Biomedical Engineering Reference
In-Depth Information
u
(
t
)
0
t
FIGURE 10.5
:
Unit step function,
u
(
t
). The step function is similar to throwing a switch
The value
f
3
for any particular instant of time
t
is the area under the product of
f
1
(
λ
) and
). To use the convolution technique efficiently, one should be able to visualize
the two functions. If convolution is a new topic to the reader, then a quick review of wave
form synthesis is essential for visualization. Let us begin with a
step function
,
u
(
t
), which
is initiated at
t
f
2
(
t
−
λ
0 (Fig. 10.5).
Figure 10.6 shows the unit step function translated,
u
(
t
=
−
a
), along the time axis,
+
a
, whereas Fig. 10.7 shows the unit step function translated
backward along the time axis,
t
, by some length,
+
a
.
In addition, the unit step function may be rotated about the
y
-axis, which is referred
to as
folding
about some axis. A folded (flipped) unit step function about the
y
-axis is
written as,
u
(
−
t
, by some length,
−
t
), and is shown in Fig. 10.8.
Figure 10.9 shows the result of positive translation of a folded unit step function.
Note that this function is written as,
u
(
a
−
t
).
In addition to rotation about the
y
-axis, the unit step function may also be folded
about the
x
-axis as shown in Fig. 10.10; note the negative sign in front of the unit step
function,
−
u
(
t
).
Figure 10.11 shows the translation of a unit step function folded about the
x
-axis
displaced by some time value,
a
. This step function is written as
−
−
u
(
t
−
a
); note the
negative signs.
u
(
t
-
a
)
a
+t
FIGURE 10.6
:
Positive translation. The unit step function of Fig. 10.5 is translated along the
time axis,
+
t
, by some length,
+
a
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