The problem of antenna array synthesis for radiation pattern defined on a planar surface will be considered in this chapter. This situation could happen when the electric field r-decay factor effect cannot be neglected, for example, an antenna array mechanically tilted and a pattern defined in terms of Cartesian coordinates, as in the electronic toll collection (ETC) scenario. Two possible approaches will be presented: the first one aims at the precise synthesis of the pattern in the case both a constant power-bounded area and a sidelobe suppression region are defined and required to be synthesized. The second approach instead devotes at stretching the coverage area toward the travel length (without considering a precise definition of the communication area) to increase the available identification time with an iterative methodology. For the latter, an antenna prototype has been fabricated, and measurement results have confirmed the approach validity.
- antenna arrays
- radiation pattern synthesis
- linear programming
- electronic toll collection (ETC)
- radio frequency identification (RFID)
Most parts of literature on antenna array describe the synthesis of the
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
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 , it is possible to approximate the maximum available time to perform the toll transaction as a function of the vehicle speed , as shown in Figure 2 . Obviously, the vehicle speed increase reduces the available transaction time, making the ETC system design more challenging. Nonetheless, the length of the coverage area 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.
Motivated by the above considerations, the problem of antenna array synthesis when the electric field
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  and reads
where is the single antenna electric field vector and is the array factor. By assuming that the single antenna beamwidth is broader than the desired one, only the term can be considered in the design. Nonetheless, the single antenna radiation pattern can also be included in the synthesis process. In fact, if can be decomposed as
where is the wavenumber, and if the maximum absolute value of the electric field components is , then it is possible to define the function:
and to include it into the synthesis process, that is, the function that has to be synthesized becomes . The function is usually called
It is now clear that the distance
where is the electrical steering direction and 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 which can be mechanically tilted by an angle (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 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
The antenna array reference system can be obtained by a rototranslation of the coverage area reference system . Particularly, the following relations can be obtained:
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 constant is herein described. In this case, it is straightforward to understand that , and then also the normalized function in (4) becomes .
3. Optimization problem and antenna array synthesis
A generic planar array of
where , , is the
and a coverage area , where it is desired that the normalized electric field is larger than a certain bound value (expressed in dB), that is,
it is possible to derive the generic optimization problem as
Some additional constraints are included to better define the function trend within the area of interest. In particular, the constraint fixes the function value on the coverage area bound
The mechanical tilt 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 and the last inequality in (9) lead to a nonlinear optimization problem with a non-convex constraint, and according to , 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 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 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 in order to synthesize the array factor as much symmetrical as possible .
The second hypothesis, instead, addresses the most part of practical cases.
Based on the choice of the steering direction , two main practical cases can be distinguished: the broadside array and the steered array .
3.1.1. Broadside array
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:
In this case, the amplitude excitations , and, consequently, the function are real. In this way, the lower bound inequality in (9) can be rewritten as a convex constraint. In fact, since the real function is close to its maximum value in the bounded area
The optimization problem (9) for the broadside direction can now be written as
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 to be lower than within the coverage area.
The optimization problem in (11) can now be written in the form of a linear program as described in  with the great advantage of a lower computational complexity.
3.1.2. Non-broadside array
When the mechanical tilt 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 . For this reason, another simplification of the problem is herein proposed. In fact, if a particular shape of the weights is chosen, that is, , where , , and , the synthesis function (4) reads
which is again a real function. As for the broadside array case, the optimization problem (9) in the known direction can be simplified as (11) and written in the form of a linear program as described in .
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 and m. A rectangular array of elements with uniform interelement distance is synthesized, with antenna elements as microstrip patch antennas (theoretical formula of the radiation pattern has been taken as reported in [2, 19]).
The linear problem has been solved by the function
The antenna array normalized electric field when , m, and dB is shown in Figure 5(a), achieved with the broadside optimization and an array of size . 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.
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 and , with °, m, and dB.
The curve 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 constraint.
The influence of the mechanical tilt choice on the optimization result is herein investigated. Broadside optimization along with the coordinate for different mechanical tilt is depicted in Figure 7(a). It is worth noting that the coverage area center position has been progressively increased to be the points on the coverage area plane which corresponds to the broadside direction, that is, m with , m with , and m with , and that the array is assumed to be of minimum size.
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.
4. Coverage area synthesis for RFID-based ETC system
After the description of a general optimization procedure for a pattern defined on a planar surface (which can be used for the synthesis of a high data-rate service area), in this section we will consider the coverage area synthesis problem from the ETC application point of view. As briefly described in Section 1, the objective of a coverage area synthesis in this context should be the maximization of the communication area length in the travel direction and not the synthesis of a specific pattern geometry. For this case, an optimization procedure similar to the one described in Section 3 might also be derived. Nonetheless, channel phenomena, for example, fading , are known and, together with other possible implementation tolerances, might lead to suboptimal solutions in spite of the synthesis effort.
