Z peretvorennyam peretvorennyam Lorana nazivayut zgortannya vihidnogo signalu zadanogo poslidovnistyu dijsnih chisel u chasovij oblasti v analitichnu funkciyu kompleksnoyi chastoti Yaksho signal yavlyaye impulsnu harakteristiku linijnoyi sistemi to koeficiyenti Z peretvorennya pokazuyut vidguk sistemi na kompleksni eksponenti E n z n r n e i w n displaystyle E n z n r n e i omega n tobto na garmonijni oscilyaciyi z riznimi chastotami i shvidkostyami narostannya zagasannya Zmist 1 Viznachennya 1 1 Vlastivosti 1 2 Dvostoronnye Z peretvorennya 1 3 Odnostoronnye Z peretvorennya 2 Zvorotne Z peretvorennya 3 Tablicya deyakih Z peretvoren 4 Div takozhViznachennyared Diskretna funkciya x t displaystyle x t nbsp ye funkciyeyu yaka viznachena u diskretni momenti chasu t l T l 0 1 2 displaystyle t lT l 0 1 2 nbsp Taku funkciyu mozhna zapisati u viglyadi x l T displaystyle x lT nbsp de t displaystyle t nbsp neperervna zminna Cya funkciya x l T displaystyle x lT nbsp harakterizuyetsya tim sho vona viznachayetsya neperervnoyu funkciyeyu neperervnogo argumentu x t displaystyle x t nbsp j primaye yiyi znachennya u momenti t l T l 0 1 2 displaystyle t lT l 0 1 2 nbsp Taka funkciya nazivayetsya gratchastoyu funkciyeyu Krim togo vikoristovuyets zmishena gratchasta funkciya x l e T 0 lt e lt 1 displaystyle x l varepsilon T 0 lt varepsilon lt 1 nbsp yaka prijmaye znachennya neperervnoyi funkciyi u momenti t l e T l 0 1 2 displaystyle t l varepsilon T l 0 1 2 nbsp z displaystyle z nbsp peretvorennya ce spivvidnoshennya 1 X z l 0 x l T z l displaystyle X z sum l 0 infty x lT z l nbsp yake stavit u vidpovidnist diskretnij funkciyi x l T displaystyle x lT nbsp funkciyu kompleksnoyi zminnoyi X z displaystyle X z nbsp Pri comu x l T displaystyle x lT nbsp nazivayetsya originalom a X z displaystyle X z nbsp zobrazhennyam abo z displaystyle z nbsp zobrazhennyam z displaystyle z nbsp peretvorennya takozh umovno zapisuyetsya u viglyadi X z Z x l T displaystyle X z Z x lT nbsp a zvorotne z displaystyle z nbsp peretvorennya u viglyadi x l T Z 1 X z displaystyle x lT Z 1 X z nbsp z displaystyle z nbsp peretvorennya iz zmishenoyu gratchastoyu funkciyeyu x l e T z l displaystyle x l varepsilon T z l nbsp tobto spivvidnoshennya X z e l 0 x l e T z 1 displaystyle X z varepsilon sum l 0 infty x l varepsilon T z 1 nbsp nazivayut modifikovanim z displaystyle z nbsp peretvorennyam Ce peretvorennya takozh zapisuyetsya u viglyadi X z e Z x l e T Z e x l T displaystyle X z varepsilon Z x l varepsilon T Z varepsilon x lT nbsp Napriklad nehaj potribno viznachiti z displaystyle z nbsp zobrazhennyam zmishenoyi gratchastoyi funkciyi x l T 1 l T displaystyle x lT 1 lT nbsp ta zmishenoyi gratchastoyi funkciyi x l e T 1 l e T displaystyle x l varepsilon T 1 l varepsilon T nbsp Oskilki za usih l 0 1 l T 1 l e T 1 displaystyle l geq 0 quad 1 lT 1 l varepsilon T 1 nbsp to X z X z e l 0 z l displaystyle X z X z varepsilon sum l 0 infty z l nbsp Po formuli neskinchenno spadayuchoyi geometrichnoyi progresiyi Z 1 l T Z 1 l e T 1 1 z 1 z z 1 z gt 1 displaystyle Z 1 lT Z 1 l varepsilon T