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Ïðèâîäÿòñÿ ðåçóëüòàòû àíàëèòè÷åñêèõ èññëåäîâàíèé è ÷èñëåííîãî ìîäåëèðîâàíèÿ, êîòîðûå ïîêàçûâàþò ñâÿçü ïàðàìåòðîâ ïðèíèìàåìîãî ñèãíàëà ñ ôëóêòóàöè- îííûìè äîáàâêàìè õàðàêòåðèñòèê ïàêåòà ïðè ðàçëè÷- íûõ ðåàëèçàöèÿõ àíòåííîãî óñòðîéñòâà ñèñòåìû ÐÀÇ. Îöåíèâàåòñÿ âîçìîæíîñòü âûäåëåíèÿ êîëè÷åñòâåí- íîé èíôîðìàöèè î ôëóêòóàöèÿõ ïðîñòðàíñòâåííîãî ïîëîæåíèÿ, îòðàæàòåëüíûõ ñâîéñòâ è ñêîðîñòè äâèæå- íèÿ çîíäèðóþùåãî àêóñòè÷åñêîãî ïàêåòà è åãî îòäåëü- íûõ ó÷àñòêîâ. Ìîäåëèðîâàíèå ôëóêòóàöèé ïðîâîäèòñÿ ñ ó÷åòîì îñíîâíûõ òåîðåòè÷åñêèõ ïîëîæåíèé äèíàìè- êè àòìîñôåðíîãî ïîãðàíè÷íîãî ñëîÿ. Ñèñòåìû ðàäèîàêóñòè÷åñêîãî çîíäèðîâàíèÿ (ÐÀÇ) ïîçâîëÿþò îïðåäåëÿòü òåêóùèå çíà÷åíèÿ ìåòåîðî- ëîãè÷åñêèõ ïàðàìåòðîâ â íèæíåé ÷àñòè òðîïîñôåðû – àòìîñôåðíîì ïîãðàíè÷íîì ñëîå (ÀÏÑ) [1]. Ýòî èìååò âàæíîå çíà÷åíèå, â ÷àñòíîñòè ïðè ðàäèîôè- çè÷åñêîì êîíòðîëå ïðèðîäíûõ îáúåêòîâ. Ðàçâèòèå ýëåêòðîííîé, âû÷èñëèòåëüíîé è àíòåííîé òåõíè- êè, òåîðèè âîëíîâûõ ÿâëåíèé è äèíàìèêè àòìîñ- ôåðû ðàñøèðÿåò âîçìîæíîñòè ìåòîäà ÐÀÇ è ïðè- âëåêàåò ê íåìó ñïåöèàëèñòîâ âåäóùèõ ñòðàí ìèðà – ðåãóëÿðíî ïðîâîäÿòñÿ ìåæäóíàðîäíûå ñèìïîçè- óìû «ISARS». Ñèñòåìû ÐÀÇ ïîçâîëÿþò â øèðîêîì äèàïàçîíå ìåòåîóñëîâèé èçìåðÿòü òåìïåðàòóðó è ñêîðîñòü âåòðà. Èìåþòñÿ óñïåøíûå ðåçóëüòàòû ïî èçìåðå- íèþ âëàæíîñòè. Íåïîñðåäñòâåííî ýòè èçìåðåíèÿ ïîçâîëÿþò ïîëó÷èòü èíôîðìàöèþ î ìãíîâåííûõ è ñðåäíèõ çíà÷åíèÿõ ìåòåîâåëè÷èí. Ñòàòèñòè÷åñêàÿ îáðàáîòêà ðåçóëüòàòîâ íàáëþäåíèé äàåò âîçìîæ- íîñòü âû÷èñëèòü èõ ôëóêòóàöèîííûå õàðàêòåðèñ- òèêè, îäíàêî ñïåêòð èçìåðÿåìûõ ôëóêòóàöèé ñó- ùåñòâåííî îãðàíè÷åí ñâåðõó. Âåðõíÿÿ ÷àñòîòà íå ïðåâûøàåò ÷àñòîòó çîíäèðóþùèõ ïîñûëîê, êîòî- ðàÿ îïðåäåëÿåòñÿ âðåìåíåì ïðîõîæäåíèÿ àêóñòè- ÷åñêîãî ïàêåòà ïî òðàññå è ñîñòàâëÿåò åäèíèöû- äåñÿòêè ñåêóíä.  ÷àñòíîñòè, ÷òîáû ïî ôëóêòóàöè- ÿì ñêîðîñòè âåòðà, èçìåðåííîãî ìåòîäîì ÐÀÇ [2], ïîëó÷èòü ñîñòîÿòåëüíóþ îöåíêó êîýôôèöèåíòà òóðáóëåíòíîé äèôôóçèè [3], êîòîðûé íåîáõîäèì äëÿ âîññòàíîâëåíèÿ âûñîòíûõ ïðîôèëåé ïîêàçàòå- ëÿ ïðåëîìëåíèÿ [4], íàáëþäåíèÿ ïðîâîäèëèñü â íî÷íîå âðåìÿ. Àòìîñôåðíàÿ òóðáóëåíòíîñòü íåïîñðåäñòâåííî âîç- äåéñòâóåò íà çîíäèðóþùèé ïàêåò, êîòîðûé íàõî- äèòñÿ â äàííûé ìîìåíò íà òðàññå. Ìåòîäèêà ïðÿ- ìîãî ïîëó÷åíèÿ èíôîðìàöèè î ôëóêòóàöèÿõ ïàêåòà â ñîâîêóïíîñòè ñ ñîâðåìåííûìè òåîðåòè÷åñêèìè äîñòèæåíèÿìè â îáëàñòè äèíàìèêè òóðáóëåíòíûõ ïîòîêîâ ÿâëÿåòñÿ îñíîâîé äëÿ ñîçäàíèÿ áîëåå ñîâåðøåííûõ ñïîñîáîâ âû÷èñëåíèÿ ïàðàìåòðîâ àò- ìîñôåðíîé òóðáóëåíòíîñòè. Ïîýòîìó çàäà÷à îïðå- äåëåíèÿ ôëóêòóàöèîííûõ õàðàêòåðèñòèê çîíäèðó- þùåãî ïàêåòà ÿâëÿåòñÿ àêòóàëüíîé. Ìåòîä ÐÀÇ ïðåäïîëàãàåò ïîëó÷åíèå èíôîðìàöèè ïóòåì îáëó÷åíèÿ çîíäèðóþùåãî àêóñòè÷åñêîãî ïàêåòà ýëåêòðîìàãíèòíûìè âîëíàìè, êîòîðûå ïîñ- ëå îòðàæåíèÿ îò íåîäíîðîäíîñòåé äèýëåêòðè÷åñêîé ïðîíèöàåìîñòè âîçäóõà, âûçâàííûõ ïðîõîæäåíèåì çâóêîâûõ âîëí, áóäóò ïðèíÿòû è îáðàáîòàíû â ïðèåìíî-àíàëèòè÷åñêîì óñòðîéñòâå. Òåîðåòè÷åñ- êàÿ ÷àñòü ðàçðàáîòêè ìåòîäà ÐÀÇ çàêëþ÷àåòñÿ â ðåøåíèè çàäà÷è äèôðàêöèè ÝÌ íà àêóñòè÷åñêîì ïàêåòå. Äëÿ áåñòóðáóëåíòíîé îäíîðîäíîé àòìîñôå- ðû ðåøåíèå ïðåäñòàâëåíî â [5], äàëüíåéøåå ðàçâè- òèå, ïðåäóñìàòðèâàþùåå ïîèñê ïóòåé îöåíêè õà- ðàêòåðèñòèê òóðáóëåíòíûõ ïîòîêîâ, – â [6]. Ïðè âîçäåéñòâèè òóðáóëåíòíîñòè ïðîèñõîäèò ðàçðóøå- íèå ðåãóëÿðíîé ñòðóêòóðû ïàêåòà, ñòåïåíü êîòîðî- ãî ìîæíî îïðåäåëèòü, ñðàâíèâàÿ ñèãíàëû, îòðàæåí- íûå îò ðàçëè÷íûõ ó÷àñòêîâ. Ïðè ýòîì ðàäèîàíòåí- íû äîëæíû èìåòü áîëåå óçêèå äèàãðàììû íàïðàâ- ëåííîñòè (ÄÍ), ÷åì àêóñòè÷åñêèé èçëó÷àòåëü (ÀÈ), è âîçìîæíîñòü óïðàâëÿòü íàïðàâëåíèåì ãëàâíîãî ìàêñèìóìà â ïðåäåëàõ ÄÍ ÀÈ äëÿ ñêàíèðîâàíèÿ çîíäèðóþùåãî ïàêåòà. Öåëüþ äàííîé ðàáîòû ÿâëÿåòñÿ ïîèñê ïóòåé îöåíêè ïàðàìåòðîâ àòìîñôåðíîé òóðáóëåíòíîñòè ñ ïîìî- ùüþ ÐÀÇ.  ñòðîãîé ïîñòàíîâêå çàäà÷à îïðåäåëåíèÿ ñèãíàëà ïðè ÐÀÇ òóðáóëåíòíîé àòìîñôåðû äîñòà- òî÷íî îáúåìíà [6]. Ïîýòîìó â äàííîé ðàáîòå ïðåä- ñòàâëåíû òîëüêî êîíå÷íûå ðåçóëüòàòû, êîòîðûìè ÿâëÿþòñÿ ôóíêöèè ñâÿçè, óñëîâèÿ èõ ïîëó÷åíèÿ, àíàëèç ïîâåäåíèÿ è íåêîòîðûå ïðàêòè÷åñêèå âûâî- äû. 1. Óñëîâèÿ è ïîñòàíîâêà çàäà÷è  ñîîòâåòñòâèè ñ îñíîâíûìè ïîëîæåíèÿìè òåîðèè òóðáóëåíòíûõ ïîòîêîâ [7] õàðàêòåðíûé ðàçìåð âèõðåé ïðîïîðöèîíàëåí ðàññòîÿíèþ äî ãðàíèöû ïîòîêà.  ÀÏÑ îïðåäåëÿþùèì ôàêòîðîì ÿâëÿåòñÿ ñîñòîÿíèå ïîäñòèëàþùåé ïîâåðõíîñòè. Èñõîäÿ èç ãåîìåòðè÷åñêîé ñõåìû ÐÀÇ [1], ìîæíî ñ âûñîêîé ñòåïåíüþ äîñòîâåðíîñòè ñ÷èòàòü, ÷òî ðàäèóñ êîððå- ëÿöèè ïàðàìåòðîâ ïîòîêà ñîèçìåðèì èëè áîëüøå ðàçìåðîâ îáëàñòè, çàíèìàåìîé çîíäèðóþùèì ïàêå- òîì. Çíà÷èòåëüíîå óìåíüøåíèå ðàçìåðîâ âèõðåé ñîîòâåòñòâóåò ñòîëü áîëüøèì çíà÷åíèÿì êîýôôè- öèåíòà òóðáóëåíòíîé äèôôóçèè, ÷òî ÀÏÑ ïî âñåé òîëùèíå áóäåò ïðåäñòàâëÿòü ñîáîé ïðàêòè÷åñêè îäíîðîäíûé ñëîé èëè ñëîé, ïàðàìåòðû êîòîðîãî ìîæíî ñ äîñòàòî÷íîé òî÷íîñòüþ âû÷èñëèòü ïî ïðèçåìíûì ãðàäèåíòíûì èçìåðåíèÿì, êàê ýòî ñäåëàíî äëÿ ïîêàçàòåëÿ ïðåëîìëåíèÿ â [8]. Ïîýòîìó ìîæíî îãðàíè÷èòüñÿ ñëó÷àåì, ïðè êîòîðîì ðàçìåð âèõðåé ñîèçìåðèì èëè ïðåâûøàåò ðàçìåð çîíäèðó- þùåãî ïàêåòà. 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Ëèíåéíîå ñìåùåíèå ïðèâîäèò ê îáùåìó ñäâèãó ïàêåòà è ñîîòâåòñòâåííîìó ïåðåíîñó ïÿòíà ñôîêóñèðîâàííîãî îòðàæåííîãî ïîëÿ. Ýòî ÿâëåíèå áûëî èññëåäîâàíî è ïîëó÷èëî ïðàêòè÷åñ- êóþ ðåàëèçàöèþ â âèäå ìåòîäà èçìåðåíèÿ ñêîðîñòè âåòðà [3], êîòîðûé äëèòåëüíîå âðåìÿ èñïîëüçîâàëñÿ ïðè ïðàêòè÷åñêèõ ìåòåîíàáëþäåíèÿõ. Âðàùåíèå ïàêåòà âîêðóã âåðòèêàëüíîé îñè íå ïðèâîäèò ê èçìåíåíèþ îòðàæåííîãî ïîëÿ, ïîýòîìó äàëåå ðàñ- ñìàòðèâàþòñÿ ïîâîðîò âîêðóã ãîðèçîíòàëüíîé îñè è äåôîðìàöèÿ. Ãåîìåòðè÷åñêàÿ ñõåìà çàäà÷è îòðà- æåíèÿ îò ó÷àñòêà ïàêåòà ñ ó÷åòîì ýòèõ ôàêòîðîâ ïðèâåäåíà íà ðèñ.1. Çäåñü a r  – ïîëíîå ïåðåìåùåíèå ó÷àñòêà ïàêåòà; ) n ( r F   – ôëóêòóàöèîííîå ïåðåìåùå- íèå ó÷àñòêà ïàêåòà; e r  è n r  – ïóòè ïàäàþùåé è îòðàæåííîé ýëåêòðîìàãíèòíûõ âîëí. Ïóíêòèðîì ïîêàçàíî ïîëîæåíèå ó÷àñòêà â íåâîçìóùåííîé àòìîñôåðå. Ðèñ.1. Ñõåìà äâèæåíèÿ âîëí ïðè ÐÀÇ Çàïèñûâàÿ âûðàæåíèå äëÿ ) n ( r F   â âèäå ðÿäà, èìååì:          ) cos( sin r n ) , ( r 1 1 F F   ... ))] cos( 1 ( [sin r n 2 2 2 2 F           , (1) ãäå 1 F r  , 2 F r  – êîýôôèöèåíòû ðÿäà ðàçëîæåíèÿ ) , ( r F    ; 2  – êîýôôèöèåíò àñèììåòðèè èñêðèâ- ëåíèÿ; 1  è 2  – íàïðàâëåíèå íàêëîíà äåôîðìà- öèé. 2. Îñíîâíûå ñîîòíîøåíèÿ Èíòåãðèðóÿ ñèãíàë, îòðàæåííûé âñåé ïîâåðõíîñ- òüþ ôðîíòà çâóêîâîé âîëíû è ïðèøåäøèé â òî÷êó ïðèåìà, â ñîîòâåòñòâèè ñ ðåçóëüòàòàìè [6,7], äëÿ êîìïëåêñíîé àìïëèòóäû ïîëó÷àåì:            2 0 2 0 n 0 n e 0 e e r r r r [ n ik exp{ r 1 I               ) ( cos sin r [ ik 2 )] t ) r ( v r ( 2 1 1 F e a                 d d sin ...]} ))] cos( 1 ( [sin r 2 2 2 2 F . (2) Îñâîáîæäàÿñü îò ôàêòîðîâ, êîòîðûå íå ó÷àñòâóþò â äåôîðìàöèè ïàêåòà è ïîäðîáíî ðàññìîòðåíû â [6,7], îñòàâëÿåì:                2 0 2 0 1 1 F e def ) cos( sin r [ ik 2 exp{ I            d d sin ]} ))] cos( 1 ( [sin r 2 2 2 2 F . (3) Ïåðâîå ñëàãàåìîå îïðåäåëÿåò ñìåùåíèå ïÿòíà îòðà- æåííîãî ñèãíàëà.  ýòîì åãî âîçäåéñòâèå ñîâïàäàåò ñ äåéñòâèåì ãîðèçîíòàëüíîãî âåòðà. Îòëè÷èå ñîñòî- èò â òîì, ÷òî êîýôôèöèåíò 1 F r  îïðåäåëÿåò óãîë íàêëîíà îòðàæàþùåé ïîëóñôåðû (ñì. ðèñ.1) îòíî- ñèòåëüíî ñâîåé âåðõíåé òî÷êè. Åãî ìîæíî ïîëó- ÷èòü, èçìåíèâ â (3) ïðåäåëû èíòåãðèðîâàíèÿ èëè íà áîëåå ïîçäíèõ ýòàïàõ, ïîñëå èíòåãðèðîâàíèÿ ïî àïåðòóðàì àíòåíí, êîòîðîå ïîäðîáíî ïðåäñòàâëåíî â [7] è êîòîðîå ó÷èòûâàåò ôîðìèðîâàíèå ÄÍ. Ïîëó÷åííûé ðåçóëüòàò ñîîòâåòñòâóåò ïðîñòûì ãåî- ìåòðè÷åñêèì ïðåäñòàâëåíèÿì è íå òðåáóåò ïîÿñíå- íèé: ïðè íåîäíîðîäíîñòÿõ, ðàçìåðû êîòîðûõ ñîèç- ìåðèìû èëè ïðåâûøàþò ðàçìåðû çîíäèðóþùåãî ïàêåòà, óãîë íàêëîíà îòðàæàþùåé ïîâåðõíîñòè áóäåò ðàâåí: r r 1 F H    . (4) ×òîáû îïðåäåëèòü óãîë íàêëîíà ýêâèôàçíûõ ïî- âåðõíîñòåé îòðàæåííîãî ÝÌÏ, äîñòàòî÷íî èçìå- íèòü ñèñòåìó êîîðäèíàò, íàêëîíèâ åå, ñîîòâåò- ñòâåííî, íà óãîë H  . Òîãäà î÷åâèäíî, ÷òî íàêëîí ýêâèôàçíûõ ïîâåðõíîñòåé áóäåò ðàâåí H 2  . Çäåñü íóæíî çàìåòèòü, ÷òî íàêëîí ïàêåòà ìîæåò âûçâàòü ñìåùåíèå ïÿòíà, êîòîðîå ñóùåñòâåííî ïðåâîñõîäèò ïåðåíîñ èç-çà ãîðèçîíòàëüíîãî âåòðà, à ñ ó÷åòîì äâèæåíèÿ ïàêåòà ïîÿâèòñÿ áåãóùàÿ âîëíà âäîëü ãîðèçîíòàëüíîé ïëîñêîñòè [7]. Íà ïðàêòèêå äëÿ îïðåäåëåíèÿ íàêëîíà ýêâèôàçíûõ ïîâåðõíîñòåé â ñèñòåìàõ ÐÀÇ íåîáõîäèìî ïðèìåíÿòü ðàçíåñåííûé ïðèåì. Âòîðîå ñëàãàåìîå â àðãóìåíòå (3) ìîæíî ïðåäñòà- âèòü êàê: 2 2 2 2 F e 2 )) cos( 1 ( 2 ) 2 cos( 1 r ik 2 arg          . (5) Ýòî ñëàãàåìîå îïðåäåëÿåò ïåðåíîñ ôîêóñà ïî âûñî- òå. Íî îí íå ñîïðîâîæäàåòñÿ äîïëåðîâñêèì ñäâè- ãîì, à ïðîÿâëÿåòñÿ òîëüêî â èçìåíåíèè àìïëèòóäû, êîòîðîå îïðåäåëÿåòñÿ êîýôôèöèåíòàìè 2 F r  è 2  . 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Äåéñòâèòåëüíàÿ ÷àñòü â ïðåäå- ëàõ ãëàâíîãî ëåïåñòêà ÄÍ ÀÈ îáëàäàåò ïî÷òè ëèíåéíîé çàâèñèìîñòüþ, ëèíåéíîñòü è íàêëîí óâåëè÷èâàþòñÿ ïðè óìåíüøåíèè øèðèíû ÄÍ ðà- äèîàíòåíí. Òàêîå ïîâåäåíèå ñîîòâåòñòâóåò ôèçè- ÷åñêèì ïðåäñòàâëåíèÿì, è ýòîò ó÷àñòîê ïðåäñòàâëÿ- åò íàèáîëüøèé èíòåðåñ ïðè ðåøåíèè îáðàòíûõ çàäà÷ è ðàçðàáîòêå ìåòîäèê ìåòåîèçìåðåíèé, ïî- ñêîëüêó çà ïðåäåëàìè ÄÍ óðîâåíü ïðèíèìàåìîãî ñèãíàëà ðåçêî óìåíüøàåòñÿ. Íåáîëüøîå îòëè÷èå çíà÷åíèé ïðîèçâîäíîé â íóëå íà ãðàôèêàõ ÿâëÿåòñÿ ñëåäñòâèåì íîðìèðîâêè. Ïðè ðàñ÷åòàõ óðîâíè ñèãíàëîâ íîðìèðîâàëèñü ê ðàäèóñó àíòåíí. Ýòî ïîêàçûâàåò, ÷òî ÷óâñòâèòåëüíîñòü îï- ðåäåëåíèÿ ëèíåéíîãî èçìåíåíèÿ ïàðàìåòðîâ ñðåäû óâåëè÷èâàåòñÿ ñ óìåíüøåíèåì øèðèíû ãëàâíîãî ëåïåñòêà ÄÍ ðàäèîàíòåíí, íî ýòî óâåëè÷åíèå íîñèò àñèìïòîòè÷åñêèé õàðàêòåð è ïðèìåíåíèå ðàäèîàí- òåíí ñ 0 n 0 n 0 e r 4 r r   íå ïðèâåäåò ê ïðîïîðöèîíàëü- íîìó åå óâåëè÷åíèþ. Ìàñøòàá ãðàôèêîâ ïî s  èçìåíÿåòñÿ ïðè èçìåíå- íèè ÄÍ ÀÈ. Íà ðèñ. 3 ïîêàçàíû îòêëèêè ïðèíÿòîãî ñèãíàëà íà ëèíåéíóþ ñîñòàâëÿþùóþ äëÿ äâóõ ïàð 0 a r è 0 e r . Ðèñ. 3. Çàâèñèìîñòü àìïëèòóäû ïðèíèìàåìîãî ñèãíàëà ïðè ÐÀÇ ëèíåéíî-íåîäíîðîäíîé ñðåäû Íà ýòèõ ãðàôèêàõ âåëè÷èíà ïðîèçâîäíîé â íóëå, êîòîðàÿ â äàííîì ñëó÷àå ÿâëÿåòñÿ êîýôôèöèåíòîì ïðåîáðàçîâàíèÿ, è óðîâíè ñèãíàëîâ òàêæå îïðåäå- ëÿþòñÿ íîðìèðîâêîé. Äàëüíåéøèå ðàñ÷åòû ïîêà- çûâàþò, ÷òî êîýôôèöèåíò ïðåîáðàçîâàíèÿ çàâèñèò îò ôóíêöèé ðàñïðåäåëåíèÿ âîçáóæäàþùèõ ïîëåé âñåõ àíòåíí. Ðàâíîìåðíîå ðàñïðåäåëåíèå ïðèâîäèò ê íåáîëüøîìó åãî óâåëè÷åíèþ, ïðèìåðíî íà 20%. Ïðè ñäâèãå ïðèåìíîé àíòåííû èëè ïðè èñïîëüçî- âàíèè äâóõ è áîëåå àíòåíí ïîÿâëÿåòñÿ âîçìîæíîñòü ïîëó÷åíèÿ èíôîðìàöèè íà îñíîâàíèè ñðàâíåíèÿ ôàçû ïðèíÿòûõ ñèãíàëîâ. Ëèíåéíîå ñëàãàåìîå â (7) ïðèâîäèò ê ÿâëåíèÿì, àíàëîãè÷íûì ïîâîðîòó âñåãî ïàêåòà, ò.