Introduction
Array antennas are the fundamental technologies of a modern communication system that has a broad range of applications in radar, remote sensing, satellite navigation, biomedical imaging, microwaves, and several other fields [Reference Hansen1]. Since its introduction, antenna arrays have evolved substantially [Reference Mailloux2] and incorporated to multi-domain research prospect related to electromagnetics, electronics, computer science, control systems, etc. [Reference Chakraborty, Ram and Mandal3]. Phased arrays are the advancement over traditional antenna arrays usually considered for electronic beam scanning and beam steering [Reference Rocca, Oliveri, Mailloux and Massa4]. Generally, the phased array antennas are equipped with transmit and receive modules that permit the independent control of amplitude and phase of the transmitted or received signals with the help of attenuators and phase shifters [Reference Rocca, Oliveri, Mailloux and Massa4]. Cutting-edge communication system has thrusted stringent stipulation on phased arrays like multi-functionalities, reconfigurabilities, lesser weight, more compact, and reduced costs [Reference Rocca, Oliveri, Mailloux and Massa4]. Further, the requirement of beamforming, beam scanning, and beam steering in advance communication systems has significantly increased the research concern in the domain of phased arrays. So, the exploration and exploitation of newer techniques for alternative solution is needed.
An unorthodox alternative to phased antennas has got the attention of antenna research community when Shanks et al. introduced the concept of fourth-dimension—“Time” to control the radiation patterns of antenna arrays [Reference Shanks and Bickmore5]. Immediately after their path-breaking research, Shanks proposed a method for electronic scanning by utilizing the idea of “Time-modulation” [Reference Shanks6]. Although the “Time-modulated” array (TMA) was initially postulated for achieving ultra-low sidelobe radiation patterns [Reference Kummer, Villeneuve, Fong and Terrio7], eventually they have become a popular choice for several other applications like electronic beam scanning [Reference Haupt8], beam shaping [Reference Yang, Chen and Nie9], beam steering [Reference Chakraborty, Ram and Mandal10], and so on. TMAs may be classified as the “Time-domain” counterparts of traditional antenna arrays where a synchronized sequence is implemented with swiftly turning ON and OFF the RF switches, attached with each antenna element [Reference Rocca, Yang, Poli and Yang11]. By adding and removing the elements from beamformer network (BFN), a desirable time-averaged radiation pattern can be generated [Reference Maneiro-Catoira, Brégains, García-Naya and Castedo12]. The turn-ON period and switch-ON instants of each radiators can be controlled to obtain a sequence equivalent to the current and phase tapering in conventional arrays [Reference Maneiro-Catoira, Brégains, García-Naya and Castedo12]. Hence, the need of costly phase shifters and erroneous attenuators may be eradicated, and the quantization error and insertion loss that occurred in traditional arrays may be avoided [Reference Fondevila, Brégains, Ares and Moreno13]. However, the added advantages of TMAs are also accompanied by the inherent drawback of infinite number of harmonic pattern generation due to the periodicity of the switches [Reference Yang, Gan and Tan14]. Harmonic radiations are generally considered unwanted because a major part of the dissipated power is transferred from the main beam to the sidebands [Reference Brégains, Fondevila-Gómez, Franceschetti and Ares15]. The sideband radiation (SR) should be reduced to improve the total efficiency of the TMAs [Reference Yang, Gan and Qing16]. The primary research work in this field was focused only on sidelobe levels (SLLs) and SR reduction [Reference Yang, Can and Tan17]. However, in due course, a new dimension has evolved with the notion of exploiting the unwanted SRs as an advantage. In this regard, TMAs for direction finding has been introduced [Reference Tennant and Chambers18]. The possibilities have been enhanced after Li et al. proposed a technique to steer the beam in some specific directions [Reference Li, Yang, Chen and Nie19]. From then on, various research works have been reported with efficiently steered and scanned performances [Reference Tong and Tennant20–Reference Ram, Panduro, Reyna, Kar and Mandal25]. This paper exploited the notion further by proposing efficient multi-patterns such as sum–difference pattern for monopulse radars and scanned beam and shaped beam patterns for multifunctional radars (MFRs).
Monopulse-based searching and tracking radar require an array system that can produce a sum and a difference pattern simultaneously [Reference Barton26]. Tracking radar automatically keeps the main array beam aligned to the target object, producing a narrow and highly directive radiation pattern [Reference Sherman27]. Monopulse arrays have been broadly employed in radar tracking systems to improve the poor speed and complicated BFNs experienced by a conical or sequential scanning [Reference Sherman27]. The spatially independent and simultaneously generated harmonic beams can be used with appropriate switching sequence to deal with the drawbacks of compromised sum–difference patterns [Reference Chakraborty, Ram and Mandal28]. This article comes up with a suitable optimized switch scheme to produce the simultaneous monopulse pattern (sum–difference pattern) in a 20-element time-modulated linear array (TMLA). Further research on electronic beam scanning beneficial for the MFRs has been addressed by designing the suitable optimal time schemes [Reference Chakraborty, Ram and Mandal29]. Steering the consecutive harmonic patterns to prespecified angles and simultaneously lowering the SLLs is extremely beneficial for radars [Reference Chakraborty, Ram and Mandal30–Reference Chakraborty, Ram and Mandal32]. In this article, a 20-element TMLA is used to produce the scanned harmonic pattern at ±30° from the broadside. Furthermore, a shaped flat-top pattern is also targeted with a 16-element TMLA. To obtain the desired optimal radiation patterns, a memetic quasi-Newton optimization algorithm is employed. Here the limitations of traditional Newton method are eliminated by incorporating a Broyden’s good method (BGM)-based direction-updating equation in the exploitation stage, along with the advantages of finding potential trust regions at the exploration stage using genetic algorithm (GA).
