Ask seic me ton Metasqhmatismì Laplace

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1 Ask seic me ton Metasqhmatismì Laplace Epimèleia: Gi rgoc Kafentz c Upoy. Didˆktwr Tm m. H/U Panepist mio Kr thc 8 IounÐou 4. 'Estw to s ma { A, t T x(t), alloô () (aþ) Na upologðsete to metasq. Fourier tou. (bþ) Na upologðsete to metasq. Laplace tou. Epibebai ste to apotèlesma tou prohgoômenou erwt matoc, qrhsimopoi ntac ton metasq. Laplace. (aþ) (bþ) ( t T ) x(t) Arect X(f) AT sinc(ft )e jπt/f AT sinc(ft)e jπft () T T Ae st A s e st T A s (e st ) A s ( est ) (3) Epeid to s ma eðnai peperasmènhc diˆrkeiac, to pedðo sôgklis c tou eðnai ìlo to s-epðpedo. Epeid to pedðo sôgklishc perilambˆnei to fantastikì ˆxona, ac broôme to metasq. Fourier mèsw tou metasq. Laplace pou mìlic upologðsame. X(s) σ A jπf ( e jπft ) A jπf e jπft (e jπft e jπft ) A jπf e jπft j sin(πft ) A πf e jπft sin(πft ) AT πft e jπft sin(πft ) AT sinc(ft )e jπft (4) pou eðnai to Ðdio akrib c apotèlesma me autì pou upologðsame parapˆnw.

2 . UpologÐste to metasq. Laplace tou s matoc x(t) e αt u(t) + e αt u( t), α > (5) UpologÐzetai gia to s ma autì o metasq. Fourier mèsw tou metasq. Laplace? An nai, breðte ton. An ìqi, exhgeðste. x(t)e st e (a s)t + e (a s)t a s e(a s)t a s e(a s)t a s a s + a s a s a s + a s a s a + s (a s)(a s) a (s a)(s a) (6) an a R{s} < kai a R{s} > a < R{s} < a. Ta duo pedða sôgklishc proèkuyan ap' touc gnwstoôc periorismoôc sta oloklhr mata, ìtan t ±, ste autˆ na sugklðnoun. Epeid èqoume ˆjroisma shmˆtwn, to pedðo sôgklishc ja eðnai h tom twn epimèrouc pedðwn sôgklishc. O metasqhmatismìc Fourier den mporeð na upologisteð mèsw tou metasq. Laplace, giatð to pedðo sôgklishc den perilambˆnei to fantastikì ˆxona R{s} σ. 3. DeÐxte ìti o metasq. Laplace tou s matoc x(t) te αt u(t), α > eðnai o DÐnetai ìti, R{s} > α (s + α) te αt ) α e αt( αt

3 x(t)e st e (a+s)t ( ) + (a + s) (a + s)t ( te (a+s)t lim t + a + s te (a+s)t lim t + a + s lim t + a + s (a + s) lim t + + te at u(t)e st e (a+s)t ) (a + s) + e (a+s)t lim t + t e (a+s)t + (a + s) (a + s) + (a + s) (a + s) e (a+s)t + (a + s) te (a+s)t (a + s), an R{s} + a > R{s} > a. (7) (a + s) Ed, to pedðo sôgklishc proèkuye apì touc gnwstoôc periorismoôc, ste ta ìria na fjðnoun sto mhdèn. EpÐshc, sto teleutaðo ìrio, efarmìsame ton kanìna tou De L Hospital gia na to lôsoume. 4. DeÐxte ìti o metasq. Laplace tou s matoc x(t) tu(t) eðnai DÐnetai ìti, R{s} >. s te αt ) α e αt( αt te st s lim t + tu(t)e st te st e st s ( st ) ( te st s ) e st s e st s lim te st + t + s s lim t t + e st + s se st + s + s, R{s} >. (8) s Ed kai pˆli qrhsimopoi same ton kanìna tou De L Hospital, en to pedðo sôgklishc prokôptei katˆ ta gnwstˆ (plèon :-) ). 5. DeÐxte ìti o metasq. Laplace tou s matoc eðnai o DÐnetai ìti s (e s e s ) s e 4s te αt ) α e αt( αt 3