frac 1 1 z 1 frac z z 1 quad z gt 1 nbsp Vlastivostired isnuyut dodatni chisla M displaystyle M nbsp ta q displaystyle q nbsp taki sho x l T lt M q t l 0 displaystyle x lT lt Mq t quad forall l geq 0 nbsp x l T 0 l lt 0 displaystyle x lT 0 quad forall l lt 0 nbsp Persha vlastivist ye neobhidnoyu dlya isnuvannya oblasti zbizhnosti ryadu u pravij chastini a druga vlastivist vikoristovuyetsya dlya vivodu deyakih vlastivostej z displaystyle z nbsp peretvorennya Funkciyi yaki zadovilnyayut vkazanim dvom vlastivostyam nazivayut fukciyami originalami Linijnist Modifikovane z displaystyle z nbsp peretvorennya vid linijnoyi kombinaciyi diskretnih funkcij dorivnyuye linijnij kombinaciyi yih modifikovanih z displaystyle z nbsp peretvoren Z i 1 n a i x i l e T i 1 n a i Z x i l e T displaystyle Z begin Bmatrix sum i 1 n a i x i l varepsilon T end Bmatrix sum i 1 n a i Z x i l varepsilon T nbsp Tut a i i 1 n displaystyle a i quad i overline 1 n nbsp konstanti Zapiznyuvannya Modifikovane z displaystyle z nbsp peretvorennya vid funkciyi iz zapiznyuvanim argumentom x l m T displaystyle x l m T nbsp viznachayetsya yak Z e x l m T z m Z e x l T z m X z e displaystyle Z varepsilon x l m T z m Z varepsilon x lT z m X z varepsilon nbsp Viperedzhennya Modifikovane z displaystyle z nbsp peretvorennya vid funkciyi iz viperedzhuyuchim argumentom x l m T displaystyle x l m T nbsp viznachayetsya yak Z e x l m T z m X z e k 0 m 1 x x e T z k displaystyle Z varepsilon x l m T z m begin bmatrix X z varepsilon sum k 0 m 1 x x varepsilon T z k end bmatrix nbsp Yaksho x e T x 1 e T x m 1 e T 0 displaystyle x varepsilon T x 1 varepsilon T x m 1 varepsilon T 0 nbsp pochatkovi umovi nulovi to Z e x l m T z m X z e displaystyle Z varepsilon x l m T z m X z varepsilon nbsp Zgortannya Dobutok zobrazhen X 1 z e displaystyle X 1 z varepsilon nbsp ta X 2 z e displaystyle X 2 z varepsilon nbsp dorivnyuye z displaystyle z nbsp peretvorennyu vid zgortannya yih originaliv x 1 l e T displaystyle x 1 l varepsilon T nbsp ta x 2 l e T displaystyle x 2 l varepsilon T nbsp X 1 z e X 2 z e Z k 0 l x 1 k e T x 2 l k e T Z k 0 l x 2 k e T x 1 l k e T displaystyle X 1 z varepsilon X 2 z varepsilon Z sum k 0 l x 1 k varepsilon T x 2 l k varepsilon T Z sum k 0 l x 2 k varepsilon T x 1 l k varepsilon T nbsp Mezhevi znachennya Pochatkove znachennya gratchastoyi funkciyi x l T displaystyle x lT nbsp po yiyi zvichajnomu ta modifikovanomu z displaystyle z nbsp zobrazhennyu viznachayetsya yak x e T lim z X z e x 0 lim z X z displaystyle x varepsilon T lim z rightarrow infty X z varepsilon quad x 0 lim z rightarrow infty X z nbsp Granicya z lim l x l T displaystyle z infty lim l rightarrow infty x lT nbsp za umovi sho vona isnuye viznachayetsya yak x lim z 1 z 1 X z e lim z 1 z 1 X z displaystyle x infty lim z rightarrow 1 z 1 X z varepsilon lim z rightarrow 1 z 1 X z nbsp Z peretvorennya yak i bagato integralnih peretvoren mozhe buti yak odnostoronnye tak i dvostoronnye Dvostoronnye Z peretvorennyared Dvostoronnye Z peretvorennya X z diskretnogo chasovogo signalu x n zadayetsya yak X z Z x