The work proposed in this article have manifold prospects briefly pointed out as follows:
• First of all, multiple patterns have been addressed by exploiting the concept of time-modulation.
• Second, for all the beam patterns, interference rejection is considered of the utmost importance.
• Third, to cancel the interference from unwanted sources, SLLs of all the patterns are minimized.
• Fourth, the unwanted higher-order harmonics apart from the first sideband are all suppressed.
• Fifth, the arrays are designed to cater the need of both the narrowband beam patterns using scanned beams and monopulse beams, as well as the wideband patterns using flat-top and cosecant beams.
The rest of the article is described as follows: The theoretical and mathematical insights of TMLA are presented in Section 2. The memetic algorithm used to optimize the control parameters is briefly discussed in Section 3. In Section 4, the outcomes of the proposed methods are analyzed, and a comparison and thorough discussion on future prospect is furnished. Finally, conclusive remarks are given in Section 5.
Theoretical background
Time-modulated linear arrays
A TMLA consists of N isotropic radiating antennas attached with same number of Single-Pole-Single-Through (SPST) Radio-Frequency (RF) switches, and it is controlled by a complex programmable logic device (CPLD), which is exhibited in Fig. 1. The array factor (AF) for the same may be given as [Reference Chakraborty, Ram and Mandal31]
\begin{equation}{\textrm{AF}}\left( {\theta ,t} \right) = {{\textrm{e}}^{j\left( {2\pi {f_0}} \right)t}} \sum\limits_{n = 1}^N {I_n}{U_n}\left( t \right){{\textrm{e}}^{jk\left( {n - 1} \right)d\,\cos \theta }}\end{equation}N-element array connected through switches and managed by a CPLD.
where In denotes the amplitude of the nth antenna, k is the constant of propagation, d represents the uniform spacing between antennas, θ is the angle of the impinging signal, f 0 is the operating frequency with T 0 as the period, and Un(t) is the time-switching function.
Due to periodicity of switches, Un(t) can be given as [Reference Chakraborty, Ram and Mandal32]
\begin{equation}{U_n}\left( t \right) = \sum \limits_{m = - \infty }^\infty {a_{mn}}{{\textrm{e}}^{jm\left( {2\pi {f_p}} \right)t}}\end{equation}The Fourier coefficient (amn) of nth element for mth frequency term, inherently generated because of time-modulation, can be presented as [Reference Bhattacharya, Saha and Bhattacharyya33]
\begin{equation}{a_{mn}} = {1 \over {{T_p}}} \smallint \limits_0^{{T_p}} {U_n}\left( t \right){{\textrm{e}}^{ - jm\left( {2\pi {f_p}} \right)t}}\,{\textrm{d}}t \end{equation}Hence, the AF of uniformly excited (In = 1) TMLA can be expressed as
\begin{equation}{\textrm{AF}}\left( {\theta ,t} \right) = \sum \limits_{m = - \infty }^\infty \sum \limits_{n = 1}^N {a_{mn}}\left\{ {{{\textrm{e}}^{jk\left( {n - 1} \right)d\,\cos \theta }}} \right\}{{\textrm{e}}^{j2\pi \left( {{f_0} + m{f_p}} \right)t}}\end{equation}The array pattern at mth harmonics may be simplified as [Reference Chakraborty, Ram and Mandal34]
\begin{equation}{\textrm{A}}{{\textrm{F}}_m}\left( {\theta ,t} \right) = {{\textrm{e}}^{j2\pi \left( {{f_0} + m{f_p}} \right)t}} \sum \limits_{n = 1}^N {a_{mn}}{{\textrm{e}}^{jk\left( {n - 1} \right)d\,\cos \theta }}\end{equation}The main beam pattern can be represented with m = 0, and SR patterns are at the multitude of modulation frequencies (mfp) where m = ±1, ±2, ±3, …, ±∞.
Time sequences for diverse applications of TMLAs may be designed, and to explore that, harmonic beam steering in TMLA is inscribed in the paper by suitably optimized shifted time sequences. The general ON–OFF time sequence for the nth antenna is given in Fig. 2(a), in which the radiator is turned on for the duration τn (0 ≤ τn ≤Tp). The pulse initiated at 
$\tau _n^1 = 0$ and terminated at 
$\tau _n^2$ having a normalized ON-time duration 
$\{ \left( {\tau _n^2 - \tau _n^1} \right)/{T_p}\} $. The switching function and the Fourier coefficient are given as [Reference Chakraborty, Ram and Mandal34]
\begin{equation}{U_n}\left( t \right) = \left\{ \begin{matrix}
{1, \tau _n^1 \le t \le \tau _n^2 \le {T_p}\ {\textrm{where}}\ \tau _n^1 = 0 } \\
{0, {\textrm{otherwise}}} \\
\end{matrix}\right.{\textrm{ }}\end{equation}
\begin{equation}{a_{mn}} = {{{\tau _n}} \over {{T_p}}}\left\{ {{\textrm{sinc}}\left( {m\pi {f_p}{\tau _n}} \right)} \right\}{{\textrm{e}}^{ - jm\pi {f_p}\left( {{\tau _n}} \right)}}\end{equation}(a) Simple switching sequence and (b) shifted switching sequence for nth radiator in TMLA.
