Introduction
The desired antenna array pattern with a specified sidelobe level (SLL) and a first null beamwidth (FNBW) can be achieved by controlling the three basic parameters: amplitude, phase, and spacing between radiating elements [Reference Sallam and Attiya1–Reference Elliott5]. However, this requires a high dynamic range of parameters with the use of costly and complicated feed network of attenuators/amplifiers and phase shifters having high insertion loss. Moreover, the inter-element spacing cannot be controlled in real-time applications.
To avoid the aforementioned problems of conventional antenna arrays, “time modulation” is introduced [Reference Shanks and Bickmore6, Reference Haupt7]. In time-modulated antenna arrays, “time” is considered as an additional dimension or degree of freedom to synthesize the radiation pattern simply by periodic on–off switching of array elements in a pre-defined sequence [Reference Varma, Ram and Kumar8–Reference Rocca, Yang, Poli and Yang10]. In this way, the radiation pattern at the fundamental frequency can be controlled to achieve low SLL with a small sacrifice in FNBW without the need of complex and expensive attenuators/amplifiers and phase shifters responsible for high insertion loss [Reference Kummer, Villeneuve, Fong and Terrio11]. Moreover, time can be controlled electronically through radio-frequency (RF) switches which make it easier and more accurate to be implemented in real-time operation. Consequently, the proposed approach is a good candidate for achieving multiple goals such as SLL reduction and sideband level (SBL) suppression by utilizing solely time modulation.
However, due to the periodic switching of array elements, an infinite number of harmonics or sidebands is produced at multiples of time modulation frequency at either side of the carrier or fundamental frequency which is considered as waste of power reducing the radiation efficiency at the fundamental frequency and may interfere with other communication systems [Reference Bregains, Fondevila-Gomez, Franceschetti and Ares12, Reference Yang, Gan and Tan13]. Several evolutionary optimization algorithms have been adopted to minimize SBLs, and hence reducing sideband radiation such as genetic algorithm (GA) [Reference Yang, Gan, Qing and Tan14], differential evolution [Reference Yang, Gan and Qing15], particle swarm optimization [Reference Poli, Rocca, Manica and Massa16], simulated annealing [Reference Fondevila, Bregains, Ares and Moreno17], and artificial bee colony [Reference Nakanishi, Kihira, Takahashi, Konishi and Chiba18]. These optimization algorithms have been extended to other approaches such as nonuniform element spacing [Reference Li, Yang, Huang and Nie19], pulse shifting [Reference Poli, Rocca, Manica and Massa20], and pulse splitting [Reference Aksoy and Afacan21] to mitigate sideband radiation. In [Reference Chakraborty, Ram and Mandal22], improved harmony search algorithm is applied for simultaneous lowering of SLL of fundamental pattern and SBLs of sidebands. However, these methods, beside time modulation, employ other forms of modulation (weighting) such as amplitude and/or phase or spacing which is impractical as explained above.
In this paper, a simultaneous reduction of SLL of fundamental pattern and a suppression of SBLs for the harmonic patterns is obtained by solely using time modulation via pulse splitting. In pulse splitting, each element’s on-pulse is divided into multiple sub-pulses. In pulse splitting, the on–off switching sequence of each radiating element is characterized by multiple rectangular sub-pulses within the modulation period which increase the degrees of freedom in order to better synthesize the desired fundamental pattern with simultaneous suppression of harmonic or sideband radiation. Thanks to pulse splitting, the harmonic power is spread over a larger number of frequencies. This improves sidebands suppression performance as a result [Reference Yang, Gan, Qing and Tan23–Reference Tong and Tennant25].
In this paper, each element’s on-pulse is composed of two sub-pulses within time modulation period with each has its own on and off switching instants. 16-element linear array is taken as an example. Thus, a total of 64 time variables should be optimized. GA is chosen to optimize these time variables for obtaining a fundamental pattern with specified SLL and FNBW and sideband patterns with specified SBL (the same as the specified SLL of fundamental pattern). These multiple objectives are taken care of using a certain mask with the specified SLL (or SBL) and FNBW. The time variables are optimized by minimizing the error between this mask and fundamental, first positive sideband, and second positive sideband patterns. The obtained results show that the SBLs of other sidebands are also suppressed although they are not taken into consideration in the optimization. To validate the effectiveness of the proposed approach, the numerical results are compared to other published results utilized different optimization algorithms. The proposed approach outperforms other previous approaches in terms of the achieved SLL and SBLs validating the effectiveness of the proposed approach.