For this reason, a simple iterative methodology for synthesizing a planar antenna array with both the aim of stretching the coverage area toward the longitudinal direction and confining it within a roadlane is described. This method has the advantage of providing acceptable results with a reduced number of antenna elements with respect to the optimization presented in Section 3.
4.1. RFID-based DSRC system
RFID technology is usually employed for the implementation of DSRC because of its well-known advantages of excellent accuracy and the possibility to be read at high speed . A RFID-based DSRC system is basically realized by means of a RSU beacon reader, raised installed in order to guarantee sufficient visibility, and some OBU transponders. Moreover, antennas are constrained to radiate with circular polarization (CP) for two main reasons: CP reduces polarization mismatch due to reciprocal rotation between RSU and OBU devices and improves immunity to multipath effect . The latter is a fundamental characteristic which guarantees the validity of a free space propagation loss model .
The coverage area definition is based on the threshold power which, in the case of a monostatic backscatter  RFID-based system, can be interpreted as the tag sensitivity threshold and the reader sensitivity . Therefore, according to the free space propagation model, the communication area is defined as the set of coordinates in the reference plane in which
where is the transmitted power, represents the antenna array gain pattern (which includes the normalized synthesis function), is the tag gain, and is the modulation factor (for a passive tag, ).
4.2. Antenna array synthesis with iterative method
Let us consider the normalized synthesis function in (6) for a rectangular planar array of elements with uniform interelement distances and which can be rewritten as
The synthesis problem is basically the definition of:
Number of antenna elements and
Interelement distances and
A simple iterative method to synthesize the coverage area with the objective of stretching its length toward the travel direction is described . In this case, complex coefficients are taken as in (12), that is, , with based on Tschebyscheff coefficients and and the Tschebyscheff design sidelobe level .
Then, the synthesis process can be performed according to the following steps:
Initialize the steering direction toward broadside , dB, and the parameters and according to the antenna design requirements, for example, directivity, mutual coupling, size constraints, etc.
Starting from a minimum size and , increase the antenna array dimension to cover a little bit more than the required transverse width.
Adjust the sidelobe level according to the required horizontal power margin requirements.
Increase the antenna array dimension in accordance with the antenna gain requirements.
Adjust the sidelobe level to obtain the required vertical power margin.
Choose the best steering elevation in the sense of maximizing the length of the coverage area along with the coordinate (with starting coordinate ).
Each step is iteratively executed to compare the Tschebyscheff synthesis results and verify the conditions in (13) and then determine the coefficients .
4.3. Synthesis example and experimental results
A coverage area synthesis is herein described for the case of a reader height m with mechanical tilt °. and OBU height m. System parameters are chosen according to the standard EPC Gen2 for UHF RFID  for the carrier frequency 920 MHz which limits the
Following the synthesis process described above, the optimized coverage area for a 6 m road width is achieved as depicted in Figure 8(a), with the following synthesis parameters: , , , (with evaluated at 920 MHz), °, dB, dB, and the coefficients as in .
The achieved coverage area is 8 m long, covers the required transversal direction width, and presents very low lateral sidelobes. Figure 8(b) also presents the achieved coverage area at m (it could represent the tag height of a truck) and m (that can represent the tag height of a motorcycle) along with the coordinate , and it shows that the higher the tag height , the shorter the coverage area. This is acceptable because the speed of a truck is usually lower than the speed of a common vehicle, so the available transaction time will be longer.
In order to confirm the simulation results, the synthesized antenna array has been designed and manufactured, as shown in Figure 9(a). The design process of the CP microstrip patch antenna array is described in . Furthermore, the 12 dBi RHCP gain antenna prototype has been fixed at the height m with a metallic scaffolding and used for collecting experimental results, as depicted in Figure 9(b). A commercial Impinj Speedway R420 UHF reader  ( dBm) and a tag device with dBm have been employed. A compact CP UHF antenna with gain dBi has been used as tag antenna.
The transmitted power has been regulated in the range 5 ÷ 30 dBm for each position of the tag device in the road plane to determine the minimum value which activates the tag, that is, , under the condition that . After that, in a similar way to what has been described in , the power when a limited dBm is applied and a specific dBm is chosen has been inferred as (cable losses have been compensated during the power evaluation). Experimental results are shown in Figure 10(a) and (b).
Good correspondence among simulations (SR), antenna measurement projection on the road plane (AM), and experimental results (ER) is visible, and only few discrepancies arise. These are mainly due to the 1 dB tag antenna gain reduction with respect to the design parameter, the tag antenna radiation pattern (not taken into account), and other possible errors in fixing the antenna mechanical tilt .
The optimization of an antenna array pattern when the electric field
This work was supported in part by the National Natural Science Foundation of China (Nos. 61601093, 61701082, and 61701116), in part by Sichuan Provincial Science and Technology Planning Program of China (Nos. 2016GZ0061 and 18HH0034), in part by the fundamental research funds for the Central Universities (No. ZYGX2016Z011), and in part by Science and Technology on Electronic Information Control Laboratory.
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this chapter.