The generalized shifted scheme of the nth antenna is given in Fig. 2(b), in which the ON-time duration of Fig. 2(a) is moved to 
$\tau _n^1$ (
$\tau _n^1 \ne 0$) and the nth antenna remains ON up to 
$\tau _n^2$. The switching function and excitation coefficient for the shifted sequence can be given as
\begin{equation}{U_n}\left( t \right) = \left\{ \begin{matrix}
{1, 0 \lt \tau _n^1 \le t \le \tau _n^2 \le {T_p}\ {\textrm{where}}\ \tau _n^1 \ne 0 } \\
{0, {\textrm{otherwise}} } \\
\end{matrix} \right.{\textrm{ }}\end{equation}
\begin{equation}{a_{mn}} = {{\left( {\tau _n^2 - \tau _n^1} \right)} \over {{T_p}}}{\ }\left[ {{\textrm{sinc}}\left\{ {m\pi {f_p}\left( {\tau _n^2 - \tau _n^1} \right)} \right\}} \right]{\ }{{\textrm{e}}^{ - jm\pi {f_p}\left( {\tau _n^1 + \tau _n^2} \right)}}\end{equation} For steering the first sideband patterns (m = ±1) in prespecified direction (
${\theta _0}$), the switching instants for each array element can be modified as [Reference Rocca, Zhu, Bekele, Yang and Massa35]
\begin{equation}\tau _n^1 = \left[ {{{\left( {n - 1} \right)kd\ {\textrm{cos}{\theta _0}}} \over {2\pi }} - {{{\tau _n}} \over 2}} \right]{\textrm{ mod }}1{\textrm{ }}\end{equation}The amount of radiated power for the fundamental pattern (P 0), the total power dissipated by harmonic patterns including the fundamental one (P T), and the directivity can be given as [Reference Poddar, Paul, Chakraborty, Ram and Mandal36]
\begin{equation}{P_0} = \smallint\limits_{2\pi }^0 \smallint\limits_\pi ^0 {\left| {A{F_0}\left( {\theta ,\phi } \right)} \right|^2}\sin \theta {\textrm{d}}\theta {\textrm{d}}\phi \end{equation}
\begin{equation}{P_T} = \sum\limits_\infty ^{m = - \infty } \smallint\limits_{2\pi }^0 \smallint\limits_\pi ^0 {\left| {A{F_m}\left( {\theta ,\phi } \right)} \right|^2}\sin \theta {\textrm{d}}\theta {\textrm{d}}\phi \end{equation}
\begin{equation}D = {{4\pi {{\left| {A{F_0}\left( {{\theta _0},{\phi _0}} \right)} \right|}^2}} \over {\sum\nolimits_{m = - \infty }^\infty \smallint\nolimits_0^{2\pi }\smallint\nolimits_0^\pi {{\left| {A{F_m}\left( {\theta ,\phi } \right)} \right|}^2}\sin \theta {\textrm{d}}\theta {\textrm{d}}\phi }}\end{equation}Objective functions
To manage the radiation characteristics of the central beam pattern and the steered SR patterns, the SLLs are lowered. The undesired higher sidebands are minimized. To obtain the coveted outcomes, an appropriate objective function (OF1) is designed as
\begin{equation}{\textrm{O}}{{\textrm{F}}_{\textrm{1}}} = {\left. {({\textrm{SLL}}_0^{\left( i \right)})} \right|_{{f_0}}} + {\left. {({\textrm{SLL}}_1^{\left( i \right)})} \right|_{{f_0} \pm {f_p}}} + {\ }({\left. {{\textrm{SBL}}_m^{\left( i \right)})} \right|_{{f_0} + m{f_p}}}{\textrm{ }}\end{equation}where i denotes the current iteration, SLL0 and SLL1 represents the maximum SLLs at the fundamental (f 0) and first harmonic frequencies (
${f_0} \pm {f_p}$), and SBLm is the maximum level of sideband patterns (
${f_0} + m{f_p}$ for m = ±2, ±3, …, ±∞).
