Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 85

Fig. 6.11 Impulse a and
frequency b responses of
low-pass FIR filter designed
with use of complex Fourier
series method; (1) ideal filter,
(2) real filter, (3) response
after application of
smoothing Hanning window
(Example 6.6)

(a)

0.4

h ( n )

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1

0 5 10 15 20

n

(b)

1.2

H ( j ω )

1

1

2

3

0.8

0.6

0.4

0.2

0

0 100 200 300 400 500

ω

Solution According to ( 6.42 ) one can write:

Z Z

X 1 X 2

h ð n Þ¼ 1
2p

exp ½ jX ð n k Þ dX þ 1
2p

exp ½ jX ð n k Þ dX

X 2 X 1

and after calculation of integrals and rearrangements, taking as before the group
delay equal 0 : 5 ð N 1 Þ (symmetry conditions), one obtains:

1
p ½ n 0 : 5 ð N 1 Þ f sin ½ X 2 ð n 0 : 5 ð N 1 ÞÞ sin ½ X ð n 0 : 5 ð N 1 ÞÞg:

h ð n Þ¼

For the specified filter parameters one can calculate:

0 : 5 ð N 1 Þ¼ 9 : 5; X 1 ¼ 2pf 1 = f S ¼ 2p50 = 1000 ¼ 0 : 1p; X 2 ¼ 0 : 3p