# Optimum Synthesis of Pencil Beams with Constrained Dynamic Range Ratio

Authors: Marko Matijascic; Maja Jurisic Bellotti; Mladen Vucic; Goran Molnar
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

In antenna array design, low dynamic range ratio (DRR) of excitation coefficients is important because it simplifies array’s feeding network and enables better control of mutual coupling. Optimization-based synthesis of pencil beams allows explicit control of DRR. However, incorporating DRR into an optimization problem leads to nonconvex constraints, making its solving challenging. In this paper, a framework for global optimization of linear pencil beams with constrained DRR is presented. By using this framework, the methods for synthesis of pencil beams with minimum sidelobe level and minimum sidelobe power are developed. Both methods utilize convex problems suitable for the synthesis of pencil beams whose coefficients’ signs are known in advance. By incorporating these problems into a branch and bound algorithm, the procedures for global optimizations are formed which systematically search the space of all coefficient signs. The method for minimization of sidelobe power is further analyzed in the context of beam efficiency. It is shown that this method can be utilized in an approximate and at the same time global design of pencil beam arrays with maximum beam efficiency and constrained DRR. Based on this approach, a method for the design of pencil beam arrays with minimum DRR and specified beam efficiency is proposed.


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## Body

## 1. Introduction

Pencil beam arrays are widely used in wireless communications [1–15] and wireless power transmission [16–23]. In both applications, they are utilized to direct the radiated energy into a specified spatial region. Therefore, the main concern in a pencil beam synthesis is shaping its radiation pattern. However, the implementation aspects of antenna arrays introduce additional requirements. One of them is low dynamic range ratio (DRR) of excitation coefficients. A low DRR simplifies the design of array’s feeding network and enables better control of mutual coupling between antenna elements.Many analytical [1–6] and optimization-based methods [7–14] provide pencil beams with low DRR. Analytical methods are based on popular windows and polynomials, for example, Gaussian [2] and ultraspherical [3] windows, as well as the Chebyshev [4], Gegenbauer [5], and Kaiser-Hamming [6] polynomials. The methods in [2–4] provide a low DRR inherently, whereas [1, 5, 6] incorporate the requirement for low DRR as a design objective.Optimization-based synthesis allows explicit control of DRR. Such a synthesis utilizes one of the following approaches: DRR minimization while other pencil beam parameters are kept within given requirements [7, 8], or optimization of other pencil beam parameters while DRR is kept below a predefined value [9–14]. The optimization-based methods utilize different solving techniques such as alternating direction method of multipliers [8, 13], genetic algorithms [9], and convex programming [7, 10, 11]. Generally, the convex programming is preferable because it provides global solution. Unfortunately, incorporating DRR into an optimization problem leads to nonconvex constraints, making its solving challenging. Recently, efficient methods for coping with this problem have been proposed in [7, 8, 13] and [14].Clearly, pencil beams with low sidelobe levels are desirable. Classic example of such a beam is obtained by Dolph-Chebyshev method, which provides the narrowest main lobe for a given maximum sidelobe level. The Dolph-Chebyshev synthesis is analytical. However, it can also be formulated as a second-order cone programming problem [24]. In such a problem, it is simple to constrain the DRR, though with the loss of convexity. In [10], a global minimization of the sidelobe level with specified DRR has been introduced. This work has been extended for sparse arrays in [11], where some coefficients are allowed zero values.In wireless power transmission, the goal is to maximize the amount of energy emitted in the spatial region of interest. This is often evaluated by using the beam efficiency, which is defined as a ratio between the energy emitted in a desired region and the total radiated energy. For linear arrays, the optimum in terms of beam efficiency is found analytically, via discrete prolate spheroidal sequences (DPSS) [16]. For planar arrays, the optimum is also found analytically, as a solution of the generalized eigenvalue problem [17]. The synthesis of pencil beams with high beam efficiency may also take into account other requirements. In [18], a method based on evolutionary programming is proposed, which yields pencil beams with high beam efficiency and constrained maximum sidelobe level. The methods in [20, 21] also consider beam efficiency and the sidelobe constraints, but the solution is found by using convex programming. In [22], a deterministic algorithm has been developed to reduce the sidelobe power pattern for linear arrays having uniform excitation and unequal element spacing. Heuristic approach to this problem has been considered in [23]. Recently, analytical method [5] based on Gegenbauer polynomials has been proposed, which maximizes the beam efficiency for a given sidelobe level. DRR constraints in power pattern synthesis have been used in [19].In this paper, we consider methods for global optimization of pencil beams with constrained dynamic range ratio of excitation coefficients. We cover the optimization of sidelobe level (SLL), sidelobe power (SLP), and beam efficiency (BE). In this context, the paper offers the following contributions:(1)
A method for global optimization of pencil beams with constrained DRR is proposed. The method is based on branch and bound optimization framework which we introduced in [10]. Here, we generalize this framework for the minimization of sidelobe level and sidelobe power. In addition, we present a detailed analysis of the influence of DRR constraints on the behavior of optimum array patterns.(2)
The method for minimization of sidelobe power is discussed in the context of beam efficiency. Numerical evaluation is provided showing that this method can be utilized in an approximate and at the same time global design of pencil beam arrays with maximum beam efficiency. Moreover, in a design of highly-efficient arrays, the approximation error is negligible.(3)
A method for the design of pencil beam arrays with minimum dynamic range ratio of excitation coefficientsand specified beam efficiency is proposed. The behavior of optimum arrays is illustrated by examples, showing that significant reduction in DRR is possible with negligible loss in efficiency, when compared to the arrays with maximum possible beam efficiency.The paper is organized as follows. Section2 describes the global optimization of sidelobe level and sidelobe power with constrained dynamic range ratio of excitation coefficients. Section 3 discusses the behavior of optimum pencil beams in the context of DRR constraints. Section 4 provides examples of pencil beams with minimum sidelobe level. The sidelobe power as a criterion for maximization of beam efficiency is considered in Section 5. One application of this approach is described in Section 6, which presents a method for the design of pencil beam arrays with minimum DRR and specified beam efficiency. Section 7 concludes the paper.


## 2. Synthesis of Pencil Beams with Constrained DRR

### 2.1. Basic Optimization Problem

Linear array withN isotropic elements uniformly spaced on x axis of the coordinate system has the far-field radiation pattern(1)fa,θ=∑k=1Nakej2π/λxksinθ,where a = [a1, a2, ..., aN]T, a ∈ RN, is the vector of excitation coefficients of antenna elements placed at the positions xk, k = 1, 2, ..., N, λ is the wavelength of transmitted signal, and θ is azimuth direction angle in range −π/2 ≤ θ ≤ π/2. In the synthesis of pencil beams, the objective is to minimize an error in the sidelobe region and keep the unity amplitude at θ = 0. Such a synthesis can be expressed by the optimization problem(2)minimizeaεa,θSsubject to∑k=1Nak=1,where ε(a, θs) is sidelobe error and θs is the beginning of the sidelobe region. In further text, we will first consider the problem in (2) and then the definition of ε(a, θs).