The exploitation of the harmonic radiations is concerned of highly directive radiation pattern generation for narrowband communication. On the other hand, the shaped pattern synthesis is aimed toward designing a flat-top pattern and the cosecant-squared pattern at the first sideband for specific wideband applications. The problems are cast into suitable OFs (OF2 for flat-top and OF3 for cosecant-squared patterns) as
\begin{align}{\textrm{O}}{{\textrm{F}}_{\textrm{2}}} & = ({\textrm{SLL}}_{{\textrm{max}}}^{\left( i \right)}){|_{{f_0}}} + ({\textrm{Ripple}}_{{\textrm{max}}}^{\left( i \right)}){|_{{f_0} + {f_p}}}\nonumber \\ & \qquad + {{\left( {\beta _{{\textrm{Trans}}}^{\left( i \right)}} \right)} \over {{90}^{\circ}}}{|_{{f_0} + {f_p}}}{\textrm{ + }}({\textrm{SBL}}_m^{\left( i \right)}){|_{{f_0} + m{f_p}}}{\textrm{ }}\end{align}
\begin{align}{\textrm{O}}{{\textrm{F}}_{\textrm{3}}} & = ({\textrm{SLL}}_{{\textrm{max}}}^{\left( i \right)}){|_{{f_0}}} + ({\textrm{Ripple}}_{{\textrm{max}}}^{\left( i \right)}){|_{{f_0} + {f_p}}}\nonumber \\ & \qquad + {{\left( {\beta _{{\textrm{Trans}}}^{\left( i \right)}} \right)} \over {{90}^{\circ}}}{|_{{f_0} + {f_p}}}{\textrm{ + }}(\beta _C^{\left( i \right)} - {\beta _0}){|_{{f_0} + {f_p}}}{\textrm{ }}\end{align}Quasi-Newton memetic optimization
Evolutionary optimization methods are population-based computational algorithms that help to unfold real-world technical issues by imitating the behavioral orientations of natural phenomenon [Reference De Jong, Fogel, Schwefel, Baeck, Fogel and Michalewicz37]. Bioinspired computation methods have evolved as suitable candidates for solving electromagnetics problems [Reference Del Ser, Osaba, Molina, Yang, Salcedo-Sanz, Camacho, Das, Suganthan, Coello and Herrera38]. Researchers have found these algorithms useful for synthesizing the radiation patterns of linear arrays [Reference Ram, Mandal, Kar and Ghoshal39, Reference Durmus and Kurban40], circular arrays [Reference Rattan, Patterh and Sohi41, Reference Das, Mandal and Kar42], planar arrays [Reference Dib43, Reference Bogdan, Godziszewski and Yashchyshyn44], concentric circular arrays [Reference Dib and Sharaqa45, Reference Ram, Mandal, Kar and Ghoshal46], hexagonal arrays [Reference Misra and Mahanti47], etc. Memetic algorithms are the population-based metaheuristics techniques, where the strengths of global search and the fine-tuning properties of local search techniques are combined [Reference Neri and Cotta48]. GA is a familiar global search, stochastic optimization method based on natural selection and evolution [Reference Weile and Michielssen49]. GA has an excellent global search ability but requires a higher number of iterations for convergence [Reference Weile and Michielssen49]. On the other hand, deterministic QNM is the advanced version of the Newton method where a modified Hessian matrix is incorporated to minimize the computation cost and complexities of the Newton method. The hybridization of both can be regarded as a memetic computation method for solving nonlinear optimization problems [Reference Zhang, Lin, Zhang and Li50]. In this paper, QNM-based memetic GA (QNM-GA) is employed.
The genetic algorithm
GA is a kind of probabilistic search method that utilizes the idea of natural selection and evolution. At each iteration, it preserves a population of individuals, also called as Chromosomes, coded from possible solutions (xk) such as [Reference Zhang, Lin, Zhang and Li50]:
\begin{equation}{x_k} = {x_{{\textrm{min}}}} + {r_k}\left( {{x_{{\textrm{max}}}} - {x_{{\textrm{min}}}}} \right)\end{equation}where x min and x max are the lower and upper boundaries of the search space and rk denotes some positive random values within the range of 0 and 1. Chromosomes are constructed over the binary range [0, 1] so that the Chromosome values can be uniquely mapped onto the decision variable domain. Each Chromosome is then evaluated by the fitness function, which is nothing but the representation of the optimization problem in hand. Based on that, elite strings are generated and crossover and mutation operations are done hereafter to generate the offsprings. In crossover operation, GA makes use of the following equation to produce two parent solutions xs and xt based on the crossover rate Cr as [Reference Zhang, Lin, Zhang and Li50]:
\begin{equation}{x_{kj}} = \left\{ \begin{matrix}
{{x_{sj}}, {\textrm{if}}\ {C_r} \lt {r_{j0}} } \\
{{x_{tj}}, {\textrm{otherwise}}} \\
\end{matrix} \right.{\textrm{ }}\end{equation}where rj 0 is a random number within the range of 0 and 1. Then the mutation operation is done based on mutation rate parameter Pm as [Reference Zhang, Lin, Zhang and Li50]:
\begin{equation}{x_{kj}} = \left\{ \begin{matrix}
{{x_{kj}} + {r_{j2}}\left( {{x_{kj}} - {x_{mj}}} \right), {\textrm{if}}\ {P_m} \lt {r_{j1}} } \\
{{x_{kj}}, {\textrm{otherwise}}} \\
\end{matrix} \right.\end{equation}where rj 1 and rj 2 are the random numbers within the range of 0 and 1. In this way, the genetic cycle updating continues iteratively until the completion of exploration stage for finding out the potential trust regions.