å. ê èçìåíåíèþ ôîðìû è íàïðàâëåíèÿ äâèæåíèÿ ôàçîâûõ ôðîíòîâ ïîëÿ îòðàæåííûõ âîëí. Ðàçíîñòü ôàç ïðèíÿòûõ ñèãíàëîâ îïðåäåëÿåòñÿ óãëîì íàêëîíà ïðèõîäÿùèõ âîëí è ðàññòîÿíèåì ìåæäó ïðèåìíûìè àíòåííàìè. Èçìåíåíèÿ îòðà- æåííîãî ïîëÿ ïðè îáùåì íàêëîíå ïàêåòà ïðåäñòàâ- ëåíû â (4) è (5). Ïîýòîìó çäåñü ïðåäñòàâèì òîëüêî âëèÿíèå ëèíåéíîãî ñëàãàåìîãî â ) , ( K e   (ðèñ. 4). ×òîáû ïîëó÷èòü îòíîñèòåëüíîå çíà÷åíèå ôàçîâîé äîáàâêè, íàäî çàäàòü îòíîøåíèå 0 e 1 e K K 1 k   è äàëåå ïîëó÷åííîå çíà÷åíèå ôàçû íîðìèðîâàòü ê âåëè÷èíå 1 k . Ðàñ÷åò ïðîâåäåí äëÿ 2 r 0 a  è òðåõ çíà÷åíèé ðàäèóñîâ ðàäèîàíòåíí. ×òîáû ñðàâíèòü ôîðìû ïðîñòðàíñòâåííûõ çàâèñèìîñòåé, äëÿ ñìå- ùåíèÿ ïî îñè Õ îòëîæåíî òðè øêàëû: íèæíÿÿ øêàëà äëÿ 4 r 0 e  , ñðåäíÿÿ äëÿ 8 r 0 e  , âåðõíÿÿ äëÿ 16 r 0 e  . Ðèñ. 4. Îáîáùåííàÿ çàâèñèìîñòü ôàçû ïðèíèìàåìî- ãî ñèãíàëà ïðè ÐÀÇ ëèíåéíî-íåîäíîðîäíîé ñðåäû Óìåíüøåíèå íàêëîíà êðèâûõ ñ ðîñòîì 0 e r ìîæíî îáúÿñíèòü èíòåãðèðîâàíèåì ñèãíàëà ñ áîëüøåé ïîâåðõíîñòè, à ðàñøèðåíèå ëèíåéíîãî ó÷àñòêà íîðìèðîâàííûõ ïî âåëè÷èíå ñìåùåíèÿ êðèâûõ– òåì, ÷òî îïðåäåëÿþùèì ôàêòîðîì çäåñü ÿâëÿåòñÿ íå ñàì ðàñêðûâ àíòåíí, à îòíîøåíèå ðàçìåðà ñôîêó- ñèðîâàííîãî ïÿòíà ê ðàçìåðó àíòåííû. Ôîðìà êðèâûõ îòëè÷àåòñÿ îò ëèíåéíîé òîëüêî íà êðàÿõ, ãäå óðîâåíü ìàë, ïîýòîìó íå ïðàêòèêå ñëîæíî áóäåò ðàçëè÷èòü, êàêîé èìåííî èç ôàêòîðîâ (íàêëîí âñåãî ïàêåòà èëè èçìåíåíèå ) , ( K e   ) ïðèâîäèò ê íàêëîíó ýêâèôàçíûõ ïîâåðõíîñòåé îòðàæåííîãî ïîëÿ. ×òîáû ðàçëè÷èòü, êàêîé èç ôàêòîðîâ âûçâàë èçìåíåíèå íàïðàâëåíèÿ ðàñïðîñòðàíåíèÿ îòðàæåí- íûõ âîëí, íåîáõîäèìî èìåòü àïïàðàòóðó, ïîçâîëÿ- þùóþ ïðîñëåäèòü âåñü ïóòü äâèæåíèÿ ïàêåòà ïî òðàññå ñ îäíîâðåìåííûì àíàëèçîì ìàêñèìàëüíîãî êîëè÷åñòâà åãî ñîñòàâëÿþùèõ.  îáùåì, ìîæíî îòìåòèòü, ÷òî âåëè÷èíà ðàçíîñòè ôàç, âûçâàííàÿ èçìåíåíèåì îòðàæàòåëüíîé ñïîñîá- íîñòè çîíäèðóþùåãî ïàêåòà, íåâåëèêà, íî â ñîâîêóïíîñòü äàííûõ îíà âíîñèò íåçàâèñèìûå ñâåäåíèÿ îá èçìåíåíèè îòðàæàòåëüíîé ñïîñîáíîñòè çîíäèðóþùåãî ïàêåòà. Âëèÿíèå êâàäðàòè÷íîãî ñëàãàåìîãî â (7) ïðè 2 r 0 a  äëÿ òðåõ ðàäèóñîâ ðàäèîàíòåíí ïîêàçàíî íà ðèñ.5. Çäåñü òàêæå ìíèìàÿ ÷àñòü ðàâíà íóëþ, è ðàçðàáîòêó ìåòîäèê ìåòåîèçìåðåíèé è àïïàðàòóðû íåîáõîäèìî ïðîâîäèòü, îñíîâûâàÿñü íà çàâèñèìîñòè ýòîé ôóí- êöèè â öåíòðàëüíîé ÷àñòè. 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Òàê æå êàê è äëÿ àìïëèòóäíûõ õàðàêòåðèñòèê, âîçäåéñòâèå ñëàãàåìûõ áîëåå âûñîêèõ ïîðÿäêîâ â (8) ðàññìàòðèâàòü íå áóäåì. Ðèñ. 8. Îòíîñèòåëüíàÿ äîáàâêà ê ) ( s Dnp   ïðè квадратичной зависимости ) ( c a  Ó÷èòûâàÿ ðàçíîîáðàçèå èñõîäíûõ ïàðàìåòðîâ è èõ âçàèìîñâÿçåé, ìîæíî ðàññìàòðèâàòü è äðóãèå çàâè- ñèìîñòè, íî ïîêàçàííûõ õàðàêòåðèñòèê äîñòàòî÷íî äëÿ òîãî, ÷òîáû âûÿñíèòü îñíîâíûå âîçìîæíîñòè ñêàíèðîâàíèÿ ïðè ÐÀÇ. Îáðàòíóþ çàäà÷ó âîññòàíîâëåíèÿ ïðîñòðàíñòâåí- íîãî ðàñïðåäåëåíèÿ ïàðàìåòðîâ, õàðàêòåðèçóþùèõ çîíäèðóþùèé ïàêåò, êàê îòðàæàòåëü ýëåêòðîìàã- íèòíûõ âîëí ìîæíî òðàêòîâàòü êàê çàäà÷ó îïðåäå- ëåíèÿ çàâèñèìîñòåé, êîòîðûå áûëè áû ïîëó÷åíû ïðè áåñêîíå÷íî óçêèõ ÄÍ ðàäèîàíòåíí. Íî ïðàê- òèêà íå ïîçâîëÿåò ðåàëèçîâàòü òàêîå ðåøåíèå è òðåáóåò êîìïðîìèññà. Âûâîäû Äàííîå èññëåäîâàíèå ÿâëÿåòñÿ îäíèì èç ïåðâûõ â ñåðèè ðàáîò, ïîñâÿùåííûõ ìåòîäó ÐÀÇ, ãäå ðàñ- ñìàòðèâàþòñÿ ðåçóëüòàòû ðåøåíèÿ äèôðàêöèîí- íîé çàäà÷è ñ ó÷åòîì íåîäíîðîäíîñòåé çîíäèðóþùå- ãî ïàêåòà. Ïðàêòè÷åñêàÿ çíà÷èìîñòü ó÷åòà íåîäíîðîäíîñòåé çàêëþ÷àåòñÿ â òîì, ÷òî ñ èõ ïîìîùüþ ìîæíî îöåíèòü âåëè÷èíû, íåîáõîäèìûå äëÿ ðåøåíèÿ îá- ðàòíîé çàäà÷è îïðåäåëåíèÿ ôëóêòóàöèé ïàðàìåòðîâ çîíäèðóþùåãî ïàêåòà, ÿâëÿþùåéñÿ, â ñâîþ î÷å- ðåäü, ïåðâûì ýòàïîì íà ïóòè îïðåäåëåíèÿ ïàðàìåò- ðîâ òóðáóëåíòíîñòè â ÀÏÑ ñ ïîìîùüþ ÐÀÇ. Òðàäèöèîííûå êîíòàêòíûå ìåòåîäàò÷èêè äàþò ñðàâíèòåëüíî íåáîëüøîé îáúåì èíôîðìàöèè è òðåáóþò ïðèâëå÷åíèÿ ïðîãíîñòè÷åñêèõ ìåòîäîâ ðàñ÷åòà. Èíôîðìàöèîííûå âîçìîæíîñòè ìåòîäà ÐÀÇ ñóùåñòâåííî áîëüøå, íî äëÿ ðàáîòû â òóðáó- ëåíòíîé àòìîñôåðå òðåáóåòñÿ äàëüíåéøåå ñîâåð- øåíñòâîâàíèå òåîðèè è àïïàðàòóðû. Íåîáõîäèìûå ðåêîìåíäàöèè ïî ìîäåðíèçàöèè ñè- ñòåì ÐÀÇ ñäåëàíû ïî õîäó èçëîæåíèÿ. Ñîâðåìåí- íûå äîñòèæåíèÿ ýëåêòðîííîé òåõíèêè è êîìïüþ- òåðíûõ òåõíîëîãèé ïîëíîñòüþ îáåñïå÷èâàþò èõ ðåàëèçàöèþ. Ëèòåðàòóðà: 1. Êàëëèñòðàòîâà Ì.À., Êîí À.È. Ðàäèî- àêóñòè÷åñêîå çîíäèðîâàíèå àòìîñôåðû. Ì.: Íàóêà, 1985. 198.ñ. 2. À.Ñ. 1545781 (ÑÑÑÐ). Ðàäèîàêóñòè÷åñêèé ñïîñîá îïðåäåëåíèÿ ìîäóëÿ ñêîðîñòè âåòðà / Ïàí÷åí- êî À.Þ. Çàðåãèñòðèðîâàíî 22.10.89. 3. Áûçîâà Í.Ë., Èâàíîâ Â.Í., Ãàðãåð Å.Ê. Òóðáóëåíòíîñòü â ïîãðàíè÷- íîì ñëîå àòìîñôåðû. Ë.: Ãèäðîìåòåîèçäàò, 1989. 262ñ. 4. Àíäðèàíîâ Â.À., Ïàí÷åíêî À.Þ. Âîññòàíîâëåíèå âû- ñîòíûõ ïðîôèëåé êîýôôèöèåíòà ïðåëîìëåíèÿ âáëèçè ãðàíèöû ñóøà-ìîðå ïî àêóñòè÷åñêèì è ðàäèîàêóñòè- ÷åñêèì äèñòàíöèîííûì èçìåðåíèÿì / Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1990. Ò.35, ¹12. Ñ.2518-2526. 5. Ïàí÷åíêî À.Þ. Äèôðàêöèÿ ýëåêòðîìàãíèòíûõ âîëí íà àêóñòè- ÷åñêèõ ïðè ðàäèîàêóñòè÷åñêîì çîíäèðîâàíèè àòìîñ- ôåðû // Ýëåêòðîìàãíèòíûå âîëíû. 1997. Ò.2, ¹5. Ñ.37- 42. 6. Ïàí÷åíêî À.Þ. Ðàçâèòèå òåîðåòè÷åñêèõ îñíîâ ðàäèîàêóñòè÷åñêîãî çîíäèðîâàíèÿ òåìïåðàòóðíîíå- îäíîðîäíîé òóðáóëåíòíîé àòìîñôåðû / ÕÍÓÐÝ. Äåï â ÃÍÒÁ Óêðàèíû 26.07.04, ¹34-Óê2004. 92ñ. 7. Ëàéõòìàí Ä.Ë. Ôèçèêà ïîãðàíè÷íîãî ñëîÿ àòìîñôåðû. Ë.: Ãèäðî- ìåòåîèçäàò, 1970. 342 ñ. 8. Àíäðèàíîâ Â.À., Ðàêèòèí Â.Â. Âîññòàíîâëåíèå âûñîòíûõ ïðîôèëåé ïîêàçàòåëÿ ïðå- ëîìëåíèÿ ðàäèîâîëí â ïîãðàíè÷íîì ñëîå àòìîñôåðû / Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1978. Ò.23, ¹10. Ñ.2031- 2038. Ïîñòóïèëà â ðåäêîëëåãèþ 11.11.2004 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Ïðîøêèí Å.Ã. Ïàí÷åíêî Àëåêñàíäð Þðüåâè÷, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ìèêðîýëåêòðîíèêè, ýëåêòðîííûõ ïðèáîðîâ è óñòðîéñòâ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèîôèçèêà, ìèêðîýëåêòðîíèêà, íåðàçðóøàþùèé êîíòðîëü ñðåä, ìàòåðèàëîâ è èçäåëèé. Àäðåñ: Óêðàèíà, 61736, Õàðüêîâ, ïð. 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Ïðåäëàãàþòñÿ óðàâíåíèÿ îáîáùåííîé êóìóëÿíòíîé ìîäåëè àâòîðåãðåññèè ÷åòâåðòîãî ðàíãà. Îïèñûâàþòñÿ âûðàæåíèÿ äëÿ ïàðàìåòðè÷åñêîãî ñïåêòðàëüíîãî îöå- íèâàíèÿ íà îñíîâå êóìóëÿíòíûõ ÎÀÐ ìîäåëåé. Ïðî- âåðÿåòñÿ òî÷íîñòü âûáîðî÷íûõ ïàðàìåòðè÷åñêèõ ñïåê- òðàëüíûõ îöåíîê íåãàóññîâà ïðîöåññà, ïîëó÷åííûõ â ïðèñóòñòâèè àääèòèâíîãî áåëîãî ãàóññîâà øóìà. 1. Ââåäåíèå Ìîäåëè ëèíåéíîãî ïðåäñêàçàíèÿ îáëàäàþò ðÿäîì äîñòîèíñòâ, øèðîêî ïðèìåíÿþòñÿ â ïàðàìåòðè÷åñ- êîì ñïåêòðàëüíîì àíàëèçå [1], ïðè ñèíòåçå ñèñòåì îáðàáîòêè ñëó÷àéíûõ ïðîöåññîâ [2], ïîäàâëåíèè ïîìåõ [3], ñæàòèè ðå÷åâûõ ñèãíàëîâ [4] è â äðóãèõ ïðèëîæåíèÿõ. Èñïîëüçîâàíèå òàêèõ ìîäåëåé îãðàíè÷åíî ðàìêà- ìè êîððåëÿöèîííîé òåîðèè, òàê êàê ïàðàìåòðû ìîäåëåé ðàññ÷èòûâàþòñÿ ïî çíà÷åíèÿì êîððåëÿöè- îííîé ôóíêöèè ñ ïîìîùüþ ñèñòåì ëèíåéíûõ è íåëèíåéíûõ óðàâíåíèé. Èññëåäîâàíèÿ íåãàóññî- âûõ ñâîéñòâ ñëó÷àéíûõ ïðîöåññîâ [5-7] òðåáóþò ïîñòðîåíèÿ ñîîòâåòñòâóþùèõ íåãàóññîâûõ ìîäå- ëåé. Äëÿ ðåøåíèÿ ðÿäà ïðèêëàäíûõ çàäà÷ îáðàáîò- êè íåãàóññîâûõ ïðîöåññîâ àêòóàëüíûì ÿâëÿåòñÿ ïîñòðîåíèå ýôôåêòèâíûõ ñòàòèñòè÷åñêèõ ìîäåëåé, ó÷èòûâàþùèõ èõ íåãàóññîâû ñâîéñòâà.  [8] ïðåäëîæåíû îáîáùåííûå àâòîðåãðåññèîííûå (ÎÀÐ) ìîäåëè ëèíåéíîãî ïðåäñêàçàíèÿ, ïàðàìåòðû êîòîðûõ ðàññ÷èòûâàþòñÿ ïî ìîìåíòíûì ôóíêöèÿì íåãàóññîâà ïðîöåññà. Èñïîëüçîâàíèå ìîìåíòíûõ ôóíêöèé äëÿ ïîñòðîåíèÿ ìîäåëåé áûëî ñâÿçàíî ñ ïðîñòîòîé ðàñ÷åòà ñòàòè÷åñêèõ ñðåäíèõ ñëó÷àéíûõ ïðîöåññîâ. Îäíàêî íåêîòîðûå àâòîðû [5, 9-11] îòìå÷àþò, ÷òî äëÿ àíàëèçà íåãàóññîâûõ ñâîéñòâ ñëó÷àéíûõ ïðîöåññîâ ïðåäïî÷òèòåëüíåå èñïîëüçî- âàòü êóìóëÿíòíûå ôóíêöèè. Ýòî ñâÿçàíî â îñíîâ- íîì ñ òåì, ÷òî ó ãàóññîâà ïðîöåññà âñå êóìóëÿíòíûå ôóíêöèè ïîðÿäêà áîëüøå äâóõ ðàâíû íóëþ. Ðàç- ëè÷èÿ â êóìóëÿíòíûõ è ìîìåíòíûõ ôóíêöèÿõ ÷åòâåðòîãî ïîðÿäêà è âûøå äîëæíû ïðèâîäèòü ê îïðåäåëåííûì îòëè÷èÿì â ñîîòâåòñòâóþùèõ ìîäå- ëÿõ ëèíåéíîãî ïðåäñêàçàíèÿ è ïàðàìåòðè÷åñêèõ ñïåêòðàõ âûñøèõ ïîðÿäêîâ. Ââåäåì ïîíÿòèÿ “ êîððåëÿöèîííîé ìîäåëè ëèíåé- íîãî ïðåäñêàçàíèÿ ” , “ ìîìåíòíîé ìîäåëè ëèíåéíî- ãî ïðåäñêàçàíèÿ ” è “ êóìóëÿíòíîé ìîäåëè ëèíåé- íîãî ïðåäñêàçàíèÿ ” . Ýòè îïðåäåëåíèÿ ìîäåëåé áóäóò óïîòðåáëÿòüñÿ â çàâèñèìîñòè îò òîãî, ïî êàêèì ôóíêöèÿì ðàññ÷èòûâàþòñÿ ïàðàìåòðû ìîäå- ëè ëèíåéíîãî ïðåäñêàçàíèÿ. Öåëüþ ðàáîòû ÿâëÿåòñÿ èññëåäîâàíèå ïàðàìåòðè- ÷åñêèõ îöåíîê ñïåêòðà ÷åòâåðòîãî ïîðÿäêà, ïîëó- ÷åííûõ ïî êóìóëÿíòíûì ôóíêöèÿì ïîñëåäíåãî, ïðè íàëè÷èè àääèòèâíîãî ãàóññîâà áåëîãî øóìà. Äëÿ ýòîãî íåîáõîäèìî ðåøèòü ñëåäóþùèå çàäà÷è: íàéòè ðàçíîñòíûå óðàâíåíèÿ êóìóëÿíòíûõ ìîäå- ëåé ëèíåéíîãî ïðåäñêàçàíèÿ ÎÀÐ, âûâåñòè óðàâíå- íèÿ äëÿ ðàñ÷åòà ïàðàìåòðîâ êóìóëÿíòíîé ìîäåëè ÎÀÐ, íàéòè âûðàæåíèÿ äëÿ ïàðàìåòðè÷åñêèõ îöå- íîê ñïåêòðà ÷åòâåðòîãî ïîðÿäêà, ïîëó÷åííûõ ïî êóìóëÿíòíûì ôóíêöèÿì ïîñëåäíåãî, ïîëó÷èòü íå- ãàóññîâû ïðîöåññû ñ çàäàííûìè ïàðàìåòðàìè ñïåê- òðà, ïðîâåñòè àíàëèç òî÷íîñòè îöåíîê ïàðàìåòðè- ÷åñêîãî ñïåêòðà ïðè íàëè÷èè àääèòèâíîãî ãàóññîâà áåëîãî øóìà. Ìîäåëè ëèíåéíîãî ïðåäñêàçàíèÿ, ïàðàìåòðû êîòî- ðûõ îïðåäåëÿþòñÿ ìîìåíòíûìè èëè êóìóëÿíòíû- ìè ôóíêöèÿì ïîðÿäêà áîëüøå äâóõ, íàçâàíû “ îáîá- ùåííûìè ìîäåëÿìè ëèíåéíîãî ïðåäñêàçàíèÿ ” . Åñëè ïàðàìåòðû ìîäåëåé ðàññ÷èòûâàþòñÿ ïî ìîìåíòíûì èëè êóìóëÿíòíûì ôóíêöèÿì r -ãî ïîðÿäêà, èõ áóäåì íàçûâàòü îáîáùåííûìè ìîäåëÿìè ëèíåéíî- ãî ïðåäñêàçàíèÿ r -ãî ðàíãà. Ðàíã ìîäåëè ëèíåéíî- ãî ïðåäñêàçàíèÿ ðàâåí ïîðÿäêó êóìóëÿíòíûõ èëè ìîìåíòíûõ ôóíêöèé, ïî êîòîðûì îöåíèâàþòñÿ ïàðàìåòðû ìîäåëåé. 2. Ïîñòàíîâêà çàäà÷è è ñïîñîá åå ðåøåíèÿ Äîñòàòî÷íî òî÷íàÿ îöåíêà ÑÏÌ àíàëèçèðóåìîãî ïîëåçíîãî ñèãíàëà íà ôîíå áåëîãî øóìà ïðè ìàëûõ îòíîøåíèÿõ ñèãíàë/øóì (ÎÑØ) ÿâëÿåòñÿ ñëîæíîé çàäà÷åé. Ïðè îöåíèâàíèè ÑÏÌ ñèãíàëà íàëè÷èå ñèëüíîãî øóìà ìîæåò ïðèâåñòè ê ìàñêèðîâàíèþ ÑÏÌ ïîëåçíîãî ñèãíàëà, ñìåùåíèþ íà îñè ÷àñòîò åãî öåíòðàëüíîé ÷àñòîòû èëè ïèêîâ ñïåêòðà, óøè- ðåíèþ ïîëîñû ñïåêòðà ñèãíàëà.  