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

FIR Filters 18-43TABLE 18.11AlgorithmError Function for Designing F(v) Using the RemezE F (v) ¼ W F (v) [F(v) D F (v)],whereD F (v) ¼ [u(v) þ l(v)]=2, W F (v) ¼ 2=[u(v) l(v)]withu(v) ¼ min(C 1 (v) þ c 1 (v), C 2 (v) þ c 2 (v))l(v) ¼ max(C 1 (v) c 1 (v), C 2 (v) c 2 (v))C k (v) ¼ D H[h k (v)]G 1 [h k (v)]G 2 [h k (v)]G 2 [h k (v)] , k ¼ 1, 21=W H [h k (v)]c k (v) ¼jG 1 [h k (v)] G 2 [h k (v)] j , k ¼ 1, 2and(2lp v)=L for v 2 [0, u]h 1 (v) ¼ (2lp þ v)=L, h 2 (v) ¼[2(l þ 1)p v]=L for v 2 [f, p]for Case A and(2lp þ v)=L for v 2 [0, u]h 1 (v) ¼ (2lp v)=L, h 2 (v) ¼[2(l 1)p þ v]=L for v 2 [f, p]for Case BThe design of F(Lv) can be performed conveniently using linear programming [3]. Another, computationallymore efficient, alternative is to use the Remez algorithm [10]. Its use is based on the fact thatjE H (v) j 1 for v 2 V (F)p[ V (F)s(18:70a)whereE H (v) ¼ W H (v)[H(v) D H (v)] (18:70b)is satisfied when F(v) is designed such that the maximum absolute value of the error function given inTable 18.11 becomes less than or equal to unity on [0, u] [ [f, p].For step 2 of the above algorithm, D H (v) ¼ 1 and W H (v) ¼ 1=d p on V p (F) , whereas D H (v) ¼ 0 andW H (v) ¼ 1=d s on V s (F) , giving for k ¼ 1, 21 for v 2 [0, u]D H [h k (v)] ¼0 for v 2 [f, p]for Case A and1 for v 2 [0, u]D H [h k (v)] ¼0 for v 2 [f, p]and W H [h k (v)] ¼ 1=d p for v 2 [0, u]1=d s for v 2 [f, p]and W H [h k (v)] ¼ 1=d s for v 2 [0, u]1=d p for v 2 [f, p](18:71a)(18:71b)for Case B. Even though the resulting error function looks very complicated, it is straightforward to usethe subroutines EFF and WATE in the Remez algorithm described in Ref. [5] for optimally designing F(z).The order of G 1 (z) can be considerably reduced by allowing larger ripples on those regions of G 1 (z)where F(Lv) has one of its stopbands. As a rule of thumb, the ripples on these regions can be selected tobe 10 times larger [3]. Similarly, the order G 2 (z) can be decreased by allowing (ten times) larger ripples onthose regions where F(Lv) has one of its passbands.