Digital Signal Processing Reference
In-Depth Information
Several properties of the Laplace transform have been developed. These
properties are useful in generating tables of Laplace transforms and in applying
the Laplace transform to the solutions of linear differential equations with con-
stant coefficients. Because we prefer to model continuous-time physical systems
with linear differential equations with constant coefficients, these properties are
useful in both the analysis and design of linear time-invariant physical systems.
Table 7.3 gives the derived properties for the Laplace transform, plus some addi-
tional properties. The derivations of some of these additional properties are
given as problems at the end of this chapter, or are derived later when the properties
are used.
TABLE 7.3
Laplace Transform Properties
Name
Property
1. Linearity, (7.10)
l[a
1
f
1
(t) + a
2
f
2
(t)] = a
1
F
1
(s) + a
2
F
2
(s)
df(t)
dt
= sF (s) - f(0
+
)
2. Derivative, (7.15)
l
B
R
d
n
f(t)
dt
n
= s
n
F(s) - s
n- 1
f(0
+
)
3.
n
th-order derivative, (7.29)
l
B
R
-
Á
-sf
(n- 2)
(0
+
) - f
(n- 1)
(0
+
)
t
F(s)
s
B
R
4. Integral, (7.31)
l
f(t)dt
=
L
0
l[f(t - t
0
)u(t - t
0
)] = e
-t
0
s
F(s)
5. Real shifting, (7.22)
6. Complex shifting, (7.20)
7. Initial value, (7.36)
l[e
-at
f(t)] = F(s + a)
t:0
+
f(t) =
lim
lim
s:q
sF(s)
8. Final value, (7.39)
lim
t: q
f(t) =
lim
s:0
sF
(s)
dF(s)
ds
9. Multiplication by
t
, (7.34)
l[tf(t)] =-
e
-sb/a
a
s
a
¢
≤
10. Time transformation, (7.42)
(a 7 0; b G 0)
l[f(at - b)u(at - b)] =
F
t
l
-1
[F
1
(s)F
2
(s)] =
L
11. Convolution
f
1
(t - t)f
2
(t) dt
0
t
=
L
f
1
(t)f
2
(t - t)dt
0
1
1 - e
-sT
F
1
(s),
12. Time periodicity
l[f(t)] =
where
T
f(t)e
-st
dt
[f(t) = f(t + T)], t G 0
F
1
(s) =
L
0
