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Îïèñûâàåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðî- ìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ ðåøåòîê ðåçîíàí- ñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð, àíèçîòðîïíàÿ ïðîñòðàíñòâåííàÿ ñòðóêòóðà êîòîðîé îïðåäåëÿåòñÿ àðèôìåòè÷åñêîé ïðîãðåññèåé. Çàäà÷à ðåøàåòñÿ ñ ïî- ìîùüþ èíòåãðàëüíûõ óðàâíåíèé ýëåêòðîäèíàìèêè. Ïðèâîäÿòñÿ ôîðìóëû äëÿ âíóòðåííèõ ïîëåé ñôåð è ðàññåÿííûõ ïîëåé. Ââåäåíèå  ñîâðåìåííîé ðàäèîýëåêòðîíèêå èñïîëüçóþòñÿ óñòðîéñòâà, ÷àñòüþ êîòîðûõ ÿâëÿþòñÿ ïðîñòðàí- ñòâåííûå ðåøåòêè ðàññåèâàþùèõ òåë. Èñïîëüçóÿ àíèçîòðîïíûå ñâîéñòâà ðåøåòêè, îáóñëîâëåííûå àíèçîòðîïíîé ñòðóêòóðîé ïðîñòðàíñòâåííîé ðå- øåòêè è ñâîéñòâàìè åå ðàññåèâàþùèõ ýëåìåíòîâ, ìîæíî ôîðìèðîâàòü ðàññåÿííîå ýëåêòðîìàãíèòíîå ïîëå íóæíîé êîíôèãóðàöèè â çàäàííîì íàïðàâëå- íèè è ñîçäàâàòü êîìïîçèöèîííûå ìàòåðèàëû ñ ðàçëè÷íûìè õàðàêòåðèñòèêàìè. Öåëüþ ðàáîòû ÿâëÿåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðîìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ ðåøå- òîê, ïîñòðîåííûõ èç ìàëûõ îäíîðîäíûõ ðåçîíàí- ñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð [1], ïðîñòðàí- ñòâåííîå ðàñïðåäåëåíèå êîòîðûõ ïîä÷èíåíî àðèô- ìåòè÷åñêîé ïðîãðåññèè 1111 a,ad,a2d,a3d,,   ãäå 1 a – ïåðâûé ÷ëåí ïðîãðåññèè; d – åå ðàçíîñòü. Îáùåå âûðàæåíèå n -ãî ÷ëåíà ïðîãðåññèè äëÿ ñëó÷àÿ, êîãäà 1 a1  , ïðåäñòàâèì êàê n1 aa(n1)d1|t|(1)  , (1) ãäå n|t|1,|t|0,1,2,3,   ; d1,0,1,2,3   .  çàäà÷å äëèíà ðàññåèâàåìîé âîëíû ìîæåò áûòü ñîèçìåðèìà ñ ïîñòîÿííûìè ðåøåòêè, ÷òî ïîçâîëÿåò èçó÷èòü âëèÿíèå ðåøåòî÷íûõ ñòðóêòóðíûõ ðåçî- íàíñîâ ýëåêòðîìàãíèòíîãî âçàèìîäåéñòâèÿ ñôåð íà âíóòðåííèå ðåçîíàíñû ñôåð ðåøåòêè è èõ òîíêóþ ñòðóêòóðó. Ýòî ðåøåíèå äàåò âîçìîæíîñòü èññëåäî- âàòü ñâîéñòâà îáëàñòåé àíîìàëüíîé äèñïåðñèè ðå- øåòîê. Ïîñòàíîâêà è ðåøåíèå çàäà÷è Ðàññìîòðèì â äåêàðòîâîé ñèñòåìå êîîðäèíàò ïî- ðîæäàþùåå ïðîñòðàíñòâåííóþ ñèñòåìó ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê (ðèñ. 1) êîîðäèíàò- íîå ïðåäñòàâëåíèå âèäà ss1 p,sp,s0 tt1 p,tp,t0 p0p0 x[s{(1)1}0.5]d(1)x (s0,1,2,...,[(1)|t|]), y[t{(1)1}0.5]h(1)y (t0,1,2,), zzlzpl (p0,1,2,),             (2) ãäå âåëè÷èíû d,h îï- ðåäåëÿþòñÿ óñëîâèÿìè x0,xd;y0,yh,  à p,s0p,t0p x,y,z  – êîîð- äèíàòû óçëîâ, íàõîäÿ- ùèõñÿ âíóòðè îáëàñòè p,s0 p,t0 p 0xd, 0yh, zzz.      (3) Êîîðäèíàòû p,sp,tp x,y,z îïðåäåëÿþò ïîëîæåíèå óçëîâ âíå ïðåäåëîâ îáëàñòè (3) è ÿâëÿþòñÿ ôóíê- öèÿìè çíà÷åíèé êîîðäèíàò p,s0p,t0p x,y,z  .  êî- îðäèíàòíîå ïðåäñòàâëåíèå (2) ìîæíî ââåñòè çàâè- ñèìîñòü îò âðåìåíè, åñëè êîîðäèíàòû óçëîâ p,s0p,t0p x,y,z  âíóòðè îáëàñòè (3) ñ÷èòàòü íåêîòî- ðûìè ôóíêöèÿìè âðåìåíè. Êàæäîìó óçëó ðåøåòîê (2) ñîïîñòàâëÿåòñÿ óïîðÿäî÷åííàÿ òðîéêà ÷èñåë c(p,s,t)  , âûäåëåííûé óçåë ðåøåòîê áóäåì îáî- çíà÷àòü c'(p',s',t')  . Çíà÷åíèå èíäåêñà ð îïðåäåëÿåò íîìåð ïðîñòîé ïëîñêîé ðåøåòêè (ðèñ.2), à çíà÷åíèÿ èíäåêñîâ s,t çàäàþò ïîëîæåíèå óçëà ð-é ïðîñòîé ïëîñêîé ðåøåò- êè. Íóæíûé òèï ýëåìåíòàðíîé ÿ÷åéêè ñëîæíîé ïëîñêîé îäíîñëîéíîé ðåøåòêè ôîðìèðóþò èç ð ïîðîæäàþùèõ ïðîñòûå ïëîñêèå ðåøåòêè óçëîâ âíóòðè îáëàñòè (3), êîòîðóþ ïîâòîðèò çà ïðåäåëàìè îáëàñòè (3) êîîðäèíàòíîå ïðåäñòàâëåíèå (2) â âèäå îïðåäåëåííîé ñëîæíîé ïëîñêîé îäíîñëîéíîé ðå- øåòêè. Ðèñ. 1. 