### 2.2. DRR Constraints

The problem in (2) does not take into account the DRR of the excitation coefficients. However, in many applications, DRR should be kept below a specified value, D, as in(3)max1≤k≤Nakmin1≤k≤Nak≤D.The constraint in (3) can be incorporated into the problem in (2) by using additional inequality constraints and auxiliary variable t, as in(4)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k=1,2,…,Nak≥t,k=1,2,…,Nt≥0.Unfortunately, the constraints |ak| ≥ t, k = 1, 2, ..., N are not convex and the problem in (4) is difficult to solve globally. However, if the signs of coefficients ak are known in advance, the problem in (4) can be solved globally [10]. Assuming S+ ⊆ {1, 2, ..., N} and S− ⊆ {1, 2, ..., N} are the sets of indexes corresponding to positive and negative coefficients, the optimization problem can be written in the form(5)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k∈S+−ak≤Dt,k∈S−ak≥t,k∈S+−ak≥t,k∈S−t≥0.The problem in (5) is convex. Moreover, if all coefficients are positive, the problem in (5) solves the problem in (4) globally. In general case, the signs of the coefficients are not known in advance. Therefore, to find global solution of the problem in (4), one may take an exhaustive search over 2N possible combinations of signs and solve the optimization problem in (5) for each of them. This procedure can be accomplished in an acceptable time only if the number of antenna elements is small. To accomplish the design of larger arrays, we utilize the branch and bound method from [10], which globally solves the problem in (4) while reducing the design time for several orders of magnitude.


### 2.3. Global Solving of Optimization Problem

The proposed optimization procedure explores a tree that contains all combinations of the coefficients’ signs. An example of the tree forN = 4 is shown in Figure 1. At the root, all coefficients are considered positive. At the first branching, the coefficients as, s = 1, 2, ..., N, are one by one assigned negative values. Negative values are further assigned at each branching. They are placed at the positions s = r+1, r + 2, ..., N, where r denotes the position at which negative value has been assigned previously. The value r = 0 is assumed at the root.Solving the problem in (5) for each node would correspond to an exhaustive search. Instead, we cut several branches using the following procedure. Let εopt and aopt denote the optimum sidelobe error value and the corresponding coefficients. At the beginning of the optimization procedure, it is assumed εopt = ∞. Then, the tree is explored by using a depth-first algorithm. At each node, two steps are made. First, the node is tested by solving the optimization problem in (5). If the problem is feasible, a∗ and ε∗ are obtained as the optimum solution. If ε∗ <  εopt, an update εopt = ε∗ and aopt = a∗ is made. Second, the branches leaving the node are tested by using the following optimization problem(6)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k∈S+&k≤rak≥t,k∈S+&k≤r−ak≤Dt,k∈S−&k≤r−ak≥t,k∈S−&k≤rak≤Dt,k=r+1,r+2,...,N−ak≤Dt,k=r+1,r+2,...,Nt≥0.Figure 1
Example of tree containing all combinations of coefficients' signs in array with four elements. Signs which are uniquely specified in particular nodes are shown in red, and their count is denoted byr.The above problem is obtained by relaxing the problem in (4). As shown in Figure 1, at each node, the signs of the first r coefficients are uniquely specified, whereas N−r coefficients remain to be assigned one of 2N−r possibilities. The first possibility (containing N−r positive signs) has already been processed in the first step (the node testing), whereas 2N−r−1 possibilities wait to be explored along the branches growing from the node. Therefore, upper- and lower-bound constraints are used in (6) only for coefficients ak, k = 1, 2, ..., r. The remaining coefficients are constrained only in their maximum absolute value, that is, |ak| ≤ Dt, k = r+1, r + 2, ..., N, whereas the corresponding nonconvex constraints |ak| ≥ t are omitted. Clearly, the solution of such an optimization problem might not satisfy (3). However, if ε∗ > εopt is obtained, there is no possibility to find better solution by further branching- and the node is pruned.The sidelobe error does not change if the antenna elements reverse their order. Therefore, the node testing might be performed only once for a pair of nodes with symmetrically placed coefficient signs (for example, +−++ and ++−+). The redundant check can be detected easily. The node value is expressed in a binary form, assuming − and + correspond to 0 and 1. LetB and Br denote the binary values of the original and the reverse ordering of the node being tested. If B > Br, the node testing is redundant.


### 2.4. Sidelobe Error

The error in (2) can be expressed as the integral of p-powered radiation pattern, calculated in the sidelobe region, θs ≤ |θ| ≤ π/2, as in(7)εpa,θs=4π∫θsπ/2fa,θpcosθdθ1/p.By using substitutionω = sin(θ), the error in (7) can be written in a compact form(8)εpa,θs=4π∫sinθs1fa,ωpdω1/p,where(9)fa,ω=∑k=1Nakej2π/λxkω.The expression in (8) is recognized as the Lp-norm of |f(a, ω)| calculated on the interval [sin(θs), 1]. In further text, we consider only the cases p ⟶ ∞ and p = 2, which lead to sidelobe-level and sidelobe-power minimization.


### 2.5. Minimization of Sidelobe Level

Forp ⟶ ∞, the error in (8) represents the L∞-norm, which can also be obtained as(10)ε∞a,θs=maxsinθS≤ω≤1fa,ω.The minimization ofε∞ results in beam patterns which approximate the sidelobe level in a minimax sense.Commonly,ε∞(a,θs) is approximated by evaluating |f(a,ω)| on a finite grid ωq, q = 1, 2, ..., Q, thus taking the form(11)ε∞a,θs≈maxA1a,A2a,…,AQa,where ||⋅|| denotes the L2 norm, and(12)Aq=cosuqx1cosuqx2⋯cosuqxNsinuqx1sinuqx2⋯sinuqxN.(13)uq=2πλωq,q=1,2,...,Q.The branch and bound method described in Section 2.3 utilizes two optimization problems − one for testing a node and the other for testing the branches leaving the node. Their general forms are given in (5) and (6). Here, they utilize the error in (11), thus taking the forms(14)minimizea,δ,tδsubject to∑k=1Nak=1Aqa≤δ,q=1,2,...,Qak≤Dt,k∈S+−ak≤Dt,k∈S−ak≥t,k∈S+−ak≥t,k∈S−t≥0,(15)minimizea,δ,tδsubject to∑k=1Nak=1Aqa≤δ,q=1,2,...,Qak≤Dt,k∈S+&k≤rak≥t,k∈S+&k≤r−ak≤Dt,k∈S−&k≤r−ak≥t,k∈S−&k≤rak≤Dt,k=r+1,r+2,...,N−ak≤Dt,k=r+1,r+2,...,Nt≥0.