BGM-based QNM
Newton’s method is an iterative method often used to solve the nonlinear equations due to its capability of rapid convergence from a sufficiently good starting position [Reference Li, Qi and Roshchina51]. The limitation of Newton’s method is the direct computation of Jacobian, which is computationally expensive [Reference Li, Qi and Roshchina51]. In addition, the Newton’s method is not suitable when the size is too large [Reference Zhou and Zhang52]. On the other hand, QNM is an updated version of Newton’s method where the computation of Jacobian is skipped without affecting the locally super linear convergence characteristics [Reference Zhou and Zhang52]. QNM is based on the gradient method with double differentiation, implying better local search capability and faster convergence characteristic [Reference Fang, Ni and Zeng53]. However, double differentials of antenna array are not possible as it is a simulation-based large-scale model [Reference Fang, Ni and Zeng53]. Hence, a derivative-free QNM is adopted where BGM is used for updating the trust regions to get the best solutions for the problems in hand [Reference Fang, Ni and Zeng53]. The brief overview of BGM-based QNM is discussed below:
Let us consider a set of nonlinear equations: 
$F\left( x \right) = 0, x \epsilon {\ }{\Re ^n}$, where 
$F:{\ }{\Re ^n} \to {\ }{\Re ^n}$ is continuously differentiable and its Jacobian is denoted by (
${J_{{x_k}}}$). Here, 
${x_k}$ is the vector of variables such that 
$F\left( {{x_k}} \right)$ is the best cost value of the objective function at kth iteration. In a conventional Newton’s method, calculation of Hessian matrix and the inverse of it is computationally expensive, especially when the dimensions are large. The Hessian of function 
$F:{\ }{\Re ^n} \to {\ }{\Re ^n}$ is the Jacobian of its gradient. In QNM, the Hessian is approximated by positive definite n × n symmetric matrix 
${B_k}$ that is updated iteratively such that the Hessian approximate (
${B_k}$) of Jacobian (
${J_{{x_k}}}$) must satisfy the quasi-Newton condition (secant equation) [Reference Li, Qi and Roshchina51]:
\begin{equation}{B_{k + 1}}\left( {{x_{k + 1}} - {x_k}} \right) = \nabla F\left( {{x_{k + 1}}{\ }} \right) - \nabla F\left( {{x_k}{\ }} \right){\textrm{ }}\end{equation}where the gradient vector is obtained from the first-order Taylor expansion of 
$\nabla F\left( {{x_{k + 1}}{\ }} \right)$ about 
$\nabla F\left( {{x_k}{\ }} \right)$ computed by finite difference method (FDM). The quasi-Newton condition of eq. 20 can be rewritten more succinctly by letting 
$\nabla F\left( {{x_{k + 1}}{\ }} \right) - \nabla F\left( {{x_k}{\ }} \right) = {y_k}$ and 
$\left( {{x_{k + 1}} - {x_k}} \right) = {s_k}$, so that we have [Reference Zhou and Zhang52]
\begin{equation}{B_{ + 1}}{s_k} = {y_k}{\textrm{ }}\end{equation}For the derivative-free quasi-Newton equation, Broyden has further extended the updating equation, which is known as BGM. The update equation of BGM is as follows [Reference Fang, Ni and Zeng53]:
\begin{equation}{B_{k + 1}} = {B_k} + {{\left( {{y_k} - {B_k}{s_k}} \right)s_k^T} \over {s_k^T{s_k}}}{\textrm{ }}\end{equation}Finally, the derivative-free quasi-Newton equation for unconstrained optimization can be represented by incorporating the Broyden’s method and direct line search method as [Reference Krishna Chaitanya, Raju, Raju and Mallikarjuna Rao54]:
\begin{equation}x_k^{{\textrm{new}}} = {x_k} + {\alpha _k}{d_k}\end{equation}where 
${\alpha _k}$ is the step size chosen to satisfy the Wolfe conditions, which is a set of inequalities used for line search in QNM. Here, 
${d_k} = - {{\bf{H}}_k}{y_k}$ denotes the Broyden-based direction, 
${y_k}$ is the gradient vector, and 
${{\textbf{H}}_k}$ is the Hessian that is approximated by BGM.
The gradient vector 
${y_k}$ in QNM is calculated using FDM, which is based on the idea of replacing derivatives with finite differences. It is not suitable for exploration as it may not be able to approximate the function derivatives [Reference Krishna Chaitanya, Raju, Raju and Mallikarjuna Rao54]. Instead of that, GA can gradually converge to global optima and can be better fitted for the exploration stage to find out the potential trust regions in a multidimensional search space. That means step size will be smaller with the iterations going on. Then FDM can be used to exploit the trust regions in later evolution stage to produce the accurate results. The memetic method can be performed with a step-by-step process starting with the initialization of GA by setting the parameter values and population size for the concerned problem. After evaluating the fitness value of each population agent, crossover and mutation operation is executed iteratively to find the best possible trust regions. Then, QNM is applied to perform the local search within the potential trust regions for a finely tuned optimal solution. The whole process is repeated until the stopping criteria are reached [Reference Krishna Chaitanya, Raju, Raju and Mallikarjuna Rao54]. The flow chart of the QNM-GA is shown in Fig. 3.
Flow chart of QNM-based memetic GA optimization.
Switching sequence obtained with QNM-GA to get the sum–difference beam pattern in a 20-element TMLA for MTR.
