Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 79

(a)

(b) A ( j Ω )

1

ak

c ()

A ( j Ω )

0

1

k

0

0.8

-1

a

0 N 0 21 − N 0 1 −

0.6

1 a S

ak

s ()

0.4

k

0

0.2

-1 0

0 N 0 21 − N 0 1 −

0 2 4 6 8

Ω Ω

0

Fig. 6.10 Characteristics of full cycle sine/cosine 2nd harmonic filters: a impulse responses
b frequency spectra

j A s ð jX a Þ j sin ½ 5 ð 0 : 25p 0 : 1p Þ

sin ½ 0 : 5 ð 0 : 25p 0 : 1p Þ 0 : 5

sin ½ 5 ð 0 : 25p þ 0 : 1p Þ
sin ½ 0 : 5 ð 0 : 25p þ 0 : 1p Þ

j ¼ 0 : 10 : 5 j

A s ð jX 1 Þ

¼ 0 : 05 ð 4 : 27 1 : 91 Þ¼ 0 : 118 ;

j A c ð jX b Þ j j A s ð jX b Þ j

j ¼ 0 : 207

and

j ¼ 0 : 0535 : j A c ð jX 1 Þ j A s ð jX 1 Þ

(2) Sampling frequency 600 Hz
Number of samples per cycle is N 1 ¼ f S = f 1 ¼ 12 and:

X a ¼ 2pf a = f S ¼ 2p125 = 600 ¼ 0 : 417p ; X b ¼ 0 : 583p ; X 1 ¼ 0 : 167p :

In this case one gets:

j A c ð jX a Þ j j A s ð jX a Þ j

A s ð jX 1 Þ

j ¼ 0 : 323 ; j ¼ 0 : 113 ; j A c ð jX 1 Þ j

j A c ð jX b Þ j j A s ð jX b Þ j

A s ð jX 1 Þ j ¼ 0 : 227 ; j ¼ 0 : 047 : j A c ð jX 1 Þ j

One can see that the cosine and sine filters have different abilities of suppres-
sion of oscillating components (it results also from the filters spectra shown), the
latter ones are more effective in this respect. The sampling frequency has some
(limited) effects on the filters characteristics.

• Recursive algorithm of a pair of orthogonal filters

Algorithms of filters having sine, cosine windows are more complex and require
more calculations and more computational burden than those applying Walsh
functions. This is due to variable coefficients, however, it can be reduced using