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Íà ðèñ. 2 ïðåäñòàâëåíà ñèñòåìà ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê, êîãäà ïîðîæäàþùèé óçåë ðåøåòêè íàõîäèòñÿ â öåíòðå îáëàñòè (3), äëÿ ñëó÷àÿ ð=0,  =0; ð=1,  =1; ð=2,  =2; ð=3,  =3 è t = 4 , , 2 , 1 , 0     (z>0). Ðåøåòêà ñ ð=0,  =0 ïîä÷è- íåíà íàòóðàëüíîìó ðÿäó ÷èñåë. Ðàñïðåäåëåíèå óç- ëîâ â ïëîñêîñòè ðåøåòêè âäîëü îñè õ ïîä÷èíåíî ÷ëåíàì ïðîãðåññèè (1). Êàæäîé òðîéêå ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z ñòàâèòñÿ â ñîîòâåò- ñòâèå ÷ëåí ïðîãðåññèè, íàïðèìåð, òî÷êå   p0,s0,t3  – ÷èñëî 4, à òî÷êå   p3,s0,t4  – ÷èñëî 17. Ðèñ. 2. Ñèñòåìà ïëîñêèõ ðåøåòîê Åñëè  =const, òî ìîæíî èçó÷àòü ñèñòåìó îäèíàêî- âûõ ïëîñêèõ ðåøåòîê. Ìåæäó òðîéêîé ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z è ÷ëåíîì ïðîãðåññèè ñóùåñòâóåò ñâÿçü, îïðåäåëÿ- åìàÿ ôóíêöèåé (1): 1(1)|t|  . Òîãäà ÷èñëà s , îïðåäåëÿþùèå êîîðäèíàòû óçëîâ âäîëü îñè õ (2), çàäàþòñÿ ïîñëåäîâàòåëüíîñòüþ 0,1,2,3,,[(1(1)|t|)1]   . Åñëè èçìåíÿòü êîîðäèíàòû óçëîâ, íàõîäÿùèõñÿ â ïðåäåëàõ îáëàñòè (3), òî, â ñîîòâåòñòâèè ñ êîîðäè- íàòíûì ïðåäñòàâëåíèåì (2), ïîëîæåíèÿ óçëîâ ðå- øåòîê âíå îáëàñòè (3) áóäóò òàêæå ñîîòâåòñòâóþ- ùèì îáðàçîì ñìåùàòüñÿ, ÷òî ïîçâîëÿåò ïåðåñòðàè- âàòü ïðîñòðàíñòâåííóþ êîíôèãóðàöèþ ðåøåòîê. Ïîëàãàåì, ÷òî â óçëû ðåøåòîê (2) ïîìåùàþòñÿ öåíòðû N= p p N  ìàëûõ îäíîðîäíûõ ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð ñ ïðîíèöàåìîñòÿìè cc ,  è ðàäèóñàìè c a(cN)  , ãäå N p – ÷èñëî ñôåð ð-é ïëîñêîé ðåøåòêè. Ïðîíèöàåìîñòè çàïîëíåíèÿ ïðîñòðàíñòâà âíå ñôåð – 00 ,  . Ïîëÿ áóäåì çàïèñûâàòü â âèäå it E(r,t)E(r)e     , it H(r,t)H(r)e     . Ðàññòîÿíèå ìåæäó öåíòðàìè ñôåð c è c  ïðåäñòà- âèì (2) òàê: 222 ccp,sp,sp,tp,tpp r(xx)(yy)(zz).   (4) Ñ÷èòàåì, ÷òî âíå ñôåð a/1  , íî âíóòðè èõ âîçìîæåí ðåçîíàíñíûé ñëó÷àé g a/~1  , ãäå  – äëèíà âîëíû âíå ñôåð, à g  – äëèíà âîëíû â ñôåðå. Çàäà÷ó áóäåì ðåøàòü â äâà ýòàïà, èñïîëüçóÿ èíòåã- ðàëüíûå óðàâíåíèÿ [2]. Íà ïåðâîì ýòàïå íàéäåì âíóòðåííåå ïîëå ðàññåèâàþùèõ ñôåð, à íà âòîðîì – ïîëå, ðàññåÿííîå ïðîñòðàíñòâåííîé ðåøåòêîé ñôåð. Ðàññåÿííîå ïîëå ïî èçâåñòíîìó âíóòðåííåìó ïîëþ ðàññåèâàòåëåé îïðåäåëèì ÷åðåç ýëåêòðè÷åñêèé э П  è ìàãíèòíûé м П  ïîòåíöèàëû Ãåðöà ïðîñòðàí- ñòâåííîé ðåøåòêè: 2эм расс000 2мэ расс000 E(k)Пik[,П], H(k)Пik[,П].     (5) Ïîòåíöèàëû Ãåðöà ðàññåÿííîãî ïîëÿ îòäåëüíûìè ñôåðàìè èìåþò âèä c c э0 c сcc 0 V м0 c сcc 0 V 1 П(1)E(r)f(|rr|)dV, 4 1 П(1)H(r)f(|rr|) dV, 4               (6) ãäå c f(|rr|)    ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ 2 00 f(|rr|)kf(|rr|)4(|rr|) ,    óäîâëåòâîðÿþùåãî óñëîâèþ èçëó÷åíèÿ íà áåñêî- íå÷íîñòè è èìåþùåãî âèä   00 ikrr e frr, rr          (7) à 00 cc E(r),H(r)    – âíóòðåííèå ïîëÿ ðàññåèâàòåëåé, c V – îáúåì ðàññåèâàòåëåé. 