Parametric Analysis of Linear Periodic Arrays Generating Flat-Top Beams

: Several synthesis techniques are available to optimize amplitude and phase excitations of periodic linear arrays to generate ﬂat-top beams. Clearly, the optimal tapering depends on design parameters such as the array length, the number of array elements, the beam ﬂatness, the beam width, the side lobe levels, and others. In this paper, in order to derive useful guidelines and rule of thumb for the synthesis of periodic array antennas, relations between these parameters are derived employing linear programming techniques, which guarantee optimality of the solutions. Such relations are then plotted and used in some design examples.


Introduction
In this paper, the canonical problem of generating a flat-top beam with a linear array of equally spaced identical elements is investigated. A flat-top beam is a radiation pattern having an intensity profile that is flat over a required region of interest. Several antenna applications, such as reconnaissance and search RADARs, and wide-area broadcasting communication systems, require flat-top beams characterized by well-defined beamwidths. For such applications, key radiation pattern requirements are the ripple within the main-beam and the level of sidelobes after a transition zone from the main beam. Well-established synthesis procedures such as Fourier [1] or Woodward-Lawson [2,3] exist but are not particularly well suited due to the Gibbs phenomenon. Inspired by Butterworth filters [4], Ksienski introduced an analytical procedure [5] (extended in [6]) which maximizes the smoothness of the radiation patter at the expenses of an enlarged transition zone. Notwithstanding their speed, these techniques result suboptimal due to the unnecessary constraint on the realness of the voltage radiation pattern. In this respect, linear programming techniques allow for finding the global optimal solution to the flat-top power pattern problem [7].
Given an array geometry (i.e., the elements positions), the problem of the optimization of array excitations to satisfy arbitrary upper and lower bounds for the main beam and sidelobes is casted in the equivalent problem of the optimization of the real and imaginary parts of the autocorrelation of the excitations, which become the new unknowns. In this auxiliary space, the power pattern is a linear transformation of the unknowns (as the voltage pattern is a linear function of the array excitations) and the power pattern constraints can be described as linear constraints on the unknowns (an additional linear constraint on the positivity of the power pattern must be added). The goodness of this transposition stays in the fact that the minimization of the sidelobes and/or of the mainbeam ripple become convex problems admitting a unique global solution, which can be found with linear programming techniques. Although the formulation [7] used in the following sections leverages on the framework developed by L. R. Rabiner in 1972 for 2 of 12 Finite-Impulse-Response (FIR) filters [8,9], the application to array antennas and optimality aspects can be found in [10].
Concerning the computational speed of linear programming solvers, while the theoretical analyses of their complexity remain an open problem, the simplex method or the interior point methods have been shown to exhibit typical polynomial (in the number of unknowns and constraints) running times (which correspond to few seconds in the instances of the addressed problem) [11]. Notwithstanding the global optimality of the linear programming solution for a specified flat-top-beam problem, the sensitivity of the solution to the input parameters has not been addressed in the technical literature (to the best knowledge of the authors). This paper analyzes the performance relationships between the following design parameters: array length, number of array elements, beam width, maximum allowed beam ripple, and width of the transition region between the main beam and the side lobes. Adopting a linear programming approach, the optimal value of the maximum side lobe level is derived and graphically illustrated as a function of the other parameters. The study presented in this paper does not aim to propose a new array synthesis algorithm but rather to deepen the analyses presented in [7] to provide antenna engineers with some design curves giving useful insights in the problem and data for the preliminary phase of synthesis.
In addition to the analytical methods [1][2][3][4][5][6], the literature offers a great variety of numerical methods for the array synthesis. As an example, in [12][13][14], deterministic procedures are proposed based on alternating projection approaches; in [15], a linear programming optimization plus a polynomial factorization is conceived; and [16] proposes a fast and versatile deterministic algorithm that iteratively minimizes a suitable weighted cost function. Furthermore, many global optimization techniques have been proposed, such as those presented in [17,18], based on genetic algorithms; the one proposed in [19], based on the differential evolution algorithm; and that in [20], which exploits the particle swarm optimization approach. Although this canonical problem has been already investigated, to the knowledge of the authors, attention has never been focused on the relations between the abovementioned design parameters. The authors believe that such an analysis may be very useful in the preliminary phase of design. For example, using the curves shown in Section 3, the antenna designer can estimate the length of the linear array necessary to achieve certain assigned performances or can anticipate the expected pattern characteristics of a flat-top beam realizable with an array having a given length.
The paper is organized as follows. In Section 2 the problem is formulated. In Section 3 the dependence of the maximum side lobe level on various design parameters is presented and discussed along with an asymptotical analysis aimed to verify the results with respect to the number of pattern samples utilized to derive the relations between the design parameters. In Section 4 some numerical examples are presented in order to describe how the reference curves can be used and to prove the superiority of the linear programming compared with conventional methods for the synthesis of shaped beams. The conclusions are proposed in Section 5.

