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Convolution 41
2
Input Sequence
1.5
x(n)
1
0.5
0
−5 0 5 10 15 20 25 30 35 40 45
n
2
Impulse Response
1.5
h(n)
1
0.5
0
−5 0 5 10 15 20 25 30 35 40 45
n
FIGURE 2.5 The input sequence and the impulse response in Example 2.7
Solution
The input x(n) and the impulse response h(n) are shown in Figure 2.5. From
(2.14)
y(n) =
9∑
9∑
(1) (0.9) (n−k) u(n − k) =(0.9) n (0.9) −k u(n − k) (2.17)
k=0
The sum in (2.17) is almost a geometric series sum except that the term u(n−k)
takes different values depending on n and k. There are three possible conditions
under which u(n − k) can be evaluated.
CASE i n