Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 357

Note that we can also determine the unit-gain factor K by substituting z ¼ e j0 ¼ 1 in the transfer function
HðzÞ¼ ð z þ 1Þ

ðz aÞ , then finding the DC gain. Set the scale factor to be a reciprocal of the DC gain. This can be easily
done as follows:

z¼1 ¼

z þ 1
z 0:9215

1 þ 1
1 0:9215 ¼ 25:4777

DC gain ¼

Hence, K ¼ 1=25:4777 ¼ 0:03925.

8.7.4 First-Order Highpass Filter Design
Similar to the lowpass filter design, the pole-zero placements for the first-order highpass filters in two
cases are shown in Figure 8.36(a) and 8.36(b).
Formulas for designing highpass filters using the pole-zero placement are listed in the following
equations:
When f c < f s = 4,

a z 1 2 ðf c =f s Þp; good for 0 : 9 r < 1 (8.53)

When f c > f s = 4,
a z ð 1 p þ 2 ðf c =f s ÞpÞ; good for 1 < r 0 : 9 (8.54)

HðzÞ¼ Kðz 1 Þ
ðz aÞ

(8.55)

K ¼ ð 1 þ aÞ
2

(8.56)

(a)

f s /2

0

f

0

f s /2

f c

(b)

f s /2

0

f

0

f s /2

f c

FIGURE 8.36

(a) Pole-zero placement for the first-order highpass filter with f c < f s =4. (b) Pole-zero placement for the first-

order highpass filter with f c > f s =4.