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

18-58 Passive, Active, and Digital FiltersInH 1 (z^L 1) H2 (z^L 2) H3 (z^L 3) HR (z^L R)I 2 I 3I 4I R(a)In+ z −m3 + + z −mR Outz −m2 +z −mR z −m3z −m2(b)I R I 4 I 3 I 2H R (z^L R) + + H3 (z^L 3) + H2 (z^L 2) + H1 (z^L 1)H r (z^L r)OutInH r(1) (z K r^Lr)H r(2) (z^Lr)Out(c)FIGURE 18.33 Implementations for a filter synthesized using the Jing–Fam approach. (a) Basic structure.(b) Transposed structure. (c) Structure for the subfilter H r (z Lr).The remaining problem is to select R, the L r ’s, the K r ’s, and the ripple values such that the filtercomplexity is minimized. The following example illustrates this.Example 18.5Consider the specifications of Example 18.1, that is, v p ¼ 0.4p, v s ¼ 0.402p, d p ¼ 0.01, and d s ¼ 0.001. Inthis case, the only alternative is to select L 1 ¼ 2. The resulting passband and stopband regions for G 1 (z)are (the substitution of Equation 18.109 is used)V (1)p¼ [0, 0:4p] and V(1) s ¼ [0:5987p, p]For ^H 2 (z), the edges become v (2)p ¼ p L 1vs ¼ 0:196p and v (2)s ¼ p L 1vp ¼ 0:2p. For L 2 , there aretwo alternatives to make the edges of ^H 3 (z), v (3)p ¼ p L(2) 2vs and v(3) s ¼ p L (2)2vp , less than p=2. Theseare L 2 ¼ 3 and L 2 ¼ 4. For R ¼ 5 stages, there are the following four alternatives to make all the v s (r)’ssmaller than p=2:L 1 ¼ 2, L 2 ¼ 4, L 3 ¼ 3, L 4 ¼ 2L 1 ¼ 2, L 2 ¼ 4, L 3 ¼ 4, L 4 ¼ 4L 1 ¼ 2, L 2 ¼ 3, L 3 ¼ 2, L 4 ¼ 4L 1 ¼ 2, L 2 ¼ 3, L 3 ¼ 2, L 4 ¼ 3Among these alternatives, the first one results in an overall filter with the minimum complexity. In thiscase, the edges of ^H 3 (z), ^H 4 (z), and ^H 5 (z)þG 5 (z) become as shown in Table 18.15. The correspondingpassband and stopband regions for G 2 (z), G 3 (z), G 4 (z), and G 5 (z) are