Abstract
The design of an effective sidelobe-suppressed beamformer array antenna using ‘time modulation’ is proposed in this research for beam scanning and anti-jamming applications in radar. Time-modulated array (TMa) is the substitute of traditional phased antenna array, where the impinging signal’s phases are controlled by managing the ON and OFF timings of the array radiators. This research aims an optimal TMa beam scanner, targeting a wide scanned region of 90° in the broadside. The scanning patterns are created using delta potential well-based quantum-optimized timing sequences, and the ON durations of the radiators are optimized to create sidelobe-suppressed radiation properties. The beamformer is developed in an 8-element TMa, and the inclusion of modulated schemes enables simultaneous scanned beams over the desired sector. The scanning patterns with suppressed sidelobes are then managed to place strategic nulls pointing to the jamming signal’s directions. The beamformer is developed to point single nulls and multi-nulls in prespecified sectors for anti-jammer purposes, which shows the efficiencies, adaptiveness, and robustness of the TMa beamformer. The directional optimized anti-jammer method is also investigated in terms of convergence speed. The sidelobe-suppressing beamformer demonstrates decent performance, minimizing the sidelobes of the main beams and scanned beams around − 35 dB and − 25 dB, respectively, for all cases. The uniquely designed objective function (OFN) helps to mitigate the unwanted interferences and suppressed higher-order sidebands below − 20 dB. The proposed beamformer also creates deep nulls around − 75 dB and − 60 dB for single-point and multi-point anti-jamming applications, respectively. The radiation efficiencies are maintained above 70%. The proposed beamformer’s superiorities are also established over the recent works in this domain, as well as with a detailed statistical comparison for the quantum-based optimization algorithm.
1 Introduction
The demand for effective and alternate techniques of designing phased arrays for advanced requirements such as beamforming and steering has escalated due to the rapid advancement in modern antenna engineering. A time-modulated array (TMa) has become a potential option compared to the traditional ones due to the augmentation of ‘Time’ as an added control, along with the previously existing parameters like excitation, distance, and phase [1]. TMa controls the ON and OFF timings of the radiators to create the required phase and amplitude taperings [2]. The modern antenna array usages such as adaptable beamforming, achieved by scanned beams in particular regions, may be addressed using shifted switch ON timings in TMa [3]. The radiating properties of the array can be managed by controlling them electronically rather than old mechanically steered approaches [4, 5]. The TMa also provides an inherent advantage of lowering sidelobes by optimized schemes [6]. Because of the inclusion of ‘Time’ and the translation of time coefficients into the frequency spectrum, multiple sidebands or harmonics are created [7]. The sidebands resemble power wastage in unwanted paths but can also be managed to produce concurrent radiation patterns [8]. The utilization of sidebands was proposed for scanning, and with that, the prospect of concurrent beampattern creation has come up [9, 10]. Early research has observed the beampattern synthesis by reducing the sidelobes [11,12,13]. The design of time framework has been addressed with efficient variable aperture sizes and phase-center-motion-based schemes [11]. The minimum element excitation method [12] and time duration optimization [13] also play important roles in sidelobe reduction. Although the sidelobe reduction is responsible for mitigating the interferences, higher-level harmonics also need to be lowered for efficient transmission and reception in TMa [14,15,16]. In this regard, enhanced sideband-suppressed TMa was proposed [14]. TMa gain improvement [15] and efficiency enhancement [16] have also been addressed by suppressing the harmonics. After the evolution of optimization algorithm, managing the sideband radiations by reducing power in unwanted sectors has become an interesting research prospect [17,18,19,20,21]. The shifted and split time durations in TMa also aided to lower the sideband radiations [22, 23]. The combination of shifting pulse and optimization technique, simultaneous minimization of sidelobe and sideband radiations were reported [23,24,25]. Optimization methods have also aided to lower the higher sideband radiations along with the main beam sidelobes [26,27,28]. In this way, beampattern synthesis with interference rejection has become a challenging prospect for the researchers [29,30,31]. The beampattern synthesis with a flat-top mainlobe and broadened nulls has been reported by combining the transmit and receive weight vectors [29]. The clutter suppression has also been addressed using synthesized subarray beampattern [30]. The constant modulus-based beamforming also helped to achieve interference suppression for radars [31]. Different algorithms had been proposed to optimize the shifted and split pulse timing frameworks for sideband minimization [32,33,34]. On the other hand, using the sideband radiation for creating harmonic patterns also evolved [35]. The creation of concurrent beampatterns apart from the central beam as multi-transmitting array has gained thrust in TMa research [36,37,38]. The moving array phase center proposal has mathematically exploited the harmonics and opened up new possibilities of beam steering in TMa [39,40,41,42]. The automatic beam steering capabilities of TMa made them suitable for radar and direction-of-arrival estimation applications [43, 44]. Monopulse patterns with concurrent sum-difference beams also proposed with TMa allowing a single beamformer to create both beams using harmonics [45, 46]. Further, the idea has been exploited to create several scanned beampatterns having decent scanning coverages for multi-utility radars [47]. To explore the applicabilities of the harmonic beams in radars, broad null placing for interference rejection was investigated [48, 49]. TMa was adopted for more complicated purposes, such as beampattern shaping [50, 51] and for cognitive radios by managing the harmonic radiations [52, 53]. Due to the adaptabilities and reconfigurable time frames, TMa has evolved as an effective and superior substitute to the conventional antennas.
The proposed research further explored the TMa idea by developing optimal timing frameworks to create an efficient sidelobe-suppressed beamforming network (BFN) for radar. The first goal is to develop an adaptive BFN that can point the radiating patterns in predetermined directions and cancel the sidelobes or lower them to reject interferences. The pulse-shift and pulse-split approaches, along with optimized time frameworks, are applied to produce the radar-scanning beams. The second goal is to point a null strategically in the specific direction to block the transmit and receive signals in that direction, creating a no-signal region. The third goal is an expanded version of the second one, where dual nulls are pointed using a single BFN by managing the lower-level sidebands. The goals are framed for real-time radar usage, where automated scanning, null control, sidelobe level (SLL), and sideband level (SBL) reductions have critical roles. The predetermined goals of this research and the proposed techniques to attain them are illustrated as:
First, the TMa beamformer is developed to discard all the interference signals at the sidelobes and generate concurrent scanned patterns in the predetermined directions. The previous research works have independently aimed either beam scanning or SLL suppression with TMa, but not both objectives at the same time. The proposed method aims to achieve both objectives at a time by controlling the first sidebands for scanning along with suppression of SLLs for the main beams and the sidebands. Further, the proposed objective function (OFN) is developed in such a way that the higher-order SBLs other than the first ones are also suppressed to reduce unwanted interferences. These multiple objectives are tackled at once by merging them into a weighted single-objective mathematical model and optimizing the model with a delta potential well-based optimization algorithm. The first aim is to create multi-scanning beams at several directions covering 45° to 135° providing wide-angle scan coverages of 90° toward broadside. To achieve this, the time sequences of the 8-element TMa are optimized such that the optimal start times of each radiator generate the cumulative phase-shifted radiating pattern. The mathematical prospect behind this concept is also derived. The TMa beamformer can suppress the SLLs, and with the beamformer, the SLLs of the scanned beams are targeted under − 20 dB. Through derivation, it is proven that the central pattern consumes the major portion of total power, and due to translation, the first sidebands with negative and positive harmonics have the second-most power in comparison to high-level side beams. The scan coverage is achieved in the beamformer by pointing the first-negative sidebands along 75°, 60°, and 45° and the first-positive sidebands to 105°, 120°, and 135°, covering 30°, 60°, and 90° angular sectors. The beamformer also minimizes the SLLs of the scanned first-order sidebands and the central pattern under the satisfactory levels for radar application. Further, the higher sidebands are also suppressed to minimize the unwanted interferences.
Second, the adaptiveness of the 8-element TMa beamformer is further tested by placing deep nulls strategically in the prespecified jamming angles. The previous research works have addressed broad null placement in TMa by suppressing the SLL to a particular level, but pointing a null at a specific direction for practical anti-jamming application has never been addressed before. In practical jamming scenario, the requirement is to nullify one or multiple signals coming from specific directions. The proposed single-point and multi-point anti-jammers using TMa aim to address the practical jamming scenario and also shows the adaptability of the model capable of pointing the nulls strategically in any specified directions. The strategic deep nulls create a no-signal region in the angular sector, letting interference signals or jammers to be efficiently impeded by concentrating only on the desired signals. To achieve this, the switch ON and OFF instants of the TMa are optimized in the angular radiation region of the array to point the deep nulls at the prespecified directions. The TMa beamformer is developed to point nulls in specific directions, and also in multi-directions, for jammer suppression. The first harmonics are used to point the single nulls in TMa radiating plane. The single-point deep nulls are aimed at 70°, 60°, 50°, and 40° with the first-positive harmonics. Similarly, the first-negative harmonics are aimed to place deep nulls at 110°, 120°, 130°, and 140°. To illustrate it further and to check the adaptability of the TMa, pair of nulls are aimed at 75° and 40°, as well as 105° and 140° using first-negative and first-positive harmonics, respectively, to impede multi-interfering signals. The TMa beamformer also suppresses the SLLs of all the patterns under the satisfactory level. In this process, the central array pattern is always focused in the broadside region ensuring effective power dissipation. To achieve all these objectives, optimized time instants and ON times of the TMa are the crucial parameters for efficient BFN design. So, an effective optimization technique is needed. Particle Swarm Optimizer (PSO) had been popularly adopted to tackle engineering challenges and also employed for antennas and electromagnetics [54]. But, PSO lacks effectiveness when several goals are aimed at the same time [55]. So, a robust quantum-inspired delta potential well-based PSO (quantum DP-PSO) is adopted to aim multiple goals with TMa. The quantum DP-PSO gives efficient outcomes for the BFN, which resembles a prominent solution for the practical radar challenges such as canceling out the sidelobes, the higher sidebands, and pointing strategic nulls for anti-jamming scenario.