Numerical results
This section discusses several applications of TMLAs through distinct examples where QNM-GA method is adopted to obtain the optimized switching schemes for achieving the desired goals. The array is operating at 3 GHz (f 0) with 1 GHz of modulating frequency (fp). The uniformly excited (In = 1) TMLA with a fixed inter-element spacing (d = λ/2) has these properties: −13.2 dB SLL, 6.48° half-power beamwidth (HPBW), 14.4° first-null beamwidth (FNBW), and a directivity of 12.04 dB. The numerical outcomes are simulated with MATLAB. The best-reported parameters of QNM-GA are used for the optimization process and also reported in Table 1 [Reference Krishna Chaitanya, Raju, Raju and Mallikarjuna Rao54]. The proposed approach can also be considered with active elements using active element pattern method through pattern multiplication [Reference Morabito, Di Carlo, Di Donato, Isernia and Sorbello55, Reference Palmeri, Isernia and Morabito56]. But, for mathematical simplicity, only isotropic electromagnetic radiators are considered in this article. Further investigation is always possible by using different active elements with the proposed concept.
Control parameters of QNM-based memetic GA optimization
Monopulse scanned beam patterns
The simultaneous sum–difference beam for monopulse scanning is targeted using a 20-element TMLA. To achieve this target, an in-phase switching scheme for the first 10 radiators and an out-of-phase time scheme for the others with respect to array phase center is implemented. The notion is to design a switching scheme so that the OFF periods of the first 10 elements shall be the ON periods for the other 10. Hence, an in-phase and a reversed-phase tapering may be initiated. As a consequence, sum and difference beam with deep null toward the main beampattern (broadside direction) may be obtained at the same time. The QNM-GA-based switching scheme for the simultaneous monopulse scanned pattern is shown in Fig. 4, and the relative power patterns acquired by the switching sequence are given in Fig. 5. The desirable aim is to produce a sum pattern at the central frequency and a squint difference beam pattern at the first harmonics.
The corresponding sum pattern at central frequency and the squinted difference pattern at first harmonic obtained with QNM-GA-based optimal in and out-of-phase time sequence presented in Fig. 4.
The suggested technique using a 20-element TMLA has achieved an ultra-low SLL0 of −40.89 dB at fundamental frequency. The offset (SBL) in difference beam is calculated as −11.29 dB. The SLL1 of the difference beam is obtained as −22.54 dB. The HPBW and FNBW of the relative power patterns attained at f 0 and (
${f_0} \pm {f_p}$) are 7.53° and 21.59°, and 4.89° and 10.1°, respectively. Out of total dissipated power, 71.8152% is utilized to produce the desired beams and 28.1848% is dissipated in higher sidebands. Thus, an efficient, controlled, and simultaneous monopulse scanned pattern is achieved with the suggested method beneficial for monopulse tracking radars (MTR). The optimized ON times and starting times are reported in Table 2. The results and the comparison with other published results are presented in Tables 3 and 4, respectively.
Optimized ON-time period and starting times for multi-pattern synthesis
Obtained simulation results for multi-pattern synthesis using QNM-GA
Comparison of the proposed work for multiple pattern generation
Note: θ 0, direction of the main beam; θ 1, scan angle for first positive harmonics; θ −1, scan angle for first negative harmonics; and NR, not reported.
Simultaneous scanned beam patterns
The abilities of TMLA are further explored by exploiting the lower sideband patterns for scanning multiple beams. The undesired SR at harmonic frequencies is utilized by suitable time scheme to acquire the scanned beam patterns in some specific directions. The estimated direction of arrival observed for this problem is ±30° from the broadside.
The QNM-GA-based optimal time scheme for ±30° scanned beam patterns are given in Fig. 6, and the corresponding radiation patterns are presented in Fig. 7. The desired objective is to obtain the scanned beam patterns at ±30° from broadside. The first negative (f 0 − fp) and first positive (f0 + fp) sideband beams are optimized to get the scanned harmonic patterns pointed at 60° (–30° shifted) and 120° (+30° shifted) while the fundamental beam at f0 is pointing toward 90°. The central frequency beam has acquired a lowered SLL0 of −33.28 dB. The SBL of first-order sideband beam (|m| = 1) is −1.45 dB. The SLL1 of −23.58 dB is also acquired for the scanning patterns comparing with the highest level (i.e., the SBL). 73.5913% of total power is utilized in the desired beams and 26.4087% is dissipated in harmonics. The radiated power in the fundamental, first positive, and negative sideband beams are calculated as 31.2099%, 21.1907%, and 21.1907% of the total power. Thus, an alternative yet less costly solution for electronic scanning is proposed where the disadvantages of traditional phased arrays are eradicated. The optimized ON times and starting times are presented in Table 2. The results and the comparison with other published results are presented in Tables 3 and 4, respectively.
Switching sequence obtained with QNM-GA to generate scanned beam patterns in a 20-element TMLA for MFRs.
The corresponding scanned beam patterns at the first sideband frequencies (|m| = 1) with ±30° shifting from the broadside direction and the unaltered fundamental pattern, obtained with the QNM-GA-based time scheme presented in Fig. 6.
