Transcript UNIT STEP FUNCTION

UNIT STEP FUNCTION
Example:
L{u (t  a)}  ?

Solution:
L{u (t  a)}   e  stu (t  a)dt
0

  e  st .1 dt
a
e  st 
e  as

 0 ( 
)
s ta
s

e
 as
s
Ex: Write the following function in terms of the unit step function


2
if 0  t  1 


1
1


f (t )   t 2 if 1  t   
2 
2
1


cos
t
if
t




2


1
2
Ex1: L{ t 2 u (t  1)}  ?
1 2
1
Ex2: L{ t u (t   )}  ?
2
2
Ex3:
=?
• Ex: Show that L{ f(t)u(t-a) }=e-asL{ f(t+a) }
Proof: L(f(t-a)u(t-a)}=e-asF(s) (1)
Let f(t-a)=g(t) then f(t)=g(t+a), put in (1)
L{g(t)u(t-a)}=e-asL{g(t+a)}, change from g to f simply
L{f(t)u(t-a)}=e-asL{f(t+a)},
Ex:
1 2
L{ t u (t  1)}  ?
2
Impulse Function
Define the function fk (t-a) as
In terms of unit step functions
Dirac delta function or unit impulse function
Mathematical expression for the unit impulse function
Some properties of the unit impulse function
a)
c)
b)
y(0)=0, y’(0)=0
• Ex: Solve
Solution:
Taking the laplace transform of both sides
Convolution
• Convolution of f(t) and g(t), h(t) is defined as:
Convolution Theorem
Let H(s), F(s), and G(s) denote the laplace transforms
of h(t), f(t), and g(t).
If h is the convolution of f and g, h=f
H(s)=F(s)G(s)
h(t)=L-1{F(s)G(s)}
* g then
Example:
Laplace transform of tf(t)
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