Physics-Informed Deep Neural Network for Low Sidelobe Cosecant-Squared Pattern Time-Modulated Antenna Array with Harmonic Suppression

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J. Electromagn. Eng. Sci. 2026;26(1):68-78

Publication date (electronic) : 2026 January 31

doi : https://doi.org/10.26866/jees.2026.1.r.340

Tarek Sallam^1,, Qun Wang¹, Ahmed M. Attiya²

¹School of Computer Science and Technology, Shandong Xiehe University, Jinan, Shandong, China

²Department of Microwave Engineering, Electronics Research Institute (ERI), Cairo, Egypt

^*Corresponding Author: Tarek Sallam (e-mail: tarek.sallam@feng.bu.edu.eg)

Received 2025 February 28; Revised 2025 May 14; Accepted 2025 September 11.

Abstract

I. Introduction

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.

II. Theory and Problem Formulation

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:

Fig. 1

The switching scheme of a time pulse.

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. 2

Initial phases of the 30-element TMLA.

Fig. 3

Initial switching sequence of the 30-element TMLA.

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.

Fig. 4

Initial radiation patterns of the 30-element TMLA.

Fig. 5

Initial sideband levels of the 30-element TMLA for |m|≤10.

III. Loss Function for the Proposed Optimization

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:

Fig. 6

The cosecant-squared pattern mask showing the csc²-shaped region and the desired SLL or SBL of −40 dB.

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:

IV. PIDNN-based CSC² Pattern Synthesis for TMLA

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.

Fig. 7

Architecture of the PIDNN for TMLA csc² pattern synthesis.

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).

V. Results and Discussion

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.

1. 30-Element TMLA

In this section, the 30-element TMLA, whose initial characteristics are presented in Figs. 2–5, is employed as an example. The value of θ_o, as depicted in Fig. 6, was taken as −9°, while SLL_mask and SBL_mask were chosen to be −40 dB. Fig. 8 shows the convergence curve of the PIDNN, where the loss function Loss, calculated using Eq. (13), is plotted against the number of iterations during optimization. It is observed that the PIDNN attains fast convergence—it nearly converges after only 200 iterations.

Fig. 8

Convergence curve of the PIDNN for csc² pattern synthesis of the 30-element TMLA.

Fig. 9 depicts the optimized element phases of the 30-element TMLA, and Fig. 10 presents the corresponding optimized switching sequence. Meanwhile, Fig. 11 illustrates the optimized radiation patterns for the fundamental and first sideband frequencies along with the mask. It is noted that SLL and SBL₁ are well suppressed under SLL_mask. The optimized SLL and SBL₁ are −39.54 dB and −45.28 dB, respectively. Furthermore, the optimized average deviation of the fundamental pattern Δ_csc² in the csc²-shaped region is 0.6765 dB, with very low ripples observed in the shaped main beam. The final value of the cost function after optimization is 0.3237 dB, which can mainly be attributed to Δ_csc² of the shaped main beam.

Fig. 9

Optimized phases of the 30-element TMLA.

Fig. 10

Optimized switching sequence of the 30-element TMLA.

Fig. 11

Optimized radiation patterns of the 30-element TMLA.

Fig. 12 shows the optimized SBLs for the first 10 positive and negative harmonics. It is observed that all SBLs are below −42 dB after optimization, with the maximum SBL_max being −42.27 dB. The higher-order SBLs, except for the first positive harmonic, are also found to be well suppressed, although they were not included in the formulation of the cost function that was optimized.

Fig. 12

Optimized SBLs of the 30-element TMLA for |m|≤10.

The total array efficiency of a TMLA depends largely on its power efficiency, while the suppression of sidebands is generally regarded as an improvement upon this efficiency. However, the OFF states of the array elements imply that high-speed RF switches absorb some part of the radiated energy in the feed network even when power is supplied. Therefore, the total array efficiency can be expressed as

(14) ηTotal=ηSwitch·ηPower

where η_Switch indicates the switching network efficiency and η_Power refers to the total power efficiency of the TMLA, defined as the ratio of the power used for generating the desired (fundamental) pattern (P₀) to the total radiated power (P_T). Notably, the switching network efficiency for the array can be formulated as [27]:

(15) ηSwitch={(∑n=1Nτn)/N}

For the 30-element TMLA, the power utilized to generate the fundamental pattern (η_Power) was found to be 99.98% of the total radiated power, while the rest was wasted in the first 10 positive and negative harmonics (dispersed power at the higher-order sidebands was negligible since they have a decaying nature). Overall, these results imply that the PIDNN ensures efficient power usage since the power wasted in harmonics is negligible in comparison to that utilized for the fundamental pattern. Furthermore, the switching network efficiency η_Switch of the optimized pulse sequence, as shown in Fig. 10, was calculated to be 57.51%. Thus, the total array efficiency, based on Eq. (14), was 57.5%.

