The cosecant-squared (csc²) radiation pattern is commonly used in both airborne and ground-based radars to maintain constant echo signal power from targets, thus ensuring that the level of reflected energy is not dependent on the target altitude [1–4]. The desired csc² pattern, along with a specified sidelobe level (SLL) and small deviations (ripples), can be achieved in the shaped beam region by controlling the excitation amplitude and phase weights of array elements [4]. However, varying the amplitudes of array elements requires a wide dynamic range, which in turn necessitates a complicated feeding network of attenuators.

To avoid the problems associated with complicated feeding networks, time modulation was introduced [5]. In time-modulated antenna arrays (TMAAs), time is considered an alternative to amplitude for producing radiation patterns, which are generated simply through periodic off-switching of the array elements in a pre-defined sequence [6]. Therefore, the shaped beampattern is produced by time-modulation phase-only synthesis [7]. In this way, the csc² radiation pattern at the fundamental frequency can be controlled to achieve low SLL and low ripples in the csc²-shaped region without the need for complex and expensive attenuators. Moreover, switch-on time can also be controlled electronically using radio-frequency (RF) switches. Overall, time modulation is easier and more accurate to implement csc² radiation pattern in real-time operations. Consequently, this approach is promising for achieving multiple goals—SLL reduction, sideband level (SBL) suppression, and beam shaping—solely through time and phase modulation.

However, periodic switching of array elements produces an infinite number of harmonics or sidebands at multiples of the time modulation frequency. This causes a loss of power, which reduces the radiation efficiency at the fundamental frequency. These harmonics may also interfere with other communication systems [8]. Notably, several evolutionary optimization algorithms have been proposed to minimize SBLs and, in turn, sideband radiation. These include genetic algorithm (GA) [9, 10], differential evolution (DE) [11], particle swarm optimization 12], simulated annealing [13], and artificial bee colony [14]. These optimization algorithms have also been employed to mitigate sideband radiation in approaches such as non-uniform element spacing [15], pulse shifting [16], and pulse splitting [17]. Furthermore, in [18], a hybrid approach combining the DE algorithm and convex programming was proposed for the synthesis of shaped beams along with simultaneous lowering of SLL of the fundamental pattern and SBLs of sidebands using time and phase modulation. Nonetheless, all the abovementioned methods rely on iterative processes that require repeated array pattern computations to quantify the mismatch between the synthesized patterns and the desired ones.

In [19], an artificial neural network (ANN) was developed to synthesize shaped beampatterns along with simultaneous reduction in the SLL and SBLs using time and phase modulation. However, since this ANN was data-driven [20–22], it required tens of thousands of training data pairs to capture the nonlinear input–output relationship. Moreover, the training data had to be generated either through numerical simulation or full-wave electromagnetic simulation, which is impractical when a large amount of training data is required. The collection of sufficient training data to cover the input/output parameter range is also an extremely resource-intensive and time-consuming task. Moreover, a large amount of data makes the training itself both difficult and time-consuming.

In this paper, a physics-informed deep neural network (PIDNN) is implemented to realize a csc²-shaped beampattern along with simultaneous reduction in the SLL of the fundamental pattern and SBLs of the harmonic patterns, achieved using only phase tapering and time modulation. Notably, the physics-informed neural networks introduced by Raissi et al. [23] successfully integrated physical knowledge into neural networks. Drawing on this, the PIDNN employed in this study uses the deviation between the desired and actual radiation patterns as the physically driven loss function to train the network parameters and identify the element switching sequence and phases coinciding with the desired radiation pattern. The most significant advantage of this procedure is that the training process does not require any data (indirect training), thus doing away with the burden of collecting a large database of input–output pairs.

To demonstrate the proposed technique, the antenna array is structured as a linear array. The spacing between the antenna elements is assumed to be half the free-space wavelength of the operating frequency. The unknowns in the problem investigated in this study are switching-off time (assuming that the on-time for all elements is zero) and the phase for each element, which means that the number of unknowns is two times the number of array elements. These unknowns need to be optimized to obtain the fundamental csc² pattern with specific SLL and SBLs. This multi-objective optimization is conducted using a PIDNN. The multiple objectives are achieved using a radiation mask based on the specified SLL, SBL, and csc²-shaped region. The variables are optimized by minimizing the error (loss) between the mask and the fundamental pattern, as well as the first positive sideband pattern, which represents the physically informed loss function of the PIDNN. In this context, it should be noted that a PIDNN was utilized in [24] to achieve a low sidelobe TMLA with harmonic suppression using solely time modulation, with the first and second positive sidebands involved in the optimization process. However, in this paper, the PIDNN is used to realize a csc²-shaped beampattern using both time and phase modulation, with only the first positive sideband being involved in the optimization. It is found that minimizing only the first positive sideband is enough to ensure that all remaining sidebands are also suppressed, even when they are not included in the optimization process. The numerical results reveal that the proposed approach outperforms previously reported optimization approaches in terms of SLL and maximum SBL, thus validating its effectiveness. Furthermore, the PIDNN is compared to GA, which is taken as an example of evolutionary optimization algorithms, revealing that the former is significantly more computationally efficient than the latter.