Formulation of the Problem
The array factor of a linear antenna array of N identical equally spaced elements is given by the following: where E n is the excitation of the n-th element; j = √ −1; k 0 = 2π/λ, with λ the wavelength; d is the inter-element spacing; and u = sin θ, where θ is the angle from broadside. In the sequel, the power pattern |F(ϑ)| 2 will be considered.
A flat-top beam pattern can be characterized by the main beam edge u MB , the ripple δ MB in the main beam, the maximum side lobe level δ SL , and the side lobe edge u SL , as depicted in Figure 1. The additional parameters ∆u = u SL − u MB and K = δ MB /δ SL are also introduced.  The aim of the investigation presented in this paper is to derivie the relations between the above design parameters, exploiting them in the synthesis of flat-top beams. With this in mind, the desired values of , (or, alternatively, ∆ ), and (or, alternatively, ) are imposed, and the power pattern is calculated in such a way as to satisfy the constraints imposed on these parameters, simultaneously minimizing the maximum side lobe level . This problem is solved by using a linear programming approach, as briefly explained below (see [8,9] for further details). At first, the power pattern is written as follows: with the asterisk denoting complex conjugate. By (3), − = * . Therefore, (2) can be written in the following form: with = + and in particular 0 = ∑ | | 2 −1 =0 = 0 . The design parameters , , , and are then used to define a mask for the power pattern in (2). Such a mask is illustrated in Figure 1. Thanks to the auxiliary variables in (2) and (3), the power pattern synthesis is formulated as a linear programming problem, where the objective function to be minimized is . The problem can be formulated as follows: subject to the following constraints: The aim of the investigation presented in this paper is to derivie the relations between the above design parameters, exploiting them in the synthesis of flat-top beams. With this in mind, the desired values of u MB , u SL (or, alternatively, ∆u), and δ MB (or, alternatively, K) are imposed, and the power pattern is calculated in such a way as to satisfy the constraints imposed on these parameters, simultaneously minimizing the maximum side lobe level δ SL . This problem is solved by using a linear programming approach, as briefly explained below (see [8,9] for further details). At first, the power pattern is written as follows: where with the asterisk denoting complex conjugate. By (3), R −n = R * n . Therefore, (2) can be written in the following form: with R n = A n + jB n and in particular The design parameters u MB , u SL , δ MB , and δ SL are then used to define a mask for the power pattern in (2). Such a mask is illustrated in Figure 1. Thanks to the auxiliary variables R n in (2) and (3), the power pattern synthesis is formulated as a linear programming problem, where the objective function to be minimized is δ SL . The problem can be formulated as follows: minimize{δ SL } subject to the following constraints: where L(u) and U(u) are two positive functions representing, respectively, the lower and the upper bound of the mask, and are defined as Note that the constraints in (5) impose that the power pattern belong to the mask specified by L(u) and U(u) (conditions (5a) and (5b)), in which δ SL ≥ 0 (condition (5c)), and impose a desired value to δ MB or, alternatively, to K (condition (5d)). The solution of the linear programming problem in (5) provides the values of the variables R n of an equi-ripple power pattern, which perfectly fits between the lower and upper masks and exhibits the minimum attainable δ SL .
In the next section, several curves representing the relations between the design parameters are illustrated. Such results are obtained by sampling the synthesized radiation patterns on a set of equally spaced points in the visibility interval [−1; 1]. Subsequently, the asymptotic behavior of the results is numerically analyzed by adopting an increasing number of samples. Three examples of array synthesis are presented in Section 4, and are solved with the aid of the obtained curves involving the design parameters.

Maximum Sidelobe Level vs. Design Parameters
In the sequel, a linear array with inter-element spacing d = λ/2 will be considered, and the dependence of the maximum side lobe level on the other design parameters is derived and plotted for some test cases, by using the above described linear programming approach.

