Infinite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 63

therefore sought partial expansion of the filter transfer function is:

:

H ð s Þ¼ x dC
j

1
s s 1

1
s s 2

p 2

Thus the transfer function of designed digital filter amounts (according to
( 5.27 )):

:

H ð z Þ¼ x dC
j

1
1 z 1 exp ð s 1 T S Þ

1
1 z 1 exp ð s 2 T S Þ

p 2

Introducing the values of poles, one obtains after simple rearrangements:

p 2

x dC exp ð b Þ sin ð b Þ z 1
1 2z 1 exp ð b Þ cos ð b Þþ z 2 exp ð 2b Þ ;

H ð z Þ¼

where

b ¼ x dC T S

p : 2

Finally, for assumed cut-off and sampling frequencies, one gets:

H ðÞ¼ 244 : 92z 1

1 1 : 158z 1 þ 0 : 412z 2 :
From the above results that filter gains for low frequencies are approximately
equal to the value of sampling frequency. In order to get filter gain equal to unity,
one can rescale the filter, multiplying its transfer function by the sampling period,
which gives:

H ð z Þ T S ¼ H 1 ð z Þ¼ 0 : 244z 1

1 1 : 158z 1 þ 0 : 412z 2 :
From the above one can derive sought difference equation of digital filter:

y ð n Þ¼ 0 : 244xn 1 ð Þ 1 : 158yn 1 ð Þ 0 : 412yn 2 ð Þ:

The impulse and frequency responses of designed filter are presented in
Fig. 5.5 . It is clear that the assumed value of sampling frequency is too low, having
significant effects in the rejection region (overlapping of replicas of analog filter
frequency responses). One can reduce this effect selecting higher value of sam-
pling frequency. The results for f S = 4000 Hz can be seen in Fig. 5.6 . It becomes
evident that with higher sampling rate the resulting frequency response of designed
digital filter are better and better, closer to the analog prototype characteristics.
The above examples show clearly that bilinear transformation gives better
results than impulse invariance method. The first one is simpler and more accurate
as far as analog origin and digital realization are concerned.