Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 354

Compute the unit-gain scale factor as
K ¼ ð1 0:9215Þ

p

1 2 0:9215 cos2 45 þ 0:9215 2

¼ 0:0755

2jsin 45 j

Finally, the transfer function is given by

0:0755 z 2 1
z 2 2 0:9215zcos 45 þ 0:9215 2 ¼

0:0755 0:0755z 2
1 1:3031z 1 þ 0:8491z 2

HðzÞ¼

8.7.2 Second-Order Bandstop (Notch) Filter Design
For this type of filter, the pole placement is the same as the bandpass filter ( Figure 8.33 ) . The zeros are
placed on the unit circle with the same angles with respect to poles. This will improve passband
performance. The magnitude and the angle of the complex conjugate poles determine the 3 dB
bandwidth and center frequency, respectively.
Design formulas for bandstop filters are given in the following equations:

r z 1 ðBW 3 dB =f s Þp; good for 0 : 9 r < 1 (8.45)

f 0
f s

360 q ¼ (8.46)

HðzÞ¼ K z e jq z þ e jq

ðz re jq Þðz re jq Þ ¼ K z

2 z cos q þ 1

2

(8.47)

ðz

2

2 rz cos q þ r

2

Þ

The scale factor to adjust the bandstop filter so it has a unit passband gain is given by

1 2 r cos q þ r

2

K ¼ (8.48)

ð 2 2 cos qÞ

r

f s /2

0

f

0

f 0 f s /2

FIGURE 8.33
Pole-zero placement for a second-order notch filter.