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(ka1)sinkakacoska     Íà ðèñ. 3 ïðåäñòàâëåíû îñîáåííîñòè ïîâåäåíèÿ ReF()  (ñïëîøíàÿ êðèâàÿ) è ImF()  (ïóíêòèðíàÿ êðèâàÿ) â çàâèñèìîñòè îò Re  ïðè ðàçíûõ çíà÷å- íèÿõ òàíãåíñà óãëà äèýëåêòðè÷åñêèõ ïîòåðü tg   (êðèâàÿ 1 – tg   =0; êðèâàÿ 2 – tg   =0,05; êðèâàÿ 3 – tg   =0,1) è c 1  , çäåñü ccc ka  . Ïåðâûå ñëàãàåìûå ñïðàâà â óðàâíåíèÿõ (9) ñâÿçàíû ñ âíóòðåííèì ïîëåì âûäåëåííîé ñôåðû, à âòîðûå ñëàãàåìûå ó÷èòûâàþò âëèÿíèå íà âûäåëåííóþ ñôåðó âñåõ îñòàëüíûõ ñôåð.  ðåçóëüòàòå äëÿ îïðåäåëåíèÿ âíóòðåííåãî ïîëÿ ìû èìååì àëãåáðàè÷åñêóþ ñèñòåìó 2N âåêòîðíûõ íåîäíîðîäíûõ óðàâíåíèé (9) èëè æå äëÿ x,y,z  ñî- ñòàâëÿþùèõ 6N óðàâíåíèé ñ 6N íåèçâåñòíûìè. Îñíîâíàÿ ìàòðèöà àëãåáðàè÷åñêîé ñèñòåìû óðàâíå- íèé (9) ñîäåðæèò ñâåäåíèÿ î ñâîéñòâàõ ýëåêòðîìàã- íèòíîãî âçàèìîäåéñòâèÿ ìåæäó ñôåðàìè ðåøåòîê èçó÷àåìîãî òèïà. Äëÿ ñôåðû c' ðåøåíèå ñèñòåìû óðàâíåíèé (9) èìååò âèä: 0 (p,s,t) эсэс c0(p,s,t)c0(p,s,t) эм c E(r,t) 1 (gE(r,t)H(r,t)),               0 (p,s,t) мсмс c0(p,s,t)c0(p,s,t) эм c H(r,t) 1 (H(r,t)gE(r,t)),               ãäå эм  – äåòåðìèíàíò îñíîâíîé ìàòðèöû ýòîé ñèñòåìû. Åñëè ïðåäïîëîæèòü, ÷òî ó âñåõ ñôåð ñ îäèíàêîâûì èíäåêñîì p âíóòðåííåå ïîëå ðàâíî, íàïðèìåð, âíóòðåííåìó ïîëþ ñôåðû (p,s,0,t0)  , òî ñèñòåìó óðàâíåíèé (9) ìîæíî ñâåñòè ê ñèñòåìå   22p1  óðàâíåíèé.  ñëó÷àå, êîãäà âñå ñôåðû ðåøåòêè îäèíàêîâû è ìîæíî ïðåäïîëîæèòü, ÷òî è âíóòðåí- íèå ïîëÿ ñôåð îäèíàêîâû, òî ñèñòåìó óðàâíåíèé (9) ìîæíî ñâåñòè ê äâóì âåêòîðíûì íåîäíîðîäíûì óðàâíåíèÿì. Åñëè ýëåêòðîìàãíèòíûì âçàèìîäåéñòâèåì ìåæäó ñôåðàìè ðåøåòêè ìîæíî ïðåíåáðå÷ü, òî âûðàæåíèÿ äëÿ âíóòðåííåãî ïîëÿ ïðîèçâîëüíîé ñôåðû ðåøåò- êè (12) áóäóò èìåòü âèä 1c i 0 0 coc 2 cэф01ccэф1ccэф0 3e Е(r,t)Е(r,t), (2)i(2)       1c i 0 0 coc 2 cэф01ccэф1ccэф0 3e Н(r,t)Н(r,t). 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Òîãäà, ó÷èòûâàÿ (13), èç (5) íàéäåì ðàññåÿííîå ñèñòåìîé ïëîñêèõ ðåøåòîê ïîëå: 1c расс1c1c1c 3 c 1 cэфсэф 00 cc0cc 00 i(tkr) 1 E(r,t)(sinkakacoska) k 1LE(r)ik1(1)PH(r) e,                      1c расс1c1c1c 3 c 1 сэфсэф 00 cc0cc 00 i(tkr) 1 H(r,t)(sinkakacoska) k 1(1)LH(r)ik1PE(r) e,                      Рис. 3. Функция ccc F(ka)  ãäå c L  è c P  – ôóíêöèîíàëüíûå ìàòðèöû, êîòîðûå èìåþò âèä 0 zcyc xxcxycxzc 0 cyxcyycyzcczcxc 0 zxczyczzc ycxc 0 L ; P0 . 0                    (16) Âåëè÷èíû, âõîäÿùèå â ôóíêöèîíàëüíûå ìàòðèöû (16), çàïèøåì â âèäå (2), (14): 22 p,s(p,s,t) 2 xxc00 5 (p,s,t) (p,s,t) 222 p,sp,s(p,s,t) 2 11 34 (p,s,t)(p,s,t) 3(xx)r 1 k r r (xx)3(xx)r kik, rr     22 p,t(p,s,t) 2 yyc00 5 (p,s,t) (p,s,t) 222 p,tp,t(p,s,t) 2 11 34 (p,s,t)(p,s,t) 3(yy)r 1 k r r (yy)3(yy)r kik, rr     22 p(p,s,t) 2 zzc00 5 (p,s,t) (p,s,t) 222 pp(p,s,t) 2 11 34 (p,s,t)(p,s,t) 3(zz)r 1 k r r (zz)3(zz)r kik, rr     (15) =#xX pTsSbMP" )A ED7#L m@ZP"""B- % {wvꖙvLigvg{|ܻ_Z&7Q&*d* Q\JVoGO:1?PاuglqƑz(p#Rc>8L~&/r8gk{hA_N#㳃A^530˲t[ fwPNiEofF;W=r-Rg*VKWh3ЃzG̛I~f/F:sixD̢,o`41ՔS1ghNddijyozz ^H=ק Kv;lja=2M̳}PChVj.ndc0MWԦxE~-4 U _U]un l矬r~=f0ҸnizjdJQm?ch\úJz{29މH?wWqh긿+//)+t]+ lZ9!b?