UploadFile_6417

42 Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS
8
Output Sequence
6
y(n)
4
2
0
−5 0 5 10 15 20 25 30 35 40 45
n
FIGURE 2.6 The output sequence in Example 2.7
CASE iii
In this case the impulse response h(n) partially overlaps the input x(n).
n ≥ 9: Then u(n − k) =1, 0 ≤ k ≤ 9 and from (2.17)
9∑
y(n) =(0.9) n (0.9) −k
k=0
=(0.9) n 1 − (0.9) −10
1 − (0.9) = −1 10(0.9)n−9 [1 − (0.9) 10 ], n ≥ 9 (2.20)
In this last case h(n) completely overlaps x(n).
The complete response is given by (2.18), (2.19), and (2.20). It is shown in
Figure 2.6 which depicts the distortion of the input pulse.
□
This example can also be done using a method called graphical convolution,
in which (2.14) is given a graphical interpretation. In this method,
h(n − k) isinterpreted as a folded-and-shifted version of h(k). The output
y(n) isobtained as a sample sum under the overlap of x(k) and h(n − k).
We use an example to illustrate this.
□ EXAMPLE 2.8 Given the following two sequences
x(n) =[3, 11, 7, 0, −1, 4, 2], −3 ≤ n ≤ 3; h(n) =[2, 3, 0, −5, 2, 1], −1 ≤ n ≤ 4
↑
↑
determine the convolution y(n) =x(n) ∗ h(n).
Solution
In Figure 2.7 we show four plots. The top-left plot shows x(k) and h(k), the
original sequences. The top-right plot shows x(k) and h(−k), the folded version
of h(k). The bottom-left plot shows x(k) and h(−1 − k), the folded-and-shiftedby-
−1 version of h(k). Then
∑
x(k)h(−1 − k) =3× (−5)+11× 0+7× 3+0× 2=6=y(−1)
k
The bottom-right plot shows x(k) and h(2 − k), the folded-and-shifted-by-2
version of h(k), which gives
∑
x(k)h(2−k) =11×1+7×2+0×(−5)+(−1)×0+4×3+2×2 =41=y(2)
k
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.