Laplace

1 - Sendo f(t)=sem(wt), encontrar a transfomada Z[f[n]]

Z[f[n]]= z [ ejwn-e-jwn2j ] = 12 [ejwn-e-jwnj].Z
Z[n] = n=o∞ejwn . z-n
Z[n] = ejw0 . z0 + ejw1 . z-1 + ejw2 . z-2 + ejw3 . z-3 ...
Z[n] = (ejw)0 . ( z-1)0 + (ejw)1 . ( z-1)1 + (ejw)2 . ( z-1)2 + (ejw)3 . ( z-1)3 ...
= a1-r a=1, r=ejw . z-1, = 11- ejw . z-1 = 11-ejwnz = 1z- ejwnz = zz- ejwn

Z[n] = n=o -∞e-jwn . z-n
Z[n] = n=o - ∞e-jwn . z-n
Z[n] =(-e-jw0 . z0 )+(- e-jw1 . z-1 )+(- e-jw2 . z-2 )+(- e-jw3 . z-3 ) ...
Z[n] = (-1)(e-jw)0 . ( z-1)0 + (-1)(ejw)1 . ( z-1)1 + (-1)(ejw)2 . ( z-1)2 + (-1)(-ejw)3 . (z-1)3 ...

a1-r a=-1, r=-e-jw . z-1, = -11- ejw . z-1 = -11-e-jwnz = -1z- e-jwnz = -zz- e-jwn

Z[n] =12j [ zz- ejwn - -zz- e-jwn ]
Z[n] =12j [z-e-jw- (- z-ejw)z2-ze-jw-ze-jw+e-jw.ejw]
Z[n] =12j[ z(ejw-e-jw)z2-2zejw+e-jw+1]
Z[n] =12j[zsen(jw)z2-2zcosjw+1]

2 - Sendo f(t)=cos(wt), encontrar a transfomada Z[f[n]]
Z[f[n]]= z [ ejwn+e-jwn2 ] = 12 [ejwn+e-jwn].Z
Z[n] = n=o∞ejwn . z-n
Z[n] = ejw0 . z0 + ejw1 . z-1 + ejw2 . z-2 + ejw3 . z-3 ...
Z[n] = (ejw)0 . ( z-1)0 + (ejw)1 . ( z-1)1 + (ejw)2 . ( z-1)2 + (ejw)3 . ( z-1)3 ...
= a1-r a=1, r=ejw . z-1, = 11- ejw . z-1 = 11-ejwnz = 1z- ejwnz = zz- ejwn

Z[n] = n=o ∞e-jwn . z-n
Z[n] = n=o ∞e-jwn . z-n
Z[n] =e-jw0 . z0 + e-jw1 . z-1 + e-jw2 . z-2 + e-jw3 . z-3 ...
Z[n] = (e-jw)0 . ( z-1)0 + (ejw)1 . ( z-1)1 + (ejw)2 . ( z-1)2 +(-ejw)3 . (z-1)3 ...

a1-r a=1, r=e-jw . z-1, = 11- ejw . z-1 = 11-e-jwnz = 1z- e-jwnz = zz- e-jwn

Z[n] =12 [ zz- ejwn + zz- e-jwn ]
Z[n] =12 [ze-jw+ z(ejw)z2-ze-jw-zejw+e-jw.ejw]
Z[n] =12[ z(ejw+e-jw)z2-2zejw+e-jw+1]
Z[n] =12[zcos(jw)z2-2zcosjw+1]

01 y’’(t) + 2y’(t) + 4y(t) = 0 y(0)=1 , y’(0)=1
Y =-b±b2-4ac2a = -2±22-4.1.42.2 = -2±4-164 = x’=-1-3 , x’’=-1-3
Suloção geral.
Y=e(-1∓3)
Y= e-x(c1cos3.x+(c2sen3.x))
Y(0) = 1