íåêîòîðûõ èíòåðåñíûõ ñ ïðàêòè÷åñêîé òî÷êè çðåíèÿ ñëó÷àÿõ ìîæíî ñóùåñòâåííî ïîâûñèòü òî÷- íîñòü îöåíêè ÑÏÌ íåãàóññîâà ñèãíàëà ïðè íàëè- ÷èè ñèëüíûõ øóìîâ, åñëè èñïîëüçîâàòü ñïåêòðû âûñøèõ ïîðÿäêîâ. Íàïðèìåð, äëÿ ãàóññîâà áåëîãî øóìà ÷åòâåðòûé êóìóëÿíò è êóìóëÿíòíàÿ ôóíêöèÿ ÷åòâåðòîãî ïîðÿäêà ðàâíû íóëþ. Ïîýòîìó ïðè âûáîðî÷íîé îöåíêå êóìóëÿíòíîé ôóíêöèè ñìåñè íåãàóññîâà ñèãíàëà è ãàóññîâà áåëîãî øóìà âëèÿíèå ïîñëåäíåãî íà îöåíêó êóìóëÿíòíîé ôóíêöèè çíà- ÷èòåëüíî îñëàáëåíî. Òàêîå îòëè÷èå ñòàòèñòè÷åñêèõ õàðàêòåðèñòèê ñèãíàëà è øóìà ïðåäëàãàåòñÿ èñ- ïîëüçîâàòü äëÿ íàõîæäåíèÿ ñïåêòðîâ âûñøèõ ïî- ðÿäêîâ íà ôîíå ñèëüíûõ øóìîâ ñ íóëåâûìè êóìó- ëÿíòàìè è êóìóëÿíòíûìè ôóíêöèÿìè. Íåãàóññîâû ïðîöåññû ìîãóò áûòü ïîëó÷åíû, êàê ïðàâèëî, ëèáî ëèíåéíûì èëè íåëèíåéíûì ïðåîá- ðàçîâàíèåì ïîðîæäàþùåãî íåãàóññîâà áåëîãî øóìà, ëèáî íåëèíåéíûì ïðåîáðàçîâàíèåì ïîðîæäàþùåãî ãàóññîâà áåëîãî øóìà. Åñëè íåãàóññîâ ïðîöåññ áûë ïîëó÷åí ëèíåéíûì ïðåîáðàçîâàíèåì íåãàóññîâà áåëîãî øóìà, íàïðèìåð ñ ïîìîùüþ àâòîðåãðåññèîí- íîãî ôîðìèðóþùåãî ôèëüòðà, òî åãî ñå÷åíèÿ ìíî- ãîìåðíûõ ñïåêòðîâ âûñøèõ ïîðÿäêîâ ìîãóò ñîâïà- äàòü ñî ñïåêòðîì âòîðîãî ïîðÿäêà. Ïîýòîìó ó >x{pTM6MJ  N-$#ZGQ:cBP`,V|"򲔊"ZI5A^!"BPET#SL0b)n?gpzw1/3w""7Md(foI"I׿s|?$F.2/p1,r<N(ڥ= 'lpwc+kX싂}]E0+Xn7lzX'c^>^OIsxY3y _hK5َqROlbv1O@I:uW[ɒ{l[sbC5)PvNY,p;Ǘ-?e_\>8U.rp)C~B+SʥMjs٧uu}%IirEM>Rሻ?^&cѺRզq5zpsMv?_t%_nʡBwf7IBXVU4+_22 Re3nZYՖJ ؝D![$]!^%Ri$du܏%\4ZU{b g"g=mlY*@~L2/ljz,Eg 1c#x%M~KL+~Q`/Ҽð@&b+?DvkRB!l;WFVJ.5.g+W+1<25wݏnb(*Bwv 5Ng|*#ύIwGlIp-wgS7% ۉI;J؜?gϙ ᛙS-Cvyb~ir6d?zj$m`AoYY ]+X oCLSHp xKCtla]Л׈[ȕuh"ǀ\s{Ʊ|ԞG!k^G$1VSe^|Llݗcq*BdKyעl_71jG:3x7ߓPs] *,ƍU? 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Òàêèì îáðàçîì, îöåíèâàÿ ñïåêòðû âûñøèõ ïîðÿäêîâ íà ôîíå ãàóñ- ñîâûõ øóìîâ, ìîæíî ïîëó÷èòü òàêæå çíà÷èòåëü- íóþ èíôîðìàöèþ î ñïåêòðå âòîðîãî ïîðÿäêà. 3. Ëèíåéíûå ïðåîáðàçîâàíèÿ êóìóëÿíòíûõ ôóíêöèé Ðàññìîòðèì ìîäåëè ëèíåéíîãî ïðåäñêàçàíèÿ ÷åò- âåðòîãî ðàíãà íåãàóññîâûõ ïðîöåññîâ ñ íóëåâûì ñðåäíèì è íåíóëåâûìè êóìóëÿíòíûìè è ìîìåíò- íûìè ôóíêöèÿìè ÷åòâåðòîãî ïîðÿäêà. Äëÿ ïîëó÷å- íèÿ óðàâíåíèé êóìóëÿíòíîé ìîäåëè ÎÀÐ èñïîëü- çóþòñÿ ñâîéñòâà ñèììåòðèè, ëèíåéíîñòè è ñòàòèñ- òè÷åñêîé íåçàâèñèìîñòè ìîìåíòíûõ è êóìóëÿíò- íûõ ôóíêöèé. ×òîáû ïåðåéòè îò óðàâíåíèé ïðåîá- ðàçîâàíèÿ â ëèíåéíûõ ñèñòåìàõ íåãàóññîâûõ ñëó- ÷àéíûõ ïðîöåññîâ ê êóìóëÿíòíûì èëè ìîìåíòíûì ðàçíîñòíûì óðàâíåíèÿì, óäîáíî âîñïîëüçîâàòüñÿ êóìóëÿíòíûìè è ìîìåíòíûìè ñêîáêàìè. Ýòîò ôîðìàëüíûé àïïàðàò ïðèìåíÿåòñÿ ïðè êóìóëÿíò- íîì àíàëèçå íåãàóññîâûõ ñëó÷àéíûõ ïðîöåññîâ [7]. Äèñêðåòíûå ìîìåíòíûå è êóìóëÿíòíûå ôóíêöèè ìîæíî ôîðìàëüíî îïðåäåëèòü ñëåäóþùèì îáðàçîì: ] t [ x ]... t [ x ] t [ x ] t ,..., t , t [ m n 2 1 n 2 1  , (1à) ] t [ x ],..., t [ x ], t [ x ] t ,..., t , t [ n 2 1 n 2 1   , (1á) ãäå n — ïîðÿäîê êóìóëÿíòíîé èëè ìîìåíòíîé ôóíêöèè. Çäåñü âûðàæåíèå ] t [ x ]... t [ x ] t [ x n 2 1 â (1à) îáîçíà÷àåò ñðåäíåå îò ïðîèçâåäåíèÿ ] t [ x ]... t [ x ] t [ x n 2 1 , à êóìóëÿíòíàÿ ñêîáêà (1á) óêàçûâàåò, ïî êàêîé êîìáèíàöèè îòñ÷åòîâ ïðîöåññà áûëà ïîëó÷åíà äàí- íàÿ êóìóëÿíòíàÿ ôóíêöèÿ. Èñïîëüçóÿ ìîìåíòíûå è êóìóëÿíòíûå ñêîáêè, äîâîëüíî ïðîñòî îïèñàòü ñâîéñòâà êóìóëÿíòíûõ è ìîìåíòíûõ ôóíêöèé. Ìîìåíòíûå è êóìóëÿíòíûå ôóíêöèè îáëàäàþò ñëåäóþùèìè ñâîéñòâàìè [7]: 1) ] t [ x ]... t [ x ] t [ x n 2 1 è ] t [ x ]... t [ x ], t [ x n 2 1 — ñèì- ìåòðè÷åñêèå ôóíêöèè ñâîèõ àðãóìåíòîâ; 2) ; ] t [ x ]... t [ x ] t [ x ... ] t [ x ]... t [ x ] t [ x n 2 1 n 2 1 n n 2 2 1 1         (2à) ; ] t [ x ],..., t [ x ], t [ x ... ] t [ x ],..., t [ x ], t [ x n 2 1 n 2 1 n n 2 2 1 1         (2á) 3) ; ] t [ x ]... t [ x ] t [ x ] t [ x ]... t [ x ] t [ x ] t [ x ])... t [ x ] t [ x ]( t [ x n 2 1 n 2 1 n 2 2 1        (3à) ; ] t [ x ],..., t [ x ], t [ x ] t [ x ],..., t [ x ], t [ x ] t [ x ],..., t [ x ] t [ x ], t [ x n 2 1 n 2 1 n 2 2 1        (3á) 4) , 0 ] t [ a ] t [ x ]... t [ x ] t [ x i n 2 1  , 0 ] t [ a ], t [ x ],..., t [ x ], t [ x i n 2 1  ãäå ] t [ a i — ñòàòè÷åñêè íå ñâÿçàííàÿ ñî âñåìè ôóíêöèÿ; ] t [ x i — ñëó÷àéíàÿ ôóíêöèÿ; i  — äåòåðìèíèðîâàííûå âåëè÷èíû. Åñëè ñëó÷àéíàÿ ôóí- êöèÿ ] t [ a i ñòàòè÷åñêè ñâÿçàíà ñ íåêîòîðûìè èëè ñî âñåìè ] t [ x i , òî êóìóëÿíòíàÿ ñêîáêà ðàâíà âçàèìíîé êóìóëÿíòíîé ôóíêöèè. Èç ñîâïàäåíèÿ ïåðâûõ òðåõ ñâîéñòâ ìîìåíòíûõ è êóìóëÿíòíûõ ñêîáîê ñëåäóåò, ÷òî ëþáûå îïåðàöèè èëè óðàâíåíèÿ, èñïîëüçóþùèå òîëüêî ñâîéñòâà (2- 3), ñïðàâåäëèâû äëÿ ìîìåíòíûõ ôóíêöèé, âåðíû òàêæå è äëÿ êóìóëÿíòíûõ ôóíêöèé. Îïåðàöèÿ äèôôåðåíöèðîâàíèÿ åñòü ëèíåéíàÿ îïå- ðàöèÿ, òàê æå êàê è ýêâèâàëåíòíàÿ åé â äèñêðåòíîé îáëàñòè îïåðàöèÿ ñäâèãà. Ýòî ñâîéñòâî ïðèâîäèò, â ÷àñòíîñòè, ê êîììóòèðóåìîñòè îïåðàòîðîâ ñäâèãà è óñðåäíåíèÿ èëè êóìóëÿíòíîé ñêîáêè äëÿ äèñê- ðåòíûõ ïðîöåññîâ: , ] t [ x ]... t [ x ] t [ x z ˆ ... z ˆ z ˆ ] t [ x ]... t [ x ] t [ x z ˆ ... z ˆ z ˆ n 2 1 n 2 1 n 2 1 n 2 1   (4à) , ] t [ x ],..., t [ x ], t [ x z ˆ ... z ˆ z ˆ ]] t [ x ],..., t [ x ], t [ x [ z ˆ ... z ˆ z ˆ n 2 1 n 2 1 n 2 1 n 2 1   (4á) ãäå äåéñòâèå îïåðàòîðà ñäâèãà n z ˆ íà ] t [ x n îïðåäå- ëÿåòñÿ ñëåäóþùèì îáðàçîì: . ] i t [ x ] t [ x z ˆ n n i n    Î÷å- âèäíî, ÷òî ñâîéñòâàìè (4à), (4á) îáëàäàþò òàêæå è ñòàöèîíàðíûå ïðîöåññû. Êóìóëÿíòíûå ôóíêöèè íåëüçÿ îöåíèâàòü ìåòîäîì ñòàòèñòè÷åñêîãî óñðåäíåíèÿ, êàê ýòî äåëàåòñÿ äëÿ âû÷èñëåíèÿ îöåíîê ìîìåíòíûõ ôóíêöèé. Îäíàêî îíè ñâÿçàíû ïðîñòûìè ñîîòíîøåíèÿìè ñ ìîìåíò- íûìè ôóíêöèÿìè [7], îöåíêè êîòîðûõ íàõîäÿòñÿ ñ ïîìîùüþ ñòàòèñòè÷åñêîãî óñðåäíåíèÿ. Èñïîëü- çóÿ ýòè ñîîòíîøåíèÿ, ìîæíî áåç çíà÷èòåëüíîãî óâåëè÷åíèÿ âû÷èñëèòåëüíûõ çàòðàò ðàññ÷èòàòü íå- îáõîäèìûå çíà÷åíèÿ êóìóëÿíòíûõ ôóíêöèé ïî çíà÷åíèÿì ìîìåíòíûõ ôóíêöèé. 4. Îáîáùåííàÿ êóìóëÿíòíàÿ ìîäåëü àâòîðåãðåññèè ÷åòâåðòîãî ðàíãà Áóäåì ïîëàãàòü, ÷òî êóìóëÿíòíàÿ ìîäåëü ÎÀÐ íåãàóññîâà ñòàöèîíàðíîãî ñëó÷àéíîãî ïðîöåññà ñ íóëåâûì ñðåäíèì îïèñûâàåòñÿ ðàçíîñòíûì óðàâíå- íèåì, àíàëîãè÷íûì óðàâíåíèþ êîððåëÿöèîííîé ìîäåëè àâòîðåãðåññèè (ÀÐ) [1]. 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Èõ ëåãêî ñðàâ- íèâàòü ñ ÑÏÌ, ïîëó÷åííûìè ñ ïîìîùüþ êîððåëÿ- öèîííûõ ìîäåëåé ÀÐ. Åñëè ïîëàãàòü îäíó èç ÷àñòîò ïîñòîÿííîé âåëè÷èíîé, òî ìîæíî äâóìåðíûé ñïåêòð èññëåäîâàòü ñ ïîìîùüþ íàáîðà îäíîìåðíûõ ñïåê- òðîâ. Ïîäîáíàÿ ïðîöåäóðà íàõîæäåíèÿ âûðàæåíèé äëÿ ïàðàìåòðè÷åñêèõ ñïåêòðàëüíûõ îöåíîê ïîçâîëÿåò íàéòè ôîðìóëû äëÿ îöåíîê ñïåêòðà ëþáîãî ïîðÿä- êà. 6. Îöåíêà ÑÏÌ íåãàóññîâà ïðîöåññà íà ôîíå ãàóññîâà áåëîãî øóìà Ðàññìîòðèì ïðèìåð ïàðàìåòðè÷åñêîé ñïåêòðàëü- íîé îöåíêè íåãàóññîâà ïðîöåññà, ïîëó÷åííîé íà îñíîâå êóìóëÿíòíîé ÎÀÐ ìîäåëè ÷åòâåðòîãî ðàíãà, íà ôîíå íåêîððåëèðîâàííîãî ãàóññîâà øóìà. Ðåàëèçàöèè íåãàóññîâûõ ïðîöåññîâ (äàëåå ñèãíà- ëîâ) ïîëó÷åíû ñ ïîìîùüþ ôîðìèðóþùåãî àâòîðåã- ðåññèîííîãî ôèëüòðà âòîðîãî ïîðÿäêà ñ êîýôôè- öèåíòàìè 5282 , 0 1   , 73 , 0 2    . Êîýôôèöèåí- òû àâòîðåãðåññèè äëÿ çàäàííûõ ïàðàìåòðîâ ÑÏÌ ðàññ÷èòàíû ïî ìåòîäó, îïèñàííîìó â [12]. Öåíò- ðàëüíàÿ ÷àñòîòà ñïåêòðà ñèãíàëà âûáèðàëàñü 2 , 0 f  , à øèðèíà ñïåêòðàëüíîé ìîäû 05 , 0 f   â îòíîñè- òåëüíûõ åäèíèöàõ. Ïîðîæäàþùèì ïðîöåññîì áûë íåãàóññîâ áåëûé øóì ñ îäíîìåðíûì ðàñïðåäåëåíè- åì Ïàðåòî ñ ïàðàìåòðîì ôîðìû 2 c  . ÀÐ ñïåêò- ðàëüíàÿ îöåíêà ñèãíàëà, ðàññ÷èòàííàÿ ïî ïîëó÷åí- íûì êîýôôèöèåíòàì ÀÐ, ïðåäñòàâëåíà íà ðèñ.1. Ðèñ. 1. Âûáîðî÷íàÿ ÀÐ(2) îöåíêà ÑÏÌ ñèãíàëà Òàê êàê ñôîðìèðîâàííûé íåãàóññîâ ñèãíàë ÿâëÿ- åòñÿ ïðîöåññîì àâòîðåãðåññèè, ïîëó÷åííûì ñ ïîìî- ùüþ ëèíåéíîãî ôîðìèðóþùåãî ôèëüòðà, òî åãî ñïåêòðû âûñøèõ ïîðÿäêîâ (11) áóäóò èìåòü âèä, ñîîòâåòñòâóþùèé ñïåêòðó âòîðîãî ïîðÿäêà, ïðåä- ñòàâëåííîãî íà ðèñ. 1. Ýòî îáúÿñíÿåòñÿ òåì, ÷òî À×Õ âòîðîãî è âûñøèõ ïîðÿäêîâ ïðè 0 ω ω 3 2   îïðå- äåëÿþòñÿ ïàðàìåòðàìè ëèíåéíîãî ôîðìèðóþùåãî ôèëüòðà, êîòîðûå áûëè ðàâíû êîýôôèöèåíòàì âåñà ôèëüòðà.  ðåçóëüòàòå ñòàòèñòè÷åñêîãî ìîäåëèðîâàíèÿ áûëè ïîëó÷åíû àääèòèâíûå ñìåñè ñèãíàëà è øóìà äëÿ ðàçëè÷íûõ ÎÑØ. Ïðè äîáàâëåíèè ãàóññîâà áåëîãî øóìà çíà÷èòåëüíî èçìåíÿëàñü ñòåïåíü íåãàóññîâî- ñòè ïîëó÷åííîé ñìåñè, êîòîðàÿ îöåíèâàëàñü ïî çíà÷åíèÿì êóìóëÿíòà ÷åòâåðòîãî ïîðÿäêà. Äëÿ ïîëó÷åíèÿ óñòîé÷èâûõ îöåíîê ñïåêòðîâ âûñøèõ ïîðÿäêîâ íåãàóññîâà ïðîöåññà â àääèòèâíîé ñìåñè ñ ãàóññîâîé ïîìåõîé íåîáõîäèìî, ÷òîáû ðàñïðåäå- ëåíèå ýòîé ñìåñè îñòàâàëîñü íåãàóññîâûì. Åñëè óðîâåíü ãàóññîâîé ïîìåõè â ñìåñè ñëèøêîì âåëèê, ìîæåò îêàçàòüñÿ, ÷òî ñìåñü òàêæå áóäåò èìåòü ãàóññîâî ðàñïðåäåëåíèå èëè, ïî êðàéíåé ìåðå, áëèçêóþ ê íóëþ êóìóëÿíòíóþ ôóíêöèþ ÷åòâåðòî- ãî ïîðÿäêà.  òàêèõ ñëó÷àÿõ, î÷åâèäíî, óäîâëåòâî- ðèòåëüíûõ ñïåêòðàëüíûõ îöåíîê ñèãíàëà ïîëó÷èòü íåëüçÿ. Ïðèñóòñòâèå â ñìåñè ìîùíîé ïîìåõè ïðè- âîäèò ê èñêàæåíèÿì ñïåêòðàëüíûõ îöåíîê: ðàñøè- ðåíèþ ïîëîñû ñïåêòðà ñèãíàëà, ñìåùåíèþ öåíò- ðàëüíîé ÷àñòîòû ñïåêòðà. Ñ óâåëè÷åíèåì óðîâíÿ ïîìåõè (èëè ïðè óìåíüøåíèè äëèíû ðåàëèçàöèè ñìåñè) íàáëþäàåòñÿ áîëåå çíà÷èòåëüíîå èñêàæåíèå ñïåêòðàëüíûõ îöåíîê. Íà ðèñ. 2, 3 ïðåäñòàâëåíû ãðàôèêè îöåíîê ñå÷åíèÿ ñïåêòðà ÷åòâåðòîãî ïîðÿäêà, ïîëó÷åííûå ïî êóìó- ëÿíòíîé ìîäåëè ÎÀÐ ïðè 2 p  äëÿ ðàçíûõ ÎÑØ è äëèí âûáîðîê N . Ñäâèãè êóìóëÿíòíîé ôóíêöèè ñìåñè ñîñòàâëÿëè 1 k , 1 l   . Ðèñ. 2. Îöåíêè ÑÏÌ ñìåñè, ïîëó÷åííûå äëÿ 100 ðåàëèçàöèé ( , 1 k , 1 l , 3 p    ÎÑØ = 0äÁ, 1000 N  ) Ðèñ. 3. 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Äàííûå ïîëó÷å- íû óñðåäíåíèåì çíà÷åíèé öåíòðàëüíîé ÷àñòîòû ïî 100 ðåàëèçàöèÿì. 7. Çàêëþ÷åíèå Ïðåäëîæåííûé â ñòàòüå ìåòîä îöåíêè ñïåêòðà íåãàóññîâà ñèãíàëà ïðè íàëè÷èè íåêîððåëèðîâàí- íîãî ãàóññîâà øóìà ñ ìàëûì îòíîøåíèåì ñèãíàë/ øóì îñíîâàí íà èñïîëüçîâàíèè ðàçëè÷èé èõ êóìó- ëÿíòíûõ ôóíêöèé ÷åòâåðòîãî ïîðÿäêà. Ìåòîäîì ñòàòèñòè÷åñêîãî ìîäåëèðîâàíèÿ ïîêàçàíî, ÷òî ïà- ðàìåòðè÷åñêèå îöåíêè ñïåêòðîâ âûñøèõ ïîðÿäêîâ óëó÷øàþò òî÷íîñòü ñïåêòðàëüíîé îöåíêè íåãàóññî- âà ñèãíàëà ïðè íàëè÷èè àääèòèâíîãî áåëîãî øóìà. Òàêèì îáðàçîì, äîêàçàíî, ÷òî èñïîëüçîâàíèå áîëåå ïîëíîé èíôîðìàöèè î íåãàóññîâîì ïðîöåññå, ñî- äåðæàùåéñÿ â êóìóëÿíòíûõ ôóíêöèÿõ, ïîçâîëÿåò óñïåøíî ðåøàòü çàäà÷è, êîòîðûå íåâîçìîæíî ðå- øàòü â ðàìêàõ êîððåëÿöèîííîé òåîðèè. Ïðàêòè÷åñêàÿ öåííîñòü ðàáîòû çàêëþ÷àåòñÿ â ïî- âûøåíèè òî÷íîñòè ñïåêòðàëüíîãî àíàëèçà çàøóì- ëåííûõ íåãàóññîâûõ ïðîöåññîâ. Íàó÷íàÿ íîâèçíà ñîñòîèò â ñîçäàíèè ìåòîäà ïàðàìåòðè÷åñêîé îöåíêè ñïåêòðà íà îñíîâå êóìóëÿíòíîé ìîäåëè àâòîðåãðåñ- ñèè òðåòüåãî ðàíãà.  