1 Harmonic Beam Steering in Time–Modulated Arrays with Simultaneous Sidelobe Control Ertugrul Aksoy, Member, IEEE Abstract—In this study, a methodological beam steering ap- proach in time-modulated linear arrays with simulateneous beam sidelobe control is presented. The method is based on the difference between pulse amplitude in order to provide necessary phase shift to steer the main beam and control the sidelobe level in harmonic frequencies. In order to illustrate the asserted idea a 16 element linear array placed alog z-axis with zero phase is considered. The results show that the proposed approach is an effective method in both beam steering and sidelobe level control. Index Terms—4D arrays, time-modulation, linear arrays, beam steering. I. I NTRODUCTION B ESIDE the switched array concept which is originally proposed in late 50’s by Shank and Bickmore [1] brings an additional degree of freedom in low/ultralow sidelobe array design, due to periodic switching of array elements, a seperation of radiated power between the main operation frequency and the harmonic frequencies called sideband radi- ations becomes inevitable. At first, these radiations are tried to be reduced as much as possible in order to shift the radiated power into main operation frequency via evolutionary techniques [2]–[4] and theoretical researches have conducted on the calculation and the control of these radiations [5]–[11]. In the mentioned above works, the sidebands are tried to be suppressed since they are regarded as power loss, however, Tennant and Chambers asserted that these radiations may be used in direction finding (DF) applications [12] and this idea is verified in [13]. In this way, the idea of exploiting the sidebands comes forward. Since the sideband usage in DF applications has been shown in [12], the pattern shaping in these frequencies for the usage of these radiations becomes in- evitable. Hence, in order to communicate over the fundamental radiation, an evolutionary approach has been introduced by Li et. al. to shape and steer the fundamental harmonic beam [14]. Also, a scale and shift method has been applied to the same problem of harmonic beam steering by Tong and Tennant [15]. In this study, a harmonic beam steering methodological technique with sidelobe control based on pulse difference is introduced and in order to show the effectiveness of the technique an 16 element linear array is given as an explanatory example. Ertugrul Aksoy is with the Department of Electrical&Electronics Engineer- ing, Engineering Faculty, Gazi University, 06570, Maltepe, Ankara, Turkey (phone: +90-312-5823356; e mail: ertugrulaksoy@ gazi.edu.tr). II. TIME MODULATION AND PULSE DIFFERENCE Suppose that each element of an antenna array is switched periodically by some simple on-off switches. For the variable aperture size (VAS), this switching action switching consisting of ideal pulses in one switching period may be modelled as: ()= { 1 , 0 < ≤ ≤ 0 , otherwise , (1) where represents the finishing instant of the pulse and represents the modulation period. Since, the switching process in periodic in time the pulse train can be decomposed into complex Fouries series (CFS) and for the VAS time scheme CFS is given by: ()= ∞ ∑ =−∞ (2) where is the angular switching frequency (i.e. = 2/ ) and is the complex Fourier coefficients (CFC) defined as: ()= 1 ∫ 0 () − . (3) Additionally, for VAS time scheme, the CFCs may be written as: = 1 2 [ 1 − −2 ] , (4) where represents the harmonic number and represents the normalized switch-on duration of element number (i.e. = / ). It can easily be shown that at a far field observation point, the Poynting vector of a time modulated array tends to un- modulated array’s Poynting vector multiplied by a CFC factor, if the carrier frequency is much bigger than the switching frequency (i.e. 0 / ≫ 1). Hence, the total time averaged array factor of an element time modulated linear array consisting of isotropic sources and positioned along positive −axis may be written as: ()= ∞ ∑ =−∞ ∑ =1 cos() (5) where , and represent the complex excitation ampli- tude, the relative distance between ℎ element and phase center of the array and wavenumber (i.e. =2/ 0 ) at operating frequency, respectively. Additionally, represents the harmonic number and =0 term represents array factor