Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 77

y ()

10

5

n

0 1 2 3 4 5

FIGURE 3.23

Plot of the convolution sum in Example 3.11.

sum of product of xðkÞ and hðkÞ: yð0Þ¼3 3 ¼ 9
sum of product of xðkÞ and hð1 kÞ: yð1Þ¼1 3 þ 3 2 ¼ 9
sum of product of xðkÞ and hð2 kÞ: yð2Þ¼2 3 þ 1 2 þ 3 1 ¼ 11
sum of product of xðkÞ and hð3 kÞ: yð3Þ¼2 2 þ 1 1 ¼ 5
sum of product of xðkÞ and hð4 kÞ: yð4Þ¼2 1 ¼ 2
sum of product of xðkÞ and hð5 kÞ: yðnÞ¼0 for n > 4, since sequences xðkÞ and hðn kÞ do not overlap.

Finally, we sketch the output sequence yðnÞ in Figure 3.23 .
b. Applying Equation (3.20) with zero initial conditions leads to
yðnÞ¼xð0ÞhðnÞþxð1Þhðn 1Þþxð2Þhðn 2Þ
n ¼ 0, yð0Þ¼xð0Þhð0Þþxð1Þhð1Þþxð2Þhð2Þ¼3 3 þ 1 0 þ 2 0 ¼ 9
n ¼ 1, yð1Þ¼xð0Þhð1Þþxð1Þhð0Þþxð2Þhð1Þ¼3 2 þ 1 3 þ 2 0 ¼ 9
n ¼ 2, yð2Þ¼xð0Þhð2Þþxð1Þhð1Þþxð2Þhð0Þ¼3 1 þ 1 2 þ 2 3 ¼ 11
n ¼ 3, yð3Þ¼xð0Þhð3Þþxð1Þhð2Þþxð2Þhð1Þ¼3 0 þ 1 1 þ 2 2 ¼ 5
n ¼ 4, yð4Þ¼xð0Þhð4Þþxð1Þhð3Þþxð2Þhð2Þ¼3 0 þ 1 0 þ 2 1 ¼ 2
n 5, yðnÞ¼xð0ÞhðnÞþxð1Þhðn 1Þþxð2Þhðn 2Þ¼3 0 þ 1 0 þ 2 0 ¼ 0

In simple cases such as this example, it is not necessary to use the graphical or formula methods. We can compute
the convolution by treating the input sequence and impulse response as number sequences and sliding the
reversed impulse response past the input sequence, cross-multiplying, and summing the nonzero overlap terms at
each step. The procedure and calculated results are listed in Table 3.4 .

Table 3.4 Convolution Sum Using the Table Method
k

:

L 2 L 1

0 1 2 3 4 5

3 1 2

xðkÞ :

1 2 3

hðkÞ : yð 0 Þ¼ 3 3 ¼ 9

1 2 3

hð 1 kÞ yð 1 Þ¼ 3 2 þ 1 3 ¼ 9

1 2 3

hð 2 kÞ yð 2 Þ¼ 3 1 þ 1 2 þ 2 3 ¼ 11

1 2 3

hð 3 kÞ yð 3 Þ¼ 1 1 þ 2 2 ¼ 5

1 2 3

hð 4 kÞ yð 4 Þ¼ 2 1 ¼ 2

1 2 3

hð5 kÞ yð5Þ¼0 (no overlap)