The advancements of this research over other contemporary TMa are in multifold: The research addresses the anti-jamming scenarios with single- and multi-point anti-jammer BFNs using TMa. The anti-jammer BFNs are designed and validated for prespecified jamming directions. The strategic deep nulls of more than − 75 dB and − 60 dB are placed using the single- and multi-point anti-jammers that resemble complete interference rejections in the prespecified directions. The BFN is designed also to reduce the SLLs below − 35 dB and − 25 dB for the single- and multi-point anti-jammers, which makes it suitable for practical radar application. The sidelobe-suppressed TMa beamformer is aimed for efficient beam scanning purposes, where a wide scanning coverage of 90° in the broadside direction is tested. The ability to scan any prespecified direction range resembles that the proposed beamscanner can be pointed to any direction by only modifying the switching schemes of the TMa. The SLLs are reduced below − 35 dB and all the higher-order SBLs are suppressed below − 20 dB for interference mitigation and efficiency enhancement. The mathematical prospect of the BFN is established by pulse-shifted switching schemes, and a detailed mathematical derivation for controlling the harmonic sidebands has been established. An unique OFN is formulated considering the necessary SLL and SBL reduction aspect in TMa, and the beamformer switching schemes are optimized, tested, and verified for beam scanning, single-point, and multi-point anti-jamming scenarios. The adoption of a quantum-based metaheuristics produces the much-needed computational superiority over conventional algorithms by balancing between the exploration and exploitation stages of the optimization process. The statistical robustness of the proposed algorithm is also investigated and presented in the ‘Results and Discussion’ section.
The article is decorated as follows: The TMa beamformer design and in-depth mathematical derivations behind the concept are given in Sect. 2, where the fundamental basics of the TMa antenna and the inclusion of time function into the radiating response are discussed. The design of an asymmetric time framework and the effect of its asymmetric nature on radiation response and antenna effectiveness are explained. The quantum DP-PSO technique is discussed in Sect. 3. TMa beamforming performance as sidelobe suppressor, sideband canceler, single-, and multi-pointed anti-jammers is discussed in Sect. 4. The important conclusions and the future research directives are given in Sect. 5.
2 TMa beamformer background
TMa includes the additional control, ‘Time,’ in the beampattern synthesis of antenna arrays and modulates the antenna array response mathematically. Specifically, the translation between time and frequency domain using a timing function gives an added advantage in beampattern synthesis. The mathematical modeling of the \(N\)-element TMa attached to radio-frequency (RF) switches, illustrated in Fig. 1, can be presented as [2]
The TMa beamformer displays an ‘\(N\)’ number of radiators attached to the combiner through the RF switching network. The controlling unit gives the required tapering by altering the ON and OFF instants of the radiators, and added-up element radiations generate the array-radiating pattern. In Eq. (1), \({U}_{n}\left(t\right)\) is the TMa timing function for all the radiators from \(n=1\) to \(N\), \({I}_{n}\) represents the excitation, \({f}_{0}\) is operating frequency, \(\beta\) stands for wave constant, \(d\) is the distance among the radiators, signals impinged from \(\theta\) direction, \(t\) is the time instant inside a modulating period, and \({\text{AF}}\left( {\theta ,t} \right)\) is the overall TMa response after incorporation of timing framework. Because of the time frame periodicity, \({U}_{n}\left(t\right)\) may be modified as [2]
Here, \({a}_{mn}\) denotes the excitation coefficient transformed between frequency and time domain and extended with ‘\(m\)’ number of frequencies for the \(n{\text{th}}\) antenna. These inherently produced frequencies are abbreviated as harmonics, where \(m=0\) gives the main pattern, and \(m=\pm 1, \pm 2, \pm 3, \dots , \pm \infty\) are the harmonic side beams produced at several bidirectional harmonic frequencies.
The \({a}_{mn}\) at the modulating frequency (\({f}_{p}\)) having time period of \({T}_{p}\) (\(=1/{f}_{p}\)) may be expanded as [2]
Replacing \({a}_{mn}\) expression into the antenna response, the TMa behavior is modulated as [2]
The simplified \({\text{AF}}\left( {\theta ,t} \right)\) response for the ‘\(m\)’ harmonic sideband components is expressed as [2]
The TMa creates the central pattern at \(m=0\). The first-negative harmonics at \(m=-1\) and its first-positive counterpart at \(m=+1\) are used in this research as they contain major amount of power apart from the central beampattern. Then, the first side beams are explored for electronic beam scanner and anti-jammer purposes for radars. So, it is essential to lower all the high-level sidebands to save power and minimize the SLLs of the desired beams for effective, interference-less reception and transmission in TMa. The phase-shifted properties are applied with optimal pulse shifting that requires thorough investigation. The mathematical modeling of the transmitting power and timing function in TMa is necessary to design a multi-purpose beamformer for beam scanner and anti-jammer applications.
2.1 Sideband power derivation
The automatically created sidebands at several harmonics are typically labeled as power losses but may also be utilized for specific goals. So, precise mathematical modeling is necessary to check the SBLs of the higher-level harmonics. The closed-form derivations of a simple and a shifted time scheme are illustrated here. The possible timing frameworks are given in Fig. 2a and b, where Fig. 2a presents a simple pulse scheme for two adjacent antennas, and Fig. 2b shows the same pulse scheme with time-shifted properties. The first sequence presents the \(n{\text{th}}\) and \(s{\text{th}}\) neighbor antennas with normalized ON durations \({\xi}_{n}\) and \({\xi}_{s}\), respectively, inside a modulating period \({T}_{p}\) as: \({\xi}_{n}={\tau}_{n}/{T}_{p}\) and \({\xi}_{s}={\tau}_{s}/{T}_{p}\). The \({\tau}_{n}\) and \({\tau}_{s}\) are the actual non-normalized time periods. The second one shows shifted \(n{\text{th}}\) and \(s{\text{th}}\) sequences with normalized start timings \({\delta}_{n}^{1}\) and \({\delta}_{s}^{1}\). The elements are ON up to \({\delta}_{n}^{2}\) and \({\delta}_{s}^{2}\), respectively, within \({T}_{p}\). Figure 2 also shows overlapping region among the neighbor antennas, and its effect on radiation response is discussed throughout this article, accompanied by suitable mathematical derivations.
Taking all the ‘\(m\)’ sideband responses, the array response can be summarized from Eq. (4) as [2]
Here,
The complete power dissipated can be measured as
Here, \({P}_{T}\), \({P}_{0}\), and \(P_{{{\text{SB}}}}\) represent the complete power, the central pattern power, and the side beampattern power, respectively. The transmitting power for ‘\(m\)’ harmonics may be extended from Eq. (8) as
The first part of Eq. (9) provides the radiation in the central pattern, and the next part gives the power consumed by all the sidebands and harmonics, except for the major pattern. Due to the complicated behavior of \({a}_{mn}\) and symmetric properties of the sidebands, as the harmonics are created in pairs, like negative and positive ones, \({|{A}_{m}\left(\theta \right)|}^{2}\) may be presented as
For a contiguous \(n{\text{th}}\) and \(s{\text{th}}\) radiator pairs without shifted pulse, \({|{A}_{m}\left(\theta \right)|}^{2}\) may be generalized as
The translation of ‘\(m\)’ number of sidebands is presented here through \({a}_{mn}\) and \({a}_{ms}\) for the neighbor radiating antennas, where \({I}_{n}\), \({I}_{s}\) are the amplitudes and \({d}_{n}\), \({d}_{s}\) are the element spacings. Replacing \({|{A}_{m}\left(\theta \right)|}^{2}\) expression in Eq. (9), the total power \({P}_{T}\) becomes
Equation (13) represents the complete power in the central pattern and all the bidirectional side beam patterns. The sideband power (\(m\ne 0\)) may be calculated from the above derivation as
To understand the efficacies of multiple timing sequence-based BFNs and to simplify the power calculation in TMa, a closely derived sideband power expression is needed.
2.2 Closed-form sideband power
Considering uniform excitations (\({I}_{n}=1\)) and uniform switching function (\({U}_{n}\left(t\right)=1\)) for the shifted pulse scheme of Fig. 2b, \({a}_{mn}\) can be presented as [2]
The normalized switch ON and OFF times are denoted by \({\delta}_{n}^{1}\left(={\tau}_{n}^{1}{f}_{p}={\tau}_{n}^{1}/{T}_{p}\right)\) and \({\delta}_{n}^{2}\left(={\tau}_{n}^{2}{f}_{p}={\tau}_{n}^{2}/{T}_{p}\right)\). The closely derived side beam power can be streamlined considering the complicated nature of \({a}_{mn}\) and \({a}_{ms}\) and adding them up for the infinite sidebands as
Taking the cosine and sine derivations as even and odd, Eq. (17) can be simplified to
Considering the cosine series expansion for the periodic function, the summation of the cosine series in the above expression can further be simplified as
Putting the above expression in Eq. (18), the summation of excitation coefficients can be modified as
By expanding the above mathematical expressions of (21), Eq. (18) can finally be simplified as
The excitation coefficient expression may be further processed considering all probable combinations of start and end timings of the \({n}^{th}\) and \({s}^{th}\) nearby radiators inside the modulating period of each radiator. Considering the adjacent radiating elements and four switch ON and OFF timings, there are \({4C}_{2}=\left\{4!/2!\left(4-2\right)!\right\}\hspace{0.17em}=\hspace{0.17em}6\) possible combinations. The detailed discussion of different switching combinations is given in Appendix A. Combining all the probable conditions, Eq. (22) can be generalized as
Here, \(\overline{{\xi }_{ns}}\) shows the overlapping time duration among the nearby elements. The closely derived shifted pulse expression of the side beams can now be presented as
The closely derived sideband power approximation can be used to examine the effectiveness of the TMa. The radiating power efficacy may be calculated as the ratio of transmitted power in the desired regions to the overall power dissipated in ‘\(m\)’ harmonics, along with the central pattern and other high-level bidirectional sidebands. After obtaining the radiating power efficacy and the closely derived sideband power for the shifted sequences, the subsequent action is to develop the TMa beamformer capable of generating the shifted time instants for all the antennas to point the beams in some directions or steer to a specified sector.