### 2.6. Minimization of Sidelobe Power

To form an optimization problem which minimizes the sidelobe power, we start from the error in (8), which for p = 2 takes the form(16)ε2a,θs=4π∫sinθs1fa,ω2dω.Clearly, the above error represents the square root of SLP rather than SLP itself. However, the coefficients that minimize the SLP can also be obtained by minimizingε2(a,θs) because ε2(a,θs) ≥ 0 and, consequently,(17)argminaε22a,θs=argminaε2a,θs.The radicand in (16) can be approximated by using numerical integration. In such an approximation, the integration domain [sin(θs), 1] is discretized in a finite grid, ωq, q = 1, 2, ..., Q, where Q is an odd integer. Then, the Simpson’s 1/3 rule is applied, resulting in(18)ε2a,θs≈4π1−sinθS3Q−1∑q=1QKq2fa,ωq2,where Kq, q = 1, 2, ..., Q, is given by(19)Kq=1,forq=1andq=Q2,forq=2k+1,k=1,2,…,Q−322,otherwise.The expression (18) can be written in a matrix form, as in(20)ε2a,θs≈4π1−sinθS3Q−1Aa,where(21)A=K1sinu1x1K1sinu1x2⋯K1sinu1xNK1cosu1x1K1cosu1x2⋯K1cosu1xNK2sinu2x1K2sinu2x2⋯K2sinu2xNK2cosu2x1K2cosu2x2⋯K2cosu2xN⋮⋮⋮KQsinuQx1KQsinuQx2⋯KQsinuQxNKQcosuQx1KQcosuQx2⋯KQcosuQxN,and uq, q = 1, 2, ..., Q, is obtained by using (13).The error in (18) is now incorporated into (5) and (6) to form the branch and bound procedure. The problem for node testing is obtained in the form(22)minimizea,δ,tδ,subject to∑k=1Nak=1,Aa≤14π1−sinθS/3Q−1δ,ak≤Dt,k∈S+−ak≤Dt,k∈S−,ak≥t,k∈S+,−ak≥t,k∈S−,t≥0.Branches leaving the nodes are tested by solving the problem(23)minimizea,δ,tδ,subject to∑k=1Nak=1,Aa≤14π1−sinθS/3Q−1δ,ak≤Dt,k∈S+&k≤r,ak≥t,k∈S+&k≤r,−ak≤Dt,k∈S−&k≤r,−ak≥t,k∈S−&k≤r,ak≤Dt,k=r+1,r+2,…,N,−ak≤Dt,k=r+1,r+2,…,N,t≥0.


### 2.7. Efficiency of Proposed Method

Table 1
Number of subproblems,Nsub, and design time, tdesign, estimated for exhaustive search and obtained in proposed minimization of SLL and SLP for arrays with various numbers of elements, N, DRR constraints, D, and beamwidth of 12°. Number of points Q = 10 N and Q = 10 N + 1 are used in minimization of SLL and SLP.
NDExhaustive searchSLL minimizationSLP minimizationNsubtdesignNsubtdesignNsubtdesign2031.049e617.4 min200.42 s200.31 s21.049e617.4 min200.42 s200.39 s11.049e617.4 min1122.10 s200.28 s3031.074e9298 hours602.17 s300.70 s21.074e9298 hours1425.10 s300.71 s11.074e9298 hours159549.4 s300.74 s4031.100e1234.9 years804.53 s401.57 s21.100e1234.9 years804.58 s401.60 s11.100e1234.9 years210495.4 s401.24 sAs shown in Section2.2, global optimization of pencil beam array with constrained dynamic range ratio has combinatorial complexity. Design based on the exhaustive search would require solving of 2N subproblems given in (5). However, the proposed branch and bound approach significantly reduces the number of subproblems. Table 1 shows the number of subproblems and the design time estimated for the exhaustive search, as well as the number of subproblems and the design time obtained in the proposed minimization of SLL and SLP. An average design time of 1 ms per subproblem is assumed in the exhaustive search. The minimizations of SLL and SLP have been performed on a personal computer with quad-core Intel i7 processor operating at the clock of 3 GHz. Clearly, the exhaustive search is unsuitable for the design of arrays with more than 20 elements. On the other hand, the proposed approach enables global design of low and medium-size arrays in a few seconds.


## 2.1. Basic Optimization Problem

Linear array withN isotropic elements uniformly spaced on x axis of the coordinate system has the far-field radiation pattern(1)fa,θ=∑k=1Nakej2π/λxksinθ,where a = [a1, a2, ..., aN]T, a ∈ RN, is the vector of excitation coefficients of antenna elements placed at the positions xk, k = 1, 2, ..., N, λ is the wavelength of transmitted signal, and θ is azimuth direction angle in range −π/2 ≤ θ ≤ π/2. In the synthesis of pencil beams, the objective is to minimize an error in the sidelobe region and keep the unity amplitude at θ = 0. Such a synthesis can be expressed by the optimization problem(2)minimizeaεa,θSsubject to∑k=1Nak=1,where ε(a, θs) is sidelobe error and θs is the beginning of the sidelobe region. In further text, we will first consider the problem in (2) and then the definition of ε(a, θs).


## 2.2. DRR Constraints

The problem in (2) does not take into account the DRR of the excitation coefficients. However, in many applications, DRR should be kept below a specified value, D, as in(3)max1≤k≤Nakmin1≤k≤Nak≤D.The constraint in (3) can be incorporated into the problem in (2) by using additional inequality constraints and auxiliary variable t, as in(4)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k=1,2,…,Nak≥t,k=1,2,…,Nt≥0.Unfortunately, the constraints |ak| ≥ t, k = 1, 2, ..., N are not convex and the problem in (4) is difficult to solve globally. However, if the signs of coefficients ak are known in advance, the problem in (4) can be solved globally [10]. Assuming S+ ⊆ {1, 2, ..., N} and S− ⊆ {1, 2, ..., N} are the sets of indexes corresponding to positive and negative coefficients, the optimization problem can be written in the form(5)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k∈S+−ak≤Dt,k∈S−ak≥t,k∈S+−ak≥t,k∈S−t≥0.The problem in (5) is convex. Moreover, if all coefficients are positive, the problem in (5) solves the problem in (4) globally. In general case, the signs of the coefficients are not known in advance. Therefore, to find global solution of the problem in (4), one may take an exhaustive search over 2N possible combinations of signs and solve the optimization problem in (5) for each of them. This procedure can be accomplished in an acceptable time only if the number of antenna elements is small. To accomplish the design of larger arrays, we utilize the branch and bound method from [10], which globally solves the problem in (4) while reducing the design time for several orders of magnitude.


## 2.3. Global Solving of Optimization Problem

The proposed optimization procedure explores a tree that contains all combinations of the coefficients’ signs. An example of the tree forN = 4 is shown in Figure 1. At the root, all coefficients are considered positive. At the first branching, the coefficients as, s = 1, 2, ..., N, are one by one assigned negative values. Negative values are further assigned at each branching. They are placed at the positions s = r+1, r + 2, ..., N, where r denotes the position at which negative value has been assigned previously. The value r = 0 is assumed at the root.Solving the problem in (5) for each node would correspond to an exhaustive search. Instead, we cut several branches using the following procedure. Let εopt and aopt denote the optimum sidelobe error value and the corresponding coefficients. At the beginning of the optimization procedure, it is assumed εopt = ∞. Then, the tree is explored by using a depth-first algorithm. At each node, two steps are made. First, the node is tested by solving the optimization problem in (5). If the problem is feasible, a∗ and ε∗ are obtained as the optimum solution. If ε∗ <  εopt, an update εopt = ε∗ and aopt = a∗ is made. Second, the branches leaving the node are tested by using the following optimization problem(6)minimizea,tεa,θSsubject to∑k=1Nak=1ak≤Dt,k∈S+&k≤rak≥t,k∈S+&k≤r−ak≤Dt,k∈S−&k≤r−ak≥t,k∈S−&k≤rak≤Dt,k=r+1,r+2,...,N−ak≤Dt,k=r+1,r+2,...,Nt≥0.Figure 1
Example of tree containing all combinations of coefficients' signs in array with four elements. Signs which are uniquely specified in particular nodes are shown in red, and their count is denoted byr.The above problem is obtained by relaxing the problem in (4). As shown in Figure 1, at each node, the signs of the first r coefficients are uniquely specified, whereas N−r coefficients remain to be assigned one of 2N−r possibilities. The first possibility (containing N−r positive signs) has already been processed in the first step (the node testing), whereas 2N−r−1 possibilities wait to be explored along the branches growing from the node. Therefore, upper- and lower-bound constraints are used in (6) only for coefficients ak, k = 1, 2, ..., r. The remaining coefficients are constrained only in their maximum absolute value, that is, |ak| ≤ Dt, k = r+1, r + 2, ..., N, whereas the corresponding nonconvex constraints |ak| ≥ t are omitted. Clearly, the solution of such an optimization problem might not satisfy (3). However, if ε∗ > εopt is obtained, there is no possibility to find better solution by further branching- and the node is pruned.The sidelobe error does not change if the antenna elements reverse their order. Therefore, the node testing might be performed only once for a pair of nodes with symmetrically placed coefficient signs (for example, +−++ and ++−+). The redundant check can be detected easily. The node value is expressed in a binary form, assuming − and + correspond to 0 and 1. LetB and Br denote the binary values of the original and the reverse ordering of the node being tested. If B > Br, the node testing is redundant.