îòëè÷èå îò äðóãèõ ðàáîò äëÿ óìåíüøåíèÿ âëèÿíèÿ øóìîâ íà òî÷íîñòü ñïåêò- ðàëüíûõ îöåíîê ïðåäëîæåíî èñïîëüçîâàòü ñïåêòðû âûñøèõ ïîðÿäêîâ. Òàêèå ñïåêòðû ìåíåå ÷óâñòâè- òåëüíû ê ãàóññîâûì ïîìåõàì. Ïðèìåíÿÿ îïèñàí- íûé â ñòàòüå ìåòîä, ìîæíî òàêæå íàõîäèòü îöåíêè ñïåêòðà íåãàóññîâà ñèãíàëà íà ôîíå áîëåå øèðîêîãî êëàññà ïîìåõ ñ ðàçëè÷íûìè ðàñïðåäåëåíèÿìè. Ëèòåðàòóðà: 1. Ìàðïë - ìë. Ñ. Ë. Öèôðîâîé ñïåêòðàëü- íûé àíàëèç è åãî ïðèëîæåíèÿ. Ì.: Ìèð, 1990. 584 ñ. 2. Êóðåøè Ø.Ó.Õ. Àäàïòèâíàÿ êîððåêöèÿ // ÒÈÈÝÐ, 1985. Ò. 73, ¹9. Ñ. 5-49. 3. Haykin S. Radar signal processing / / IEEE ASSP Magazine 1985. Vol. 2. P. 2-18. 4. Êîðîòàåâ Ã.À. Ýôôåêòèâíûé àëãîðèòì êîäèðîâàíèÿ ðå÷åâîãî ñèãíàëà íà ñêîðîñòè 4,8 êáèò/ñ è íèæå // Çàðóáåæíàÿ ðàäèîýëåêòðîíèêà, 1996. ¹3. Ñ. 57-68. 5. Êóí÷åíêî Þ.Ï. Íåëèíåéíàÿ îöåíêà ïàðàìåòðîâ íåãàóññîâñêèõ ðàäèî- òåõíè÷åñêèõ ñèãíàëîâ. Ê.: Âûùà øêîëà, 1987. 191ñ. 6. Âàëååâ Â.Ã., Äàíèëîâ Â.À. Îïòèìàëüíîå îáíàðóæåíèå ñèãíàëîâ íà ôîíå íåãàóññîâñêèõ êîððåëèðîâàííûõ ðàäèîïîìåõ // Èçâ. âóçîâ. Ðàäèîýëåêòðîíèêà. 1991. ¹7. Ñ. 30-34. 7. Ìàëàõîâ À.Í. Êóìóëÿíòíûé àíàëèç ñëó÷àé- íûõ íåãàóññîâûõ ïðîöåññîâ è èõ ïðåîáðàçîâàíèé. Ì.: Ñîâ. ðàäèî, 1978. 376 ñ. 8. Òèõîíîâ Â.À. Îáîáùåííàÿ ìîäåëü àâòîðåãðåññèè íåãàóññîâûõ ïðîöåññîâ // Ðà- äèîòåõíèêà. 2003. ¹ 132. Ñ.78-82. 9. Æóðáåíêî È.Ã. Àíàëèç ñòàöèîíàðíûõ è îäíîðîäíûõ ñëó÷àéíûõ ñèñ- òåì. Ì.: Èçä-âî Ìîñê. óí-òà, 1987. 240 ñ. 10. Áðèëëèíä- æåð Ä.Ð. Âðåìåííûå ðÿäû. Îáðàáîòêà äàííûõ è òåî- ðèÿ. Ì.: Ìèð, 1980. 536 ñ. 11. Øåëóõèí Î.È., Áåëÿåâ È.Â. Íåãàóññîâñêèå ïðîöåññû. ÑÏá.: Ïîëèòåõíèêà, 1992. 312 ñ. 12. Òèõîíîâ Â.À., Ðóñàíîâñêèé Ä.Å., Òèõîíîâ Ä.Â. Ãåíåðàöèÿ óçêîïîëîñíûõ èìèòàöèîííûõ ñëó÷àéíûõ ïðîöåññîâ // Ðàäèîýëåêòðîíèêà è èíôîðìàòèêà. 1999. ¹4. Ñ. 83-85. Ïîñòóïèëà â ðåäêîëëåãèþ 24.04.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Ëó÷àíèíîâ À.È. Òèõîíîâ Âÿ÷åñëàâ Àíàòîëüåâè÷, êàíä. òåõí. íàóê, äî- öåíò êàôåäðû ÐÝÑ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: òåî- ðèÿ ëèíåéíîãî ïðåäñêàçàíèÿ, íåãàóññîâû ïðîöåññû, ðàñïîçíàâàíèå è êîäèðîâàíèå ðå÷è, ýêîíîìè÷åñêàÿ ñòàòèñòèêà. Àäðåñ: Óêðàèíà, 61736, Õàðüêîâ, ïð. Ëåíè- íà, 14, òåë. 702-15-87. Íåòðåáåíêî Êîíñòàíòèí Âëàäèìèðîâè÷, àñïèðàíò êà- ôåäðû ÐÝÑ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàñïîçíàâà- íèå è êîäèðîâàíèå ðå÷è, íåãàóññîâû ïðîöåññû, òåîðèÿ ëèíåéíîãî ïðåäñêàçàíèÿ. Àäðåñ: Óêðàèíà, 61736, Õàðü- êîâ, ïð. Ëåíèíà, 14, òåë. 702-15-87. N ОСШ 1000 500 300 200 0,63 1,16 2,16 2,6 0 дБ 19,96 19,81 19,68 19,74 2,23 6,93 9,09 10,61 – 3 дБ 19,93 19,9 19,16 19,47 14,08 16,09 – 6 дБ 21,31 24,0 — — ÓÄÊ 621.372.832 ÝÍÅÐÃÅÒÈ×ÅÑÊÈÅ ÕÀÐÀÊÒÅÐÈÑÒÈÊÈ ÌÍÎÃÎÝËÅÌÅÍÒÍÛÕ ÐÅÇÎÍÀÒÎÐÍÎ-ÙÅËÅÂÛÕ ÈÇËÓ×ÀÒÅËÅÉ ÊÀÒÐÈ× Â.À. Ìåòîäîì íàâåäåííûõ ìàãíèòîäâèæóùèõ ñèë ðåøàåòñÿ çàäà÷à î ðàñïðåäåëåíèè ìàãíèòíûõ òîêîâ â ðåçîíàòîð- íî-ùåëåâîé ñòðóêòóðå: “ ùåëè ñâÿçè â ñòåíêå áåñêî- íå÷íîãî ïðÿìîóãîëüíîãî âîëíîâîäà - ïðÿìîóãîëüíûé îáúåìíûé ðåçîíàòîð - èçëó÷àþùàÿ ùåëü â áåñêîíå÷- íîì ýêðàíå ” . Âûïîëíÿþòñÿ ÷èñëåííûå ðàñ÷åòû è ïðèâîäÿòñÿ ãðàôèêè ýíåðãåòè÷åñêèõ õàðàêòåðèñòèê òàêèõ ñèñòåì. 1. Ââåäåíèå  íàñòîÿùåå âðåìÿ â òåõíèêå ÑÂ× øèðîêî èñïîëü- çóþòñÿ âîëíîâîäíî-ùåëåâûå èçëó÷àòåëè â êà÷åñòâå ýëåìåíòîâ ñèíôàçíûõ è ñêàíèðóþùèõ àíòåííûõ ðåøåòîê, îáëó÷àòåëåé îñòðîíàïðàâëåííûõ çåðêàëü- íûõ è ëèíçîâûõ àíòåíí, à òàêæå óñòðîéñòâ ñâÿçè ýëåêòðîäèíàìè÷åñêèõ îáúåìîâ [1-4]. Îäèíî÷íûå ùåëåâûå èçëó÷àòåëè õàðàêòåðèçóþòñÿ ñóùåñòâåí- íîé øèðîêîïîëîñíîñòüþ, ÷òî â óñëîâèÿõ, íàïðè- ìåð, ñëîæíîé ýëåêòðîìàãíèòíîé îáñòàíîâêè ìîæåò ïðèâåñòè ê íàðóøåíèþ ðàáîòû ðàäèîýëåêòðîííûõ ñèñòåì (ÐÝÑ).  ñâÿçè ñ ýòèì çàäà÷è àíàëèçà, ñèíòåçà è óïðàâëåíèÿ äèàïàçîííûìè õàðàêòåðèñòè- êàìè ùåëåâûõ àíòåíí ïðåäñòàâëÿþò íåñîìíåííûé èíòåðåñ äëÿ ïðàêòèêè, â ÷àñòíîñòè, ñ òî÷êè çðåíèÿ îáåñïå÷åíèÿ ýëåêòðîìàãíèòíîé ñîâìåñòèìîñòè ðàç- ëè÷íûõ êîìïîíåíòîâ ÐÝÑ. @x pǿ! 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Çàêëþ÷åíèå Òàêèì îáðàçîì, â äàííîé ðàáîòå äëÿ ðåøåíèÿ ñâÿçàííûõ èíòåãðàëüíûõ óðàâíåíèé îòíîñèòåëüíî ìàãíèòíûõ òîêîâ â ìíîãîýëåìåíòíîé ðåçîíàòîðíî- ùåëåâîé ñòðóêòóðå èñïîëüçîâàíû íîâûå áàçèñíûå ôóíêöèè, àäåêâàòíûå ðåàëüíîìó ôèçè÷åñêîìó ïðîöåññó, ÷òî ñîñòàâëÿåò íàó÷íóþ íîâèçíó èññëåäî- âàíèé. Ðàññìîòðåííûå èçëó÷àòåëè ìîãóò áûòü ïîëåçíû êàê ïðè ðàçðàáîòêå íîâûõ àíòåííûõ ïðèåìî-ïåðåäàþ- ùèõ ñèñòåì ÑÂ×, òàê è äëÿ ìîäåðíèçàöèè ñóùå- ñòâóþùèõ íà îñíîâå èìåþùåéñÿ ýëåìåíòíîé áàçû ïóòåì íåçíà÷èòåëüíîãî èçìåíåíèÿ êîíñòðóêöèè, ÷åì îïðåäåëÿåòñÿ ïðàêòè÷åñêàÿ çíà÷èìîñòü ïðîâå- äåííûõ èññëåäîâàíèé. Ëèòåðàòóðà: 1. Schaik H.J. The performance of an iris- loaded planar phased array antenna of rectangular waveguides with external dielectric sheet // IEEE Trans. 1978. AP-26, ¹ 3. P. 413-419. 2. Íåñèììåòðè÷íûå êðåñ- òîîáðàçíûå ùåëåâûå èçëó÷àòåëè â òîðöàõ âîëíîâîäîâ / Ãîðîáåö Í.Í., Æèðîíêèíà À.Â., Çäîðîâ À.Ã., ßöóê Ë.Ï. // Ñá.: Àíòåííû. 1979. ¹ 27. Ñ. 159-166. 3. ßöóê Ë.Ï., Êàòðè÷ Â.À. Ùåëåâûå ýëåìåíòû ñâÿçè ñ ÷àñòè÷- íûì äèýëåêòðè÷åñêèì çàïîëíåíèåì // Èçâ. âóçîâ. Ðà- äèîýëåêòðîíèêà. 1979. Ò. 22, ¹ 2. Ñ. 60-65. 4. Êàòðè÷ Â.À., Íåñòåðåíêî Ì.Â. Ýëåêòðîìàãíèòíàÿ ñâÿçü ïðÿìî- óãîëüíûõ âîëíîâîäîâ ÷åðåç óçêèå ùåëè â îáùèõ ñòåí- êàõ // Ðàäèîòåõíèêà. 2002. Âûï. 127. Ñ. 77-84. 5. Áåðäíèê Ñ.Ë., Êàòðè÷ Â.À., Ëÿùåíêî Â.À. Ýíåðãåòè÷åñêèå è íàïðàâëåííûå õàðàêòåðèñòèêè êîìáèíèðîâàííûõ âîë- íîâîäíî-ùåëåâûõ èçëó÷àòåëåé // ³ñí. Õàðê³â. íàö. óí-òó. Ðàä³îô³çèêà òà åëåêòðîí³êà. 2002. ¹ 544. Ñ. 39- 42. 6. Êàòðè÷ Â.À., Íåñòåðåíêî Ì.Â., Áåðäíèê Ñ.Ë. Ìåòîä íàâåäåííûõ ìàãíèòîäâèæóùèõ ñèë äëÿ ñèñòåìû ïðî- äîëüíûõ ùåëåé â ñòåíêàõ âîëíîâîäîâ // ³ñí. Õàðê³â. íàö. óí-òó. Ðàä³îô³çèêà òà åëåêòðîí³êà. 2004. ¹ 622. Ñ. 95-101. 7. Chen T.S. Waveguide resonant-iris filters with very wide passband and stopbands // Int. J. Electronics. 1966. V. 21, ¹ 5. P. 401-424. 8. Ýëåêòðîäèíàìè÷åñêèé ñèíòåç è àíàëèç øèðîêîïîëîñíûõ âîëíîâîäíûõ ôèëü- òðîâ íà ðåçîíàíñíûõ äèàôðàãìàõ / Êèðèëåíêî À.À., Ðóäü Ë.À., Ñåíêåâè÷ Ñ.Ë., Òêà÷åíêî Â.È.// Èçâ. âóçîâ. Ðàäèîýëåêòðîíèêà. 1997. Ò. 40, ¹ 11. Ñ. 55-62. 9. Êàòðè÷ Â.À., Íåñòåðåíêî Ì.Â. Ìåòîä íàâåäåííûõ ìàãíèòîäâè- æóùèõ ñèë äëÿ ðåçîíàòîðíî-ùåëåâûõ èçëó÷àòåëåé è îòâåðñòèé ñâÿçè // Èçâ. âóçîâ. Ðàäèîýëåêòðîíèêà. 2004. Ò. 47, ¹ 1. Ñ. 12-20. 10. Êàòðè÷ Â.À., Íåñòåðåíêî Ì.Â., Áåðäíèê Ñ.Ë. Ìåòîä íàâåäåííûõ ìàãíèòîäâèæóùèõ ñèë äëÿ ñèñòåìû ïîïåðå÷íûõ ùåëåé â øèðîêîé ñòåíêå ïðÿìîóãîëüíîãî âîëíîâîäà // Ðàäèîòåõíèêà. 2003. Âûï. 131. Ñ. 76-82. 11. Êàòðè÷ Â.À., Íåñòåðåíêî Ì.Â., Õèæíÿê Í.À. Ñðàâíèòåëüíûé àíàëèç àíàëèòè÷åñêèõ ìåòîäîâ ðåøåíèÿ èíòåãðàëüíûõ óðàâíåíèé äëÿ ìàãíèòíîãî òîêà â ùåëåâûõ èçëó÷àòåëÿõ // Çàðóáåæíàÿ ðàäèîýëåê- òðîíèêà. Óñïåõè ñîâðåìåííîé ðàäèîýëåêòðîíèêè. 2002. ¹ 12. Ñ. 15-25. 12. Warne L.K. Eddy current power dissipation at sharp corners: closely spaced rectangular conductors // Journal of Electromagnetic Waves and Applications. 1995. Vol. 9, ¹ 11-12. P. 1441-1458. Ïîñòóïèëà â ðåäêîëëåãèþ 09.12.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Áóëãàêîâ À.À. Êàòðè÷ Âèêòîð Àëåêñàíäðîâè÷, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ïðèêëàäíîé ýëåêòðîäèíàìèêè ðàäèî- ôèçè÷åñêîãî ôàêóëüòåòà Õàðüêîâñêîãî íàöèîíàëüíîãî óíèâåðñèòåòà èì. Â.Í. Êàðàçèíà. Íàó÷íûå èíòåðåñû: èçëó÷åíèå è ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí ùåëå- âûìè íåîäíîðîäíîñòÿìè â ïðÿìîóãîëüíûõ è ïîëîñêî- âûõ âîëíîâîäàõ. Àäðåñ: Óêðàèíà, 61077, Õàðüêîâ, ïë. 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Ðåøàåòñÿ ãðàíè÷íàÿ ýëåêòðîäèíàìè÷åñêàÿ çàäà÷à äëÿ ðåçîíàòîðà ñ äâóìÿ íàñòðîå÷íûìè ýëåìåíòàìè è ñâî- áîäíîé ïðèîñåâîé îáëàñòüþ. Âûâîäèòñÿ äèñïåðñèîí- íîå óðàâíåíèå îòíîñèòåëüíî ñîáñòâåííûõ ÷àñòîò. Ïðèâîäèòñÿ ôîðìóëà ðàñ÷åòà äîáðîòíîñòè. 1. Ââåäåíèå Ôóíäàìåíòàëüíûå èññëåäîâàíèÿ ïî ãåíåðàöèè, èçëó÷åíèþ, ðàññåÿíèþ è ðàñïðîñòðàíåíèþ ýëåêò- ðîìàãíèòíûõ âîëí ñ ïðèìåíåíèåì ïîëó÷åííûõ ðåçóëüòàòîâ ê îñâîåíèþ äèàïàçîíîâ âîëí, à òàêæå èññëåäîâàíèÿ ñòàöèîíàðíûõ êîëåáàíèé è íåñòàöè- îíàðíûõ ýëåêòðîìàãíèòíûõ ïîëåé ñâÿçàíû ñ ïëà- íàìè, îáîçíà÷åííûìè Ãåíåðàëüíîé Àññàìáëååé URSI åùå â 1996 ã. [1]. Ñîñòàâíîé ÷àñòüþ ìíîãèõ ñîâðåìåííûõ òåõíè÷åñêèõ óñòðîéñòâ ÿâëÿþòñÿ ðå- çîíàòîðû.  ñâÿçè ñ èññëåäîâàíèåì ýëåêòðîìàãíèò- íûõ êîëåáàíèé â çàêðûòûõ ðåçîíàíñíûõ îáúåìàõ ïðîäîëæàåòñÿ èõ èçó÷åíèå è óñîâåðøåíñòâîâàíèå. Ñîçäàíèå ðåçîíàòîðîâ ñ çàäàííûìè õàðàêòåðèñòè- êàìè ñâÿçàíî ñ ðàññìîòðåíèåì ñòðóêòóð ñî ñëîæíîé ãåîìåòðèåé âíóòðåííåé ïîâåðõíîñòè. Ïðîáëåìà çàêëþ÷àåòñÿ â âûáîðå êîíôèãóðàöèè ðåçîíàíñíîé ñèñòåìû è ïîñòðîåíèè àäåêâàòíîé ìàòåìàòè÷åñêîé ìîäåëè. Èçó÷åíèþ ñîáñòâåííûõ êîëåáàíèé è âîçáóæäåíèþ îáúåìíûõ ðåçîíàòîðîâ ðàçíîé ãåîìåòðè÷åñêîé êîí- ôèãóðàöèè óäåëÿåòñÿ âíèìàíèå âî ìíîãèõ ðàáîòàõ, íà÷èíàÿ îò ôóíäàìåíòàëüíûõ, ñòàâøèõ óæå êëàññè- ÷åñêèìè [2-6], è çàêàí÷èâàÿ ðÿäîì ñîâðåìåííûõ ïóáëèêàöèé [7-9]. Îáû÷íî ñíà÷àëà ðåøàåòñÿ çàäà÷à íà ñîáñòâåííûå çíà÷åíèÿ, êîòîðàÿ ñâîäèòñÿ ê íà- õîæäåíèþ ñîáñòâåííûõ ïîëåé è ÷àñòîò ïîëîãî ðåçîíàòîðà. Ïðè ýòîì ñëåäóåò íàéòè ðåøåíèå óðàâ- íåíèé Ìàêñâåëëà â îáúåìíîé îáëàñòè V, îãðàíè÷åí- íîé ïîâåðõíîñòüþ S, êîòîðîå óäîâëåòâîðÿåò ñîîò- âåòñòâóþùèì óñëîâèÿì íà ãðàíèöå. Íà ñòðóêòóðó è ñâîéñòâà ýëåêòðîìàãíèòíîãî ïîëÿ â îáúåìíîì ðåçîíàòîðå âëèÿþò åãî ãåîìåòðè÷åñêèå ðàçìåðû, ñïîñîá âîçáóæäåíèÿ, à òàêæå ñâîéñòâà ñðåäû, êîòî- ðîé ìîæåò çàïîëíÿòüñÿ ðåçîíàòîð.  ñâÿçè ñ èññëåäîâàíèåì ýëåêòðîìàãíèòíûõ õàðàê- òåðèñòèê ïîëóïðîâîäíèêîâûõ, ìàãíèòíûõ è ñâåðõ- ïðîâîäÿùèõ èçîòðîïíûõ è àíèçîòðîïíûõ ìàòåðè- àëîâ ïðèìåíÿåòñÿ ðåçîíàòîðíûé ìåòîä èçìåðåíèé.  