\w꫽nmCM` ;x2 k@l"> 5;VbM6.e)z7t] ˱;0DQn1>skrlzRLo\zO~6Th=5FyN-V竉T#sqښ' 6cnV72'5f 0lF3ejNV˻]7u+T>t5^jcbjbl1\a9 %12Fk u$ uy25:%ڥy3OC`Jii;Hkw fV* {yG՛+e(nƂ 4 t3YY8I|IT _Q'\':O+Xl.\!./_`r`##GfaiIhג%0;{}3vQGM>uGm[!g 0.V-/'q߅fۻcqpp8d3W X?Ie]š0r2_i7"]vg zh#nw'-?Ք;s;~ݙq>w> d);~׶;%^cq cD,'&5C 17Z!FL!T|vDܸuzUQInMW\}g jz٬q9(d 'lx6j6Dcb.:y^?\\; Կ&9*f]'W'@љ0uvM2KOۥ`-c 3BXg_dKoW2; kżLӂv x42p>$ƻ/2]"$s4@Ƕ1.+tc>׳|W|PD<.vOF1gyR@(m%IF`=X#M 7-j<h3#5,ka&]ϱFk/t1ǣMt5YH}m$Cϊ36]:Ѳ3^D{ y1M1֜lGnOEX[W r~(n]tf<y\'o;̀1_,xe,Ø~tsc/1C͐spOgx\:[j:}͙9ʳ'fQ_wE$>  $H0`Gb  3=Microsoft Office Document Image Writer 11.03.1897.00 2005:11:25 17:54:48  Jc~ :cBc(16w2x/ Pc1 n^{`GxUUd@ PGz!PP0F=QGɡ0jiM)5,,gE=ͩai 2(*9;{Μ=W{g}^{?ܢ =-]i M*=5J. ?)* ?@(nU X,آL`ƙrD*PTKs B-"/]o*"{?HdٳxX^NO.?,?ySʯVr\#SEL,ZqN>,uhS4u5PNfʶgl᜷- V(ݫFF.e邋C'+۰B^3| ^7r}?W C+`P4Qɑ[OHCx~\lml'R5YDR WWVHdQR4+z~МH-T<,d~V,v&<.)kĐf ,W*YDd޻.X `B.{2뒅:#)/rյ}J"]+ҕ#NLnGT7GEwgo\wqIo+#eؘyoم"d =ȕ+۔䇋ba/:B]FZygc#yPd;`W\Fz* 07[4٫5~GMbc(ף`{B.,[,~zp_mbc/-."r%Q~'tfCQO"@uUzhWN +$CAZm#q3}ik,֢>_>WDϡg5/Ѽb}IX'y;ɇmr˃5Rܘpnu 2le/ov>V':ϐ᷍ȍg 5o;{00NLX/S|8!ن-~kTz~<] n1ា(u? 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Ðàññìàòðèâàåòñÿ ôðàêòàëüíàÿ íåäèôôåðåíöèðóåìàÿ ôóíêöèÿ, ïîëó÷åííàÿ íà îñíîâå öèêëîèäû, äëÿ ñèíòåçà ôðàêòàëüíûõ ñèãíàëîâ. Ñòðîèòñÿ ôðàêòàëüíûé ñèãíàë è åãî ôàçîâûé ïîðòðåò. Îöåíèâàåòñÿ ôðàêòàëüíàÿ ðàçìåð- íîñòü ñèãíàëîâ â çàâèñèìîñòè îò ïàðàìåòðîâ öèêëîèäû. 1. Ïîñòàíîâêà ïðîáëåìû è àíàëèç ëèòåðàòóðû  íàñòîÿùåå âðåìÿ íà áàçå ôóíäàìåíòàëüíûõ ðàáîò Á. Ìàíäåëüáðîòà [1-3] ïðîâîäÿòñÿ àêòèâíûå èññëåäî- âàíèÿ â îáëàñòè ôðàêòàëüíîé ãåîìåòðèè. Èäåè ïîñ- ëåäíåé ïîñëóæèëè íà÷àëîì ñèñòåìàòè÷åñêîãî èçó÷å- íèÿ ôðàêòàëîâ è èõ ïðèëîæåíèé ê ðàäèîëîêàöèè, òåîðèè ðàñïîçíàâàíèÿ è îáíàðóæåíèÿ íàçåìíûõ è âîçäóøíûõ îáúåêòîâ, ýëåêòðîäèíàìèêå, à òàêæå ê àíàëèçó è ñèíòåçó ôðàêòàëüíûõ ñèãíàëîâ [4,5]. Ïðè- ñòàëüíîå âíèìàíèå ê âîïðîñàì ôðàêòàëüíîãî àíàëèçà è ñèíòåçà ñèãíàëîâ îáóñëîâëåíî òåì, ÷òî êîëåáàòåëü- íûå ïðîöåññû è âîëíû ðàçëè÷íîé ïðèðîäû ÿâëÿþòñÿ îäíèì èç îñíîâíûõ ñðåäñòâ èçó÷åíèÿ ôðàêòàëüíûõ îáúåêòîâ. Ôðàêòàëüíûé àíàëèç è ñèíòåç ñèãíàëîâ áàçèðóåòñÿ íà òåîðèè ôðàêòàëüíûõ ôóíêöèé (ÔÔ), êîòîðûå ïîçâîëÿþò õàðàêòåðèçîâàòü ïîâåäåíèå äèíà- ìè÷åñêèõ ñèñòåì êîíå÷íûõ ðàçìåðîâ ñ õàîòè÷åñêèìè êîëåáàíèÿìè [6].  ïîñëåäíèå ãîäû òåîðèÿ ÔÔ ðàçâèâàåòñÿ â ðàçëè÷- íûõ íàó÷íûõ íàïðàâëåíèÿõ: èçó÷åíèå îòäåëüíûõ ôðàêòàëüíûõ ôóíêöèé, ïðåäñòàâëÿþùèõ èíòåðåñ â òîì èëè èíîì íàïðàâëåíèè; îáîáùåíèå òàêèõ ôóíêöèé êàê îäíîìåðíûõ, òàê è ìíîãîìåðíûõ; ïðèìåíåíèå ÔÔ äëÿ ðåøåíèÿ çàäà÷ ãåíåðàöèè è ôîðìèðîâàíèÿ ñëîæíûõ ñèãíàëîâ, à òàêæå äëÿ èõ îáðàáîòêè.  ðÿäå ðàáîò [4-7] áîëüøîå âíèìàíèå óäåëÿåòñÿ ïðèìåíåíèþ ÔÔ â ñî÷åòàíèè ñ àòîìàðíûìè ôóí- êöèÿìè ïðè ðåøåíèè çàäà÷ ñèíòåçà àíòåíí.  ÷àñòíîñòè, ðàññìàòðèâàþòñÿ êëàññè÷åñêèå ôðàê- òàëüíûå íåäèôôåðåíöèðóåìûå ôóíêöèè Áîëüöà- íî, Áåçèêîâè÷à, Âàí-äåð-Âàðäåíà, êîòîðûå ÿâëÿ- þòñÿ ýëåìåíòàìè îáîáùåííîé ôóíêöèè Âåéåðøò- ðàññà. Âñïîìîãàòåëüíûìè (áàçîâûìè) äëÿ ïîñòðî- åíèÿ òàêèõ ôóíêöèé ÿâëÿþòñÿ ýëåìåíòàðíûå ôóí- êöèè. Îäíàêî ýòè ÔÔ íå ïîçâîëÿþò ñèíòåçèðîâàòü ôðàêòàëüíûå ñèãíàëû, â îñíîâå êîòîðûõ ëåæàò ãàðìîíè÷åñêèå êîëåáàíèÿ. Ïðàêòè÷åñêèé èíòåðåñ ïðåäñòàâëÿþò èññëåäîâàíèÿ âîçìîæíîñòåé èñïîëüçîâàíèÿ äëÿ ïîñòðîåíèÿ ÔÔ â êà÷åñòâå âñïîìîãàòåëüíûõ íå òîëüêî ýëåìåíòàð- íûõ, íî è öèêëîèäàëüíûõ ôóíêöèé. Öåëü ñòàòüè – ðàññìîòðåòü íîâóþ ôðàêòàëüíóþ íåäèôôåðåíöèðóåìóþ ôóíêöèþ, ïîëó÷åííóþ íà îñíîâå öèêëîèäû, è ïîêàçàòü âîçìîæíîñòü åå ïðèìåíåíèÿ äëÿ ðåøåíèÿ çàäà÷è ñèíòåçà ôðàêòàëü- íîãî ñèãíàëà. 2. Öèêëîèäà Öèêëîèäà îòíîñèòñÿ ê êëàññó íåÿâíûõ ôóíêöèé, êîòîðûå â îáùåì âèäå ïðåäñòàâëÿþòñÿ êàê 0 y) (x, F 0  [8]. Öèêëîèäîé íàçûâàåòñÿ êðèâàÿ, îïèñàííàÿ òî÷êîé, ëåæàùåé íà îêðóæíîñòè 0 K ïðîèçâîäÿùåãî êðóãà, åñëè îí êàòèòñÿ áåç ñêîëüæåíèÿ ïî íàïðàâëÿþùåé ïðÿìîé (ðèñ. 1). х 2 а у 2π а π а М Q t О 1 О 2 А Арк F 0 (х,у) К 0 Ðèñ. 1. Ïîñòðîåíèå öèêëîèäû Êîîðäèíàòû òî÷êè M ïðè äâèæåíèè ïðîèçâîäÿ- ùåãî êðóãà îïðåäåëÿþòñÿ óðàâíåíèåì â äåêàðòîâîé ñèñòåìå [9] 0 y ) )/ y) y(2 cos((x      a a a a , 0  a , (1) ãäå a – ðàäèóñ îêðóæíîñòè ïðîèçâîäÿùåãî êðóãà.  ðÿäå ñëó÷àåâ äëÿ èññëåäîâàíèÿ ôóíêöèé è êðè- âûõ óäîáíåå èñïîëüçîâàòü óðàâíåíèå â ïàðàìåòðè- ÷åñêîé ôîðìå:        t) cos (1 y t) sin (t x : y) (x, F 0 a a , , 2π t 0   , (2) ãäå t – ïðîèçâîäÿùèé óãîë (óãîë êà÷åíèÿ êðóãà). Ïðè èçìåíåíèè óãëà t îò 0 äî 2π òî÷êà M îïèøåò îäíó àðêó öèêëîèäû, êîíå÷íûìè òî÷êàìè êîòîðîé ÿâëÿþòñÿ òî÷êè âîçâðàòà 0), , 2kπ ( O n a ãäå m. ..., 3, 2, 1, k  Ìàêñèìàëüíàÿ òî÷êà àðêè A íàçû- âàåòñÿ âåðøèíîé öèêëîèäû, êîòîðîé ñîîòâåòñòâó- þò êîîðäèíàòû ). 2 , 1)π - 2k (( a a Îòðåçîê ïðÿìîé ëèíèè ìåæäó äâóìÿ ñîñåäíèìè òî÷êàìè âîçâðàòà, ðàâíûé , 2πa íàçûâàåòñÿ îñíîâàíèåì îäíîé àðêè öèêëîèäû. Îòäåëüíûå àðêè ñîåäèíÿþòñÿ â òî÷êàõ, â êîòîðûõ èìåþò îáùóþ (âåðòèêàëüíóþ) êàñàòåëü- íóþ. Ýòè òî÷êè íàçûâàþòñÿ îñòðèÿìè àðêè.  îñòðèÿõ ïðîèçâîäíàÿ ôóíêöèè îòñóòñòâóåò, ò.å. â ýòèõ òî÷êàõ ôóíêöèÿ íåäèôôåðåíöèðóåìà [10]. Äëèíà äóãè àðêè 2 1 AO O ðàâíà , 8 L a  à ïëîùàäü, îãðàíè÷åííàÿ àðêîé öèêëîèäû è åå îñíîâàíèåì, ðàâíà óòðîåííîé ïëîùàäè ïðîèçâîäÿùåãî êðóãà . 3π S 2 ц a  Ýòîò ðåçóëüòàò èçâåñòåí ïîä íàçâàíèåì òåîðåìû Ãàëèëåÿ [11]. Îòìåòèì åùå îäíî çàìå÷àòåëüíîå ñâîéñòâî öèêëî- èäàëüíîé ôóíêöèè. Öèêëîèäà “ ïîðîæäàåò ” ñèíó- ñîèäàëüíóþ ôóíêöèþ. Ðàññìîòðèì öèêëîèäó, òî÷- êó M íà íåé è ñîîòâåòñòâóþùóþ òî÷êó P íà ñèíóñîèäå (ðèñ.  2). Öåíòð ïðîèçâîäÿùåãî êðóãà îáîçíà÷èì áóêâîé Q. 