Infinite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 59

(a) (b)

1 0.3

h ()

H ( jf )

d

0.9

0.25

0.8

0.2

0.7

0.6

0.15

0.5

0.1

0.4

0.3

0.05

0.2

0

0.1

f

dC

0 -0.05

f d n

0 100 200 300 400 500 0 5 10 15 20

Fig. 5.2 Impulse response a and frequency response b of digital filter from Example 5.1

As a result the transfer function of digital filter equals:

1

H ð z Þ¼

2 þ 2 þ 1

p

3 : 078 1 z 1
1 þ z 1

3 : 078 1 z 1
1 þ z 1

and after simple rearrangements:

H ð z Þ¼ 0 : 067 1 þ 2z 1 þ z 2
1 1 : 143z 1 þ 0 : 412z 2
Therefore the filter difference equation has the form:

y ð n Þ¼ 0 : 067x ð n Þþ 0 : 134x ð n 1 Þþ 0 : 067x ð n 2 Þþ 1 : 143y ð n 1 Þ 0 : 412y ð n 2 Þ

The impulse response and frequency characteristics of designed filter are shown
in Fig. 5.2 .

Example 5.2 Design a bandpass IIR filter, having cutoff frequencies 50 and
150 Hz. Assume sampling rate 1000 Hz and analog prototype approximation of
Butterworth II order.

Solution According to ( 5.16 - 5.18 ) one should substitute for s in the filter transfer
function

1
s 2 þ 2

H ð s Þ¼

p

s þ 1

appropriate form, with the constant coefficients as follows:

B ¼ ctg ½ 0 : 5 ð x dCH x dCL Þ T S Þ¼ ctg ð 0 : 5 ð 150 50 Þ 2p 0 : 001 Þ¼ 3 : 08 ;
a ¼ cos ½ 0 : 5 ð x dCH þ x dCL Þ T S Þ

cos ½ 0 : 5 ð x dCH x dCL Þ T S Þ ¼ cos ð 0 : 5 ð 150 þ 50 Þ 2p 0 : 001 Þ cos ð 0 : 5 ð 150 50 Þ 2p 0 : 001 Þ ¼ 0 : 85 :