Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 355

EXAMPLE 8.18
A second-order notch filter is required to satisfy the following specifications:
• Sampling rate ¼ 8,000 Hz
• 3 dB bandwidth: BW ¼ 100 Hz
• Narrow passband centered at f 0 ¼ 1; 500 Hz
Find the transfer function using the pole-zero placement approach.
Solution:
We first calculate the required magnitude of the poles
r z 1 ð100=8; 000Þp ¼ 0:9607

which is a good approximation. We use the center frequency to obtain the angle of the pole location:

1; 500
8; 000

360 ¼ 67:5 q ¼

The unit-gain scale factor is calculated as

1 2 0:9607 cos 67:5 þ 0:9607 2
ð2 2 cos 67:5 Þ

K ¼ ¼ 0:9620

Finally, we obtain the transfer function:

0:9620 z 2 2z cos 67:5 þ 1

z 2 2 0:9607z cos 67:5 þ 0:9607 2 ¼ 0:9620 0:7363 z 1 þ 0:9620 z 2

HðzÞ¼

1 0:7353z 1 þ 0:9229

8.7.3 First-Order Lowpass Filter Design
The first-order pole-zero placement can be utilized in two cases. The first situation is when the cutoff
frequency is less than f s = 4. Then the pole-zero placement is shown in Figure 8.34 .
As shown in Figure 8.34 , the pole z ¼ a is placed in the real axis. The zero is placed at z ¼ 1to
ensure zero gain at the folding frequency (Nyquist limit). When the cutoff frequency is above f s = 4, the
pole-zero placement is adopted as shown in Figure 8.35 .
Design formulas for lowpass filters using the pole-zero placement are given in the following
equations.

f s /2

0

f

0

f s /2

f c

FIGURE 8.34
Pole-zero placement for the first-order lowpass filter with

f c < f s =4.