Case of Equality between Main Beam Width and Side Lobe Region Width
First, let us analyze the case where the main beam width (u MB ) is equal to the side lobe region width (1 − u SL ), that is u MB + u SL = 1, and in (5d) K des = δ MB /δ SL = 1 (i.e., δ MB = δ SL = δ). Figure 2 shows δ as a function of the transition width ∆u. In the figure, different curves refer to different values of N. As expected, higher values of ∆u correspond to lower values of δ as well as higher values of N correspond to lower values of δ. Thus, lower side lobes can be obtained with larger arrays or by allowing for a larger transition region.
Note that the constraints in (5) impose that the power pattern belong to th specified by ( ) and ( ) (conditions (5a) and (5b)), in which ≥ 0 (conditi and impose a desired value to or, alternatively, to (condition (5d)). The of the linear programming problem in (5) provides the values of the variables equi-ripple power pattern, which perfectly fits between the lower and upper ma exhibits the minimum attainable . In the next section, several curves representing the relations between the de rameters are illustrated. Such results are obtained by sampling the synthesized ra patterns on a set of equally spaced points in the visibility interval [−1; 1]. Subseq the asymptotic behavior of the results is numerically analyzed by adopting an inc number of samples. Three examples of array synthesis are presented in Section 4, solved with the aid of the obtained curves involving the design parameters.

Maximum Sidelobe Level vs. Design Parameters
In the sequel, a linear array with inter-element spacing = /2 will be con and the dependence of the maximum side lobe level on the other design param derived and plotted for some test cases, by using the above described linear progra approach.

Case of Equality between Main Beam Width and Side Lobe Region Width
First, let us analyze the case where the main beam width ( ) is equal to lobe region width (1 − ), that is + = 1, and in (5d) Figure 2 shows as a function of the transition width Δ . In the different curves refer to different values of . As expected, higher values of Δ spond to lower values of as well as higher values of correspond to lower v . Thus, lower side lobes can be obtained with larger arrays or by allowing for transition region.  From now on, unlike the previous case, flat-top beams are considered with u MB and 1 − u SL possibly different, and δ MB possibly different from δ SL (i.e., K des = 1).

Dependence on
The dependence of the maximum side lobe level on (= / ) is analyzed here. In the next three figures, the same main beam edge = 0.5 is considered, and the maximum side lobe level decreases as increases. In Figure 4a, which corresponds to Δ = 0.2, all curves are practically linear. Instead, in Figure 4b, corresponding to Δ = 0.15, for high values of , the curve obtained for = 20 is nonlinear. In Figure 4c, corresponding to Δ = 0.1, the behavior is quasi-linear only for = 40.   Figure 5 shows the maximum side lobe level as a function of , through the parameter = (1 + )/(1 − ) expressed in dB (the parameter is used in agreement with several other papers, for example [9]). As expected, the maximum side lobe level is lower for higher values of : in other words, when relaxing the requirements

Dependence on K
The dependence of the maximum side lobe level δ SL on K(= δ MB /δ SL ) is analyzed here. In the next three figures, the same main beam edge u MB = 0.5 is considered, and the maximum side lobe level decreases as K increases. In Figure 4a, which corresponds to ∆u = 0.2, all curves are practically linear. Instead, in Figure 4b, corresponding to ∆u = 0.15, for high values of K, the curve obtained for N = 20 is nonlinear. In Figure 4c, corresponding to ∆u = 0.1, the behavior is quasi-linear only for N = 40.

Dependence on
The dependence of the maximum side lobe level on (= / ) is analyzed here. In the next three figures, the same main beam edge = 0.5 is considered, and the maximum side lobe level decreases as increases. In Figure 4a, which corresponds to Δ = 0.2, all curves are practically linear. Instead, in Figure 4b, corresponding to Δ = 0.15, for high values of , the curve obtained for = 20 is nonlinear. In Figure 4c, corresponding to Δ = 0.1, the behavior is quasi-linear only for = 40.   Figure 5 shows the maximum side lobe level as a function of , through the parameter = (1 + )/(1 − ) expressed in dB (the parameter is used in agreement with several other papers, for example [9]). As expected, the maximum side lobe level is lower for higher values of : in other words, when relaxing the requirements  Figure 5 shows the maximum side lobe level as a function of δ MB , through the parameter τ MB = (1 + δ MB )/(1 − δ MB ) expressed in dB (the parameter τ MB is used in agreement with several other papers, for example [9]). As expected, the maximum side lobe level is lower for higher values of δ MB : in other words, when relaxing the requirements for the maximum allowed ripple amplitude in the main beam, a reduction in the maximum side lobe level is observed. Moreover, the different curves refer to different values of ∆u. All curves have nearly the same shape and show that larger transition regions permit achieving better side lobe levels. All curves refer to the case u MB = 0.46 and N = 20.

