Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 78

We can see that the calculated results using all the methods are consistent. The steps using the table method
are summarized in Table 3.5 .

Table 3.5 Digital Convolution Steps via the Table Method

Step 1. List the index k covering a sufficient range.

Step 2. List the input .

Step 3. Obtain the reversed sequence

xðkÞ

hðkÞ , and align the rightmost element of hðn kÞ to the leftmost

element of .

Step 4. Cross-multiply and sum the nonzero overlap terms to produce

xðkÞ

yðnÞ .

Step 5. Slide to the right by one position.

Step 6. Repeat Step 4; stop if all the output values are zero or if required.

hðn kÞ

EXAMPLE 3.12
Convolve the following two rectangular sequences using the table method:

<
:

0 n ¼ 0

( 1

n ¼ 0; 1; 2

xðnÞ¼ and hðnÞ¼ 1 n ¼ 1; 2

0 otherwise

0 otherwise

Solution:
Using Table 3.5 as a guide, we list the operations and calculations in Table 3.6 . Note that the output should show
the trapezoidal shape.

Table 3.6 Convolution Sum in Example 3.12
k :

L 2 L 1012345

111

xðkÞ :

1 1 0

hðkÞ : yð 0 Þ¼ 0 (no overlap)

1 1 0

hð 1 kÞ yð 1 Þ¼ 1 1 ¼ 1

110

hð 2 kÞ yð 2 Þ¼ 1 1 þ 1 1 ¼ 2

110

hð 3 kÞ yð 3 Þ¼ 1 1 þ 1 1 ¼ 2

110

hð4 kÞ yð4Þ¼1 1 ¼ 1

110yðnÞ¼ 0 , n 5 (no overlap)
Stop

hðn kÞ