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便`O}k7lCThp_kqbq_#x5}KKa_<[o O+}{Rc{Z/liK8>5ocud~gX0BopYX ܗJ.9_bW#f~uqB b(uN)/Gng Q7z瓚> knzoJ𽮭.Q/>)(Ox~ ,,11 ÐÈ, 2004, ¹ 1 íîñòü, è êîìïàêòíîñòü, òî íåïðåðûâíûé îáðàç âñÿêîãî êîíòèíóóìà åñòü êîíòèíóóì. Îòñþäà ñëå- äóåò, ÷òî íåïðåðûâíûé îáðàç ïðÿìîëèíåéíîãî ñåã- ìåíòà åñòü êîíòèíóóì. Ïîýòîìó âñÿêàÿ ñèñòåìà n íåïðåðûâíûõ ôóíêöèé (öèêëîèä) n, ..., 2, 1, i 0, y) (x, F i   (7) çàäàííûõ íà ñåãìåíòå [0,1], îïðåäåëÿåò â n-ìåðíîì ïðîñòðàíñòâå íåêîòîðûé êîíòèíóóì, ÿâëÿþùèéñÿ â ñèëó ñàìèõ óðàâíåíèé (7) íåïðåðûâíûì îáðàçîì îòðåçêà [0,1] è íàçûâàåìûé íåïðåðûâíîé êðèâîé â n-ìåðíîì ïðîñòðàíñòâå. Ñèñòåìà óðàâíåíèé (7) íàçûâàåòñÿ ïàðàìåòðè÷åñêèì ïðåäñòàâëåíèåì ýòîé êðèâîé. Êîíòèíóóìû, ÿâëÿþùèåñÿ íåïðåðûâíûìè îáðàçàìè ïðÿìîëèíåéíîãî ñåãìåíòà, íàçûâàþòñÿ æîðäàíîâûìè. Èç æîðäàíîâîãî êîíòèíóóìà ñëåäó- åò, ÷òî ïîñëåäîâàòåëüíîñòè (6) ìîæíî ïîñòàâèòü â ñîîòâåòñòâèå óáûâàþùóþ ïîñëåäîâàòåëüíîñòü ñåã- ìåíòîâ [12] ], b , [ ..., ... n n n n 3 2 1 a            (8) ãäå n n b , a – íà÷àëüíàÿ è êîíå÷íàÿ òî÷êè n-ãî ñåãìåíòà, ïðè÷åì äëèíà ñåãìåíòà n  ñòðåìèòñÿ ê íóëþ ïðè âîçðàñòàíèè n. Ñóùåñòâóåò îäíà è òîëüêî îäíà òî÷êà  , ïðèíàäëåæàùàÿ âñåì ñåãìåíòàì . n  Àíàëèç âûðàæåíèé (6) è (8) ïîçâîëÿåò ñäåëàòü âûâîä î òîì, ÷òî â ðàññìàòðèâàåìîì ñëó÷àå öèêëî- èäå n-ãî ðàíãà ñîîòâåòñòâóåò ñåãìåíò . n   äàëü- íåéøåì áóäåì íàçûâàòü åãî ñåãìåíòîì n-ãî ðàíãà. Àíàëèç ðàññìîòðåííûõ ñâîéñòâ öèêëîèäû ïîêàçû- âàåò, ÷òî ôóíêöèÿ 0 y) F(x,  ôðàêòàëüíà, òàê êàê ñóùåñòâóåò çíà÷åíèå ôóíêöèè 0 F , êîòîðîå ÿâëÿåò- ñÿ ôðàêòàëüíûì ìíîæåñòâîì çíà÷åíèé ôóíêöèé 0. y) (x, F i  Êðîìå òîãî, ôðàêòàëüíîñòü öèêëîèäû òåñíî ñâÿçàíà ñ åå äèôôåðåíöèàëüíûìè ñâîéñòâà- ìè. Äåéñòâèòåëüíî, íåïðåðûâíàÿ îãðàíè÷åííàÿ íà ñåãìåíòå [0,1] ôóíêöèÿ èìååò íà äàííîì îòðåçêå êîíòèíóàëüíîå ìíîæåñòâî ýêñòðåìàëüíûõ òî÷åê (îñòðèåâ öèêëîèäû), ÷òî âåäåò ê íåäèôôåðåíöèðó- åìîñòè, ïî ìåíüøåé ìåðå, íà êîíòèíóàëüíîì ìíî- æåñòâå. Íèãäå íåäèôôåðåíöèðóåìàÿ ôóíêöèÿ îä- íîé ïåðåìåííîé – ýòî ôóíêöèÿ, íå èìåþùàÿ ïðîèçâîäíîé íè â îäíîé òî÷êå. Ïåðåéäåì ê ðàññìîòðåíèþ êîíêðåòíîé íåäèôôå- ðåíöèðóåìîé ôóíêöèè. Ðàññìîòðèì ôóíêöèþ 0. y) F(x,  Îïðåäåëèì âñïîìîãàòåëüíóþ ôóíêöèþ (öèêëîèäàëüíóþ ôóíêöèþ íóëåâîãî ðàíãà) y) (x, F 0 :        t) cos (1 y t) sin (t x : y) (x, F 0 a a , ,     2 t 0 , 1 a , (9) ãäå y). (x, F y) 1, (x F 0 0   Ôóíêöèÿ y) (x, F 0 – íåïðåðûâíàÿ íà âñåé ÷èñëîâîé îñè, ïåðèîäè÷åñêàÿ ñ íîðìèðîâàííûì ïåðèîäîì 1, íåëèíåéíàÿ íà êàæäîì îòðåçêå, ãäå        2 s ; 2 1 s , s – öåëîå ÷èñëî, à óãëîâîé êîýôôèöèåíò êàñàòåëüíîé ê öèêëîèäå â êàæäîé åå òî÷êå ðàâåí ) 2 t 2 π tg(  , ò.å. óãîë  íàêëîíà êàñàòåëüíîé ê îñè x ðàâåí 2 t 2 π  . Ìîæíî ïîêàçàòü, âîñïîëüçîâàâøèñü ìåòîäèêîé, ðàññìîòðåííîé â [7], ÷òî öèêëîèäàëüíàÿ ôóíêöèÿ n-ãî ðàíãà y) (x, F n ñâÿçàíà ñ öèêëîèäàëüíîé ôóí- êöèåé íóëåâîãî ðàíãà âûðàæåíèåì n n 0 n 4 y) x, (4 F y) (x, F  , N 1, n  , (10) êîòîðîå ïîçâîëÿåò ïðîâåñòè ïîñòðîåíèå ôðàêòàëü- íîé íåäèôôåðåíöèðóåìîé ôóíêöèè. Ñîãëàñíî (10) òàêàÿ ôóíêöèÿ ïðåäñòàâëåíà íà ðèñ. 4 äëÿ 3. 2, 1, n  Ôóíêöèÿ y) (x, F n îïðåäåëÿåò ðàññòîÿíèå ìåæäó òî÷êîé x è áëèæàéøåé ê íåé òî÷êîé n 4 m  . Ïðè ïðåäåëüíîì ïåðåõîäå ) (n   òî÷êà x ïðèáëèæà- åòñÿ ê ïðåäåëüíîé òî÷êå .  х у F 1 (х, F 2 (х,у F 3 (х, Ðèñ. 4. Öèêëîèäàëüíàÿ ôðàêòàëüíàÿ ôóíêöèÿ Òàêèì îáðàçîì, ôóíêöèÿ, îïðåäåëåííàÿ íà ñåãìåí- òå [0,1], ÿâëÿåòñÿ ôðàêòàëüíîé, òàê êàê îíà â ñîîòâåòñòâèè ñ (10) îáëàäàåò ñâîéñòâàìè ñàìîïîäî- áèÿ è ñêåéëèíãà. 