Theory and problem formulation
Assume N-element time-modulated linear array (TMLA) consisting of uniform amplitude, phase, and spacing isotropic elements placed along the positive z-axis, and controlled via RF switches. In this case, the array factor of TMLA is defined as
\begin{equation}AF\left( {\theta ,t} \right) = {e^{j2\pi {f_0}t}}\mathop \sum \limits_{n = 1}^N {U_n}\left( t \right){e^{j\beta \left( {n - 1} \right)d\cos \theta }}\end{equation} where 
$\beta = 2\pi /\lambda $ is the propagation constant with 
$\lambda $ being the wavelength at the fundamental frequency f\!0, and d is the uniform spacing between array elements. The switching modulation function of nth element is given by Un(t), n = 1, 2, …, N. θ is the elevation angle of the array.
Since the modulation function is periodic with time modulation frequency fp (fp ≪ f\!0), it can be expressed by Fourier series, given by
\begin{equation}{U_n}\left( t \right) = \mathop \sum \limits_{m = - \infty }^\infty {a_{mn}}{e^{j2\pi m{f_p}t}}\end{equation} where amn is the complex Fourier coefficient of nth element at m harmonic mode (
$m = 0, \pm 1, \pm 2, \ldots , \pm \infty $) with m = 0 represents array fundamental frequency, while the rest values of m represent the harmonic frequencies emerged as a result of time modulation. amn is given by
\begin{equation}{a_{mn = \frac{1}{{{T_p}}}\mathop{\smallint}\limits_0^{{T_p}} {U_n}\left( t \right){e^{ - j2\pi m{f_p}t}}dt}}\end{equation}where Tp is the time modulation period, Tp = 1/fp. Now, the array factor of TMLA can be expressed as
\begin{equation}AF\left( {\theta ,t} \right) = \mathop \sum \limits_{m = - \infty }^\infty \mathop \sum \limits_{n = 1}^N {a_{mn}}{e^{j\beta \left( {n - 1} \right)d\cos \theta }}{e^{j2\pi \left( {{f_0} + m{f_p}} \right)t}}\end{equation}The array factor for mth order harmonic frequency can be further simplified as
\begin{equation}A{F_m}\left( {\theta ,t} \right) = {e^{j2\pi \left( {{f_0} + m{f_p}} \right)t}}\mathop \sum \limits_{n = 1}^N {a_{mn}}{e^{j\beta \left( {n - 1} \right)d\cos \theta }}\end{equation} where harmonic radiation patterns occur at 
$m = \pm 1, \pm 2,$ 
$\ldots , \pm \infty $, while the fundamental radiation pattern occurs at m = 0. In this study, the by-product harmonics represent power loss in unintended directions that should be suppressed as possible.
The switching scheme of pule-split.
Pulse splitting
The switching scheme of pulse-split is shown in Fig. 1. The y-axis represents the switching modulation function for nth element Un(t), while the x-axis represents time normalized to modulation period τ = t/Tp. The on-time pulse for nth element is divided into two on-time sub-pulses with off state between them. The first sub-pulse has switching on and off instants 
$\tau _n^1$ and 
$\tau _n^2$, respectively. The switching on and off instants of second sub-pulse are 
$\tau _n^3$ and 
$\tau _n^4$, respectively. There is an off state between 
$\tau _n^2$ and 
$\tau _n^3$, i.e., 
$\tau _n^3 - \tau _n^2 = 0$. The modulation function for pulse-split within modulation period (i.e.,
$\tau _n^1 \lt \tau _n^2 \lt \tau _n^3 \lt \tau _n^4 \lt 1$) is expressed by
\begin{equation}{U_n}\left( t \right) = \left\{ \begin{array}{*{20}{c}}
{1,{\text{ }}\tau _n^1 \lt \tau \lt \tau _n^2} \\
{1,{\text{ }}\tau _n^3 \lt \tau \lt \tau _n^4} \\
{0,{\text{ }}otherwise{\text{ }}}
\end{array}\right.\end{equation}The corresponding Fourier coefficient can be derived as
\begin{equation}{a_{mn}} = (\tau _n^2 - \tau _n^1){\text{sinc}}\left( {m\pi (\tau _n^2 - \tau _n^1} \right)){e^{ - jm\pi \left( {\tau _n^1 + \tau _n^2} \right)}}\end{equation}
\begin{equation*} \qquad{}\qquad{}+ (\tau _n^4 - \tau _n^3){\text{sinc}}\left( {m\pi (\tau _n^4 - \tau _n^3} \right)){e^{ - jm\pi \left( {\tau _n^3 + \tau _n^4} \right)}}\end{equation*}Initial switching sequence for 16-element TMLA.
Initial radiation patterns for 16-elelemt TMLA.