2.3 Modeling of pulse-shifted instants
To precisely measure the start timings of all antennas for the required phase deviation, the TMa excitation \({a}_{mn}\) can be modeled from Eq. (16) as
The derived expression of \(a_{mn}\) for the shifted timing sequence may affect the radiating response as
The expression in (27) implies that a shifted time-modulated pulse sequence incorporates a special behavior in the overall response, i.e., \(e^{{j\left[ {\left\{ {\beta \left( {n - 1} \right)d\cos \theta } \right\} - \left\{ {m\pi \left( {2\delta_{n}^{1} + \xi_{n} } \right)} \right\}} \right]}}\), which provides a static phased distribution including delays. Thus, a shift in phase may be incorporated by controlling \(\delta_{n}^{1}\) and \(\xi_{n}\) (\(= \delta_{n}^{2} - \delta_{n}^{1}\)) and maintaining the design standards for other antenna parameters. To illustrate the concept and calculate the start instants (\(\delta_{n}^{1}\)) of all the antennas, the phased response behavior is set equal to zero, i.e., \(e^{0} = {1}\). The mathematical derivation offers an additional advantage, as a phase-shifting mechanism is not needed to scan or steer the beams. The start times (\(\delta_{n}^{1}\)) may be measured by solving \(e^{{j\left[ {\left\{ {\beta \left( {n - 1} \right)d\cos \theta } \right\} - \left\{ {m\pi \left( {2\delta_{n}^{1} + \xi_{n} } \right)} \right\}} \right]}} = e^{0} = 1\) that gives a static and uniform phase behavior. The modeling of \(\delta_{n}^{1}\) to scan the harmonics at predetermined \(\theta_{0}\) can now be calculated as
According to Eq. (28), the time shift instants of each radiator may be determined to scan or steer the \(m^{th}\) harmonic patterns toward angle \(\theta_{0}\). This research concentrates on developing a TMa beamformer by exploring the first-level sidebands, as the first-level harmonics consume maximum power next to the central pattern. So, the above derivation can be minimized considering the first-positive and first-negative sidebands (\(m = \pm {1}\)) as
Here, \({\text{mod}}\) stands for the modulo operation included due to the symmetric behavior of the sideband (\(m = \pm 1\)). After modeling the start times to steer the patterns at \(\theta_{0}\), the consequent step is to optimize (\(\xi_{n} = \delta_{n}^{2} - \delta_{n}^{1}\)) or the ON duration of all array elements. Optimizing \(\xi_{n}\) will reduce the SLLs of the beampatterns and to optimize \(\xi_{n}\), a quantum DP-PSO algorithm is adopted. The higher-level sidebands can also be canceled out if the BFN is appropriately designed.
The radiating power efficiency of the beamformer can be measured as the ratio of radiated power for the desired beams to the power emitted by the central beampattern and all the sideband patterns. For all the applications addressed in this research, such as beam scanning with sidelobe cancelation, as well as the single- and multi-point jammers, the fixed central pattern and the first side beam patterns are considered as the desired ones, and all other higher-level side beams or harmonic patterns are regarded as power wasted in unwanted directions. Thus, the power efficiencies can be mathematically presented as
To incorporate the non-ideal switching scheme properties, the switch efficiencies of the proposed beamformer TMa are also considered. The non-ideal switching characteristics do not represent the exact physical switch non-idealities such as insertion loss, leakage times, rise or fall times of practical switches which can only be addressed through hardware implementation of the proposed concept, which is a future research direction of this research. The switching efficiency only accounts for the loss due to the OFF time duration of each switch and measures the efficiency of the beamformer in simulation stage. The switching efficiency of the beamformer is calculated by considering the overall switch ON durations of all the elements in the array. The switch OFF time periods for each element reduce the switching, as well as the overall efficiency of the TMa beamformer for practical implementation scenario. The mathematical representation of switch efficiency can be given as
The overall efficiency of the TMa beamformer can be calculated by multiplying the non-ideal switching factor with the radiating power efficiency as
3 Quantum-based metaheuristics
The quantum-based metaheuristics technique is adopted here to develop the optimal beamformer to cancel out interferences by minimizing the SLL and SBLs. The beamformer is developed especially for scanner and anti-jamming scenarios in radar. So, the minimization of SLLs for all the scanning or steering beams is required. Parameters such as SLLs of the central pattern and the first sidebands, along with all the high-level SBLs, are aimed to be lowered with quantum DP-PSO method. The popular and widely known swarm intelligence-based PSO has long been used for solving real-life engineering challenges and also rigorously employed for array beampattern synthesis [55]. PSO has been adopted for antenna array optimization challenges due to its population-centric, stochastic approach to solve dynamic, nonlinear, and multi-modal optimization problems. The antenna array radiation pattern synthesis deals with optimizing several parameters like excitation control, phase control, switching control, antenna positioning and spacing to improve the sidelobes, beamwidths, and harmonic radiations. These real-time objectives lead to highly nonlinear objective function with multiple local optimal solutions for the optimization process that cannot be solved with classical deterministic gradient-based optimization methods. The swarm intelligence-based PSO algorithm can solve dynamic, nonlinear, non-deterministic, and non-differentiable objective functions as it can stabilize between the exploration and exploitation stages, making it suitable for the electromagnetic optimization problems. Due to this, several researchers have utilized PSO for antenna array optimization. However, PSO lacks effectiveness when several goals are aimed simultaneously. For this purpose, many modifications have also been made in recent years to improve its characteristics, and several variants of PSO have been reported for real-time engineering problems [56]. However, the challenges, such as balance between the exploitation and exploration phases, remain. So, a powerful and more effective quantum-based DP-PSO is adopted to tackle the multiple goals of the beamformer by considering the standard trade-offs of an optimization process.
PSO was first proposed in 1995 by Kennedy and Eberhart [55]. The population-based swarm intelligence technique was inspired from the nature of bird flocking or fish schooling. The population particles having size ‘\(P\)’ generally move along a multi-dimensional search region (\(D\)) for optimum and suboptimum solutions based on a mathematical model. It analyzes the flight paths of all particles to find their velocities and positions. In standard PSO, the position (\(x_{i}\)) and the velocity (\(v_{i}\)) vectors of the \(i{\text{th}}\) particle (\(1 \le i \le P\)) at \(k{\text{th}}\) iteration are expressed as [55]
The velocities and positions of the particles are updated as per their trajectories as follows [55]
Here, \(w\) represents the inertial weight; the constantly modified velocities and positions of the \(i{\text{th}}\) particles for the \(d{\text{th}}\left( {1 \le d \le D} \right)\) dimensions are given as \(v_{i,d} \left( {k + 1} \right)\) and \(x_{i,d} \left( {k + 1} \right)\). The learning constants are \(c_{1}\), \(c_{2}\), \({\text{rand}}_{1}\), \({\text{rand}}_{2}\) are the randomness factors \(\in \left( {0,1} \right)\), \(p{\text{best}}_{i,d} \left( t \right)\) and \(g{\text{best}}_{d} \left( k \right)\) are the personal and group bests after each iteration. All the particles must converge to their best positions if the maximum limit \(\varphi_{k,d}\) is suitably chosen. The best confluence for the particles may be expressed as [56]
Here, \(p_{i,d} \left( k \right) = \frac{{\left\{ {\left( {\varphi_{1,d} *p{\text{best}}_{i,d} \left( k \right)} \right) + \left( {\varphi_{2,d} *g{\text{best}}_{d} \left( k \right)} \right)} \right\}}}{{\left( {\varphi_{1,d} + \varphi_{2,d} } \right)}}\) and \(\varphi_{1,d}\), \(\varphi_{2,d} \in \left( {0,1} \right)\) follow uniform distribution [56].
