Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 80

recursive procedures. It is possible to find many such procedures but perhaps the
most effective and the simplest result can be obtained using complex notation of a
pair of orthogonal sine, cosine filters. Such representation for the filters given by
Eqs. 6.26 and 6.27 is the following ð cos ð x Þþ j sin ð x Þ¼ exp ð jx ÞÞ :

X 0 :

y ð n Þ¼ y c ð n Þþ jy s ð n Þ¼ X

N 1

N 1
2

x ð n k Þ exp j k ð 6 : 35 Þ

k ¼ 0

Rearranging ( 6.35 ) yields:

X

N 1

j N 1
2

y ð n Þ¼ exp X 0 x ð n k Þ exp ð jkX 0 Þ ð 6 : 36 Þ

k ¼ 0

and writing each component of the filters, output at instant n we obtain:

x ð n Þþ x ð n 1 Þ exp ð jX 0 Þþþ x ð n N þ 1 Þ

j N 1
2

y ð n Þ¼ exp X 0 f

exp j ð N 1 Þ X 0 ½ g: ð 6 : 37 Þ

Doing the same at instant n - 1 one gets:

x ð n 1 Þþ x ð n 2 Þ exp ð jX 0 Þþþ x ð n N Þ

j N 1
2

y ð n 1 Þ¼ exp X 0 f

exp j ð N 1 Þ X 0 ½ g: ð 6 : 38 Þ

Multiplying the last equation by adequately chosen complex function and
subtracting it from Eq. 6.37 one can reduce many components simplifying the final
equation to the form:

x ð n Þ x ð n N Þ exp ð jX 0 Þ

j N 1
2

y ð n Þ y ð n 1 Þ exp ð jX 0 Þ¼ exp X 0 ½ :

ð 6 : 39 Þ

Comparing real and imaginary parts of the equation one can get finally the
recursive algorithm of a pair of orthogonal sine, cosine FIR filters.

Y ð n Þ CY ð n 1 Þþ DX ; ð 6 : 40 Þ

where

; X ¼ ; C ¼ ;

Y ð n Þ¼ y c ð n Þ
y s ð n Þ

x ð n Þ
x ð n N Þ

cos ð X 0 Þ sin ð X 0 Þ

sin ð X 0 Þ cos ð X 0 Þ

cos ð a Þ cos ð b Þ
sin ð a Þ

D ¼

sin ð b Þ

a ¼ 0 : 5 ð N 1 Þ X 0 ; b ¼ 0 : 5 ð N þ 1 Þ X 0 :