êà÷åñòâå ìîäåëåé ðåçîíàòîðîâ ñ èçìåíÿþùåéñÿ ÷àñòîòîé êîëåáàíèé èñïîëüçóþòñÿ, â ÷àñòíîñòè, çàìêíóòûå öèëèíäðè÷åñêèå ðåçîíàòîðû ñëîæíîé ôîðìû êàê ñ êîàêñèàëüíî ðàñïîëîæåííûìè ýëå- ìåíòàìè [10], òàê è ñî ñâîáîäíîé ïðèîñåâîé îáëà- ñòüþ.  [11] ðàññìàòðèâàåòñÿ ýëåêòðîäèíàìè÷åñêàÿ ñèñ- òåìà òèïà êîàêñèàëüíûé ðåçîíàòîð ñ äèàãíîñòèðó- åìûì èçîòðîïíûì âåùåñòâîì, ðàçìåùåííûì â îáëàñòè óêîðà÷èâàþùåé åìêîñòè. Ïðèìåíèòåëüíî ê äàííîé ñèñòåìå èñïîëüçóåòñÿ ìîäèôèöèðîâàí- íûé ìåòîä Òðåôòöà. Ïðåäëîæåííûé âû÷èñëèòåëü- íûé àëãîðèòì ïîçâîëèë ïðîâåñòè àíàëèç âëèÿíèÿ ýëåìåíòîâ êîíôèãóðàöèè ðåçîíàòîðà è ïàðàìåòðîâ ñðåäû íà ðåçîíàíñíóþ ÷àñòîòó îñíîâíîãî òèïà è åãî äîáðîòíîñòü. Íåñìîòðÿ íà óñîâåðøåíñòâîâàíèå íåêîòîðûõ ìàòå- ìàòè÷åñêèõ ïîäõîäîâ ïî îòíîøåíèþ ê ðàñ÷åòàì ðåçîíàòîðîâ ñëîæíîé ñòðóêòóðû, êëàññè÷åñêèå ìåòîäû îñòàþòñÿ âîñòðåáîâàííûìè [12-16]. 2. Ïîñòàíîâêà è ãåîìåòðèÿ çàäà÷è Îáúåêòîì èññëåäîâàíèÿ ÿâëÿåòñÿ ýëåêòðîìàãíèò- íîå ïîëå â öèëèíäðè÷åñêîì ðåçîíàòîðå ñî ñëîæíîé ôîðìîé âíóòðåííåé îãðàíè÷èâàþùåé ïîâåðõíîñ- òè. Ýòî äàåò âîçìîæíîñòü ñôîðìóëèðîâàòü ãðàíè÷- íóþ çàäà÷ó äëÿ óðàâíåíèé Ìàêñâåëëà, ðåøåíèå êîòîðîé ïîçâîëèò îïðåäåëèòü ñîáñòâåííûå ÷àñòîòû â ñîîòâåòñòâèè ñ ãåîìåòðè÷åñêèìè ïàðàìåòðàìè ðåçîíàòîðà. Ðàññìàòðèâàåìàÿ ñòðóêòóðà ïðåäñòàâ- ëåíà íà ðèñóíêå. Îáëàñòè ðåçîíàòîðà îïðåäåëÿþòñÿ ñëåäóþùèìè èíòåðâàëàìè èçìåíåíèÿ êîîðäèíàò: d r d , z 0 : ) I (     l , t g g , b r d , z g : ) II ( 1 1         l l , a r d , g z 0 : ) III (     è ìîãóò çàïîëíÿòüñÿ ìàòåðèàëàìè ñ ðàçëè÷íûìè äèýëåêòðè÷åñêèìè êîíñòàíòàìè ) j ( 0          . 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(3) Ïîòåíöèàëüíûå ôóíêöèè ) z , r ( ) m (  ÿâëÿþòñÿ ðå- øåíèÿìè ñêàëÿðíîãî îäíîðîäíîãî óðàâíåíèÿ Ãåëü- ìãîëüöà è äëÿ òðåõ îáëàñòåé ðåçîíàòîðà îïðåäåëÿ- þòñÿ ñëåäóþùèì îáðàçîì:                          ; III область , ) z k cos( ) r k ( Z C ; II область , )) g z ( k cos( ) r k ( Z B ; I область , ) z k cos( ) r k ( Z A z , r 0 i III zi III ri III 0 i 0 i 1 II zi II ri II 0 i 0 i I zi I ri I 0 i m (4) ãäå l / i k I zi   , 1 II zi / i k l   , g / i k III zi   , 2 ) m ( zi 2 ) m ( ri ) k ( k k   . Öåëü ðàáîòû – ïîëó÷èòü óðàâíåíèå ñîáñòâåííûõ ÷àñòîò è ôîðìóëó âû÷èñëåíèÿ äîáðîòíîñòè öèëèí- äðè÷åñêîãî ðåçîíàòîðà ñî ñâîáîäíîé ïðèîñåâîé îáëàñòüþ è äâóìÿ íåçàâèñèìûìè ýëåìåíòàìè íà- ñòðîéêè. 3. Ìåòîä ðåøåíèÿ è îñíîâíîå ñîäåðæàíèå ðàáîòû  ñâÿçè ñ ïîòðåáíîñòÿìè ïðàêòèêè âîçíèâàåò íåîá- õîäèìîñòü èññëåäîâàíèÿ âñå áîëåå ñëîæíûõ ãðà- íè÷íûõ çàäà÷. Ó÷åò íîâûõ ýëåêòðîäèíàìè÷åñêèõ è ãåîìåòðè÷åñêèõ ôàêòîðîâ ïðèâîäèò ê óñëîæíåíèþ âèäà óðàâíåíèé, ïðè ýòîì òî÷íûå ðåøåíèÿ óäàåòñÿ ïîëó÷èòü ðåäêî.  ýòèõ ñëó÷àÿõ èñïîëüçóþòñÿ ïðèáëèæåííûå ìåòîäû, ñðåäè êîòîðûõ êâàçèñòàöè- îíàðíûé, ìåòîä “ âîçìóùåíèé ” , ìåòîä ýêâèâàëåí- òíûõ ñõåì èëè èìïåäàíñíûé, âàðèàöèîííûé, ìåòîä ôàêòîðèçàöèè [12-17]. Îäíàêî äëÿ ó÷åòà ãåîìåòðèè îáëàñòåé íà àíàëèòè÷åñêîì óðîâíå ïðèìåíÿåòñÿ ìåòîä ÷àñòè÷íûõ îáëàñòåé. Îí îñíîâûâàåòñÿ íà ðàçäåëåíèè ðåçîíàíñíûõ îáúåìîâ íà îáëàñòè áîëåå ïðîñòîé ôîðìû, äëÿ êîòîðûõ èçâåñòíû ñîáñòâåí- íûå ôóíêöèè, è ñòðîãîì óäîâëåòâîðåíèè óñëîâèé íåïðåðûâíîñòè ïîëÿ íà ïîâåðõíîñòè ðàçäåëà ÷àñ- òè÷íûõ îáëàñòåé. Äëÿ ðàññìàòðèâàåìîãî ðåçîíàòîðà íåîáõîäèìî ó÷åñòü ñëåäóþùèå óñëîâèÿ: íà ãðàíèöå ðàçäåëà ÷àñòè÷íûõ îáëàñòåé ýëåêòðè÷åñêîå è ìàãíèòíîå ïîëÿ äîëæíû áûòü íåïðåðûâíûìè; êàñàòåëüíûå ñîñòàâëÿþùèå âåêòîðà ýëåêòðè÷åñêîãî ïîëÿ äîëæíû îáðàùàòüñÿ â íóëü íà èäåàëüíî ïðîâîäÿùèõ ñòåíêàõ ðåçîíàòîðà, ò.å.                      , g z 0 , E , z g , g z g , 0 , ) ( z g , E E ) III ( z 1 1 1 1 ) II ( z ) I ( z l l l l (5)                . g z 0 , H , ) ( z g , H H ) III ( 1 ) II ( ) I ( l l (6) Ñîñòàâëÿþùèå z E è  H , îïðåäåëåííûå ôîðìóëà- ìè (1)-(3), ïîä÷èíÿþòñÿ ãðàíè÷íûì óñëîâèÿì (5), (6).  ðåçóëüòàòå ïîëó÷àåòñÿ ñèñòåìà ôóíêöèîíàëü- íûõ óðàâíåíèé, îòêóäà ñëåäóåò áåñêîíå÷íàÿ ñèñòå- ìà ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé îòíîñè- òåëüíî êîýôôèöèåíòîâ i A , ðàâíûé íóëþ îïðåäå- ëèòåëü êîòîðîé ïðåäñòàâëÿåò ñîáîé èñêîìîå óðàâ- íåíèå ñîáñòâåííûõ ÷àñòîò:     )) x ( Z 2 /( ) x ( Z x ) 1 ( )[ x ( Z x { det I j I 1 I j I 0 I j 0 j ji I i I 1 I i          ) x ( Z ) x ( Z x ) 1 ( ) n , i ( i ) n , j ( i 2 II n II 1 II n II 0 II n 0 n 1 0 n 2 2 2 1 l 0 ]} ) x ( Z ) x ( Z x g ) 1 ( ) n , i ( i ) n , j ( i 2 III n III 1 III n III 0 III n 0 n 0 n 1 1 3 1          , (7) ãäå ij  – ñèìâîë Êðîíåêåðà; d k x I rj I j  , d k x II rj II j  , d k x III rj III j  . Ôóíêöèè I s Z , II s Z , III s Z â ñîîòâåòñòâó- þùèõ ÷àñòè÷íûõ îáëàñòÿõ ðåçîíàòîðà îïðåäåëÿþò- ñÿ êàê: ) r k ( J ) r k ( Z I ri s I ri I s  , ) r k ( N ) b k ( J ) b k ( N ) r k ( J ) r k ( Z II ri s II ri s II ri s II ri s II ri II s   , ) r k ( N ) a k ( J ) a k ( N ) r k ( J ) r k ( Z III ri s III ri s III ri s III ri s III ri III s   , ...; , 2 , 1 , 0 j , i  . 1 , 0 s  Ïðè ìíèìûõ çíà÷åíèÿõ r k , ò.å. 0 k 2 r  , èìååò ìåñòî 2 2 z r r k k j k j k       , òîãäà s Z ÿâëÿþòñÿ ìîäèôèöèðîâàííûìè ôóíêöè- ÿìè Áåññåëÿ è Íåéìàíà: ) r k ( I ) 2 jn exp( 2 ) r k ( Z I r n I r I n       ,         ) b k ( K ) r k ( I [ ) 2 jn exp( 2 ) r k ( Z II r 0 II r n II r II n )] r k ( K ) b k ( I ) 1 ( II r 0 II r 0 n     ,         ) a k ( K ) r k ( I [ ) 2 jn exp( 2 ) r k ( Z III r 0 III r n III r III n 2 , 1 , 0 n , )] r k ( K ) a k ( I ) 1 ( III r 0 III r 0 n      . Ñîñòàâëÿþùèå ) n , j ( i 1 è ) n , j ( i 2 âû÷èñëÿþòñÿ êàê îïðåäåëåííûå èíòåãðàëû: dz ) z k ( cos ) z k ( cos ) i , j ( i III zj g 0 I zj 1   , dz ) g z ( k cos ) z k ( cos ) i , j ( i 1 II zj g I zj 2 1      l l . 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Âûðàæåíèÿ äëÿ ðàñ÷åòà õàðàêòåðèñòèê ðåçîíàòîðà Äîáðîòíîñòü ðåçîíàòîðà îïðåäåëÿåòñÿ ôîðìóëîé: , V 0 P W Q   ãäå 0  – ðåçîíàíñíàÿ ÷àñòîòà; W – ñóììà ýëåêò- ðè÷åñêîé è ìàãíèòíîé ýíåðãèé m e W W W   ; V P – ïîòåðè ñîáñòâåííûõ êîëåáàíèé íà ÷àñòîòå  0 . Ýëåêòðè÷åñêàÿ è ìàãíèòíàÿ ýíåðãèè ñîãëàñíî [17] âû÷èñëÿþòñÿ òàê:              2 I ri 0 i 0 i 2 i 1 2 0 0 2 m e ) k ( ) 1 {( A l { 4 A W      ))] x ( Z ) x ( Z ( ) x ( ) / 2 [( I i 2 I 1 I i 2 I 0 2 I i 2      ))]} x ( Z ) x ( Z ) x ( Z ( ) x ( ) / 2 [( ) k ( I i I 2 I i I 0 I i 2 I 1 2 I i 2 2 I zi            2 II ri 0 i 0 i 2 i 2 ) k ( ) 1 ( B l        ] / 2 )) x ( Z ) x ( Z ( ) x [( 2 II i 2 II 1 II i 2 II 0 2 II i      ]} ) / 2 ( )) x ( Z ) x ( Z ) x ( Z ( ) x [( ) k ( 2 II i II 2 II i II 0 II i 2 II 1 2 II i 2 II zi            2 III ri 0 i 0 i 2 i 3 ) k ( ) 1 ( C g    ))] x ( Z ) x ( Z ( ) x [( III i 2 III 1 III i 2 III 0 2 III i      III I j ej III i III 2 III i III 0 III i 2 III 1 2 III i 2 III zi W ˆ ))]} x ( Z ) x ( Z ) x ( Z ( x [ ) k ( , (8)           0 i 0 i 2 i 2 m 0 m ) 1 ( A { 4 A W l      ))] x ( Z ) x ( Z ) x ( Z ( ) x ( ) / 2 [( I i I 2 I i I 0 I i 2 I 1 2 I i 2         0 i 0 i 2 i 1 ) 1 ( B l      ] ) / 2 ( )) x ( Z ) x ( Z ) x ( Z ( ) x [( 2 II i II 2 II i II 0 II i 2 II 1 2 II i         ) 1 ( C g 0 i 0 i 2 i ))]} x ( Z ) x ( Z ) x ( Z ( ) x [( III i III 2 III i III 0 III i 2 III 1 2 III i   . (9) Çäåñü êîýôôèöèåíò m A , ,... 2 , 1 , 0 m  – ïðîèçâîëü- íûé íîðìèðóþùèé ìíîæèòåëü; i A  , i B  , i C  – íîðìèðîâàííûå êîýôôèöèåíòû i A , i B , i C , ñîîò- âåòñòâåííî: ,... 2 , 1 , 0 j , ) x ( Z x ) 1 ( ) j , i ( i ) x ( Z x A 2 B II j II 1 II j 1 0 j 0 i 2 I 1 I 1 I i i j          l , ,... 2 , 1 , 0 j , ) x ( Z gx ) 1 ( ) j , i ( i ) x ( Z x A 2 C III j III 1 III j 0 j 0 i 1 I 1 I 1 I i i j          . Ïîòåðè â ðåçîíàòîðå îïðåäåëÿþòñÿ ñóììîé äèýëåê- òðè÷åñêèõ ïîòåðü vd P è ïîòåðü íà íàãðåâàíèå vs P : vs vd V P P P   , ãäå vd P è vs P çàäàþòñÿ ôîðìóëà- ìè: ej III 1 j 0 vd W ˆ P                , çäåñü ej W ˆ îïðåäåëåíû ôîðìóëîé (8),            0 i 0 j I j I 1 I j I i I 1 I i 3 4 j i 2 m vs ) x ( Z x ) x ( Z x ) j , i ( i R { A A { A P        ] R R ) 1 )[( 1 ( 5 2 j i ij     2 I j 2 I i I j I 0 2 I j I i I 1 I i I j I 1 I j I i I 0 2 I i ) x ( ) x ( ) x ( Z ) x )( x ( Z x ) x ( Z x ) x ( Z ) x (           2 / ) R R ( ) b 2 / ( ) / 2 ( 1 R [ 5 2 1 2 0 i 1 ij l      ))]]} x ( Z ) x ( Z ) x ( Z ( ) x ( ) / 2 [( I i I 2 I i I 0 I i 2 I 1 2 I i 2                0 i 4 2 j i 0 j ij j i ] R R ) 1 )[( 1 {( B B     2 II j 2 II i II j II 1 II j II i II 0 2 II i II j II 0 II j II i II 1 II i ) x ( ) x ( ) x ( Z x ) x ( Z ) x ( ) x ( Z x ) x ( Z x         2 / ) R R ( a 2 / ) / 2 )( 1 ( R [ 4 2 1 2 0 i 3 ij l      ]]} ) / 2 ( )) x ( Z ) x ( Z ) x ( Z ( ) x [( 2 II i II 2 II i II 0 II i 2 II 1 2 II i                0 i 5 4 j i 0 j ij j i ] R R ) 1 )[( 1 {( C C   ) x ( Z ) x )( x ( Z x [ III j III 0 2 III j III i III 1 III i    2 III j 2 III i III j III 1 III j III i III 0 2 III i ) x ( ) x /( )] x ( Z x ) x ( Z ) x (     2 III i 5 4 ij ) x ( 2 R R ))} x ( Z ) x ( Z ) x ( Z ( III i III 2 III i III 0 III i 2 III 1   .  âûðàæåíèè äëÿ vs P âåëè÷èíû k R , k=1,2,3,4,5 âû÷èñëÿþòñÿ ïî ôîðìóëå k 0 0 k 2 R     , ãäå k  îçíà÷àåò ïîâåðõíîñòíîå ñîïðîòèâëåíèå k-é ÷àñòè÷íîé ïîâåðõíîñòè. 5. 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Äëÿ ðàññìàòðèâàåìîé ñòðóêòóðû ïîëó÷åíî äèñ- ïåðñèîííîå óðàâíåíèå, èç êîòîðîãî îïðåäåëÿþòñÿ ñîáñòâåííûå çíà÷åíèÿ. Îíî ïðåäñòàâëÿåò ñîáîé ðàâíûé íóëþ îïðåäåëèòåëü áåñêîíå÷íîé ñèñòåìû. Åãî ïîðÿäîê çàâèñèò îò íåîáõîäèìîé òî÷íîñòè ðàñ÷åòîâ. Íà îñíîâå [17] ìîæíî îæèäàòü, ÷òî ðàñ÷åòíàÿ ÷àñòîòà óæå ïðè ïîðÿäêå ñèñòåìû 7 N  è òàêîì æå êîëè÷åñòâå ñëàãàåìûõ äîëæíà õîðîøî ñîãëàñîâûâàòüñÿ ñ ýêñïåðèìåíòàëüíîé. 3. Âûâåäåíà ôîðìóëà äëÿ îïðåäåëåíèÿ äîáðîòíîñòè è ðåçîíàíñíîãî ñîïðîòèâëåíèÿ ñòðóêòóðû. Ïîëó- ÷åííûå òî÷íûå àíàëèòè÷åñêèå ïðåäñòàâëåíèÿ äàþò âîçìîæíîñòü ïðîèçâåñòè íåîáõîäèìûå ÷èñëåííûå ðàñ÷åòû, ÷òî ïðåäñòàâëÿåò ïåðñïåêòèâó äàííîãî èññëåäîâàíèÿ. Ïðè âû÷èñëåíèè äîáðîòíîñòè ó÷èòûâàåòñÿ êîíå÷íàÿ ïðîâîäèìîñòü ðàçíûõ ó÷àñ- òêîâ ïîâåðõíîñòè ðåçîíàòîðà. Ëèòåðàòóðà: 1. Ukrainian URSI Committee // Ðàäèîôè- çèêà è ðàäèîàñòðîíîìèÿ. 1996. Ò.1, ¹1. Ñ. 125-158. 2. Âàéíøòåéí Ë.À., Ìàíåíêîâ À.Á. Êîàêñèàëüíûå ðåçîíà- òîðû // Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1973. Ò.18. Âûï. 9. Ñ. 1777-1784. 