place nulls at 0, = M", O2 = (17". ;inti O3 = 68" by using the abnvc technique with thc iiorin IIP(i)ll = [j.: I P(i)(~]~~du]~'~ (tlic projectors Ui- and U*, were implcmcntcd by using the mntheinatical resdts described in [5]). The synthcsiscd pallern is shown in 1;ig. Ih. A sccoiirl cxaiiiple refers to a circular ring array or radius X = 1.8X, consisting or A' = 30 equally spaccrl isolropic elements. The array lies in the S-JJ plane of n Cnrtcsian system O(x,y,z) and its cciitrc coinciidcs wilh 1he origin 0. The gcncric far-freltl paltern in tlie plane or the army is indicaicd by P(i)(+), where (I is the azi- tiiulli angle. Fig. 2a shows llic pallern P(io)($) (thick linc) obtaincd by approximating tlic pattern FO($) (Ihiii line) in the mean-square sensc [7]. We modified /'(io)($) so ns to placc nulls at 49", by tising tlic above technique with the norm 111'(i)ll = 1j-i lP(i)($)12d$]i'2 (the projectors U, and U, were implemented hy tlie techniques descrikd in [3] and [7], rcspectively). The sytithe- sised pattern is shown in Fig. 2A. Notc that the palleriis in Fig. Ih and Fig. 26 have deep nulls in the required directions and sutisfrictorily approximate the patterns in Fig. la arid Fig. 2cr, rccspcctivcly. In monclusion, we have described an itcrahe leclmiqiie that allows nulls to be placed in thc radialioii pattern of an antenna army by modifying only thc excitation phases. The technicpc is based 011 the mcthod of projcclions, is easy to iinplcnicnt and gives reasonably satisfactory resulls in nll of the cam cxamincd, including thosa illuslraled here. $1 = 49", $2 = 48", $3 = - 47", $4 = 47", $5 = 48", aiId $6 = Rcfcrcnces 1 PRRSAD, s~, and CCIARAN, IC.: 'On the constrained synthesis oC array pitterns with applic.arioiis to circular and IC lirrays', LWfl Trm?,~., 1984, AP-32, (7). pp. 725-730 2 NG, u.P., ER, M.II., and KOT. c.: 'Lincar w i l y geonielry synthcsis with minimum sidc Iobc lcvel and null control', Pvnc,, M/c~(JIv. Anrennm Propug., 1994, 141, (3, pp. 162-166 wscovo. R.: 'I'atfcrn synthcsis with null constraitits for circular arrays of cqually ~pliced isotropic elenicntp', ILL PI.oE., Yicroiv. Anfewwl.x Proprig., 1996, 143, (Z), pp, 103 -106 IIAUIT. K.I..: 'Phasc-only adnptive nulling with a gcnctic algorithm', imui "VIS., 1997, AP-45, (GI, pp. 1009-ioi~ VFSCOVD. R.: 'Null control io linear array in the presence of 1111 uppcr hound for the cxcittltion dynamics'. Fimc. Ini. Conf. E1cctroin:ig. in Advaucetl Applications, Torino, Italy, September 1999, pp. 433-435 6 LEVI. A., ~iid SI',\lw, H.: 'Iinagc rcvloriition by the mcthod of generalized projections with npplicatioti lo rcstorution l'rom mngnitudc', J. Opt. SUC. Atwr:, 1984, I, (9), pp. 932-943 (Pnrl A) 7 v~scovo, R.: 'Constrained nnd unconstraiiicd synthesis or miry Factor Tor circular arrays', IEE'nB Trotis., 1995, AP-43, (]I), pi). 3 4 5 1405- I4 10 Chaotic dynamics of oscillators based on circuits with VCO and nonlinear delayed feedback L. Largcr, V.S. Udaltsov, 9.P. Goedgcbucr and W.T. Rhodes An cleclronic oscillator that produccs clws with n diniensioiiality st least 20 rims Ihighcr tliaii the dimensionality obtaincd with prcviously publishcd elcctroiit cliilos gceiicratnrs is rcportcd hlro&c/ion: ?'he idea of hiding a messagc sigiial in chaos was first proposcd carly in the 1990s [l]. Sincc thcn, chaos-based coding ant1 decoding have been dc~nonstraltd successfiilly [2, 31. ldcally, ihe chaos generators uscd in thcse systems should havc thc follow- ing ctiaractcristics: (i) llie chaotic signal should be of tlie highest possiblc dimcnsionality (hypercliaosj, in ordcr lo ensure the gent- cst complexity possible and, tlicrcforc, as high a degree of sccurity ELECTRONICS LETTERS 3rd February 2000 Vo/. 36 as possible; (ii) the bwndwidth of the chaotic signml must bc suita- ble Tor limited-baiidwidtli communication chaniiels; (iii) the chaos gcncration must bc of such 21 nature as to allow chnotic synchroni- sation at the receiver and extraction of the mcssiigc itironnation Most clcctronic chaos generators developed thus fir have used nonlincar circiiils wilh Cliua diodes or hy.qteresis-type elements (as an example, scc IS, 61). Thcy generate rather low-dimetision cha- otic signals: tl~ere itre usually less than three non-negirtivc Lyapu- nov cxponenls, corresponding to n dimension of < 5. The low complcxily of the chaos yields a low degree of confideddily, and an eavesdropper can defeat the chaotic key relativcly easily. In contrast, oscillators einploying dclaycd nonliiicar feedback caii generate high-dimcnsional clinos ( h c nuiiibcr of positive Lyapu- nov cxponciits can be > 10). Optical systems with nonlincar fed- hick, oiie employing R ring fibre laser [2] and thc other a wavelength tunable laser diode 171, have recently bccn demon- slrnled. The dimension d of the chaos thus produccd is given by rl = 0.4&71~, where Tis die time deltry introduccd by thc feedback loop, T is its response timc, and p is thc bifurcation parameter. In the experiment dcscribcd in [7] thc maximum dimension was reported to bc 250, i.a. the numkr of posilive Lyapitiiov cxpu- iicnts WAY -125. In this Lctter wc report B nonlinear-feedback chaotic electronic oscillator that has potcnlial application Lo secure radiocomimini- calions. [41. nonllnearily detector Q A Y -1lowpass tilter] I tlmedelay b Big. 1 8twk otid schcrruiir diagrams oJ clrriotic osciliaror n Block diagram h Sclicinatic diagr:iin Pritic/ji/c of operation: A funclional block diagram for tlic systcm and il sclicinatic diagram Tor the circuit are shown in Fig. 1. The saurcc is R voltage-controlled oscillator (VCO), which produces a frcqii~ii~lq-motlulate~ (FM) sigiial with arnplitude A. nild fie- qiiency/(t). This FM signal is sent into ii filtcr NL (nonlinear in frequency) and an envelope detector, which detccts the envelope of the output signal from the fillcr. This cnvclopc signal is delnycd by T and wves as tlic driving signal V(r) of the VCO. The responsc time of thc lowpnss filler corresponds to the feedback loop rime conshiill t. Driven ns shown through tlie feedback circuit, thc VCO gener- ates a chaotic FM signal. The frequency fof the signal is iclatcd to thc dtiviug volkgc V(t) of the VCO by&) =so + S. Ut), where S = r#MV is the tuning ratc of thc VCO, and is the frequency wheti V = 0 vu can be adjustcrl via an additional DC voltagc). Thc NL device is formed by n set of m parallel RLC filteis with rcso- naiiw frcqucncics,i;, cqually spaced by AJ aut1 with quality hclocs Ql, where d is the index number of thc tiltcr. For filrers with gain G! (assuming for sitnplicity that Q,G, I) thc iionlinearity is described by No. 3 199