Digital Signal Processing Reference
In-Depth Information
Figure 7.5(c) shows a plot of the function
f(t - t
0
),
where
t
0
is the amount of the
shift in time, with
t
0
7 0.
The function
f(t - t
0
)u(t)
is shown in Figure 7.5(d), and
the function
f(t - t
0
)u(t - t
0
)
is given in Figure 7.5(e). For this last function,
f(t - t
0
),
t 7 t
0
b
f(t - t
0
)u(t - t
0
) =
.
0,
t 6 t
0
The reader should note carefully the differences in the functions in Figure 7.5.
Because we have defined the Laplace transform for only, the Laplace trans-
form of requires the function in Figure 7.5(b). Notice that the function of Fig-
ure 7.5(e) differs from the function of Figure 7.5(b) only by a time-shift. We now
derive a property that relates the Laplace transform of the function of Figure 7.5(e)
to that of the function of Figure 7.5(b).
The Laplace transform of the function of Figure 7.5(e) is given by
t Ú 0
f(t)
q
f(t - t
0
)u(t - t
0
)e
-st
dt
l
[f(t - t
0
)u(t - t
0
)] =
L
0
q
f(t - t
0
)e
-st
dt.
=
L
t
0
We make the change of variable
(t - t
0
) = t.
Hence,
t = (t + t
0
), dt = dt,
and it
follows that
q
f(t)e
-s(t+ t
0
)
dt
l
[f(t - t
0
)u(t - t
0
)]
=
L
0
q
= e
-t
0
s
f(t)e
-st
dt.
(7.21)
L
0
Because is the variable of integration and can be replaced with
t
, the integral on
the right side of (7.21) is
t
F(s).
Hence, the Laplace transform of the shifted time
function is given by
l
[f(t - t
0
)u(t - t
0
)] = e
-t
0
s
F(s),
(7.22)
where and This relationship, called the
real-shifting,
or
real-
translation, property,
applies only for a function of the type shown in Figure 7.5(e);
it is necessary that the shifted function be zero for time less than
t
0
G 0
l
[f(t)] = F(s).
t
0
,
the amount of
the shift. Three examples are now given to illustrate this property.
Laplace transform of a delayed exponential function
EXAMPLE 7.4
Consider the exponential function shown in Figure 7.6(a), which has the equation
f(t) = 5e
-0.3t
,
where
t
is in seconds. This function delayed by 2 s and multiplied by
u(t - 2)
is shown in
Figure 7.6(b); the equation for this delayed exponential function is given by
f
1
(t) = 5e
-0.3(t - 2)
u(t - 2).