4 Sq ma : Sq ma 'Askhshc 5.5 oc trìpoc: (t )e st + e st s ( st ) e s s + e s s 4 + e st s e 4s s e st e st s 4 te st e st + 4 e st s (e s e s ) s e 4s (9) To pedðo sôgklishc eðnai ìlo to s-epðpedo, afoô to x(t) eðnai peperasmèno. oc trìpoc: Sq ma : Parˆgwgoc 'Askhshc 5.5 dx(t) ( t 3 ) rect δ(t 4) L{ dx(t) } s e 3 s (e s e s ) e 4s s e s s e s e 4s () 4

5 IsqÔei ìti L{ dx(t) } sx(s) x( ) sx(s) 'Ara s (e s e s ) e 4s sx(s) x( ) s (e s e s ) s e 4s () O deôteroc trìpoc lôshc eðnai pio eôkoloc, me thn proôpìjesh ìti ja paragwgisteð swstˆ to sq ma kai ja efarmìsete swstˆ thn idiìthta. 6. Na upologisteð o metasq. Laplace tou s matoc pou faðnetai sto sq ma 3. Sq ma 3: Sq ma 'Askhshc 5.6 ParagwgÐzontac, èqoume to s ma tou sq matoc 4. Sq ma 4: Parˆgwgoc sq matoc 'Askhshc 5.6 sx(s) x( ) L{ dx(t) } L{δ(t ) rect ( t 3 ) } + rect e s s ( e s ) + s ( e s )e s e 3s s e s s e s + s e s + s e s s e 3s e 3s ) + e s( s + ) e 3s( ) s s + e s( s e s (s ) s ( t 5 ) δ(t 3)} + e s s e 3s (s + ) s () 5

6 To s ma eðnai peperasmènhc diˆrkeiac, ˆra to pedðo sôgklishc eðnai ìlo to s-epðpedo. 7. 'Estw 3s + 5 s + 3s + me ROC : < R{s} <. BreÐte ton antðstrofo metasq. Laplace, x(t). Profan c de ja qrhsimopoi soume ton orismì gia na broôme ton antistr. metasq. Laplace, allˆ ja qrhsimopoi soume dh gnwstˆ mac zeôgh metasqhmatism n, spˆzontac to megˆlo klˆsma se mikrìtera. IsqÔei ìti h tˆxh tou poluwnômou tou arijmht eðnai mikrìterh ap' thn antðstoiqh tou paronomast, ˆra mporoôme na efarmìsoume anˆptugma se merikˆ klˆsmata. : 3s + 5 s + 3s + 3s + 5 (s + )(s + ) A s + + A s + A X(s)(s + ) 3s + 5 σ s + σ A X(s)(s + ) 3s + 5 σ s + σ 'Ara ja eðnai To pedðo sôgklishc dðdetai ìti eðnai s + + s + ROC : < R{s} < R{s} > R{s} < 'Ara, gnwrðzontac ìti to X(s) eðnai ˆjroisma duo shmˆtwn thc parapˆnw morf c, me pedða sôgklishc ta duo parapˆnw, jèloume na broôme ta duo s mata sto qrìno. Apì touc pðnakec twn zeug n metasqhmatism n, èqoume ìti: e t u(t), R{s} > s + e t u( t), R{s} < s + 'Ara telikˆ x(t) e t u(t) e t u( t) (3) eðnai to s ma pou yˆqnoume. 8. Na upologisteð o antðstrofoc metasq. Laplace tou s matoc ìtan: (aþ) ROC : < R{s} < (bþ) ROC : R{s} > 3 (s + )(s ) 6

7 (gþ) ROC : R{s} < Se kˆje perðptwsh, sqediˆste thn perioq sôgklishc. me 3 (s + )(s ) A s + + A s A X(s)(s + ) 3 s s s A X(s)(s ) 3 s s + s Anˆloga me ta pedða sôgklishc pou dðnontai, ja kajoristeð kai to s ma sto qrìno pou ja prokôyei. (aþ) 'Ara s + s <R{s}<{ <R{s}} {R{s}<} x(t) e t u(t) + e t u( t) (4) L To pedðo sôgklishc faðnetai sto sq ma 5. Sq ma 5: o PedÐo SÔgklishc 'Askhshc 5.8 (bþ) 'Ara s + s R{s}>{R{s}> } {R{s}>} x(t) e t u(t) e t tu(t) (5) L (gþ) 'Ara To pedðo sôgklishc faðnetai sto sq ma 6. s + s R{s}< {R{s}< } {R{s}<} x(t) e t u( t) + e t tu( t) (6) L To pedðo sôgklishc faðnetai sto sq ma 7. 7

