Fourier and Laplace
347
R.4.82 The most important time (t)–frequency (s) properties and relations of the unilateral
LT are given as follows (similar to the FT properties):
a. Linearity
Let
£[
]
and £
f t
F s
f t
F s
1
1
2
2
( )
( )
[ ( )]
( )
ϭ
ϭ
then
a f t
a f t
a F s a F s
1 1
2 2
1 1
2 2
( )
( )
( )
( )
ϩ
ϩ
← →

b. Time scaling
f at
a
F
s
a
( ) ↔

 

 
1
c. Time shifting
f t t
F s e
st
(
)
( )
Ϫ
Ϫ
0
0
↔
d. s-Shifting
f t e
F s s
s t
( )
(
)
0
0
↔
Ϫ
TABLE 4.2
Transform Pairs
Signal/System—f(t)
Transform—F(s)
δ(t)
1
u(t)
1 / s
tu(t)
1 / s
2
t
n u(t)
n!/s
n+1
e
−at u(t)
1/(s + a)
te
−at u(t)
1/(s + a)
2
cos(ω a t)u(t)
s/(s + ω 0
2 )
sin(ω a t)u(t)
ω 0 /(s
2 – ω 0
2 )
e
−at cos(ω a t)u(t)
( s + a)/ ( (s + a)
2 + ω 0
2 )
e
−at sin(ω a t)u(t)
( s + ω 0 )/ ( (s + a)
2 + ω 0
2 )
cosh(ω 0 t)u(t)
s/(s
2 − ω 0
2 )
sinh(ω 0 t)u(t)
ω 0 /(s
2 + ω 0
2
)
1
1
1
(
)!
( )
n
e u t
n
a t
−
−
t
1
(
)
s a
n
−
1
a b
e
e ut
at
bt
−
−
(
)()
1
(
)(
)
s a s b
−
−
(1 − e
at )u(t)
−
−
a
s s a
(
)
[1 − cos (w 0 t)]u(t)
w
s s
a
0
2
2
2
(
)
+
[sin (w 0 t + θ)]u(t)
s
w
s
w
sin( )
cos( )
+
+
0
2
0
2
[cos (w 0 t + θ)]u(t)
s
w
s
w
cos( )
sin( )
−
+
0
2
0
2
CRC_47760_CH004.indd 347
CRC_47760_CH004.indd 347
7/28/2008 12:25:57 PM
7/28/2008 12:25:57 PM