Finite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 83

H ð X Þ¼ X

N 1

h i ð n Þ exp ð jnX Þ: ð 6 : 44 Þ

n ¼ 0

From ( 6.44 ) one cannot find the relation between frequency responses of ideal
and real filter. To get that one should use general Laplace and Fourier transform
relationships. A product of two functions in time domain results in the convolution
of their transforms in frequency domain (and the other way round):

Z

p

H ð X Þ¼ H i ð X Þ W ð X Þ¼ 1
2p

H i ð X 0 Þ W ð X X 0 Þ dX 0 ;

ð 6 : 45 Þ

p

where H i ð X Þ; W ð X Þ are frequency responses of ideal filter and rectangular win-
dow, respectively.
It may be concluded that frequency responses of real filters obtained in such a
way will have oscillatory bands (ripples) in both pass and rejection regions. It
results from the known frequency response of rectangular window (zero order
Walsh function presented before). This oscillation has the biggest peak values in
the region of transition period of the filter characteristic. The phenomenon is
known as the Gibbs effect. To limit the effect we can use different than rectangular
cutting windows, designed to get small values of coefficients at both window ends.
Known examples of such windows are: Blackman, Bartlett, Hamming, Hanning,
Kaiser, etc. Their impulse responses of some of them are following [ 5 , 7 ]:

• Hanning window

;

2pn
N 1

h ð n Þ¼ 0 : 51 cos for 0 n N 1 ;

• Hamming window

;

2pn
N 1

h ð n Þ¼ 0 : 54 0 : 46 cos for 0 n N 1 ;

• Blackman window

þ 0 : 08 cos ;

2pn
N 1

4pn
N 1

h ð n Þ¼ 0 : 42 0 : 5 cos for 0 n N 1 :

All these windows decrease oscillations of frequency responses of the filters
both at pass and stop bands, however, they increase the transition band. These
effects are presented in Fig. 6.11 for an example of low-pass filter and application
of rectangular window and the other windows as above. It is seen that the noise at
stop band is decreased more effectively (in the case of Blackman window even by
80 dB), which is an evident advantage, however, the transition period becomes