Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 81

The recursive procedure can be applied to the filters having different parame-
ters. They influence matrices and constants of Eq. 6.40 and can be calculated
substituting:

• for full cycle filters: N ¼ N 0 ; X 0 ¼ 2p = N 0 ; a ¼ p X 0 = 2; b ¼ p þ X 0 = 2;
• for

half cycle filters: N ¼ N 0 = 2; X 0 ¼ 2p = N 0 ; a ¼ p = 2 X 0 = 2; b ¼ p = 2 þ

X 0 = 2;
• for two cycle filters: N ¼ 2N 0 ; X 0 ¼ 2p = N 0 ; a ¼ X 0 = 2; b ¼þ X 0 = 2;
• for half cycle filter having two times greater natural frequency:

N ¼ N 0 = 2; X 0 ¼ 4p = N 0 ; a ¼ p X 0 = 2; b ¼ p þ X 0 = 2 :

Example 6.5 Derive recursive equations of the full cycle sine, cosine filters.
Assume sampling frequency 1000 Hz and fundamental frequency 50 Hz.

Solution Number of samples per cycle N 1 ¼ f S = f 1 ¼ 20 and X 0 ¼ X 1 ¼ 0 : 1p
The matrices C and D of Eq. 6.40 are then equal:

¼ ;

cos ð 0 : 1p Þ sin ð 0 : 1p Þ 0 : 951 0 : 309

C ¼

sin ð 0 : 1p Þ cos ð 0 : 1p Þ 0 : 309 0 : 951

¼

cos ð a Þ cos ð b Þ
sin ð a Þ

cos ð 0 : 05p Þ cos ð 0 : 05p Þ
sin ð 0 : 05p Þ; sin ð 0 : 05p Þ

D ¼

cos ð b Þ

;

0 : 988 0 : 988
0 : 156 0 : 156

¼

where a ¼ p X 0 = 2 ¼ 2 ¼ p 0 : 05p; b ¼ p þ X 0 = 2 ¼ p þ 0 : 05p :
Substituting them to Eq. 6.40 , after rearrangement, the following recursive
equations are reached:

y c ð n Þ¼ 0 : 951y c ð n 1 Þþ 0 : 309y s ð n 1 Þþ 0 : 988 ½ x ð n 20 Þ x ð n Þ;

y s ð n Þ¼ 0 : 309y c ð n 1 Þþ 0 : 951y s ð n 1 Þ 0 : 156 ½ x ð n 20 Þ x ð n Þ:
The above formulae can be directly used for practical implementation and
realization of filtration. Analogously one can also derive recursive procedures for
other sampling frequencies and/or other window lengths.

6.3 Synthesis of FIR Filters

FIR filter analysis was presented in the preceding section. Time and frequency
domain characteristics of the orthogonal filters applied in power system protection
were discussed. Some general conditions allowing to get orthogonal filters were
also given. Among many filters basing on sine, cosine and Walsh windows it is
possible to choose in almost all cases an adequate one for given application.
However, sometimes special requirements concerning for instance the shape or