2. 40-Element TMLA

To better describe the generalizability of the proposed method, it was applied to a different array size with different SLL and SBL requirements. For this experiment, the number of elements of the TMLA was increased to 40, and the desired SL_mask or SBL_mask was chosen to be −30 dB, with θ_o being 0°. Fig. 13 shows the convergence curve of the PIDNN, indicating fast convergence after only 100 iterations. Moreover, the PIDNN took only 2 minutes and 17 seconds to reach 250 iterations.

Fig. 13

Convergence curve of the PIDNN for csc² pattern synthesis of the 40-element TMLA.

Fig. 14 shows the optimized element phases of the 40-element TMLA, and Fig. 15 presents the corresponding optimized switching sequence. In addition, Fig. 16 displays the optimized radiation patterns for the fundamental and first sideband frequencies along with the mask. As expected, it is observed that SLL and SBL₁ are well suppressed under SLL_mask, with the optimized SLL and SBL₁ being −29.28 dB and −35.02 dB, respectively. The optimized average deviation of the fundamental pattern Δ_csc² in the csc²-shaped region is 0.3128 dB, with almost no ripples in the shaped mainbeam. The final value of the loss function after optimization is 0.0443 dB. Fig. 17 presents the optimized SBLs for the first 10 positive and negative harmonics. It is noted that all SBLs are below −32 dB after optimization, with the maximum SBL_max being −32.02 dB.

Fig. 14

Optimized phases of the 40-element TMLA.

Fig. 15

Optimized switching sequence of the 40-element TMLA.

Fig. 16

Optimized radiation patterns of the 40-element TMLA.

Fig. 17

Optimized SBLs of the 40-element TMLA for |m|≤10.

With regard to power calculation, the power utilized to generate the fundamental pattern (η_Power) was estimated to be 99.12% of the total radiated power, while the rest was wasted in sidebands. This again implies that the PIDNN ensures efficient power usage, given that the wasted power is very small. The switching network efficiency η_Switch of the optimized pulse sequence was calculated to be 39.54%. Therefore, the total array efficiency was 39.19%.

3. Comparison of Accuracy

To highlight the efficiency of the proposed approach, the optimized results for SLL and SBL_max were compared to those attained by other published approaches employing different evolutionary optimization techniques for a 30-element TMLA. Table 1 clarifies that the proposed approach outperforms the other approaches in terms of SLL and maximum SBL, emphasizing the superiority of PIDNN over other optimization techniques. In particular, the optimized SLL of the fundamental pattern is observed to be considerably lower at −39.54 dB when using the proposed approach compared to the best result of −34.60 dB achieved in [19] using the ANN approach. The proposed technique also achieved an improved maximum SBL of −42.27 dB compared to the best result of −38.90 dB attained in [19].

Table 1

Comparison of the optimized SLL and SBL_max between the proposed approach and other approaches for a 30-element TMLA

Furthermore, GA was employed in [28] to realize a low side-lobe TMLA along with harmonic suppression using solely time modulation. In [29], it was used to realize csc²-shaped beampattern using time modulation along with phase modulation. However, while GA helps attain relatively good sideband suppression performance, it results in very low SLL, as shown in Table 1. Moreover, as evident from the comparison conducted for a 30- element TMLA in Table 2, GA led to an average deviation of 1.0485 dB in the csc²-shaped region of fundamental pattern Δ_csc², indicating high ripples in the shaped main beam. Table 2 also shows that GA took 47.13 min to produce the final results, while PIDNN took only 2.87 minutes. Moreover, the PIDNN required only 250 array pattern calculations (250 iterations) to reach a final Loss value of 0.3237 dB, while GA required 10,000 array pattern calculations (a population size of 10 and 1,000 iterations) to reach a final value of 1.1584 dB. These results indicate that the PIDNN is not only substantially more computationally efficient than evolutionary optimization algorithms but also superior in terms of loss function minimization.

Table 2

Comparison of GA and PIDNN in terms of Δ_csc², final loss value, number of iterations, and computation time for a 30-element TMLA

VI. Conclusion

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.

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