Comments on Frequency Diverse Array Antenna Using Time-Modulated Optimized Frequency Offset to Obtain Time-Invariant Spatial Fine Focusing Beampattern

Accepted for publication in IE EE Transactions on An tennas and Propagation 1  In the recent papers [1 -4] including above paper time modulated frequency diverse array s (FDAs) have been presented to obtain tim e inv ariant spatial patterns. The presented FDAs in [1-3] h ave the f eature of tim e-invari ant spatial focusing which means they have a constant ma ximum in a time duration, T , at a point with desired range,  , and angle,  . In [4] the patter n is ti me i nvariant, yet, not focused. However, in th is communication it is indicated that the patterns are ob tained using incorrect d efinition of time in some equations. The equatio n syste m o f [ 3] is explained here th at can be extended to [1, 2, 4], explicitly. For the a ntenna ar ray with 󰇛  󰇜 elements, each labeled by an integer number,  (   󰇟   󰇠 ), the array factor (AF) is given by [3 , 5 ]  󰇛      󰇜       󰇡   󰇢     󰇛     󰇜        (1) where   is frequency of the ce nter ele ment a nd 󰇛      󰇜 is the frequenc y of  -th element w ith the assu mptions,      and     .  is the speed of electr om agnetic ( EM) wave p ropagation a nd  is t he spacin g between eleme nts.   is a constant phase difference co rresponding  -th ele ments. The AF as a function of time ,  , indicates a wave moving with velocity,  , while for  time shifting  movement in ran ge,  , is required to obtain the same AF . No w , t he practicabilit y of compensating th e w ave speed by defining each   as a function of time is disc ussed in the follo wing explanations. It is ob vious tha t each   can b e defined as a function o f time at the position of  -th element. However, relocating from its position, at the sa me time, the o bserved   is for the past. For example, i n t he distance  from the array, the observed   is for   b efore the present time. T herefore,   in the AF cannot be a function of time and the propagation delay should be considered. In particular, each   can be defined as a function of 󰇛      󰇜 . This is while in [1-4] the observed frequenc y difference s ar e defined as the functio ns of time in the AF . T his ass um ption means that t he freque ncy of each element is a f unction of time and range, which is not feasible. Consequently, in Eq. (3) of [3],   󰇛 󰇜 should be replaced by   󰇛   󰇜 (admitting that    ) and therefore it is   󰇛   󰇜 that should be set to Eq . (6), leading to a Manuscript re ceived Ju ly 27, 2017 , accepted Mar. 30, 2018. M. Farto okzadeh is with Depar tment of Electrical and Ele ctronics Engineering, Malek Ashtar University , P. O. B ox 1774 -15875, Tehran, Iran (Mahdi.fartoo kzadeh@gmail. com ). dependence of   on  . The same ap plies to Eq . (1 2) and (13) of [1]. An easy-understa nding evidence for observing incorrectness of this assumption is the power pattern plots in [1 -4]. For example, it can be observed in F ig. 8 (a) of [ 3] that the maximum o f AF is obtained in the range     m, at    using the freque ncy offsets                           󰇟   󰇠  (2) where      is the angle of focus,   is an op tim ized factor for  -th ele m ent and    ns. This EM power should have been sent fro m the ante nna array at 50 ns b efore ; if we agree that the EM wave travels in space- ti me on light -cone with t he velocit y       m /s. It can be observed th at there is no infor mation from the frequencie s of the elements in the past and the excitation has been started at    . Therefore, the po wer could not have ap peared at the sa me time to the range,     m. Nevertheless, it is worth no ting that the opti m ization method in [1-3] is i nteresting and useful. For exa mple, similar spatial power p attern with Fig. 6 of [ 3] can be ob tained for the FDA with    ,     GHz, and         at    using the same opti mized factors, ( 󰇟   󰇠  󰇟   󰇠  󰇟      󰇠 , reading fro