IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 11, 2012 133 Calculation of Sideband Power Radiation in Time-Modulated Arrays With Asymmetrically Positioned Pulses Ertugrul Aksoy and Erkan Afacan, Member, IEEE Abstract—In this letter, the effect of shifting pulses of a time- modulated array (TMA) on sideband radiations is examined, and the total power formulation is generalized to asymmetric cases. Also, it is shown theoretically that shifting pulses changes the total power radiated in harmonics. Index Terms—Antenna arrays, sideband radiation, time modu- lation. I. INTRODUCTION T IME-MODULATED antenna arrays were proposed and investigated for the first time in 1959 [1], and scientific interest in this topic has increased after the work published by Yang et al. in 2002 [2]. For more information, the references given in [3] and [4] may be followed. However, to give an overview on time modulation and related applications, some re- cent advances in pattern synthesis may be summarized as fol- lows: In [5], time modulation is applied to a monopulse sub- arrayed antenna in order to synthesize compromised sum–dif- ference patterns with the aid of particle swarm optimization and contiguous partition method. In [6], it is shown that first harmonic of a time-modulated linear array may be used along with the fundamental radiation to obtain multiple-beam patterns using the same aperture. In [7], adaptive nulling in linear arrays is achieved by time modulation, and in [8], a multistage subarray optimization is given to shape the overall radiation pattern. In early research on time modulation such as [2], sideband radia- tions were considered as power loss. However, in some recent published works, it is shown and experimentally verified that harmonic radiations may be also used on demand. Two recent publications may be given as reference to time-modulated di- rection-finding applications: [9] and [10]. In time-modulated antenna arrays, total harmonic radiation may be calculated by using the formulation originally given in [11]. Since there are infinite radiation harmonics, [11] brought simplicity to the calculation of the time-average total power by presenting a straightforward closed-form equation. However, all pulses are assumed to be symmetric around the time-axis for the sake of simplicity, so the formulation developed in that paper Manuscript received October 05, 2011; revised November 22, 2011 and De- cember 12, 2011; accepted January 15, 2012. Date of publication January 26, 2012; date of current version March 19, 2012. The authors are with the Department of Electrical and Electronics Engi- neering, Gazi University, 06570 Ankara, Turkey (e-mail: ertugrulaksoy@gazi. edu.tr). Digital Object Identifier 10.1109/LAWP.2012.2185916 depends on the symmetry of pulses. After that work, the formu- lation given in [11] is extended to the case of a planar rectangular grid by Poli et al. [12], and the joint formulation for an arbitrary linear and planar array is followed and given in [13]. Here, it must be noted that also in [12] and [13], pulses are assumed to be symmetric as in [11]. In a recently published work by Poli et al. [14], it is shown that the sideband level may be reduced by shifting pulses, and it is assumed that the total radiated power only depends on pulse durations and remains constant. This assumption is correct for the half-wavelength interelement spacing (for shifted pulses), as taken in [14] and for the cases in which all pulses start at the origin of the modulation period. However, if interelement spacing is different from half-operating wavelength and pulses do not start at the origin of the modulation period, usage of ex- pression given in [11] may produce inappropriate results. In this letter, total harmonic radiation for asymmetric pulse distribution (e.g., shifted pulses) in time-modulated antenna ar- rays is investigated. The formulation is extended to asymmetric cases, and it is shown that the total sideband radiation is not only pulse-duration-dependent, but also pulse-position-dependent. II. FORMULATION Consider an array of identical radiating elements along the -axis of a Cartesian coordinate system, and let each of them be energized periodically by the function otherwise. (1) This function represents a symmetric switching sequence around the beginning of the period, and by using this function, the total power radiated in harmonics can be calculated via a closed-form formulation [11] (2) Here, represents the real part of a complex quantity, represents the minimum of and , represents the distance between the th element and coordinate system origin, denotes the complex excitations, and is the 1536-1225/$31.00 © 2012 IEEE