Introduction to systems - Continuous and Discrete Time Signals and Systems - page 82

5

4

3

2

x [ k ] y [ k ]

1 1 1 1

k k

−1 −2 0 12345678910 −1 −2 0 12345678910

(a) (b)

7

6

5

4

x [ k − 2]

y 2 [ k ]

1 1 1 1

k

k

−1 −2 0 1 234 56 78910 −1 −2 012345678910

(c) (d)

and

Fig. 2.15. Input-output pairs of
the DT time-varying system
specified in Example 2.5(ii). The
output y 2 [ k ] for the time-shifted
input x 2 [ k ] = x [ k − 2] is
different in shape from the
output y [ k ] obtained for input
x [ k ]. Therefore the system is
time-variant. Parts (a)-(d) are
discussed in the text .

x [ k − k 0 ] → 3( x [ k − k 0 ] − x [ k − k 0 − 2]) = y [ k − k 0 ] .

Therefore, the system in Eq. (2.44) is a time-invariant system.
(ii) From Eq. (2.45), it follows that:

x [ k ] → kx [ k ] = y [ k ]

and

x [ k − k 0 ] → kx [ k − k 0 ] = y [ k − k 0 ] = ( k − k 0 ) x [ k − k 0 ] .

Therefore, system II is not time-invariant. In Fig. 2.15, we plot the outputs of
the DT system in Eq. (2.45) for input x [ k ], shown in Fig. 2.15(a) and a shifted
version x [ k − 2] of the input, shown in Fig. 2.15(c). The resulting outputs are
plotted, respectively, in Figs. 2.15(b) and (d). As expected, the Fig. 2.15(d) is
not a delayed version of Fig. 2.15(b) since the system is time-variant.