Infinite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 58

:

1
x dCH x dCL

s 1 þ x dCH x dCL
s 1

s ) ð 5 : 16 Þ

Then one should apply bilinear transformation:

s 1 ) A 1 z 1

1 þ z 1 :

ð 5 : 17 Þ

These two steps can be joined, giving the required substitution:

s ) B 1 2az 1 þ z 2
1 z 1

; ð 5 : 18 Þ

where

;

ð x dCH x dCL Þ T S
2

a ¼ cos ½ 0 : 5 ð x dCH þ x dCL Þ B ¼ ctg

cos ½ 0 : 5 ð x dCH x dCL Þ :

Synthesis of a band rejection digital IIR filter

Transfer function of digital band rejection digital IIR filter one obtains using the
following substitution:

1 z 2
1 2az 1 þ z 2 ;

s ) C ð 5 : 19 Þ

where

:

ð x dCH x dCL Þ T S
2

C ¼ tg ð 5 : 20 Þ

Example 5.1 Applying the bilinear transformation method that perform synthesis
of a lowpass IIF filter, having the cutoff frequency x dC ¼ 628 ; taking the
Butterworth II order low-pass approximation ð x aC ¼ 1 Þ as analog filter prototype.

1
s 2 þ 2

H ð s Þ¼ p

s þ 1

Assume sampling frequency f S ¼ 1 = T S ¼ 1000 Hz :
Solution The bilinear transformation method allows easy obtaining of the IIR
filter transfer function and its difference equation. First one calculates the constant
coefficient of bilinear transformation. For a lowpass digital filter this factor
amounts to (according to ( 5.9 )):

ctg ð 0 : 5x dC T S Þ¼ ctg ð 0 : 5 628 0 : 001 Þ¼ 3 : 078

and in transfer function of analog filter one substitutes for s the expression:

s ) 3 : 078 1 z 1

1 þ z 1 :