As per classical mechanics, trajectories of an agent or particle in a search space are established by position (\(\vec{x}\)) velocity (\(\vec{v}\)) and vectors. Particles may move along the trajectories following the Newtonian mechanics, but, as per quantum mechanics, particles cannot have fixed trajectories, as simultaneously finding the \(\vec{x}\) and \(\vec{v}\) is not feasible because of the uncertainty rule [57]. The particles in quantum PSO travel inside a potential field, resembling a confined state behavior. Between several field models in quantum mechanics, the delta potential well is the most simple and fine model that guarantees one confined state having finite energy and a localized wave function, enclosed to a particular region [58]. The quantum nature of a particle traveling in Hilbert space may be presented with a wave function \(\psi \left( {x,k} \right)\). The delta potential well creates a sharp, localized effect, with a single-bound state, because of the typical behavior of a delta function, which permits the existence of only one state. More specifically, the value of the delta function \(\delta \left( x \right)\) is zero at all places apart from \(x = 0\), and the integration over the entire range is unity that can be expressed as [57]
Considering a particle in the one-dimensional region, the ideal form of delta potential can be shown as [57]
Here, a positive coefficient \(\alpha\) shows potential field strength equivalent to depth of the delta potential well, and \(\delta \left( x \right)\) is Dirac delta function, which gives a finite-strength potential with narrow infinite response for \(x = 0\). To attain a bound state within a potential well, the particle’s energy should be less than zero (\(E < 0\)). In a bound state, the particles are enclosed and trapped in the well and cannot escape without the externally supplied energy. Because of the negative energy in the bound state, particles are captive, and the wave function \(\psi \left( {x,t} \right)\) is confined within its potential. The \(\psi \left( x \right)\) decays exponentially away from \(x = 0\) at a particular instant, which resembles that the particles gained highest probability of appearance close to potential well, and the probability reduces when away from the well. The probability density function \(\left| {\psi \left( x \right)} \right|^{2}\) also decays similarly, showing a localized bound state as [57]
The one-dimensional representation of the time-independent Schrödinger equation for a particle with mass \(\tilde{m}\), energy \(E\), and potential \(V\left( x \right)\) can be represented by [57]
Here, \(\hbar\) is the reduced Planck’s constant. Putting the delta potential \(V\left( x \right)\) in the above expression
The above expression can further be simplified as [57]
The delta potential does not affect a general solution at \(x \ne 0\), and the above expression can be given as [57]
The general solution of the above-simplified expression can be given as [57]
In the above equation, the exponential terms resemble negative energies in a bound state. From the expressions of the simplified Schrödinger equation and the general solution at \(x \ne 0\), \(\gamma\) can be derived as \(\gamma = \sqrt { - 2\tilde{m}E/\hbar^{2} }\) so that the energy in the bound state satisfies the condition (\(E < 0)\) as \(E = - \hbar^{2} \gamma^{2} /2\tilde{m}\). As per the Dirac delta function properties, wave function \(\psi \left( x \right)\) is continuous and symmetric at \(x = 0\), which implies \(\psi \left( {0^{ + } } \right) = \psi \left( {0^{ - } } \right) = \psi \left( 0 \right)\) and \(A = B = C\), where \(C\) is a normalization constant. The delta potential well introduces discontinuities in the derivative of the wave function \(\psi \left( x \right)\) at \(x = 0\). For this, the Schrödinger equation must be integrated across an infinitesimally small interval between \(- \varepsilon\) and \(\varepsilon\) as [57]
Due to the continuity of the wave function \(\psi \left( x \right)\) at \(x = 0\), the infinitesimal integral over energy vanishes, and the above expression can be simplified as [57]
To solve Eq. (52), the exponential derivatives can be presented as [57]
Since \(A = B = C\), the solution may be simplified by considering the normalization constant \(C \ne 0\) as [57]
Thus, the energy in the bound state becomes [57]
To ensure that \(\psi \left( x \right)\) is normalized in the quantum potential well, it has to satisfy the condition
Now, from Eq. (55), \(\psi \left( x \right)\) can be represented as [57]
By solving \(\mathop \smallint \limits_{ - \infty }^{\infty } e^{ - 2\gamma \left| x \right|} = 2/2\gamma = 1/\gamma = L\), where \(L\) denotes the characteristic length of the delta potential well. Equating \(\int_{{ - \infty }}^{\infty } {\left| {\psi \left( x \right)} \right|^{2} {\text{d}}x} = \left| C \right|^{2} /\gamma = 1\), the normalization constant can be derived as \(C = \sqrt \gamma = 1/\sqrt L\). Now, the normalized wave function \(\psi \left( x \right)\) can be presented as [57]
The probability density function \(\left| {\psi \left( x \right)} \right|^{2}\) now becomes [57]
Now, considering that PSO agents or particles fall within a quantum well and attracted to the center \(p_{i,d} \left( k \right)\) to converge well, the modified position vector of the \(i{\text{th}}\) particle sampled around a probabilistically distributed \(p_{i,d} \left( k \right)\) may be derived as [58]
Here, \(u_{i,d} \left( k \right)\) gives a uniformly randomized value \(\in \left( {0,1} \right)\), and \(L_{i,d} \left( k \right)\) is the dynamic length of the potential well around the \(i{\text{th}}\) particle’s best mean position in the quantum well, expressed as [58]
Here, \(\tilde{\beta }\) denotes the contraction–expansion parameter. The value of \(\tilde{\beta }\) slowly reduces with time to assure better exploration in the starting stage and excellent exploitation in the final stage of a search operation, expediting an optimized fine-tuned solution. In this regard, DP-PSO manages the wave function collapse in quantum well by making \(\tilde{\beta }\) values adaptive. The collapsing indicates that at the time of measuring a particle’s position, the wave function abruptly falls into a local state or a single eigenstate having delta-like spike focused to a confined measuring point. In quantum well, the collapsing can be established through Copenhagen interpretation, which suggests that a wave function stays in a local state rather being present in an amalgamation of different states bounded by Schrödinger equation. The practical relevance of these phenomena in quantum PSO is the bounded exploration and undesirable premature converging to local minima. Thus, a corrective measure is deployed to stabilize the exploration–exploitation stages by slowly reducing the \(\tilde{\beta }\) values as
where \(\tilde{\beta }\left( k \right)\) represents the decaying \(\tilde{\beta }\) values for \(k{\text{th}}\) iteration, \(\tilde{\beta}_{\max }\) is the maximum value of \(\tilde{\beta }\) that implies more step size at the time of exploring the solution region in the starting phases of optimization, \(\tilde{\beta}_{\min }\) is the minimum value of \(\tilde{\beta }\) that implies less step size exploited for a better convergency in the final stages of optimization operation, and \(c\) is the decay rate to control the adaptive \(\tilde{\beta }\). This adaptive fine tuning eventually tries to evade the collapse in quantum well for DP-PSO. Putting the expression of (57) in (56), the final position updating in DP-PSO can be obtained as [58]
The above equation resembles that, in DP-PSO, only the \(\tilde{\beta }\) values need to be managed, compared to \(w\), \(c_{1}\), and \(c_{2}\) as in standard PSO. The \(m{\text{best}}_{d} \left( k \right)\) term denotes the mean best of the personal best positions of each swarm in the population. The major advantage is that no velocity updating equation is needed to update the particle’s position, which again simplifies the optimization operation by giving quicker outcomes and also avoiding the situation of being trapped in a local minimum. The main advantages of quantum DP-PSO over standard PSO are: In traditional PSO, the movement of the particles in a search space depends upon the velocity and position of the particles. So, it experiences limited search space due to the velocity constraints. But, in the proposed quantum PSO, the particles follow quantum behavior and move probabilistically around the delta potential well. Due to this, the particles are allowed to search much larger area in the solution space which resembles a diverse search behavior and increases the probability of finding a global optimal result. Traditional PSO often suffers from converging into a local optimum because of less diversified search space. On the other hand, the quantum PSO allows particles to explore diverse search space minimizing the chances of getting trapped at local optimum and also avoids the premature convergence. The optimization accuracies are higher in quantum PSO compared to its traditional counterpart due to the algorithm’s capability to find better solutions, attaining lower OFN values (for minimization problems) that gives accurate results. In traditional PSO, many parameters such as weight of inertia (w), cognitive and social learning coefficients (c1, c2), and velocity limits (vmax) need to be tuned to get the near-optimum results. But, in quantum PSO, only the contraction–expansion parameter (\(\tilde{\beta }\)) needs to be managed, which is easier to implement and also increases the robustness of the optimization process due to less trial and errors. Quantum PSO also provides a better exploration–exploitation balance due to quantum probability-distributed search mechanism with contraction–expansion fine tuning in delta potential well. The exploration–exploitation balance improves the convergence and makes it suitable for multimodal and high-dimensional real-life engineering problems. To achieve all these superiorities over traditional PSO, the quantum optimization model may require slightly more time as it searches much larger and diverse solution space for the accurate and optimum solutions, which can be regarded as a trade-off. But these advantages eventually help to achieve the multiple objectives more accurately compared to the traditional PSO.
This research on TMa deals with different goals, such as suppressing the SLLs for the central and side patterns, steering the lowest sidebands effectively such that all higher sidebands are oppressed, and pointing single and multi-nulls in prespecified areas for anti-jammer while preserving the aimed SLL for the desired beampatterns. In this context, the time instants and durations of all radiators in the TMa have crucial purposes, and the mathematical derivation provided in the last section also proves that. To optimize the problem with a quantum DP-PSO technique, all the targeted goals of the proposed research are added into a single OFN as follows:
The first part of the OFN aims to minimize the SLL of the central pattern (\(m = 0\)) and the first sidebands (\(m = \pm 1\)) through iterations (\(k = 1, 2, 3, ..., K\)), where \(\widetilde{{{\text{SLL}}}}\) denotes the coveted level of sidelobe and \({\text{SLL}}\xi_{n}\) gives the SLLs produced after a specific \(k{\text{th}}\) iteration for\(m = 0, \pm 1\). \(H\left( \cdot \right)\) is the Heaviside function to manage the abrupt alterations in SLLs. The second part addresses the oppression of higher sidebands created at infinitely generated harmonic frequencies apart from\(m = 0, \pm 1\), i.e., for all the frequencies from \(m = \pm 2\) to \(\propto\). The parameters \(w_{{{\text{SLL}}}}\) and \(w_{{{\text{SBL}}}}\) are the two weightage constants used to minimize the OFN given in Eq. (60). The significance of these weights lies in their emphasis on suppressing the SLLs and the sideband power or SBLs in the optimization process. The weights are considered as 1 for the optimization problem at hand, as equal weightage has been given to reduce the SLLs for the desired beam patterns and suppress the undesired SBLs. The goal is to target several objectives at a time with a single-OFN optimization framework operating with the quantum DP-PSO.
