This preview shows page 1. Sign up to view the full content.
Unformatted text preview: um=[2 T 4 ]; d en=[eonv(eonv([1 1 ],[1 2 ]),[1 2J)J;
[r, p , k ]= r esidue(num,den); = 3, r z [ Hence, 2, 1 = and f (t) 2t = ( 3e + 2 te 2t  et)u(t) ( c) r = 3 .5  0.4811i. 3 .5 + 0 .4811i. 1 .00
P =  0.5 + 2 .5981i.  0.5  2.5981i.  2.00
k=[1
a ngle =  0.1366. 0 .1366. 0
mag = 3 .5329. 3 .5329. 1 .00
1
F (s) =  s +2 3.5329 e jO.1366 + s + 0.5  j2.5981 3.5329 e jO.1366 + ==,::::c~
8 + 0.5 + j 2.5981 and
f (t) = [ e 2t o E xercise E 6.2
(i) Show that the Laplace transform of l Oe 3t cos (4t lOa from Table 6.1.
( ii) Find the inverse Laplace transform of: (a) ( e) [3e 2t 321
s + 2 + (8 + 2)2  s + 1 n um=[8 2 1 1 9]; d en=[conv([O 1 2 ],[1 1 7])];
[r, p , k j= r esidue(num,den)
[ angle,mag] = eart2pol(real(r) , imag(r» Hence, 379 82 + 53.13°) is 6s  14 .
82 + 68 + 25 Use Pair + 17 ( b)
38  5
+ 485
( 8+ 1 )(82 + 2s+5) 8 Answers:(a) (3e t 2e 5t )u(t) ( b) [2et+~etcos(2t36.87°)Ju(t) 1
F (s) T he Laplace Transform 168 + 43
( c) (8 _ 2)(8 + 3)2 P =  2.  2.  1
k 6.1 + l .766eO. 5 tcos(2.598lt  0.1366)Ju(t) 0 C omputer E xample C 6.2 Find (a) t he direct Laplace transform of sin a t + cos bt (b) the inverse Laplace
transform of (as 2)/(s2 + b2).
Here we shall use S ymbolic Math Toolbox, which is a collection of functions for
MATLAB used for manipulating and solving symbolic expressions.
( a)
f =sym('sin(a*t)+eos(b*t)'};
F =laplace(f) F =(a*s2+b2*a+s3+s*a2)/(s2+a2)/(s2+b2) Thus, ( b)
F =sym('( a *s 2)/(s 2+b 2);
f =invlaplace(F)
F =a*dirac(t)a*b*sin(b*t) Thus,
f (t) = a6(t) + absin(bt)u(t) 0 + (t  3)e 3t Ju (t) \l A Historical Note: Marquis PierreSimon De Laplace ( 17491827)
T he Laplace transform is named after t he g reat French mathematician a nd
a stronomer Laplace, who first presented t he t ransform a nd its applications to differential equations in a p aper published in 1779.
Laplace developed t he foundations of potential theory a nd m ade i mportant
c ontributions t o special functions, probability theory, astronomy, a nd celestial mechanics. In his E xposition du s ysteme du Monde (1796), Laplace formulated a
nebular hypothesis o f cosmic origin a nd t ried t o explain t he universe as a pure
mechanism. In his Traite de M echanique Celeste (celestial mechanics), which completed t he work of Newton, Laplace used mathematics a nd physics t o s ubject t he
s olar system a nd all heavenly bodies t o t he laws o f m otion a nd t he principle of
gravitation. Newton had been unable to explain t he irregularities of some heavenly
bodies; in desperation, he concluded t hat God himself must intervene now a nd t hen
t o prevent some catastrophes, such as J upiter eventually falling into t he s un (and
t he moon into t he e arth) as predicted by Newton's calculations. Laplace proposed
t o show t hat these irregularities would correct themselves periodically, a nd t hat a
l ittle p atiencein J upiter's case, 929 y earswould see everything returning a utomatically t o order, a nd t here was no reason why t he solar a nd t he s tellar systems
could not continue to...
View
Full
Document
This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

Click to edit the document details