Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 73

Fig. 6.5 Normalized
frequency responses of the
remaining two Walsh
functions from Fig. 6.1

W ( j Ω )

W ( j Ω )

1

N

w 3

0.8

w 4

0.6

0.4

0.2

0

0 1 2 3 4 5

Ω Ω

N

selectivity and transient region, however the filters have the highest gain for the
frequencies two times higher than the filters analyzed before. They could be
considered as orthogonal filters of the second harmonic of the power system
nominal frequency.
Application of Walsh functions to orthogonal FIR filters can be concluded in
the following way. The most important seem to be band-pass filters applying 1st
and 2nd order Walsh functions. In such a case the filter window lengths N are
equal to number of samples N 1 in one period of fundamental component and that is
why the filters are called full cycle filters. Taking into account frequency responses
obtained before, one can get for such filters:

2
sin ð p = N 1 Þ ;

j W 1 ð jX 1 Þ j ¼ W 2 ð jX 1 Þ j j ¼ ð 6 : 24a Þ

¼ p

2 N 1 1

X 1 ¼ p

2 þ p

arg W 1 ð jX 1 Þ ½

N 1 ;

ð 6 : 24b Þ

2

¼ N 1 1
2

X 1 ¼ p þ p arg W 2 ð jX 1 Þ

½

N 1 :

ð 6 : 24c Þ

Another important case is application of Walsh functions in the filters having
''half period window''. This is the case when N ¼ N 1 = 2 ; i.e. filter window length is
equal to number of samples within half period of fundamental frequency com-
ponent. Similarly as before one can calculate filter magnitudes and phase shifts,
remembering that now Walsh functions of the zero and first order are applied (as a
result of shortening of the window). Calculated values are following:

1
sin ð p = N 1 Þ ;

j W 0 ð jX 1 Þ j ¼ W 1 ð jX 1 Þ j j ¼ ð 6 : 25a Þ

¼ N 1 = 2 1
2

X ¼ p

2 þ p

arg W 0 ð jX 1 Þ ½

N 1 ;

ð 6 : 25b Þ