## 2.4. Sidelobe Error

The error in (2) can be expressed as the integral of p-powered radiation pattern, calculated in the sidelobe region, θs ≤ |θ| ≤ π/2, as in(7)εpa,θs=4π∫θsπ/2fa,θpcosθdθ1/p.By using substitutionω = sin(θ), the error in (7) can be written in a compact form(8)εpa,θs=4π∫sinθs1fa,ωpdω1/p,where(9)fa,ω=∑k=1Nakej2π/λxkω.The expression in (8) is recognized as the Lp-norm of |f(a, ω)| calculated on the interval [sin(θs), 1]. In further text, we consider only the cases p ⟶ ∞ and p = 2, which lead to sidelobe-level and sidelobe-power minimization.


## 2.5. Minimization of Sidelobe Level

Forp ⟶ ∞, the error in (8) represents the L∞-norm, which can also be obtained as(10)ε∞a,θs=maxsinθS≤ω≤1fa,ω.The minimization ofε∞ results in beam patterns which approximate the sidelobe level in a minimax sense.Commonly,ε∞(a,θs) is approximated by evaluating |f(a,ω)| on a finite grid ωq, q = 1, 2, ..., Q, thus taking the form(11)ε∞a,θs≈maxA1a,A2a,…,AQa,where ||⋅|| denotes the L2 norm, and(12)Aq=cosuqx1cosuqx2⋯cosuqxNsinuqx1sinuqx2⋯sinuqxN.(13)uq=2πλωq,q=1,2,...,Q.The branch and bound method described in Section 2.3 utilizes two optimization problems − one for testing a node and the other for testing the branches leaving the node. Their general forms are given in (5) and (6). Here, they utilize the error in (11), thus taking the forms(14)minimizea,δ,tδsubject to∑k=1Nak=1Aqa≤δ,q=1,2,...,Qak≤Dt,k∈S+−ak≤Dt,k∈S−ak≥t,k∈S+−ak≥t,k∈S−t≥0,(15)minimizea,δ,tδsubject to∑k=1Nak=1Aqa≤δ,q=1,2,...,Qak≤Dt,k∈S+&k≤rak≥t,k∈S+&k≤r−ak≤Dt,k∈S−&k≤r−ak≥t,k∈S−&k≤rak≤Dt,k=r+1,r+2,...,N−ak≤Dt,k=r+1,r+2,...,Nt≥0.


## 2.6. Minimization of Sidelobe Power

To form an optimization problem which minimizes the sidelobe power, we start from the error in (8), which for p = 2 takes the form(16)ε2a,θs=4π∫sinθs1fa,ω2dω.Clearly, the above error represents the square root of SLP rather than SLP itself. However, the coefficients that minimize the SLP can also be obtained by minimizingε2(a,θs) because ε2(a,θs) ≥ 0 and, consequently,(17)argminaε22a,θs=argminaε2a,θs.The radicand in (16) can be approximated by using numerical integration. In such an approximation, the integration domain [sin(θs), 1] is discretized in a finite grid, ωq, q = 1, 2, ..., Q, where Q is an odd integer. Then, the Simpson’s 1/3 rule is applied, resulting in(18)ε2a,θs≈4π1−sinθS3Q−1∑q=1QKq2fa,ωq2,where Kq, q = 1, 2, ..., Q, is given by(19)Kq=1,forq=1andq=Q2,forq=2k+1,k=1,2,…,Q−322,otherwise.The expression (18) can be written in a matrix form, as in(20)ε2a,θs≈4π1−sinθS3Q−1Aa,where(21)A=K1sinu1x1K1sinu1x2⋯K1sinu1xNK1cosu1x1K1cosu1x2⋯K1cosu1xNK2sinu2x1K2sinu2x2⋯K2sinu2xNK2cosu2x1K2cosu2x2⋯K2cosu2xN⋮⋮⋮KQsinuQx1KQsinuQx2⋯KQsinuQxNKQcosuQx1KQcosuQx2⋯KQcosuQxN,and uq, q = 1, 2, ..., Q, is obtained by using (13).The error in (18) is now incorporated into (5) and (6) to form the branch and bound procedure. The problem for node testing is obtained in the form(22)minimizea,δ,tδ,subject to∑k=1Nak=1,Aa≤14π1−sinθS/3Q−1δ,ak≤Dt,k∈S+−ak≤Dt,k∈S−,ak≥t,k∈S+,−ak≥t,k∈S−,t≥0.Branches leaving the nodes are tested by solving the problem(23)minimizea,δ,tδ,subject to∑k=1Nak=1,Aa≤14π1−sinθS/3Q−1δ,ak≤Dt,k∈S+&k≤r,ak≥t,k∈S+&k≤r,−ak≤Dt,k∈S−&k≤r,−ak≥t,k∈S−&k≤r,ak≤Dt,k=r+1,r+2,…,N,−ak≤Dt,k=r+1,r+2,…,N,t≥0.


## 2.7. Efficiency of Proposed Method

Table 1
Number of subproblems,Nsub, and design time, tdesign, estimated for exhaustive search and obtained in proposed minimization of SLL and SLP for arrays with various numbers of elements, N, DRR constraints, D, and beamwidth of 12°. Number of points Q = 10 N and Q = 10 N + 1 are used in minimization of SLL and SLP.
NDExhaustive searchSLL minimizationSLP minimizationNsubtdesignNsubtdesignNsubtdesign2031.049e617.4 min200.42 s200.31 s21.049e617.4 min200.42 s200.39 s11.049e617.4 min1122.10 s200.28 s3031.074e9298 hours602.17 s300.70 s21.074e9298 hours1425.10 s300.71 s11.074e9298 hours159549.4 s300.74 s4031.100e1234.9 years804.53 s401.57 s21.100e1234.9 years804.58 s401.60 s11.100e1234.9 years210495.4 s401.24 sAs shown in Section2.2, global optimization of pencil beam array with constrained dynamic range ratio has combinatorial complexity. Design based on the exhaustive search would require solving of 2N subproblems given in (5). However, the proposed branch and bound approach significantly reduces the number of subproblems. Table 1 shows the number of subproblems and the design time estimated for the exhaustive search, as well as the number of subproblems and the design time obtained in the proposed minimization of SLL and SLP. An average design time of 1 ms per subproblem is assumed in the exhaustive search. The minimizations of SLL and SLP have been performed on a personal computer with quad-core Intel i7 processor operating at the clock of 3 GHz. Clearly, the exhaustive search is unsuitable for the design of arrays with more than 20 elements. On the other hand, the proposed approach enables global design of low and medium-size arrays in a few seconds.


