Digital Signal Processing Reference
In-Depth Information
x
(
t
)
1
t
1
0
1
2
4
2
0
2
t
2
2
(a)
y
(
t
)
x
(1
t
/2)
1
2
0
2
4
t
(b)
Figure 2.4
Signals for Example 2.1.
This transformation has reversal, scaling, and shifting. First, we solve the transformation for
the variable
t
:
t
2
Q t = 2 - 2t.
t = 1 -
The
t
-axis is shown below the time axis in Figure 2.4(a), and Figure 2.4(b) gives the desired
plot of the transformed signal. As always, we should check our work. For any particular value
of time,
t = t
0
,
we can write, from (2.7),
t
0
a
-
b
a
¢
≤
y(t
0
) = x(at
0
+ b) and x(t
0
) = y
.
x(t)
Choosing
t
0
= 1
as an easily identifiable point in
from Figure 2.4(a), we calculate
x(1) = y(2 - 2(1)) = y(0).
Again choosing
t
0
= 1,
we calculate
1
2
1
2
¢
≤
¢
≤
y(1) = x
1 -
= x
.
Both calculated points confirm the correct transformation.
■
A general approach for plotting transformations of the independent variable
is as follows:
1.
On the plot of the original signal, replace
t
with
t.
t
a
-
b
a
.
2.
Given the time transformation
t = at + b,
solve for t =