3. Ãàïîíîâ À.Â. 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Òèõîíîâ À.Í., Ñàìàðñêèé À.À. Óðàâíåíèÿ ìàòå- ìàòè÷åñêîé ôèçèêè. Ì.: Íàóêà, 1972. 735 ñ. 17. Kuhn E. Untersuchung eines kapazitiv belasteten koaxialen Hohlraumresonators//Archiv Der Elektrischen Ubertragung (A.E.U.). 1968. Band 22. ¹12. S.557–566. Ïîñòóïèëà â ðåäêîëëåãèþ 24.12.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò íàóê, ïðîô. ×óðþìîâ Ã.È. Ñëèï÷åíêî Íèêîëàé Èâàíîâè÷, êàíä. òåõí. íàóê, ïðî- ôåññîð, ïðîðåêòîð ïî íàó÷íîé ðàáîòå ÕÍÓÐÝ. Íàó÷- íûå èíòåðåñû: ðàäèîôèçèêà è ýëåêòðîíèêà. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. 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A new version of the software package KinFitSim (version 2.1) for kinetic simulation and experimental data fitting is presented. In addition to the possibilities offered by the previous version of KinFitSim, which was described before (Computers and Chemistry, V.26, 2002, pp. 379- 386), new features have been added to the program that include a robust automatic regime for kinetic simulation which simplifies the operation with KinFitSim for the user. KinFitSim 2.1 allows simultaneous fitting of several experimental curves corresponding to the same reaction mechanism that permits better quality of the results due to the better use of input information. The program can be used for both research and educational purposes. 1. Introduction Homogeneous reactions are of constant interest among chemists and biochemists due to the diversity of situations where they may take place. These range from simple laboratory experiments to extremely complex reactions occurring in living organisms and ecological systems. Despite their different nature all homogeneous chemical reactions can be represented by (systems of) stoichiometric equations. This general approach allows for relatively simple construction of mathematical models of homogeneous chemical processes which is implemented in a number of existing software packages offering kinetic simulation capabilities. The most known and frequently used software are briefly outlined below. Encora [1] is a program for enzymatic kinetic parameter fitting using progress curve analysis. It was developed by R.J.W. Slats et al. at Delft University of Technology. The program can handle a limited number of reaction mechanisms which constrains its applicability. Differential equations are solved using the fourth order Runge-Kutta method which means that the program would fail to solve a stiff system of ordinary differential equations. The Nelder-Mead method is used for fitting. EZ-Fit [2] is designed to simplify nonlinear regression analysis of enzyme kinetic data. After entering the data the program may suggests a fit or allow the user to select an equation from the menu of commonly used inhibition models. Thus the program is not able to simulate an arbitrary kinetic scheme. KINETICS [3] uses the Gear method for the numerical integration of multistep rate equations. Systems containing up to 100 reversible reactions and 50 chemical compounds can be examined. Each individual reaction can contain as many as 5 reactants and 5 products, allowing most systems of interest to be examined. However, the program does not permit fitting experimental data. KINSIM and FITSIM [4] are perhaps the most popular programs for kinetic simulation and data fitting. This is due to their universal nature: KINSIM can model any system of reactions. However, these programs have severe limitations because they were written for the MS- DOS operating system. The drawback is the absence of a modern user-friendly interface. The mechanism must be written using a separate text editor and compiled before use. The Chemical Kinetic Simulator (CKS) [5] is based on a stochastic simulation technique, has advanced user interface and a comprehensive manual for users. However, it lacks tools for kinetic parameter determination which makes it useful only for purely theoretical observations or teaching. Facsimile [6] models complex steady state and time dependent processes. Features are available to allow users to solve both differential and algebraic equations and fit parameters to experimental data. Kinetic simulations are performed using the Gear method with automatic time stepping. The disadvantages of Facsimile are complex model writing procedure and the need to use command language. As follows from its name, KinFitSim [7] was designed to carry out the kinetic simulation of any given reaction mechanisms and fit experimental data in order to determine the values of kinetic parameters governing the behaviour of a system under study. Both tasks can be performed with ease due to intrinsic straightforwardness of the operation with this software package. With KinFitSim, there are no limitations on the number of chemical steps in the mechanism or the number of species taking part in it. The program has been successfully employed in scientific research in the area of photochemistry [8]. The new version of the program has a number of important features such as the automatic simulation regime and simultaneous fitting of several datasets which make it an essential tool for the investigation of homogeneous chemical and biochemical processes. These features are discussed in the following sections. 2. Kinetic simulation The core of the KinFitSim package is represented by the kinetic simulator (KS). The Kinetic Simulator parses the entered reaction mechanism and generates its mathematical model which is a system of ordinary differential equations (ODEs). KS comprises various algorithms and methods aimed at efficient simulation of reaction mechanisms. The latter can be performed in either the automatic regime or by manual setting of simulation parameters depending on the user ’ s choice. 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Then the automatic simulation procedure analyses the mathematical model and solves it numerically ensuring that the results are physically meaningful and precise. This is achieved by applying appropriate methods for the solution of systems of ordinary differential equations and choosing the necessary accuracy of numerical integration. To this end, a specially designed algorithm of discrimination between stiff and non-stiff systems of ODEs [9] is implemented in KinFitSim. This permits the program to automatically select the appropriate numerical integration method, i.e. the Gear method for stiff systems and the Adams- Moulton method for non-stiff ones. In some cases it may happen that according to a priori information an ODE system appears to be non-stiff but then becomes stiff in the course of the simulation (i.e. for some t > 0). This may lead to erroneous solutions because of wrong choice of the solution method. However, the algorithm implemented in the automatic simulation subroutine of KinFitSim is capable of overcoming such situations and choosing an appropriate method in any case. The accuracy of a numerical method (defined by the limit of local integration error on integration time step) is also chosen automatically based on the previously proven theorem and criterion [9] which ensure that the resulting concentration-time curves do not suffer from non-physical oscillations. This means that KS begins calculations with reasonably high accuracy and then augments it if necessary. The importance of this issue is demonstrated by the following example. Thus the automatic simulation regime makes KinFitSim an extremely convenient and robust instrument for modelling homogeneous reaction mechanisms which can be exploited by the user with any level of mathematical skill. At the same time an interested user can benefit from the manual simulation regime where he/she can learn more about the numerical methods used for the solution of ODEs by choosing and tuning them at their discretion. The program can be conveniently switched to the manual simulation regime where the user can choose between the Runge-Kutta and Gear methods (if the latter is chosen and the ODE system is non-stiff the Adams-Moulton procedure is applied), set the desired accuracy of integration and other simulation parameters [7]. By manually setting the simulation method parameters the user will be able to observe the effects of different simulation parameters on the results of computation. These settings are especially important when a stiff system of ODEs is to be solved. In this case the program may fail when running in a manual simulation regime due to numerical solutions of stiff ODE systems being very sensitive to applied numerical methods and their accuracy. Therefore in this case the program suggests the user to apply the automatic regime which is a recommended option for any ODE system regardless of its stiffness. 