## 3. Properties of Pencil Beams with Constrained DRR

Constraining dynamic range ratio of excitation coefficients reduces design freedom and, consequently, causes the deterioration of radiation pattern. However, a small deterioration is acceptable if it is accompanied by a significant improvement in DRR. In further text, we discuss such cases. Unless otherwise specified, we consider pencil beam arrays with half-wavelength element spacing.Figure2 shows sidelobe levels and dynamic range ratios of the arrays with 30 elements obtained by minimization of SLL. The comparison is made between the arrays with unconstrained DRR [24] and the proposed arrays with DRR constrained to D = 2. Clearly, constraining DRR increases SLL more in wider beams. However, the improvement obtained in DRR is significant. Different behavior is encountered for narrow beams. The improvement in DRR is somewhat lower, but it is here obtained with a negligible increase of SLL.Figure 2
(a) Sidelobe level and (b) Dynamic range ratio of pencil beam arrays with 30 elements obtained by minimization of SLL and DRR constrained toD = 2, compared to DRR-unconstrained minimax design [24].
(a)(b)Figure3 compares the beams obtained via discrete prolate spheroidal sequences [16], whose DRR is unconstrained, and the proposed beams with minimum sidelobe power and DRR constrained to D = 2.5. The former ensure maximum beam efficiency. Therefore, we calculated the efficiency of the latter, as well. As shown in the figure, the proposed beams exhibit lower efficiency than those obtained in [16]. However, the decrease does not exceed 1.2%. This is acceptable because the reduction in DRR of up to 10 times has been obtained.Figure 3
(a) Beam efficiency and (b) Dynamic range ratio of pencil beam arrays with 30 elements obtained by minimization of SLP and DRR constrained toD = 2.5, compared to DRR-unconstrained design with maximum beam efficiency [16].
(a)(b)To illustrate how pencil beam pattern is influenced by DRR constraints, we designed arrays with 30 elements, beamwidth of 12°, and DRRs constrained to several values between one and the values achieved in DRR-unconstrained designs. Figure4 shows the radiation patterns obtained by minimization of SLL. Clearly, for DRRs which are close to 5.7, the sidelobes exhibit regular, nearly equiripple behavior. On the other hand, for DRRs close to one, the sidelobe equality vanishes. For DRR ≥ 1.94, all optimum coefficients are positive, whereas for DRR < 1.94, some coefficients take negative values, as well. The latter cases also differ in the shape of far-out sidelobes. Figure 5 shows the distribution of all-positive as well as positive-and-negative coefficients in optimum arrays with various numbers of elements and dynamic range ratios. Clearly, the coefficients containing positive and negative values appear in arrays with lower DRRs.Figure 4
Radiation patterns of pencil beam arrays with 30 elements and beamwidth of 12°, obtained by minimization of SLL and DRR constrained to various values betweenD = 1 and D = 5.7.Figure 5
Distribution of all-positive (blue) as well as positive-and-negative (red) coefficients in pencil beam arrays with various numbers of elements and beamwidth of 12°, obtained by minimization of SLL and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design.Different behavior is encountered in Figures6 and 7, which show the radiation patterns obtained by minimization of sidelobe power and the corresponding distribution of coefficient signs. In all arrays, the coefficients take positive values. Consequently, all beam patterns have similar shapes, with sidelobe levels which decrease with an increase in DRR.Figure 6
Radiation patterns of pencil beam arrays with 30 elements and beamwidth of 12°, obtained by minimization of SLP and DRR constrained to various values betweenD = 1 and D = 12.6.Figure 7
Distribution of all-positive (blue) as well as positive-and-negative (red) coefficients in pencil beam arrays with various numbers of elements and beamwidth of 12°, obtained by minimization of SLP and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design.


## 4. Examples of Minimax Pencil Beams with Constrained DRR

Tables2 and 3 show two examples, comparing the sidelobe level, directivity, and dynamic range ratio of the arrays with equiripple sidelobes and unconstrained DRR [24], the arrays having minimum DRR for specified SLL and all positive coefficients [7], and the proposed arrays. As expected, the arrays in [24] exhibit the lowest SLL and the highest DRR. Constraining DRR in [7] deteriorates SLL only slightly, whereas the improvement in DRR is high. Further improvement in DRR is achieved by the proposed arrays. This is a consequence of the presence of positive and negative coefficients, which are both allowed in the presented design.Table 2
Properties of proposed pencil beam array with 16 elements and beamwidth of 10°, compared with array having equiripple sidelobes and unconstrained DRR [24] and with array having minimum DRR for given SLL and all positive coefficients [7]. Improvement is highlighted in bold.
Figure of merit[24][7]Proposed,D = 1.8Sidelobe level, dB−12.0−11.3−11.3Directivity, dB11.111.610.8Dynamic range ratio3.62.01.8Table 3
Properties of proposed pencil beam array with 30 elements and beamwidth of 6°, compared with array having equiripple sidelobes and unconstrained DRR [24], and with array having minimum DRR for given SLL and all positive coefficients [7]. Improvement is highlighted in bold.
Figure of merit[24][7]Proposed,D = 2.4Sidelobe level, dB−14.8−14.4−14.4Directivity, dB13.614.113.9Dynamic range ratio5.13.02.4Figure8 shows the radiation patterns of the arrays from Table 3. As clear from the figure, equiripple sidelobes are obtained in DRR-unconstrained design, whereas introducing constraints causes less regular behavior in the sidelobe region.Figure 8
Radiation patterns of proposed pencil beam array, array having equiripple sidelobes and unconstrained DRR [24], and array having minimum DRR for given SLL and all positive coefficients [7]. All arrays have 30 elements and beamwidth of 6°.Numerical values of the proposed coefficients are shown in Tables4 and 5, for convenience.Table 4
Optimum coefficients of proposed array from Table2.
kak10.09843020.09555830.0984304−0.05468350.09300260.05468370.05468380.05470090.061373100.054683110.054683120.054683130.058629140.058988150.063726160.098430Table 5
Optimum coefficients of proposed array from Table3.
kak10.06733220.06350630.0424574−0.02805550.03867360.02805570.02805580.02805590.028055100.044739110.028055120.028055130.036311140.035011150.028055160.037935170.037873180.028055190.039736200.028220210.028055220.028055230.028055240.028055250.028055260.028055270.028055280.028055290.040053300.067332