m Fig. 4 of [ 3]) as indicated in Fig. 1(a) b y removing  from   in (2). However, at    the po wer p attern will be changed a nd the focus point will be at    m as indicated in Fig. 1(b) . Projection of power- pattern on the time -range a xes for the constant   is indicated in Fig. 1(c). It should b e noted that the maximum p ower is observed again at    on    m f or the cons tant   as can be seen in Fig. 1(c). The zero time is conventional, w hile the frequencies of propagated signals of t he elements are constant with time. In fact, the prop agation has b egun fro m t he earlier time for cons tructing this p ower p attern. Therefore, th e resulting power patter n at this range is due to excitat ion i n the past. For exa mple, if t he ex citation began from     ns the po w er patter n would be as indicated in Fi g. 2( a). Ho w ever, if it is d esired to begin the excitation at    , one can change the frequency offsets as                      (3) where   is the desired time of appearing the m aximum po wer at the lo cation of array. I n the previous case   w as i n fact - 50 ns. T he power pattern of a s imilar FDA with     ns is indicated in Fig 2(b), assuming that the excita tion has beg un from    . Comments on ‘Frequency Diverse Array Antenna Using Time-Modulated Optimized Frequency Offset to Obtain Time-Invariant Spatial Fine Focusing Beampattern’ M. Fartookzadeh © 2018 IEEE. Pe rsonal use of th is material is per mitted. Permiss ion from IEEE m ust be o btained for all other uses, in any cur rent or fut ure media, incl uding reprinting/re publishing this ma terial for advertising or promo tional purpose s, creating ne w colle ct ive works, for r esale o r redistribution to serve rs or lists, or reuse of any copyrig hted component of this w ork in other w orks. Accepted for publication in IE EE Transactions on An tennas and Propagation 2 (a) (b) (c) Fig. 1 . Projection of power p attern o f the FDA with constant frequency offse ts using the optim ized values of   in [3] (a) on the r ange-angle axes at    , (b ) on the range-angle axes at    ns and (c) on the time-range axe s at     . (a) (b) Fig. 2. Projection of powe r pa ttern of FDA with constant frequency offsets using the optimized values of   in [3 ] on the time-range axes at     ; (a) similar pow er patter n with Fig. 1(c) when the excitat ion begins from     ns , (b ) the power pat tern when   is d efined by (3),     ns, and the beg inning of excitatio n is at    . In addition, constant r ange for the focus point of FDAs with     , appears to b e unattainable us ing a ny method [6 -8], since t he velocity o f EM wave is independent from its frequency, amplitude, etc. Thus, the power deli vered by the array to the range,   , will appear in the range     after elapsing t he time,  . Consequently, the focusing po int of antennas al ways moves with the velocit y of wave propagation in farfield. Ho wever, consta nt nearfield focu sing a ntennas ar e accessible [9- 11 ], since the nearfield range is not similar fo r elements o f an arra y or parts of an a ntenna. Further m ore, it is not in contradiction with movement of the EM wa ve in space - time on t he light -cone, since the EM po w er has been reached to the focusing point from different locations. R EFERENCES [1] A. -M. Yao, W. Wu, and D.- G. F ang, “Fre quency Diverse Array Antenna Using Time-Modulated Optimized Frequency Offset to O btain Time- Invariant Spatial F ine F ocusing Be ampattern,” IEEE Tr ansactions o n Antennas and P ropagation, vo l. 64, no. 10, pp. 4434 – 4446, Oct. 2016. [2] A. -M. Yao, W. Wu, and D. - G . Fan g, “Solutions of Time -Invariant Spatial Focusing for Mu lti-Tar gets Using Time Modulated Frequency Diverse Antenna A rrays,” I EEE Tr ansactions on Antennas and Propagation, vol . 65, no. 2, p p. 552 – 566, Fe b. 2017. [3] A. - M. Yao, P. Rocca, W. W u, A. Massa, and D.- G. 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