4 Results and discussion
The research discussed here targets to develop and explore a sidelobe-suppressed beamforming antenna array by incorporating the time modulation idea for beam scanner and anti-jammer radar applications. The detailed mathematical modeling of a TMa beamformer is presented in the last section. The major control aspects of the TMa framework are the normalized switch times (\(\delta_{n}^{1}\)) and the normalized ON durations (\(\xi_{n}\)) of each antenna. The mathematical prospect of the shifted switching times of each element is adapted for the 8-element TMa beamformer, and the time durations of each antenna are optimized with the quantum DP-PSO. The radiating characteristics of the TMa beamformer are investigated over a standard conventional array antenna without time modulation. In a conventional linearly placed array antenna, the amplitude (\(I_{n}\)) and the distance between elements (\(d\)) are generally regarded as uniform, i.e., the elements are always ON (\(I_{n} = {1}\)) and distance among them is same. The distance among nearby elements has a major role in suppressing the mutual coupling and significantly affects the array-radiating performance. The distance lower than a half-wavelength generates non-negligible amount of coupling among the transmitting/receiving signals from each element and perturbs the communication. On the contrary, distance more than a half-wavelength produces harmful grating lobes. So, a half-wavelength element distance is regarded as the standard uniform spacing for a conventional array antenna, and the same is adopted here. The radiation characteristics of a traditional 8-element array antenna have SLL of − 12.80 dB, 13° beamwidth for the half power (HPBW) angle, 28° beamwidth among its first nulls (FNBW), and 9.03 dB of directivity. The simulations are carried out with MATLAB software in a computing machine consisting of a 3.2 GHz AMD RYZEN 7 series CPU with 16 GB RAM, which shows the computing power and memory requirement for the simulation.
4.1 Optimization parameter settings
The simulation configurations for the 8-element TMa under test are considered, with an operating frequency of 3 GHz in the S-band, having uniform excitation and spacing. The equal amplitude (\(I_{n} = {1}\)) streamlines the general BFN as it removes the crucial challenges of dynamic ratio ranges (DRRs) faced in traditional tapering. The uniform half-wavelength distance (\(d = {0}{\text{.5}}\lambda\)) improves the antenna response, constraining the creation of grating lobes and mutual coupling. The propagating constant number (\(\beta\)) is calculated as \({20}\pi\) radian per meter having 0.1m wavelength. The phases of the transmitted and received signals are constants, such that external phase shifters are not required. All the conventional control aspects of the TMa are made as non-variable parameters, so that no attenuating or phase-shifting elements are needed for the TMa beamformer. The only controlling entity is ‘Time,’ and by altering the normalized switch timings (\(\delta_{n}^{1}\)) and the ON durations (\(\xi_{n}\)) of each antenna, desired excitation and phase tapering is generated. The switch ON timings (\(\delta_{n}^{1}\)) can be managed using (29), and a desired beampattern adaptively steered to predefined regions can be created. To taper the normalized ON durations (\(\xi_{n}\)) for excitation tapering, the DP-PSO is applied along with the proposed OFN (\(\Psi \xi_{n}\)). The goal is to produce a set of optimized \(\xi_{n}\) values for \(n = {\text{1 to 8}}\) elements. To measure the adaptability of the quantum DP-PSO, the method has been investigated with several dimension and population sizes, and it has been proved that the technique converges adequately when the population size is more than 20 [58]. So, the population size (\(P\)) is chosen as 30, determined by running the DP-PSO 100 times individually for 500 iterations (\(k\)) over standardized benchmark equations for engineering challenges reported in [58]. These tried-and-tested parameter values are also employed in this work while running the optimization technique to solve the OFN in (60). So, we have a population (\(P\)) of 30 with 8 variables, due to the normalized ON times of 8 antenna elements in the proposed TMa. A populating matrix of size 30 × 8 is used in conjunction with the OFN to obtain an optimum set of time periods that minimize the SLL (first part of the OFN) and SBLs (second part of the OFN). The iteration number (\(k\)) is set to 300 as it is observed in each run of the optimization process that the convergence has been reached within 300 iterations (also shown later in Sect. 4.3). The DP-PSO reduces multiple weighting factors used in standard PSO (i.e., \(w\), \(c_{1}\), and \(c_{2}\)) and includes a single \(\tilde{\beta }\) that entails simpler computing operations. For the proposed minimization challenge, the \(\tilde{\beta }\) values are ranged from 1 to 0.5 to balance the exploration–exploitation stages of the optimization operation. The maximum limit is considered as \(\tilde{\beta}_{\max }\) (1) and the lower limit is considered as \(\tilde{\beta}_{\min }\) (0.5) with a decay rate (\(c\)) of 0.01. These parameter settings are standard to solve real engineering challenges with quantum PSO, as these settings are already validated over standard benchmark equations and also compared against other algorithms based on their average rankings according to Friedman’s statistics reported in [58]. The performance of the proposed 8-element TMa beamformer in attaining different goals is discussed in the following subsections. The parameters used for the optimization process are reported in Table 1.
4.2 Quantum-optimized beam scanner
The quantum-optimized 8-element TMa beam scanner is developed to suppress the interfering SLLs SBLs and produce concurrent scanned patterns at predefined sectors from 45° to 135°, covering a large area of 90°. To develop the beam scanner, the derivation of the shifting pulse schemes is used to determine the ON timings (\(\delta_{n}^{1}\)) of each antenna, and the ON durations (\(\xi_{n}\)) are optimized considering a single OFN designed in Eq. (60). The level of aimed \(\widetilde{{{\text{SLL}}}}\) for the desired beams is fixed to − 20 dB. The optimal parameters for each antenna establish the optimal beam scanner time sequence that eventually creates a cumulative phased shifting to the beampatterns. The scanner is targeted at exploring the first sidebands, while the central one is fixed at 90˚. The scanning patterns are aimed at 105° and 75°, i.e., 15° steered from 90° with the first-positive and first-negative sidebands (\(m = \pm 1\)). To demonstrate the adaptability of the proposed beam scanner, the first-positive and first-negative sidebands are also pointed to 120°, 135°, and 60°, 45°, respectively. The optimized BFNs for different pairs of scanning beampatterns are shown in Fig. 3a, b, and c. The sidelobe-suppressing beampatterns created with the optimized BFNs are shown in Fig. 4a, b, and c. In Fig. 3a, the scanner is designed to scan the first sidebands to 75° and 105°, and the respective scanned beampatterns, given in Fig. 4a, indicate that the goal is attained with the first-negative sideband at 75.01° and the first-positive sideband at 105°. In the same manner, the aimed scan regions pointing at 60° and 120° are obtained by the optimal BFN shown in Fig. 3b. The scanned patterns are pointing at 60.01° and 120°, as depicted in Fig. 4b. The third scan region, at 45° and 135°, is also attained with the optimal BFN given in Fig. 3c, and the respective scan beampatterns are observed in Fig. 4c. The SLLs of the central patterns produced in the scanner BFNs are minimized to − 34.78 dB, which displays a significant enhancement over − 12.80 dB SLL in a uniformly radiating pattern (without optimizing \(\xi_{n}\)). The SLLs of the first sidebands are also suppressed to − 24.50 dB, which gives a decisive performance over the aimed goal of − 20 dB \(\widetilde{{{\text{SLL}}}}\) for interference mitigation. Only the central beam and the first-level sidebands are exploited to produce the desired beams. The trade-off associated with SLL reduction is the widening of beamwidth of the antenna radiation patterns, which adversely affects the directionality of the array. The directivity of the beam scanning array is measured as 7.46 dB compared to 9.03 dB of the uniform array without optimization. Thus, an improvement of − 21.98 dB and − 11.70 dB SLLs has been obtained for the central and scanned patterns having a minor deterioration of 1.57 dB in directivity, which is considerable. The comparison of the beam scanner in terms of SLL, radiating power, and effectiveness is described in Table 2.
To demonstrate that the optimized beam scanner design achieves power-effective scanner performance, all the higher-order SBLs are oppressed. The high-level SBLs are minimized as an outcome of the later part of the designed OFN, and the dissipated radiations in 10 higher SBLs are measured using the closed-form sideband power statement developed in Eq. (24). The minimized power in the higher SBLs, along with the central and first side beampatterns, is shown in Fig. 5a. The figure indicates that the central pattern and all the first side patterns consume the major share of the total radiated power, i.e., 30.115% and 20.923%, respectively. The other higher-order SBLs such as the second, third, and fourth consumes 6.829%, 2.126%, and 1.842% of power, respectively. After that, less than 1% power is consumed by all the higher harmonic patterns. So, the exploitation of the central beam and the first side beampatterns for scanning purpose is also justified as the other higher-order beams are almost minimized to reduce unwanted interference. To demonstrate the nature of the sidebands, all the suppressed higher-order SBLs are also shown in Fig. 5b, which shows that almost all the harmonic SBLs are suppressed below the –20 dB targeted level. The power efficiency, wasted power in higher-level sidebands, non-ideal switching factor, and the overall efficiency of the proposed beamformer are calculated using Eqs. (30), (31), (32) and (33). The power efficiency of the TMa scanner is 71.958%, and the power wasted in higher frequencies is 28.042%. The other properties of the TMa, like HPBWs of the major patterns for all three cases, are measured as 17.28°. The HPBWs for the ± 45°, ± 30°, and ± 15° scanned beams are measured as 24.12°, 19.26°, and 17.28°, respectively. To incorporate the switching characteristics, the switching and overall efficiencies of the TMa beam scanner are measured using Eqs. (32) and (33). The switching efficiency is calculated as 60.957% and the overall array efficiency is 43.864%. The detailed discussion unveils that an effective TMa beam scanner is developed in this research to manage the much-required sidelobe cancelation and scanned beam applications in radars.
