Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 385

Solution:
For key 7, we have f L ¼ 852 Hz and f H ¼ 1;209 Hz.
a. Using Equation (8.79) , we get

852
8; 000 205 z 22 and

k H ¼ 1; 209 k L ¼

8; 000 205 z 31

2p 22
205

2p 31
205

b. Since 2cos ¼ 1:5623, and 2cos ¼ 1:1631, it follows that

1
1 1:5623z 1 þ z 2

H 22 ðzÞ¼

and

1
1 1:1631z 1 þ z 2

H 31 ðzÞ¼

The DSP equations are therefore given by
v 22 ðnÞ¼1:5623v 22 ðn 1Þv 22 ðn 2ÞþxðnÞ with xð205Þ¼0; for n ¼ 0; 1; / ; 205

v 31 ðnÞ¼1:1631v 31 ðn 1Þv 31 ðn 2ÞþxðnÞ with xð205Þ¼0; for n ¼ 0; 1; / ; 205

c. The amplitude spectral values are determined by
jX ð22Þj 2 ¼ðv 22 ð205ÞÞ 2 þðv 22 ð204ÞÞ 2 1:5623ðv 22 ð205ÞÞ ðv 22 ð204ÞÞ

q

jX ð22Þj 2 2

A 22 ¼

205

and

jX ð31Þj 2 ¼ðv 31 ð205ÞÞ 2 þðv 31 ð204ÞÞ 2 1:1631ðv 31 ð205ÞÞ ðv 31 ð204ÞÞ

q

jX ð31Þj 2 2

A 31 ¼

205

The MATLAB simulation for decoding key 7 is shown in Program 8.21. Figure 8.56 ( a) shows the
frequency responses of the second-order Goertzel bandpass filters. The input is generated as shown in
Figure 8.52 . After filtering, the calculated spectral values and threshold value for decoding key 7 are
displayed in Figure 8.56 ( b), where only two spectral values corresponding to the frequencies of 770 Hz
and 1,209 Hz are above the threshold, and are encoded as logic 1. According to the key information in
the Figure 8.55 , the final logic operation decodes the key as 7.
The principle can easily be extended to transmit the ASCII (American Standard Code for Infor-
mation Interchange) code or other types of code using the parallel Goertzel filter bank. If the calculated
spectral value is larger than the threshold value, then the logic operation outputs logic 1; otherwise,