Dependence on δ MB
Electronics 2021, 10, x FOR PEER REVIEW 6 of 13 for the maximum allowed ripple amplitude in the main beam, a reduction in the maximum side lobe level is observed. Moreover, the different curves refer to different values of Δ . All curves have nearly the same shape and show that larger transition regions permit achieving better side lobe levels. All curves refer to the case = 0.46 and = 20.  Figure 6a shows the maximum side lobe level as a function of for a maximum allowed beam ripple = 0.05 and a transition region Δ varying from 0.05 to 0.2. As expected, larger transition regions globally allow for lower maximum side lobe levels, but does not vary monotonically as a function of : all curves show the same behavior, with alternating maxima and minima. However, the average slope of the curves and the differences between maxima and minima tend to increase when increasing of Δ . Figure 6b shows the maximum side lobe level as a function of the main beam edge for an assigned beam ripple = 0.05. All of the curves in the figure refer to a transition region Δ = 0.15 and to different values of , varying from = 18 to = 22. It is interesting to note that all curves show a similar behavior. As expected, lower side lobe levels can be obtained with higher values of . However, for certain values of , corresponding to the minima along the curve obtained with elements, the corresponding values of are nearly coincident with the values obtained with + 1 elements. Moreover, for low values of , the maximum of the curve corresponding to elements is higher compared with the closest minimum of the curve with + 1 elements.  Figure 6a shows the maximum side lobe level δ SL as a function of u MB for a maximum allowed beam ripple δ MB = 0.05 and a transition region ∆u varying from 0.05 to 0.2. As expected, larger transition regions globally allow for lower maximum side lobe levels, but δ SL does not vary monotonically as a function of u MB : all curves show the same behavior, with alternating maxima and minima. However, the average slope of the curves and the differences between maxima and minima tend to increase when increasing of ∆u. It is interesting to note that all curves show a similar behavior. As expected, lower side lobe levels can be obtained with higher values of N. However, for certain values of u MB , corresponding to the minima along the curve obtained with N elements, the corresponding values of δ SL are nearly coincident with the values obtained with N + 1 elements. Moreover, for low values of u MB , the maximum of the curve corresponding to N elements is higher compared with the closest minimum of the curve with N + 1 elements.

Asymptotical Analysis
The above numerical results have been obtained by sampling the power pat a grid of = 800 equally spaced points in the interval [−1; 1]. In order to ve asymptotic stability of the results, we applied the adopted method of synthesis calculated the above curves using different numbers of samples. For examp reference to the already discussed Figure 7, which corresponds to = 20 and 0.16, we set = 0.1 and analyzed the dependence of on Δ for goi 200 to 3200. Figure 8 shows the results obtained.

Asymptotical Analysis
The above numerical results have been obtained by sampling the power patterns on a grid of N s = 800 equally spaced points in the interval [−1; 1]. In order to verify the asymptotic stability of the results, we applied the adopted method of synthesis and re-calculated the above curves using different numbers N s of samples. For example, with reference to the already discussed Figure 7, which corresponds to N = 20 and u MB = 0.16, we set δ MB = 0.1 and analyzed the dependence of δ SL on ∆u for N s going from 200 to 3200. Figure 8 shows the results obtained.
Different curves correspond to different values of N s . As it can be seen, the results nearly independent of the number of samples. In order to better illustrate the convergence, Figure 9 shows the maximum side lobe level as a function of the number of samples for N = 20, u MB = 0.16, and ∆u = 0.7. Note that the curve tends to saturate rapidly when increasing the number of samples. Thus, the results prove the effectiveness of the adopted method to calculate the relations between the design parameters, showing a behavior quasi-independent of the number of points on which the power patterns have been sampled. Different curves correspond to different values of . As it can be seen, the results nearly independent of the number of samples. In order to better illustrate the convergence, Figure 9 shows the maximum side lobe level as a function of the number of samples for = 20, = 0.16, and Δ = 0.7. Note that the curve tends to saturate rapidly when increasing the number of samples. Thus, the results prove the effectiveness of the adopted method to calculate the relations between the design parameters, showing a behavior quasi-independent of the number of points on which the power patterns have been sampled.

Examples
In this section, three significant examples are proposed, which show how the above derived curves can help in the preliminary design phase of linear antenna arrays radiating flat-top beams.

First Example: Case of Equality between Main Beam Width and Side Lobe Region Width
Let us assume that a linear array composed by = 30 elements is required to radiate a flat-top beam, with the main beam region and the side lobe region having the same  Different curves correspond to different values of . As it can be seen, the results nearly independent of the number of samples. In order to better illustrate the convergence, Figure 9 shows the maximum side lobe level as a function of the number of samples for = 20, = 0.16, and Δ = 0.7. Note that the curve tends to saturate rapidly when increasing the number of samples. Thus, the results prove the effectiveness of the adopted method to calculate the relations between the design parameters, showing a behavior quasi-independent of the number of points on which the power patterns have been sampled.

