Digital Signal Processing Reference
In-Depth Information
Hence, the equation agrees with the figure. Each term in
f(t),
(7.23), is of the form required
by the real-shifting property:
l
[f(t - t
0
)u(t - t
0
)] = e
-t
0
s
F(s).
[eq(7.22)]
The Laplace transform of
f(t)
is, then, from (7.23) and Table 7.2,
10e
-s
s
2
10e
-2s
s
2
3e
-2s
s
7e
-3s
s
F(s) =
-
-
-
.
so, These terms can be combined to yield
10e
-s
- 10e
-2s
- 3se
-2s
- 7se
-3s
F(s) =
.
■
s
2
We make two points relative to this example. First, complicated Laplace
transforms can occur for complex waveforms. For signals of this type, the function
should be written as a sum of terms such that each term is of the form required
in the real-shifting property, (7.22). Otherwise, the definition, (7.4), must be inte-
grated to find the Laplace transform.
As the second point, note from Table 7.2 that all transforms listed are ratios of
two polynomials in
s
. A ratio of polynomials is called a
rational function
. Any sum
of the signals in Table 7.2 generally yields a rational function of higher order. The
appearance of an exponential function of
s
in a Laplace transform generally results
from delayed time functions.
f(t)
We next consider two of the most useful properties of the Laplace transform, which
are related to differentiation and integration. The differentiation property was de-
rived in Section 7.2 and is, from (7.15),
df(t)
dt
= sF(s) - f(0
+
).
B
R
l
(7.24)
Property (7.24) is now extended to higher-order derivatives. The Laplace
transform of the second derivative of
f(t)
can be expressed as
d
2
f(t)
dt
2
df
¿
(t)
dt
df(t)
dt
B
R
B
R
l
=
l
,
f¿(t) =
.
(7.25)
Then, replacing
f(t)
with
f¿(t)
in (7.24), we can express (7.25) as
d
2
f(t)
dt
2
= s
l
[f¿(t)] - f¿(0
+
),
B
R
l