n n x n z n displaystyle X z Z x n sum n infty infty x n z n nbsp de n cile z kompleksne chislo z A e j f displaystyle z Ae j varphi nbsp de A amplituda a f displaystyle varphi nbsp kutova chastota u radianah na vidlik Odnostoronnye Z peretvorennyared U vipadkah koli x n viznachena tilki dlya n 0 displaystyle n geqslant 0 nbsp odnostoronnye Z peretvorennya zadayetsya yak X z Z x n n 0 x n z n displaystyle X z Z x n sum n 0 infty x n z n nbsp Zvorotne Z peretvorennyared Zvorotne Z peretvorennya viznachayetsya napriklad tak x n Z 1 X z 1 2 p j C X z z n 1 d z displaystyle x n Z 1 X z frac 1 2 pi j oint limits C X z z n 1 dz nbsp de C kontur sho ohoplyuye oblast zbizhnosti X z Kontur povinen mistiti vsi vidrahuvannya X z Poklavshi v poperednij formuli z r e j f displaystyle z re j varphi nbsp otrimayemo ekvivalentne viznachennya x n r n 2 p p p X r e j f e j n f d f displaystyle x n frac r n 2 pi int limits pi pi X re j varphi e jn varphi d varphi nbsp Tablicya deyakih Z peretvorenred Poznachennya 8 n displaystyle theta n nbsp funkciya Gevisajda d n displaystyle delta n nbsp delta funkciya Diraka Signal x n displaystyle x n nbsp Z peretvorennya X z displaystyle X z nbsp Oblast zbizhnosti 1 d n displaystyle delta n nbsp 1 displaystyle 1 nbsp z displaystyle forall z nbsp 2 d n n 0 displaystyle delta n n 0 nbsp 1 z n 0 displaystyle frac 1 z n 0 nbsp z 0 displaystyle z neq 0 nbsp 3 8 n displaystyle theta n nbsp z z 1 displaystyle frac z z 1 nbsp z gt 1 displaystyle z gt 1 nbsp 4 a n 8 n displaystyle a n theta n nbsp 1 1 a z 1 displaystyle frac 1 1 az 1 nbsp z gt a displaystyle z gt a nbsp 5 n a n 8 n displaystyle na n theta n nbsp a z 1 1 a z 1 2 displaystyle frac az 1 1 az 1 2 nbsp z gt a displaystyle z gt a nbsp 6 a n 8 n 1 displaystyle a n theta n 1 nbsp 1 1 a z 1 displaystyle frac 1 1 az 1 nbsp z lt a displaystyle z lt a nbsp 7 n a n 8 n 1 displaystyle na n theta n 1 nbsp a z 1 1 a z 1 2 displaystyle frac az 1 1 az 1 2 nbsp z lt a displaystyle z lt a nbsp 8 cos w 0 n 8 n displaystyle cos omega 0 n theta n nbsp 1 z 1 cos w 0 1 2 z 1 cos w 0 z 2 displaystyle frac 1 z 1 cos omega 0 1 2z 1 cos omega 0 z 2 nbsp z gt 1 displaystyle z gt 1 nbsp 9 sin w 0 n 8 n displaystyle sin omega 0 n theta n nbsp z 1 sin w 0 1 2 z 1 cos w 0 z 2 displaystyle frac z 1 sin omega 0 1 2z 1 cos omega 0 z 2 nbsp z gt 1 displaystyle z gt 1 nbsp 10 a n cos w 0 n 8 n displaystyle a n cos omega 0 n theta n nbsp 1 a z 1 cos w 0 1 2 a z 1 cos w 0 a 2 z 2 displaystyle frac 1 az 1 cos omega 0 1 2az 1 cos omega 0 a 2 z 2 nbsp z gt a displaystyle z gt a nbsp 11 a n sin w 0 n 8 n displaystyle a n sin omega 0 n theta n nbsp a z 1 sin w 0 1 2 a z 1 cos w 0 a 2 z 2 displaystyle frac az 1 sin omega 0 1 2az 1 cos omega 0 a 2 z 2 nbsp z gt a displaystyle z gt a nbsp Div takozhred Peretvorennya Laplasa Gratchasta funkciya nbsp Cya stattya ye zagotovkoyu Vi mozhete dopomogti proyektu dorobivshi yiyi Ce povidomlennya varto zaminiti tochnishim Kim D P Teoriya avtomaticheskogo upravleniya tom 1 Otrimano z https uk wikipedia org w index php title Z peretvorennya amp oldid 34421371