## 5. Optimization of Beam Efficiency

### 5.1. Sidelobe Power as Criterion for Maximization of Beam Efficiency

Beam efficiency is defined as a ratio of the power radiated in a spatial region of interest and the total radiated power. In a pencil beam, the efficiency is obtained as(24)BEa,θs=PMLa,θsPMLa,θs+PSLa,θs,where PML(a,θs) and PSL(a,θs) are powers radiated in the main lobe and the sidelobe region, respectively. Assuming(25)argmaxaBEa,θs=argmina1BEa,θs,the coefficients maximizing BE can be found by solving the problem(26)minimizeaPSLa,θsPMLa,θs,subject to4π∑k=1Nak2=1,or(27)minimizeaPSLa,θsPMLa,θs,subject to∑k=1Nak=1.The constraint in (26) keeps the total radiated power equal to one, whereas its counterpart in (27) keeps the unity amplitude at θ = 0. However, neither of these constraints does influence the shape nor the efficiency of the optimum beam pattern.The constraint in (26) normalizes the total radiated power, thus ensuring PML(a, θs) + PSL(a, θs) = 1. By using this identity, the objective function can be written as PSL(a, θs)/(1 − PSL(a, θs)). Clearly, in this function, numerator’s minimizer is equal to denominator’s maximizer. Therefore, this objective function takes its minimum at the same point as PSL(a, θs) itself. The sidelobe power can be obtained as PSLa,θs=ε22a,θs, with ε2 defined in (16). By using (17), the coefficients maximizing BE can be found by solving the problem(28)minimizeaε2a,θs,subject to4π∑k=1Nak2=1.As shown in (20), ε2(a,θs) can be expressed in a convex form. Unfortunately, the constraint in (28) is not convex and the problem is difficult to solve globally.The constraint in (27) does not force the total radiated power to unity. Therefore, the above considerations cannot be applied. However, it is expected that minimization of the sidelobe power while keeping the unity amplitude at θ = 0 also results in efficient beam patterns. Therefore, we approximate the problem in (27) by(29)minimizeaε2a,θs,subject to∑k=1Nak=1.Note that this problem is convex. Moreover, it is equivalent to the problem in (22), with the exception of the constraints for DRR.To evaluate the quality of approximation in (29), we compare the beam efficiencies of the patterns obtained by solving (29) and those obtained analytically, via discrete prolate spheroidal sequences [16]. The relative approximation error is shown in Figure 9 for the arrays with 30 elements and various θs. It is clear that the proposed optimization results in a small approximation error. Moreover, the error becomes negligible in arrays with high beam efficiency.Figure 9
Relative approximation error of beam efficiency in pencil beam arrays with 30 elements, obtained by minimization of SLP. Selected values of obtained beam efficiencies,BEs, are added to graph.By adding constraints for DRR, the problem in (29) takes the form of the problem in (22), thus becoming suitable for the synthesis of efficient beam patterns with constrained dynamic range ratio of excitation coefficients. Clearly, the problem in (22) assumes the coefficients’ signs are known in advance. This is convenient because the analysis in Section 3 shows that coefficients take positive values at the optimum, regardless of DRR. However, if a global solution or global proof is required, the branch and bound search described in Sections 2.3 and 2.6 can be applied.


### 5.2. Beam Efficiency of Arrays with Minimum Sidelobe Power and Constrained DRR

Figure10 shows the beam efficiency of pencil beam arrays with the beamwidth of 12°, obtained by minimization of sidelobe power and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design. As clear from the figure, the increase in beam efficiency is accompanied by an increase in DRR. Nevertheless, after reaching a certain value, small or even negligible improvement in efficiency is achieved with the price of high increase in DRR. Apparently, in the design of highly-efficient pencil beam arrays, constraining the DRR might pay off.Figure 10
Beam efficiency of pencil beam arrays with various numbers of elements and beamwidth of 12°, obtained by minimization of SLP and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design.Tables6 and 7 show two examples, comparing the beam efficiency and dynamic range ratio of the arrays with maximum efficiency [16] and the proposed arrays with minimum sidelobe power and constrained DRR. It is clear from the tables that the proposed arrays offer high improvement in DRR and small loss of efficiency.Table 6
Properties of proposed pencil beam arrays with 16 elements and beamwidth of 20°, compared with array having maximum efficiency and unconstrained DRR [16]. Improvements are highlighted in bold.
Figure of merit[16]Proposed,D = 4Proposed,D = 2Beam efficiency, %99.899.698.0Directivity, dB11.111.311.8Dynamic range ratio7.34.02.0Table 7
Properties of proposed pencil beam arrays with 30 elements and beamwidth of 12°, compared with array having maximum efficiency and unconstrained DRR [16]. Improvements are highlighted in bold.
Figure of merit[16]Proposed,D = 6Proposed,D = 2.5Beam efficiency, %99.999.898.8Directivity, dB13.613.814.3Dynamic range ratio13.56.02.5Figure11 illustrates the radiation patterns of the arrays in Table 7. Numerical values of the proposed coefficients obtained for D = 2 and D = 2.5 are shown in Tables 8 and 9, for convenience.Figure 11
Radiation patterns of proposed pencil beam arrays with minimum SLP and DRR constrained toD = 6 and D = 2.5, compared to array with maximum efficiency [16]. All arrays have 30 elements and beamwidth of 12°.Table 8
Optimum coefficients of proposed array from Table6, obtained for D = 2.
kak10.04066020.04066030.04854840.05963050.06974260.07811770.08132180.08132190.081321100.081321110.078117120.069742130.059630140.048548150.040660160.040660Table 9
Optimum coefficients of proposed array from Table7, obtained for D = 2.5.
kak10.01834920.01834930.01834940.02116750.02468560.02819970.03163280.03490690.037948100.040689110.043070120.045038130.045872140.045872150.045872160.045872170.045872180.045872190.045038200.043070210.040689220.037948230.034906240.031632250.028199260.024685270.021167280.018349290.018349300.018349


### 5.3. Beam Efficiency of Unequally Spaced Arrays

In previous examples, we considered pencil beam arrays with half-wavelength element spacing. However, the presented method also allows the optimization of the arrays with unequally spaced elements, assuming their positions are known in advance. In [22], the positions of unequally spaced elements which minimize the sidelobe power of the arrays with uniform excitation have been provided. Here, we employed these positions to optimize the SLP assuming nonuniform coefficients with constrained DRR.Figure12 shows the beam efficiency of optimum arrays with 32 elements and the beamwidth of 6°, obtained for element spacing from the last column of Table 2 in [22] and for half-wavelength element spacing. Positive coefficients are obtained in all cases. As shown in the figure, higher beam efficiency is obtained for unequal element spacing. The improvement is the highest for D = 1 and decreases with an increase in D. It is expected since the used unequal positions are optimal for D = 1 and suboptimal for D > 1.Figure 12
Beam efficiency of pencil beam arrays with minimum SLP and constrained DRR, obtained for unequal element spacing from [22] and for equal element spacing of λ/2. Both arrays have 32 elements and beamwidth of 6°.Table10 shows the optimum coefficients obtained for the array with unequal element spacing and D = 2, for convenience.Table 10
Optimum coefficients of array with unequal element spacing obtained forD = 2.
kak10.01701920.02162930.02596940.02998750.03337260.03403770.03267880.03403790.033545100.034037110.033714120.034037130.033864140.034037150.034000160.034037170.034037180.034000190.034037200.033864210.034037220.033714230.034037240.033545250.034037260.032678270.034037280.033372290.029987300.025969310.021629320.017019


