'$7$352&(66,1* 352&(66,1* /HWXVVWDUWZLWKDQXPHULFDOH[DPSOH 7\SLFDOOLQHRI∆υ=+]= (π7± Æ 7 π±≈V TXHVWLRQ LGHDODFTXLVLWLRQWLPHWD" $W\SH),'LVREWDLQHGWUXQFDWHG PHPR LIWDWRRVKRUW Æ %W\SH),'LVREWDLQHGWRRPXFKQRLVH LIWDWRRORQJ Æ DQVZHUDFTXLVLWLRQXQWLOWKHLQWHQVLW\RIWKH),'LVRIWKHLQLWLDOYDOXHRI6 6W 6W H[SW7DQG7≈V PHPR W\SLFDOWPD[!7 Æ Æ H[SW Æ W V W LQWKLVFDVHWKHVXJJHVWHG PD[!V TXHVWLRQ VL]HRIWKHGDWDYHFWRUUW" PHPR GZ 6Z± HJ 6Z N+] Æ GZ µV $FTXLULQJGDWDIRUW V $QHZSRLQWDWHYHU\µVÆ≈SRLQWVWREHVWRUHG DQVZHU UWVKRXOGEHNSRLQWV FRQFOXVLRQ GLJLWDOUHVROXWLRQ+]SRLQW TXHVWLRQ ZKDWLIWPD[V FRQVHTXHQFHWUXQFDWHGGDWD TXHVWLRQ ZKDWLIWPD[!V FRQVHTXHQFHQRLVHLVFROOHFWHG DQVZHUWRLPSURYHGLJLWDOUHVROXWLRQ+]SRLQW ]HURILOO HJ≈+]SRLQW ]HURILOOXSWRN 352&(66,1* 7\SLFDOSURFHVVLQJVWHSVRQHGLPHQVLRQDORSHUDWLRQVDORQJWKHUHFRUGHG'GDWD $WLPHWDFTGRPDLQFRPSOH[YHFWRUUW ↓ VXSSUHVVLRQRIVROYHQWDQGUHPRYH'&RIIVHW ↓ DSRGL]H ↓ ]HURILOO ↓ VFDOHILUVWSRLQW ↓ GLVFUHWH)RXULHUWUDQVIRUP ↓ SKDVHFRUUHFWLRQ ↓ EDVHOLQHFRUUHFWLRQ ↓ $IUHTXHQF\GRPDLQIXQFWLRQIν 7\SLFDOSURFHVVLQJVWHSVDORQJD'GDWDVHW $WLPHWDFTGRPDLQFRPSOH[YHFWRUUWW SURFHVVLQJDORQJW0FRPSOH[YHFWRUV VXSSUHVVLRQRIVROYHQWDQGUHPRYH'&RIIVHW DSRGL]H ]HURILOO VFDOHILUVWSRLQW GLVFUHWH)RXULHUWUDQVIRUP SKDVHFRUUHFWLRQ EDVHOLQHFRUUHFWLRQ UWν SURFHVVLQJDORQJW1FRPSOH[YHFWRUV DSRGL]H ]HURILOO VFDOHILUVWSRLQW GLVFUHWH)RXULHUWUDQVIRUP SKDVHFRUUHFWLRQ Iνν 352&(66,1* WLPHGRPDLQVLJQDO6 6W PHPR)7^6W` !6υ ∫6WH[SLπυWGWπυ ω PHPR,)7^6υ` !6W ∫6υH[SLπυWGυ PHPR6υDQG6WDUH)RXULHUSDLUV PHPR)7DQG,)7DUHOLQHDURSHUDWLRQVWKHUHIRUH )7^F6W` F)7^6W`DQG)7^6W4W` )7^6W`)7^4W` WKHRUHPVRI)7UHOHYDQWIRU105GDWDSURFHVVLQJ $ VLPLODULW\WKHRUHP )7^6FW` |F|6ωF |F|6πυF GRPDLQ % %URDGHQLQJWKHIXQFWLRQLQWKHWLPHGRPDLQLVQDUURZLQJLWLQWKHIUHTXHQF\ WLPHVKLIWLQJWKHRUHP )7^6Wτ` H[SLπυτ6υ 'HOD\LQJLQWKHWLPHGRPDLQZLOOLQGXFHDIUHTXHQF\GHSHQGHQWSKDVHVKLIW & IUHTXHQF\VKLIWLQJWKHRUHP )7^6WH[SLπυRτ` 6υυR IUHTXHQFLHVFDQEHVKLIWHGDIWHUDFTXLVLWLRQ ' FRQYROXWLRQWKHRUHP 4W6W ∫4τ6WτGτ LW V)7 )7^4W6W` 4πυ6πυ FRQYROXWLRQRIWZRIXQFWLRQVHJUHPRYDORIUHVLGXDOVROYHQWIUHTXHQFLHV ( 3DUVHYDO VWKHRUHP ∫|6W|GW ∫|6πυ|Gυ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υ` 6’υ +υ6’υ DQVZHULQSUDFWLFH PXOWLSOLFDWLRQRIWKH),'ZLWKRQHRUPRUHZLQGRZIXQFWLRQVLQWKH WLPHGRPDLQ 352&(66,1* $1RWVKLIWHG+W functions $ DQH[SRQHQWLDO+W=H[S±π&W ([DPSOH7KHRULJLQDO6W H[S±W7H[SLπυDW DIWHU)7KDVD/RUHQW]LDQOLQHVKDSHZLWK∆υ=BZKHUH% π7± 0HPR &RQYROXWLRQLQυLVPXOWLSOLFDWLRQLQW LI+W=H[S±π&WLVWREHXVHGDQH[SRQHQWLDO>)LJXUH$D@WKHQWKHQHZIXQFWLRQLV 6’W H[S±W7H[SLπυDWH[S±π&W $IWHUPXOWLSOLFDWLRQ 6’W H[S±^π&+7W`H[SLπυDW $IWHU)76’WKDVD/RUHQW]LDQOLQHVKDSHZLWK∆υ=B+CZKHUH&π7± FRQFOXVLRQ WKHOLQHKDVEURDGHQHGE\&EXW61KDVLPSURYHGQRLVHLVEURDGHQHGRXW ([DPSOH LI+W=H[Sπ&WLV WR EH XVHG DQ H[SRQHQWLDO IRU UHVROXWLRQ HQKDQFHPHQW WKHQ WKH QHZIXQFWLRQLV 6’W H[S±W7H[SLπυDWH[Sπ&W $IWHUPXOWLSOLFDWLRQ 6’W H[S±^±π&+7W`H[SLπυDW $IWHU)7WKH6’WKDVD/RUHQW]LDQOLQHVKDSHZLWK∆υ=B–CZKHUH&±π7± FRQFOXVLRQ 7KHOLQHLVQDUURZHUWRWKHH[SHQVHRI61 ,QWKHRU\WKHOLQHVKDSHFDQEHUHPRYHG $ *DXVVLDQ+W=H[S^±NW` %6KLIWHG+W functions % % % 6KLIWHG*DXVVLDQ /RUHQW]LDQWR*DXVVWUDQVIRUPDWLRQ 6LQEHOO +W=H[S^±&NWNW` +W=H[S^π&W`H[S^NW` +W=sin^π[t+t0]\[tmax+t0])} LGHDODSRGL]DWLRQDJDLQVWWUXQFDWLRQDUWLIDFW SUREOHP 352&(66,1* GDWDLVRIWHQWUXQFDWHG FRQVHTXHQFH WKHWLPHGRPDLQVLJQDOLVQRWUHGXFHGVPRRWKO\WR]HUR!WKHIUHTXHQF\ GRPDLQVLJQDOFRQWDLQVWUXQFDWLRQULSSOHV DQVZHU XVHDSRGL]DWLRQ WKHLGHDOILOWHULQJIXQFWLRQ 'ROSK&KHE\VKHYIXQFWLRQ ,)7^FRV>1FRV^]RFRVω∆W`@FRVK>1FRV^]R`@` ZKHUH 1 VDPSOHGSRLQWV ∆W VDPSOLQJSHULRG ]R >FRVδ∆W@ SUREOHP WRRFRPSOH[LQLWVQDWXUH DSSOLFDWLRQ XVHDVEHQFKPDUNIRUHYDOXDWLQJRWKHUILOWHULQJIXQFWLRQVHJ+DPPLQJIXQFW .DLVHU)XQFW PD[LPL]HVLJQDOWRQRLVH LIWPD[!7WKHQDQH[SRQHQWLDOOLQHEURDGHQLQJIXQFWLRQLVXVHGLQWKHWLPHGRPDLQ H[SλW SUREOHPV ZKHUHλ≈7λLVWKHIXOOZLWKDWKDOIKHLJKWRIWKH/RUHQW]LDQ H[SλWFDQRSWLPLVHRQO\WKHVLJQDODWλIUHTX )RUDOORWKHUIUHTXLWLVMXVWDQDSSURDFK PDNHVWKHLQWHJUDWLRQRIWKHDSSRGL]HGIXQFWLRQLQFRUUHFW RIWHQDSSOLHGIRUDFTXLVLWLRQGLPHQVLRQ 352&(66,1* OLQHDUSUHGLFWLRQ
© Copyright 2026 Paperzz