Cost function
First of all, a mask is defined with a specified SLL which is the desired SLL (or SBL). Also, the mask has a specified FNBW which is the desired FNBW of fundamental pattern. To achieve the desired fundamental pattern with a simultaneous suppression of sideband patterns, the switching on/off instants of all sub-pulses for all elements should be determined through an optimization (minimization) of a properly defined cost function that covers the design specifications. The cost function is considered as the total error between mask and fundamental pattern along with first two positive sideband patterns. Thus, the total error consists of three components. The first component ε 0 is the error between the array factor of fundamental pattern AF 0 and mask, given by
\begin{equation}{\varepsilon _0}\left( \theta \right) = \frac{{1 + {\text{sgn}}\left( {A{F_0}\left( \theta \right) - mask\left( \theta \right)} \right)}}{2}\left[ {A{F_0}\left( \theta \right) - mask\left( \theta \right)} \right]\end{equation}Initial SBLs for 16-elemnt TMLA for 
$\left| m \right| \leq 10$.
Optimized switching sequence for 16-element TMLA.
The sign function sgn(θ) indicates that the error is expressed by “don’t exceed” criterion meaning that only the pattern points of array factor those are greater than the mask contribute to the score of the error. Note that this error is computed at a certain combination of switching on and off instants of sub-pulses, so the time variable is dropped from array factor. The second and third components ε 1 and ε 2 are defined as the “don’t exceed” errors between SLL of mask SLL mask and array factors of 1st and 2nd positive sidebands AF 1 and AF 2, respectively, defined as
\begin{equation}{\varepsilon _{1,2}}\left( \theta \right) = \frac{{1 + {\text{sgn}}\left( {A{F_{1,2}}\left( \theta \right) - SL{L_{mask}}} \right)}}{2}\left[ {A{F_{1,2}}\left( \theta \right) - SL{L_{mask}}} \right]\end{equation}Optimized radiation patterns for 16-element TMLA.
The total error thus given by
\begin{equation}\varepsilon \left( \theta \right) = {w_0}*{\varepsilon _0}\left( \theta \right) + {w_1}*{\varepsilon _1}\left( \theta \right) + {w_2}*{\varepsilon _2}\left( \theta \right)\end{equation} where w 0, w 1, and w 2 are weighting factors for the three components of the total error. The cost function is the total error averaged over all samples of the elevation angle 
$\Theta$
\begin{equation}CF = \frac{1}{\Theta }\mathop \sum \limits_{i = 1}^\Theta \varepsilon \left( {{\theta _i}} \right)\end{equation}Optimized SBLs for 16-elemnt TMLA for 
$\left| m \right| \leq 10$.
GA is employed to minimize the cost function to get the optimum switching on and off instants with each chromosome (individual) of the population has a dimension of 4 N which is the same as the dimension of the optimization problem (the total number of switching on and off instants). Detailed discussions on GA can be found in [Reference Holland26–Reference Haupt and Werner29], and its applications to antenna array synthesis are reported in [Reference Sallam and Attiya1, Reference Sallam and Attiya2, Reference You, Liu, Xu, Zhu and Liu30–Reference Reyna, Panduro, Covarrubias and Mendez33].
Results and discussion
Consider 16-element uniform amplitude, phase, and spacing TMLA. In this case, we have a total of 64 switching on and off instants to be optimized. After many simulation trials, the inter-element spacing d is chosen as 0.88
$\lambda $ which leads to better results than the usual spacing 
$\lambda /2$ The fundamental frequency f\!0 is 10 GHz and time modulation frequency fp = 1 MHz. All numerical results are obtained with the aid of MATLAB R2023a. The GA is also implemented in MATLAB using the “ga” function with a population size of 200. The optimization is terminated when average change in value of the cost function is less than 0.01.
Comparison of optimized SLL and SBL1,2 between proposed approach and other approaches
The optimization starts with an initial set of all variables (switching on and off instants) defined randomly within the normalized modulation period (from 0 to 1), and also constrained within this period during optimization process. Figure 2 shows the initial (before optimization) switching sequence for all array elements.
Figure 3 shows the mask along with initial fundamental (m = 0) and first two positive (m = 1, 2) sideband patterns. After several trials, the SLL of mask SLL mask is chosen to be −43 dB, as shown in Fig. 3, which represents the desired SLL of fundamental pattern or desired maximum SBL of sideband patterns. This mask’s SLL is chosen such that it is the lowest level the SLL of fundamental pattern can reach with an acceptable error. The FNBW of mask (the desired FNBW of fundamental pattern) is 12.
