Introduction to signals - Continuous and Discrete Time Signals and Systems - page 45

1.25 1.25

1 1

0.75 0.75

0.5 0.5

0.25 0.25

k k

0 0

−8 −6 −4 −2 −8 −6 −4 −2

(a) (b)

The time-reversed signal x ( − t ) is plotted in Fig. 1.28(b). Signal inversion can
also be performed graphically by simply flipping the signal x ( t ) about the
y -axis.

Fig. 1.29. Time inversion of the
DT signal in Example 1.20.
(a) Original CT sequence x [ k ].
(b) Time-inverted version x [− k ].

Example 1.20
Sketch the time-inverted version of the following DT sequence:

1

− 4 ≤ k ≤− 1

x [ k ] = 0 . 25 k 0 ≤ k ≤ 4 (1.57)

0 elsewhere ,

which is plotted in Fig. 1.29(a).

Solution
To

derive the expression for the time-inverted signal x [ − k ], substitute

k =− m in Eq. (1.57). The resulting expression is given by

1 − 4 ≤− m ≤− 1

x [ − m ] = − 0 . 25 m 0 ≤− m ≤ 4

0 elsewhere.

Simplifying the above expression and expressing it in terms of the independent
variable k yields

1 1 ≤ m ≤ 4

− 0 . 25 m − 4 ≤− m ≤ 0
0 elsewhere .
The time-reversed signal x [ − k ] is plotted in Fig. 1.29(b).

x [ − m ] =

1.3.4 Combined operations

In Sections 1.3.1-1.3.3, we presented three basic time-domain transformations.
In many signal processing applications, these operations are combined. An
arbitrary linear operation that combines the three transformations is expressed
as x ( α t + β ), where α is the time-scaling factor and β is the time-shifting
factor. If α is negative, the signal is inverted along with the time-scaling and
time-shifting operations. By expressing the transformed signal as

t + β
α

x ( α t + β ) = x α , (1.58)