Assuming that a time-modulated linear array (TMLA) composed of N isotropic elements with no mutual coupling is placed along the z-axis, and the amplitude and spacing between the elements is uniform, the array factor (AF) of the TMLA can be expressed as follows:

where t is the time variable and β =2π/λ refers to the propagation constant, with λ being the free-space wavelength at the fundamental frequency f₀. Furthermore, d denotes the spacing between the array elements. The switching modulation function of the nth element is expressed as U_n(t), n = 1, 2, …, N, while θ is the elevation angle of the array and α_n is the phase of the nth array element.

Since the modulation function is periodic with respect to the time modulation frequency f_p (where f_p ≪ f₀), it can be expressed in terms of a Fourier series as follows:

where a_mn is the complex Fourier coefficient of the nth element at m harmonic mode (m=0,±1,±2,…,±∞), with m = 0 being the fundamental frequency and the remaining values of m denoting the harmonic frequencies resulting from time modulation. In this context, a_mn can be formulated as follows:

where T_p is the time modulation period, expressed as T_p = 1/f_p. By combining Eqs. (1) and (2), the AF of the TMLA can be derived as follows:

Along these lines, the AF for the mth order harmonic radiation can be simplified into the following equation:

where m=±1, ±2,…,±∞, with the fundamental radiation pattern occurring at m = 0. In this study, the byproduct harmonics represent the power loss in unintended directions that should be suppressed as much as possible.

Fig. 1 illustrates the switching scheme of a time pulse. The y-axis represents the switching modulation function for nth element U_n(t), while the x-axis indicates the time normalized to the modulation period τ =t/T_p. Notably, since the on-time for all elements begins at 0, the switching-on and switching-off instants of the time pulse are 0 and τ _n, respectively, where τ_n denotes the OFF time of the nth element. Therefore, the modulation function for the time pulse within the modulation period can be expressed as follows:

Furthermore, the corresponding Fourier coefficient can be derived as follows:

Initially, the phase and switching-off times for each element were assumed randomly. Subsequently, these variables were optimized to achieve the required radiation pattern for the fundamental harmonic and to suppress the higher-order harmonics. In the present study, a 30-element TMLA with uniform amplitude and spacing is employed. As a result, 30 switching-off instants and 30 phases had to be optimized. For the optimization process, the inter-element spacing d was set to 0.5λ, while the fundamental frequency f₀ was 3 GHz and the time modulation frequency f_p was 1 MHz. The process began with an initial set of element phases ranging from −π to +π, which were allowed to vary continuously within this range. Fig. 2 depicts the initial phases of the array elements (before optimization). The initial switching-off instants for all elements were also assumed randomly within the normalized modulation period (from 0 to 1), as shown in Fig. 3.

Fig. 4 illustrates the initial fundamental AF (m = 0) and the first positive harmonic sideband AF (m = 1). Meanwhile, Fig. 5 presents the peak values of the different sideband AFs normalized to the peak of the fundamental AF. It is observed that higher-order harmonic AFs are substantially lower than the first harmonic AF. As a result, suppression of the first harmonic AF is expected to reduce all other harmonic AFs. Notably, this property serves to simplify the optimization process while also reducing required costs by requiring the suppression of only the first harmonic AF.

The loss function for the radiation AF comprises three main parts. The first part is the csc² pattern, the second is the specified SLL for the fundamental harmonic, and the third is the desired level of the SBL. Fig. 6 presents a combination of these three parts. The csc²-shaped region has an angular span of θ_min ≤ θ ≤ θ_max. The required SLL is −40 dB within the ranges −90° ≤ θ ≤ θ_o and θ_max < θ ≤ 90°, while the range θ_o < θ < θ_min indicates a transition region. Accordingly, the csc² mask function can be defined in dB as follows:

To achieve the required fundamental pattern, along with the simultaneous suppression of sideband patterns, the switching-off instants of the pulses and phases for all elements had to be determined through optimization (minimization) based on a properly defined loss function that covers the design specifications. The loss function can be calculated as the total error between the mask and the fundamental pattern, as well as between the mask and the first positive sideband pattern. Therefore, the total error consists of two parts. The first part e₀ pertains to the error between the AF of the fundamental pattern AF₀ and the required Mask, given by:

where w₁ and w₂ are the weighting factors of the SLL and the csc²-shaped region, respectively. The Δ_SLL can be defined as follows:

Here, the error criterion of the SLL is obtained using the following equation:

Notably, the sign function sgn(·) indicates that the error is expressed in terms of the “don’t exceed” criterion. This means that only the pattern points of the AFs greater than the SLL of the mask contribute to the SLL error. On the other hand, the loss caused by the error due to deviations (ripples) in the csc²-shaped region can be expressed as follows:

