Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 67

The plus sign in ( 6.8 ) appears for even symmetry of filter coefficients and
minus—for odd symmetry. In the first case one obtains:

X X X ¼ H e ð jX Þ

N = 2 1

N 1
2

N 1
2

H ð jX Þ¼ 2 exp j a ð k Þ cos k

k ¼ 0

ð 6 : 9 Þ

and in the second case:

X X X ¼ H o ð jX Þ:

N = 2 1

N 1
2

N 1
2

H ð jX Þ¼ 2j exp j a ð k Þ sin

k

k ¼ 0

ð 6 : 10 Þ

Since in both cases the sums are real, the filter arguments are given by:

¼ N 1
2

arg H e ð jX Þ ½ X ; ð 6 : 11 Þ

¼ N 1
2

X þ p arg H o ð jX Þ ½

2 :

ð 6 : 12 Þ

Concluding, one can stress that if the FIR filters have even and odd impulse
responses, respectively, their transfers function have linear phase and are
orthogonal, i.e. the difference of their phases equals p = 2. The filters applied in
protection systems should have that features, which may be very helpful in sim-
plifying substantially development of algorithms for criterion values measurement.

6.2 Analysis of Standard FIR Filters

6.2.1 Filters with Walsh Windows

The first five so called Walsh windows are shown in Fig. 6.1 . Their impulse
responses have coefficients being equal either plus or minus one [ 1 , 3 , 7 ]. This
feature gives the simplest possible equations of filter realization, which is

important in fast protection systems.
Digital FIR filter realized applying Walsh function of zero order, called also
rectangular window, produces output signal being the sum of N most recent
samples of the input signal, which is described by very simple equation:

y ð n Þ¼ X

N 1

x ð n k Þ: ð 6 : 13 Þ

k ¼ 0