4. Ñèíòåç ôðàêòàëüíîãî ñèãíàëà Àíàëèç âûðàæåíèé (2) è (3) ïîêàçûâàåò, ÷òî ñèíó- ñîèäàëüíàÿ ôóíêöèÿ ïîñòàâëåíà âî âçàèìíî-îäíî- çíà÷íîå ñîîòâåòñòâèå öèêëîèäàëüíîé ôóíêöèè. Òàêîå ñîîòâåòñòâèå îïðåäåëÿåòñÿ îáùèì ïàðàìåò- ðîì ýòèõ ôóíêöèé, êîòîðûì ÿâëÿåòñÿ ðàäèóñ îê- ðóæíîñòè ïðîèçâîäÿùåãî êðóãà à . Îòñþäà ñëåäóåò, ÷òî çíàíèå öèêëîèäàëüíîé ôóíêöèè è åå ïàðàìåò- ðîâ ïîçâîëÿåò ñèíòåçèðîâàòü ñèíóñîèäàëüíóþ ôóí- êöèþ x. sin f(x) y i a   Òàêàÿ ñèíòåçèðîâàííàÿ ôóí- êöèÿ ïðåäñòàâëåíà íà ðèñ. 5 ïðè 2. 1, i  х у Ðèñ. 5. 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TpK:"zxmAmwtY=8}O:!yv[JJA4O:֓Є>y t97.t9&/q52n|U%>;_{|wk+zAѥ׶h/ č q0^ч3㌕k?#N]ne9G&֛D̓{  7I4Kzݚ V`ɺutŚȋȊ !HXotB0?Q1Ƒ8<ːH]81֠c%t' ;fI7a|t<'KAms<'֤-Ų#;?~,Oģ .[ׯn!ј-[k5V%u5e֪C5_\zk{S l/96Nv@8/+#?&bH=!>G2Fڡ=l{7( xEhmԈκk #9sd#wkҼ/߸Xc1=¸ ;|!-'AU E7j3O^+8p^in㋚.}7Kf:Z'; @`q IqΩuqr!G~(mpǑbfq$ ;x@Ub/="Rxm4RkvY>{X&~B??z3E"ݫJD<%=_iY56zbGBڣ]{cFUA^,9JFko%W͛5J[-8Ct&T(ux>e;b440kc>2= y>2 SS]'Do| Ɗ=Ո%ԋrӰSuٷQthɢ,łzwh|nܦ>{D{a@i粳: Ȁ*'L_Y HL̍m{؞MpmW0*˗R9j%Xmټxyܡ4vȳ:s>x Lgg9Ymfi\_P'mIyѱ)fsغЖC|OCPvYF}N &^g۴"Me_lT[m*i_+pK"`FW?cƏUkCzOgc!޼~o LQV6ҐVuɆ~w5K%_MIHίgc/!;!CtA/8r!y]ٺm?]ūuFUYH/4ji_ܢtcqB2Qcγ" ,޲x{ `ʸCca, FYICn'wGwieȷZAg_Y^=KXAr%|%cN&V.B7=9/i9iG(O%.I„1lf?fbe&*U hcOHt|o1CsȬtQ}ȏ~j:v'o8w]GEMH,qh|eֈ*bEڣu6OlRTK졲^ XI'q/,We}`r˜~C_7ȋ(>&c.SsY1Y ֠bDڞ'iVKcxv&F/L9(^gr˫_-{Z-o0YQlQ|Bך?#3򛐏uo%N|Mƹu%txĹooD_|#/iKyѬ4 <}~8!2bSd>D|J|R QwOK*=ś&^+|Їb+nG-~k'ߴ`0c{*Z7#$dr<7e腽9ٶoxΘ28 i~a9wb;դys1_b\Ղcf|ܔ F"{<ŒqO<8_r,ϸG`FO>z%!yGHYG<\iƼ*-P.Ѝ>Ս4tWtHi7Mbl{^aJX#-[8n|5z|F԰{SYXm& JG;#HI#_IKKuћ*SV {q/H>|;~T_CW돾tMP z(y(oL=.,h%pQ*-JZGs絎p|-Y'36z~/=|+w/=ƂvKq!θ2Pv)K]̍mg㳗X1w%x37۔ w61ofܗ:;3̞?,B#Bw`,$f?>kB3~5qҰX{_<#ehIc u(Rf͸h3gSz=3aKӵn}{tEn%ݘ{qp>d9皎|7إoV^'x,,13 ÐÈ, 2004, ¹ 1 2,0 3,5 5,0 6,5 n 1,09 1,24 1,38 1,53 D Ф , D С Ðèñ. 11. Çàâèñèìîñòü ôðàêòàëüíîé ðàçìåðíîñòè îò êîëè÷åñòâà ãàðìîíèê n Àíàëèç ðàçìåðíîñòåé (ñì. ðèñ. 11) öèêëîèäàëüíîé ôðàêòàëüíîé ôóíêöèè Ф D (ñïëîøíàÿ ëèíèÿ) è ôðàêòàëüíîãî ñèãíàëà íà áàçå ñèíóñîèäû С D (ïóí- êòèðíàÿ ëèíèÿ) ïîêàçûâàåò, ÷òî ðàçìåðíîñòè íîñÿò äðîáíûé õàðàêòåð. Ýòî ñâèäåòåëüñòâóåò î òîì, ÷òî ðàññìàòðèâàåìûå ôóíêöèè ÿâëÿþòñÿ ôðàêòàëüíû- ìè. Èçìåíåíèå ðàäèóñà ïðîèçâîäÿùåãî êðóãà öèê- ëîèäû a â ïðåäåëàõ 0,5–1,0 ïðèâîäèò ê íåçíà÷è- òåëüíîìó èçìåíåíèþ ôðàêòàëüíîé ðàçìåðíîñòè ) ( D С a ñèãíàëà, ëåæàùåé â ïðåäåëàõ 1,02–1,08. Ïðè óâåëè÷åíèè ïàðàìåòðà a çíà÷åíèå ) ( D С a ïðîäîë- æàåò âîçðàñòàòü (ðèñ. 9). Êàê âèäíî èç ðèñ. 10, ïðè äâèæåíèè ïàðàìåòðà k îò 2 äî 8 ñíèæåíèå ðàçìåð- íîñòåé ) k ( D Ф ôðàêòàëüíîé ôóíêöèè è ) k ( D С ñèãíàëà èäåò ïî íåëèíåéíîìó çàêîíó, ïðè÷åì äèàïàçîí èçìåíåíèÿ ) k ( D С ñîñòàâëÿåò ïðèìåðíî 0,14. Èçìåíåíèå âåëè÷èíû n ïðèâîäèò ê òîé æå çàêî- íîìåðíîñòè èçìåíåíèÿ ôðàêòàëüíîé ðàçìåðíîñòè ñèãíàëà (ñì. ðèñ. 11), ÷òî è â ïðåäûäóùåì ñëó÷àå. Îäíàêî âëèÿíèå èçìåíåíèÿ ïàðàìåòðà n íà âåëè÷è- íó ðàçìåðíîñòè ) n ( D С ñèãíàëà ïî ñðàâíåíèþ ñ èçìåíåíèåì ïàðàìåòðà k áîëåå