## 5.1. Sidelobe Power as Criterion for Maximization of Beam Efficiency

Beam efficiency is defined as a ratio of the power radiated in a spatial region of interest and the total radiated power. In a pencil beam, the efficiency is obtained as(24)BEa,θs=PMLa,θsPMLa,θs+PSLa,θs,where PML(a,θs) and PSL(a,θs) are powers radiated in the main lobe and the sidelobe region, respectively. Assuming(25)argmaxaBEa,θs=argmina1BEa,θs,the coefficients maximizing BE can be found by solving the problem(26)minimizeaPSLa,θsPMLa,θs,subject to4π∑k=1Nak2=1,or(27)minimizeaPSLa,θsPMLa,θs,subject to∑k=1Nak=1.The constraint in (26) keeps the total radiated power equal to one, whereas its counterpart in (27) keeps the unity amplitude at θ = 0. However, neither of these constraints does influence the shape nor the efficiency of the optimum beam pattern.The constraint in (26) normalizes the total radiated power, thus ensuring PML(a, θs) + PSL(a, θs) = 1. By using this identity, the objective function can be written as PSL(a, θs)/(1 − PSL(a, θs)). Clearly, in this function, numerator’s minimizer is equal to denominator’s maximizer. Therefore, this objective function takes its minimum at the same point as PSL(a, θs) itself. The sidelobe power can be obtained as PSLa,θs=ε22a,θs, with ε2 defined in (16). By using (17), the coefficients maximizing BE can be found by solving the problem(28)minimizeaε2a,θs,subject to4π∑k=1Nak2=1.As shown in (20), ε2(a,θs) can be expressed in a convex form. Unfortunately, the constraint in (28) is not convex and the problem is difficult to solve globally.The constraint in (27) does not force the total radiated power to unity. Therefore, the above considerations cannot be applied. However, it is expected that minimization of the sidelobe power while keeping the unity amplitude at θ = 0 also results in efficient beam patterns. Therefore, we approximate the problem in (27) by(29)minimizeaε2a,θs,subject to∑k=1Nak=1.Note that this problem is convex. Moreover, it is equivalent to the problem in (22), with the exception of the constraints for DRR.To evaluate the quality of approximation in (29), we compare the beam efficiencies of the patterns obtained by solving (29) and those obtained analytically, via discrete prolate spheroidal sequences [16]. The relative approximation error is shown in Figure 9 for the arrays with 30 elements and various θs. It is clear that the proposed optimization results in a small approximation error. Moreover, the error becomes negligible in arrays with high beam efficiency.Figure 9
Relative approximation error of beam efficiency in pencil beam arrays with 30 elements, obtained by minimization of SLP. Selected values of obtained beam efficiencies,BEs, are added to graph.By adding constraints for DRR, the problem in (29) takes the form of the problem in (22), thus becoming suitable for the synthesis of efficient beam patterns with constrained dynamic range ratio of excitation coefficients. Clearly, the problem in (22) assumes the coefficients’ signs are known in advance. This is convenient because the analysis in Section 3 shows that coefficients take positive values at the optimum, regardless of DRR. However, if a global solution or global proof is required, the branch and bound search described in Sections 2.3 and 2.6 can be applied.


## 5.2. Beam Efficiency of Arrays with Minimum Sidelobe Power and Constrained DRR

Figure10 shows the beam efficiency of pencil beam arrays with the beamwidth of 12°, obtained by minimization of sidelobe power and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design. As clear from the figure, the increase in beam efficiency is accompanied by an increase in DRR. Nevertheless, after reaching a certain value, small or even negligible improvement in efficiency is achieved with the price of high increase in DRR. Apparently, in the design of highly-efficient pencil beam arrays, constraining the DRR might pay off.Figure 10
Beam efficiency of pencil beam arrays with various numbers of elements and beamwidth of 12°, obtained by minimization of SLP and constrained DRR. Black curve indicates maximum DRR, which is achieved in DRR-unconstrained design.Tables6 and 7 show two examples, comparing the beam efficiency and dynamic range ratio of the arrays with maximum efficiency [16] and the proposed arrays with minimum sidelobe power and constrained DRR. It is clear from the tables that the proposed arrays offer high improvement in DRR and small loss of efficiency.Table 6
Properties of proposed pencil beam arrays with 16 elements and beamwidth of 20°, compared with array having maximum efficiency and unconstrained DRR [16]. Improvements are highlighted in bold.
Figure of merit[16]Proposed,D = 4Proposed,D = 2Beam efficiency, %99.899.698.0Directivity, dB11.111.311.8Dynamic range ratio7.34.02.0Table 7
Properties of proposed pencil beam arrays with 30 elements and beamwidth of 12°, compared with array having maximum efficiency and unconstrained DRR [16]. Improvements are highlighted in bold.
Figure of merit[16]Proposed,D = 6Proposed,D = 2.5Beam efficiency, %99.999.898.8Directivity, dB13.613.814.3Dynamic range ratio13.56.02.5Figure11 illustrates the radiation patterns of the arrays in Table 7. Numerical values of the proposed coefficients obtained for D = 2 and D = 2.5 are shown in Tables 8 and 9, for convenience.Figure 11
Radiation patterns of proposed pencil beam arrays with minimum SLP and DRR constrained toD = 6 and D = 2.5, compared to array with maximum efficiency [16]. All arrays have 30 elements and beamwidth of 12°.Table 8
Optimum coefficients of proposed array from Table6, obtained for D = 2.
kak10.04066020.04066030.04854840.05963050.06974260.07811770.08132180.08132190.081321100.081321110.078117120.069742130.059630140.048548150.040660160.040660Table 9
Optimum coefficients of proposed array from Table7, obtained for D = 2.5.
kak10.01834920.01834930.01834940.02116750.02468560.02819970.03163280.03490690.037948100.040689110.043070120.045038130.045872140.045872150.045872160.045872170.045872180.045872190.045038200.043070210.040689220.037948230.034906240.031632250.028199260.024685270.021167280.018349290.018349300.018349


## 5.3. Beam Efficiency of Unequally Spaced Arrays

In previous examples, we considered pencil beam arrays with half-wavelength element spacing. However, the presented method also allows the optimization of the arrays with unequally spaced elements, assuming their positions are known in advance. In [22], the positions of unequally spaced elements which minimize the sidelobe power of the arrays with uniform excitation have been provided. Here, we employed these positions to optimize the SLP assuming nonuniform coefficients with constrained DRR.Figure12 shows the beam efficiency of optimum arrays with 32 elements and the beamwidth of 6°, obtained for element spacing from the last column of Table 2 in [22] and for half-wavelength element spacing. Positive coefficients are obtained in all cases. As shown in the figure, higher beam efficiency is obtained for unequal element spacing. The improvement is the highest for D = 1 and decreases with an increase in D. It is expected since the used unequal positions are optimal for D = 1 and suboptimal for D > 1.Figure 12
Beam efficiency of pencil beam arrays with minimum SLP and constrained DRR, obtained for unequal element spacing from [22] and for equal element spacing of λ/2. Both arrays have 32 elements and beamwidth of 6°.Table10 shows the optimum coefficients obtained for the array with unequal element spacing and D = 2, for convenience.Table 10
Optimum coefficients of array with unequal element spacing obtained forD = 2.
kak10.01701920.02162930.02596940.02998750.03337260.03403770.03267880.03403790.033545100.034037110.033714120.034037130.033864140.034037150.034000160.034037170.034037180.034000190.034037200.033864210.034037220.033714230.034037240.033545250.034037260.032678270.034037280.033372290.029987300.025969310.021629320.017019


## 6. Pencil Beams with Minimum DRR and Specified Beam Efficiency

In a design at hand, it might be interesting to know the lowest value to which the DRR can be constrained while still achieving desired beam efficiency,BEdes. As shown in Figure 10, for a given number of elements, the beam efficiency decreases monotonically with a decrease in DRR. Apparently, the minimum value of DRR is uniquely determined by BEdes. We found this minimum by using bisection method.


