1. Introduction

Most parts of literature on antenna array describe the synthesis of the array factor [1]. In fact, when the antenna array elements are the same, and assuming that the single antenna beamwidth is broader than that of the final array, it is possible to observe that the magnitude of the array factor is proportional to that of the total radiated electric field. Lots of synthesis methodologies have been presented over the years for both simple array structures, for example, linear uniform arrays [2], and more complex geometries (which require the use of optimization algorithms) [3–7]. Nevertheless, there exist other situations which require the whole electric field behavior control and, in particular, its r-decay behavior. In literature, these problems are called beam-shaped pattern or contoured pattern synthesis. Possible applications for these methods are in the field of satellite communications where small antennas or antenna arrays are employed for illuminating a profiled reflector, as described in [8]. Besides the use of a profiled reflector, other techniques have been developed and proposed [9–12]. The minimum least square error (MLSE) criteria are used in [9] for the synthesis of a desired contoured pattern specified with points in the angular domain. Moreover, a discrete Fourier transform (DFT) shape of the synthesis function is assigned to provide a better radiation control. In [10], a successive projection

method (SPM) procedure is developed which exploits a new set of basis functions instead of the DFT. This reduces the number of optimization variables with respect to the conventional SPM [11]. Another example of synthesis technique which minimizes the difference with a desired pattern in an iterative fashion can be found in [12].

Besides the case of satellite communications, there exist other applications in the context of vehicular communications and, in particular, for vehicle-to-infrastructure connection and vice versa, in which the electric field r-decay behavior affects the beam pattern, for example, a road side unit (RSU) equipped with an antenna array which has to radiate toward a specific area, defined on the road surface, for dedicated short-range communications (DSRC). This specific problem is usually not addressed as a beam-shaped pattern problem because nowadays electronic toll collection (ETC) is still performed with low-speed dedicated corridor sufficient to guarantee the automatic vehicle identification (AVI). However, in the futuristic envision of multilane free flow (MLFF) in which vehicles will perform tolling operation without reducing travel speed [13], an efficient beam pattern synthesis will become fundamental. In order to better highlight this point, let us consider Figure 1, in which a MLFF situation is depicted. In this example, each roadlane is managed by a dedicated RSU antenna array which radiates a certain beam pattern on the road surface.

If this beam pattern is synthesized for guaranteeing the correct communication between RSU and on-board unit (OBU) within a certain coverage area of length lca, it is possible to approximate the maximum available time to perform the toll transaction τtrav as a function of the vehicle speed vcar, as shown in Figure 2 [14]. Obviously, the vehicle speed increase reduces the available transaction time, making the ETC system design more challenging. Nonetheless, the length of the coverage area lca is also fundamental to increase the available transaction time and relax the ETC system requirements, and for this reason, the antenna array beam pattern synthesis should be carefully optimized.

direction will be investigated with the objective of increasing the available transaction time for radio frequency identification (RFID)-based DSRC, and a simple iterative approach will be presented [16]. Both the presented methodologies will be analyzed with the aid of numerical results. Moreover, the second approach will be

Let us consider the design of an antenna array. The total electric field radiated by

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

r

s

where k<sup>0</sup> is the wavenumber, and if the maximum absolute value of the electric

E<sup>ϕ</sup> E0 � �<sup>2</sup>

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

Etotð Þ¼ r; ϕ; θ E0ð Þ� r; ϕ; θ AFð Þ ϕ; θ (1)

<sup>E</sup><sup>ϕ</sup> � <sup>ϕ</sup>^ <sup>þ</sup> <sup>E</sup><sup>θ</sup> � <sup>θ</sup>

Eθ E0 � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ

^ � � (2)

(3)

an array of identical antenna elements can be written by using the well-known

2. Problem statement and reference system description

Maximum available transaction time as function of the vehicle speed.

<sup>E</sup>0ð Þ¼ <sup>r</sup>; <sup>ϕ</sup>; <sup>θ</sup> <sup>e</sup>�jk0<sup>r</sup>

field components is E0, then it is possible to define the function:

fð Þ¼ ϕ; θ

confirmed by experimental results.

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

Figure 2.

pattern multiplication property [2] and reads

fact, if E0ð Þ r; ϕ; θ can be decomposed as

pattern.

33

Motivated by the above considerations, the problem of antenna array synthesis when the electric field r-decay effect cannot be neglected is treated in this chapter, with particular emphasis on the context of vehicular communications where a RSU equipped with a mechanically tilted antenna array has to radiate a beam pattern defined on the road surface. The problem will be addressed in two different ways: firstly, a generic optimization problem will be presented for the case of a precise pattern definition; a circular objective area will be considered and synthesized together with a suppression surrounding area (useful for guaranteeing a minimum sidelobe level margin) [15]; and then, the coverage area stretching toward the travel

#### Figure 1.

Example of three-roadway highway tolling system with RFID antenna reader.

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

method (SPM) procedure is developed which exploits a new set of basis functions instead of the DFT. This reduces the number of optimization variables with respect to the conventional SPM [11]. Another example of synthesis technique which minimizes the difference with a desired pattern in an iterative fashion can be found in [12]. Besides the case of satellite communications, there exist other applications in the context of vehicular communications and, in particular, for vehicle-to-infrastructure connection and vice versa, in which the electric field r-decay behavior affects the beam pattern, for example, a road side unit (RSU) equipped with an antenna array which has to radiate toward a specific area, defined on the road surface, for dedicated short-range communications (DSRC). This specific problem is usually not addressed as a beam-shaped pattern problem because nowadays electronic toll collection (ETC) is still performed with low-speed dedicated corridor sufficient to guarantee the automatic vehicle identification (AVI). However, in the futuristic envision of multilane free flow (MLFF) in which vehicles will perform tolling operation without reducing travel speed [13], an efficient beam pattern synthesis will become fundamental. In order to better highlight this point, let us consider Figure 1, in which a MLFF situation is depicted. In this example, each roadlane is managed by a dedicated RSU

antenna array which radiates a certain beam pattern on the road surface.

synthesis should be carefully optimized.

Array Pattern Optimization

Example of three-roadway highway tolling system with RFID antenna reader.

Figure 1.

32

If this beam pattern is synthesized for guaranteeing the correct communication between RSU and on-board unit (OBU) within a certain coverage area of length lca, it is possible to approximate the maximum available time to perform the toll transaction τtrav as a function of the vehicle speed vcar, as shown in Figure 2 [14]. Obviously, the vehicle speed increase reduces the available transaction time, making the ETC system design more challenging. Nonetheless, the length of the coverage area lca is also fundamental to increase the available transaction time and relax the ETC system requirements, and for this reason, the antenna array beam pattern

Motivated by the above considerations, the problem of antenna array synthesis when the electric field r-decay effect cannot be neglected is treated in this chapter, with particular emphasis on the context of vehicular communications where a RSU equipped with a mechanically tilted antenna array has to radiate a beam pattern defined on the road surface. The problem will be addressed in two different ways: firstly, a generic optimization problem will be presented for the case of a precise pattern definition; a circular objective area will be considered and synthesized together with a suppression surrounding area (useful for guaranteeing a minimum sidelobe level margin) [15]; and then, the coverage area stretching toward the travel

Figure 2. Maximum available transaction time as function of the vehicle speed.

direction will be investigated with the objective of increasing the available transaction time for radio frequency identification (RFID)-based DSRC, and a simple iterative approach will be presented [16]. Both the presented methodologies will be analyzed with the aid of numerical results. Moreover, the second approach will be confirmed by experimental results.
