Infinite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 55

X ð z Þ ¼ H ð z Þ¼ P N 1

k ¼ 0 a ð k Þ z k

Y ð z Þ

1 þ P k ¼ 1 b ð k Þ z k :

ð 5 : 2 Þ

It is possible to get frequency response of the filters by substituting e ð jxT S Þ

for

z in Eq. 5.2 . Then for IIR filter holds:
H ð e jxT s Þ¼ H ð jx Þ¼ P N 1

k ¼ 0 a ð k Þ e jkxT s
1 þ P k ¼ 1 b ð k Þ e jkxT s

: ð 5 : 3 Þ

There are three fundamental tasks concerning digital filters. These are: filter
analysis, synthesis and realization. Filter analysis means checking its performance
in time and frequency domains for known coefficients a(k) and b(k). Time domain
filter performance concerns first of all adequately defined impulse response time or
response for unit step (resulting from Eq. 5.1 ). In frequency domain one usually
finds the magnitude and phase frequency responses. Synthesis is just an opposite
process—assuming magnitude frequency response and some additional conditions
concerning for instance phase shift frequency response or time response one
should design the filter, i.e. calculate its coefficients a(k) and b(k). Filter realization
means writing a program transmitting difference Eq. 5.1 to given digital system
and starting its operation.
IIR digital filters are described in time domain with Eq. 5.1 , using transfer
function ( 5.2 ) or frequency response ( 5.3 ). As it was said, filter analysis in time
domain means in general calculating either impulse or step response for known
filter coefficients, whereas frequency domain analysis is understood as calculation
of magnitude and phase shift as functions of frequency. Both calculations are easy
for known filter coefficients and can be realized using handy calculator.
Filter synthesis is a little bit more difficult and means calculation of filter coeffi-
cients for assumed frequency response. There are many methods of synthesis. Most
of them rely on normalized transfer functions of different types of standard filter
approximations (e.g. Butterworth, Bessel, Tschebyshev) allowing to get required
filter features. Next step is frequency transformation allowing to get required fre-
quency response. The final step is transformation from analog to digital filter. Among
several known methods of designing digital IIR filters below two selected ones will
be presented: bilinear transformation and impulse time invariant methods [ 1 , 5 ].

5.2 Synthesis of IIR Filters

5.2.1 Application of Bilinear Transformation

The starting point in filter synthesis is a transfer function of low pass normalized
analog filter prototype. Next step is transformation of that filter to given type and
given cut off frequency according to methods described below. The analog filter