can be required to better address a specific coverage area requirement on a planar surface). In this case, both the array electrical steering direction ϕ<sup>0</sup> ð Þ ; θ<sup>0</sup> and the mechanical tilt steer the beam pattern. Figure 3(a) describes this scenario. The coverage area (herein defined as the region where the normalized total electric field on the road surface is larger than a certain threshold value) could be arbitrarily assigned in shape, even if circular or elliptical is a more realistic hypothesis. A coverage area might be required for high-power reception within a high data-rate service spatial area or to guarantee signal reception as it will be described later for the specific case of RFID-based ETC. Furthermore, the synthesized beam sidelobe-level control might also be important to avoid signal interference with other coverage zones illuminated by other RSUs as in Figure 3(b). Finally, other situations could require to limit the coverage area extension toward a specific direction in order to avoid possible overlap with other

Figure 4 depicts the antenna array reference system in spherical coordinates r, ϕ, and θ and in Cartesian coordinates x, y, and z and the coverage area reference system in Cartesian coordinates xR, yR, and zR. Moreover, a RSU is placed on a hA height pole (or a highway gate tolling station), and the coverage area is defined on the road plane, that is, zR ¼ 0. However, the presented methodology can also address the case in which zR ¼ htag . It is worth noting that htag which represents the OBU height (usually installed on the vehicle windshield) depends on the vehicle model and a univocal solution for the coverage area at a fixed height cannot be specified. For this reason, a reference OBU height can be defined for carrying out the synthesis process of the antenna array, and then synthesis results at different

Example of RSU displacement and coverage areas. (a) RSU with a coverage area defined on a road surface and

� AFð Þ ϕ; θ (4)

reason, the normalized function that has to be optimized becomes

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

tion).

Array Pattern Optimization

coverage areas.

Figure 3.

34

heights htag should be verified.

(b) RSUs with close coverage areas.

r

The antenna array reference system can be obtained by a rototranslation of the coverage area reference system [17]. Particularly, the following relations can be obtained:

$$\begin{cases} \begin{aligned} \mathbf{x}\_R &= \mathbf{y} \\ \mathbf{y}\_R &= \cos(\theta\_A)\mathbf{z} - \sin(\theta\_A)\mathbf{x} \\ \mathbf{z}\_R &= \sin(\theta\_A)\mathbf{z} + \cos(\theta\_A)\mathbf{x} + h\_A \end{aligned} \end{cases}$$

$$\begin{cases} \mathbf{x} = \sin(\theta\_A)\mathbf{y}\_R + \cos(\theta\_A)(\mathbf{z}\_R - h\_A) \\\\ \mathbf{y} &= \mathbf{x}\_R \end{aligned}$$

$$\begin{cases} \begin{aligned} r &= \sqrt{\mathbf{x}\_R^2 + \mathbf{y}\_R^2 + (\mathbf{z}\_R - h\_A)^2} \\ \begin{Bmatrix} \phi = \arctan\left\{\frac{\mathbf{x}\_R}{\sin(\theta\_A)\mathbf{y}\_R + \cos(\theta\_A)(\mathbf{z}\_R - h\_A)}\right\} \end{Bmatrix} \end{cases} \tag{5}$$

$$\theta = \arccos\left\{\frac{\cos(\theta\_A)\mathbf{y}\_R - \sin(\theta\_A)(\mathbf{z}\_R - h\_A)}{\sqrt{\mathbf{x}\_R^2 + \mathbf{y}\_R^2 + (\mathbf{z}\_R - h\_A)^2}}\right\}$$

It should be noted that other synthesis surfaces could be considered with the method herein presented. For the sake of comprehension simplification, and also because it represents a practical situation, the case zR constant is herein described. In this case, it is straightforward to understand that r ¼ rð Þ ϕ; θ , and then also the normalized function F rð Þ ; ϕ; θ in (4) becomes F ¼ Fð Þ ϕ; θ .