### 6.1. Synthesis of Pencil Beams with Minimum DRR and Specified Beam Efficiency

In pencil beam array, the lowest value of DRR is one. The highest value of DRR in array that maximizes beam efficiency is achieved by using discrete prolate spheroidal sequences. We denote this value byDRRDPSS.The search for optimum DRR starts with specifying the interval of its possible values, which is given by [DRRlow, DRRhigh] = [1, DRRDPSS]. The interval’s middle point is then found as D = (DRRlow + DRRhigh)/2. Using D as DRR constraint, a pencil beam synthesis described in Section 2.6 is performed. Either method can be chosen—the convex optimization with all-positive coefficients or the global search covering all coefficient signs. The beam efficiency of the obtained pencil beam, BE, is then calculated. If BE < BEdes, the lower bound of DRR is updated as DRRlow = D. Otherwise, the upper bound is updated as DRRhigh = D. In the next step, new middle point is found and the procedure is repeated until the interval [DRRlow, DRRhigh] has been tightened below specified tolerance, εtol, what indicates that D represents the optimum DRR with sufficient accuracy. In practice, εtol = 1e−3 is appropriate.A systematic search performing the above procedure is given by Algorithm 1. Its input parameters are the number of elements,N, beamwidth, BW, desired beam efficiency, BEdes, and tolerance, εtol. To ensure that BEdes is possible to achieve with given N and BW, an initial check is performed, which compares BEdes with its highest as well as with its lowest possible value, denoted as BEhigh and BElow.


### 6.2. Examples of Pencil Beams with Minimum DRR and Specified Beam Efficiency

Table11 shows the beam efficiency and dynamic range ratio of the arrays from [16] and the proposed arrays with minimum DRR and specified beam efficiency. The efficiency of the arrays from [16] exceeds 99.9%, whereas DRR >11. In the proposed design, the efficiency is specified to somewhat lower values, BEdes, of 99.9%, 99.5%, and 99.0%. As shown in the table, the coefficients obtained have significantly lower DRRs, which take the values down to 2.59.Table 11
Beam efficiency and dynamic range ratio of DPSS pencil beam arrays [16] and proposed arrays with minimum DRR and BEdes of 99.9%, 99.5%, and 99.0%, together with design time of proposed arrays, tdesign. Improvements are highlighted in bold.
NBW, degDPSS pencil beams [16]Proposed pencil beamsBEDPSS, %DRRDPSSBEopt, %DRRopttdesign, s152599.9511.3799.906.642.199.503.532.099.002.592.0301299.9313.5499.907.9711.699.503.6810.999.002.7010.740999.9314.6799.908.0624.999.503.6924.899.002.7124.8The proposed optimization can be performed in a short time. It is illustrated in the last column of Table11, which shows the time required for each design. The time is obtained by using Algorithm 1 utilizing the global search in Steps 2 and 8.


## 6.1. Synthesis of Pencil Beams with Minimum DRR and Specified Beam Efficiency

In pencil beam array, the lowest value of DRR is one. The highest value of DRR in array that maximizes beam efficiency is achieved by using discrete prolate spheroidal sequences. We denote this value byDRRDPSS.The search for optimum DRR starts with specifying the interval of its possible values, which is given by [DRRlow, DRRhigh] = [1, DRRDPSS]. The interval’s middle point is then found as D = (DRRlow + DRRhigh)/2. Using D as DRR constraint, a pencil beam synthesis described in Section 2.6 is performed. Either method can be chosen—the convex optimization with all-positive coefficients or the global search covering all coefficient signs. The beam efficiency of the obtained pencil beam, BE, is then calculated. If BE < BEdes, the lower bound of DRR is updated as DRRlow = D. Otherwise, the upper bound is updated as DRRhigh = D. In the next step, new middle point is found and the procedure is repeated until the interval [DRRlow, DRRhigh] has been tightened below specified tolerance, εtol, what indicates that D represents the optimum DRR with sufficient accuracy. In practice, εtol = 1e−3 is appropriate.A systematic search performing the above procedure is given by Algorithm 1. Its input parameters are the number of elements,N, beamwidth, BW, desired beam efficiency, BEdes, and tolerance, εtol. To ensure that BEdes is possible to achieve with given N and BW, an initial check is performed, which compares BEdes with its highest as well as with its lowest possible value, denoted as BEhigh and BElow.


## 6.2. Examples of Pencil Beams with Minimum DRR and Specified Beam Efficiency

Table11 shows the beam efficiency and dynamic range ratio of the arrays from [16] and the proposed arrays with minimum DRR and specified beam efficiency. The efficiency of the arrays from [16] exceeds 99.9%, whereas DRR >11. In the proposed design, the efficiency is specified to somewhat lower values, BEdes, of 99.9%, 99.5%, and 99.0%. As shown in the table, the coefficients obtained have significantly lower DRRs, which take the values down to 2.59.Table 11
Beam efficiency and dynamic range ratio of DPSS pencil beam arrays [16] and proposed arrays with minimum DRR and BEdes of 99.9%, 99.5%, and 99.0%, together with design time of proposed arrays, tdesign. Improvements are highlighted in bold.
NBW, degDPSS pencil beams [16]Proposed pencil beamsBEDPSS, %DRRDPSSBEopt, %DRRopttdesign, s152599.9511.3799.906.642.199.503.532.099.002.592.0301299.9313.5499.907.9711.699.503.6810.999.002.7010.740999.9314.6799.908.0624.999.503.6924.899.002.7124.8The proposed optimization can be performed in a short time. It is illustrated in the last column of Table11, which shows the time required for each design. The time is obtained by using Algorithm 1 utilizing the global search in Steps 2 and 8.


## 7. Conclusion

Method for global optimization of linear pencil beams with specified DRR and minimum sidelobe level or minimum sidelobe power was described. The analysis of optimum pencil beams shows that minimization of sidelobe level results in positive coefficients if higher values of DRR are allowed, whereas positive and negative coefficients are obtained for lower values of DRR. Different behavior is obtained in minimization of sidelobe power, where optimum coefficients are all positive.The proposed method for minimization of sidelobe power can be utilized in approximate and at the same time global design of pencil beam arrays with maximum beam efficiency. In such a design, all coefficients can be assumed positive, which leads to convex optimization. However, if the global solution or global proof is required, the branch and bound search can be applied.The minimization of sidelobe power was utilized in a design of pencil beam arrays with minimum DRR and specified beam efficiency. In comparison with the arrays having maximum efficiency, the proposed arrays exhibit significantly lower DRRs and negligible loss of efficiency.Algorithm 1: Search for pencil beam array with minimum DRR and specified beam efficiency.
(1)
specifyN, BW, BEdes, and εtol(2)
for specifiedN and BWcalculatecoefficientsaDPSS of DPSS arraycoefficientsa1 by using method in Section 2.6 with D = 1(3)
calculateBEhigh as beam efficiency of array with coefficients aDPSSBElow as beam efficiency of array with coefficients a1(4)
ifBEdes > BEhighthenerror:BEdes is too large for given N and BWend if(5)
ifBEdes < BElowthenerror:BEdes is too small for given N and BWend if(6)
calculateDRRhigh as dynamic range ratio of aDPSS(7)
setDRRlow = 1 // dynamic range ratio of a1(8)
while (DRRhigh − DRRlow) > εtoldocalculateD = (DRRlow + DRRhigh)/2calculate coefficients a by using method in Section 2.6calculateBE as beam efficiency of array with coefficients aifBE < BEdesthenDRRlow = DelseDRRhigh = Dend ifend while(9)
set aopt = a, DRRopt = D, and BEopt = BE(10)
return aopt, DRRopt, and BEopt



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*Source: 3664607-2022-11-16.xml*