Bilateral Laplace Transform (6A) Young Won Lim 2/23/15

Transcription

1 Bilterl Lplce Trnsform (6A)

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3 An Improper Integrtion F (s) = f (t)e s t dt Complex Number Rel Number Rel Number s = + i t Integrtion Vrible R{s} rel prt I {s} img prt The improper integrl converges if the limit defining it exists. Bilterl Trnsform (6A) 3

4 Lplce trnsforms of nd exp( t) L s F(s) = e s t dt b = lim b s e st] = lim b s e sb + ] s e s s < lim e s b = s > F(s) = b s e t L s+ F(s) = e t e s t dt = lim b e (s+ = lim b (s+) b )t] (s+) e (s+ )b + (s+) e (s+ )] (s+) < lim e (s+ )b = s > F(s) = (s+) b Bilterl Trnsform (6A) 4

5 Lplce trnsforms of exp(+t) nd exp( t) e t L s+ F(s) = e t e s t dt = lim b e (s+ = lim b (s+) b )t] (s+) e (s+ )b + (s+) e (s+ )] (s+) < lim e (s+ )b = s > F(s) = (s+) b e +t L s F (s) = e + t e s t dt = lim b e t] b = lim b e (s )b + e ] < lim e b = s > + F (s) = b Bilterl Trnsform (6A) 5

6 Converging Improper Integrls e +k t dt = + k k > ek t] = + k (ek e +k ) = k k < e k t dt = k k < e k t] = k (e k e k ) = k k > Bilterl Trnsform (6A) 6

7 Existence of Lplce Trnsforms Right-sided function e t < M e α t Right-sided function e t < M e α t α > + > e t u(t) e +t u(t ) α α Left-sided function Left-sided function e t < M e βt e t < M e βt β β e t u( t ) β < < β < < e +t u( t) Bilterl Trnsform (6A) 7

8 ROC nd Exponentil Order Right-sided function Right-sided function exponentil order β < s > exponentil order α > s > + e t u(t) Lplce trnsform exists β < β < e +t u(t ) α < + < α Left-sided function exponentil order β < Left-sided function exponentil order α > β α e t u( t ) β < < s < e +t u( t) < α < s < + Bilterl Trnsform (6A) 8

9 Improper Integrls of f(t)u(+t) nd f(t)u( t) f (t ) L + dt F (s) F (s) = f (t ) e s t dt ( < t < +) f (t) t f (t )u(+t ) right side of f(t) is used f (t ) ^L dt G(s) G(s) = f (t ) e s t dt ( < t < + ) f (t) t f (t )u( t ) left side of f(t) is used Bilterl Trnsform (6A) 9

10 Improper Integrls of e t u(+t) nd e t u( t) e t u(t ) e t u( t) e t u( t) t t t F (s) = e + t e s t dt G(s) = e + t e s t dt F (s) = e +t e s t dt = e t] = e t] = e t] s > + s < + s < + e +t e s t dt = e + t e s t dt = e +t e s t dt = G(s) = F (s) e +t e s t dt = F (s) Bilterl Trnsform (6A)

11 Two functions of s : G(s) = -F(s) f (t )u(t ) f (t )u( t ) f (t)u( t) F (s) = f (t) e s t dt G(s) = f (t) e s t dt F (s) = f (t) e s t dt s > + s < + s < + f (t ) L F (s) f (t ) ^L G(s) f (t) ^L F (s) dt s > + s < + dt dt the sme function of s the different ROC's F (s) s > + s < + Bilterl Trnsform (6A)

12 Improper Integrls of One-Sided Functions Right-sided function exponentil order α > e + t u(+t) L s > e +t u(t ) F (s) = e +t e s t dt s > + F (s) = Left-sided function exponentil order α > e +t u( t ) ^L s > +e + t u( t) G(s) = e + t e s t dt s < + G(s) = = F (s) Bilterl Trnsform (6A) 2

13 The Sme Formul with Different ROCs Right-sided function exponentil order α > s > + > e +t u(t ) s < + Left-sided function Left-sided function exponentil order α > exponentil order α > +e + t u( t) e +t u( t) Bilterl Trnsform (6A) 3