8 Sq ma 6: o PedÐo SÔgklishc 'Askhshc 5.8 Sq ma 7: 3o PedÐo SÔgklishc 'Askhshc O metasq. Laplace dðnetai apì th sqèsh s + s s 3 (aþ) Gia ìla ta dunatˆ pedða sôgklishc, breðte ton antðstrofo metasq. Laplace, x(t). (bþ) Se poiˆ perðptwsh upologðzetai o metasq. Fourier? UpologÐste ton. (aþ) Oi pìloi tou paronomast eðnai oi s 3, s, ìpwc eôkola diapist noume. 'Ara s + (s 3)(s + ) A s 3 + A s + 8

9 Ta A, A dðnontai apì tic sqèseic A X(s)(s 3) s + 5 s3 s + s3 4 A X(s)(s + ) s + s s 3 s 4 'Ara èqoume me pijanˆ pedða sôgklishc ta 4 s s 3 R{s} <, ROC R{s} > 3, < R{s} < 3 Gia thn perðptwsh R{s} <, qreiazìmaste h tom twn pedðwn sôgklishc twn duo shmˆtwn na mac dðnei to hmiepðpedo σ <. Autì gðnetai mìnon an {R{s} < 3} {R{s} < }. Gi' autˆ ta duo pedða sôgklishc, ja èqoume 4 e t u( t) 5 4 e3t u( t) 'Ara to s ma sto qrìno ja eðnai to R{s}< L R{s}<3 L 5 4 s + 4 s 3 x(t) 4 e t u( t) 5 4 e3t u( t) (7) Gia thn perðptwsh R{s} > 3, qreiazìmaste h tom twn pedðwn sôgklishc twn duo shmˆtwn na mac dðnei to hmiepðpedo σ > 3. Autì gðnetai mìnon an {R{s} > 3} {R{s} > }. Gi' autˆ ta duo pedða sôgklishc, ja èqoume 4 e t u(t) 5 4 e3t u(t) 'Ara to s ma sto qrìno ja eðnai to R{s}> L R{s}>3 L 5 4 s + 4 s 3 x(t) 4 e t u(t) e3t u(t) (8) Gia thn perðptwsh < R{s} < 3, qreiazìmaste h tom twn pedðwn sôgklishc twn duo shmˆtwn na mac dðnei to hmiepðpedo < σ < 3. Autì gðnetai mìnon an {R{s} < 3} {R{s} > }. Gi' autˆ ta duo pedða sôgklishc, ja èqoume 4 e t u(t) 5 4 e3t u( t) 'Ara to s ma sto qrìno ja eðnai to R{s}> L R{s}<3 L 5 4 s + 4 s 3 x(t) 4 e t u(t) 5 4 e3t u( t) (9) 9

10 (bþ) O metasqhmatismìc Fourier upologðzetai mìno sthn perðptwsh ROC < R{s} < 3, giatð mìno se autì to pedðo perilambˆnetai o ˆxonac twn fantastik n, σ. 'Ara gia σ, ja èqoume X(s) σ. UpologÐste th lôsh thc diaforik c exðswshc d y(t) me arqikèc sunj kec y(), dy(t) (jπf + ) (jπf + )(jπf 3) + 7 dy(t) + y(t) x(t) kai x(t) u(t). t PaÐrnoume to metasq. Laplace twn duo mer n thc exðswshc: () L{ d y(t) } + 7 L{ dy(t) } + L{y(t)} L{x(t)} s Y (s) sy( ) y () + 7sY (s) 7y( ) + Y (s) X(s) s Y (s) ( ) + 7sY (s) + Y (s) X(s) Y (s)(s + 7s + ) X(s) Y (s) s s(s + 7s + ) Y (s) A s + B s C s + 3 () Ja eðnai A Y (s)s s B Y (s)(s + 4) 9 s 4 4 C Y (s)(s + 3) 7 s 3 3 'Ara Y (s) s s s + 3 y(t) u(t) e 4t u(t) 7 3 e 3t u(t) ( e 4t 7 3 e 3t) u(t) () Parat rhsh: Qrhsimopoi same thn idiìthta L{ dn x(t) n } sn X(s) s n x( ) x (n ) ( )