Examples
In this section, three significant examples are proposed, which show how the above derived curves can help in the preliminary design phase of linear antenna arrays radiating flat-top beams.

First Example: Case of Equality between Main Beam Width and Side Lobe Region Width
Let us assume that a linear array composed by = 30 elements is required to radiate a flat-top beam, with the main beam region and the side lobe region having the same

Examples
In this section, three significant examples are proposed, which show how the above derived curves can help in the preliminary design phase of linear antenna arrays radiating flat-top beams.

First Example: Case of Equality between Main Beam Width and Side Lobe Region Width
Let us assume that a linear array composed by N = 30 elements is required to radiate a flat-top beam, with the main beam region and the side lobe region having the same width (i.e., u MB + u SL = 1), and K = 1 (i.e, δ MB = δ SL = δ). We want to find the maximum beam width (or, equivalently, the minimum transition region ∆u), which guarantees a maximum side lobe level δ SL ≤ −15 dB. Using the curve corresponding to N = 30 in Figure 2, we can observe that a value ∆u = 0.055 can give a power pattern satisfying the required specifications. In fact, using the linear programming with u MB = 0.5 − ∆u 2 = 0.4725, u SL = 0.5 + ∆u 2 = 0.5275, and K = 1, we obtain the radiation pattern of Figure 10a

Second Example: Estimation of the Number of Elements
As a second example, we want to produce a flat-top beam with the following valu for the design parameters: = 0.375, ∆ = 0.1, = 0.02, and ≤ −30 dB. W the help of Figure 3b, we note that an array of = 27 elements can achieve the requir

Second Example: Estimation of the Number of Elements
As a second example, we want to produce a flat-top beam with the following values for the design parameters: u MB = 0.375, ∆u = 0.1, δ MB = 0.02, and δ SL ≤ −30 dB. With the help of Figure 3b, we note that an array of N = 27 elements can achieve the required performances. In fact, using the linear programming with these parameters yields the power pattern of Figure 10b, which satisfies the constraints. The CPU time for this example is slightly lower than 0.1 s.

Third Example: Estimation of the Width of the Transition Region
Let us consider a linear antenna array composed by N = 20 elements. We want to radiate a flat-top beam with a beamwidth u MB = 0.46, a maximum side lobe level of −30 dB and a maximum allowed ripple in the main beam region δ MB = 0.0575 (τ MB = 0.5 dB). The curves in Figure 5 suggest that a possible value of width of the transition region that allows us to satisfy the requirement is ∆u = 0.125. In fact, using the linear programming with these parameters yields the power pattern in Figure 10c, which satisfies all of the constraints. The CPU time for this example is slightly lower than 0.1 s.

Result Comparison
This final subsection aims to prove the superiority of the adopted linear programming in terms of constraint compliance for the flat-top beam synthesis. Precisely, the previous three examples have been solved with two well-known methods, i.e., the Fourier Transform approach [1] and the Woodward-Lawson technique [2]. Figure 11 shows the patterns obtained by the three methods. As it can be seen, only the linear programming always satisfies the imposed constraints, whereas the classical methods may fail to meet the requirements. On the other hand, both the Woodward-Lawson technique and the Fourier Transform approach have the advantage of using closed-form expressions for the addressed problems. Thus, the CPU times are quite low (approximately 1 ms or even less).

Conclusions
The power synthesis of flat-top beams for linear periodic arrays is a well-known canonical problem in the antenna community. The shape of the synthesized pattern depends on a number of design parameters, such as the array length, the beam width, the maximum allowed ripple in the main beam region, the width of the transition region, and the maximum side lobe level.
The aim of this paper was to derive the relations between these design parameters and to use them for design purposes. Adopting a linear programming technique that guarantees the optimality of the solutions, several case studies were analyzed, obtaining a set of relations between the above parameters, which are represented graphically by curves that can be useful in the array design. Three examples of array design were proposed, and solved with the aid of these curves, obtaining results fully compliant with the assigned masks. In particular, the curves show that, also for the simple case of linear equally spaced arrays, the relations between the design parameters can be nonlinear. This is an interesting and not completely expected result, which must be taken into account in the array design.
Author Contributions: P.A. and G.B.: conceptualization, review and editing, G.B.: conceptualization, software, and original draft preparation, G.T.: conceptualization, writing and validation, R.V.: conceptualization. All authors have read and agreed to the published version of the manuscript.