Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 86

and after substituting to the impulse response equation h(n) one gets the final form:

1
p ð n 9 : 5 Þ

h ð n Þ¼ f sin 0 : 3p ð n 9 : 5 Þ ½ sin 0 : 1p ð n 9 : 5 Þ ½ g:

The filter impulse response, shown in Fig. 6.12 a, is symmetrical, thus the
designed filter has linear phase. From the filter frequency response (Fig. 6.12 b)
calculated with use of FFT it is seen that designed filter is quite good band-pass
filter. Narrowing its pass band to the range 75-125 Hz (separate design performed)
results in increase of ripples of the frequency characteristics (Fig. 6.12 c). One can
counteract this by increasing the number of filter coefficients (longer impulse
response). In Fig. 6.12 d the new characteristic is shown for N = 100, representing
very good band-pass filter with steep transition bands. Almost ideal results are
obtained after superimposing the smoothing Hanning window (Fig. 6.12 e).

Example 6.8 Design high-pass FIR filter with the cut-off frequency 150 Hz,
applying the complex Fourier series method. Assume sampling frequency
1000 Hz. Determine filter frequency response for the number of coefficients
N = 20 and 100.

Solution The high-pass digital filter can be treated as a band-pass one with the
higher cut-off frequency equal to half of sampling rate. According to Eq. 6.42 and
preceding example designed with the same approach, one can write now:

Z Z

X p = 2

X

h ð n Þ¼ 1
2p

exp ½ jX ð n ka Þ dX þ 1
2p

exp ½ jX ð n k Þ dX ;

X p = 2 X 0

where

k ¼ N 1
N

; X 0 ¼ 2p f 0

X S

2 ¼ 2p 0 : 5f S f S ;

¼ p :

f S

Finally, one obtains:

1
p ½ n 0 : 5 ð n 1 Þ f sin ½ p ð n 0 : 5 ð N 1 Þ sin ½ X 0 ð n 0 : 5 ð N 1 Þg:

h ð n Þ¼

The filter impulse response for N = 20 is shown in Fig. 6.13 a, while its fre-
quency response is presented in Fig. 6.13 b. Superimposing the smoothing window
one gets the resulting filter spectrum as seen in Fig. 6.13 c. The steep slope of the
transition band can be improved (increased) taking more filter coefficients
(Fig. 6.13 d). The final filter characteristic with N = 100 and additional smoothing
window is shown in Fig. 6.13 e. One should understand, however, that taking
higher number of N results in longer transient response of the filter in time domain.
One can shorten it (by the same number of coefficients) when higher sampling
frequency is selected, if possible.