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

FIR Filters 18-51In this case, the overall transfer function can be written in the following simplified form [6,8]:H(z) ¼ F(z L )G(z) (18:90)where the orders of both F(z) and G(z) can be freely selected to be either even or odd. As shown in Figure18.28, the role of G(z) is to provide the desired attenuation on those regions where F(z L ) has extraunwanted passband and transition band regions, that is, onbL=2cV s (L, v s ) ¼ [ k 2p Lk¼1v s , min k 2p L þ v s, p(18:91)There exist two ways of designing the subfilters F(z L ) and G(z). In the first case, they are determined, bymeans of the Remez algorithm, to satisfywhere1 d (F)p1 d (G)p F(v) 1 þ d (F)pfor v 2 [0, Lv p ] (18:92a)d s F(v) d s for v 2 [Lv s , p] (18:92b) G(v) 1 þ d (G)pfor v 2 [0, v p ] (18:92c)d s G(v) d s for v 2 V s (L, v s ) (18:92d)d (G)pþ d (F)p¼ d p (18:92e)The ripples d p (F) and d p (G) can be selected, e.g., to be half the overall ripple d p . In the above specifications,both F(z L ) and G(z) have [0, v p ] as a passband region.Another approach, leading to a considerable reduction in the order of G(z), is to design simultaneouslyF(v) to satisfy1 d p F(v)G(v L) 1 þ d p for v 2 [0, Lv p ](18:93a)d s F(v)G(v= L) d s for v 2 [Lv s , p](18:93b)and G(v) to satisfyG(0) ¼ 1(18:94a)d s jleF(Lv)G(v) d s for v 2 V s (L, v s ) (18:94b)The desired overall solution can be obtained by iteratively determining, by means of the Remezalgorithm, F(z) to meet the criteria of Equations 18.93a and b and G(z) to meet the criteria of Equations18.94a and b. Typically, only three to five designs of both of the subfilters are required to arrive at asolution that does not change if further iterations are used. For more details, see Ref. [8] or [10]. Figure18.29 shows typical responses for G(z) and F(z L ) and for the overall optimized design. As seen in thisfigure, G(z) has all its zeros on the unit circle concentrating on providing the desired attenuation for theoverall response on V s (L, v s ), whereas F(z L ) makes the overall response equiripple in the passband and inthe stopband portion [v s , p=L].