Digital Signal Processing Reference
In-Depth Information
7.6.
(a)
Sketch the time functions given.
(i) (ii)
(iii) (iv)
(b)
Use the definition of the Laplace transform, (7.4), and the table of integrals in
Appendix A to calculate the Laplace transforms of these time functions.
(c)
Use the real-shifting property and the transform table to find the Laplace trans-
forms.
(d)
Compare the results of parts (b) and (c).
3e
-5t
u(t - 2)
-3e
-5t
u(t - 1)
-5e
-at
u(t - b)
-5e
-a(t -b)
u(t - c)
7.7.
Use the real shifting property (7.22) and the transform table to find the Laplace trans-
forms of the time functions given. Manipulate the time functions as required. Do not
use the defining integral (7.4). Let
a 7 0, b 7 0, a 6 b.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
5u(t - 2)u(3 - t)
3tu(t - 2)
3u(t - 3)u(t - 2)
3t[u(t - 1) - u(t - 3)]
3t[u(t - a) - u(t - b)]
2e
-3t
u(t - 5)
2e
-at
u(t - b)
7.8.
Consider the triangular voltage waveform
v(t)
shown in Figure P7.8.
(a)
Express mathematically.
(b)
Use the real-shifting property to find
(c)
Sketch the first derivative of
(d)
Find the Laplace transform of the first derivative of
(e)
Use the results of part (d) and the integral property, (7.31), to verify the results of
parts (b) and (d).
v(t)
l
[v(t)].
v(t).
v(t).
(f)
Use the derivative property, (7.15), and the result of (b) to verify the results of (b)
and (d).
v
(
t
) (V)
5
0
2
4
t
(
s
)
Figure P7.8
7.9.
Given the Laplace transform
s
(s + 1)(s + 2)
,
V(s) =
v(t), v(0
+
),
(a)
Find the initial value of
by
(i)
the initial value property (Property 7 of Table 7.3);
finding v(t) = l
-1
[V(s)].
(ii)
