UNIT STEP FUNCTION. Solution: Example : Ex: Write the following function in terms of the unit step...
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Transcript of UNIT STEP FUNCTION. Solution: Example : Ex: Write the following function in terms of the unit step...
UNIT STEP FUNCTION
Solution:
?)}({ atuL
0
)()}({ dtatueatuL st
Example:
)s
e(0
ats
e asst
a
st dt e 1.
s
e as
Ex: Write the following function in terms of the unit step function
2
1t if t
2
1t1 if t
t0 if
tf
cos
2
1
12
)( 2
Ex1: ?)}1(2
1{ 2 tutL
Ex2: ?)}2
1(
2
1{ 2 tutL
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
1{ 2 tutL
Impulse FunctionDefine 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) b)
c)
• Ex: Solve Solution: Taking the laplace transform of both sides
y(0)=0, y’(0)=0
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* g then
H(s)=F(s)G(s)
h(t)=L-1{F(s)G(s)}
Example:
Laplace transform of tf(t)
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