As shown in Fig. 3, the initial FNBW of fundamental pattern is 8°. The initial SLL of fundamental pattern SLL, SBL of 1st sideband SBL 1, and SBL of 2nd sideband SBL 2 are −10.99 dB, −19.75 dB, and −28.27 dB, respectively. As can be noticed, SBL 2 is the lowest level among the three levels. Thus, it is expected that 2nd sideband pattern will be easier to be optimized than the fundamental and 1st sideband patterns. That is why the weighting factor for 2nd sideband w 2 should be chosen to be less than the other weighting factors of cost function w 0 and w 1. After several runs, the three factors are selected as w 0 = 1, w 1 = 1, and w 2 = 0.25. In this case, the initial value of cost function is 35.41 dB. Figure 4 shows the initial SBLs for first positive and negative ten harmonics. As can be shown, all initial SBLs are under the level of 1st sideband which is −19.75 dB and also above a level of about −40 dB.
Figure 5 shows the optimized (after optimization) switching sequence for 16-element TMLA. As can be shown, the total on-time duration decreases from center to outer elements with center elements have the longest total on-time duration, while the outer elements have the shortest ones. This ensures a high feeding network efficiency [Reference Zhu, Yang, Yao and Nie34]. Figure 6 shows the optimized radiation patterns for fundamental and first two sideband frequencies along with mask. It is can be noticed that SLL, SBL 1, and SBL 2 are well suppressed under SLL mask. The optimized SLL, SBL 1, and SBL 2 are −42.84 dB, −43.33 dB, and −43.08 dB, respectively. The optimized FNBW of fundamental pattern is 16° exceeding the desired FNBW by 4° which is a little sacrifice compared to the high gain in reducing SLL of fundamental pattern. The final value of cost function after optimization is 0.32 dB.
Fig. 7 shows the optimized SBLs for first positive and negative ten harmonics. As can be shown, all SBLs after optimization become under a level of −41 dB. The higher-order SBLs are also well suppressed, although they are not included in the formulation of cost function that was optimized.
To highlight the efficacy of the proposed approach, the optimized results for SLL and SBL 1,2 are compared with other published approaches for 16-element TMLA and presented in Table 1. The proposed approach outperforms other approaches in terms of SLL and SBLs. The optimized SLL of the fundamental pattern is lowered to −42.84 dB compared to the best result of −40.31 dB [Reference Chakraborty, Ram and Mandal22]. The first and second positive SBLs are considerably improved to −43.33 and −43.08 dB from the best results of −29.47 and −32.59 dB [Reference Chakraborty, Ram and Mandal22], respectively.
Conclusion
In this paper, a simultaneous reduction of SLL of fundamental pattern and suppression of SBLs of generated harmonics of TMLAs is presented with optimized switching sequences. A pulse splitting technique is employed to give more degrees of freedom to simultaneously realize the desired radiation patterns in a GA-based approach. This is achieved by controlling only the periodic pulse-split sequences, which also eliminates the need for attenuators and/or phase shifters. The proposed approach outperforms other previous approaches in terms of the achieved SLL and SBLs validating the effectiveness of the proposed approach. The proposed method also maintains a narrow FNBW, which implies a highly directive main beam at the fundamental pattern. Consequently, the proposed approach is promising for achieving multiple goals such as SLL reduction, SBL suppression, and directivity maximization all by utilizing merely time modulation.
Competing interests
The authors declare none.
Tarek Sallam was born in Cairo, Egypt, in 1982. He received the B.S. degree in electronics and telecommunications engineering and the M.S. degree in engineering mathematics from Benha University, Cairo, Egypt, in 2004 and 2011, respectively, and the Ph.D. degree in electronics and communications engineering from Egypt-Japan University of Science and Technology, Alexandria, Egypt, in 2015. In 2006, he joined the Faculty of Engineering at Shoubra, Benha University. In 2019, he joined Huaiyin Institute of Technology, Huai’an, China. In 2022, he joined Qujing Normal University, Qujing, China. In 2024, he joined the School of Computer Science and Technology, Shandong Xiehe University, Jinan, China, where he is currently an Associate Professor. He was a Visiting Researcher with the Electromagnetic Compatibility Lab, Osaka University, Osaka, Japan. His research interests include evolutionary optimization, neural networks and deep learning, phased array antennas with array signal processing and adaptive beamforming.
Ahmed M. Attiya M.Sc. and Ph.D. Electronics and Electrical Communications, Faculty of Engineering, Cairo University at 1996 and 2001 respectively. He joined Electronics Research Institute as a Researcher Assistant in 1991. In the period from 2002 to 2004 he was a Postdoc in Bradley Department of Electrical and Computer Engineering at Virginia Tech. In the period from 2004 to 2005 he was a Visiting Scholar in Electrical Engineering Dept. in University of Mississippi. In the period from 2008 to 2012 he was a Visiting Teaching Member in King Saud University. He is currently Full Professor and the Head of Microwave Engineering Dept. in Electronics Research Institute. He is also the Founder of Nanotechnology Lab. in Electronics Research Institute.
