Digital Signal Processing Reference
In-Depth Information
TABLE 7.4
Key Equations of Chapter 7
Equation Title
Equation Number
Equation
q
f(t)e
-st
dt
Bilateral Laplace transform
(7.1)
l
b
[f(t)] = F
b
(s) =
L
-
q
c+ j
q
1
2pj
L
f(t) = l
-1
[F(s)] =
F(s)e
st
ds, j =
Inverse Laplace transform
(7.2)
2
-1
c- j
q
q
f(t)e
-st
dt
Unilateral Laplace transform
(7.4)
l[f(t)] = F(s) =
L
0
+
Á
+ b
1
s + b
0
b
n
s
n
+ b
n- 1
s
n- 1
Y(s)
X(s)
=
Transfer function
(7.49)
H(s) =
+
Á
+ a
1
s + a
0
a
n
s
n
+ a
n- 1
s
n- 1
7.1.
Sketch the time functions given. Then use the definition of the Laplace transform,
(7.4), and the table of integrals in Appendix A to calculate the Laplace transforms of
the time functions.
(a)
(b)
(c)
(d)
5u(t - 2)
3[u(t - 2) - u(t - 3)]
- 5u(t - 2)u(3 - t)
- 5u(t - a)u(b - t),
where
b 7 a 7 0
7.2.
Use the definition of the Laplace transform, (7.4), and an integral table to verify the
following Laplace transforms:
2bs
(s
2
+ b
2
)
2
(a)
l[t sin bt] =
s
(b)
l[cos
bt] =
s
2
+ b
2
1
s - a
l[e
at
] =
(c)
1
(s - a)
2
l[te
at
] =
(d)
1
s
2
(e)
l[t] =