14 ROCs nd one-sided functions e t u( t) e t u(t ) e t (t < ) e t (t > ) e + t e s t dt t < t > e + t e st dt s < s < I{s} s > s > R{s} Bilterl Trnsform (6A) 4

15 Improper Integrls of f(t)u(+t) nd f(-t)u(+t) f (t )u(t ) f (t )u( t ) f ( t)u(t ) L {f (t )u(t)} = F (s) + = f (t) e st dt L { f (+ v) } = F (+s) L { f ( v) } = G ( s) ^L {f (t )u( t )} = G(s) = f (t ) e s t dt L {f ( t )u(t )} = H (s) = G( s) G(s) = H ( s) L {f ( t )u(t )} = H (s) + = = = = G( s) f ( t ) e s t dt f (v) e s( v) dv f (v) e s v dv Bilterl Trnsform (6A) 5

16 Improper Integrls : e t u(+t), e t u( t), e t u(+t) e + t u(+t) L dt s F (s) = e + t e s t dt = lim b e t] b = lim b < lim e b = s > + F (s) = b b > e (s )b + e ] e +t u( t ) ^L dt s G(s) = e +t e s t dt = lim b < lim e b = s < + G(s) = b b < e t] b = lim b e + e b] = F (s) L { f (+v) } = F (+s) L{e + t } = F (+s) = ^L{e +t } = G(+s) = L { f ( v) } = G( s) G(+s) = L{e t } = G( s) = (s+) ( s+) = Bilterl Trnsform (6A) 6

17 Unilterl nd Bilterl Lplce Trnsform Unilterl Lplce Trnsform + F (s) = f (t )e st dt f (t ) = + j 2π j j F(s)e +s t ds Bilterl Lplce Trnsform + F (s) = f (t )e st dt f (t ) = + j 2π j j F(s)e +s t ds includes ll includes ll singulrities singulrities (t > ) ROC (t > ) ROC includes ll singulrities (t < ) Bilterl Trnsform (6A) 7

18 ROCs nd two-sided functions e t u( t) e t u(t ) e t (t < ) e t (t > ) t < t > Bilterl Lplce Trnsform + e + t e st dt no overlpping ROC No Convergence e + t e s t dt s < I{s} s > e t e st dt s < R{s} s > Bilterl Trnsform (6A) 8

19 ROCs nd two-sided functions e t u( t) e t u(t ) e t (t < ) e t (t > ) t < t > Bilterl Lplce Trnsform + e + t e st dt = ( s+) ( s ) = 2 s 2 2 e + t e s t dt I{s} ROC e t e st dt s < R{s} (s+) s > Bilterl Trnsform (6A) 9

20 ROCs nd two-sided functions e t u( t) e t u(t ) e t (t < ) e t (t > ) t < t > Bilterl Lplce Trnsform + e + t e st dt no overlpping ROC No Convergence e t e s t dt s < I{s} ROC s > e t e st dt (s+) s < R{s} s > Bilterl Trnsform (6A) 2

21 ROCs nd two-sided functions e t u( t) e b t u(t) e b t u( t ) e t u(t ) t < t > t < t > (s+b) (s+b) s < s > b s < b s > I{s} ROC I{s} ROC b R{s} b R{s} < b < < b < Bilterl Trnsform (6A) 2

22 ROCs nd two-sided functions e b t u( t ) e t u(t) e t u( t) e bt u(t ) t < t > t < t > (s+b) (s+) (s b) s < b s > s < s > b I{s} ROC I{s} ROC b R{s} b R{s} < b < < b < Bilterl Trnsform (6A) 22

23 References ] 2] 3] M.L. Bos, Mthemticl Methods in the Physicl Sciences 4] E. Kreyszig, Advnced Engineering Mthemtics 5] D. G. Zill, W. S. Wright, Advnced Engineering Mthemtics 6] T. J. Cvicchi, Digitl Signl Processing 7] F. Wleffe, Mth 32 Notes, UW 22/2/ 8] J. Nering, University of Mimi 9]