4.3 Quantum-optimized single-point jammer
The adaptiveness of the proposed array is further explored for enhanced radar conditions, such as pointing the strategic nulls in jammers’ directions. The concept is to produce a no-signal area in the angular sector, allowing the interference signals to be efficiently hindered in the sector. The TMa BFN of the anti-jammer is developed to point single nulls by steering the first sidebands of the 8-element TMa for jammer signal blocking. The first-positive sidebands are used to point nulls at 70°, 60°, 50°, and 40°. In the same way, the first-negative sidebands are pointed to create nulls at 110°, 120°, 130°, and 140°. First, the optimized BFN scheme for a single-null anti-jammer is developed to create nulls at 140° and 40° with the negative and positive sidebands, respectively. The optimal BFN for this purpose is given in Fig. 6, and the respective radiating patterns attained are presented in Fig. 7a and b. In Fig. 7a, the first-positive sideband is steered, creating a deep null of − 75.3 dB at 40.1°. Similarly, the first-negative sideband creates a single deep null of equal level at 139.9°. Then, a single-null anti-jammer performance is aimed by placing nulls at 130° and 50° with the first-negative and first-positive sidebands. The optimized BFN is shown in Fig. 8, and the respective anti-jammers are given in Fig. 9a and b. The deep nulls of − 73.5 dB are attained at 129.98° and 50.02° with the first-negative and first-positive sidebands, respectively. The next set of single-null scenario is aimed at 120° and 60°. The optimized TMa BFN is shown in Fig. 10, and the anti-jamming beampatterns are presented in Fig. 11a and b. The attained results show that deep nulls of − 75.8 dB are placed at 60.2° and 119.8° with the first-positive and first-negative sidebands. The last directions for the single-null anti-jammers are targeted to place nulls at 110° and 70˚ using the first-negative and first-positive sidebands. The optimal BFN scheme is presented in Fig. 12, and the anti-jammer beampatterns are shown in Fig. 13a and b. The deep nulls of − 75.3 dB magnitude is obtained at the specified targeted directions. The central beampatterns are untouched in the 90° direction. Additionally, the same SLLs, i.e., − 34.78 dB and − 24.99 dB, are achieved for the central and side patterns, respectively. The measured power in the central beampattern is 30.115%, and the first side patterns have 20.923% of power. The power efficiency of the beamformer is measured as 71.958% considering the main beam and the first side beams as the desired patterns using Eq. (39). The non-ideal switching factors are also considered as per Eq. (32), and the switching efficiency is calculated to be 60.957%. Further, the overall efficiency of the beamformer is calculated as 43.864% by multiplying the power efficiency with the non-ideal switching factor as per Eq. (33). The directivity for the single-point jamming application is calculated as 8.01 dB compared to the 9.03 dB directivity of uniform array. Thus, an improvement of − 21.98 dB and − 12.18 dB SLLs has also been obtained for the jamming application with a minor degradation of 1.02 dB in directivity. The proposed TMa performance for jammer signal suppression demonstrates the much-needed adaptiveness and improved reception of the desired signals while blocking the undesired ones.
4.4 Quantum-optimized multi-point jammer
The concept of the TMa anti-jammer is exploited here. To investigate whether the proposed method is adaptable to suppress different interfering signals or not, multiple nulls are pointed at 75°, 40°, and 140°, 105° with the first-negative and first-positive sidebands. The beamformer assures that the SLLs of all beampatterns are lower than the target levels. The optimized beamforming scheme for multi-null pointing is shown in Fig. 14. The created beampatterns for multi-null anti-jammer are given in Fig. 15a and b. The first-negative beampattern in Fig. 15a indicates dual deep nulls of − 58.7 dB pointed at 105.08° and 140.02°, respectively. In the same manner, the first-positive beampattern also gets − 58.7 dB deep nulls pointed at 39.98° and 74.92°, as given in Fig. 15b. The SLLs of the major and first side beampatterns are minimized below − 24.98 and − 22.78 dB, respectively, demonstrating a decisive performance in achieving the aimed goal of below − 20 dB \(\widetilde{{{\text{SLL}}}}\) for interference suppression. The directivity of the TMa for multi-point jamming application is measured as 8.57 dB, which shows a degradation of 0.46 dB from the uniform array directivity of 9.03 dB can be considered as minor trade-off to reduce the SLLs. The TMa multi-null anti-jammer also shows effective power usage with 26.992% and 20.592% power transmitted by the central beam and each of the first sidebands, shown in Fig. 16a. The other higher-order SBLs such as the second, third, fourth, and fifth ones consume 8.910%, 2.583%, 1.598%, and 1.558% of power, respectively. After that, less than 1% power is consumed by all the higher harmonic patterns. So, the exploitation of the main beam and the first side beams for the anti-jamming purpose is also justified as the other higher-order beams are almost minimized to reduce unwanted interference. To demonstrate the nature of the sidebands, all the suppressed higher-order SBLs are also shown in Fig. 16b, which shows that almost all the harmonic SBLs are suppressed below the − 20 dB targeted level. The power efficiency of the beamformer is measured as 68.176% considering the main beam and the first side beams as the desired patterns using Eq. (30). The wasted power in higher frequencies is 31.824%. The non-ideal switching factors are also considered as per Eq. (32), and the switching efficiency is calculated as 69.825%. Further, the overall efficiency of the TMa beamformer is calculated as 47.604% by multiplying the power efficiency with the non-ideal switching characteristics as per Eq. (33).
4.5 Comparisons and accuracy
Throughout the result analysis for the beam scanner and anti-jammer applications, the optimal balance between exploration and exploitation has been established by reducing both the SLLs of the desired patterns and the higher-order SBLs of the harmonics. In this regard, the designing of the OFN and balancing the SLL and SBL suppression play the most important role. A detailed observation has been carried out by changing the weightage constants (\(w_{{{\text{SLL}}}}\) and \(w_{{{\text{SBL}}}}\)) in the range of 0 to 2. The weightage constant \(w_{{{\text{SLL}}}}\) is gradually increased from 0 to 2, while the other weightage function \(w_{{{\text{SBL}}}}\) is gradually decreased from 2 to 0 at the same time to understand the effect. The detailed comparison is presented in Table 3, which shows that the best optimally balanced result is obtained when both the constants are equally weighted (1, 1). The table shows the SLLs for the desired beams i.e., the main beam (\(m = {0}\)) and the first side beam (\(m = {1}\)) along with the higher-order SBLs up to 10th harmonics (\(m = {2}\) to 10) as designed in the OFN in Eq. (60). To show the superior performance of the proposed quantum-inspired DP-PSO algorithm in designing the optimal TMa beamformer, a brief comparison with the conventional PSO and its modified variants, such as improved PSO and adaptive PSO, is presented in Table 4. The comparison is carried out in terms of execution time, optimized minimum OFN values (Min OFN), maximum OFN values (Max OFN), average OFN values of all runs (Mean OFN), and the standard deviations values for all the PSO variants over 300 iterations (\(K\)). The convergence plots for the optimization process involved in single-point jammer and multi-point anti-jamming applications are also presented in Fig. 17a and b. The convergence plots show that the OFN is well-converged for all PSO variants within the iteration range of 300 and the best convergence is achieved with quantum DP-PSO.
To further investigate the algorithmic complexities and the superiorities of the DP-PSO algorithm over others, statistical t test analysis is performed. In this regard, a two-sample t test with equal sample or population size of 30 having unequal variances or standard deviations is implemented. The performance of the proposed DP-PSO with other variants of PSO algorithms and its algorithmic superiority can be tested with this two-sample statistical investigation. The t test and degree of freedom (DoF) can be calculated as
Here, \(\overline{\alpha}_{11}\) and \(\overline{\alpha}_{22}\) are the mean OFN values of the comparing algorithms; \(\sigma_{1}\) and \(\sigma_{2}\) are the unequal standard deviations (\(\sigma_{1}^{2} \ne \sigma_{2}^{2}\)) of the first and second algorithm; and \(n_{A}\), \(n_{B}\) denote the sample or population sizes of the first and second algorithm, respectively. Considering \(n_{A} = n_{B} = {30}\) for the 8-element TMa beamformer, the two-sample t tests are investigated for different DoFs comparing the performance of DP-PSO over all other PSO variants. The comparison of t test values for different DoFs calculated using (61) and (62) is reported in Table 5. The threshold t test values for different DoFs with 99% confidence level are also mentioned to show that the calculated t test values are superior than the standard critical values, which resembles that the first algorithm (i.e., DP-PSO) performs better than its conventional counterparts. The p values are calculated less than the significance level 0.01 (\(\alpha\)), which shows that null hypothesis can be rejected with \({100}\left( {{1} - \alpha } \right)\%\) or 99% confidence. So, it can be concluded that DP-PSO is performing better than other PSO variants with 99% confidence level for all applications.
