Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

23-18 Passive, Active, and Digital FiltersH S (z 1 , z 2 ) ¼ H A (z 1 , z 2 ) [H SL (z 1 , z 2 )] 2(b) Decomposition of H 24 (z 1 , z 2 ): Similarly, H 24 (z 1 , z 2 ) can be decomposed into the cascade of atwo quadrant filter G 24 (z 1 , z 2 ) [as is shown in Figure 23.7b], and a bandpass filter H B (z 1 , z 2 )which can be realized by using two 1-D bandpass filters for both directions.H 24 (z 1 , z 2 ) ¼ G 24 (z 1 , z 2 )H B (z 1 , z 2 )The final configuration of the desired filter H(z 1 , z 2 ) is illustrated in Figure 23.12d, with thefinal transfer function being of the formH(z 1 , z 2 ) ¼ H 13 (z 1 , z 2 ) þ H 24 (z 1 , z 2 ) H 13 (z 1 , z 2 )H 24 (z 1 , z 2 )where the purpose of the term H 13 (z 1 , z 2 )H 24 (z 1 , z 2 ), is to remove the overlap that may becreated by adding H 13 (z 1 , z 2 ) and H 24 (z 1 , z 2 ).(2) Design of all the subfilters: At this point, the problem is to derive the two quadrant subfiltersG 13 (z 1 , z 2 ) and G 24 (z 1 , z 2 ), the low-pass subfilters H L (z 1 , z 2 ) and H SL (z 1 , z 2 ), the allpass subfilterH A (z 1 , z 2 ), and the bandstop filter H B (z 1 , z 2 ).Note the symmetry of the given characteristics, the identical 1-D sections can be used todevelop all the above 2-D subfilters, and the given specifications can easily be combined intothe designs of all the 1-D sections.(3) By connecting all the 2-D subfilters in cascade or parallel as specified in Figure 23.12d, therequired 2-D filter is obtained. The 3-D plot of the amplitude response of the final resulting 2-D filter is depicted in Figure 23.12e.23.4 Design of Circularly Symmetric Filters23.4.1 Design of LP FiltersAs mentioned in Section 23.2.1, rotated filters can be used to design circularly symmetric filters. Costa andVenetsanopoulos [14] and Goodman [15] proposed two methods of this class, based on transforming ananalog transfer function or a discrete one by rotated filter transformation, respectively. The two methodslead to filters that are, theoretically, unstable but by using an alternative transformation suggested byMendonca et al. [16], this problem can be eliminated.23.4.1.1 Design Based on 1-D Analog Transfer FunctionCosta and Venetsanopoulos [14] proposed a method to design circularly symmetric filters. In their method,a set of 2-D analog transfer functions is first obtained by applying the rotated filter transformation inEquation 23.9a for several different values of the rotation angle b to a 1-D analog low-pass transferfunction. A set of 2-D discrete low-pass functions are then deduced through the application of the bilineartransformation. The design is completed by cascading the set of 2-D digital filters obtained. The stepsinvolved are as follows.Step 1. Obtain a stable 1-D analog low-pass transfer functionH Al (s) ¼ Ns Q MDs ¼ K i¼1 (s0 Q Ni¼1 (sz ai)p ai)(23:50)wherez ai and p ai for i ¼ 1, 2, . . . , are the zeros and poles of H A1 (s), respectivelyK 0 is a multiplier constant