tailieunhanh - Digital Filters Part 7

Tham khảo tài liệu 'digital filters part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | New Design Methods for Two-Dimensional Filters Based on 1D Prototypes and Spectral Transformations 111 where the coefficients b0 b1 b2 a1 a2 may take the values in 74 . Since function 81 can be described by the templates Bf Af corresponding to Bf z1 z2 Af z1 z2 we have Bf b2 B B b1 A B b0 A A Af a2 B B a1 A B A A 82 where denotes matrix convolution. For our filter the templates Bf and Af result of size 9 X 9 . In a the two-lobe filter frequency response is shown. Very Selective Four-Lobe Filter The design of a very selective four-lobe filter in polar coordinates was presented in Matei 2009 b and is briefly reconsidered as follows. Let us consider the radial function Hr ọ 1 p B ọ - p 1 83 where B ọ is a periodic function let B ọ cos 4ọ . We use this function to design a 2D filter with four narrow lobes in the frequency plane. Using trigonometric identities we get Hr ọ 1Ị 1 8p cosọ 2 -8p cosọ 4 84 plotted for ọe -K k in d . This periodic function has the period Q K4 and the shape of a comb filter. In order to control the shape of this function we introduce another parameter k such that the radial function p ọ becomes p ọ k Hr ọ . We get using 70 CC2 F o1 C2 4 2 8p o2o2 2 k i C2 85 4 2 2 4 2 2 2 01 82 81 2 8p s182 82 k 81 82 86 a b c d Fig. 9. a Frequency response of the 2-lobe filter b c frequency response and contour plot for a narrow 4-lobe filter d contour plot of a rotated 4-lobe filter 112 Digital Filters As in the previous example we find the transformation of the same form 79 where Z1 and Z2 are the vectors given by 27 and B A are the 5 X 5 matrices B 8 -1 - 2p 0 4p - 2 0 -1 - 2p 0 8 0 8 0 4p - 2 0 -8p - 20 0 4p - 2 0 8 0 8 0 -1 - 2p 0 4p - 2 0 -1 - 2p 1 0 1 1 2 1 A k 0 -4 0 2 4 2 .1 0 1_ . 1 2 1_ k A1 A2 87 Using the prototype 74 we get a transfer function H z1 z2 similar to 81 and the templates result from 82 . The designed filter has the frequency response and contour plot as in Fig. 9 b c . We remark that the filter is very selective .