Shaped flat-top beam patterns
To obtain the shaped flat-top pattern, uniformly excited (In = 1) 16-element TMLA with equally spaced (d = 0.5λ) elements is optimized by QNM-GA, and the optimal switching scheme and the obtained radiation beam patterns are given in Figs. 8 and 9, respectively. The synthesized shaped pattern at the first sideband shows a wideband flat-top beam pattern of more than 30° width. A sharp transition of 7.56° is achieved with SLLs of −20.62 dB and −20.32 dB for the fundamental pattern and first sideband pattern, respectively. The power dissipated in flat-top shaped beam is measured as 21.2770% of total dissipated power, whereas the fundamental pattern used 34.2112% of the radiated power. 76.7652% of radiated power is utilized for the desired synthesized patterns, and 23.2348% of the power is dissipated in higher-order harmonics. The optimized ON times and starting times are presented in Table 2, and the results are shown in Table 3.
Switching sequence obtained with QNM-GA to generate the flat-top beam pattern in a 16-element TMLA for shaped pattern synthesis.
The corresponding shaped beam patterns at the first sideband (m = 1), obtained with the QNM-GA-based time scheme shown in Fig. 8.
Cosecant-squared beam patterns
To obtain the cosecant-squared pattern, uniformly excited (In = 1) 20-element TMLA with equally spaced (d = 0.5λ) elements is optimized by QNM-GA, and the optimal switching scheme and the obtained radiation beam patterns are given in Figs. 10 and 11, respectively. The synthesized cosecant-squared beam pattern at the first sideband as well as the fundamental pattern has also been considered for SLL reduction. The SLLs of −20.17 dB and −18.6 dB is achieved for the fundamental pattern and cosecant-shaped first sideband pattern, respectively. The power dissipated in desired patterns is measured as 84.18% of total dissipated power, whereas the remaining 15.82% is wasted in higher sidebands. The optimized ON times and starting times are presented in Table 2, and the results are shown in Table 3.
Switching sequence obtained with QNM-GA to generate the cosecant-squared beam pattern in a 20-element TMLA for cosecant pattern synthesis.
The corresponding cosecant-squared beam pattern at the first sideband (m = 1), obtained with the QNM-GA-based time scheme shown in Fig. 10.
The optimized ON times and starting times are reported in Table 1. The results and the comparison with other published results for the monopulse and scanned beam patterns are presented in Tables 2 and 3, respectively. The SLLs achieved for all the multi-patterns are relatively smaller than the other reported works. It is observed that the SLL0 of monopulse and steered patterns are reduced to −40.89 dB and −33.12 dB compared to the best-reported results of −40 dB [Reference Yang, Chen and Nie9] and −21.7 dB [Reference Rocca, Zhu, Bekele, Yang and Massa35], respectively. The proposed method also shows improvement in terms of SLL1 as −22.54 dB and −23.58 dB is achieved compared to −20.1 dB [Reference Yang, Chen and Nie9] and −21.6 dB [Reference Rocca, Zhu, Bekele, Yang and Massa35] of the best-reported works. From the comparison, it is clear that multiple patterns can be efficiently achieved with the proposed method. The total radiated power used by the synthesized multi-beams and the wasted power in other harmonic frequencies for all the proposed applications are presented in Fig. 12.
Dissipated power in proposed multi-beams and unused power in higher harmonics obtained with QNM-GA-based switching schemes.
Conclusion
This work has proposed and verified an unorthodox but efficient substitute of traditional antenna arrays for multi-pattern synthesis. Multiple radiation patterns for monopulse and MFRs systems are investigated with fourth-dimensional (4D) antenna arrays, where the radiation patterns of the array are synthesized by controlling the ON times of each element. The monopulse scanning patterns are investigated by creating concurrent sum–difference beams with controlled radiating properties. The “fourth-dimensional” concept is further extended for MFRs where simultaneously scanned and shaped patterns are beneficial. Apart from the narrowband patterns for MFRs, the same concept is also exploited for wideband applications like shaped flat-top and cosecant-squared beam patterns. For all the desired beam patterns, utmost care has been taken to minimize the wanted inter-references. Furthermore, the higher-order harmonics inherently generated due to time-modulation are also suppressed to save power. All the examples are supported with simulated results and also compared with its counterpart results whenever applicable (and available) to show the enormous possibilities toward an alternate solution of phased antenna arrays. The “time-modulated” arrays can be implemented with high-speed switches eliminating costly phase shifters required in traditional phased arrays. The erroneous attenuators can also be discarded with uniform excitations. The dynamic range ratio obtained is 1, which is an added advantage of this work compared to traditional arrays. The SLLs of the monopulse beams are lowered below the ultra-low level (–40 dB). The SLLs of the scanned beams are also suppressed below −30 dB, whereas for wideband patterns, all the SLLs are suppressed below −20 dB. The power wasted in unwanted directions for all the patterns are also suppressed to improve the efficiencies. Thus, a cost-effective alternate to phased arrays for multi-pattern synthesis is proposed in this article, and the idea can be exploited further in near future.
Financial Support
This work is endorsed under the grant of EEQ/2021/000700 provided by SERB-DST.
Competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Avishek Chakraborty passed his B.Tech. degree in Electronics and Communication Engineering. He received his M.Tech. degree in Radio Physics and Electronics with a specialization in Space Science and Microwaves from the University of Calcutta, West Bengal, India, in 2017. He has completed his Ph.D. in Electronics and Communication Engineering from National Institute of Technology (NIT) Durgapur in 2021. He has also successfully completed a DST-SERB sponsored research project by serving as DST-SERB Research Fellow in his Ph.D. tenure. He is presently working as an Assistant Professor at the Department of Electrical, Electronics & Communication Engineering (EECE), GITAM University, Bangalore. He has authored several research papers published in peer-reviewed international journals, conferences, and book chapters. He has also served as a reviewer of IEEE Access, International Journal of RF and Computer-Aided Engineering, Circuits Systems and Signal Processing, The Journal of Supercomputing, etc. His current research interests include antenna array synthesis, phased arrays, antenna array optimization using soft computing and artificial intelligence, and signal processing for antenna arrays and radars.
Ravi Shankar Saxena received his B.E. degree in Electronics and Communication Engineering from Rohilkhand University, India, in 2000. He has completed his M.Tech. from MNNIT Allahabad in 2004, and D.Phil. in Communication Technology from Department of Electronics and Communication, University of Allahabad in 2016. He has published more than 16 research papers in different national and international journals and conference proceedings. His areas of interests include Patch antenna, Experimental nanoscience including Nanostructure fabrication, Optoelectronics properties, and Energy technologies. He is currently working as a Professor in GMRIT, Razam, Andhra Pradesh, India.
Anshoo Verma is working as Assistant Professor in Department of Civil Engineering, IES Institute of Technology and Management at IES University, Bhopal, since last 3 years. He has experience of more than 3 years in academics and research. Dr. Verma has completed his Master of Technology in Construction Technology and Management from Oriental Institute of Science & Technology in 2021.
Ashima Juyal has been working in the Department of Electronics & Communication Engineering at Uttaranchal Institute of Technology, Uttaranchal University, Dehradun, India. She has experience of more than 5 years in research and academics. Ms. Ashima is currently deputed at Research and Innovations Division at Uttaranchal University. Her areas of interests are Internet of Things, fuzzy logics, blockchain, and deep learning.
Sumit Gupta is working as Assistant Professor in the Department of Electronics and Communication at SR University, boasts over 13 years of experience in the field of educational profession and research, having previously served as a faculty member at KL University. He completed Ph.D. in Optical Multiplexing Technique in Fiber Communication from the prestigious National Institute of Technology, Bhopal, completed in 2017. He published his work in various reputed journals. Dr. Gupta currently focuses his research efforts on the cutting-edge fields of quantum communication and free space optical technology, exploring advanced multiplexing techniques.
Indrasen Singh is working as an Assistant Professor in the School of Electronics Engineering, VIT Vellore, Tamil Nadu, India. He received his B.Tech. and M.Tech. Degrees in electronics and communication engineering from Uttar Pradesh Technical University, Lucknow, India, in 2006, and 2010, respectively. He obtained his Ph.D. degree in electronics and communication engineering from National Institute of Technology Kurukshetra, Haryana, India, in 2019. He has more than 12 years of teaching, research experience in various reputed technical institutes or universities. He is the editorial board member of AJECE, Science Publishing Group, USA. He is the reviewer of many international journals and served as TPC member and reviewer in different conferences. He has published more than 20 research papers in national/international journals/conferences of repute and many are under review. His research interests are in the area of cooperative communication, stochastic geometry, modeling of wireless, heterogeneous networks, millimeter wave communications, device-to-device, and 5G/6G communication.
Gopi Ram completed his B.E. degree in Electronics and Telecommunication Engineering from Government Engineering College, Jagdalpur, Chhattisgarh, India, in the year 2007. He received his M.Tech. degree in Telecommunication Engineering from National Institute of Technology Durgapur, West Bengal, India, in the year 2011. He joined as a full-time Institute Research Scholar in the year of 2012 at National Institute of Technology, Durgapur, to carry out research for Ph.D. degree. He received the scholarship from the Ministry of Human Resource and Development (MHRD), Government of India, for the period 2009–2011 (M. Tech) and 2012–2015 (Ph.D). He is currently working as an Assistant Professor at the Department of Electronics and Communication Engineering, National Institute of Technology, Warangal, Telangana, India. His research interests include evolutionary optimization-based radiation pattern synthesis of antenna arrays, time-modulated antenna arrays, and applications of soft computing techniques in electromagnetics. He has published more than 50 research papers in international journals and conferences.
Durbadal Mandal completed his B.E. degree in Electronics and Communication Engineering from Regional Engineering College, Durgapur, West Bengal, India, in the year 1996. He received M.Tech. and Ph.D. degrees from National Institute of Technology, Durgapur, West Bengal, India, in the year 2008 and 2011, respectively. Presently, he is attached with National Institute of Technology Durgapur, West Bengal, India, as Associate Professor in the Department of Electronics and Communication Engineering. He has supervised 16 doctoral students and has completed 2 DST-SERB Sponsored Research Projects as Project Investigator till date. His research interest includes array antenna design; filter optimization via evolutionary computing techniques. He has published more than 350 research papers in international journals and conferences.
