Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 365

Design method: Pole-zero placement

Bandpass filter specifications:
Passband frequency range: 0.25-40 Hz

Passband ripple: 0.5 dB

Filter type: Chebyshev fourth order

Design method: Bilinear transformation method

DSP sampling rate: 600 Hz

Let us carry out the 60-Hz eliminator design and determine the transfer function and difference
equation for each notch filter and bandpass filter. For the first section with the notch frequency of 60
Hz, applying Equations (8.45) to (8.48) leads to

r ¼ 1 ð 4 = 600 Þp ¼ 0 : 9791

60
600

360 ¼ 36

q ¼

We calculate 2 cos ð 36 Þ¼ 1 : 6180, 2 r cos ð 36 Þ¼ 1 : 5842, and

1 2 r cos q þ r

2

K ¼ ¼ 0 : 9803

ð 2 2 cos qÞ

Hence it follows that

0 : 9803 1 : 5862 z 1 þ 0 : 9803 z 2

H 1 ðzÞ¼

1 1 : 5842 z 1 þ 0 : 9586 z 2

y 1 ðnÞ¼ 0 : 9803 xðnÞ 1 : 5862 xðn 1 Þþ 0 : 9802 xðn 2 Þþ 1 : 5842 y 1 ðn 1 Þ 0 : 9586 y 1 ðn 2 Þ
Similarly, we obtain the transfer functions and difference equations for the second section and third
section as follows:
Second section:

0 : 9794 0 : 6053 z 1 þ 0 : 9794 z 2

H 2 ðzÞ¼

1 0 : 6051 z 1 þ 0 : 9586 z 2

y 2 ðnÞ¼ 0 : 9794 y 1 ðnÞ 0 : 6053 y 1 ðn 1 Þþ 0 : 9794 y 1 ðn 2 Þþ 0 : 6051 y 2 ðn 1 Þ 0 : 9586 y 2 ðn 2 Þ

Third section:

0 : 9793 þ 0 : 6052 z 1 þ 0 : 9793 z 2

H 3 ðzÞ¼

1 þ 0 : 6051 z 1 þ 0 : 9586 z 2

y 3 ðnÞ¼ 0 : 9793 y 2 ðnÞþ 0 : 6052 y 2 ðn 1 Þþ 0 : 9793 y 2 ðn 2 Þ 0 : 6051 y 3 ðn 1 Þ 0 : 9586 y 3 ðn 2 Þ

The cascaded frequency responses are plotted in Figure 8.44 . As we can see, the rejection for each
notch frequency is below 50 dB.