The second component e₁ is defined as the “don’t exceed” error between the peak of the first harmonic SBL (AF₁) and the mask SBL_mask, formulated as follows:

where

Finally, the total loss function, averaged over all samples within the elevation angle ranging from −90° to +90°, can be expressed as follows:

To achieve efficient csc² pattern synthesis for TMLA, the function y⁻¹(·) must be estimated so that [τ ;α] = y⁻¹(Mask), where τ is a vector of the OFF instants related to all array elements, α denotes the vector of the element phases, and Mask refers to the sampling vector of the mask (the desired radiation pattern). Drawing upon this estimation, a PIDNN-based framework is proposed for TMLA csc² beampattern synthesis in this study. As shown in Fig. 7, the PIDNN consists of an input layer, a number of hidden layers, and an output layer. The input—Mask=[Mask(θ_p);p=1,2,…,P]—of the PIDNN is the sampling vector of the mask, where θ_p(p=1,2,…,P) refers to the sampling angles, while its outputs— τ =[τ _n;n=1,…,N] and α=[α_n;n=1,…,N]—denote the element off- switching instants and phases, respectively, corresponding to Mask.

The PIDNN was designed as an array synthesizer. Notably, during antenna array synthesis, only the desired radiation pattern was known a priori, while the corresponding off-switching instants and phases were unknown. However, unlike most ANNs that require many input–output training pairs to learn unknown nonlinear relationships, the proposed PIDNN requires only one unlabeled input training sample—the desired radiation pattern Mask—thus eliminating the need for corresponding off-switching instants and phases.

The input sample Mask was trained by the PIDNN to predict τ and α. By minimizing the deviation of the actual array patterns from Mask, the PIDNN was finally able to acquire a set of OFF instants and phases that satisfied the requirements of Mask, thereby addressing the mask-constrained array synthesis problem. Subsequently, the loss function Loss of the PIDNN, as noted in Eq. (13), was set according to this deviation. Considering this perspective, the PIDNN can actually be considered a DNN-based optimizer—in dynamically adjusting its own weights and biases based on the criterion of minimizing the loss function Loss, it indirectly optimizes the array switch-off instants and phases.

To validate the efficiency of the PIDNN, it was compared to the GA. In GA, the loss function, as noted in Eq. (13), was minimized to identify the optimum switching-off instants and phases, with each chromosome (individual) in the population having a dimension of 2N—same as the dimension of the optimization problem (the total number of switching-off instants and phases).

The DNN employed in this study comprises an input layer of 351 nodes (P) and four hidden layers comprising 100 neurons each. After extensive tuning through a number of simulation trials, it was confirmed that selected values of the number of hidden layers and the number of neurons in them enable the achievement of the highest accuracy (least Loss). The number of neurons in the output layer was twice the number of array elements, 2N, representing the elements’ OFF instants and phases. The activation function used in all layers was hyperbolic tangent (Tanh). Furthermore, the Adam (adaptive momentum) optimization algorithm [25] was implemented to update the network weights and biases. The initial value of the learning rate was 0.01. Notably, during training, the learning rate decreased at a decay rate of 0.0005. The batch size was 1 (since there was only one input training sample), while the number of epochs or iterations (since each epoch had only one iteration) was 250. In addition, based on the guidelines provided in [26], a two-point crossover was chosen for the GA. The mutation probability was chosen to be 0.04, and roulette wheel selection was applied to solve the presented problem. The population size was taken as 10, with the termination condition being a maximum iteration of 1,000. All numerical results were obtained using MATLAB R2024a. In particular, the PIDNN was implemented using the Deep Learning Toolbox in MATLAB.

With reference to the findings reported in Fig. 6, θ_min = 5° and θ_max = 30° were considered for the optimization procedure. Moreover, since realizing a csc² beam is more computationally intensive than reducing the SLL, the weighting factors applied to the cost function for evaluating the fundamental AF in Eq. (9) were selected as w₁ = 0.1 and w₂ = 0.9.

In this paper, the synthesis of a csc²-shaped fundamental beampattern, along with simultaneous reduction in SLL and suppression of SBLs of the generated harmonics, in a TMLA was achieved using optimized switching sequences and element phases. A PIDNN-based technique was employed to simultaneously realize the desired radiation patterns. This was made possible by controlling the periodic pulse sequence and phase of each antenna element, thereby eliminating the need for complex attenuators. In addition, the PIDNN training process did not rely on external training data; instead, it utilized a physics-informed loss function derived from the deviation between the desired and actual beampatterns. The results confirm that the proposed method outperforms previous approaches in terms of the obtained SLL and maximum SBL, validating its effectiveness. A small csc²-main beam ripple was also observed, indicating the high shaping quality of the fundamental beampattern.

Consequently, the proposed approach can be implemented to achieve multiple goals, such as SLL reduction, SBL suppression, and beam shaping, merely through time and phase modulation. Moreover, in contrast to evolutionary optimization algorithms, the proposed PIDNN-based TMAA synthesis method significantly reduces computational time, making it a highly promising solution for real-time TMAA synthesis.