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

Two-Part Synthesis by Ladder Development 8-11N αNN βFIGURE 8.13Parallel connection of two ladder networks.hand, provides a conduit for multiple path signal transmission, so that the signals arriving at the outputthrough the various paths may cancel one another, resulting in the zero output for a finite input.Therefore, they are capable of producing complex zeros of transmission. This structure was firstsuggested by Guillemin [1].Figure 8.13 is the parallel connection of the ladder networks N a and N b . The y-parameters y ij of thecomposite two-port N can be expressed in terms of those y 0 ija and y0 ijb of the component two-ports N a andN b by the equationy ij ¼ yija 0 þ y0 ijb, i, j ¼ 1, 2 (8:34)Thus, to realize y 21 (s) and y 22 (s), we may separate them into pairs like y21a 0 , y0 22a and y0 21b , y0 22brealize an individual pair as an LC or RC ladder. Then connect these individual ladders in parallel torealize y 21 (s) and y 22 (s). In order for the procedure to succeed, we must resolve the following problem.Recall than in the Cauer development of LC and RC ladders, y 21 (s) is realized only within themultiplicative constant k. Thus, the transfer admittances realized by the component two-ports actuallywill be k a y 0 21aand k b y 0 21b. The sum of these two functions will not result in the desired ky 21 unlessk ¼ k a ¼ k b . To circumvent this difficulty, we introduce an additional degree of freedom by adjusting theadmittance level of the a-ladder N a by a factor b a and the b-ladder N b by b b . Then the functions of theresulting realizations becomey 0 21a ¼ b ak a y 21a , y 0 22a ¼ b ay 22a (8:35a)y 0 21b ¼ b bk b y 21b, y 0 22b ¼ b by 22b (8:35b)where y ij ¼ y ija þ y ijb , i, j ¼ 1, 2. Substituting these in Equation 8.34 givesy 21 ¼ b a k a y 21a þ b b k b y 21by 22 ¼ b a y 22a þ b b y 22b(8:36a)(8:36b)Our objective is to choose b a and b b to satisfy the above equations, once k a and k b are known. One way tomeet these requirements is to let y 0 22a and y0 22b have the same zeros and poles as y 22 but different scalefactors such that y 0 22a ¼ b ay 22 and y 0 22b ¼ b by 22 , obtaining from Equation 8.36b