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
$$AF\left(\theta ,t\right)={e}^{j2\pi {f}_{0}t}\sum _{n=1}^{N}{U}_{n}\left(t\right){e}^{j\beta \left(n-1\right)d\text{cos}\theta }$$
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where\(\beta =2\pi /\lambda\) is the propagation constant with \(\lambda\) being the wavelength at the fundamental frequency f0, 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 < < f0), it can be expressed by Fourier series, given by
$${U}_{n}\left(t\right)=\sum _{m=-\infty }^{\infty }{a}_{mn}{e}^{j2\pi m{f}_{p}t}$$
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where amn is the complex Fourier coefficient of nth element at m harmonic mode (\(m=0,\pm 1,\pm 2,\dots , \pm \infty\)) 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
$${a}_{mn=\frac{1}{{T}_{p}}{\int }_{0}^{{T}_{p}}{U}_{n}\left(t\right){e}^{-j2\pi m{f}_{p}t}dt}$$
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where Tp is the time modulation period, Tp = 1/fp. Now, the array factor of TMLA can be expressed as
$$AF\left(\theta ,t\right)=\sum _{m=-\infty }^{\infty }\sum _{n=1}^{N}{a}_{mn}{e}^{j\beta \left(n-1\right)d\text{cos}\theta }{e}^{j2\pi \left({f}_{0}+m{f}_{p}\right)t}$$
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The array factor for mth order harmonic frequency can be further simplified as
$${AF}_{m}\left(\theta ,t\right)={e}^{j2\pi \left({f}_{0}+m{f}_{p}\right)t}\sum _{n=1}^{N}{a}_{mn}{e}^{j\beta \left(n-1\right)d\text{cos}\theta }$$
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where harmonic radiation patterns occur at \(m=\pm 1,\pm 2,\dots , \pm \infty\), 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.
2.1 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.,\({\tau }_{n}^{1}<{\tau }_{n}^{2}<{\tau }_{n}^{3}<{\tau }_{n}^{4}<1\)) is expressed by
$${U}_{n}\left(t\right)=\left\{\begin{array}{c}1, {\tau }_{n}^{1}<\tau <{\tau }_{n}^{2}\\ 1, {\tau }_{n}^{3}<\tau <{\tau }_{n}^{4}\\ 0, otherwise \end{array}\right.$$
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The corresponding Fourier coefficient can be derived as
$${a}_{mn}={(\tau }_{n}^{2}-{\tau }_{n}^{1}\left)\text{s}\text{i}\text{n}\text{c}\right(m\pi {(\tau }_{n}^{2}-{\tau }_{n}^{1}\left)\right){e}^{-jm\pi \left({\tau }_{n}^{1}+{\tau }_{n}^{2}\right)}$$
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$$+{(\tau }_{n}^{4}-{\tau }_{n}^{3}\left)\text{s}\text{i}\text{n}\text{c}\right(m\pi {(\tau }_{n}^{4}-{\tau }_{n}^{3}\left)\right){e}^{-jm\pi ({\tau }_{n}^{3}+{\tau }_{n}^{4})}$$
2.2 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 ε0 is the error between the array factor of fundamental pattern AF0 and mask, given by
$${\epsilon }_{0}\left(\theta \right)=\frac{1+\text{s}\text{g}\text{n}({AF}_{0}\left(\theta \right)-mask\left(\theta \right))}{2}[{AF}_{0}\left(\theta \right)-mask\left(\theta \right)]$$
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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 ε1 and ε2 are defined as the “don’t exceed” errors between SLL of mask SLLmask and array factors of 1st and 2nd positive sidebands AF1 and AF2, respectively, defined as
$${\epsilon }_{\text{1,2}}\left(\theta \right)=\frac{1+\text{s}\text{g}\text{n}({AF}_{\text{1,2}}\left(\theta \right)-{SLL}_{mask})}{2}[{AF}_{\text{1,2}}\left(\theta \right)-{SLL}_{mask}]$$
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The total error thus given by
$$\epsilon \left(\theta \right)={w}_{0}*{\epsilon }_{0}\left(\theta \right)+{w}_{1}*{\epsilon }_{1}\left(\theta \right)+{{w}_{2}*\epsilon }_{2}\left(\theta \right)$$
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where w0, w1, and w2 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 Θ
$$CF=\frac{1}{\varTheta }\sum _{i=1}^{\varTheta }\epsilon \left({\theta }_{i}\right)$$
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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 4N 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 [26–29], and its applications to antenna array synthesis are reported in [1], [2], [30], [31].