11 . Na brejeð o antðstrofoc metasq. Laplace thc 3s + s + s + 3s + s + s + 3s + (s ( + 3j))(s ( 3j)) A s ( + 3j) + A s ( 3j) Ta A, A dðnontai apì A X(s)(s ( + 3j)) s +3j 3s + 3 s ( 3j) s +3j + j 6 Opìte A 3 j 6 ( 3 6) + j ( 3 s ( + 3j) + j ) 6 s ( 3j) Oi pijanoð pìloi eðnai oi s 3j, s s + 3j, oi opoðoi brðskontai pˆnw sthn Ðdia eujeða, σ. 'Ara ta pijanˆ pedða sôgklishc eðnai ta R{σ} >, R{σ} <. Anˆloga me autˆ ta pedða sôgklishc, ja èqoume kai ta antðstoiqa x(t). BreÐte ta! :-). Gia èna s ma kai to metasq. Laplace tou gnwrðzete ìti: (aþ) to s ma sto qrìno eðnai pragmatikì kai ˆrtio (bþ) èqei 4 pìlouc kai kanèna mhdenikì sto migadikì epðpedo (gþ) ènac pìloc brðsketai sto s ej π 4 (dþ) x(t) 4 BreÐte to X(s). Profan c to X(s) ja eðnai thc morf c: A (s s )(s s )(s s 3 )(s s 4 ) DÐdetai ìmwc ìti to s ma eðnai pragmatikì kai ˆrtio, ˆra ja eðnai x(t) x (t) X (s ) kai x(t) x( t) X( s) 'Ara an èqei ènan pìlo sth jèsh s k, ja èqei epðshc ènan pìlo sth jèsh s k (apì th sqèsh tou ˆrtiou s matoc). 'Omoia, an èqei ènan pìlo sth jèsh s k ja èqei epðshc ènan pìlo sth jèsh s k (apì th sqèsh tou pragmatikoô s matoc). GnwrÐzoume ìti èqei ènan pìlo s, ˆra ja èqei ki ènan s, ki ènan s kai ènan s. Opìte to s ma ja grˆfetai: A (s s )(s s )(s + s )(s + s )

12 Mènei na broôme to A. DÐnetai ìti 'Omwc xèroume ìti x(t) 4 X() 4 x(t)e t x(t) 4 A ( s )( s )( + s )( + s ) A s s A 4 4 A 6 A 4 'Ara to s ma pou yˆqnoume eðnai to 4 (s ej π 4 )(s + ej π 4 )(s e j π 4 )(s + e j π 4 ) (3) 3. Sac dðnontai ta parakˆtw stoiqeða gia èna pragmatikì s ma x(t) kai to metasq. Laplace tou. (aþ) To X(s) èqei akrib c duo pìlouc (bþ) To X(s) den èqei kanèna mhdenikì (gþ) To X(s) èqei pìlo sto s + j (dþ) To e t x(t) den eðnai apolôtwc oloklhr simo (eþ) X() 8 BreÐte to X(s) kai thn perioq sôgklishc. LÔste to! :-) Hint: BreÐte pr ta th majhmatik morf tou X(s). BreÐte ta pijanˆ pedða sôgklishc. Tèloc, to (d') stoiqeðo axiopoi ste to gia na breðte to pedðo sôgklishc. ErmhneÔste swstˆ ti shmaðnei to e t x(t) den eðnai apolôtwc oloklhr simo. 4. ApodeÐxte ìti èna anti-aitiatì s ma eðnai eustajèc an kai mìno an ìloi oi pìloi tou brðskontai sto dexiì migadikì hmiepðpedo. GenikeÔste gia èna mh-aitiatì sôsthma, apodeiknôontac ìti to pedðo sôgklis c tou prèpei na perilambˆnei ton fantastikì ˆxona. Gia èna anti-aitiatì sôsthma ja èqoume H(s) Y (s) N A k h(t) s s k k A k e skt u( t) (4) k

