Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 90

The procedure of FIR filter design using sampling in frequency domain is the
following:

• assume given continuous frequency response of the digital filter,
• assume sampling frequency f S and number of filter coefficients N (according to
general rules of the sampling theorem),
• present the filter frequency response in the range of frequency from zero to
sampling frequency and realize the sampling in frequency domain (Df ¼ f S = N),
• use the values to calculate discrete inverse Fourier transform in order to receive
filter coefficients (impulse response),
• check the resulting continuous frequency response and in case of high oscilla-
tions either in pass or in stop band choose and introduce special smoothing
window.

Example 6.9 This course of design is illustrated with an example of low-pass filter
synthesis. It is assumed that the filter should have unity magnitude of frequency
response from zero to 200 Hz and zero gain for other frequencies. Further
parameters are: N ¼ 20 and f S ¼ 1000 Hz. Thus the assumed digitalized frequency
response of the filter is as shown in Fig. 6.14 , which can also be expressed by:

1
0

k ¼ 0 ; 1 ; 2 ; 3 ; 4 and 16 ; 17 ; 18 ; 19 ;
other values of k ;

j H ð k Þ j ¼ for

arg ½ H ð k Þ ¼
exp ½ jkp ð N 1 Þ= N ¼ exp ð jpk19 = 20 Þ
exp ½ j ð N k Þ p ð N 1 Þ= N ¼ exp ½ j ð 20 k Þ p19 = 20

k ¼ 0 ; 1 ; 2 ; 3 ; 4
k ¼ 16 ; 17 ; 18 ; 19 :

for

In the next step inverse digital Fourier transform is performed, which yields
coefficients of designed FIR filter h(n), shown in Fig. 6.15 a. The filter can now be
realized and its frequency response calculated. The resulting frequency charac-
teristic is presented in Fig. 6.15 b, compared with the one obtained after application
of

a smoothing window that helps to remove the ripples in original filter

characteristic.

Example 6.10 Applying the method of sampling in frequency domain design the
low-pass linear phase FIR filter with unity gain between 50 and 150 Hz, zero
outside of this range. Assume sampling frequency 1000 Hz and number of filter
coefficients N = 20.

Solution From the specified parameters it results that the distance between con-
secutive discrete frequency points of the filter frequency response equals:
Df k ¼ f S = N ¼ 1000 = 20 ¼ 50 Hz :

The filter spectrum is symmetrical with respect to the Nyquist frequency and in
the range between 0 and 950 Hz there exist 20 points of this spectrum, given as
follows: