UNIT STEP FUNCTION
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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 ta
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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