5 Conclusion
This work focuses the development an effective sideband-suppressed time-modulated array (TMa) beamformer for cutting-edge radar scenarios, exploring the much-required beam scanner and anti-jammer characteristics. Steering a beam toward predetermined directions is a critical characteristic of radar so that the position of the desired signal can be detected properly. The interrelated difficulties, such as the minimization of sidelobe levels (SLLs), sideband levels (SBLs), and strategic null creation while preserving the efficacies of the beamformer, are addressed in this research. The TMa beamforming network (BFN) is developed for a wide-angle scanning coverage and also gives anti-jamming capabilities by pointing deep nulls in the desired prespecified directions. In this regard, these multiple objectives are incorporated into a unique single-objective weighted mathematical model or objective function (OFN). The complete mathematical derivation of switching sequences necessary to achieve the desired goals is proposed. In addition to that, the mathematical modeling of the shifted switching scheme, responsible for a cumulative phase-shifted behavior, is also presented in this research. The OFN is investigated and validated for an 8-element TMa by optimizing its parameters with a quantum-inspired PSO algorithm. The adoption of delta potential well-based PSO (DP-PSO) helps to find the near-optimum solutions for the problems in hand. The proposed TMa achieves good scan coverage of 90˚ in the broadside region. The concurrent patterns of the TMa beamformer also show decent sidelobe-suppressed properties lower to − 25 dB SLLs for all the scenarios. The SLLs of the harmonic beams are also suppressed along with the central pattern, which implies a simple BFN that can control both the harmonics and the desired beams at the same time. The scanned patterns show that the first-negative and positive sidebands are explored to scan different directions, such as 135°, 120°, 105°, 75°, 60°, and 45°. The broad scanned sector between 135° and 45° with SLLs lowered to − 34.78 and − 24.50 dB for interfering signal rejection useful for scan radars. Further, the concept is explored pointing the single nulls in the direction of jammers, and the outcomes show that the anti-jamming single nulls are placed at 40°, 50°, 60°, 70°, 110°, 120°, 130°, and 140° along with SLLs suppression. The TMa is extended for a multi-null anti-jammer, where dual nulls at 40°, 75°, and 105°, 140° is pointed by minimizing the SLLs below − 25 dB. The power efficacies of all the proposed scenarios show that more than 70% power is utilized for scanning and anti-jamming. The outcomes are also compared with contemporary works to show its superior performance. The detailed statistical comparisons are also provided to prove the algorithmic superiorities of the DP-PSO. Overall, a unique mathematical antenna array model is proposed and validated for beam scanner and anti-jammer applications that can be further extended with practical phased radar implementations.
Data availability
No new data are produced or analyzed during this research. Data sharing is not applicable to this research.
References
Maneiro-Catoira R, Brégains J, García-Naya J, Castedo L (2017) Time modulated arrays: from their origin to their utilization in wireless communication systems. Sensors (Basel) 17:590. https://doi.org/10.3390/s17030590
Rocca P, Yang F, Poli L, Yang S (2019) Time-modulated array antennas – theory, techniques, and applications. J Electromagn Waves Appl 33:1503–1531. https://doi.org/10.1080/09205071.2019.1627251
Tong Y, Tennant A (2012) A two-channel time modulated linear array with adaptive beamforming. IEEE Trans Antennas Propagat 60:141–147. https://doi.org/10.1109/TAP.2011.2167936
Li H, Chen Y, Yang S (2021) Design and analysis of an amplitude-phase weighting module for harmonic beamforming in time-modulated antenna arrays. AEU Int J Electron Commun 138:153835. https://doi.org/10.1016/j.aeue.2021.153835
Chakraborty A, Mandal D (2025) Mathematical modelling of harmonics-generated side beam power radiation in an unconventional phased antenna array with time-modulation. AEU Int J Electron Commun 200:155936. https://doi.org/10.1016/j.aeue.2025.155936
Yang S, Gan YB, Qing A, Tan PK (2005) Design of a uniform amplitude time modulated linear array with optimized time sequences. IEEE Trans Antennas Propag 53:2337–2339. https://doi.org/10.1109/TAP.2005.850765
Poli L, Rocca P, Oliveri G, Massa A (2011) Harmonic beamforming in time-modulated linear arrays. IEEE Trans Antennas Propag 59:2538–2545. https://doi.org/10.1109/TAP.2011.2152323
Bregains JC, Fondevila-Gomez J, Franceschetti G, Ares F (2008) Signal radiation and power losses of time-modulated arrays. IEEE Trans Antennas Propag 56:1799–1804. https://doi.org/10.1109/TAP.2008.923345
Shanks H (1961) A new technique for electronic scanning. IRE Trans Antennas Propag 9:162–166. https://doi.org/10.1109/TAP.1961.1144965
Li G, Yang S, Chen Y, Nie ZP (2009) A novel electronic beam steering technique in time modulated antenna array. PIER 97:391–405. https://doi.org/10.2528/PIER09072602
Yang S, Gan YB, Tan PK (2004) Comparative study of low sidelobe time modulated linear arrays with different time schemes. J Electromagn Waves Appl 18:1443–1458. https://doi.org/10.1163/1569393042954910
Ma Y, Miao C, Li Y, Wu W (2020) Time-modulated sparse linear array synthesis with minimum elements and a specific sidelobe. IET Microw Antennas Propag 14:1595–1598. https://doi.org/10.1049/iet-map.2020.0459
Poddar S, Paul P, Chakraborty A et al (2022) Design optimization of linear arrays and time-modulated antenna arrays using meta-heuristics approach. Int J Numer Model 35:e3010. https://doi.org/10.1002/jnm.3010
Maneiro-Catoira R, Bregains JC, Garcia-Naya JA, Castedo L (2017) Enhanced time-modulated arrays for harmonic beamforming. IEEE J Sel Top Signal Process 11:259–270. https://doi.org/10.1109/JSTSP.2016.2627178
Zhu Q, Yang S, Yao R, Nie Z (2012) Gain improvement in time-modulated linear arrays using SPDT switches. Antennas Wirel Propag Lett 11:994–997. https://doi.org/10.1109/LAWP.2012.2213292
Chen J, Liang X, He C et al (2017) Efficiency improvement of time modulated array with reconfigurable power divider/combiner. IEEE Trans Antennas Propag 65:4027–4037. https://doi.org/10.1109/TAP.2017.2712811
Yang S, Gan YB, Qing A (2002) Sideband suppression in time-modulated linear arrays by the differential evolution algorithm. Antennas Wirel Propag Lett 1:173–175. https://doi.org/10.1109/LAWP.2002.807789
Aksoy E, Afacan E (2014) A comparative study on sideband optimization in time-modulated arrays. Int J Antennas Propag 2014:1–14. https://doi.org/10.1155/2014/290737
Poli L, Rocca P, Manica L, Massa A (2010) Handling sideband radiations in time-modulated arrays through particle swarm optimization. IEEE Trans Antennas Propag 58:1408–1411. https://doi.org/10.1109/TAP.2010.2041165
Poli L, Rocca P, Manica L, Massa A (2010) Time modulated planar arrays—analysis and optimisation of the sideband radiations. IET Microw Antennas Propag 4:1165–1171. https://doi.org/10.1049/iet-map.2009.0379
Poli L, Rocca P, Massa A (2012) Sideband radiation reduction exploiting pattern multiplication in directive time-modulated linear arrays. IET Microw Antennas Propag 6:214–222. https://doi.org/10.1049/iet-map.2011.0159
Poli L, Rocca P, Manica L, Massa A (2010) Pattern synthesis in time-modulated linear arrays through pulse shifting. IET Microw Antennas Propag 4:1157–1164. https://doi.org/10.1049/iet-map.2009.0042
Poli L, Moriyama T, Rocca P (2014) Pulse splitting for harmonic beamforming in time-modulated linear arrays. Int J Antennas Propag. https://doi.org/10.1155/2014/797590
Zhu Q, Yang S, Yao R et al (2013) Unified time- and frequency-domain study on time-modulated arrays. IEEE Trans Antennas Propag 61:3069–3076. https://doi.org/10.1109/TAP.2013.2253538
Tan J, Hu J, Dong X et al (2021) Research on pattern synthesis of time modulated sparse array based on discrete variable convex optimization. Wirel Commun Mob Comput. https://doi.org/10.1155/2021/6622168
Chakraborty A, Ram G, Mandal D (2020) Optimal pulse shifting in timed antenna array for simultaneous reduction of sidelobe and sideband level. IEEE Access 8:131063–131075. https://doi.org/10.1109/ACCESS.2020.3010047
Chakraborty A, Ram G, Mandal D (2022) Time-modulated linear array synthesis with optimal time schemes for the simultaneous suppression of sidelobe and sidebands. Int J Microw Wirel Technol 14:768–780. https://doi.org/10.1017/S175907872100088X
Chakraborty A, Ram G, Mandal D (2021) Pattern synthesis of timed antenna array with the exploitation and suppression of harmonic radiation. Int J Commun 34:e4727. https://doi.org/10.1002/dac.4727
Lan L, Liao G, Xu J et al (2023) Beampattern synthesis based on novel receive delay array for mainlobe interference mitigation. IEEE Trans Antennas Propag 71:4470–4485. https://doi.org/10.1109/TAP.2023.3247916
Wan F, Xu J, Xu Y et al (2025) Clutter suppression for STAP-based radar using synthesized subarray beampattern. IEEE Trans Aerosp Electron Syst 61:4507–4525. https://doi.org/10.1109/TAES.2024.3506506
Liang J, Wang T, Liu W et al (2024) Constant modulus waveform estimation and interference suppression via two-stage fractional program-based beamforming. IEEE Trans Signal Process 72:2348–2363. https://doi.org/10.1109/TSP.2024.3392363