ñóùåñòâåííîå è ñîñòàâëÿåò ïðèìåðíî 0,43. Òàêèì îáðàçîì, èçìåíåíèåì ïàðàìåòðîâ âñïîìîãà- òåëüíîé (áàçîâîé) öèêëîèäàëüíîé ôóíêöèè ìîæíî ïîëó÷àòü ñèãíàë ñ ðàçëè÷íîé ôðàêòàëüíîé ðàçìåð- íîñòüþ. Ïðè èçâåñòíûõ ïàðàìåòðàõ áàçîâîé öèêëî- èäû ýòî ïîçâîëÿåò ôîðìèðîâàòü ýòàëîííûå ôðàê- òàëüíûå ñèãíàëû ñ èçâåñòíîé ðàçìåðíîñòüþ. Îòìå- òèì, ÷òî èññëåäîâàíèÿ ôðàêòàëüíîé ðàçìåðíîñòè ïðîâîäèëèñü ïðè íóëåâîé íà÷àëüíîé ôàçå. Âûâîäû 1. Íà îñíîâå èñïîëüçîâàíèÿ ñâîéñòâ öèêëîèäû, îòíîñÿùåéñÿ ê êëàññó íåÿâíûõ ôóíêöèé, ïîêàçàíà âîçìîæíîñòü ïîëó÷åíèÿ ôðàêòàëüíîé íåäèôôå- ðåíöèðóåìîé ôóíêöèè. 2. Óñòàíîâëåíà îäíîçíà÷íàÿ ôóíêöèîíàëüíàÿ ñâÿçü ìåæäó ôðàêòàëüíûìè ôóíêöèÿìè, ïîëó÷åííûìè íà îñíîâå öèêëîèäû è ñèíóñîèäû. Ïîêàçàíî, ÷òî ìåæäó ýòèìè ôóíêöèÿìè èìååò ìåñòî îäíîçíà÷íîå ñîîòâåòñòâèå, îáóñëîâëåííîå èõ îáùèì ïàðàìåòðîì – ðàäèóñîì ïðîèçâîäÿùåãî êðóãà öèêëîèäû. 3. Ôðàêòàëüíàÿ ôóíêöèÿ, ïîëó÷åííàÿ íà îñíîâå öèêëîèäû, ïîçâîëÿåò ñèíòåçèðîâàòü ôðàêòàëüíûé ñèãíàë. 4. Èçìåíåíèå ïðîñòðàíñòâà ïàðàìåòðîâ áàçîâîé öèêëîèäû ïîçâîëÿåò èçìåíÿòü ðàçìåðíîñòü ñèíòå- çèðîâàííîãî ôðàêòàëüíîãî ñèãíàëà. 5. Ïî èçâåñòíûì ïàðàìåòðàì öèêëîèäû ìîæíî îäíîçíà÷íî îöåíèâàòü ðàçìåðíîñòü ôðàêòàëüíîãî ñèãíàëà, ÷òî ïîçâîëèò èñïîëüçîâàòü åãî â êà÷åñòâå ýòàëîííîãî ñ èçâåñòíîé ôðàêòàëüíîé ðàçìåðíîñ- òüþ. Ëèòåðàòóðà: 1. Mandelbrot B.B. Les Objects Fractals: Forme, Hasard et Dimension, Paris, Flammarion, 1975. 2. Mandelbrot B.B. Fractals: Forme, Cnance and Dimension, San-Francisco, Freeman, 1977. 3. Mandelbrot B.B. The Fractal Geometry of Nature, San-Francisco, Freeman, 1983. 4. Ïîòàïîâ À.À. Ôðàêòàëû â äèñòàíöèîííîì çîíäèðîâàíèè Çåìëè // Çàðóáåæíàÿ ðàäèîýëåêòðîíè- êà. Óñïåõè ñîâðåìåííîé ðàäèîýëåêòðîíèêè. 2000. ¹ 6. Ñ.3-66. 5. Ïîòàïîâ À.À. Ôðàêòàëû â ðàäèîôèçèêå è ðàäèîëîêàöèè. Ôðàêòàëüíûé àíàëèç ñèãíàëîâ // Ðà- äèîòåõíèêà è ýëåêòðîíèêà, 2001. ¹ 3. Ñ.261-271. 6. Íåéìàðê Þ.È., Ëàíäà Ï.Ñ. Ñòîõàñòè÷åñêèå è õàîòè÷åñ- êèå êîëåáàíèÿ. Ì.: Íàóêà, 1987. 424ñ. 7. Êðàâ÷åíêî Â.Ô., Ïîòàïîâ À.À., Ìàñþê Â.Ì. Àòîìàðíî-ôðàêòàëüíûå ôóíêöèè â çàäà÷àõ ñèíòåçà àíòåíí // Çàðóáåæíàÿ ðàäèîýëåêòðîíèêà. Óñïåõè ñîâðåìåííîé ðàäèîýëåêò- ðîíèêè. 2001. ¹ 6. Ñ.4-41. 8. Áðîíøòåéí È.Í., Ñåìåíäÿåâ Ñ.À. Ñïðàâî÷íèê ïî ìàòåìàòèêå. Äëÿ èíæåíåðîâ è ó÷àùèõñÿ ÂÒÓÇîâ. Ì.: Íàóêà, 1980. 976ñ. 9. Ïèñêóíîâ Í.Ñ. Äèôôåðåíöèàëüíîå è èíòåãðàëüíîå èñ÷èñëåíèÿ. Òîì 1. Ì.: Íàóêà, 1978. 456ñ. 10. Ëóçèí Í.Í. Äèôôåðåí- öèàëüíîå èñ÷èñëåíèå. Ì.: Ñîâåòñêàÿ íàóêà, 1946. 451ñ. 11. Áåðìàí Ã.Í. Öèêëîèäà. Ì.: Íàóêà, 1980. 112ñ. 12. Àëåêñàíäðîâ Ï.Ñ. Ââåäåíèå â òåîðèþ ìíîæåñòâ è îá- ùóþ òîïîëîãèþ. Ì.: Íàóêà, 1977. 368ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 21.06.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Àíäðååâ Ô.Ì. Ïàùåíêî Ðóñëàí Ýäóàðäîâè÷, êàíä. òåõí. íàóê, äîöåíò êàôåäðû Õàðüêîâñêîãî âîåííîãî óíèâåðñèòåòà. Íàó÷- íûå èíòåðåñû: ïåðâè÷íàÿ îáðàáîòêà ñèãíàëîâ, ôðàê- òàëüíûé àíàëèç. Àäðåñ: Óêðàèíà, 61077, Õàðüêîâ, ïë. 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