13 ìpou s k oi pìloi tou metasqhmatismoô Laplace, ja eðnai h(t) A k e skt u( t) k A k k k e σ kt e jπft A k e σkt (5) to opoðo kai shmaðnei ìti to sôsthma eðnai eustajèc mìno an σ k >. 'Ara ìloi oi pìloi tou antiaitiatoô sust matoc prèpei na brðskontai sto dexiì migadikì hmiepðpedo. GenikeÔontac gia èna opoiod pote s ma, ja èqoume H(s) Y (s) N A k + s s k k L k B k s λ k h(t) ìpou s k, λ k oi pìloi tou metasqhmatismoô Laplace, ja eðnai A k e skt u(t) k L B k e λkt u( t) (6) k h(t) < ( N A k e skt u(t) + k A k e skt u(t) + k A k k A k k L k e σ kt e jπft + e σ kt + L B k k ) B k e λkt u( t) k L B k e λkt u( t) L B k e ζ kt e jπft k e ζkt (7) To pr to olokl rwma sugklðnei ìtan R{s k } σ k < kai to deôtero ìtan R{λ k } ζ k >. Autì shmaðnei ìti ìloi oi pìloi s k tou aitiatoô tm matoc tou s matoc brðskontai sto aristerì migadikì hmiepðpedo, en ìloi oi pìloi λ k tou anti-aitiatoô tm matoc tou s matoc brðskontai sto dexiì megadikì hmiepðpedo. 'Estw s r o dexiìteroc pìloc tou sunìlou twn pìlwn s k kai λ l o aristerìteroc pìloc tou sunìlou twn pìlwn λ k. To pedðo sôgklishc ja eðnai σ r < R{s} < ζ l, kai afoô σ r < kai ζ l >, to pedðo sôgklishc perilambˆnei to fantastikì ˆxona. 'Ara krit rio eustˆjeiac gia ta sust mata eðnai h perðlhyh tou fantastikoô ˆxona mèsa sto pedðo sôgklishc tou metasq. Laplace pou to perigrˆfei. 5. DÐnetai h parakˆtw grammik diaforik exðswsh hc tˆxhc: me arqikèc sunj kec d y(t) + 5 dy(t) y( ), dy(t) + 6y(t) x(t) t 3

14 kai x(t) e t u(t) BreÐte to y(t). GnwrÐzoume ìti dx(t) sx(s) x( ) d x(t) s X(s) sx( ) x ( ) d y(t) + 5 dy(t) + 6y(t) x(t) s Y (s) sy( ) y ( ) + 5sY (s) 5y( ) + 6Y (s) X(s) s Y (s) s + 5sY (s) + 6Y (s) X(s) Y (s)(s + 5s + 6) X(s) + s + Y (s) s+ + s + s + 5s + 6 Y (s) s + 3s + (s + )(s + )(s + 3) AnalÔoume se merikˆ klˆsmata (mporoôme kateujeðan, giatð h tˆxh tou arijmht eðnai mikrìterh aut c tou paronomast ): me Y (s) s + 3s + (s + )(s + )(s + 3) A s + + B s + + C s + 3 A (s + )Y (s) s + 3s + s (s + )(s + 3) s B (s + )Y (s) s + 3s + 6 s (s + )(s + 3) s C (s + 3)Y (s) s + 3s + 9 s 3 (s + )(s + ) s 3 'Ara telikˆ, Y (s) s s + 9 s + 3 y(t) e t u(t) + 6e t u(t) 9 e 3t u(t) (8) pou eðnai kai to zhtoômeno. 4

15 6. 'Estw to s ma x(t) e t u(t) + e t cos(3t)u(t) (aþ) Na brejeð o metasq. Laplace (bþ) Na brejeð h arqik sunj kh x( + ) mèsw tou metasq. Laplace (gþ) Na upologisteð to ìrio lim t x(t) mèsw tou metasq. Laplace (dþ) Epibebai ste ta apotelèsmata twn parapˆnw duo erwthmˆtwn upologðzontac analutikˆ ta ìria (aþ) L{e t u(t)} + L{e t cos(3t)u(t)} s + + s + (s + ), R{s} > + 9 kai kˆnontac lðgec prˆxeic, èqoume s + 5s + s 3 + 5s + 4s + (bþ) (gþ) (dþ) Dikì sac. :-) x( + ) lim s s lim s s s + 5s + s 3 + 5s + 4s + s 3 + 5s + s lim s s 3 + 5s + 4s + (De L Hospital) 6s + s + lim s 3s + s + 4 (De L Hospital) s + lim s 6s + (De L Hospital) lim s 6 x( + ) (9) s 3 + 5s + s lim x(t) lim s lim t s s s 3 + 5s lim + 4s + x(t) t 5