Aksoy E, Afacan E (2010) Thinned nonuniform amplitude time-modulated linear arrays. Antennas Wirel Propag Lett 9:514–517. https://doi.org/10.1109/LAWP.2010.2051312
Zhu Q, Yang S, Zheng L, Nie Z (2012) Design of a low sidelobe time modulated linear array with uniform amplitude and sub-sectional optimized time steps. IEEE Trans Antennas Propag 60:4436–4439. https://doi.org/10.1109/TAP.2012.2207082
Chakraborty A, Singh I, Gupta S et al (2023) Sideband power control in time-modulated antenna arrays for bidirectional harmonic beamforming and beam scanning. AEU Int J Electron Commun 170:154788. https://doi.org/10.1016/j.aeue.2023.154788
Chakraborty A, Ram G, Mandal D (2021) Multibeam steered pattern synthesis in time-modulated antenna array with controlled harmonic radiation. Int J RF Microw Comput Aided Eng. https://doi.org/10.1002/mmce.22597
Chakraborty A, Ram G, Mandal D (2021) Time-modulated multibeam steered antenna array synthesis with optimally designed switching sequence. Int J Commun 34:e4828. https://doi.org/10.1002/dac.4828
Gassab O, Azrar A, Dahimene A et al (2020) Efficient electronic beam steering method in time modulated linear arrays. IET Microw Antennas Propag 14:402–408. https://doi.org/10.1049/iet-map.2019.0673
Ram G (2021) Multi-beam steered harmonic pattern synthesis in timed antenna array with optimized and pre-defined RF switching. Int J Numer Model 34:e2912. https://doi.org/10.1002/jnm.2912
Chakraborty A, Ram G, Mandal D (2022) Power pattern synthesis of a moving phase center time modulated antenna array using symmetrically and asymmetrically positioned time schemes. Int J RF Mic Comp-Aid Eng. https://doi.org/10.1002/mmce.23442
Chakraborty A, Ram G, Mandal D (2023) Phase center motion based time modulated arrays with preprocessed time schemes for selective harmonic beamforming in B5G communication systems. Trans Emerging Telecommun Technol 34:e4754. https://doi.org/10.1002/ett.4754
Yang S, Gan Y-B, Tan PK (2005) Linear antenna arrays with bidirectional phase center motion. IEEE Trans Antennas Propagat 53:1829–1835. https://doi.org/10.1109/TAP.2005.846754
He C, Yi G, Chen J et al (2019) A novel radar based on two-element time-modulated array. IEEE Geosci Remote Sensing Lett 16:524–528. https://doi.org/10.1109/LGRS.2018.2877973
Li G, Yang S, Nie Z (2010) Direction of arrival estimation in time modulated linear arrays with unidirectional phase center motion. IEEE Trans Antennas Propagat 58:1105–1111. https://doi.org/10.1109/TAP.2010.2041313
Rocca P, Manica L, Poli L, Massa A (2009) Synthesis of compromise sum-difference arrays through time-modulation. IET Radar Sonar Navig 3:630–637. https://doi.org/10.1049/iet-rsn.2009.0058
Rocca P, Poli L, Oliveri G, Massa A (2011) Synthesis of sub-arrayed time modulated linear arrays through a multi-stage approach. IEEE Trans Antennas Propagat 59:3246–3254. https://doi.org/10.1109/TAP.2011.2161535
Chakraborty A, Ram G, Mandal D (2023) Time modulation based unconventional phased antenna arrays for monopulse and multifunction radar systems. Int J Numer Model 36:e3074. https://doi.org/10.1002/jnm.3074
Poli L, Rocca P, Oliveri G, Massa A (2011) Adaptive nulling in time-modulated linear arrays with minimum power losses. IET Microw Antennas Propag 5:157–166. https://doi.org/10.1049/iet-map.2010.0015
Chakraborty A, Mishra A, Singh I et al (2024) Optimal design of smart antenna arrays for beamforming, direction finding, and null placement using the soft computing method. Int J Numer Model 37:e3302. https://doi.org/10.1002/jnm.3302
Chakraborty A, Saxena RS, Verma A et al (2024) Multi-pattern synthesis in fourth-dimensional antenna arrays using BGM-based quasi-Newton memetic optimization method. Int J Microw Wirel Technol 16:284–294. https://doi.org/10.1017/S1759078723000910
Chakraborty A, Singh I, Bhattacharya S et al (2024) Multiple pattern synthesis in four-dimensional antenna arrays using the quasi-Newton memetic optimization method. IETE J Res 70:6055–6068. https://doi.org/10.1080/03772063.2023.2294867
Li G, Yang S, Huang M, Nie Z (2010) Shaped patterns synthesis in time-modulated antenna arrays with static uniform amplitude and phase excitations. Front Electr Electron Eng China 5:179–184. https://doi.org/10.1007/s11460-010-0005-2
Rocca P, Zhu Q, Bekele ET et al (2014) 4-D arrays as enabling technology for cognitive radio systems. IEEE Trans Antennas Propag 62:1102–1116. https://doi.org/10.1109/TAP.2013.2288109
Bhattacharya R, Saha S, Bhattacharyya TK (2017) Mutated IWO optimized 4-D array for femtocell cognitive radio. Antennas Wirel Propag Lett 16:2614–2617. https://doi.org/10.1109/LAWP.2017.2735999
Robinson J, Rahmat-Samii Y (2004) Particle swarm optimization in electromagnetics. IEEE Trans Antennas Propag 52:397–407. https://doi.org/10.1109/TAP.2004.823969
Eberhart, Shi Y (2001) Particle swarm optimization: developments, applications and resources. In: Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546). IEEE, Seoul, Korea (South), vol. 1, pp 81–86
Shami TM, El-Saleh AA, Alswaitti M et al (2022) Particle swarm optimization: a comprehensive survey. IEEE Access 10:10031–10061. https://doi.org/10.1109/ACCESS.2022.3142859
Sun J, Feng B, Xu W (2004) Particle swarm optimization with particles having quantum behavior. In: Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753). IEEE, Portland, OR, USA, pp 325–331
Agrawal RK, Kaur B, Agarwal P (2021) Quantum inspired particle swarm optimization with guided exploration for function optimization. Appl Soft Comput 102:107122. https://doi.org/10.1016/j.asoc.2021.107122
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Avishek Chakraborty helped in drafting, investigation, preparing main manuscript. Ravi Shankar Saxena and Indrasen Singh contributed to simulation and resources. Jatinder Kaur and Manjula M helped in mathematical model and vision. Johar MGM and Gaganjot Kaur helped in editing and supervision. Shimpee Seema prepared illustrations and figures.
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Appendix A
Appendix A
The detailed derivation of Eq. (23) combining all possible asymmetric switching scheme combinations is discussed here: The first possible combination can be designed in such a way that the first element of the two consecutive elements is switched ON first and remains in the ON state for the stipulated time period. After the first element is switched OFF, the second element will be turned ON. The probable combinations and their mathematical derivations to simplify Eq. (22) are derived as follows:
Combination 1: \(\delta_{n}^{2}> \delta_{n}^{1} \ge \delta_{s}^{2}> \delta_{s}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = (\delta_{n}^{1} - \delta_{s}^{1} )\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{n}^{1} - \delta_{s}^{2} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{n}^{2} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{n}^{2} - \delta_{s}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combination 2: \(\delta_{n}^{2} \ge \delta_{s}^{2}> \delta_{s}^{1} \ge \delta_{n}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = \left( {\delta_{s}^{1} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{n}^{2} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{n}^{2} - \delta_{s}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combination 3: \(\delta_{n}^{2}> \delta_{s}^{2}> \delta_{n}^{1}> \delta_{s}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = \left( {\delta_{n}^{1} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{n}^{2} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{n}^{2} - \delta_{s}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combination 4: \(\delta_{s}^{2}> \delta_{n}^{2}> \delta_{s}^{1}> \delta_{n}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = \left( {\delta_{s}^{1} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{n}^{2} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combination 5: \(\delta_{s}^{2} \ge \delta_{n}^{2}> \delta_{n}^{1} \ge \delta_{s}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = \left( {\delta_{n}^{1} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{n}^{2} - \delta_{s}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combination 6: \(\delta_{s}^{2}> \delta_{s}^{1} \ge \delta_{n}^{2}> \delta_{n}^{1}\) which resembles \(|\delta_{n}^{1} - \delta_{s}^{1} | = \left( {\delta_{s}^{1} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{1} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{1} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{1} | = \left( {\delta_{s}^{1} - \delta_{n}^{2} } \right)\), \(|\delta_{n}^{2} - \delta_{s}^{2} | = \left( {\delta_{s}^{2} - \delta_{n}^{2} } \right)\) and the simplified expression of Eq. (22) becomes
Combining all the possible conditions, Eq. (22) can be simplified as
Here, \(\overline{{\xi_{ns} }}\) shows the overlapping time duration between the adjacent elements.
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Chakraborty, A., Saxena, R.S., Seema, S. et al. Delta potential well-based quantum-optimized time-modulated beamformer for scanning and jamming mitigation in radars. J Supercomput 82, 713 (2026). https://doi.org/10.1007/s11227-026-08831-9
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DOI: https://doi.org/10.1007/s11227-026-08831-9