3. Fitting experimental data The fitting simulator (FS) is designed for automatic fitting of theoretically predicted time dependent distributions to those obtained experimentally. This is achieved by iteratively running KS with various values of kinetic parameters in order to try and minimise the difference between experimental and theoretical curves. In the new version of the KinFitSim package, FS offers a new possibility of fitting multiple pairs of curves. This is useful in a situation when a reaction under study is characterised by several different quantities, which can be independently measured. These quantities can be, for example, solution absorbance at different wavelengths. The importance of such a feature is shown by the following example. Consider a reaction scheme: B A 1 k    , (1) D C B 2 k     , (2) F A E D 3 k      (3) with initial concentrations [A] 0  = 1M, [C] 0  = 0.5M, [E] 0  = 0.7M, [B] 0  = [D] 0  = [F] 0  = 0 and rate constants k 1  = 10 s -1 , k 2  = k 3  = 10 cm 3 mol -1 s -1 . By simulating reactions (1)-(3) with KS and adding 2% Gaussian noise to the concentrations of A, B, D and F we obtain spiky curves which replicate experimental ones measured with random fluctuations (see Figure 1, which shows a snapshot of the program window designed for loading and editing experimental data). Next we assume that the values of the rate constants are unknown to us and try to fit the ‘ experimental ’ curves from Figure 1 using FS given an initial guess for the rate constants of k 1  = 1 s -1 , k 2  = 1 cm 3 mol -1 s -1 , k 3  = 20 cm 3 mol -1 s -1 . A usual way of doing this would be to fit the experimental curve for [A] to the simulated one and repeat this for the rest of the curves. The results of this procedure are shown in Table 1, where the Nelder- Mead minimisation algorithm [10] was used with an accuracy of 10 -4 . It is clear that the results cannot be regarded as satisfactory because the resulting values of the rate constants are far from their true values (the deviation approaches 400% for k 1 !). This result was obtained because even though each experimentally measured curve contains all the information about the reaction mechanism it has different sensitivity to different kinetic parameters. For instance, the concentration distribution of species A is much more sensitive to k 1 than it is to k 2 and k 3 , which is observed in Table, i.e. the value of k 1 resulting from the fitting is much closer to the true one than those of k 2 and k 3 . Similar conclusions can be drawn from the other three fitting results which, however, are not as straightforward. Fitted curve k 1 / s -1 k 2 / cm 3 mol -1 s -1 k 3 / cm 3 mol -1 s -1 [A] 8.6099 1.0708 1.3051 [B] 5.2051 0.1428 2.3245 [D] 49.986 5.6705 9.7073 [F] 33.130 6.8927 9.7240 Fitting results for the reaction mechanism (1)-(3) tXx{pUM 7 4A!ڒh!0$@jB FT(R U  HU:BGTEKUVQtE[KN9gހ7{k?7͘iƘc2TSx1ӘHW`)y-7&[o? nfw=+H24ش34VEYoJb *ݘ̦^|ޖ*K5A=]kWuGG55"{~ yLv˭^HmZ۴/4i%٥雛*`r'|6_:?6 scdrS73)0k:tF矾xټb.mlʎ GoN6dv9g菎eF1]x)5TV7FJc'?t=MӁʯ%վ\.~fcSA\E_m~ _UfA{Ojz  y _vm^qg+sD6~[.~?͓hoqv7/H%Vg 7Y{G92T?0to^Xc̤X5M`7x$Vn7*ofNF5ļG siv|V586tx(?#DmMdǍKNy|5xD:ҕ#=EG{.lf_uly#>t4A 7MCLyڴ1Ep{-+Oq )?twiNȲIݤ A9FW ropmfJas6\ +"F$0S6ȵ5s W~-`uJȞ r} OÓu޴mV5p%< &< .joqS[Ͳ o_\[ξ 5 IznL.Oz?x 8!->׍@%+(LbN NwvPL:X֘E~~l'e·e*},XK5%[bb<ǂ. ŗElE&~e ͭk7n?5jրYZHZ*v<>m;7omKtt 8h.WT?N3nس@cl؛{co, AmvjV'I/5*i{k7zkg(tNGIN+M.)s]ɝpx |<@.OZ  +;`~_oˑuO5s.>k@>xgH$fc0|E 5^gSgۛOq[.P| 4Wsn?$ UE1ϕ*\&M=iG{s7x6< (n @1E@T|~e1Y YokK()`XG>GboZ5೨SIUCaI/ <> EG75ൻj_9*@KHhfsLc_Ag vWڴB:ŁƺW@sa2xۆHwl|xAglg~ 5uX::x*rSQ*K~(ڋo`v57΀W,90=_ G9PtO5(2ͭ @}հg|iE"gw ͫ 6:tdpFyR6ckXok:}#v_i/}˽ԽVnW8=LeHI^/_Fx^ޜn7)ԙO[%s36)oYz_n˸T!@o+orF~7+t=^L{P jΗ\P urYޱsrX9>W AZ2wG#@-At^΅@vob.shOs_i<.M]N4KcĤ 񨄵>4 .XsV/)/"F@7TOp'y<_gOz^·_=~5 so]ۺm/ɼzS[wn[K[=h|'Xf^oU)fA%q:[gǜ(gNJ,4 #W8e31D?Wn=3rO7f}g_oӸqM3=;S9AX_NfydїEs`B tK7x}etrM'TW&õ^kJ#a5>I:R?,US$pe-Oԏd֏`]},OwvKl]YJۖd>̶kpf }iN*jw޶d-K&Kf3W2֮T~e$Lv$Ӓ7~g`#Hڳ[k~`Ճi&Q-cڴu,m-Sa^Kơ[;[*A~|e/,~L_*x-=^Oo+6`S x{-2~|Õvz`:޷||xy|>i[7Q+(ߕw__0a#F @0a#F @0a#?G #F @0a#F @0a.EE7H  D  /  #TJ R (16Z 2 /' 1 ̜ :%{uxy@N?~&MhB{5 `D4h0%[n*cH26 !Jk3f]Nvt|>Y{uzy2A"(/O'TA&]T]Ai rv_&t X4)e B&/us~(%E  !$Ha"& n-2]YEDТ_zKא[]8)cHWt/gJJ?;uaT S f)P! 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In this case the fitting results are far better than those given in Table 1: k 1  = 9.990 s -1 , k 2  = 10.138 cm 3 mol - 1 s -1 , k 3  = 9.922 cm 3 mol -1 s -1 . The deviation from the true values of the rate constants does not exceed 1.5% and can be regarded as excellent given the significant amplitude of noise on the ‘ experimental ’ curves. Such a drastic difference in the fitting results with the previous approach is due to the fact that by fitting all the data simultaneously we even out the lack of sensitivity of certain curves to certain kinetic parameters (rate constants). Therefore the fitting procedure converges to the true solution in all kinetic parameters. Consequently, the possibility of fitting multiple data sets is a very important feature which allows obtaining significantly more accurate fitting results provided that several different experimental quantities characterising the same kinetic process are available. Apart from this new feature the FS has become much more reliable and robust compared to previous versions of the program. This was achieved by adding the Nelder- Mead minimisation method to the FS and optimising the work of methods which were implemented in FS previously (these are the direct search algorithm, Gradient search, Gauss-Newton and Marquardt methods). The Nelder-Mead method usually requires less computation compared to Gradient search, Gauss- Newton and Marquardt methods and therefore is typically more efficient than other methods. Although the recommended way of fitting data is the automatic fitting procedure the user can perform this task manually by adjusting the values of kinetic parameters, running KS and visually evaluating the quality of fit. This may be a good idea to find a reasonable initial fit between experimental and theoretical curves manually before running the automatic procedure, especially when the user is in possession of any additional information about possible values of kinetic parameters. This will reduce the time necessary to achieve a good fit by the FS. 4. Program interface KinFitSim has a well designed interface which is simple to understand for an untrained user (see Figure 2). The new version of the KinFitSim software includes a comprehensive user manual. Context help is available in most program windows which facilitates obtaining necessary information immediately when it is needed. The kinetic mechanism can be saved together with the simulation data and user ’ s comments for later use. This facilitates repeated simulation of the mechanism and storing research archives. The output of the program is also fully controllable by the user: it may be not only the concentrations of reacting species but also any algebraic functions of concentrations that permit direct comparison of simulated curves to experimental data. This feature significantly broadens the range of applicability of the KinFitSim software. The simulation or fitting results can be conveniently presented in the form of graphical images and reports containing comprehensive information about the initial data, solution process and its results. 5. Conclusions The new version of KinFitSim (2.1) has many advantages in comparison with previous versions of KinFitSim and other programs that are currently available. The principal advantages are the automatic simulation regime in KS, which allows simplification of the kinetic simulation for users of any mathematical skill, and a possibility to fit several pairs of theoretical and experimental curves at the same time in FS. 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All this makes KinFitSim software a valuable tool for research and chemical education. Acknowledgements The authors wish to thank the users of all previous versions of KinFitSim for their valuable comments and suggestions, which have allowed us to develop this new improved version. References: 1. Straathof A.J.J. Development of a computer program for analysis of enzyme kinetics by progress curve fitting // Journal of Molecular Catalysis B: Enzymatic. 2001. V. 11. P. 991-998. 2. http://www.jlc.net/~fperrell/webps04.htm. 3. http://www.ari.net/ars/kin.html. 4. Barshop B.A., Wrenn R.F., Frieden C. Analysis of numerical methods for computer simulation of kinetic processes: development of KINSIM – a flexible, portable system // Analytical Biochemistry. 1983. V. 130. P. 134-145. 5. http://www.almaden.ibm.com/st/ msim. 6. http://www.esm-software.com/facsimile. 7. Svir I.B., Klymenko O.V., Platz M.S. ‘ KINFITSIM ’ – a software to fit kinetic data to a user selected mechanism // Computers and Chemistry. 2002. V. 26. P. 379-386. 8. Tippmann E., Platz M., Svir I., Klymenko O. Evidence for Specific Solvation of Two Halocarbene Amides // Journal of the American Chemical Society 2004. V.126, P. 5750-5762. 9. Klymenko O.V. Methods for the determination of stiffness and required accuracy of numerical integration of mathematical models describing homogeneous chemical processes // Radioelektronika i Informatika. 2004. V. 3. P. 42-47. 10. Himmelblau D.M. Applied nonlinear programming. Moscow: Mir, 1975. 534 p. Ïîñòóïèëà â ðåäêîëëåãèþ 24.11.2004 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Õàõàíîâ Â.È. Svir Irina Borisovna, Chief Scientist, Dr. Tech. Sci., Head of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in research. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue. Email: svir@kture.kharkov.ua. Klymenko Oleksiy Victorovich, PhD, Senior Scientist of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in research, mathematical physics, programming. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue. Email: klymenko@kture.kharkov.ua. Oleinick Alexander Igorevich, Scientist of the Mathematical and Computer Modelling Laboratory, KhNURE. Scientific interests: mathematical modelling and numerical methods in research, mathematical physics, programming. Address: KhNURE, Kharkov, 61166, 14 Lenin Avenue. Email: oleinick@kture.kharkov.ua. Platz Matthew S., Distinguished Professor of Chemistry, Department of Chemistry, Newman and Wolfrom Laboratory, The Ohio State University. Scientific interests: organic chemistry and photochemistry. Address: 100 West 18th Avenue, Columbus, OH 43210-1183, USA. Fig. 2. 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Ñõåìàòè÷åñêîå èçîáðàæåíèå ÐÈÏ Äëÿ èññëåäîâàíèÿ ôóíäàìåíòàëüíûõ îñîáåííîñòåé ìîäåëè âíà÷àëå ðàññìîòðèì îäèí âàðèàíò ÿ÷ååê, ðàçäåëÿåìûõ òîíêèìè ïðîñëîéêàìè ñâîáîäíîãî ïðîñòðàíñòâà. Ýòà ìîäåëü îòðàæàåò ïîâåäåíèå äèñ- ïåðñíîãî îáúåêòà ðàçëè÷íîé ïëîòíîñòè. Èç íåîáõîäèìîñòè âûïîëíåíèÿ óñëîâèé íåïðåðûâ- íîñòè òàíãåíöèàëüíûõ Å è Í ñîñòàâëÿþùèõ ýëåê- òðîìàãíèòíîãî ïîëÿ ðàáî÷åãî âèäà êîëåáàíèé â ñîîòâåòñòâóþùåì ðåçîíàòîðå íà ãðàíèöàõ ñëîåâ ( i z ) è îáúåìîâ I, II, III âûòåêàåò ñëåäóþùàÿ ñèñòåìà ðåêóððåíòíûõ ñîîòíîøåíèé: ; )] z z ( [ th C 1 C )] z z ( [ th Y ) z ( th 1 0 2 1 1 1 0 2 2 , I 1 1          ; )] z z ( [ th C 1 C )] z z ( [ th Y )] z z ( [ th C 1 C )] z z ( [ th 2 0 3 2 2 2 0 3 3 , 2 2 0 2 1 1 2 0 2              ……………………………………(1) ; )] z z ( [ th C 1 C )] z z ( [ th Y )] z z ( [ th C 1 C )] z z ( [ th i 0 1 i i i i 0 1 i 1 i , i i 0 i 1 i 1 i i 0 i                   …………………………………………………… H th Y C III III , N N   , ãäå I 2 2 I 2 , I ~ ~ Y      ; i 1 i 1 i i 1 i , i ~ ~ Y         ; N III III N III , N ~ ~ Y      äëÿ ÐÈÏ Å-òèïà; 2 I 2 , I Y    ; 1 i i 1 i , i Y      ; III N III , N Y    äëÿ ÐÈÏ Í- òèïà; i i i i i i 2 0 кр i 0 0 0 i tg ; j ~ ; ~                                           ; 0  – êðóãîâàÿ ðåçîíàíñíàÿ ÷àñòîòà; кр  – íèæíÿÿ ïðåäåëüíàÿ ÷àñòîòà ðàñïðîñòðàíåíèÿ ðàññìàòðèâàå- ìîãî òèïà âîëíû â âîëíîâîäå, îáðàçóþùåì ÐÈÏ; 0 0 ,   – äèýëåêòðè÷åñêàÿ è ìàãíèòíàÿ ïðîíèöàå- ìîñòè âàêóóìà; i i ,      – îòíîñèòåëüíûå çíà÷åíèÿ äåéñòâèòåëüíîé è ìíèìîé ÷àñòè äèýëåêòðè÷åñêîé ïðîíèöàåìîñòè êîìïîíåíò. Ôóíäàìåíòàëüíûå âûõîäíûå ñèãíàëû èçìåðèòåëü- íîé èíôîðìàöèè ÐÈÏ 0 /    è 1 Q / Q  íàõîäÿòñÿ ÷åðåç   è    , êîòîðûå ÿâëÿþòñÿ êîðíÿìè ýòîé ñèñòåìû ðåêóððåíòíûõ ñîîòíîøåíèé. Ýôôåêòèâ- íûå ïàðàìåòðû эфф  è эфф tg  ìîäåëè â ñâîþ î÷åðåäü íàõîäÿòñÿ â âèäå êîðíåé эфф   è эфф    ñëåäóþùåãî õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ: N N N , 1 1 1 h th Y z th     , ãäå 1 эфф эфф N 1 N , 1 ) j ( Y           äëÿ ÐÈÏ Å-òèïà; N 1 N , 1 Y    äëÿ ÐÈÏ Í-òèïà;    N 1 i i N h h ; i h - òîëùèíà i-ãî ñëîÿ ìíîãîñëîéíîé ñòðóêòóðû; ; ) j ( j ) j ( 2 N N 2 кр эфф эфф 0 0 N N n                                 N N ,      – êîðíè óðàâíåíèÿ (1). Íàì ïðåäñòàâëÿåòñÿ, ÷òî òàêîé àëãîðèòì ïîçâîëÿåò îðãàíèçîâàòü êîìïàêòíûå è îäíîâðåìåííî ìíîãî- ñòîðîííèå èññëåäîâàíèÿ ïðåäñòàâëåííîé ìîäåëè. Èñïîëüçóåìîå èì è ñàìîé ìîäåëüþ îäíîìîäîâîå ïðèáëèæåíèå íå äîëæíî ñèëüíî ïîâëèÿòü íà îáù- íîñòü ôèçè÷åñêèõ âûâîäîâ. Èçíà÷àëüíî âàæíûì â ìîäåëè ÿâëÿåòñÿ óñòàíîâëå- íèå çàâèñèìîñòè эфф  îò êîëè÷åñòâà ÿ÷ååê яч 1 0 h ) z z ( N   (ñì. ðèñ. 1). Ïåðâûå ïîïûòêè óñòàíîâëåíèÿ ýòîé çàâèñèìîñòè ñ èñïîëüçîâàíèåì âîçìîæíîñòè MatLab äëÿ ðåàëèçà- öèè ÷èñëåííûõ èññëåäîâàíèé îáíàðóæèëè ïîÿâëå- íèå íåóñòîé÷èâîñòåé ïðè N>10 (îïðåäåëÿåòñÿ ñîîò- íîøåíèåì îáùåãî ðàçìåðà îáëàñòè II ñ äëèíîé âîëíû 0  ). Ìîæíî ïðåäïîëîæèòü, ÷òî âîçíèêíî- âåíèå íåóñòîé÷èâîñòåé âûçâàíî ïðîÿâëåíèåì îñî- áûõ òî÷åê â ôóíêöèÿõ ) z z ( th i 0 i   . Èçâåñòíî [11], ÷òî ïîëüçóÿñü ñîîòíîøåíèÿìè (1), ìîæíî ïîëó÷èòü áîëåå êîìïàêòíîå â çàïèñè õàðàê- òåðèñòè÷åñêîå óðàâíåíèå âèäà ... 1 ... h th h th Y 1 ... 1 ... h th Y h th h th Y 1 ... 1 ... h th h th Y 1 ... 1 ... h th Y h th Y h th Y z th 4 4 3 3 34 4 4 34 3 3 2 2 23 4 4 3 3 34 4 4 34 3 3 23 2 2 12 1 1                            . (2) Ïîÿâëåíèå íåóñòîé÷èâîñòåé ïðè ÷èñëåííîì íàõîæ- äåíèè çíà÷åíèÿ êîðíåé      , â ýòîì ñëó÷àå ñóùåñòâåííî ñîêðàòèëîñü. Îäíàêî (2) îêàçàëîñü âñå æå íåóäîáíûì â ïðîãðàììèðîâàíèè ìíîãîñòîðîí- íèõ ÷èñëåííûõ èññëåäîâàíèé. Ïîýòîìó áûë ðàññìîòðåí äðóãîé âàðèàíò ìîäèôè- êàöèè (1), èñêëþ÷àþùèé âû÷èñëåíèå ñëîæíûõ êîìáèíàöèé ôóíêöèé i i h th  , ïðèâîäÿùèõ ê âîç- íèêíîâåíèþ íóëåé â çíàìåíàòåëÿõ.  ñîîòâåòñòâèè ñ ýòèì ñîîòíîøåíèÿ (1) ïåðåïèñûâàþòñÿ â âèäå 2 2 2 2 2 2 2 , 1 1 1 h th A 1 h th A Y z th        ; 3 3 3 3 3 3 3 , 2 2 h th A 1 h th A Y A      ; (3) i i i i i i i , 1 i 1 i h th A 1 h th A Y A        ; H th Y A III III , N N   . 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Ðåçóëüòàòû ìîäåëèðîâàíèÿ  ðàáîòå ïðîâåäåíî ÷èñëåííîå èññëåäîâàíèå ìîäå- ëè, ïðåäñòàâëåííîé ðàññìîòðåííûì âûøå àíàëèòè- ÷åñêèì àëãîðèòìîì, ïðè ñëåäóþùèõ êîëè÷åñòâåí- íûõ åå ïàðàìåòðàõ. Ãåîìåòðèÿ ÐÈÏ ïîäîáðàíà òàêèì îáðàçîì, ÷òîáû åãî ðåçîíàíñíàÿ ÷àñòîòà áåç îáúåêòà ñîñòàâëÿëà 10 0 10 f  Ãö. Ïàðàìåòð 2 0 кр ) / (     ìîæåò èçìå- íÿòüñÿ îò 0,05 äî 0,95 ïóòåì èçìåíåíèÿ ðàäèóñà ðåçîíàòîðà è ðàçëè÷åí äëÿ ðàçíûõ ðàáî÷èõ âèäîâ êîëåáàíèé. Êîîðäèíàòà 1 z èçìåíÿëàñü ïðè âàðèà- öèè êîëè÷åñòâà è òîëùèíû ñëîåâ â ñîîòâåòñòâèè ñ òðåáîâàíèåì ïîñòîÿíñòâà èñõîäíîé äëèíû ðåçîíà- òîðà. Èñõîäíàÿ äîáðîòíîñòü ÐÈÏ áûëà âûáðàíà ðàâíîé 3 10 5  . Òîëùèíà ñëîåâ â ÿ÷åéêå âûáèðàëàñü èç óñëîâèÿ 1 h 0 i    , â òî âðåìÿ êàê îáùàÿ ýëåêòðè÷åñêàÿ òîëùèíà ìíîãîñëîéíîé ñòðóêòóðû (îáúåêòà) ìîæåò áûòü ñðàâíèìà ñ äëèíîé âîëíû 0  . Íà ðèñ. 2, à,á è ðèñ. 3, à,á ïðåäñòàâëåíû çàâèñèìîñòè âûõîäíûõ ñèãíàëîâ èçìåðèòåëüíîé èíôîðìàöèè 0 /    è 1 Q / Q  îò êîëè÷åñòâà ÿ÷ååê N â îáúåêòå äëÿ ÐÈÏ ñ 011 E è 011 H âèäàìè êîëåáàíèé, ñîîòâåò- ñòâåííî. Ïàðàìåòðû ÿ÷åéêè âûáðàíû ñëåäóþùèìè. 0 50 100 150 200 250 300 0 0.05 0.1 0.15 0.2 0.25  o   N 1 3 4 5 6 7 2 à N  Q Q 1 0 50 100 150 200 250 300 0 2 4 6 8 10 12 1 2 3 4 5 6 7 á Ðèñ. 2. Çàâèñèìîñòü èçìåíåíèÿ 0 /    (à) è 1 Q / Q  (á) îò êîëè÷åñòâà ÿ÷ååê N äëÿ ÐÈÏ 011 E - òèïà N  o   0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 5 4 3 2 6 7 1 à N  Q Q 1 0 50 100 150 200 250 300 0 20 40 60 80 100 1 2 3 4 5 6 7 á Ðèñ. 3. Çàâèñèìîñòü èçìåíåíèÿ 0 /    (à) è 1 Q / Q  (á) îò êîëè÷åñòâà ÿ÷ååê N äëÿ ÐÈÏ 011 H - òèïà ß÷åéêà äâóõñëîéíàÿ: 1 1   ; 0 tg 1   ; 15 2   ; 02 . 0 tg 2   . Òîëùèíà ñëîåâ 1 h è 2 h âàðüèðóåòñÿ ïðè óñëîâèè ïîñòîÿíñòâà îáùåé òîëùèíû ÿ÷åéêè 0 2 1 / ) h h (   = 0,02/30. Ïðè ýòîì ôàêòè÷åñêè ìîäåëèðóåòñÿ èçìåíåíèå ïëîòíîñòè äèñïåðñíîãî îáúåêòà ïðè îäíîâðåìåííîì èçìåíåíèè ðàçìåðà ÷àñòèö îäíîãî ñîðòà è âîçäóøíûõ ïðîìåæóòêîâ ìåæäó íèìè. Ïàðàìåòðîì â ïðåäñòàâëåííûõ íà ðèñ. 2-5 çàâèñèìîñòÿõ ÿâëÿåòñÿ 2 1 h / h (1 - 2 1 h / h = 99; 2 - 2 1 h / h = 9; 3 - 2 1 h / h = 4; 4 - 2 1 h / h = 1; 5 - 2 1 h / h = 0,25; 6 - 2 1 h / h = 1/9; 7 - 2 1 h / h = 1/99). Êà÷åñòâåííî õàðàêòåð çàâèñèìîñòåé äëÿ ÐÈÏ Í- è Å-òèïà ïðàêòè÷åñêè íå îòëè÷àåòñÿ è êîëè÷åñòâåí- íî, â ïåðâóþ î÷åðåäü, íåñåò îòïå÷àòîê ñïåöèôèêè âçàèìîäåéñòâèÿ ïîëÿ ýòèõ âèäîâ êîëåáàíèé ñ äèýëåêòðèêîì, èìåþùèì ñóùåñòâåííûå ÑÂ× ïîòå- ðè [12,13]. Íà ðèñ. 4, à,á è ðèñ. 5, à,á ïðåäñòàâëåíû âûòåêàþ- ùèå èç îöåíîê ïî ðåçóëüòàòàì ðèñ. 2 è 3 çàâèñèìî- ñòè îò N ýôôåêòèâíûõ çíà÷åíèé äèýëåêòðè÷åñêîé ïðîíèöàåìîñòè è òàíãåíñà óãëà ïîòåðü, óñòàíàâëè- âàåìûõ äëÿ ýòèõ æå ÐÈÏ ñ ïðåäñòàâëåíèåì îáúåêòà îäíîðîäíûì, ñ òîëùèíîé 0 2 1 / ) h h ( N   . 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Çàâèñèìîñòü èçìåíåíèÿ эфф  (à) è эфф tg  (á) îò êîëè÷åñòâà ÿ÷ååê N äëÿ ÐÈÏ 011 E -òèïà ïðè: 1 -  = 0,05; 2 -  = 0,1; 3 -  = 0,2; 4 -  = 0,6; 5 -  = 0,95  ðàçâèòèè ïðåäñòàâëåíèé î âîçìîæíîñòÿõ ðàññìàò- ðèâàåìîé ìîäåëè íà ðèñ. 7 à,á èçîáðàæåíû çàâèñè- ìîñòè эфф  è эфф tg  äèñïåðñíîé îäíîêîìïîíåíò- íîé ñðåäû äëÿ ðàçëè÷íûõ çíà÷åíèé 2  è 2 tg  ÷àñòèö ïðè  = 0,1 è 1 2 h / h = 4. Îáðàùàåò íà ñåáÿ âíèìàíèå îòñóòñòâèå çàâèñèìîñòè эфф  îò 2 tg  è èíâàðèàíòíîñòü эфф tg  ê 2  ïðè N > 2 10 . Åå ñóùå- ñòâîâàíèå ìîæåò îêàçàòüñÿ âåñüìà ïîëåçíûì ïðè ïîïûòêå îñóùåñòâëåíèÿ àáñîëþòíîé (áåçýòàëîí- íîé) ãðàäóèðîâêè äèýëüêîìåòðè÷åñêèõ ñðåäñòâ äèàãíîñòèêè îáúåêòîâ. Íà ðèñ. 8 ïðåäñòàâëåíû ïîëåçíûå, íà íàø âçãëÿä, çàâèñèìîñòè эфф  è эфф tg  îò ïàðàìåòðà äèñïåðñ- íîñòè 0 2 / h  ïðè ïîñòîÿííîé ïëîòíîñòè 2 1 h / h = 0.25. Èç ýòèõ îöåíîê ñëåäóåò èíâàðèàíòíîñòü эфф tg  â ðàññìîòðåííîì äèàïàçîíå 0 2 / h  ê ñòåïå- íè äèñïåðñíîñòè. Ýòîò âûâîä èìååò òàêæå âàæíîå ïðàêòè÷åñêîå çíà÷åíèå. 4. Çàêëþ÷åíèå Òàêèì îáðàçîì, àðãóìåíòèðîâàíû äîñòàòî÷íàÿ àäåê- âàòíîñòü è ýôôåêòèâíîñòü ÷èñëåííî-àíàëèòè÷åñ- êîé ìîäåëè êîìïëåêñíîé äèýëåêòðè÷åñêîé ïðîíè- öàåìîñòè ìåëêîäèñïåðñíûõ ñðåä íà ÑÂ×, áàçèðóþ- ùåéñÿ íà ïðèíöèïå ôèçè÷åñêîé ýêâèâàëåíòíîñòè âçàèìîäåéñòâèÿ äèñïåðñíûõ ñðåä ñ êâàçèîäíîðîä- íûì ïîëåì è ìíîãîñëîéíûõ ñðåä ñ ñóùåñòâåííî íåîäíîðîäíûì ïîëåì.  ýòîì ïðèíöèïå îñíîâíîå çíà÷åíèå èìååò îòðàæåíèå ðîëè íîðìàëüíîé ê ïîâåðõíîñòè ÷àñòèöû ýëåêòðè÷åñêîé ñîñòàâëÿþ- ùåé ÑÂ× ïîëÿ. N 0 50 100 150 200 250 300 2 4 6 8 10 12 14  2 =17 15 10 5  эфф à N 0 50 100 150 200 250 300 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 tg  2 =0.1 0.08 0.06 0.04 0.02  2 =5  2 =10  2  2 =15 =17 tg  эфф á Ðèñ. 7. Çàâèñèìîñòü èçìåíåíèÿ эфф  (à) è эфф tg  (á) îò êîëè÷åñòâà ÿ÷ååê N äëÿ ÐÈÏ 011 E -òèïà h 2 /  0 0 1 2 3 4 5 11.6 11.7 11.8 11.9 12.1  эфф 12.0 -3 x10 7 2.024 2.026 2.028 tg  эфф 2.02 3 -2 x10 1 2 Ðèñ. 8. Çàâèñèìîñòü эфф  (1) è эфф tg  (2) îò ïàðà- ìåòðà äèñïåðñíîñòè 0 2 / h  Äîêàçàíà íåëîêàëüíîñòü âçàèìîäåéñòâèÿ, ïðèñó- ùàÿ â îáùåì ñëó÷àå âñåì ïåðâè÷íûì èçìåðèòåëü- íûì ïðåîáðàçîâàòåëÿì, ïðèìåíÿåìûõ â ÑÂ× äè- ýëüêîìåòðèè. Íàéäåíû óñëîâèÿ ìîäåëèðîâàíèÿ è ïîñòàíîâêè ïðàêòè÷åñêèõ èçìåðåíèé, ïðè êîòîðûõ îáå ñîñòàâ- ëÿþùèå êîìïëåêñíîé äèýëåêòðè÷åñêîé ïðîíèöàå- ìîñòè ñòðóêòóðíî-íåîäíîðîäíûõ îáúåêòîâ èìåþò ñëàáûé âêëàä íåëîêàëüíîé ÷àñòè âçàèìîäåéñòâèÿ. Ïîêàçàíî, ÷òî эфф tg  ìîæåò áûòü èíâàðèàíòíûì â îòëè÷èå îò эфф  ê âàðèàöèè ïëîòíîñòè ñðåäû, i  êîìïîíåíò è èõ äèñïåðñíîñòè â íåêîòîðîé îáëàñòè. Ëèòåðàòóðà: 1. Ïðèáîðû äëÿ íåðàçðóøàþùåãî êîíòðîëÿ ìàòåðèàëîâ è èçäåëèÿ. Ñïðàâî÷íèê ïîä ðåä. ïðîô. Â.Â. Êëþåâà. Ì.:Ìàøèíîñòðîåíèå, 1986. Ò. 1, 2 . 486 ñ., 351ñ. 2. Industrial Microwave sensors. E.G. Nyfors, P.Vainikainen Artech House, Inc. 1989. 350 p. 3. Ïîòàïîâ À.À. Èçìåðå- íèå âëàæíîñòè ãàçîâ íà îñíîâå äèýëåêòðè÷åñêèõ ìåòî- äîâ // Èçìåðèòåëüíàÿ òåõíèêà, 1985. ¹6. Ñ. 48-50. 4. Áåíçàðü Â.Ê. Òåõíèêà ÑÂ× âëàãîìåòðèè. Ìèíñê:Âû- øýéøàÿ øêîëà, 1974. 349 ñ. 5. Èñòàòóëëàåâ Ï.Ð. 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